LMFDB - Embedded newform 1045.2.b.c.419.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + \cdots + 4 $ x^20 + 26x^18 + 281x^16 + 1640x^14 + 5623x^12 + 11551x^10 + 13894x^8 + 9095x^6 + 2753x^4 + 276*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.4
Root			$-2.00304i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.c.419.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00304i q^{2} -2.62978i q^{3} -2.01218 q^{4} +(0.741511 + 2.10954i) q^{5} -5.26757 q^{6} -4.24477i q^{7} +0.0243959i q^{8} -3.91575 q^{9} +O(q^{10})$	\(q-2.00304i q^{2} -2.62978i q^{3} -2.01218 q^{4} +(0.741511 + 2.10954i) q^{5} -5.26757 q^{6} -4.24477i q^{7} +0.0243959i q^{8} -3.91575 q^{9} +(4.22550 - 1.48528i) q^{10} +1.00000 q^{11} +5.29159i q^{12} -1.93096i q^{13} -8.50246 q^{14} +(5.54763 - 1.95001i) q^{15} -3.97549 q^{16} +0.203952i q^{17} +7.84342i q^{18} +1.00000 q^{19} +(-1.49205 - 4.24477i) q^{20} -11.1628 q^{21} -2.00304i q^{22} -5.43129i q^{23} +0.0641559 q^{24} +(-3.90032 + 3.12849i) q^{25} -3.86780 q^{26} +2.40823i q^{27} +8.54125i q^{28} +7.64880 q^{29} +(-3.90596 - 11.1121i) q^{30} -1.09535 q^{31} +8.01187i q^{32} -2.62978i q^{33} +0.408525 q^{34} +(8.95452 - 3.14755i) q^{35} +7.87920 q^{36} +10.6804i q^{37} -2.00304i q^{38} -5.07801 q^{39} +(-0.0514641 + 0.0180898i) q^{40} -7.00756 q^{41} +22.3596i q^{42} +10.9332i q^{43} -2.01218 q^{44} +(-2.90357 - 8.26044i) q^{45} -10.8791 q^{46} -6.63725i q^{47} +10.4547i q^{48} -11.0181 q^{49} +(6.26651 + 7.81251i) q^{50} +0.536350 q^{51} +3.88544i q^{52} +7.73991i q^{53} +4.82379 q^{54} +(0.741511 + 2.10954i) q^{55} +0.103555 q^{56} -2.62978i q^{57} -15.3209i q^{58} +9.95331 q^{59} +(-11.1628 + 3.92377i) q^{60} +5.60294 q^{61} +2.19404i q^{62} +16.6215i q^{63} +8.09714 q^{64} +(4.07344 - 1.43183i) q^{65} -5.26757 q^{66} -10.3751i q^{67} -0.410389i q^{68} -14.2831 q^{69} +(-6.30467 - 17.9363i) q^{70} +0.0644000 q^{71} -0.0955283i q^{72} -12.5405i q^{73} +21.3933 q^{74} +(8.22726 + 10.2570i) q^{75} -2.01218 q^{76} -4.24477i q^{77} +10.1715i q^{78} +8.00036 q^{79} +(-2.94787 - 8.38646i) q^{80} -5.41413 q^{81} +14.0364i q^{82} -2.68074i q^{83} +22.4616 q^{84} +(-0.430246 + 0.151233i) q^{85} +21.8997 q^{86} -20.1147i q^{87} +0.0243959i q^{88} -10.2584 q^{89} +(-16.5460 + 5.81598i) q^{90} -8.19649 q^{91} +10.9287i q^{92} +2.88054i q^{93} -13.2947 q^{94} +(0.741511 + 2.10954i) q^{95} +21.0695 q^{96} +16.2745i q^{97} +22.0697i q^{98} -3.91575 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	2.00304i		−	1.41636i	−0.706029	−	0.708182i	$-0.749515\pi$
$2$		−	2.00304i		−	1.41636i	0.706029	−	0.708182i	$-0.250485\pi$
$3$		−	2.62978i		−	1.51831i	−0.650913	−	0.759153i	$-0.725614\pi$
$3$		−	2.62978i		−	1.51831i	0.650913	−	0.759153i	$-0.274386\pi$
$4$	−2.01218			−1.00609
$5$	0.741511	+	2.10954i	0.331614	+	0.943415i
$5$	0.741511	+	2.10954i	0.331614	+	0.943415i
$6$	−5.26757			−2.15047
$7$		−	4.24477i		−	1.60437i	−0.597073	−	0.802187i	$-0.703670\pi$
$7$		−	4.24477i		−	1.60437i	0.597073	−	0.802187i	$-0.296330\pi$
$8$			0.0243959i			0.00862525i
$9$	−3.91575			−1.30525
$10$	4.22550	−	1.48528i	1.33622	−	0.469686i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$			5.29159i			1.52755i
$13$		−	1.93096i		−	0.535552i	−0.963481	−	0.267776i	$-0.913711\pi$
$13$		−	1.93096i		−	0.535552i	0.963481	−	0.267776i	$-0.0862887\pi$
$14$	−8.50246			−2.27238
$15$	5.54763	−	1.95001i	1.43239	−	0.503491i
$16$	−3.97549			−0.993873
$17$			0.203952i			0.0494657i	0.999694	+	0.0247329i	$0.00787351\pi$
$17$			0.203952i			0.0494657i	−0.999694	+	0.0247329i	$0.992126\pi$
$18$			7.84342i			1.84871i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	−1.49205	−	4.24477i	−0.333633	−	0.949160i
$21$	−11.1628			−2.43593
$22$		−	2.00304i		−	0.427050i
$23$		−	5.43129i		−	1.13250i	−0.824233	−	0.566251i	$-0.808393\pi$
$23$		−	5.43129i		−	1.13250i	0.824233	−	0.566251i	$-0.191607\pi$
$24$	0.0641559			0.0130958
$25$	−3.90032	+	3.12849i	−0.780065	+	0.625699i
$26$	−3.86780			−0.758537
$27$			2.40823i			0.463465i
$28$			8.54125i			1.61414i
$29$	7.64880			1.42035			0.710173	−	0.704027i	$-0.248617\pi$
$29$	7.64880			1.42035			0.710173	+	0.704027i	$0.248617\pi$
$30$	−3.90596	−	11.1121i	−0.713127	−	2.02879i
$31$	−1.09535			−0.196731			−0.0983657	−	0.995150i	$-0.531362\pi$
$31$	−1.09535			−0.196731			−0.0983657	+	0.995150i	$0.531362\pi$
$32$			8.01187i			1.41631i
$33$		−	2.62978i		−	0.457786i
$34$	0.408525			0.0700615
$35$	8.95452	−	3.14755i	1.51359	−	0.532032i
$36$	7.87920			1.31320
$37$			10.6804i			1.75585i	0.478801	+	0.877923i	$0.341072\pi$
$37$			10.6804i			1.75585i	−0.478801	+	0.877923i	$0.658928\pi$
$38$		−	2.00304i		−	0.324936i
$39$	−5.07801			−0.813132
$40$	−0.0514641	+	0.0180898i	−0.00813719	+	0.00286025i
$41$	−7.00756			−1.09440			−0.547198	−	0.837003i	$-0.684306\pi$
$41$	−7.00756			−1.09440			−0.547198	+	0.837003i	$0.684306\pi$
$42$			22.3596i			3.45017i
$43$			10.9332i			1.66730i	0.552292	+	0.833651i	$0.313753\pi$
$43$			10.9332i			1.66730i	−0.552292	+	0.833651i	$0.686247\pi$
$44$	−2.01218			−0.303347
$45$	−2.90357	−	8.26044i	−0.432839	−	1.23139i
$46$	−10.8791			−1.60404
$47$		−	6.63725i		−	0.968143i	−0.875028	−	0.484072i	$-0.839157\pi$
$47$		−	6.63725i		−	0.968143i	0.875028	−	0.484072i	$-0.160843\pi$
$48$			10.4547i			1.50900i
$49$	−11.0181			−1.57402
$50$	6.26651	+	7.81251i	0.886218	+	1.10486i
$51$	0.536350			0.0751040
$52$			3.88544i			0.538813i
$53$			7.73991i			1.06316i	0.847008	+	0.531579i	$0.178401\pi$
$53$			7.73991i			1.06316i	−0.847008	+	0.531579i	$0.821599\pi$
$54$	4.82379			0.656435
$55$	0.741511	+	2.10954i	0.0999853	+	0.284450i
$56$	0.103555			0.0138381
$57$		−	2.62978i		−	0.348323i
$58$		−	15.3209i		−	2.01173i
$59$	9.95331			1.29581			0.647905	−	0.761721i	$-0.275645\pi$
$59$	9.95331			1.29581			0.647905	+	0.761721i	$0.275645\pi$
$60$	−11.1628	+	3.92377i	−1.44112	+	0.506557i
$61$	5.60294			0.717383			0.358692	−	0.933456i	$-0.383223\pi$
$61$	5.60294			0.717383			0.358692	+	0.933456i	$0.383223\pi$
$62$			2.19404i			0.278644i
$63$			16.6215i			2.09411i
$64$	8.09714			1.01214
$65$	4.07344	−	1.43183i	0.505248	−	0.177596i
$66$	−5.26757			−0.648392
$67$		−	10.3751i		−	1.26752i	−0.773529	−	0.633761i	$-0.781510\pi$
$67$		−	10.3751i		−	1.26752i	0.773529	−	0.633761i	$-0.218490\pi$
$68$		−	0.410389i		−	0.0497669i
$69$	−14.2831			−1.71948
$70$	−6.30467	−	17.9363i	−0.753552	−	2.14380i
$71$	0.0644000			0.00764288			0.00382144	−	0.999993i	$-0.498784\pi$
$71$	0.0644000			0.00764288			0.00382144	+	0.999993i	$0.498784\pi$
$72$		−	0.0955283i		−	0.0112581i
$73$		−	12.5405i		−	1.46775i	−0.679284	−	0.733876i	$-0.737710\pi$
$73$		−	12.5405i		−	1.46775i	0.679284	−	0.733876i	$-0.262290\pi$
$74$	21.3933			2.48692
$75$	8.22726	+	10.2570i	0.950002	+	1.18438i
$76$	−2.01218			−0.230813
$77$		−	4.24477i		−	0.483737i
$78$			10.1715i			1.15169i
$79$	8.00036			0.900111			0.450056	−	0.893001i	$-0.351404\pi$
$79$	8.00036			0.900111			0.450056	+	0.893001i	$0.351404\pi$
$80$	−2.94787	−	8.38646i	−0.329582	−	0.937635i
$81$	−5.41413			−0.601570
$82$			14.0364i			1.55006i
$83$		−	2.68074i		−	0.294250i	−0.989118	−	0.147125i	$-0.952998\pi$
$83$		−	2.68074i		−	0.294250i	0.989118	−	0.147125i	$-0.0470020\pi$
$84$	22.4616			2.45076
$85$	−0.430246	+	0.151233i	−0.0466667	+	0.0164035i
$86$	21.8997			2.36151
$87$		−	20.1147i		−	2.15652i
$88$			0.0243959i			0.00260061i
$89$	−10.2584			−1.08739			−0.543696	−	0.839283i	$-0.682975\pi$
$89$	−10.2584			−1.08739			−0.543696	+	0.839283i	$0.682975\pi$
$90$	−16.5460	+	5.81598i	−1.74410	+	0.613058i
$91$	−8.19649			−0.859226
$92$			10.9287i			1.13940i
$93$			2.88054i			0.298698i
$94$	−13.2947			−1.37124
$95$	0.741511	+	2.10954i	0.0760774	+	0.216434i
$96$	21.0695			2.15039
$97$			16.2745i			1.65243i	0.563358	+	0.826213i	$0.309509\pi$
$97$			16.2745i			1.65243i	−0.563358	+	0.826213i	$0.690491\pi$
$98$			22.0697i			2.22938i
$99$	−3.91575			−0.393548
$100$	7.84815	−	6.29509i	0.784815	−	0.629509i
$101$	6.96737			0.693279			0.346640	−	0.937998i	$-0.387323\pi$
$101$	6.96737			0.693279			0.346640	+	0.937998i	$0.387323\pi$
$102$		−	1.07433i		−	0.106375i
$103$		−	5.30064i		−	0.522288i	−0.965300	−	0.261144i	$-0.915900\pi$
$103$		−	5.30064i		−	0.522288i	0.965300	−	0.261144i	$-0.0840997\pi$
$104$	0.0471075			0.00461927
$105$	−8.27736	−	23.5484i	−0.807788	−	2.29809i
$106$	15.5034			1.50582
$107$		−	4.65950i		−	0.450451i	−0.974307	−	0.225226i	$-0.927688\pi$
$107$		−	4.65950i		−	0.450451i	0.974307	−	0.225226i	$-0.0723119\pi$
$108$		−	4.84580i		−	0.466287i
$109$	17.7225			1.69751			0.848755	−	0.528786i	$-0.177353\pi$
$109$	17.7225			1.69751			0.848755	+	0.528786i	$0.177353\pi$
$110$	4.22550	−	1.48528i	0.402886	−	0.141616i
$111$	28.0871			2.66591
$112$			16.8751i			1.59454i
$113$		−	13.9694i		−	1.31413i	−0.753834	−	0.657065i	$-0.771798\pi$
$113$		−	13.9694i		−	1.31413i	0.753834	−	0.657065i	$-0.228202\pi$
$114$	−5.26757			−0.493353
$115$	11.4575	−	4.02736i	1.06842	−	0.375554i
$116$	−15.3908			−1.42900
$117$			7.56117i			0.699030i
$118$		−	19.9369i		−	1.83534i
$119$	0.865731			0.0793615
$120$	0.0475723	+	0.135339i	0.00434274	+	0.0123547i
$121$	1.00000			0.0909091
$122$		−	11.2229i		−	1.01608i
$123$			18.4283i			1.66163i
$124$	2.20405			0.197930
$125$	−9.49182	−	5.90808i	−0.848974	−	0.528434i
$126$	33.2936			2.96603
$127$			6.59272i			0.585009i	0.956264	+	0.292505i	$0.0944887\pi$
$127$			6.59272i			0.585009i	−0.956264	+	0.292505i	$0.905511\pi$
$128$		−	0.195164i		−	0.0172502i
$129$	28.7520			2.53147
$130$	−2.86801	−	8.15927i	−0.251541	−	0.715616i
$131$	−0.889689			−0.0777325			−0.0388663	−	0.999244i	$-0.512375\pi$
$131$	−0.889689			−0.0777325			−0.0388663	+	0.999244i	$0.512375\pi$
$132$			5.29159i			0.460574i
$133$		−	4.24477i		−	0.368069i
$134$	−20.7818			−1.79527
$135$	−5.08027	+	1.78573i	−0.437240	+	0.153691i
$136$	−0.00497560			−0.000426654
$137$		−	1.66168i		−	0.141967i	−0.997477	−	0.0709834i	$-0.977386\pi$
$137$		−	1.66168i		−	0.141967i	0.997477	−	0.0709834i	$-0.0226137\pi$
$138$			28.6097i			2.43542i
$139$	10.8055			0.916509			0.458255	−	0.888821i	$-0.348475\pi$
$139$	10.8055			0.916509			0.458255	+	0.888821i	$0.348475\pi$
$140$	−18.0181	+	6.33343i	−1.52281	+	0.535272i
$141$	−17.4545			−1.46994
$142$		−	0.128996i		−	0.0108251i
$143$		−	1.93096i		−	0.161475i
$144$	15.5671			1.29725
$145$	5.67167	+	16.1354i	0.471006	+	1.33998i
$146$	−25.1191			−2.07887
$147$			28.9752i			2.38984i
$148$		−	21.4909i		−	1.76654i
$149$	−4.41473			−0.361669			−0.180834	−	0.983514i	$-0.557880\pi$
$149$	−4.41473			−0.361669			−0.180834	+	0.983514i	$0.557880\pi$
$150$	20.5452	−	16.4796i	1.67751	−	1.34555i
$151$	−17.5209			−1.42583			−0.712916	−	0.701250i	$-0.752626\pi$
$151$	−17.5209			−1.42583			−0.712916	+	0.701250i	$0.752626\pi$
$152$			0.0243959i			0.00197877i
$153$		−	0.798627i		−	0.0645652i
$154$	−8.50246			−0.685148
$155$	−0.812217	−	2.31069i	−0.0652389	−	0.185599i
$156$	10.2179			0.818083
$157$		−	14.0489i		−	1.12123i	−0.828078	−	0.560613i	$-0.810566\pi$
$157$		−	14.0489i		−	1.12123i	0.828078	−	0.560613i	$-0.189434\pi$
$158$		−	16.0251i		−	1.27489i
$159$	20.3543			1.61420
$160$	−16.9014	+	5.94089i	−1.33617	+	0.469669i
$161$	−23.0546			−1.81696
$162$			10.8447i			0.852043i
$163$		−	11.4782i		−	0.899039i	−0.893271	−	0.449519i	$-0.851595\pi$
$163$		−	11.4782i		−	0.899039i	0.893271	−	0.449519i	$-0.148405\pi$
$164$	14.1005			1.10106
$165$	5.54763	−	1.95001i	0.431883	−	0.151808i
$166$	−5.36965			−0.416765
$167$		−	4.97097i		−	0.384665i	−0.981330	−	0.192333i	$-0.938395\pi$
$167$		−	4.97097i		−	0.384665i	0.981330	−	0.192333i	$-0.0616053\pi$
$168$		−	0.272327i		−	0.0210105i
$169$	9.27139			0.713184
$170$	0.302926	+	0.861800i	0.0232334	+	0.0660971i
$171$	−3.91575			−0.299445
$172$		−	21.9996i		−	1.67746i
$173$			11.1163i			0.845160i	0.906326	+	0.422580i	$0.138876\pi$
$173$			11.1163i			0.845160i	−0.906326	+	0.422580i	$0.861124\pi$
$174$	−40.2905			−3.05442
$175$	13.2798	+	16.5560i	1.00386	+	1.25152i
$176$	−3.97549			−0.299664
$177$		−	26.1750i		−	1.96744i
$178$			20.5481i			1.54014i
$179$	−9.22455			−0.689475			−0.344737	−	0.938699i	$-0.612032\pi$
$179$	−9.22455			−0.689475			−0.344737	+	0.938699i	$0.612032\pi$
$180$	5.84251	+	16.6215i	0.435475	+	1.23889i
$181$	2.01987			0.150136			0.0750679	−	0.997178i	$-0.476083\pi$
$181$	2.01987			0.150136			0.0750679	+	0.997178i	$0.476083\pi$
$182$			16.4179i			1.21698i
$183$		−	14.7345i		−	1.08921i
$184$	0.132501			0.00976812
$185$	−22.5307	+	7.91963i	−1.65649	+	0.582263i
$186$	5.76985			0.423066
$187$			0.203952i			0.0149145i
$188$			13.3553i			0.974039i
$189$	10.2224			0.743571
$190$	4.22550	−	1.48528i	0.306550	−	0.107753i
$191$	−26.0923			−1.88797			−0.943987	−	0.329983i	$-0.892957\pi$
$191$	−26.0923			−1.88797			−0.943987	+	0.329983i	$0.892957\pi$
$192$		−	21.2937i		−	1.53674i
$193$		−	14.1647i		−	1.01960i	−0.860293	−	0.509800i	$-0.829719\pi$
$193$		−	14.1647i		−	1.01960i	0.860293	−	0.509800i	$-0.170281\pi$
$194$	32.5985			2.34044
$195$	−3.76540	−	10.7123i	−0.269646	−	0.767121i
$196$	22.1704			1.58360
$197$		−	17.5606i		−	1.25114i	−0.780169	−	0.625569i	$-0.784867\pi$
$197$		−	17.5606i		−	1.25114i	0.780169	−	0.625569i	$-0.215133\pi$
$198$			7.84342i			0.557408i
$199$	19.2762			1.36645			0.683225	−	0.730208i	$-0.260577\pi$
$199$	19.2762			1.36645			0.683225	+	0.730208i	$0.260577\pi$
$200$	−0.0763224	−	0.0951519i	−0.00539681	−	0.00672825i
$201$	−27.2843			−1.92449
$202$		−	13.9559i		−	0.981936i
$203$		−	32.4674i		−	2.27877i
$204$	−1.07923			−0.0755614
$205$	−5.19618	−	14.7827i	−0.362917	−	1.03247i
$206$	−10.6174			−0.739750
$207$			21.2676i			1.47820i
$208$			7.67652i			0.532271i
$209$	1.00000			0.0691714
$210$	−47.1685	+	16.5799i	−3.25494	+	1.14412i
$211$	−12.6533			−0.871092			−0.435546	−	0.900166i	$-0.643445\pi$
$211$	−12.6533			−0.871092			−0.435546	+	0.900166i	$0.643445\pi$
$212$		−	15.5741i		−	1.06963i
$213$		−	0.169358i		−	0.0116042i
$214$	−9.33319			−0.638003
$215$	−23.0641	+	8.10711i	−1.57296	+	0.552900i
$216$	−0.0587510			−0.00399750
$217$			4.64953i			0.315631i
$218$		−	35.4990i		−	2.40429i
$219$	−32.9787			−2.22849
$220$	−1.49205	−	4.24477i	−0.100594	−	0.286183i
$221$	0.393824			0.0264915
$222$		−	56.2597i		−	3.77590i
$223$			21.6776i			1.45164i	0.687885	+	0.725820i	$0.258539\pi$
$223$			21.6776i			1.45164i	−0.687885	+	0.725820i	$0.741461\pi$
$224$	34.0086			2.27229
$225$	15.2727	−	12.2504i	1.01818	−	0.816694i
$226$	−27.9813			−1.86129
$227$			22.2768i			1.47856i	0.673397	+	0.739281i	$0.264835\pi$
$227$			22.2768i			1.47856i	−0.673397	+	0.739281i	$0.735165\pi$
$228$			5.29159i			0.350444i
$229$	7.64707			0.505333			0.252666	−	0.967553i	$-0.418692\pi$
$229$	7.64707			0.505333			0.252666	+	0.967553i	$0.418692\pi$
$230$	−8.06698	−	22.9499i	−0.531921	−	1.51327i
$231$	−11.1628			−0.734460
$232$			0.186599i			0.0122508i
$233$		−	3.83039i		−	0.250937i	−0.992098	−	0.125469i	$-0.959957\pi$
$233$		−	3.83039i		−	0.250937i	0.992098	−	0.125469i	$-0.0400435\pi$
$234$	15.1453			0.990082
$235$	14.0016	−	4.92160i	0.913361	−	0.321050i
$236$	−20.0278			−1.30370
$237$		−	21.0392i		−	1.36664i
$238$		−	1.73410i		−	0.112405i
$239$	8.90264			0.575864			0.287932	−	0.957651i	$-0.407032\pi$
$239$	8.90264			0.575864			0.287932	+	0.957651i	$0.407032\pi$
$240$	−22.0546	+	7.75226i	−1.42362	+	0.500406i
$241$	−16.1822			−1.04239			−0.521195	−	0.853438i	$-0.674513\pi$
$241$	−16.1822			−1.04239			−0.521195	+	0.853438i	$0.674513\pi$
$242$		−	2.00304i		−	0.128760i
$243$			21.4627i			1.37683i
$244$	−11.2741			−0.721752
$245$	−8.17005	−	23.2431i	−0.521965	−	1.48495i
$246$	36.9128			2.35347
$247$		−	1.93096i		−	0.122864i
$248$		−	0.0267222i		−	0.00169686i
$249$	−7.04977			−0.446761
$250$	−11.8341	+	19.0125i	−0.748456	+	1.20246i
$251$	12.4705			0.787128			0.393564	−	0.919297i	$-0.371242\pi$
$251$	12.4705			0.787128			0.393564	+	0.919297i	$0.371242\pi$
$252$		−	33.4454i		−	2.10686i
$253$		−	5.43129i		−	0.341462i
$254$	13.2055			0.828587
$255$	0.397710	+	1.13145i	0.0249055	+	0.0708543i
$256$	15.8034			0.987710
$257$		−	21.2619i		−	1.32628i	−0.748495	−	0.663141i	$-0.769223\pi$
$257$		−	21.2619i		−	1.32628i	0.748495	−	0.663141i	$-0.230777\pi$
$258$		−	57.5915i		−	3.58549i
$259$	45.3359			2.81703
$260$	−8.19649	+	2.88110i	−0.508325	+	0.178678i
$261$	−29.9508			−1.85391
$262$			1.78209i			0.110098i
$263$			15.6810i			0.966933i	0.875363	+	0.483467i	$0.160623\pi$
$263$			15.6810i			0.966933i	−0.875363	+	0.483467i	$0.839377\pi$
$264$	0.0641559			0.00394852
$265$	−16.3277	+	5.73923i	−1.00300	+	0.352558i
$266$	−8.50246			−0.521319
$267$			26.9774i			1.65099i
$268$			20.8766i			1.27524i
$269$	−7.87081			−0.479892			−0.239946	−	0.970786i	$-0.577130\pi$
$269$	−7.87081			−0.479892			−0.239946	+	0.970786i	$0.577130\pi$
$270$	3.57690	+	10.1760i	0.217683	+	0.619291i
$271$	4.88158			0.296535			0.148268	−	0.988947i	$-0.452630\pi$
$271$	4.88158			0.296535			0.148268	+	0.988947i	$0.452630\pi$
$272$		−	0.810811i		−	0.0491626i
$273$			21.5550i			1.30457i
$274$	−3.32841			−0.201077
$275$	−3.90032	+	3.12849i	−0.235198	+	0.188655i
$276$	28.7402			1.72996
$277$		−	0.377962i		−	0.0227096i	−0.999936	−	0.0113548i	$-0.996386\pi$
$277$		−	0.377962i		−	0.0227096i	0.999936	−	0.0113548i	$-0.00361441\pi$
$278$		−	21.6439i		−	1.29811i
$279$	4.28914			0.256784
$280$	0.0767872	+	0.218454i	0.00458891	+	0.0130551i
$281$	−9.01138			−0.537574			−0.268787	−	0.963200i	$-0.586623\pi$
$281$	−9.01138			−0.537574			−0.268787	+	0.963200i	$0.586623\pi$
$282$			34.9622i			2.08197i
$283$			2.08252i			0.123793i	0.998083	+	0.0618965i	$0.0197149\pi$
$283$			2.08252i			0.123793i	−0.998083	+	0.0618965i	$0.980285\pi$
$284$	−0.129584			−0.00768942
$285$	5.54763	−	1.95001i	0.328613	−	0.115509i
$286$	−3.86780			−0.228708
$287$			29.7455i			1.75582i
$288$		−	31.3725i		−	1.84864i
$289$	16.9584			0.997553
$290$	32.3200	−	11.3606i	1.89789	−	0.667117i
$291$	42.7984			2.50889
$292$			25.2337i			1.47669i
$293$			14.3115i			0.836086i	0.908427	+	0.418043i	$0.137284\pi$
$293$			14.3115i			0.836086i	−0.908427	+	0.418043i	$0.862716\pi$
$294$	58.0386			3.38488
$295$	7.38049	+	20.9969i	0.429709	+	1.22249i
$296$	−0.260558			−0.0151446
$297$			2.40823i			0.139740i
$298$			8.84289i			0.512255i
$299$	−10.4876			−0.606514
$300$	−16.5547	−	20.6389i	−0.955787	−	1.19159i
$301$	46.4091			2.67498
$302$			35.0951i			2.01950i
$303$		−	18.3227i		−	1.05261i
$304$	−3.97549			−0.228010
$305$	4.15464	+	11.8196i	0.237894	+	0.676791i
$306$	−1.59968			−0.0914478
$307$		−	7.46501i		−	0.426051i	−0.977047	−	0.213025i	$-0.931668\pi$
$307$		−	7.46501i		−	0.426051i	0.977047	−	0.213025i	$-0.0683317\pi$
$308$			8.54125i			0.486683i
$309$	−13.9395			−0.792993
$310$	−4.62842	+	1.62691i	−0.262877	+	0.0924021i
$311$	−10.8096			−0.612956			−0.306478	−	0.951878i	$-0.599151\pi$
$311$	−10.8096			−0.612956			−0.306478	+	0.951878i	$0.599151\pi$
$312$		−	0.123882i		−	0.00701347i
$313$			24.9719i			1.41150i	0.708463	+	0.705748i	$0.249389\pi$
$313$			24.9719i			1.41150i	−0.708463	+	0.705748i	$0.750611\pi$
$314$	−28.1406			−1.58807
$315$	−35.0637	+	12.3250i	−1.97562	+	0.694436i
$316$	−16.0982			−0.905593
$317$		−	6.15513i		−	0.345706i	−0.984948	−	0.172853i	$-0.944701\pi$
$317$		−	6.15513i		−	0.345706i	0.984948	−	0.172853i	$-0.0552986\pi$
$318$		−	40.7705i		−	2.28630i
$319$	7.64880			0.428250
$320$	6.00412	+	17.0812i	0.335640	+	0.954870i
$321$	−12.2535			−0.683923
$322$			46.1794i			2.57348i
$323$			0.203952i			0.0113482i
$324$	10.8942			0.605234
$325$	6.04100	+	7.53137i	0.335094	+	0.417765i
$326$	−22.9912			−1.27337
$327$		−	46.6064i		−	2.57734i
$328$		−	0.170956i		−	0.00943944i
$329$	−28.1736			−1.55326
$330$	−3.90596	−	11.1121i	−0.215016	−	0.611703i
$331$	8.09554			0.444971			0.222486	−	0.974936i	$-0.428583\pi$
$331$	8.09554			0.444971			0.222486	+	0.974936i	$0.428583\pi$
$332$			5.39414i			0.296042i
$333$		−	41.8218i		−	2.29182i
$334$	−9.95707			−0.544826
$335$	21.8867	−	7.69327i	1.19580	−	0.420328i
$336$	44.3778			2.42101
$337$			0.427228i			0.0232726i	0.999932	+	0.0116363i	$0.00370403\pi$
$337$			0.427228i			0.0232726i	−0.999932	+	0.0116363i	$0.996296\pi$
$338$		−	18.5710i		−	1.01013i
$339$	−36.7364			−1.99525
$340$	0.865731	−	0.304308i	0.0469509	−	0.0165034i
$341$	−1.09535			−0.0593168
$342$			7.84342i			0.424124i
$343$			17.0560i			0.920935i
$344$	−0.266726			−0.0143809
$345$	−10.5911	−	30.1308i	−0.570205	−	1.62219i
$346$	22.2665			1.19706
$347$		−	2.69469i		−	0.144658i	−0.997381	−	0.0723292i	$-0.976957\pi$
$347$		−	2.69469i		−	0.144658i	0.997381	−	0.0723292i	$-0.0230432\pi$
$348$			40.4743i			2.16965i
$349$	23.5167			1.25882			0.629409	−	0.777074i	$-0.283297\pi$
$349$	23.5167			1.25882			0.629409	+	0.777074i	$0.283297\pi$
$350$	33.1624	−	26.5999i	1.77260	−	1.42183i
$351$	4.65020			0.248210
$352$			8.01187i			0.427034i
$353$		−	21.6861i		−	1.15423i	−0.816662	−	0.577117i	$-0.804178\pi$
$353$		−	21.6861i		−	1.15423i	0.816662	−	0.577117i	$-0.195822\pi$
$354$	−52.4297			−2.78661
$355$	0.0477533	+	0.135854i	0.00253448	+	0.00721040i
$356$	20.6418			1.09401
$357$		−	2.27669i		−	0.120495i
$358$			18.4772i			0.976548i
$359$	−11.5228			−0.608153			−0.304076	−	0.952648i	$-0.598348\pi$
$359$	−11.5228			−0.608153			−0.304076	+	0.952648i	$0.598348\pi$
$360$	0.201521	−	0.0708353i	0.0106211	−	0.00373335i
$361$	1.00000			0.0526316
$362$		−	4.04589i		−	0.212647i
$363$		−	2.62978i		−	0.138028i
$364$	16.4928			0.864458
$365$	26.4546	−	9.29890i	1.38470	−	0.486727i
$366$	−29.5139			−1.54271
$367$		−	20.4552i		−	1.06775i	−0.845562	−	0.533877i	$-0.820735\pi$
$367$		−	20.4552i		−	1.06775i	0.845562	−	0.533877i	$-0.179265\pi$
$368$			21.5921i			1.12556i
$369$	27.4399			1.42846
$370$	15.8634	+	45.1300i	0.824697	+	2.34620i
$371$	32.8542			1.70570
$372$		−	5.79617i		−	0.300517i
$373$		−	7.56824i		−	0.391869i	−0.980617	−	0.195934i	$-0.937226\pi$
$373$		−	7.56824i		−	0.391869i	0.980617	−	0.195934i	$-0.0627739\pi$
$374$	0.408525			0.0211243
$375$	−15.5370	+	24.9614i	−0.802325	+	1.28900i
$376$	0.161922			0.00835048
$377$		−	14.7695i		−	0.760669i
$378$		−	20.4759i		−	1.05317i
$379$	−12.6921			−0.651951			−0.325976	−	0.945378i	$-0.605693\pi$
$379$	−12.6921			−0.651951			−0.325976	+	0.945378i	$0.605693\pi$
$380$	−1.49205	−	4.24477i	−0.0765407	−	0.217752i
$381$	17.3374			0.888223
$382$			52.2640i			2.67406i
$383$		−	23.8875i		−	1.22059i	−0.792173	−	0.610297i	$-0.791050\pi$
$383$		−	23.8875i		−	1.22059i	0.792173	−	0.610297i	$-0.208950\pi$
$384$	−0.513238			−0.0261910
$385$	8.95452	−	3.14755i	0.456365	−	0.160414i
$386$	−28.3726			−1.44413
$387$		−	42.8118i		−	2.17625i
$388$		−	32.7472i		−	1.66249i
$389$	−4.99029			−0.253018			−0.126509	−	0.991965i	$-0.540377\pi$
$389$	−4.99029			−0.253018			−0.126509	+	0.991965i	$0.540377\pi$
$390$	−21.4571	+	7.54225i	−1.08652	+	0.381917i
$391$	1.10772			0.0560200
$392$		−	0.268797i		−	0.0135763i
$393$			2.33969i			0.118022i
$394$	−35.1746			−1.77207
$395$	5.93236	+	16.8771i	0.298489	+	0.849179i
$396$	7.87920			0.395945
$397$			15.6805i			0.786984i	0.919328	+	0.393492i	$0.128733\pi$
$397$			15.6805i			0.786984i	−0.919328	+	0.393492i	$0.871267\pi$
$398$		−	38.6110i		−	1.93539i
$399$	−11.1628			−0.558841
$400$	15.5057	−	12.4373i	0.775285	−	0.621865i
$401$	14.8693			0.742540			0.371270	−	0.928525i	$-0.378922\pi$
$401$	14.8693			0.742540			0.371270	+	0.928525i	$0.378922\pi$
$402$			54.6516i			2.72578i
$403$			2.11509i			0.105360i
$404$	−14.0196			−0.697501
$405$	−4.01464	−	11.4213i	−0.199489	−	0.567531i
$406$	−65.0336			−3.22756
$407$			10.6804i			0.529408i
$408$			0.0130847i			0.000647791i
$409$	22.6583			1.12038			0.560190	−	0.828364i	$-0.310728\pi$
$409$	22.6583			1.12038			0.560190	+	0.828364i	$0.310728\pi$
$410$	−29.6104	+	10.4082i	−1.46235	+	0.514023i
$411$	−4.36985			−0.215549
$412$			10.6658i			0.525469i
$413$		−	42.2496i		−	2.07896i
$414$	42.5999			2.09367
$415$	5.65514	−	1.98780i	0.277600	−	0.0975773i
$416$	15.4706			0.758509
$417$		−	28.4161i		−	1.39154i
$418$		−	2.00304i		−	0.0979720i
$419$	17.4123			0.850648			0.425324	−	0.905041i	$-0.360160\pi$
$419$	17.4123			0.850648			0.425324	+	0.905041i	$0.360160\pi$
$420$	16.6555	+	47.3837i	0.812707	+	2.31209i
$421$	−3.47748			−0.169482			−0.0847410	−	0.996403i	$-0.527006\pi$
$421$	−3.47748			−0.169482			−0.0847410	+	0.996403i	$0.527006\pi$
$422$			25.3452i			1.23378i
$423$			25.9899i			1.26367i
$424$	−0.188822			−0.00917001
$425$	−0.638064	−	0.795480i	−0.0309506	−	0.0385864i
$426$	−0.339231			−0.0164358
$427$		−	23.7832i		−	1.15095i
$428$			9.37576i			0.453194i
$429$	−5.07801			−0.245168
$430$	16.2389	+	46.1983i	0.783108	+	2.22788i
$431$	20.9460			1.00893			0.504467	−	0.863431i	$-0.331689\pi$
$431$	20.9460			1.00893			0.504467	+	0.863431i	$0.331689\pi$
$432$		−	9.57391i		−	0.460625i
$433$			31.1980i			1.49928i	0.661846	+	0.749640i	$0.269773\pi$
$433$			31.1980i			1.49928i	−0.661846	+	0.749640i	$0.730227\pi$
$434$	9.31321			0.447048
$435$	42.4327	−	14.9152i	2.03449	−	0.715131i
$436$	−35.6609			−1.70785
$437$		−	5.43129i		−	0.259814i
$438$			66.0578i			3.15636i
$439$	12.1175			0.578336			0.289168	−	0.957278i	$-0.406621\pi$
$439$	12.1175			0.578336			0.289168	+	0.957278i	$0.406621\pi$
$440$	−0.0514641	+	0.0180898i	−0.00245346	+	0.000862399i
$441$	43.1442			2.05449
$442$		−	0.788846i		−	0.0375216i
$443$		−	4.51478i		−	0.214504i	−0.994232	−	0.107252i	$-0.965795\pi$
$443$		−	4.51478i		−	0.214504i	0.994232	−	0.107252i	$-0.0342051\pi$
$444$	−56.5163			−2.68215
$445$	−7.60674	−	21.6406i	−0.360594	−	1.02586i
$446$	43.4212			2.05605
$447$			11.6098i			0.549124i
$448$		−	34.3705i		−	1.62385i
$449$	26.9029			1.26963			0.634814	−	0.772665i	$-0.281077\pi$
$449$	26.9029			1.26963			0.634814	+	0.772665i	$0.281077\pi$
$450$	−24.5381	−	30.5919i	−1.15674	−	1.44211i
$451$	−7.00756			−0.329973
$452$			28.1089i			1.32213i
$453$			46.0762i			2.16485i
$454$	44.6214			2.09418
$455$	−6.07779	−	17.2908i	−0.284931	−	0.810607i
$456$	0.0641559			0.00300437
$457$			21.7398i			1.01695i	0.861078	+	0.508473i	$0.169790\pi$
$457$			21.7398i			1.01695i	−0.861078	+	0.508473i	$0.830210\pi$
$458$		−	15.3174i		−	0.715736i
$459$	−0.491165			−0.0229256
$460$	−23.0546	+	8.10378i	−1.07493	+	0.377841i
$461$	25.5865			1.19168			0.595841	−	0.803103i	$-0.296819\pi$
$461$	25.5865			1.19168			0.595841	+	0.803103i	$0.296819\pi$
$462$			22.3596i			1.04026i
$463$		−	20.7098i		−	0.962465i	−0.876593	−	0.481233i	$-0.840189\pi$
$463$		−	20.7098i		−	0.962465i	0.876593	−	0.481233i	$-0.159811\pi$
$464$	−30.4077			−1.41164
$465$	−6.07662	+	2.13595i	−0.281797	+	0.0990525i
$466$	−7.67244			−0.355419
$467$		−	23.0019i		−	1.06440i	−0.846618	−	0.532200i	$-0.821365\pi$
$467$		−	23.0019i		−	1.06440i	0.846618	−	0.532200i	$-0.178635\pi$
$468$		−	15.2144i		−	0.703287i
$469$	−44.0400			−2.03358
$470$	−9.85817	−	28.0457i	−0.454723	−	1.29365i
$471$	−36.9456			−1.70236
$472$			0.242820i			0.0111767i
$473$			10.9332i			0.502710i
$474$	−42.1424			−1.93567
$475$	−3.90032	+	3.12849i	−0.178959	+	0.143545i
$476$	−1.74201			−0.0798448
$477$		−	30.3076i		−	1.38769i
$478$		−	17.8324i		−	0.815633i
$479$	10.6450			0.486381			0.243191	−	0.969979i	$-0.421806\pi$
$479$	10.6450			0.486381			0.243191	+	0.969979i	$0.421806\pi$
$480$	15.6233	+	44.4469i	0.713101	+	2.02872i
$481$	20.6234			0.940348
$482$			32.4137i			1.47640i
$483$			60.6286i			2.75870i
$484$	−2.01218			−0.0914627
$485$	−34.3317	+	12.0677i	−1.55892	+	0.547967i
$486$	42.9907			1.95010
$487$		−	11.9723i		−	0.542518i	−0.962506	−	0.271259i	$-0.912560\pi$
$487$		−	11.9723i		−	0.542518i	0.962506	−	0.271259i	$-0.0874399\pi$
$488$			0.136689i			0.00618761i
$489$	−30.1851			−1.36502
$490$	−46.5570	+	16.3650i	−2.10323	+	0.739293i
$491$	−24.0347			−1.08467			−0.542335	−	0.840162i	$-0.682460\pi$
$491$	−24.0347			−1.08467			−0.542335	+	0.840162i	$0.682460\pi$
$492$		−	37.0811i		−	1.67175i
$493$			1.55999i			0.0702584i
$494$	−3.86780			−0.174020
$495$	−2.90357	−	8.26044i	−0.130506	−	0.371279i
$496$	4.35457			0.195526
$497$		−	0.273363i		−	0.0122620i
$498$			14.1210i			0.632777i
$499$	3.24663			0.145339			0.0726695	−	0.997356i	$-0.476848\pi$
$499$	3.24663			0.145339			0.0726695	+	0.997356i	$0.476848\pi$
$500$	19.0992	+	11.8881i	0.854144	+	0.531652i
$501$	−13.0726			−0.584039
$502$		−	24.9788i		−	1.11486i
$503$		−	35.8800i		−	1.59981i	−0.600126	−	0.799905i	$-0.704883\pi$
$503$		−	35.8800i		−	1.59981i	0.600126	−	0.799905i	$-0.295117\pi$
$504$	−0.405496			−0.0180622
$505$	5.16638	+	14.6979i	0.229901	+	0.654050i
$506$	−10.8791			−0.483635
$507$		−	24.3817i		−	1.08283i
$508$		−	13.2657i		−	0.588572i
$509$	−40.6670			−1.80253			−0.901266	−	0.433265i	$-0.857361\pi$
$509$	−40.6670			−1.80253			−0.901266	+	0.433265i	$0.857361\pi$
$510$	2.26635	−	0.796629i	0.100356	−	0.0352753i
$511$	−53.2315			−2.35482
$512$		−	32.0451i		−	1.41621i
$513$			2.40823i			0.106326i
$514$	−42.5885			−1.87850
$515$	11.1819	−	3.93049i	0.492734	−	0.173198i
$516$	−57.8542			−2.54689
$517$		−	6.63725i		−	0.291906i
$518$		−	90.8097i		−	3.98995i
$519$	29.2336			1.28321
$520$	0.0349307	+	0.0993752i	0.00153181	+	0.00435789i
$521$	33.1958			1.45433			0.727167	−	0.686460i	$-0.240836\pi$
$521$	33.1958			1.45433			0.727167	+	0.686460i	$0.240836\pi$
$522$			59.9927i			2.62581i
$523$			30.8129i			1.34735i	0.739025	+	0.673677i	$0.235286\pi$
$523$			30.8129i			1.34735i	−0.739025	+	0.673677i	$0.764714\pi$
$524$	1.79021			0.0782059
$525$	43.5386	−	34.9229i	1.90018	−	1.52416i
$526$	31.4097			1.36953
$527$		−	0.223400i		−	0.00973146i
$528$			10.4547i			0.454982i
$529$	−6.49893			−0.282562
$530$	11.4959	+	32.7050i	0.499351	+	1.42061i
$531$	−38.9747			−1.69136
$532$			8.54125i			0.370310i
$533$			13.5313i			0.586106i
$534$	54.0369			2.33841
$535$	9.82941	−	3.45507i	0.424963	−	0.149376i
$536$	0.253110			0.0109327
$537$			24.2585i			1.04683i
$538$			15.7656i			0.679702i
$539$	−11.0181			−0.474583
$540$	10.2224	−	3.59321i	0.439902	−	0.154627i
$541$	−23.2145			−0.998068			−0.499034	−	0.866582i	$-0.666312\pi$
$541$	−23.2145			−0.998068			−0.499034	+	0.866582i	$0.666312\pi$
$542$		−	9.77802i		−	0.420002i
$543$		−	5.31182i		−	0.227952i
$544$	−1.63404			−0.0700589
$545$	13.1415	+	37.3864i	0.562918	+	1.60146i
$546$	43.1756			1.84774
$547$			15.1083i			0.645985i	0.946402	+	0.322992i	$0.104689\pi$
$547$			15.1083i			0.645985i	−0.946402	+	0.322992i	$0.895311\pi$
$548$			3.34360i			0.142831i
$549$	−21.9398			−0.936366
$550$	6.26651	+	7.81251i	0.267205	+	0.333127i
$551$	7.64880			0.325850
$552$		−	0.348449i		−	0.0148310i
$553$		−	33.9597i		−	1.44411i
$554$	−0.757074			−0.0321650
$555$	20.8269	+	59.2509i	0.884053	+	2.51506i
$556$	−21.7426			−0.922091
$557$			28.0009i			1.18644i	0.805042	+	0.593218i	$0.202143\pi$
$557$			28.0009i			1.18644i	−0.805042	+	0.593218i	$0.797857\pi$
$558$		−	8.59133i		−	0.363700i
$559$	21.1116			0.892927
$560$	−35.5986	+	12.5130i	−1.50432	+	0.528773i
$561$	0.536350			0.0226447
$562$			18.0502i			0.761400i
$563$		−	4.07945i		−	0.171928i	−0.996298	−	0.0859641i	$-0.972603\pi$
$563$		−	4.07945i		−	0.171928i	0.996298	−	0.0859641i	$-0.0273970\pi$
$564$	35.1217			1.47889
$565$	29.4690	−	10.3585i	1.23977	−	0.435783i
$566$	4.17137			0.175336
$567$			22.9818i			0.965144i
$568$			0.00157110i			6.59217e-5i
$569$	39.5862			1.65954			0.829769	−	0.558107i	$-0.188472\pi$
$569$	39.5862			1.65954			0.829769	+	0.558107i	$0.188472\pi$
$570$	−3.90596	−	11.1121i	−0.163603	−	0.465436i
$571$	38.9955			1.63191			0.815955	−	0.578116i	$-0.196212\pi$
$571$	38.9955			1.63191			0.815955	+	0.578116i	$0.196212\pi$
$572$			3.88544i			0.162458i
$573$			68.6171i			2.86652i
$574$	59.5815			2.48688
$575$	16.9918	+	21.1838i	0.708606	+	0.883425i
$576$	−31.7064			−1.32110
$577$		−	19.3000i		−	0.803470i	−0.915756	−	0.401735i	$-0.868407\pi$
$577$		−	19.3000i		−	0.803470i	0.915756	−	0.401735i	$-0.131593\pi$
$578$		−	33.9684i		−	1.41290i
$579$	−37.2502			−1.54807
$580$	−11.4124	−	32.4674i	−0.473875	−	1.34814i
$581$	−11.3792			−0.472087
$582$		−	85.7270i		−	3.55350i
$583$			7.73991i			0.320554i
$584$	0.305936			0.0126597
$585$	−15.9506	+	5.60669i	−0.659476	+	0.231808i
$586$	28.6665			1.18420
$587$			5.79416i			0.239150i	0.992825	+	0.119575i	$0.0381533\pi$
$587$			5.79416i			0.239150i	−0.992825	+	0.119575i	$0.961847\pi$
$588$		−	58.3033i		−	2.40439i
$589$	−1.09535			−0.0451333
$590$	42.0577	−	14.7834i	1.73149	−	0.608624i
$591$	−46.1805			−1.89961
$592$		−	42.4599i		−	1.74509i
$593$			5.13007i			0.210667i	0.994437	+	0.105333i	$0.0335909\pi$
$593$			5.13007i			0.210667i	−0.994437	+	0.105333i	$0.966409\pi$
$594$	4.82379			0.197923
$595$	0.641949	+	1.82630i	0.0263174	+	0.0748708i
$596$	8.88323			0.363871
$597$		−	50.6921i		−	2.07469i
$598$			21.0071i			0.859046i
$599$	−35.2647			−1.44088			−0.720439	−	0.693518i	$-0.756060\pi$
$599$	−35.2647			−1.44088			−0.720439	+	0.693518i	$0.756060\pi$
$600$	−0.250229	+	0.200711i	−0.0102155	+	0.00819401i
$601$	28.8280			1.17592			0.587959	−	0.808891i	$-0.299932\pi$
$601$	28.8280			1.17592			0.587959	+	0.808891i	$0.299932\pi$
$602$		−	92.9594i		−	3.78874i
$603$			40.6264i			1.65444i
$604$	35.2552			1.43451
$605$	0.741511	+	2.10954i	0.0301467	+	0.0857650i
$606$	−36.7011			−1.49088
$607$			6.38028i			0.258967i	0.991582	+	0.129484i	$0.0413320\pi$
$607$			6.38028i			0.258967i	−0.991582	+	0.129484i	$0.958668\pi$
$608$			8.01187i			0.324924i
$609$	−85.3822			−3.45986
$610$	23.6752	−	8.32193i	0.958582	−	0.336945i
$611$	−12.8163			−0.518491
$612$			1.60698i			0.0649584i
$613$			25.5239i			1.03090i	0.856919	+	0.515451i	$0.172376\pi$
$613$			25.5239i			1.03090i	−0.856919	+	0.515451i	$0.827624\pi$
$614$	−14.9527			−0.603443
$615$	−38.8753	+	13.6648i	−1.56761	+	0.551019i
$616$	0.103555			0.00417235
$617$			40.7075i			1.63882i	0.573206	+	0.819411i	$0.305700\pi$
$617$			40.7075i			1.63882i	−0.573206	+	0.819411i	$0.694300\pi$
$618$			27.9215i			1.12317i
$619$	31.3746			1.26105			0.630525	−	0.776169i	$-0.282840\pi$
$619$	31.3746			1.26105			0.630525	+	0.776169i	$0.282840\pi$
$620$	1.63433	+	4.64953i	0.0656362	+	0.186730i
$621$	13.0798			0.524875
$622$			21.6521i			0.868170i
$623$			43.5447i			1.74458i
$624$	20.1876			0.808150
$625$	5.42504	−	24.4043i	0.217002	−	0.976171i
$626$	50.0198			1.99919
$627$		−	2.62978i		−	0.105023i
$628$			28.2690i			1.12805i
$629$	−2.17829			−0.0868542
$630$	24.6875	+	70.2341i	0.983575	+	2.79819i
$631$	−28.4794			−1.13375			−0.566874	−	0.823805i	$-0.691847\pi$
$631$	−28.4794			−1.13375			−0.566874	+	0.823805i	$0.691847\pi$
$632$			0.195176i			0.00776369i
$633$			33.2755i			1.32258i
$634$	−12.3290			−0.489646
$635$	−13.9076	+	4.88857i	−0.551907	+	0.193997i
$636$	−40.9565			−1.62403
$637$			21.2755i			0.842967i
$638$		−	15.3209i		−	0.606559i
$639$	−0.252175			−0.00997587
$640$	0.411705	−	0.144716i	0.0162741	−	0.00572040i
$641$	33.0519			1.30547			0.652736	−	0.757585i	$-0.273621\pi$
$641$	33.0519			1.30547			0.652736	+	0.757585i	$0.273621\pi$
$642$			24.5442i			0.968684i
$643$			48.7988i			1.92444i	0.272276	+	0.962219i	$0.412224\pi$
$643$			48.7988i			1.92444i	−0.272276	+	0.962219i	$0.587776\pi$
$644$	46.3900			1.82802
$645$	21.3199	+	60.6535i	0.839471	+	2.38823i
$646$	0.408525			0.0160732
$647$		−	17.2358i		−	0.677611i	−0.940856	−	0.338806i	$-0.889977\pi$
$647$		−	17.2358i		−	0.677611i	0.940856	−	0.338806i	$-0.110023\pi$
$648$		−	0.132083i		−	0.00518869i
$649$	9.95331			0.390702
$650$	15.0857	−	12.1004i	0.591708	−	0.474616i
$651$	12.2273			0.479224
$652$			23.0961i			0.904514i
$653$			44.9093i			1.75744i	0.477341	+	0.878718i	$0.341601\pi$
$653$			44.9093i			1.75744i	−0.477341	+	0.878718i	$0.658399\pi$
$654$	−93.3546			−3.65045
$655$	−0.659714	−	1.87684i	−0.0257772	−	0.0733340i
$656$	27.8585			1.08769
$657$			49.1054i			1.91578i
$658$			56.4330i			2.19999i
$659$	−0.147705			−0.00575376			−0.00287688	−	0.999996i	$-0.500916\pi$
$659$	−0.147705			−0.00575376			−0.00287688	+	0.999996i	$0.500916\pi$
$660$	−11.1628	+	3.92377i	−0.434513	+	0.152733i
$661$	3.87012			0.150530			0.0752651	−	0.997164i	$-0.476020\pi$
$661$	3.87012			0.150530			0.0752651	+	0.997164i	$0.476020\pi$
$662$		−	16.2157i		−	0.630241i
$663$		−	1.03567i		−	0.0402221i
$664$	0.0653992			0.00253798
$665$	8.95452	−	3.14755i	0.347242	−	0.122057i
$666$	−83.7709			−3.24606
$667$		−	41.5428i		−	1.60855i
$668$			10.0025i			0.387008i
$669$	57.0074			2.20403
$670$	−15.4099	−	43.8401i	−0.595338	−	1.69369i
$671$	5.60294			0.216299
$672$		−	89.4352i		−	3.45004i
$673$			23.6340i			0.911024i	0.890230	+	0.455512i	$0.150544\pi$
$673$			23.6340i			0.911024i	−0.890230	+	0.455512i	$0.849456\pi$
$674$	0.855756			0.0329625
$675$	−7.53415	−	9.39289i	−0.289989	−	0.361532i
$676$	−18.6557			−0.717527
$677$		−	15.0220i		−	0.577343i	−0.957428	−	0.288672i	$-0.906786\pi$
$677$		−	15.0220i		−	0.577343i	0.957428	−	0.288672i	$-0.0932136\pi$
$678$			73.5847i			2.82600i
$679$	69.0816			2.65111
$680$	−0.00368946	−	0.0104962i	−0.000141484	−	0.000402512i
$681$	58.5831			2.24491
$682$			2.19404i			0.0840142i
$683$		−	23.6347i		−	0.904358i	−0.891927	−	0.452179i	$-0.850647\pi$
$683$		−	23.6347i		−	0.904358i	0.891927	−	0.452179i	$-0.149353\pi$
$684$	7.87920			0.301269
$685$	3.50538	−	1.23215i	0.133934	−	0.0470781i
$686$	34.1638			1.30438
$687$		−	20.1101i		−	0.767250i
$688$		−	43.4650i		−	1.65709i
$689$	14.9455			0.569377
$690$	−60.3533	+	21.2144i	−2.29761	+	0.807618i
$691$	−33.1498			−1.26108			−0.630540	−	0.776157i	$-0.717166\pi$
$691$	−33.1498			−1.26108			−0.630540	+	0.776157i	$0.717166\pi$
$692$		−	22.3681i		−	0.850307i
$693$			16.6215i			0.631398i
$694$	−5.39757			−0.204889
$695$	8.01239	+	22.7946i	0.303927	+	0.864649i
$696$	0.490715			0.0186005
$697$		−	1.42921i		−	0.0541351i
$698$		−	47.1049i		−	1.78295i
$699$	−10.0731			−0.381000
$700$	−26.7212	−	33.3136i	−1.00997	−	1.25914i
$701$	13.0640			0.493420			0.246710	−	0.969089i	$-0.420650\pi$
$701$	13.0640			0.493420			0.246710	+	0.969089i	$0.420650\pi$
$702$		−	9.31456i		−	0.351555i
$703$			10.6804i			0.402819i
$704$	8.09714			0.305172
$705$	−12.9427	−	36.8210i	−0.487451	−	1.38676i
$706$	−43.4382			−1.63482
$707$		−	29.5749i		−	1.11228i
$708$			52.6689i			1.97942i
$709$	−27.5865			−1.03603			−0.518016	−	0.855371i	$-0.673329\pi$
$709$	−27.5865			−1.03603			−0.518016	+	0.855371i	$0.673329\pi$
$710$	0.272122	−	0.0956519i	0.0102126	−	0.00358975i
$711$	−31.3275			−1.17487
$712$		−	0.250263i		−	0.00937902i
$713$			5.94919i			0.222799i
$714$	−4.56030			−0.170665
$715$	4.07344	−	1.43183i	0.152338	−	0.0535474i
$716$	18.5614			0.693674
$717$		−	23.4120i		−	0.874337i
$718$			23.0807i			0.861366i
$719$	−5.86813			−0.218844			−0.109422	−	0.993995i	$-0.534900\pi$
$719$	−5.86813			−0.218844			−0.109422	+	0.993995i	$0.534900\pi$
$720$	11.5431	+	32.8393i	0.430187	+	1.22385i
$721$	−22.5000			−0.837945
$722$		−	2.00304i		−	0.0745455i
$723$			42.5558i			1.58267i
$724$	−4.06434			−0.151050
$725$	−29.8328	+	23.9292i	−1.10796	+	0.888709i
$726$	−5.26757			−0.195498
$727$			36.5169i			1.35434i	0.735828	+	0.677169i	$0.236793\pi$
$727$			36.5169i			1.35434i	−0.735828	+	0.677169i	$0.763207\pi$
$728$		−	0.199961i		−	0.00741104i
$729$	40.1998			1.48888
$730$	−18.6261	−	52.9898i	−0.689382	−	1.96124i
$731$	−2.22986			−0.0824742
$732$			29.6485i			1.09584i
$733$			8.35429i			0.308573i	0.988026	+	0.154286i	$0.0493078\pi$
$733$			8.35429i			0.308573i	−0.988026	+	0.154286i	$0.950692\pi$
$734$	−40.9727			−1.51233
$735$	−61.1244	+	21.4854i	−2.25461	+	0.792503i
$736$	43.5148			1.60398
$737$		−	10.3751i		−	0.382172i
$738$		−	54.9632i		−	2.02322i
$739$	−15.9022			−0.584974			−0.292487	−	0.956270i	$-0.594483\pi$
$739$	−15.9022			−0.584974			−0.292487	+	0.956270i	$0.594483\pi$
$740$	45.3359	−	15.9357i	1.66658	−	0.585809i
$741$	−5.07801			−0.186545
$742$		−	65.8083i		−	2.41590i
$743$			19.6036i			0.719185i	0.933109	+	0.359592i	$0.117084\pi$
$743$			19.6036i			0.719185i	−0.933109	+	0.359592i	$0.882916\pi$
$744$	−0.0702734			−0.00257635
$745$	−3.27357	−	9.31305i	−0.119934	−	0.341204i
$746$	−15.1595			−0.555029
$747$			10.4971i			0.384070i
$748$		−	0.410389i		−	0.0150053i
$749$	−19.7785			−0.722692
$750$	49.9988	+	31.1212i	1.82570	+	1.13638i
$751$	−30.8298			−1.12499			−0.562497	−	0.826799i	$-0.690159\pi$
$751$	−30.8298			−1.12499			−0.562497	+	0.826799i	$0.690159\pi$
$752$			26.3864i			0.962211i
$753$		−	32.7946i		−	1.19510i
$754$	−29.5840			−1.07739
$755$	−12.9919	−	36.9611i	−0.472825	−	1.34515i
$756$	−20.5693			−0.748099
$757$			23.6020i			0.857831i	0.903345	+	0.428915i	$0.141104\pi$
$757$			23.6020i			0.857831i	−0.903345	+	0.428915i	$0.858896\pi$
$758$			25.4229i			0.923401i
$759$	−14.2831			−0.518444
$760$	−0.0514641	+	0.0180898i	−0.00186680	+	0.000656187i
$761$	32.4511			1.17635			0.588175	−	0.808733i	$-0.299846\pi$
$761$	32.4511			1.17635			0.588175	+	0.808733i	$0.299846\pi$
$762$		−	34.7276i		−	1.25805i
$763$		−	75.2281i		−	2.72344i
$764$	52.5024			1.89947
$765$	1.68474	−	0.592191i	0.0609118	−	0.0214107i
$766$	−47.8477			−1.72881
$767$		−	19.2194i		−	0.693974i
$768$		−	41.5594i		−	1.49964i
$769$	−14.1056			−0.508660			−0.254330	−	0.967118i	$-0.581855\pi$
$769$	−14.1056			−0.508660			−0.254330	+	0.967118i	$0.581855\pi$
$770$	−6.30467	−	17.9363i	−0.227205	−	0.646379i
$771$	−55.9142			−2.01370
$772$			28.5020i			1.02581i
$773$			29.9621i			1.07766i	0.842413	+	0.538832i	$0.181134\pi$
$773$			29.9621i			1.07766i	−0.842413	+	0.538832i	$0.818866\pi$
$774$	−85.7539			−3.08236
$775$	4.27224	−	3.42681i	0.153463	−	0.123095i
$776$	−0.397031			−0.0142526
$777$		−	119.223i		−	4.27712i
$778$			9.99576i			0.358365i
$779$	−7.00756			−0.251072
$780$	7.57665	+	21.5550i	0.271288	+	0.771792i
$781$	0.0644000			0.00230441
$782$		−	2.21882i		−	0.0793448i
$783$			18.4201i			0.658280i
$784$	43.8024			1.56437
$785$	29.6368	−	10.4174i	1.05778	−	0.371814i
$786$	4.68650			0.167162
$787$			25.9877i			0.926362i	0.886264	+	0.463181i	$0.153292\pi$
$787$			25.9877i			0.926362i	−0.886264	+	0.463181i	$0.846708\pi$
$788$			35.3350i			1.25876i
$789$	41.2377			1.46810
$790$	33.8055	−	11.8828i	1.20275	−	0.422770i
$791$	−59.2969			−2.10835
$792$		−	0.0955283i		−	0.00339445i
$793$		−	10.8191i		−	0.384196i
$794$	31.4088			1.11466
$795$	15.0929	+	42.9382i	0.535291	+	1.52286i
$796$	−38.7871			−1.37477
$797$		−	9.65797i		−	0.342103i	−0.985262	−	0.171051i	$-0.945284\pi$
$797$		−	9.65797i		−	0.342103i	0.985262	−	0.171051i	$-0.0547164\pi$
$798$			22.3596i			0.791522i
$799$	1.35368			0.0478899
$800$	−25.0651	−	31.2489i	−0.886185	−	1.10482i
$801$	40.1695			1.41932
$802$		−	29.7839i		−	1.05171i
$803$		−	12.5405i		−	0.442544i
$804$	54.9009			1.93621
$805$	−17.0952	−	48.6346i	−0.602528	−	1.71415i
$806$	4.23661			0.149228
$807$			20.6985i			0.728623i
$808$			0.169975i			0.00597971i
$809$	−41.4280			−1.45653			−0.728266	−	0.685294i	$-0.759673\pi$
$809$	−41.4280			−1.45653			−0.728266	+	0.685294i	$0.759673\pi$
$810$	−22.8774	+	8.04149i	−0.803830	+	0.282549i
$811$	48.8467			1.71524			0.857619	−	0.514286i	$-0.171943\pi$
$811$	48.8467			1.71524			0.857619	+	0.514286i	$0.171943\pi$
$812$			65.3303i			2.29264i
$813$		−	12.8375i		−	0.450231i
$814$	21.3933			0.749835
$815$	24.2136	−	8.51118i	0.848167	−	0.298134i
$816$	−2.13226			−0.0746439
$817$			10.9332i			0.382505i
$818$		−	45.3855i		−	1.58687i
$819$	32.0954			1.12151
$820$	10.4556	+	29.7455i	0.365127	+	1.03876i
$821$	−42.4166			−1.48035			−0.740174	−	0.672415i	$-0.765257\pi$
$821$	−42.4166			−1.48035			−0.740174	+	0.672415i	$0.765257\pi$
$822$			8.75300i			0.305296i
$823$		−	17.1257i		−	0.596963i	−0.954415	−	0.298482i	$-0.903520\pi$
$823$		−	17.1257i		−	0.596963i	0.954415	−	0.298482i	$-0.0964802\pi$
$824$	0.129314			0.00450487
$825$	8.22726	+	10.2570i	0.286436	+	0.357103i
$826$	−84.6276			−2.94457
$827$			50.6848i			1.76248i	0.472666	+	0.881242i	$0.343292\pi$
$827$			50.6848i			1.76248i	−0.472666	+	0.881242i	$0.656708\pi$
$828$		−	42.7942i		−	1.48720i
$829$	−37.1815			−1.29137			−0.645683	−	0.763605i	$-0.723427\pi$
$829$	−37.1815			−1.29137			−0.645683	+	0.763605i	$0.723427\pi$
$830$	−3.98165	−	11.3275i	−0.138205	−	0.393183i
$831$	−0.993958			−0.0344800
$832$		−	15.6353i		−	0.542055i
$833$		−	2.24717i		−	0.0778598i
$834$	−56.9186			−1.97093
$835$	10.4865	−	3.68603i	0.362899	−	0.127560i
$836$	−2.01218			−0.0695927
$837$		−	2.63787i		−	0.0911781i
$838$		−	34.8776i		−	1.20483i
$839$	12.7017			0.438511			0.219256	−	0.975667i	$-0.429637\pi$
$839$	12.7017			0.438511			0.219256	+	0.975667i	$0.429637\pi$
$840$	0.574485	−	0.201934i	0.0198216	−	0.00696737i
$841$	29.5041			1.01738
$842$			6.96554i			0.240048i
$843$			23.6980i			0.816201i
$844$	25.4608			0.876397
$845$	6.87484	+	19.5584i	0.236502	+	0.672829i
$846$	52.0588			1.78982
$847$		−	4.24477i		−	0.145852i
$848$		−	30.7700i		−	1.05665i
$849$	5.47657			0.187955
$850$	−1.59338	+	1.27807i	−0.0546525	+	0.0438374i
$851$	58.0084			1.98850
$852$			0.340779i			0.0116749i
$853$		−	44.2940i		−	1.51660i	−0.651906	−	0.758299i	$-0.726031\pi$
$853$		−	44.2940i		−	1.51660i	0.651906	−	0.758299i	$-0.273969\pi$
$854$	−47.6388			−1.63017
$855$	−2.90357	−	8.26044i	−0.0993002	−	0.282501i
$856$	0.113673			0.00388526
$857$		−	33.6860i		−	1.15069i	−0.817910	−	0.575346i	$-0.804867\pi$
$857$		−	33.6860i		−	1.15069i	0.817910	−	0.575346i	$-0.195133\pi$
$858$			10.1715i			0.347248i
$859$	−13.5992			−0.464000			−0.232000	−	0.972716i	$-0.574527\pi$
$859$	−13.5992			−0.464000			−0.232000	+	0.972716i	$0.574527\pi$
$860$	46.4091	−	16.3130i	1.58254	−	0.556267i
$861$	78.2242			2.66587
$862$		−	41.9558i		−	1.42902i
$863$			6.37246i			0.216921i	0.994101	+	0.108461i	$0.0345921\pi$
$863$			6.37246i			0.216921i	−0.994101	+	0.108461i	$0.965408\pi$
$864$	−19.2945			−0.656411
$865$	−23.4504	+	8.24289i	−0.797337	+	0.280267i
$866$	62.4909			2.12353
$867$		−	44.5969i		−	1.51459i
$868$		−	9.35569i		−	0.317553i
$869$	8.00036			0.271394
$870$	−29.8759	−	84.9945i	−1.01289	−	2.88158i
$871$	−20.0339			−0.678824
$872$			0.432357i			0.0146415i
$873$		−	63.7269i		−	2.15683i
$874$	−10.8791			−0.367991
$875$	−25.0784	+	40.2906i	−0.847806	+	1.36207i
$876$	66.3591			2.24207
$877$			49.2321i			1.66245i	0.555936	+	0.831225i	$0.312360\pi$
$877$			49.2321i			1.66245i	−0.555936	+	0.831225i	$0.687640\pi$
$878$		−	24.2718i		−	0.819135i
$879$	37.6361			1.26943
$880$	−2.94787	−	8.38646i	−0.0993727	−	0.282708i
$881$	20.2763			0.683125			0.341562	−	0.939859i	$-0.389044\pi$
$881$	20.2763			0.683125			0.341562	+	0.939859i	$0.389044\pi$
$882$		−	86.4197i		−	2.90990i
$883$		−	7.95263i		−	0.267627i	−0.991007	−	0.133814i	$-0.957278\pi$
$883$		−	7.95263i		−	0.267627i	0.991007	−	0.133814i	$-0.0427224\pi$
$884$	−0.792444			−0.0266528
$885$	55.2173	−	19.4091i	1.85611	−	0.652429i
$886$	−9.04330			−0.303815
$887$		−	26.2463i		−	0.881265i	−0.897687	−	0.440633i	$-0.854754\pi$
$887$		−	26.2463i		−	0.881265i	0.897687	−	0.440633i	$-0.145246\pi$
$888$			0.685211i			0.0229942i
$889$	27.9846			0.938573
$890$	−43.3470	+	15.2366i	−1.45299	+	0.510733i
$891$	−5.41413			−0.181380
$892$		−	43.6192i		−	1.46048i
$893$		−	6.63725i		−	0.222107i
$894$	23.2549			0.777760
$895$	−6.84010	−	19.4596i	−0.228639	−	0.650461i
$896$	−0.828425			−0.0276757
$897$			27.5801i			0.920874i
$898$		−	53.8877i		−	1.79826i
$899$	−8.37814			−0.279427
$900$	−30.7314	+	24.6500i	−1.02438	+	0.821668i
$901$	−1.57857			−0.0525899
$902$			14.0364i			0.467362i
$903$		−	122.046i		−	4.06143i
$904$	0.340796			0.0113347
$905$	1.49776	+	4.26100i	0.0497871	+	0.141640i
$906$	92.2925			3.06621
$907$			34.8745i			1.15799i	0.815331	+	0.578995i	$0.196555\pi$
$907$			34.8745i			1.15799i	−0.815331	+	0.578995i	$0.803445\pi$
$908$		−	44.8249i		−	1.48757i
$909$	−27.2825			−0.904904
$910$	−34.6343	+	12.1741i	−1.14811	+	0.403566i
$911$	2.69271			0.0892133			0.0446066	−	0.999005i	$-0.485797\pi$
$911$	2.69271			0.0892133			0.0446066	+	0.999005i	$0.485797\pi$
$912$			10.4547i			0.346189i
$913$		−	2.68074i		−	0.0887197i
$914$	43.5458			1.44037
$915$	31.0831	−	10.9258i	1.02757	−	0.361196i
$916$	−15.3873			−0.508410
$917$			3.77653i			0.124712i
$918$			0.983824i			0.0324710i
$919$	13.8628			0.457291			0.228646	−	0.973510i	$-0.426570\pi$
$919$	13.8628			0.457291			0.228646	+	0.973510i	$0.426570\pi$
$920$	0.0982511	+	0.279517i	0.00323924	+	0.00921539i
$921$	−19.6313			−0.646875
$922$		−	51.2508i		−	1.68786i
$923$		−	0.124354i		−	0.00409316i
$924$	22.4616			0.738933
$925$	−33.4136	−	41.6570i	−1.09863	−	1.36967i
$926$	−41.4826			−1.36320
$927$			20.7560i			0.681717i
$928$			61.2812i			2.01165i
$929$	12.9794			0.425840			0.212920	−	0.977070i	$-0.431703\pi$
$929$	12.9794			0.425840			0.212920	+	0.977070i	$0.431703\pi$
$930$	4.27841	+	12.1717i	0.140295	+	0.399127i
$931$	−11.0181			−0.361104
$932$			7.70744i			0.252466i
$933$			28.4269i			0.930655i
$934$	−46.0738			−1.50758
$935$	−0.430246	+	0.151233i	−0.0140705	+	0.00494584i
$936$	−0.184461			−0.00602931
$937$			9.20310i			0.300652i	0.988636	+	0.150326i	$0.0480324\pi$
$937$			9.20310i			0.300652i	−0.988636	+	0.150326i	$0.951968\pi$
$938$			88.2141i			2.88029i
$939$	65.6707			2.14308
$940$	−28.1736	+	9.90314i	−0.918923	+	0.323005i
$941$	−19.3393			−0.630442			−0.315221	−	0.949018i	$-0.602079\pi$
$941$	−19.3393			−0.630442			−0.315221	+	0.949018i	$0.602079\pi$
$942$			74.0036i			2.41117i
$943$			38.0601i			1.23941i
$944$	−39.5693			−1.28787
$945$	7.58003	+	21.5646i	0.246578	+	0.701496i
$946$	21.8997			0.712021
$947$			20.9304i			0.680148i	0.940399	+	0.340074i	$0.110452\pi$
$947$			20.9304i			0.680148i	−0.940399	+	0.340074i	$0.889548\pi$
$948$			42.3347i			1.37497i
$949$	−24.2152			−0.786057
$950$	6.26651	+	7.81251i	0.203312	+	0.253471i
$951$	−16.1866			−0.524888
$952$			0.0211203i			0.000684513i
$953$			27.8155i			0.901034i	0.892768	+	0.450517i	$0.148760\pi$
$953$			27.8155i			0.901034i	−0.892768	+	0.450517i	$0.851240\pi$
$954$	−60.7074			−1.96547
$955$	−19.3477	−	55.0428i	−0.626078	−	1.78114i
$956$	−17.9137			−0.579371
$957$		−	20.1147i		−	0.650215i
$958$		−	21.3223i		−	0.688893i
$959$	−7.05345			−0.227768
$960$	44.9199	−	15.7895i	1.44978	−	0.509604i
$961$	−29.8002			−0.961297
$962$		−	41.3096i		−	1.33188i
$963$			18.2455i			0.587952i
$964$	32.5616			1.04874
$965$	29.8811	−	10.5033i	0.961907	−	0.338114i
$966$	121.442			3.90732
$967$		−	4.64430i		−	0.149351i	−0.997208	−	0.0746754i	$-0.976208\pi$
$967$		−	4.64430i		−	0.149351i	0.997208	−	0.0746754i	$-0.0237921\pi$
$968$			0.0243959i			0.000784114i
$969$	0.536350			0.0172300
$970$	24.1722	+	68.7679i	0.776121	+	2.20800i
$971$	−2.45881			−0.0789069			−0.0394535	−	0.999221i	$-0.512562\pi$
$971$	−2.45881			−0.0789069			−0.0394535	+	0.999221i	$0.512562\pi$
$972$		−	43.1868i		−	1.38522i
$973$		−	45.8669i		−	1.47042i
$974$	−23.9811			−0.768403
$975$	19.8059	−	15.8865i	0.634295	−	0.508776i
$976$	−22.2745			−0.712988
$977$			6.36078i			0.203499i	0.994810	+	0.101750i	$0.0324441\pi$
$977$			6.36078i			0.203499i	−0.994810	+	0.101750i	$0.967556\pi$
$978$			60.4620i			1.93336i
$979$	−10.2584			−0.327861
$980$	16.4396	+	46.7694i	0.525144	+	1.49399i
$981$	−69.3971			−2.21568
$982$			48.1425i			1.53629i
$983$			33.3797i			1.06465i	0.846541	+	0.532323i	$0.178681\pi$
$983$			33.3797i			1.06465i	−0.846541	+	0.532323i	$0.821319\pi$
$984$	−0.449576			−0.0143320
$985$	37.0447	−	13.0214i	1.18034	−	0.414895i
$986$	3.12473			0.0995115
$987$			74.0906i			2.35833i
$988$			3.88544i			0.123612i
$989$	59.3816			1.88822
$990$	−16.5460	+	5.81598i	−0.525867	+	0.184844i
$991$	−32.8103			−1.04225			−0.521127	−	0.853479i	$-0.674488\pi$
$991$	−32.8103			−1.04225			−0.521127	+	0.853479i	$0.674488\pi$
$992$		−	8.77584i		−	0.278633i
$993$		−	21.2895i		−	0.675602i
$994$	−0.547559			−0.0173675
$995$	14.2935	+	40.6638i	0.453134	+	1.28913i
$996$	14.1854			0.449482
$997$		−	39.0068i		−	1.23536i	−0.786430	−	0.617679i	$-0.788073\pi$
$997$		−	39.0068i		−	1.23536i	0.786430	−	0.617679i	$-0.211927\pi$
$998$		−	6.50313i		−	0.205853i
$999$	−25.7209			−0.813773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.c.419.4	✓	20
5.2	odd	4		5225.2.a.ba.1.17		20
5.3	odd	4		5225.2.a.ba.1.4		20
5.4	even	2	inner	1045.2.b.c.419.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.c.419.4	✓	20	1.1	even	1	trivial
1045.2.b.c.419.17	yes	20	5.4	even	2	inner
5225.2.a.ba.1.4		20	5.3	odd	4
5225.2.a.ba.1.17		20	5.2	odd	4

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00304i q^{2} -2.62978i q^{3} -2.01218 q^{4} +(0.741511 + 2.10954i) q^{5} -5.26757 q^{6} -4.24477i q^{7} +0.0243959i q^{8} -3.91575 q^{9} +O(q^{10})\)	\(q-2.00304i q^{2} -2.62978i q^{3} -2.01218 q^{4} +(0.741511 + 2.10954i) q^{5} -5.26757 q^{6} -4.24477i q^{7} +0.0243959i q^{8} -3.91575 q^{9} +(4.22550 - 1.48528i) q^{10} +1.00000 q^{11} +5.29159i q^{12} -1.93096i q^{13} -8.50246 q^{14} +(5.54763 - 1.95001i) q^{15} -3.97549 q^{16} +0.203952i q^{17} +7.84342i q^{18} +1.00000 q^{19} +(-1.49205 - 4.24477i) q^{20} -11.1628 q^{21} -2.00304i q^{22} -5.43129i q^{23} +0.0641559 q^{24} +(-3.90032 + 3.12849i) q^{25} -3.86780 q^{26} +2.40823i q^{27} +8.54125i q^{28} +7.64880 q^{29} +(-3.90596 - 11.1121i) q^{30} -1.09535 q^{31} +8.01187i q^{32} -2.62978i q^{33} +0.408525 q^{34} +(8.95452 - 3.14755i) q^{35} +7.87920 q^{36} +10.6804i q^{37} -2.00304i q^{38} -5.07801 q^{39} +(-0.0514641 + 0.0180898i) q^{40} -7.00756 q^{41} +22.3596i q^{42} +10.9332i q^{43} -2.01218 q^{44} +(-2.90357 - 8.26044i) q^{45} -10.8791 q^{46} -6.63725i q^{47} +10.4547i q^{48} -11.0181 q^{49} +(6.26651 + 7.81251i) q^{50} +0.536350 q^{51} +3.88544i q^{52} +7.73991i q^{53} +4.82379 q^{54} +(0.741511 + 2.10954i) q^{55} +0.103555 q^{56} -2.62978i q^{57} -15.3209i q^{58} +9.95331 q^{59} +(-11.1628 + 3.92377i) q^{60} +5.60294 q^{61} +2.19404i q^{62} +16.6215i q^{63} +8.09714 q^{64} +(4.07344 - 1.43183i) q^{65} -5.26757 q^{66} -10.3751i q^{67} -0.410389i q^{68} -14.2831 q^{69} +(-6.30467 - 17.9363i) q^{70} +0.0644000 q^{71} -0.0955283i q^{72} -12.5405i q^{73} +21.3933 q^{74} +(8.22726 + 10.2570i) q^{75} -2.01218 q^{76} -4.24477i q^{77} +10.1715i q^{78} +8.00036 q^{79} +(-2.94787 - 8.38646i) q^{80} -5.41413 q^{81} +14.0364i q^{82} -2.68074i q^{83} +22.4616 q^{84} +(-0.430246 + 0.151233i) q^{85} +21.8997 q^{86} -20.1147i q^{87} +0.0243959i q^{88} -10.2584 q^{89} +(-16.5460 + 5.81598i) q^{90} -8.19649 q^{91} +10.9287i q^{92} +2.88054i q^{93} -13.2947 q^{94} +(0.741511 + 2.10954i) q^{95} +21.0695 q^{96} +16.2745i q^{97} +22.0697i q^{98} -3.91575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

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Database

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