LMFDB - Embedded newform 23.15.b.b.22.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,15,Mod(22,23)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 15, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.5956626749$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.13
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.14

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+37.1439 q^{2} +4213.49 q^{3} -15004.3 q^{4} -126427. i q^{5} +156505. q^{6} +1.01652e6i q^{7} -1.16588e6 q^{8} +1.29705e7 q^{9} +O(q^{10})$	\(q+37.1439 q^{2} +4213.49 q^{3} -15004.3 q^{4} -126427. i q^{5} +156505. q^{6} +1.01652e6i q^{7} -1.16588e6 q^{8} +1.29705e7 q^{9} -4.69598e6i q^{10} -2.72237e7i q^{11} -6.32206e7 q^{12} +8.12923e6 q^{13} +3.77576e7i q^{14} -5.32697e8i q^{15} +2.02525e8 q^{16} -3.49406e8i q^{17} +4.81776e8 q^{18} -4.58813e8i q^{19} +1.89695e9i q^{20} +4.28311e9i q^{21} -1.01119e9i q^{22} +(-6.74684e8 - 3.33731e9i) q^{23} -4.91244e9 q^{24} -9.88018e9 q^{25} +3.01951e8 q^{26} +3.44982e10 q^{27} -1.52522e10i q^{28} -1.51140e10 q^{29} -1.97865e10i q^{30} -3.61827e9 q^{31} +2.66244e10 q^{32} -1.14707e11i q^{33} -1.29783e10i q^{34} +1.28516e11 q^{35} -1.94614e11 q^{36} -1.19082e9i q^{37} -1.70421e10i q^{38} +3.42524e10 q^{39} +1.47399e11i q^{40} -7.65847e9 q^{41} +1.59091e11i q^{42} -3.94536e10i q^{43} +4.08473e11i q^{44} -1.63982e12i q^{45} +(-2.50604e10 - 1.23961e11i) q^{46} +4.24788e11 q^{47} +8.53339e11 q^{48} -3.55095e11 q^{49} -3.66988e11 q^{50} -1.47222e12i q^{51} -1.21974e11 q^{52} +1.33518e12i q^{53} +1.28140e12 q^{54} -3.44180e12 q^{55} -1.18515e12i q^{56} -1.93320e12i q^{57} -5.61391e11 q^{58} +3.11541e12 q^{59} +7.99277e12i q^{60} +3.13178e12i q^{61} -1.34397e11 q^{62} +1.31848e13i q^{63} -2.32924e12 q^{64} -1.02775e12i q^{65} -4.26065e12i q^{66} +4.52513e11i q^{67} +5.24260e12i q^{68} +(-2.84278e12 - 1.40617e13i) q^{69} +4.77357e12 q^{70} +1.17106e13 q^{71} -1.51221e13 q^{72} +5.81686e11 q^{73} -4.42317e10i q^{74} -4.16300e13 q^{75} +6.88418e12i q^{76} +2.76735e13 q^{77} +1.27227e12 q^{78} +2.55469e13i q^{79} -2.56046e13i q^{80} +8.33202e13 q^{81} -2.84465e11 q^{82} +5.45097e12i q^{83} -6.42652e13i q^{84} -4.41742e13 q^{85} -1.46546e12i q^{86} -6.36825e13 q^{87} +3.17397e13i q^{88} +3.32120e13i q^{89} -6.09093e13i q^{90} +8.26355e12i q^{91} +(1.01232e13 + 5.00741e13i) q^{92} -1.52456e13 q^{93} +1.57783e13 q^{94} -5.80062e13 q^{95} +1.12182e14 q^{96} +4.07652e13i q^{97} -1.31896e13 q^{98} -3.53105e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) $ 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	37.1439			0.290187			0.145093	−	0.989418i	$-0.453652\pi$
$2$	37.1439			0.290187			0.145093	+	0.989418i	$0.453652\pi$
$3$	4213.49			1.92661			0.963304	−	0.268414i	$-0.0864995\pi$
$3$	4213.49			1.92661			0.963304	+	0.268414i	$0.0864995\pi$
$4$	−15004.3			−0.915792
$5$		−	126427.i		−	1.61826i	−0.587629	−	0.809130i	$-0.699939\pi$
$5$		−	126427.i		−	1.61826i	0.587629	−	0.809130i	$-0.300061\pi$
$6$	156505.			0.559076
$7$			1.01652e6i			1.23433i	0.786834	+	0.617164i	$0.211719\pi$
$7$			1.01652e6i			1.23433i	−0.786834	+	0.617164i	$0.788281\pi$
$8$	−1.16588e6			−0.555937
$9$	1.29705e7			2.71182
$10$		−	4.69598e6i		−	0.469598i
$11$		−	2.72237e7i		−	1.39700i	−0.715608	−	0.698502i	$-0.753850\pi$
$11$		−	2.72237e7i		−	1.39700i	0.715608	−	0.698502i	$-0.246150\pi$
$12$	−6.32206e7			−1.76437
$13$	8.12923e6			0.129553			0.0647763	−	0.997900i	$-0.479367\pi$
$13$	8.12923e6			0.129553			0.0647763	+	0.997900i	$0.479367\pi$
$14$			3.77576e7i			0.358186i
$15$		−	5.32697e8i		−	3.11775i
$16$	2.02525e8			0.754466
$17$		−	3.49406e8i		−	0.851505i	−0.904840	−	0.425753i	$-0.860009\pi$
$17$		−	3.49406e8i		−	0.851505i	0.904840	−	0.425753i	$-0.139991\pi$
$18$	4.81776e8			0.786933
$19$		−	4.58813e8i		−	0.513287i	−0.966506	−	0.256644i	$-0.917383\pi$
$19$		−	4.58813e8i		−	0.513287i	0.966506	−	0.256644i	$-0.0826166\pi$
$20$			1.89695e9i			1.48199i
$21$			4.28311e9i			2.37807i
$22$		−	1.01119e9i		−	0.405392i
$23$	−6.74684e8	−	3.33731e9i	−0.198155	−	0.980171i
$23$	−6.74684e8	−	3.33731e9i	−0.198155	−	0.980171i
$24$	−4.91244e9			−1.07107
$25$	−9.88018e9			−1.61877
$26$	3.01951e8			0.0375944
$27$	3.44982e10			3.29800
$28$		−	1.52522e10i		−	1.13039i
$29$	−1.51140e10			−0.876178			−0.438089	−	0.898932i	$-0.644345\pi$
$29$	−1.51140e10			−0.876178			−0.438089	+	0.898932i	$0.644345\pi$
$30$		−	1.97865e10i		−	0.904730i
$31$	−3.61827e9			−0.131513			−0.0657566	−	0.997836i	$-0.520946\pi$
$31$	−3.61827e9			−0.131513			−0.0657566	+	0.997836i	$0.520946\pi$
$32$	2.66244e10			0.774873
$33$		−	1.14707e11i		−	2.69148i
$34$		−	1.29783e10i		−	0.247095i
$35$	1.28516e11			1.99747
$36$	−1.94614e11			−2.48346
$37$		−	1.19082e9i		−	0.0125440i	−0.999980	−	0.00627198i	$-0.998004\pi$
$37$		−	1.19082e9i		−	0.0125440i	0.999980	−	0.00627198i	$-0.00199645\pi$
$38$		−	1.70421e10i		−	0.148949i
$39$	3.42524e10			0.249597
$40$			1.47399e11i			0.899651i
$41$	−7.65847e9			−0.0393237			−0.0196619	−	0.999807i	$-0.506259\pi$
$41$	−7.65847e9			−0.0393237			−0.0196619	+	0.999807i	$0.506259\pi$
$42$			1.59091e11i			0.690083i
$43$		−	3.94536e10i		−	0.145147i	−0.997363	−	0.0725733i	$-0.976879\pi$
$43$		−	3.94536e10i		−	0.145147i	0.997363	−	0.0725733i	$-0.0231211\pi$
$44$			4.08473e11i			1.27937i
$45$		−	1.63982e12i		−	4.38842i
$46$	−2.50604e10	−	1.23961e11i	−0.0575020	−	0.284432i
$47$	4.24788e11			0.838469			0.419234	−	0.907878i	$-0.362299\pi$
$47$	4.24788e11			0.838469			0.419234	+	0.907878i	$0.362299\pi$
$48$	8.53339e11			1.45356
$49$	−3.55095e11			−0.523566
$50$	−3.66988e11			−0.469745
$51$		−	1.47222e12i		−	1.64052i
$52$	−1.21974e11			−0.118643
$53$			1.33518e12i			1.13660i	0.822820	+	0.568302i	$0.192399\pi$
$53$			1.33518e12i			1.13660i	−0.822820	+	0.568302i	$0.807601\pi$
$54$	1.28140e12			0.957034
$55$	−3.44180e12			−2.26072
$56$		−	1.18515e12i		−	0.686209i
$57$		−	1.93320e12i		−	0.988903i
$58$	−5.61391e11			−0.254255
$59$	3.11541e12			1.25185			0.625924	−	0.779884i	$-0.284722\pi$
$59$	3.11541e12			1.25185			0.625924	+	0.779884i	$0.284722\pi$
$60$			7.99277e12i			2.85521i
$61$			3.13178e12i			0.996512i	0.867030	+	0.498256i	$0.166026\pi$
$61$			3.13178e12i			0.996512i	−0.867030	+	0.498256i	$0.833974\pi$
$62$	−1.34397e11			−0.0381634
$63$			1.31848e13i			3.34727i
$64$	−2.32924e12			−0.529608
$65$		−	1.02775e12i		−	0.209650i
$66$		−	4.26065e12i		−	0.781031i
$67$			4.52513e11i			0.0746633i	0.999303	+	0.0373316i	$0.0118858\pi$
$67$			4.52513e11i			0.0746633i	−0.999303	+	0.0373316i	$0.988114\pi$
$68$			5.24260e12i			0.779802i
$69$	−2.84278e12	−	1.40617e13i	−0.381767	−	1.88840i
$70$	4.77357e12			0.579638
$71$	1.17106e13			1.28756			0.643782	−	0.765209i	$-0.277364\pi$
$71$	1.17106e13			1.28756			0.643782	+	0.765209i	$0.277364\pi$
$72$	−1.51221e13			−1.50760
$73$	5.81686e11			0.0526537			0.0263268	−	0.999653i	$-0.491619\pi$
$73$	5.81686e11			0.0526537			0.0263268	+	0.999653i	$0.491619\pi$
$74$		−	4.42317e10i		−	0.00364009i
$75$	−4.16300e13			−3.11873
$76$			6.88418e12i			0.470064i
$77$	2.76735e13			1.72436
$78$	1.27227e12			0.0724297
$79$			2.55469e13i			1.33030i	0.746710	+	0.665149i	$0.231632\pi$
$79$			2.55469e13i			1.33030i	−0.746710	+	0.665149i	$0.768368\pi$
$80$		−	2.56046e13i		−	1.22092i
$81$	8.33202e13			3.64213
$82$	−2.84465e11			−0.0114112
$83$			5.45097e12i			0.200876i	0.994943	+	0.100438i	$0.0320243\pi$
$83$			5.45097e12i			0.200876i	−0.994943	+	0.100438i	$0.967976\pi$
$84$		−	6.42652e13i		−	2.17781i
$85$	−4.41742e13			−1.37796
$86$		−	1.46546e12i		−	0.0421196i
$87$	−6.36825e13			−1.68805
$88$			3.17397e13i			0.776647i
$89$			3.32120e13i			0.750871i	0.926849	+	0.375435i	$0.122507\pi$
$89$			3.32120e13i			0.750871i	−0.926849	+	0.375435i	$0.877493\pi$
$90$		−	6.09093e13i		−	1.27346i
$91$			8.26355e12i			0.159910i
$92$	1.01232e13	+	5.00741e13i	0.181469	+	0.897632i
$93$	−1.52456e13			−0.253374
$94$	1.57783e13			0.243312
$95$	−5.80062e13			−0.830633
$96$	1.12182e14			1.49288
$97$			4.07652e13i			0.504530i	0.967658	+	0.252265i	$0.0811755\pi$
$97$			4.07652e13i			0.504530i	−0.967658	+	0.252265i	$0.918825\pi$
$98$	−1.31896e13			−0.151932
$99$		−	3.53105e14i		−	3.78842i
$100$	1.48245e14			1.48245
$101$	−1.75954e14			−1.64115			−0.820577	−	0.571535i	$-0.806348\pi$
$101$	−1.75954e14			−1.64115			−0.820577	+	0.571535i	$0.806348\pi$
$102$		−	5.46839e13i		−	0.476056i
$103$			1.76621e14i			1.43609i	0.695997	+	0.718045i	$0.254963\pi$
$103$			1.76621e14i			1.43609i	−0.695997	+	0.718045i	$0.745037\pi$
$104$	−9.47775e12			−0.0720231
$105$	5.41499e14			3.84833
$106$			4.95938e13i			0.329827i
$107$		−	2.25775e14i		−	1.40601i	−0.711184	−	0.703006i	$-0.751841\pi$
$107$		−	2.25775e14i		−	1.40601i	0.711184	−	0.703006i	$-0.248159\pi$
$108$	−5.17622e14			−3.02028
$109$		−	1.70101e14i		−	0.930510i	−0.885177	−	0.465255i	$-0.845963\pi$
$109$		−	1.70101e14i		−	0.930510i	0.885177	−	0.465255i	$-0.154037\pi$
$110$	−1.27842e14			−0.656030
$111$		−	5.01752e12i		−	0.0241673i
$112$			2.05872e14i			0.931259i
$113$			3.35281e14i			1.42515i	0.701597	+	0.712574i	$0.252471\pi$
$113$			3.35281e14i			1.42515i	−0.701597	+	0.712574i	$0.747529\pi$
$114$		−	7.18067e13i		−	0.286966i
$115$	−4.21925e14	+	8.52981e13i	−1.58617	+	0.320667i
$116$	2.26775e14			0.802397
$117$	1.05440e14			0.351323
$118$	1.15719e14			0.363270
$119$	3.55179e14			1.05104
$120$			6.21064e14i			1.73327i
$121$	−3.61379e14			−0.951623
$122$			1.16327e14i			0.289174i
$123$	−3.22689e13			−0.0757614
$124$	5.42898e13			0.120439
$125$			4.77471e14i			1.00133i
$126$			4.89736e14i			0.971333i
$127$	3.25245e14			0.610358			0.305179	−	0.952295i	$-0.401284\pi$
$127$	3.25245e14			0.610358			0.305179	+	0.952295i	$0.401284\pi$
$128$	−5.22732e14			−0.928558
$129$		−	1.66237e14i		−	0.279641i
$130$		−	3.81747e13i		−	0.0608376i
$131$	−4.00810e13			−0.0605396			−0.0302698	−	0.999542i	$-0.509637\pi$
$131$	−4.00810e13			−0.0605396			−0.0302698	+	0.999542i	$0.509637\pi$
$132$			1.72110e15i			2.46483i
$133$	4.66394e14			0.633565
$134$			1.68081e13i			0.0216663i
$135$		−	4.36149e15i		−	5.33702i
$136$			4.07367e14i			0.473383i
$137$			7.77646e13i			0.0858496i	0.999078	+	0.0429248i	$0.0136676\pi$
$137$			7.77646e13i			0.0858496i	−0.999078	+	0.0429248i	$0.986332\pi$
$138$	−1.05592e14	−	5.22307e14i	−0.110784	−	0.547990i
$139$	−1.13762e15			−1.13474			−0.567368	−	0.823464i	$-0.692038\pi$
$139$	−1.13762e15			−1.13474			−0.567368	+	0.823464i	$0.692038\pi$
$140$	−1.92829e15			−1.82926
$141$	1.78984e15			1.61540
$142$	4.34975e14			0.373634
$143$		−	2.21308e14i		−	0.180986i
$144$	2.62686e15			2.04597
$145$			1.91081e15i			1.41788i
$146$	2.16061e13			0.0152794
$147$	−1.49619e15			−1.00871
$148$			1.78675e13i			0.0114877i
$149$			1.83283e14i			0.112413i	0.998419	+	0.0562066i	$0.0179006\pi$
$149$			1.83283e14i			0.112413i	−0.998419	+	0.0562066i	$0.982099\pi$
$150$	−1.54630e15			−0.905014
$151$	7.71582e14			0.431066			0.215533	−	0.976497i	$-0.430851\pi$
$151$	7.71582e14			0.431066			0.215533	+	0.976497i	$0.430851\pi$
$152$			5.34923e14i			0.285355i
$153$		−	4.53198e15i		−	2.30913i
$154$	1.02790e15			0.500387
$155$			4.57446e14i			0.212823i
$156$	−5.13935e14			−0.228579
$157$		−	1.50455e15i		−	0.639894i	−0.947435	−	0.319947i	$-0.896335\pi$
$157$		−	1.50455e15i		−	0.639894i	0.947435	−	0.319947i	$-0.103665\pi$
$158$			9.48912e14i			0.386035i
$159$			5.62577e15i			2.18979i
$160$		−	3.36604e15i		−	1.25395i
$161$	3.39245e15	−	6.85832e14i	1.20985	−	0.244589i
$162$	3.09484e15			1.05690
$163$	2.81812e15			0.921821			0.460910	−	0.887447i	$-0.347523\pi$
$163$	2.81812e15			0.921821			0.460910	+	0.887447i	$0.347523\pi$
$164$	1.14910e14			0.0360124
$165$	−1.45020e16			−4.35552
$166$			2.02470e14i			0.0582914i
$167$	6.16095e15			1.70072			0.850359	−	0.526202i	$-0.176384\pi$
$167$	6.16095e15			1.70072			0.850359	+	0.526202i	$0.176384\pi$
$168$		−	4.99361e15i		−	1.32205i
$169$	−3.87129e15			−0.983216
$170$	−1.64080e15			−0.399865
$171$		−	5.95105e15i		−	1.39194i
$172$			5.91974e14i			0.132924i
$173$	7.01825e14			0.151323			0.0756617	−	0.997134i	$-0.475893\pi$
$173$	7.01825e14			0.151323			0.0756617	+	0.997134i	$0.475893\pi$
$174$	−2.36542e15			−0.489850
$175$		−	1.00434e16i		−	1.99809i
$176$		−	5.51349e15i		−	1.05399i
$177$	1.31268e16			2.41182
$178$			1.23362e15i			0.217893i
$179$	3.23524e15			0.549458			0.274729	−	0.961522i	$-0.411412\pi$
$179$	3.23524e15			0.549458			0.274729	+	0.961522i	$0.411412\pi$
$180$			2.46044e16i			4.01888i
$181$		−	1.42522e15i		−	0.223940i	−0.993712	−	0.111970i	$-0.964284\pi$
$181$		−	1.42522e15i		−	0.223940i	0.993712	−	0.111970i	$-0.0357161\pi$
$182$			3.06940e14i			0.0464039i
$183$			1.31957e16i			1.91989i
$184$	7.86604e14	+	3.89092e15i	0.110162	+	0.544913i
$185$	−1.50552e14			−0.0202994
$186$	−5.66279e14			−0.0735258
$187$	−9.51210e15			−1.18956
$188$	−6.37366e15			−0.767863
$189$			3.50682e16i			4.07081i
$190$	−2.15457e15			−0.241038
$191$		−	8.58538e15i		−	0.925819i	−0.886406	−	0.462910i	$-0.846805\pi$
$191$		−	8.58538e15i		−	0.925819i	0.886406	−	0.462910i	$-0.153195\pi$
$192$	−9.81424e15			−1.02035
$193$	−4.48381e15			−0.449517			−0.224758	−	0.974415i	$-0.572159\pi$
$193$	−4.48381e15			−0.449517			−0.224758	+	0.974415i	$0.572159\pi$
$194$			1.51418e15i			0.146408i
$195$		−	4.33042e15i		−	0.403913i
$196$	5.32796e15			0.479478
$197$	2.42347e15			0.210462			0.105231	−	0.994448i	$-0.466442\pi$
$197$	2.42347e15			0.210462			0.105231	+	0.994448i	$0.466442\pi$
$198$		−	1.31157e16i		−	1.09935i
$199$		−	3.40231e15i		−	0.275298i	−0.990481	−	0.137649i	$-0.956045\pi$
$199$		−	3.40231e15i		−	0.275298i	0.990481	−	0.137649i	$-0.0439546\pi$
$200$	1.15191e16			0.899933
$201$			1.90666e15i			0.143847i
$202$	−6.53562e15			−0.476241
$203$		−	1.53637e16i		−	1.08149i
$204$			2.20896e16i			1.50237i
$205$			9.68234e14i			0.0636361i
$206$			6.56039e15i			0.416734i
$207$	−8.75101e15	−	4.32867e16i	−0.537361	−	2.65804i
$208$	1.64638e15			0.0977431
$209$	−1.24906e16			−0.717065
$210$	2.01134e16			1.11673
$211$	2.10745e16			1.13183			0.565914	−	0.824465i	$-0.308524\pi$
$211$	2.10745e16			1.13183			0.565914	+	0.824465i	$0.308524\pi$
$212$		−	2.00335e16i		−	1.04089i
$213$	4.93423e16			2.48063
$214$		−	8.38615e15i		−	0.408006i
$215$	−4.98798e15			−0.234885
$216$	−4.02209e16			−1.83348
$217$		−	3.67806e15i		−	0.162330i
$218$		−	6.31820e15i		−	0.270021i
$219$	2.45093e15			0.101443
$220$	5.16419e16			2.07035
$221$		−	2.84040e15i		−	0.110315i
$222$		−	1.86370e14i		−	0.00701302i
$223$	−3.44720e16			−1.25699			−0.628497	−	0.777812i	$-0.716329\pi$
$223$	−3.44720e16			−1.25699			−0.628497	+	0.777812i	$0.716329\pi$
$224$			2.70643e16i			0.956448i
$225$	−1.28151e17			−4.38980
$226$			1.24536e16i			0.413559i
$227$		−	4.94506e16i		−	1.59218i	−0.605181	−	0.796088i	$-0.706899\pi$
$227$		−	4.94506e16i		−	1.59218i	0.605181	−	0.796088i	$-0.293101\pi$
$228$			2.90064e16i			0.905629i
$229$			4.13694e16i			1.25265i	0.779561	+	0.626326i	$0.215442\pi$
$229$			4.13694e16i			1.25265i	−0.779561	+	0.626326i	$0.784558\pi$
$230$	−1.56719e16	+	3.16830e15i	−0.460286	+	0.0930533i
$231$	1.16602e17			3.32217
$232$	1.76211e16			0.487100
$233$	−9.59027e15			−0.257241			−0.128620	−	0.991694i	$-0.541055\pi$
$233$	−9.59027e15			−0.257241			−0.128620	+	0.991694i	$0.541055\pi$
$234$	3.91647e15			0.101949
$235$		−	5.37045e16i		−	1.35686i
$236$	−4.67447e16			−1.14643
$237$			1.07642e17i			2.56296i
$238$	1.31927e16			0.304997
$239$	−3.16172e16			−0.709803			−0.354901	−	0.934904i	$-0.615486\pi$
$239$	−3.16172e16			−0.709803			−0.354901	+	0.934904i	$0.615486\pi$
$240$		−	1.07885e17i		−	2.35224i
$241$		−	4.94119e16i		−	1.04644i	−0.852199	−	0.523218i	$-0.824731\pi$
$241$		−	4.94119e16i		−	1.04644i	0.852199	−	0.523218i	$-0.175269\pi$
$242$	−1.34230e16			−0.276148
$243$	1.86065e17			3.71895
$244$		−	4.69903e16i		−	0.912597i
$245$			4.48934e16i			0.847267i
$246$	−1.19859e15			−0.0219849
$247$		−	3.72980e15i		−	0.0664977i
$248$	4.21849e15			0.0731131
$249$			2.29676e16i			0.387008i
$250$			1.77351e16i			0.290572i
$251$			6.28614e16i			1.00154i	0.865580	+	0.500770i	$0.166950\pi$
$251$			6.28614e16i			1.00154i	−0.865580	+	0.500770i	$0.833050\pi$
$252$		−	1.97830e17i		−	3.06540i
$253$	−9.08538e16	+	1.83674e16i	−1.36930	+	0.276824i
$254$	1.20809e16			0.177118
$255$	−1.86127e17			−2.65478
$256$	1.87460e16			0.260153
$257$	9.00262e16			1.21573			0.607865	−	0.794040i	$-0.292026\pi$
$257$	9.00262e16			1.21573			0.607865	+	0.794040i	$0.292026\pi$
$258$		−	6.17470e15i		−	0.0811480i
$259$	1.21050e15			0.0154834
$260$			1.54207e16i			0.191996i
$261$	−1.96036e17			−2.37603
$262$	−1.48876e15			−0.0175678
$263$			1.39461e17i			1.60237i	0.598419	+	0.801183i	$0.295796\pi$
$263$			1.39461e17i			1.60237i	−0.598419	+	0.801183i	$0.704204\pi$
$264$			1.33735e17i			1.49629i
$265$	1.68802e17			1.83932
$266$	1.73237e16			0.183852
$267$			1.39939e17i			1.44663i
$268$		−	6.78965e15i		−	0.0683760i
$269$	1.93432e17			1.89785			0.948926	−	0.315499i	$-0.102172\pi$
$269$	1.93432e17			1.89785			0.948926	+	0.315499i	$0.102172\pi$
$270$		−	1.62003e17i		−	1.54873i
$271$	−1.75559e17			−1.63545			−0.817727	−	0.575607i	$-0.804766\pi$
$271$	−1.75559e17			−1.63545			−0.817727	+	0.575607i	$0.804766\pi$
$272$		−	7.07635e16i		−	0.642432i
$273$			3.48184e16i			0.308085i
$274$			2.88848e15i			0.0249124i
$275$			2.68975e17i			2.26143i
$276$	4.26539e16	+	2.10987e17i	0.349619	+	1.72938i
$277$	1.56846e17			1.25347			0.626737	−	0.779231i	$-0.284390\pi$
$277$	1.56846e17			1.25347			0.626737	+	0.779231i	$0.284390\pi$
$278$	−4.22557e16			−0.329285
$279$	−4.69309e16			−0.356640
$280$	−1.49834e17			−1.11047
$281$			1.34633e17i			0.973211i	0.873622	+	0.486605i	$0.161765\pi$
$281$			1.34633e17i			0.973211i	−0.873622	+	0.486605i	$0.838235\pi$
$282$	6.64816e16			0.468768
$283$		−	2.29733e17i		−	1.58023i	−0.612961	−	0.790113i	$-0.710022\pi$
$283$		−	2.29733e17i		−	1.58023i	0.612961	−	0.790113i	$-0.289978\pi$
$284$	−1.75709e17			−1.17914
$285$	−2.44408e17			−1.60030
$286$		−	8.22022e15i		−	0.0525196i
$287$		−	7.78500e15i		−	0.0485384i
$288$	3.45333e17			2.10131
$289$	4.62936e16			0.274939
$290$			7.09748e16i			0.411451i
$291$			1.71764e17i			0.972031i
$292$	−8.72781e15			−0.0482198
$293$			4.38873e15i			0.0236737i	0.999930	+	0.0118368i	$0.00376787\pi$
$293$			4.38873e15i			0.0236737i	−0.999930	+	0.0118368i	$0.996232\pi$
$294$	−5.55742e16			−0.292713
$295$		−	3.93871e17i		−	2.02582i
$296$			1.38836e15i			0.00697365i
$297$		−	9.39168e17i		−	4.60732i
$298$			6.80783e15i			0.0326208i
$299$	−5.48466e15	−	2.71298e16i	−0.0256715	−	0.126984i
$300$	6.24631e17			2.85611
$301$	4.01054e16			0.179159
$302$	2.86596e16			0.125089
$303$	−7.41381e17			−3.16186
$304$		−	9.29213e16i		−	0.387258i
$305$	3.95941e17			1.61262
$306$		−	1.68335e17i		−	0.670077i
$307$	−3.91555e17			−1.52344			−0.761718	−	0.647909i	$-0.775644\pi$
$307$	−3.91555e17			−1.52344			−0.761718	+	0.647909i	$0.775644\pi$
$308$	−4.15222e17			−1.57916
$309$			7.44191e17i			2.76678i
$310$			1.69913e16i			0.0617583i
$311$	−7.43612e16			−0.264255			−0.132128	−	0.991233i	$-0.542181\pi$
$311$	−7.43612e16			−0.264255			−0.132128	+	0.991233i	$0.542181\pi$
$312$	−3.99344e16			−0.138760
$313$			1.04358e16i			0.0354582i	0.999843	+	0.0177291i	$0.00564364\pi$
$313$			1.04358e16i			0.0354582i	−0.999843	+	0.0177291i	$0.994356\pi$
$314$		−	5.58847e16i		−	0.185689i
$315$	1.66691e18			5.41676
$316$		−	3.83315e17i		−	1.21828i
$317$	3.72320e16			0.115745			0.0578724	−	0.998324i	$-0.481568\pi$
$317$	3.72320e16			0.115745			0.0578724	+	0.998324i	$0.481568\pi$
$318$			2.08963e17i			0.635448i
$319$			4.11458e17i			1.22402i
$320$			2.94478e17i			0.857045i
$321$		−	9.51299e17i		−	2.70883i
$322$	1.26009e17	−	2.54745e16i	0.351083	−	0.0709764i
$323$	−1.60312e17			−0.437067
$324$	−1.25016e18			−3.33543
$325$	−8.03183e16			−0.209716
$326$	1.04676e17			0.267500
$327$		−	7.16718e17i		−	1.79273i
$328$	8.92889e15			0.0218615
$329$			4.31806e17i			1.03495i
$330$	−5.38660e17			−1.26391
$331$	5.53675e17			1.27192			0.635959	−	0.771723i	$-0.280605\pi$
$331$	5.53675e17			1.27192			0.635959	+	0.771723i	$0.280605\pi$
$332$		−	8.17881e16i		−	0.183960i
$333$		−	1.54456e16i		−	0.0340169i
$334$	2.28842e17			0.493526
$335$	5.72097e16			0.120825
$336$			8.67438e17i			1.79417i
$337$		−	5.71712e17i		−	1.15816i	−0.815271	−	0.579080i	$-0.803412\pi$
$337$		−	5.71712e17i		−	1.15816i	0.815271	−	0.579080i	$-0.196588\pi$
$338$	−1.43795e17			−0.285316
$339$			1.41270e18i			2.74570i
$340$	6.62804e17			1.26192
$341$			9.85027e16i			0.183725i
$342$		−	2.21045e17i		−	0.403922i
$343$			3.28467e17i			0.588076i
$344$			4.59983e16i			0.0806924i
$345$	−1.77778e18	+	3.59402e17i	−3.05593	+	0.617799i
$346$	2.60685e16			0.0439120
$347$	−9.99122e17			−1.64935			−0.824674	−	0.565609i	$-0.808641\pi$
$347$	−9.99122e17			−1.64935			−0.824674	+	0.565609i	$0.808641\pi$
$348$	9.55514e17			1.54590
$349$	2.36981e16			0.0375781			0.0187891	−	0.999823i	$-0.494019\pi$
$349$	2.36981e16			0.0375781			0.0187891	+	0.999823i	$0.494019\pi$
$350$		−	3.73052e17i		−	0.579819i
$351$	2.80444e17			0.427264
$352$		−	7.24815e17i		−	1.08250i
$353$	1.96218e17			0.287287			0.143644	−	0.989629i	$-0.454118\pi$
$353$	1.96218e17			0.287287			0.143644	+	0.989629i	$0.454118\pi$
$354$	4.87579e17			0.699878
$355$		−	1.48053e18i		−	2.08361i
$356$		−	4.98324e17i		−	0.687641i
$357$	1.49654e18			2.02494
$358$	1.20169e17			0.159445
$359$			1.09280e18i			1.42194i	0.703223	+	0.710969i	$0.251743\pi$
$359$			1.09280e18i			1.42194i	−0.703223	+	0.710969i	$0.748257\pi$
$360$			1.91184e18i			2.43969i
$361$	5.88497e17			0.736536
$362$		−	5.29383e16i		−	0.0649845i
$363$	−1.52266e18			−1.83340
$364$		−	1.23989e17i		−	0.146445i
$365$		−	7.35406e16i		−	0.0852074i
$366$			4.90141e17i			0.557126i
$367$			1.13121e18i			1.26148i	0.775993	+	0.630742i	$0.217249\pi$
$367$			1.13121e18i			1.26148i	−0.775993	+	0.630742i	$0.782751\pi$
$368$	−1.36641e17	−	6.75890e17i	−0.149501	−	0.739506i
$369$	−9.93344e16			−0.106639
$370$	−5.59207e15			−0.00589061
$371$	−1.35724e18			−1.40294
$372$	2.28749e17			0.232038
$373$		−	5.92890e17i		−	0.590218i	−0.955464	−	0.295109i	$-0.904644\pi$
$373$		−	5.92890e17i		−	0.590218i	0.955464	−	0.295109i	$-0.0953559\pi$
$374$	−3.53316e17			−0.345194
$375$			2.01182e18i			1.92917i
$376$	−4.95253e17			−0.466136
$377$	−1.22865e17			−0.113511
$378$			1.30257e18i			1.18129i
$379$			1.07976e16i			0.00961282i	0.999988	+	0.00480641i	$0.00152993\pi$
$379$			1.07976e16i			0.00961282i	−0.999988	+	0.00480641i	$0.998470\pi$
$380$	8.70344e17			0.760686
$381$	1.37042e18			1.17592
$382$		−	3.18894e17i		−	0.268660i
$383$		−	1.07534e18i		−	0.889514i	−0.895651	−	0.444757i	$-0.853290\pi$
$383$		−	1.07534e18i		−	0.889514i	0.895651	−	0.444757i	$-0.146710\pi$
$384$	−2.20253e18			−1.78897
$385$		−	3.49866e18i		−	2.79047i
$386$	−1.66546e17			−0.130444
$387$		−	5.11734e17i		−	0.393611i
$388$		−	6.11654e17i		−	0.462045i
$389$			4.46095e17i			0.330964i	0.986213	+	0.165482i	$0.0529179\pi$
$389$			4.46095e17i			0.330964i	−0.986213	+	0.165482i	$0.947082\pi$
$390$		−	1.60849e17i		−	0.117210i
$391$	−1.16607e18	+	2.35738e17i	−0.834621	+	0.168730i
$392$	4.13999e17			0.291070
$393$	−1.68881e17			−0.116636
$394$	9.00171e16			0.0610733
$395$	3.22981e18			2.15277
$396$			5.29811e18i			3.46940i
$397$	1.34898e18			0.867906			0.433953	−	0.900935i	$-0.357118\pi$
$397$	1.34898e18			0.867906			0.433953	+	0.900935i	$0.357118\pi$
$398$		−	1.26375e17i		−	0.0798878i
$399$	1.96514e18			1.22063
$400$	−2.00099e18			−1.22131
$401$		−	1.83141e18i		−	1.09844i	−0.835678	−	0.549219i	$-0.814925\pi$
$401$		−	1.83141e18i		−	1.09844i	0.835678	−	0.549219i	$-0.185075\pi$
$402$			7.08207e16i			0.0417424i
$403$	−2.94138e16			−0.0170379
$404$	2.64007e18			1.50296
$405$		−	1.05339e19i		−	5.89391i
$406$		−	5.70667e17i		−	0.313834i
$407$	−3.24185e16			−0.0175240
$408$			1.71644e18i			0.912024i
$409$	−2.73604e17			−0.142909			−0.0714545	−	0.997444i	$-0.522764\pi$
$409$	−2.73604e17			−0.142909			−0.0714545	+	0.997444i	$0.522764\pi$
$410$			3.59640e16i			0.0184663i
$411$			3.27660e17i			0.165398i
$412$		−	2.65008e18i		−	1.31516i
$413$			3.16689e18i			1.54519i
$414$	−3.25047e17	−	1.60784e18i	−0.155935	−	0.771328i
$415$	6.89148e17			0.325069
$416$	2.16436e17			0.100387
$417$	−4.79336e18			−2.18619
$418$	−4.63948e17			−0.208083
$419$		−	1.13325e18i		−	0.499838i	−0.968267	−	0.249919i	$-0.919596\pi$
$419$		−	1.13325e18i		−	0.499838i	0.968267	−	0.249919i	$-0.0804040\pi$
$420$	−8.12483e18			−3.52427
$421$		−	3.01007e18i		−	1.28411i	−0.766660	−	0.642054i	$-0.778083\pi$
$421$		−	3.01007e18i		−	1.28411i	0.766660	−	0.642054i	$-0.221917\pi$
$422$	7.82789e17			0.328441
$423$	5.50972e18			2.27377
$424$		−	1.55667e18i		−	0.631880i
$425$			3.45219e18i			1.37839i
$426$	1.83276e18			0.719846
$427$	−3.18353e18			−1.23002
$428$			3.38760e18i			1.28761i
$429$		−	9.32477e17i		−	0.348688i
$430$	−1.85273e17			−0.0681605
$431$			1.74204e18i			0.630546i	0.949001	+	0.315273i	$0.102096\pi$
$431$			1.74204e18i			0.630546i	−0.949001	+	0.315273i	$0.897904\pi$
$432$	6.98677e18			2.48823
$433$			4.69550e18i			1.64538i	0.568490	+	0.822690i	$0.307528\pi$
$433$			4.69550e18i			1.64538i	−0.568490	+	0.822690i	$0.692472\pi$
$434$		−	1.36617e17i		−	0.0471061i
$435$			8.05117e18i			2.73171i
$436$			2.55225e18i			0.852153i
$437$	−1.53120e18	+	3.09554e17i	−0.503109	+	0.101711i
$438$	9.10370e16			0.0294374
$439$	2.79640e18			0.889913			0.444956	−	0.895552i	$-0.353219\pi$
$439$	2.79640e18			0.889913			0.444956	+	0.895552i	$0.353219\pi$
$440$	4.01274e18			1.25682
$441$	−4.60577e18			−1.41981
$442$		−	1.05503e17i		−	0.0320119i
$443$	−3.11048e18			−0.928970			−0.464485	−	0.885581i	$-0.653760\pi$
$443$	−3.11048e18			−0.928970			−0.464485	+	0.885581i	$0.653760\pi$
$444$			7.52845e16i			0.0221322i
$445$	4.19888e18			1.21510
$446$	−1.28043e18			−0.364763
$447$			7.72259e17i			0.216576i
$448$		−	2.36773e18i		−	0.653711i
$449$	−4.03214e18			−1.09600			−0.548001	−	0.836477i	$-0.684611\pi$
$449$	−4.03214e18			−1.09600			−0.548001	+	0.836477i	$0.684611\pi$
$450$	−4.76003e18			−1.27386
$451$			2.08492e17i			0.0549355i
$452$		−	5.03067e18i		−	1.30514i
$453$	3.25105e18			0.830494
$454$		−	1.83679e18i		−	0.462028i
$455$	1.04473e18			0.258777
$456$			2.25389e18i			0.549768i
$457$			4.04424e18i			0.971457i	0.874110	+	0.485729i	$0.161446\pi$
$457$			4.04424e18i			0.971457i	−0.874110	+	0.485729i	$0.838554\pi$
$458$			1.53662e18i			0.363503i
$459$		−	1.20539e19i		−	2.80826i
$460$	6.33070e18	−	1.27984e18i	1.45260	−	0.293664i
$461$	−6.53435e18			−1.47671			−0.738357	−	0.674410i	$-0.764398\pi$
$461$	−6.53435e18			−1.47671			−0.738357	+	0.674410i	$0.764398\pi$
$462$	4.33105e18			0.964049
$463$	−1.41038e18			−0.309222			−0.154611	−	0.987975i	$-0.549412\pi$
$463$	−1.41038e18			−0.309222			−0.154611	+	0.987975i	$0.549412\pi$
$464$	−3.06096e18			−0.661047
$465$			1.92744e18i			0.410026i
$466$	−3.56220e17			−0.0746478
$467$			4.59904e18i			0.949399i	0.880148	+	0.474700i	$0.157443\pi$
$467$			4.59904e18i			0.949399i	−0.880148	+	0.474700i	$0.842557\pi$
$468$	−1.58206e18			−0.321738
$469$	−4.59989e17			−0.0921590
$470$		−	1.99479e18i		−	0.393743i
$471$		−	6.33939e18i		−	1.23282i
$472$	−3.63221e18			−0.695949
$473$	−1.07407e18			−0.202771
$474$			3.99823e18i			0.743738i
$475$			4.53315e18i			0.830893i
$476$	−5.32922e18			−0.962531
$477$			1.73180e19i			3.08226i
$478$	−1.17438e18			−0.205975
$479$			4.83162e18i			0.835112i	0.908651	+	0.417556i	$0.137113\pi$
$479$			4.83162e18i			0.835112i	−0.908651	+	0.417556i	$0.862887\pi$
$480$		−	1.41828e19i		−	2.41586i
$481$		−	9.68047e15i		−	0.00162510i
$482$		−	1.83535e18i		−	0.303662i
$483$	1.42941e19	−	2.88974e18i	2.33091	−	0.471226i
$484$	5.42224e18			0.871488
$485$	5.15380e18			0.816461
$486$	6.91118e18			1.07919
$487$	2.79325e18			0.429940			0.214970	−	0.976621i	$-0.431035\pi$
$487$	2.79325e18			0.429940			0.214970	+	0.976621i	$0.431035\pi$
$488$		−	3.65130e18i		−	0.553998i
$489$	1.18741e19			1.77599
$490$			1.66752e18i			0.245865i
$491$	−6.09666e18			−0.886180			−0.443090	−	0.896477i	$-0.646118\pi$
$491$	−6.09666e18			−0.886180			−0.443090	+	0.896477i	$0.646118\pi$
$492$	4.84173e17			0.0693817
$493$			5.28090e18i			0.746070i
$494$		−	1.38539e17i		−	0.0192967i
$495$	−4.46419e19			−6.13065
$496$	−7.32792e17			−0.0992223
$497$			1.19040e19i			1.58928i
$498$			8.53106e17i			0.112305i
$499$	−3.16096e18			−0.410313			−0.205157	−	0.978729i	$-0.565770\pi$
$499$	−3.16096e18			−0.410313			−0.205157	+	0.978729i	$0.565770\pi$
$500$		−	7.16413e18i		−	0.917009i
$501$	2.59591e19			3.27662
$502$			2.33492e18i			0.290633i
$503$		−	5.00310e18i		−	0.614134i	−0.951688	−	0.307067i	$-0.900652\pi$
$503$		−	5.00310e18i		−	0.614134i	0.951688	−	0.307067i	$-0.0993476\pi$
$504$		−	1.53720e19i		−	1.86087i
$505$			2.22453e19i			2.65582i
$506$	−3.37466e18	+	6.82236e17i	−0.397353	+	0.0803306i
$507$	−1.63117e19			−1.89427
$508$	−4.88008e18			−0.558961
$509$	1.26182e18			0.142552			0.0712762	−	0.997457i	$-0.477293\pi$
$509$	1.26182e18			0.142552			0.0712762	+	0.997457i	$0.477293\pi$
$510$	−6.91350e18			−0.770383
$511$			5.91297e17i			0.0649919i
$512$	9.26074e18			1.00405
$513$		−	1.58282e19i		−	1.69282i
$514$	3.34392e18			0.352789
$515$	2.23296e19			2.32397
$516$			2.49428e18i			0.256093i
$517$		−	1.15643e19i		−	1.17135i
$518$	4.49626e16			0.00449306
$519$	2.95713e18			0.291541
$520$			1.19824e18i			0.116552i
$521$		−	1.88781e19i		−	1.81173i	−0.423568	−	0.905864i	$-0.639223\pi$
$521$		−	1.88781e19i		−	1.81173i	0.423568	−	0.905864i	$-0.360777\pi$
$522$	−7.28154e18			−0.689493
$523$		−	1.44378e19i		−	1.34893i	−0.738305	−	0.674467i	$-0.764374\pi$
$523$		−	1.44378e19i		−	1.34893i	0.738305	−	0.674467i	$-0.235626\pi$
$524$	6.01389e17			0.0554417
$525$		−	4.23179e19i		−	3.84954i
$526$			5.18011e18i			0.464985i
$527$			1.26424e18i			0.111984i
$528$		−	2.32310e19i		−	2.03063i
$529$	−1.06824e19	+	4.50326e18i	−0.921469	+	0.388452i
$530$	6.26998e18			0.533747
$531$	4.04086e19			3.39478
$532$	−6.99792e18			−0.580213
$533$	−6.22575e16			−0.00509449
$534$			5.19786e18i			0.419794i
$535$	−2.85439e19			−2.27529
$536$		−	5.27578e17i		−	0.0415081i
$537$	1.36316e19			1.05859
$538$	7.18481e18			0.550731
$539$			9.66698e18i			0.731425i
$540$			6.54413e19i			4.88760i
$541$	8.14336e18			0.600375			0.300187	−	0.953880i	$-0.402951\pi$
$541$	8.14336e18			0.600375			0.300187	+	0.953880i	$0.402951\pi$
$542$	−6.52095e18			−0.474587
$543$		−	6.00516e18i		−	0.431445i
$544$		−	9.30273e18i		−	0.659809i
$545$	−2.15053e19			−1.50581
$546$			1.29329e18i			0.0894020i
$547$	−7.90232e18			−0.539316			−0.269658	−	0.962956i	$-0.586911\pi$
$547$	−7.90232e18			−0.539316			−0.269658	+	0.962956i	$0.586911\pi$
$548$		−	1.16681e18i		−	0.0786203i
$549$			4.06209e19i			2.70236i
$550$			9.99077e18i			0.656236i
$551$			6.93448e18i			0.449731i
$552$	3.31435e18	+	1.63943e19i	0.212239	+	1.04983i
$553$	−2.59690e19			−1.64203
$554$	5.82587e18			0.363741
$555$	−6.34348e17			−0.0391090
$556$	1.70693e19			1.03918
$557$		−	1.21146e19i		−	0.728323i	−0.931336	−	0.364161i	$-0.881356\pi$
$557$		−	1.21146e19i		−	0.728323i	0.931336	−	0.364161i	$-0.118644\pi$
$558$	−1.74320e18			−0.103492
$559$		−	3.20727e17i		−	0.0188041i
$560$	2.60277e19			1.50702
$561$	−4.00792e19			−2.29181
$562$			5.00079e18i			0.282413i
$563$		−	2.87541e19i		−	1.60376i	−0.597482	−	0.801882i	$-0.703832\pi$
$563$		−	2.87541e19i		−	1.60376i	0.597482	−	0.801882i	$-0.296168\pi$
$564$	−2.68553e19			−1.47937
$565$	4.23885e19			2.30626
$566$		−	8.53319e18i		−	0.458560i
$567$			8.46968e19i			4.49558i
$568$	−1.36532e19			−0.715805
$569$		−	2.63799e19i		−	1.36612i	−0.730364	−	0.683058i	$-0.760649\pi$
$569$		−	2.63799e19i		−	1.36612i	0.730364	−	0.683058i	$-0.239351\pi$
$570$	−9.07828e18			−0.464386
$571$			3.18808e19i			1.61093i	0.592642	+	0.805466i	$0.298085\pi$
$571$			3.18808e19i			1.61093i	−0.592642	+	0.805466i	$0.701915\pi$
$572$			3.32057e18i			0.165745i
$573$		−	3.61744e19i		−	1.78369i
$574$		−	2.89165e17i		−	0.0140852i
$575$	6.66600e18	+	3.29732e19i	0.320768	+	1.58667i
$576$	−3.02115e19			−1.43620
$577$	2.35676e19			1.10684			0.553421	−	0.832902i	$-0.313322\pi$
$577$	2.35676e19			1.10684			0.553421	+	0.832902i	$0.313322\pi$
$578$	1.71952e18			0.0797835
$579$	−1.88925e19			−0.866042
$580$		−	2.86704e19i		−	1.29849i
$581$	−5.54103e18			−0.247946
$582$			6.37997e18i			0.282071i
$583$	3.63485e19			1.58784
$584$	−6.78179e17			−0.0292721
$585$		−	1.33305e19i		−	0.568532i
$586$			1.63015e17i			0.00686978i
$587$	5.06490e18			0.210913			0.105457	−	0.994424i	$-0.466370\pi$
$587$	5.06490e18			0.210913			0.105457	+	0.994424i	$0.466370\pi$
$588$	2.24493e19			0.923765
$589$			1.66011e18i			0.0675040i
$590$		−	1.46299e19i		−	0.587865i
$591$	1.02113e19			0.405478
$592$		−	2.41172e17i		−	0.00946399i
$593$	−3.36450e19			−1.30478			−0.652392	−	0.757882i	$-0.726234\pi$
$593$	−3.36450e19			−1.30478			−0.652392	+	0.757882i	$0.726234\pi$
$594$		−	3.48843e19i		−	1.33698i
$595$		−	4.49040e19i		−	1.70085i
$596$		−	2.75003e18i		−	0.102947i
$597$		−	1.43356e19i		−	0.530391i
$598$	−2.03722e17	−	1.00771e18i	−0.00744953	−	0.0368490i
$599$	4.11882e19			1.48862			0.744312	−	0.667832i	$-0.232777\pi$
$599$	4.11882e19			1.48862			0.744312	+	0.667832i	$0.232777\pi$
$600$	4.85358e19			1.73382
$601$	−4.34423e18			−0.153388			−0.0766940	−	0.997055i	$-0.524436\pi$
$601$	−4.34423e18			−0.153388			−0.0766940	+	0.997055i	$0.524436\pi$
$602$	1.48967e18			0.0519894
$603$			5.86933e18i			0.202473i
$604$	−1.15771e19			−0.394766
$605$			4.56879e19i			1.53997i
$606$	−2.75378e19			−0.917530
$607$	−3.89856e19			−1.28405			−0.642027	−	0.766682i	$-0.721906\pi$
$607$	−3.89856e19			−1.28405			−0.642027	+	0.766682i	$0.721906\pi$
$608$		−	1.22156e19i		−	0.397732i
$609$		−	6.47347e19i		−	2.08361i
$610$	1.47068e19			0.467960
$611$	3.45320e18			0.108626
$612$			6.79993e19i			2.11468i
$613$			4.44811e19i			1.36758i	0.729679	+	0.683790i	$0.239669\pi$
$613$			4.44811e19i			1.36758i	−0.729679	+	0.683790i	$0.760331\pi$
$614$	−1.45439e19			−0.442081
$615$			4.07965e18i			0.122602i
$616$	−3.22641e19			−0.958637
$617$		−	7.12788e17i		−	0.0209394i	−0.999945	−	0.0104697i	$-0.996667\pi$
$617$		−	7.12788e17i		−	0.0209394i	0.999945	−	0.0104697i	$-0.00333267\pi$
$618$			2.76421e19i			0.802883i
$619$			4.70421e19i			1.35099i	0.737364	+	0.675495i	$0.236070\pi$
$619$			4.70421e19i			1.35099i	−0.737364	+	0.675495i	$0.763930\pi$
$620$		−	6.86367e18i		−	0.194901i
$621$	−2.32754e19	−	1.15131e20i	−0.653515	−	3.23260i
$622$	−2.76206e18			−0.0766833
$623$	−3.37608e19			−0.926821
$624$	6.93699e18			0.188312
$625$	6.11965e16			0.00164273
$626$			3.87627e17i			0.0102895i
$627$	−5.26289e19			−1.38150
$628$			2.25747e19i			0.586010i
$629$	−4.16080e17			−0.0106812
$630$	6.19157e19			1.57187
$631$			1.82104e19i			0.457209i	0.973519	+	0.228605i	$0.0734163\pi$
$631$			1.82104e19i			0.457209i	−0.973519	+	0.228605i	$0.926584\pi$
$632$		−	2.97848e19i		−	0.739562i
$633$	8.87972e19			2.18059
$634$	1.38294e18			0.0335876
$635$		−	4.11196e19i		−	0.987718i
$636$		−	8.44110e19i		−	2.00539i
$637$	−2.88665e18			−0.0678294
$638$			1.52831e19i			0.355196i
$639$	1.51892e20			3.49164
$640$			6.60872e19i			1.50265i
$641$		−	1.53351e19i		−	0.344889i	−0.985019	−	0.172445i	$-0.944833\pi$
$641$		−	1.53351e19i		−	0.344889i	0.985019	−	0.172445i	$-0.0551666\pi$
$642$		−	3.53350e19i		−	0.786067i
$643$			3.53762e19i			0.778457i	0.921141	+	0.389228i	$0.127258\pi$
$643$			3.53762e19i			0.778457i	−0.921141	+	0.389228i	$0.872742\pi$
$644$	−5.09014e19	+	1.02904e19i	−1.10797	+	0.223992i
$645$	−2.10168e19			−0.452531
$646$	−5.95460e18			−0.126831
$647$	2.83766e19			0.597903			0.298952	−	0.954268i	$-0.403363\pi$
$647$	2.83766e19			0.597903			0.298952	+	0.954268i	$0.403363\pi$
$648$	−9.71417e19			−2.02479
$649$		−	8.48130e19i		−	1.74884i
$650$	−2.98333e18			−0.0608567
$651$		−	1.54974e19i		−	0.312747i
$652$	−4.22840e19			−0.844196
$653$	−3.15224e19			−0.622625			−0.311312	−	0.950308i	$-0.600769\pi$
$653$	−3.15224e19			−0.622625			−0.311312	+	0.950308i	$0.600769\pi$
$654$		−	2.66217e19i		−	0.520225i
$655$			5.06731e18i			0.0979689i
$656$	−1.55103e18			−0.0296684
$657$	7.54477e18			0.142787
$658$			1.60390e19i			0.300327i
$659$			3.69496e19i			0.684560i	0.939598	+	0.342280i	$0.111199\pi$
$659$			3.69496e19i			0.684560i	−0.939598	+	0.342280i	$0.888801\pi$
$660$	2.17592e20			3.98875
$661$			3.88211e19i			0.704138i	0.935974	+	0.352069i	$0.114522\pi$
$661$			3.88211e19i			0.704138i	−0.935974	+	0.352069i	$0.885478\pi$
$662$	2.05656e19			0.369094
$663$		−	1.19680e19i		−	0.212533i
$664$		−	6.35520e18i		−	0.111674i
$665$		−	5.89646e19i		−	1.02527i
$666$		−	5.73709e17i		−	0.00987125i
$667$	1.01972e19	+	5.04400e19i	0.173619	+	0.858804i
$668$	−9.24410e19			−1.55750
$669$	−1.45248e20			−2.42173
$670$	2.12499e18			0.0350617
$671$	8.52586e19			1.39213
$672$			1.14035e20i			1.84270i
$673$	3.44455e19			0.550842			0.275421	−	0.961324i	$-0.411183\pi$
$673$	3.44455e19			0.550842			0.275421	+	0.961324i	$0.411183\pi$
$674$		−	2.12356e19i		−	0.336082i
$675$	−3.40848e20			−5.33869
$676$	5.80861e19			0.900421
$677$		−	5.03380e19i		−	0.772281i	−0.922440	−	0.386140i	$-0.873808\pi$
$677$		−	5.03380e19i		−	0.772281i	0.922440	−	0.386140i	$-0.126192\pi$
$678$			5.24733e19i			0.796765i
$679$	−4.14387e19			−0.622756
$680$	5.15020e19			0.766058
$681$		−	2.08360e20i		−	3.06750i
$682$			3.65877e18i			0.0533144i
$683$	2.26176e19			0.326213			0.163107	−	0.986608i	$-0.447849\pi$
$683$	2.26176e19			0.326213			0.163107	+	0.986608i	$0.447849\pi$
$684$			8.92915e19i			1.27473i
$685$	9.83152e18			0.138927
$686$			1.22005e19i			0.170652i
$687$			1.74309e20i			2.41337i
$688$		−	7.99035e18i		−	0.109508i
$689$			1.08540e19i			0.147250i
$690$	−6.60335e19	+	1.33496e19i	−0.886790	+	0.179277i
$691$	6.18273e19			0.821928			0.410964	−	0.911652i	$-0.365192\pi$
$691$	6.18273e19			0.821928			0.410964	+	0.911652i	$0.365192\pi$
$692$	−1.05304e19			−0.138581
$693$	3.58940e20			4.67615
$694$	−3.71113e19			−0.478619
$695$			1.43826e20i			1.83630i
$696$	7.42465e19			0.938450
$697$			2.67591e18i			0.0334844i
$698$	8.80239e17			0.0109047
$699$	−4.04085e19			−0.495602
$700$			1.50695e20i			1.82984i
$701$			8.92013e19i			1.07237i	0.844100	+	0.536185i	$0.180135\pi$
$701$			8.92013e19i			1.07237i	−0.844100	+	0.536185i	$0.819865\pi$
$702$	1.04168e19			0.123986
$703$	−5.46364e17			−0.00643865
$704$			6.34105e19i			0.739866i
$705$		−	2.26283e20i		−	2.61414i
$706$	7.28831e18			0.0833670
$707$		−	1.78861e20i		−	2.02572i
$708$	−1.96958e20			−2.20872
$709$			1.58691e20i			1.76209i	0.473028	+	0.881047i	$0.343161\pi$
$709$			1.58691e20i			1.76209i	−0.473028	+	0.881047i	$0.656839\pi$
$710$		−	5.49925e19i		−	0.604637i
$711$			3.31357e20i			3.60752i
$712$		−	3.87214e19i		−	0.417437i
$713$	2.44119e18	+	1.20753e19i	0.0260600	+	0.128905i
$714$	5.55874e19			0.587609
$715$	−2.79792e19			−0.292882
$716$	−4.85426e19			−0.503189
$717$	−1.33219e20			−1.36751
$718$			4.05909e19i			0.412627i
$719$	−1.17605e20			−1.18392			−0.591960	−	0.805968i	$-0.701645\pi$
$719$	−1.17605e20			−1.18392			−0.591960	+	0.805968i	$0.701645\pi$
$720$		−	3.32105e20i		−	3.31092i
$721$	−1.79539e20			−1.77261
$722$	2.18591e19			0.213733
$723$		−	2.08197e20i		−	2.01607i
$724$			2.13845e19i			0.205083i
$725$	1.49329e20			1.41833
$726$	−5.65577e19			−0.532029
$727$		−	2.07014e19i		−	0.192868i	−0.995339	−	0.0964339i	$-0.969256\pi$
$727$		−	2.07014e19i		−	0.192868i	0.995339	−	0.0964339i	$-0.0307436\pi$
$728$		−	9.63434e18i		−	0.0889001i
$729$	3.85465e20			3.52283
$730$		−	2.73158e18i		−	0.0247260i
$731$	−1.37853e19			−0.123593
$732$		−	1.97993e20i		−	1.75822i
$733$		−	1.64753e20i		−	1.44912i	−0.689210	−	0.724561i	$-0.742042\pi$
$733$		−	1.64753e20i		−	1.44912i	0.689210	−	0.724561i	$-0.257958\pi$
$734$			4.20176e19i			0.366066i
$735$			1.89158e20i			1.63235i
$736$	−1.79631e19	−	8.88540e19i	−0.153545	−	0.759508i
$737$	1.23191e19			0.104305
$738$	−3.68966e18			−0.0309451
$739$	−1.35568e20			−1.12628			−0.563141	−	0.826361i	$-0.690407\pi$
$739$	−1.35568e20			−1.12628			−0.563141	+	0.826361i	$0.690407\pi$
$740$	2.25893e18			0.0185900
$741$		−	1.57155e19i		−	0.128115i
$742$	−5.04132e19			−0.407115
$743$			1.21668e20i			0.973314i	0.873593	+	0.486657i	$0.161784\pi$
$743$			1.21668e20i			0.973314i	−0.873593	+	0.486657i	$0.838216\pi$
$744$	1.77746e19			0.140860
$745$	2.31718e19			0.181914
$746$		−	2.20222e19i		−	0.171273i
$747$			7.07019e19i			0.544737i
$748$	1.42723e20			1.08939
$749$	2.29505e20			1.73548
$750$			7.47267e19i			0.559818i
$751$			3.48343e19i			0.258540i	0.991609	+	0.129270i	$0.0412633\pi$
$751$			3.48343e19i			0.258540i	−0.991609	+	0.129270i	$0.958737\pi$
$752$	8.60303e19			0.632596
$753$			2.64866e20i			1.92957i
$754$	−4.56368e18			−0.0329394
$755$		−	9.75485e19i		−	0.697577i
$756$		−	5.26175e20i		−	3.72801i
$757$		−	1.69995e20i		−	1.19335i	−0.802485	−	0.596673i	$-0.796489\pi$
$757$		−	1.69995e20i		−	1.19335i	0.802485	−	0.596673i	$-0.203511\pi$
$758$			4.01063e17i			0.00278951i
$759$	−3.82812e20	+	7.73908e19i	−2.63811	+	0.533331i
$760$	6.76285e19			0.461779
$761$	1.02483e20			0.693363			0.346682	−	0.937983i	$-0.387308\pi$
$761$	1.02483e20			0.693363			0.346682	+	0.937983i	$0.387308\pi$
$762$	5.09026e19			0.341236
$763$	1.72911e20			1.14855
$764$			1.28818e20i			0.847857i
$765$	−5.72962e20			−3.73677
$766$		−	3.99421e19i		−	0.258125i
$767$	2.53259e19			0.162180
$768$	7.89861e19			0.501213
$769$			9.46898e19i			0.595413i	0.954657	+	0.297707i	$0.0962217\pi$
$769$			9.46898e19i			0.595413i	−0.954657	+	0.297707i	$0.903778\pi$
$770$		−	1.29954e20i		−	0.809757i
$771$	3.79325e20			2.34223
$772$	6.72765e19			0.411664
$773$			8.04353e19i			0.487742i	0.969808	+	0.243871i	$0.0784173\pi$
$773$			8.04353e19i			0.487742i	−0.969808	+	0.243871i	$0.921583\pi$
$774$		−	1.90078e19i		−	0.114221i
$775$	3.57492e19			0.212889
$776$		−	4.75275e19i		−	0.280487i
$777$	5.10042e18			0.0298304
$778$			1.65697e19i			0.0960412i
$779$			3.51380e18i			0.0201844i
$780$			6.49751e19i			0.369900i
$781$		−	3.18804e20i		−	1.79873i
$782$	−4.33125e19	+	8.75624e18i	−0.242196	+	0.0489633i
$783$	−5.21405e20			−2.88963
$784$	−7.19157e19			−0.395013
$785$	−1.90215e20			−1.03552
$786$	−6.27290e18			−0.0338462
$787$		−	2.83915e20i		−	1.51832i	−0.650901	−	0.759162i	$-0.725609\pi$
$787$		−	2.83915e20i		−	1.51832i	0.650901	−	0.759162i	$-0.274391\pi$
$788$	−3.63626e19			−0.192740
$789$			5.87616e20i			3.08713i
$790$	1.19968e20			0.624705
$791$	−3.40821e20			−1.75910
$792$			4.11680e20i			2.10612i
$793$			2.54590e19i			0.129101i
$794$	5.01065e19			0.251855
$795$	7.11248e20			3.54365
$796$			5.10495e19i			0.252116i
$797$		−	6.85079e19i		−	0.335377i	−0.985840	−	0.167688i	$-0.946370\pi$
$797$		−	6.85079e19i		−	0.335377i	0.985840	−	0.167688i	$-0.0536302\pi$
$798$	7.29931e19			0.354211
$799$		−	1.48423e20i		−	0.713961i
$800$	−2.63054e20			−1.25434
$801$			4.30777e20i			2.03622i
$802$		−	6.80258e19i		−	0.318752i
$803$		−	1.58356e19i		−	0.0735574i
$804$		−	2.86081e19i		−	0.131734i
$805$	−8.67074e19	−	4.28896e20i	−0.395808	−	1.95786i
$806$	−1.09254e18			−0.00494416
$807$	8.15024e20			3.65641
$808$	2.05142e20			0.912379
$809$	4.40997e19			0.194445			0.0972223	−	0.995263i	$-0.469004\pi$
$809$	4.40997e19			0.194445			0.0972223	+	0.995263i	$0.469004\pi$
$810$		−	3.91270e20i		−	1.71033i
$811$	1.34688e20			0.583690			0.291845	−	0.956466i	$-0.405731\pi$
$811$	1.34688e20			0.583690			0.291845	+	0.956466i	$0.405731\pi$
$812$			2.30522e20i			0.990421i
$813$	−7.39716e20			−3.15088
$814$	−1.20415e18			−0.00508522
$815$		−	3.56286e20i		−	1.49175i
$816$		−	2.98161e20i		−	1.23771i
$817$	−1.81018e19			−0.0745019
$818$	−1.01627e19			−0.0414703
$819$			1.07183e20i			0.433648i
$820$		−	1.45277e19i		−	0.0582774i
$821$	−2.53940e20			−1.01002			−0.505010	−	0.863114i	$-0.668511\pi$
$821$	−2.53940e20			−1.01002			−0.505010	+	0.863114i	$0.668511\pi$
$822$			1.21706e19i			0.0479964i
$823$	−3.69511e20			−1.44487			−0.722435	−	0.691439i	$-0.756977\pi$
$823$	−3.69511e20			−1.44487			−0.722435	+	0.691439i	$0.756977\pi$
$824$		−	2.05920e20i		−	0.798376i
$825$			1.13332e21i			4.35688i
$826$			1.17631e20i			0.448394i
$827$		−	1.11003e20i		−	0.419561i	−0.977748	−	0.209781i	$-0.932725\pi$
$827$		−	1.11003e20i		−	0.419561i	0.977748	−	0.209781i	$-0.0672750\pi$
$828$	1.31303e20	+	6.49488e20i	0.492110	+	2.43421i
$829$	2.15469e20			0.800762			0.400381	−	0.916349i	$-0.368878\pi$
$829$	2.15469e20			0.800762			0.400381	+	0.916349i	$0.368878\pi$
$830$	2.55976e19			0.0943307
$831$	6.60869e20			2.41495
$832$	−1.89350e19			−0.0686121
$833$			1.24072e20i			0.445819i
$834$	−1.78044e20			−0.634403
$835$		−	7.78908e20i		−	2.75221i
$836$	1.87413e20			0.656682
$837$	−1.24824e20			−0.433730
$838$		−	4.20935e19i		−	0.145046i
$839$			5.49253e20i			1.87689i	0.345431	+	0.938444i	$0.387733\pi$
$839$			5.49253e20i			1.87689i	−0.345431	+	0.938444i	$0.612267\pi$
$840$	−6.31325e20			−2.13943
$841$	−6.91264e19			−0.232312
$842$		−	1.11806e20i		−	0.372631i
$843$			5.67274e20i			1.87500i
$844$	−3.16209e20			−1.03652
$845$			4.89434e20i			1.59110i
$846$	2.04652e20			0.659818
$847$		−	3.67349e20i		−	1.17461i
$848$			2.70408e20i			0.857529i
$849$		−	9.67979e20i		−	3.04447i
$850$			1.28228e20i			0.399990i
$851$	−3.97414e18	+	8.03429e17i	−0.0122952	+	0.00248565i
$852$	−7.40348e20			−2.27174
$853$	4.70723e20			1.43259			0.716296	−	0.697797i	$-0.245836\pi$
$853$	4.70723e20			1.43259			0.716296	+	0.697797i	$0.245836\pi$
$854$	−1.18249e20			−0.356936
$855$	−7.52371e20			−2.25252
$856$			2.63227e20i			0.781654i
$857$	2.60371e20			0.766879			0.383440	−	0.923566i	$-0.374739\pi$
$857$	2.60371e20			0.766879			0.383440	+	0.923566i	$0.374739\pi$
$858$		−	3.46358e19i		−	0.101185i
$859$	2.70905e20			0.784991			0.392496	−	0.919754i	$-0.371612\pi$
$859$	2.70905e20			0.784991			0.392496	+	0.919754i	$0.371612\pi$
$860$	7.48413e19			0.215106
$861$		−	3.28020e19i		−	0.0935144i
$862$			6.47061e19i			0.182976i
$863$	−1.14752e20			−0.321873			−0.160936	−	0.986965i	$-0.551451\pi$
$863$	−1.14752e20			−0.321873			−0.160936	+	0.986965i	$0.551451\pi$
$864$	9.18495e20			2.55553
$865$		−	8.87293e19i		−	0.244881i
$866$			1.74409e20i			0.477467i
$867$	1.95057e20			0.529699
$868$			5.51868e19i			0.148661i
$869$	6.95481e20			1.85843
$870$			2.99052e20i			0.792705i
$871$			3.67858e18i			0.00967282i
$872$			1.98318e20i			0.517305i
$873$			5.28746e20i			1.36819i
$874$	−5.68747e19	+	1.14980e19i	−0.145995	+	0.0295150i
$875$	−4.85360e20			−1.23597
$876$	−3.67745e19			−0.0929006
$877$	1.48805e20			0.372923			0.186462	−	0.982462i	$-0.440298\pi$
$877$	1.48805e20			0.372923			0.186462	+	0.982462i	$0.440298\pi$
$878$	1.03869e20			0.258241
$879$			1.84919e19i			0.0456099i
$880$	−6.97052e20			−1.70564
$881$		−	2.61695e20i		−	0.635279i	−0.948212	−	0.317640i	$-0.897110\pi$
$881$		−	2.61695e20i		−	0.635279i	0.948212	−	0.317640i	$-0.102890\pi$
$882$	−1.71076e20			−0.412011
$883$	2.13998e20			0.511311			0.255655	−	0.966768i	$-0.417709\pi$
$883$	2.13998e20			0.511311			0.255655	+	0.966768i	$0.417709\pi$
$884$			4.26183e19i			0.101025i
$885$		−	1.65957e21i		−	3.90295i
$886$	−1.15535e20			−0.269575
$887$	−6.43804e20			−1.49035			−0.745175	−	0.666869i	$-0.767634\pi$
$887$	−6.43804e20			−1.49035			−0.745175	+	0.666869i	$0.767634\pi$
$888$			5.84984e18i			0.0134355i
$889$			3.30619e20i			0.753382i
$890$	1.55963e20			0.352607
$891$		−	2.26828e21i		−	5.08807i
$892$	5.17230e20			1.15114
$893$		−	1.94898e20i		−	0.430375i
$894$			2.86847e19i			0.0628475i
$895$		−	4.09020e20i		−	0.889167i
$896$		−	5.31369e20i		−	1.14615i
$897$	−2.31096e19	−	1.14311e20i	−0.0494590	−	0.244648i
$898$	−1.49769e20			−0.318045
$899$	5.46864e19			0.115229
$900$	1.92282e21			4.02014
$901$	4.66520e20			0.967824
$902$			7.74419e18i			0.0159415i
$903$	1.68984e20			0.345168
$904$		−	3.90899e20i		−	0.792293i
$905$	−1.80186e20			−0.362394
$906$	1.20757e20			0.240998
$907$			4.57941e20i			0.906898i	0.891282	+	0.453449i	$0.149807\pi$
$907$			4.57941e20i			0.906898i	−0.891282	+	0.453449i	$0.850193\pi$
$908$			7.41974e20i			1.45810i
$909$	−2.28222e21			−4.45051
$910$	3.88054e19			0.0750936
$911$		−	2.01155e20i		−	0.386280i	−0.981171	−	0.193140i	$-0.938133\pi$
$911$		−	2.01155e20i		−	0.386280i	0.981171	−	0.193140i	$-0.0618671\pi$
$912$		−	3.91523e20i		−	0.746094i
$913$	1.48395e20			0.280624
$914$			1.50219e20i			0.281904i
$915$	1.66829e21			3.10688
$916$		−	6.20720e20i		−	1.14717i
$917$		−	4.07433e19i		−	0.0747258i
$918$		−	4.47727e20i		−	0.814920i
$919$			4.09841e20i			0.740299i	0.928972	+	0.370150i	$0.120694\pi$
$919$			4.09841e20i			0.740299i	−0.928972	+	0.370150i	$0.879306\pi$
$920$	4.91916e20	−	9.94477e19i	0.881812	−	0.178271i
$921$	−1.64981e21			−2.93506
$922$	−2.42711e20			−0.428522
$923$	9.51978e19			0.166807
$924$	−1.74953e21			−3.04242
$925$			1.17655e19i			0.0203058i
$926$	−5.23871e19			−0.0897321
$927$			2.29087e21i			3.89441i
$928$	−4.02401e20			−0.678927
$929$	4.38840e20			0.734846			0.367423	−	0.930054i	$-0.380240\pi$
$929$	4.38840e20			0.734846			0.367423	+	0.930054i	$0.380240\pi$
$930$			7.15928e19i			0.118984i
$931$			1.62922e20i			0.268740i
$932$	1.43896e20			0.235579
$933$	−3.13320e20			−0.509116
$934$			1.70826e20i			0.275503i
$935$			1.20258e21i			1.92501i
$936$	−1.22931e20			−0.195313
$937$			6.13111e20i			0.966856i	0.875384	+	0.483428i	$0.160609\pi$
$937$			6.13111e20i			0.966856i	−0.875384	+	0.483428i	$0.839391\pi$
$938$	−1.70858e19			−0.0267433
$939$			4.39713e19i			0.0683140i
$940$			8.05800e20i			1.24260i
$941$			6.56084e20i			1.00423i	0.864802	+	0.502114i	$0.167444\pi$
$941$			6.56084e20i			1.00423i	−0.864802	+	0.502114i	$0.832556\pi$
$942$		−	2.35470e20i		−	0.357749i
$943$	5.16705e18	+	2.55587e19i	0.00779221	+	0.0385440i
$944$	6.30951e20			0.944477
$945$	4.43355e21			6.58763
$946$	−3.98952e19			−0.0588413
$947$	−1.27235e21			−1.86276			−0.931381	−	0.364047i	$-0.881395\pi$
$947$	−1.27235e21			−1.86276			−0.931381	+	0.364047i	$0.881395\pi$
$948$		−	1.61509e21i		−	2.34714i
$949$	4.72866e18			0.00682142
$950$			1.68379e20i			0.241114i
$951$	1.56877e20			0.222995
$952$	−4.14097e20			−0.584311
$953$		−	1.30381e21i		−	1.82627i	−0.407659	−	0.913134i	$-0.633655\pi$
$953$		−	1.30381e21i		−	1.82627i	0.407659	−	0.913134i	$-0.366345\pi$
$954$			6.43258e20i			0.894431i
$955$	−1.08542e21			−1.49822
$956$	4.74395e20			0.650032
$957$			1.73367e21i			2.35822i
$958$			1.79465e20i			0.242338i
$959$	−7.90495e19			−0.105967
$960$			1.24078e21i			1.65119i
$961$	−7.43852e20			−0.982704
$962$		−	3.59570e17i		−	0.000471583i
$963$		−	2.92842e21i		−	3.81284i
$964$			7.41393e20i			0.958317i
$965$			5.66873e20i			0.727435i
$966$	5.30937e20	−	1.07336e20i	0.676399	−	0.136744i
$967$	−3.02376e20			−0.382439			−0.191220	−	0.981547i	$-0.561244\pi$
$967$	−3.02376e20			−0.382439			−0.191220	+	0.981547i	$0.561244\pi$
$968$	4.21326e20			0.529042
$969$	−6.75472e20			−0.842056
$970$	1.91432e20			0.236926
$971$		−	9.44668e20i		−	1.16077i	−0.814344	−	0.580383i	$-0.802903\pi$
$971$		−	9.44668e20i		−	1.16077i	0.814344	−	0.580383i	$-0.197097\pi$
$972$	−2.79178e21			−3.40579
$973$		−	1.15642e21i		−	1.40064i
$974$	1.03752e20			0.124763
$975$	−3.38420e20			−0.404040
$976$			6.34265e20i			0.751835i
$977$			9.12006e20i			1.07334i	0.843793	+	0.536668i	$0.180317\pi$
$977$			9.12006e20i			1.07334i	−0.843793	+	0.536668i	$0.819683\pi$
$978$	4.41051e20			0.515367
$979$	9.04153e20			1.04897
$980$		−	6.73596e20i		−	0.775920i
$981$		−	2.20630e21i		−	2.52337i
$982$	−2.26454e20			−0.257157
$983$			6.08142e20i			0.685694i	0.939391	+	0.342847i	$0.111391\pi$
$983$			6.08142e20i			0.685694i	−0.939391	+	0.342847i	$0.888609\pi$
$984$	3.76218e19			0.0421186
$985$		−	3.06391e20i		−	0.340583i
$986$			1.96153e20i			0.216500i
$987$			1.81941e21i			1.99393i
$988$			5.59631e19i			0.0608980i
$989$	−1.31669e20	+	2.66187e19i	−0.142269	+	0.0287616i
$990$	−1.65817e21			−1.77903
$991$	1.05541e21			1.12436			0.562178	−	0.827016i	$-0.309964\pi$
$991$	1.05541e21			1.12436			0.562178	+	0.827016i	$0.309964\pi$
$992$	−9.63345e19			−0.101906
$993$	2.33290e21			2.45049
$994$			4.42162e20i			0.461187i
$995$	−4.30143e20			−0.445504
$996$		−	3.44613e20i		−	0.354419i
$997$	−4.42807e20			−0.452219			−0.226109	−	0.974102i	$-0.572601\pi$
$997$	−4.42807e20			−0.452219			−0.226109	+	0.974102i	$0.572601\pi$
$998$	−1.17411e20			−0.119067
$999$		−	4.10812e19i		−	0.0413699i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.13	✓	24
23.22	odd	2	inner	23.15.b.b.22.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.13	✓	24	1.1	even	1	trivial
23.15.b.b.22.14	yes	24	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+37.1439 q^{2} +4213.49 q^{3} -15004.3 q^{4} -126427. i q^{5} +156505. q^{6} +1.01652e6i q^{7} -1.16588e6 q^{8} +1.29705e7 q^{9} +O(q^{10})\)	\(q+37.1439 q^{2} +4213.49 q^{3} -15004.3 q^{4} -126427. i q^{5} +156505. q^{6} +1.01652e6i q^{7} -1.16588e6 q^{8} +1.29705e7 q^{9} -4.69598e6i q^{10} -2.72237e7i q^{11} -6.32206e7 q^{12} +8.12923e6 q^{13} +3.77576e7i q^{14} -5.32697e8i q^{15} +2.02525e8 q^{16} -3.49406e8i q^{17} +4.81776e8 q^{18} -4.58813e8i q^{19} +1.89695e9i q^{20} +4.28311e9i q^{21} -1.01119e9i q^{22} +(-6.74684e8 - 3.33731e9i) q^{23} -4.91244e9 q^{24} -9.88018e9 q^{25} +3.01951e8 q^{26} +3.44982e10 q^{27} -1.52522e10i q^{28} -1.51140e10 q^{29} -1.97865e10i q^{30} -3.61827e9 q^{31} +2.66244e10 q^{32} -1.14707e11i q^{33} -1.29783e10i q^{34} +1.28516e11 q^{35} -1.94614e11 q^{36} -1.19082e9i q^{37} -1.70421e10i q^{38} +3.42524e10 q^{39} +1.47399e11i q^{40} -7.65847e9 q^{41} +1.59091e11i q^{42} -3.94536e10i q^{43} +4.08473e11i q^{44} -1.63982e12i q^{45} +(-2.50604e10 - 1.23961e11i) q^{46} +4.24788e11 q^{47} +8.53339e11 q^{48} -3.55095e11 q^{49} -3.66988e11 q^{50} -1.47222e12i q^{51} -1.21974e11 q^{52} +1.33518e12i q^{53} +1.28140e12 q^{54} -3.44180e12 q^{55} -1.18515e12i q^{56} -1.93320e12i q^{57} -5.61391e11 q^{58} +3.11541e12 q^{59} +7.99277e12i q^{60} +3.13178e12i q^{61} -1.34397e11 q^{62} +1.31848e13i q^{63} -2.32924e12 q^{64} -1.02775e12i q^{65} -4.26065e12i q^{66} +4.52513e11i q^{67} +5.24260e12i q^{68} +(-2.84278e12 - 1.40617e13i) q^{69} +4.77357e12 q^{70} +1.17106e13 q^{71} -1.51221e13 q^{72} +5.81686e11 q^{73} -4.42317e10i q^{74} -4.16300e13 q^{75} +6.88418e12i q^{76} +2.76735e13 q^{77} +1.27227e12 q^{78} +2.55469e13i q^{79} -2.56046e13i q^{80} +8.33202e13 q^{81} -2.84465e11 q^{82} +5.45097e12i q^{83} -6.42652e13i q^{84} -4.41742e13 q^{85} -1.46546e12i q^{86} -6.36825e13 q^{87} +3.17397e13i q^{88} +3.32120e13i q^{89} -6.09093e13i q^{90} +8.26355e12i q^{91} +(1.01232e13 + 5.00741e13i) q^{92} -1.52456e13 q^{93} +1.57783e13 q^{94} -5.80062e13 q^{95} +1.12182e14 q^{96} +4.07652e13i q^{97} -1.31896e13 q^{98} -3.53105e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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