LMFDB - Embedded newform 6422.2.a.bq.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

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:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.02081$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.02081 q^{3} +1.00000 q^{4} +1.43823 q^{5} -1.02081 q^{6} +5.08222 q^{7} +1.00000 q^{8} -1.95796 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.02081 q^{3} +1.00000 q^{4} +1.43823 q^{5} -1.02081 q^{6} +5.08222 q^{7} +1.00000 q^{8} -1.95796 q^{9} +1.43823 q^{10} +4.14791 q^{11} -1.02081 q^{12} +5.08222 q^{14} -1.46816 q^{15} +1.00000 q^{16} +5.61847 q^{17} -1.95796 q^{18} -1.00000 q^{19} +1.43823 q^{20} -5.18796 q^{21} +4.14791 q^{22} +4.40837 q^{23} -1.02081 q^{24} -2.93149 q^{25} +5.06111 q^{27} +5.08222 q^{28} -4.53513 q^{29} -1.46816 q^{30} -6.77822 q^{31} +1.00000 q^{32} -4.23421 q^{33} +5.61847 q^{34} +7.30941 q^{35} -1.95796 q^{36} +7.83347 q^{37} -1.00000 q^{38} +1.43823 q^{40} +2.47361 q^{41} -5.18796 q^{42} -5.28236 q^{43} +4.14791 q^{44} -2.81600 q^{45} +4.40837 q^{46} +12.6201 q^{47} -1.02081 q^{48} +18.8289 q^{49} -2.93149 q^{50} -5.73536 q^{51} +10.9335 q^{53} +5.06111 q^{54} +5.96566 q^{55} +5.08222 q^{56} +1.02081 q^{57} -4.53513 q^{58} -7.26543 q^{59} -1.46816 q^{60} -7.47606 q^{61} -6.77822 q^{62} -9.95076 q^{63} +1.00000 q^{64} -4.23421 q^{66} -12.0693 q^{67} +5.61847 q^{68} -4.50009 q^{69} +7.30941 q^{70} -10.2817 q^{71} -1.95796 q^{72} -14.8354 q^{73} +7.83347 q^{74} +2.99248 q^{75} -1.00000 q^{76} +21.0806 q^{77} +14.6239 q^{79} +1.43823 q^{80} +0.707461 q^{81} +2.47361 q^{82} -3.27736 q^{83} -5.18796 q^{84} +8.08066 q^{85} -5.28236 q^{86} +4.62948 q^{87} +4.14791 q^{88} +3.71188 q^{89} -2.81600 q^{90} +4.40837 q^{92} +6.91925 q^{93} +12.6201 q^{94} -1.43823 q^{95} -1.02081 q^{96} +0.438413 q^{97} +18.8289 q^{98} -8.12143 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9	$ 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9} + q^{10} + 4 q^{11} + 18 q^{14} + 23 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 15 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} + 18 q^{28} - 20 q^{29} + 23 q^{30} + 30 q^{31} + 15 q^{32} + 36 q^{33} + 2 q^{34} + 32 q^{35} + 17 q^{36} + 35 q^{37} - 15 q^{38} + q^{40} + 15 q^{41} + 2 q^{42} + q^{43} + 4 q^{44} - 11 q^{45} + 17 q^{46} + 29 q^{49} + 8 q^{50} - q^{51} - q^{53} + 12 q^{54} - 6 q^{55} + 18 q^{56} - 20 q^{58} - 7 q^{59} + 23 q^{60} - 2 q^{61} + 30 q^{62} + 42 q^{63} + 15 q^{64} + 36 q^{66} + 34 q^{67} + 2 q^{68} - 12 q^{69} + 32 q^{70} + 4 q^{71} + 17 q^{72} + 12 q^{73} + 35 q^{74} + 31 q^{75} - 15 q^{76} - 20 q^{77} + 23 q^{79} + q^{80} + 7 q^{81} + 15 q^{82} - 3 q^{83} + 2 q^{84} + 46 q^{85} + q^{86} + 22 q^{87} + 4 q^{88} + 17 q^{89} - 11 q^{90} + 17 q^{92} + 60 q^{93} - q^{95} + 18 q^{97} + 29 q^{98} - 6 q^{99}+O(q^{100}) $ 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9 + q^10 + 4 * q^11 + 18 * q^14 + 23 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 15 * q^19 + q^20 + 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 + 18 * q^28 - 20 * q^29 + 23 * q^30 + 30 * q^31 + 15 * q^32 + 36 * q^33 + 2 * q^34 + 32 * q^35 + 17 * q^36 + 35 * q^37 - 15 * q^38 + q^40 + 15 * q^41 + 2 * q^42 + q^43 + 4 * q^44 - 11 * q^45 + 17 * q^46 + 29 * q^49 + 8 * q^50 - q^51 - q^53 + 12 * q^54 - 6 * q^55 + 18 * q^56 - 20 * q^58 - 7 * q^59 + 23 * q^60 - 2 * q^61 + 30 * q^62 + 42 * q^63 + 15 * q^64 + 36 * q^66 + 34 * q^67 + 2 * q^68 - 12 * q^69 + 32 * q^70 + 4 * q^71 + 17 * q^72 + 12 * q^73 + 35 * q^74 + 31 * q^75 - 15 * q^76 - 20 * q^77 + 23 * q^79 + q^80 + 7 * q^81 + 15 * q^82 - 3 * q^83 + 2 * q^84 + 46 * q^85 + q^86 + 22 * q^87 + 4 * q^88 + 17 * q^89 - 11 * q^90 + 17 * q^92 + 60 * q^93 - q^95 + 18 * q^97 + 29 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.02081	−0.589362	−0.294681	−	0.955596i	$-0.595213\pi$
$3$	−1.02081	−0.589362	−0.294681	+	0.955596i	$0.595213\pi$
$4$	1.00000	0.500000
$5$	1.43823	0.643197	0.321599	−	0.946876i	$-0.395780\pi$
$5$	1.43823	0.643197	0.321599	+	0.946876i	$0.395780\pi$
$6$	−1.02081	−0.416742
$7$	5.08222	1.92090	0.960449	−	0.278456i	$-0.0898228\pi$
$7$	5.08222	1.92090	0.960449	+	0.278456i	$0.0898228\pi$
$8$	1.00000	0.353553
$9$	−1.95796	−0.652652
$10$	1.43823	0.454809
$11$	4.14791	1.25064	0.625321	−	0.780367i	$-0.284968\pi$
$11$	4.14791	1.25064	0.625321	+	0.780367i	$0.284968\pi$
$12$	−1.02081	−0.294681
$13$	0	0
$13$	0	0
$14$	5.08222	1.35828
$15$	−1.46816	−0.379076
$16$	1.00000	0.250000
$17$	5.61847	1.36268	0.681339	−	0.731968i	$-0.261398\pi$
$17$	5.61847	1.36268	0.681339	+	0.731968i	$0.261398\pi$
$18$	−1.95796	−0.461495
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.43823	0.321599
$21$	−5.18796	−1.13210
$22$	4.14791	0.884338
$23$	4.40837	0.919209	0.459604	−	0.888124i	$-0.347991\pi$
$23$	4.40837	0.919209	0.459604	+	0.888124i	$0.347991\pi$
$24$	−1.02081	−0.208371
$25$	−2.93149	−0.586297
$26$	0	0
$27$	5.06111	0.974011
$28$	5.08222	0.960449
$29$	−4.53513	−0.842152	−0.421076	−	0.907025i	$-0.638347\pi$
$29$	−4.53513	−0.842152	−0.421076	+	0.907025i	$0.638347\pi$
$30$	−1.46816	−0.268047
$31$	−6.77822	−1.21741	−0.608703	−	0.793399i	$-0.708310\pi$
$31$	−6.77822	−1.21741	−0.608703	+	0.793399i	$0.708310\pi$
$32$	1.00000	0.176777
$33$	−4.23421	−0.737081
$34$	5.61847	0.963559
$35$	7.30941	1.23552
$36$	−1.95796	−0.326326
$37$	7.83347	1.28781	0.643907	−	0.765103i	$-0.277312\pi$
$37$	7.83347	1.28781	0.643907	+	0.765103i	$0.277312\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	1.43823	0.227405
$41$	2.47361	0.386313	0.193157	−	0.981168i	$-0.438127\pi$
$41$	2.47361	0.386313	0.193157	+	0.981168i	$0.438127\pi$
$42$	−5.18796	−0.800519
$43$	−5.28236	−0.805553	−0.402776	−	0.915298i	$-0.631955\pi$
$43$	−5.28236	−0.805553	−0.402776	+	0.915298i	$0.631955\pi$
$44$	4.14791	0.625321
$45$	−2.81600	−0.419784
$46$	4.40837	0.649979
$47$	12.6201	1.84084	0.920418	−	0.390935i	$-0.127848\pi$
$47$	12.6201	1.84084	0.920418	+	0.390935i	$0.127848\pi$
$48$	−1.02081	−0.147341
$49$	18.8289	2.68985
$50$	−2.93149	−0.414575
$51$	−5.73536	−0.803111
$52$	0	0
$53$	10.9335	1.50183	0.750914	−	0.660400i	$-0.229613\pi$
$53$	10.9335	1.50183	0.750914	+	0.660400i	$0.229613\pi$
$54$	5.06111	0.688730
$55$	5.96566	0.804410
$56$	5.08222	0.679140
$57$	1.02081	0.135209
$58$	−4.53513	−0.595491
$59$	−7.26543	−0.945878	−0.472939	−	0.881095i	$-0.656807\pi$
$59$	−7.26543	−0.945878	−0.472939	+	0.881095i	$0.656807\pi$
$60$	−1.46816	−0.189538
$61$	−7.47606	−0.957212	−0.478606	−	0.878030i	$-0.658858\pi$
$61$	−7.47606	−0.957212	−0.478606	+	0.878030i	$0.658858\pi$
$62$	−6.77822	−0.860835
$63$	−9.95076	−1.25368
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.23421	−0.521195
$67$	−12.0693	−1.47449	−0.737247	−	0.675623i	$-0.763875\pi$
$67$	−12.0693	−1.47449	−0.737247	+	0.675623i	$0.763875\pi$
$68$	5.61847	0.681339
$69$	−4.50009	−0.541747
$70$	7.30941	0.873642
$71$	−10.2817	−1.22021	−0.610107	−	0.792319i	$-0.708874\pi$
$71$	−10.2817	−1.22021	−0.610107	+	0.792319i	$0.708874\pi$
$72$	−1.95796	−0.230747
$73$	−14.8354	−1.73635	−0.868174	−	0.496260i	$-0.834706\pi$
$73$	−14.8354	−1.73635	−0.868174	+	0.496260i	$0.834706\pi$
$74$	7.83347	0.910623
$75$	2.99248	0.345541
$76$	−1.00000	−0.114708
$77$	21.0806	2.40236
$78$	0	0
$79$	14.6239	1.64532	0.822659	−	0.568536i	$-0.192490\pi$
$79$	14.6239	1.64532	0.822659	+	0.568536i	$0.192490\pi$
$80$	1.43823	0.160799
$81$	0.707461	0.0786068
$82$	2.47361	0.273165
$83$	−3.27736	−0.359737	−0.179868	−	0.983691i	$-0.557567\pi$
$83$	−3.27736	−0.359737	−0.179868	+	0.983691i	$0.557567\pi$
$84$	−5.18796	−0.566052
$85$	8.08066	0.876471
$86$	−5.28236	−0.569612
$87$	4.62948	0.496332
$88$	4.14791	0.442169
$89$	3.71188	0.393459	0.196729	−	0.980458i	$-0.436968\pi$
$89$	3.71188	0.393459	0.196729	+	0.980458i	$0.436968\pi$
$90$	−2.81600	−0.296832
$91$	0	0
$92$	4.40837	0.459604
$93$	6.91925	0.717493
$94$	12.6201	1.30167
$95$	−1.43823	−0.147560
$96$	−1.02081	−0.104186
$97$	0.438413	0.0445141	0.0222571	−	0.999752i	$-0.492915\pi$
$97$	0.438413	0.0445141	0.0222571	+	0.999752i	$0.492915\pi$
$98$	18.8289	1.90201
$99$	−8.12143	−0.816234
$100$	−2.93149	−0.293149
$101$	−5.15113	−0.512556	−0.256278	−	0.966603i	$-0.582496\pi$
$101$	−5.15113	−0.512556	−0.256278	+	0.966603i	$0.582496\pi$
$102$	−5.73536	−0.567885
$103$	12.6076	1.24227	0.621134	−	0.783705i	$-0.286672\pi$
$103$	12.6076	1.24227	0.621134	+	0.783705i	$0.286672\pi$
$104$	0	0
$105$	−7.46149	−0.728167
$106$	10.9335	1.06195
$107$	−13.9445	−1.34806	−0.674032	−	0.738702i	$-0.735439\pi$
$107$	−13.9445	−1.34806	−0.674032	+	0.738702i	$0.735439\pi$
$108$	5.06111	0.487005
$109$	1.44804	0.138697	0.0693485	−	0.997592i	$-0.477908\pi$
$109$	1.44804	0.138697	0.0693485	+	0.997592i	$0.477908\pi$
$110$	5.96566	0.568804
$111$	−7.99645	−0.758990
$112$	5.08222	0.480224
$113$	−7.95897	−0.748717	−0.374358	−	0.927284i	$-0.622137\pi$
$113$	−7.95897	−0.748717	−0.374358	+	0.927284i	$0.622137\pi$
$114$	1.02081	0.0956072
$115$	6.34026	0.591233
$116$	−4.53513	−0.421076
$117$	0	0
$118$	−7.26543	−0.668837
$119$	28.5543	2.61756
$120$	−1.46816	−0.134024
$121$	6.20517	0.564106
$122$	−7.47606	−0.676851
$123$	−2.52508	−0.227678
$124$	−6.77822	−0.608703
$125$	−11.4073	−1.02030
$126$	−9.95076	−0.886484
$127$	−14.8829	−1.32065	−0.660324	−	0.750981i	$-0.729581\pi$
$127$	−14.8829	−1.32065	−0.660324	+	0.750981i	$0.729581\pi$
$128$	1.00000	0.0883883
$129$	5.39227	0.474763
$130$	0	0
$131$	4.52135	0.395032	0.197516	−	0.980300i	$-0.436713\pi$
$131$	4.52135	0.395032	0.197516	+	0.980300i	$0.436713\pi$
$132$	−4.23421	−0.368541
$133$	−5.08222	−0.440684
$134$	−12.0693	−1.04263
$135$	7.27905	0.626481
$136$	5.61847	0.481779
$137$	−6.61223	−0.564921	−0.282461	−	0.959279i	$-0.591151\pi$
$137$	−6.61223	−0.564921	−0.282461	+	0.959279i	$0.591151\pi$
$138$	−4.50009	−0.383073
$139$	−11.8025	−1.00108	−0.500539	−	0.865714i	$-0.666865\pi$
$139$	−11.8025	−1.00108	−0.500539	+	0.865714i	$0.666865\pi$
$140$	7.30941	0.617758
$141$	−12.8827	−1.08492
$142$	−10.2817	−0.862822
$143$	0	0
$144$	−1.95796	−0.163163
$145$	−6.52257	−0.541670
$146$	−14.8354	−1.22778
$147$	−19.2207	−1.58530
$148$	7.83347	0.643907
$149$	21.7459	1.78149	0.890746	−	0.454501i	$-0.150182\pi$
$149$	21.7459	1.78149	0.890746	+	0.454501i	$0.150182\pi$
$150$	2.99248	0.244335
$151$	−23.0798	−1.87821	−0.939103	−	0.343637i	$-0.888341\pi$
$151$	−23.0798	−1.87821	−0.939103	+	0.343637i	$0.888341\pi$
$152$	−1.00000	−0.0811107
$153$	−11.0007	−0.889355
$154$	21.0806	1.69872
$155$	−9.74867	−0.783032
$156$	0	0
$157$	−11.9483	−0.953579	−0.476789	−	0.879018i	$-0.658200\pi$
$157$	−11.9483	−0.953579	−0.476789	+	0.879018i	$0.658200\pi$
$158$	14.6239	1.16342
$159$	−11.1609	−0.885120
$160$	1.43823	0.113702
$161$	22.4043	1.76571
$162$	0.707461	0.0555834
$163$	−3.11405	−0.243911	−0.121955	−	0.992536i	$-0.538916\pi$
$163$	−3.11405	−0.243911	−0.121955	+	0.992536i	$0.538916\pi$
$164$	2.47361	0.193157
$165$	−6.08978	−0.474089
$166$	−3.27736	−0.254372
$167$	−5.96403	−0.461510	−0.230755	−	0.973012i	$-0.574120\pi$
$167$	−5.96403	−0.461510	−0.230755	+	0.973012i	$0.574120\pi$
$168$	−5.18796	−0.400259
$169$	0	0
$170$	8.08066	0.619759
$171$	1.95796	0.149729
$172$	−5.28236	−0.402776
$173$	2.61812	0.199052	0.0995260	−	0.995035i	$-0.468267\pi$
$173$	2.61812	0.199052	0.0995260	+	0.995035i	$0.468267\pi$
$174$	4.62948	0.350960
$175$	−14.8984	−1.12622
$176$	4.14791	0.312661
$177$	7.41659	0.557465
$178$	3.71188	0.278217
$179$	19.8815	1.48602	0.743008	−	0.669282i	$-0.233398\pi$
$179$	19.8815	1.48602	0.743008	+	0.669282i	$0.233398\pi$
$180$	−2.81600	−0.209892
$181$	11.8618	0.881678	0.440839	−	0.897586i	$-0.354681\pi$
$181$	11.8618	0.881678	0.440839	+	0.897586i	$0.354681\pi$
$182$	0	0
$183$	7.63161	0.564145
$184$	4.40837	0.324989
$185$	11.2664	0.828319
$186$	6.91925	0.507344
$187$	23.3049	1.70422
$188$	12.6201	0.920418
$189$	25.7217	1.87098
$190$	−1.43823	−0.104340
$191$	13.6185	0.985397	0.492699	−	0.870200i	$-0.336011\pi$
$191$	13.6185	0.985397	0.492699	+	0.870200i	$0.336011\pi$
$192$	−1.02081	−0.0736703
$193$	7.07674	0.509395	0.254698	−	0.967021i	$-0.418024\pi$
$193$	7.07674	0.509395	0.254698	+	0.967021i	$0.418024\pi$
$194$	0.438413	0.0314763
$195$	0	0
$196$	18.8289	1.34492
$197$	−6.55682	−0.467154	−0.233577	−	0.972338i	$-0.575043\pi$
$197$	−6.55682	−0.467154	−0.233577	+	0.972338i	$0.575043\pi$
$198$	−8.12143	−0.577165
$199$	−0.996412	−0.0706338	−0.0353169	−	0.999376i	$-0.511244\pi$
$199$	−0.996412	−0.0706338	−0.0353169	+	0.999376i	$0.511244\pi$
$200$	−2.93149	−0.207287
$201$	12.3204	0.869012
$202$	−5.15113	−0.362432
$203$	−23.0485	−1.61769
$204$	−5.73536	−0.401556
$205$	3.55763	0.248476
$206$	12.6076	0.878415
$207$	−8.63139	−0.599923
$208$	0	0
$209$	−4.14791	−0.286917
$210$	−7.46149	−0.514892
$211$	−5.38932	−0.371016	−0.185508	−	0.982643i	$-0.559393\pi$
$211$	−5.38932	−0.371016	−0.185508	+	0.982643i	$0.559393\pi$
$212$	10.9335	0.750914
$213$	10.4956	0.719148
$214$	−13.9445	−0.953225
$215$	−7.59727	−0.518130
$216$	5.06111	0.344365
$217$	−34.4484	−2.33851
$218$	1.44804	0.0980736
$219$	15.1440	1.02334
$220$	5.96566	0.402205
$221$	0	0
$222$	−7.99645	−0.536687
$223$	13.0297	0.872533	0.436266	−	0.899818i	$-0.356301\pi$
$223$	13.0297	0.872533	0.436266	+	0.899818i	$0.356301\pi$
$224$	5.08222	0.339570
$225$	5.73972	0.382648
$226$	−7.95897	−0.529423
$227$	−9.04800	−0.600537	−0.300268	−	0.953855i	$-0.597076\pi$
$227$	−9.04800	−0.600537	−0.300268	+	0.953855i	$0.597076\pi$
$228$	1.02081	0.0676045
$229$	25.3521	1.67532	0.837658	−	0.546195i	$-0.183924\pi$
$229$	25.3521	1.67532	0.837658	+	0.546195i	$0.183924\pi$
$230$	6.34026	0.418065
$231$	−21.5192	−1.41586
$232$	−4.53513	−0.297746
$233$	14.3951	0.943054	0.471527	−	0.881852i	$-0.343703\pi$
$233$	14.3951	0.943054	0.471527	+	0.881852i	$0.343703\pi$
$234$	0	0
$235$	18.1507	1.18402
$236$	−7.26543	−0.472939
$237$	−14.9282	−0.969688
$238$	28.5543	1.85090
$239$	11.8226	0.764738	0.382369	−	0.924010i	$-0.375108\pi$
$239$	11.8226	0.764738	0.382369	+	0.924010i	$0.375108\pi$
$240$	−1.46816	−0.0947691
$241$	2.59870	0.167397	0.0836986	−	0.996491i	$-0.473327\pi$
$241$	2.59870	0.167397	0.0836986	+	0.996491i	$0.473327\pi$
$242$	6.20517	0.398883
$243$	−15.9055	−1.02034
$244$	−7.47606	−0.478606
$245$	27.0804	1.73010
$246$	−2.52508	−0.160993
$247$	0	0
$248$	−6.77822	−0.430418
$249$	3.34554	0.212015
$250$	−11.4073	−0.721463
$251$	−2.65611	−0.167652	−0.0838261	−	0.996480i	$-0.526714\pi$
$251$	−2.65611	−0.167652	−0.0838261	+	0.996480i	$0.526714\pi$
$252$	−9.95076	−0.626839
$253$	18.2855	1.14960
$254$	−14.8829	−0.933838
$255$	−8.24879	−0.516559
$256$	1.00000	0.0625000
$257$	−3.58554	−0.223660	−0.111830	−	0.993727i	$-0.535671\pi$
$257$	−3.58554	−0.223660	−0.111830	+	0.993727i	$0.535671\pi$
$258$	5.39227	0.335708
$259$	39.8114	2.47376
$260$	0	0
$261$	8.87958	0.549632
$262$	4.52135	0.279330
$263$	8.71634	0.537473	0.268736	−	0.963214i	$-0.413394\pi$
$263$	8.71634	0.537473	0.268736	+	0.963214i	$0.413394\pi$
$264$	−4.23421	−0.260598
$265$	15.7249	0.965972
$266$	−5.08222	−0.311611
$267$	−3.78911	−0.231890
$268$	−12.0693	−0.737247
$269$	−8.38391	−0.511176	−0.255588	−	0.966786i	$-0.582269\pi$
$269$	−8.38391	−0.511176	−0.255588	+	0.966786i	$0.582269\pi$
$270$	7.27905	0.442989
$271$	15.6311	0.949521	0.474761	−	0.880115i	$-0.342535\pi$
$271$	15.6311	0.949521	0.474761	+	0.880115i	$0.342535\pi$
$272$	5.61847	0.340669
$273$	0	0
$274$	−6.61223	−0.399460
$275$	−12.1595	−0.733248
$276$	−4.50009	−0.270873
$277$	−30.3006	−1.82058	−0.910292	−	0.413967i	$-0.864143\pi$
$277$	−30.3006	−1.82058	−0.910292	+	0.413967i	$0.864143\pi$
$278$	−11.8025	−0.707869
$279$	13.2715	0.794542
$280$	7.30941	0.436821
$281$	−2.49333	−0.148740	−0.0743698	−	0.997231i	$-0.523695\pi$
$281$	−2.49333	−0.148740	−0.0743698	+	0.997231i	$0.523695\pi$
$282$	−12.8827	−0.767154
$283$	−15.6041	−0.927566	−0.463783	−	0.885949i	$-0.653508\pi$
$283$	−15.6041	−0.927566	−0.463783	+	0.885949i	$0.653508\pi$
$284$	−10.2817	−0.610107
$285$	1.46816	0.0869661
$286$	0	0
$287$	12.5714	0.742068
$288$	−1.95796	−0.115374
$289$	14.5671	0.856891
$290$	−6.52257	−0.383018
$291$	−0.447535	−0.0262350
$292$	−14.8354	−0.868174
$293$	−5.81536	−0.339737	−0.169868	−	0.985467i	$-0.554334\pi$
$293$	−5.81536	−0.339737	−0.169868	+	0.985467i	$0.554334\pi$
$294$	−19.2207	−1.12097
$295$	−10.4494	−0.608386
$296$	7.83347	0.455311
$297$	20.9930	1.21814
$298$	21.7459	1.25971
$299$	0	0
$300$	2.99248	0.172771
$301$	−26.8461	−1.54738
$302$	−23.0798	−1.32809
$303$	5.25830	0.302081
$304$	−1.00000	−0.0573539
$305$	−10.7523	−0.615676
$306$	−11.0007	−0.628869
$307$	8.78515	0.501395	0.250697	−	0.968066i	$-0.419340\pi$
$307$	8.78515	0.501395	0.250697	+	0.968066i	$0.419340\pi$
$308$	21.0806	1.20118
$309$	−12.8699	−0.732145
$310$	−9.74867	−0.553687
$311$	−5.34058	−0.302836	−0.151418	−	0.988470i	$-0.548384\pi$
$311$	−5.34058	−0.302836	−0.151418	+	0.988470i	$0.548384\pi$
$312$	0	0
$313$	−17.5931	−0.994421	−0.497211	−	0.867630i	$-0.665642\pi$
$313$	−17.5931	−0.994421	−0.497211	+	0.867630i	$0.665642\pi$
$314$	−11.9483	−0.674282
$315$	−14.3115	−0.806362
$316$	14.6239	0.822659
$317$	1.77517	0.0997032	0.0498516	−	0.998757i	$-0.484125\pi$
$317$	1.77517	0.0997032	0.0498516	+	0.998757i	$0.484125\pi$
$318$	−11.1609	−0.625875
$319$	−18.8113	−1.05323
$320$	1.43823	0.0803997
$321$	14.2346	0.794498
$322$	22.4043	1.24854
$323$	−5.61847	−0.312620
$324$	0.707461	0.0393034
$325$	0	0
$326$	−3.11405	−0.172471
$327$	−1.47817	−0.0817428
$328$	2.47361	0.136582
$329$	64.1383	3.53606
$330$	−6.08978	−0.335232
$331$	0.416611	0.0228990	0.0114495	−	0.999934i	$-0.496355\pi$
$331$	0.416611	0.0228990	0.0114495	+	0.999934i	$0.496355\pi$
$332$	−3.27736	−0.179868
$333$	−15.3376	−0.840495
$334$	−5.96403	−0.326337
$335$	−17.3584	−0.948391
$336$	−5.18796	−0.283026
$337$	−0.675427	−0.0367928	−0.0183964	−	0.999831i	$-0.505856\pi$
$337$	−0.675427	−0.0367928	−0.0183964	+	0.999831i	$0.505856\pi$
$338$	0	0
$339$	8.12456	0.441265
$340$	8.08066	0.438235
$341$	−28.1155	−1.52254
$342$	1.95796	0.105874
$343$	60.1172	3.24603
$344$	−5.28236	−0.284806
$345$	−6.47218	−0.348450
$346$	2.61812	0.140751
$347$	−6.04837	−0.324693	−0.162347	−	0.986734i	$-0.551906\pi$
$347$	−6.04837	−0.324693	−0.162347	+	0.986734i	$0.551906\pi$
$348$	4.62948	0.248166
$349$	11.0953	0.593916	0.296958	−	0.954890i	$-0.404028\pi$
$349$	11.0953	0.593916	0.296958	+	0.954890i	$0.404028\pi$
$350$	−14.8984	−0.796355
$351$	0	0
$352$	4.14791	0.221084
$353$	25.0786	1.33480	0.667399	−	0.744700i	$-0.267408\pi$
$353$	25.0786	1.33480	0.667399	+	0.744700i	$0.267408\pi$
$354$	7.41659	0.394187
$355$	−14.7875	−0.784839
$356$	3.71188	0.196729
$357$	−29.1483	−1.54269
$358$	19.8815	1.05077
$359$	18.2808	0.964826	0.482413	−	0.875944i	$-0.339761\pi$
$359$	18.2808	0.964826	0.482413	+	0.875944i	$0.339761\pi$
$360$	−2.81600	−0.148416
$361$	1.00000	0.0526316
$362$	11.8618	0.623440
$363$	−6.33427	−0.332463
$364$	0	0
$365$	−21.3367	−1.11681
$366$	7.63161	0.398911
$367$	−3.13127	−0.163451	−0.0817254	−	0.996655i	$-0.526043\pi$
$367$	−3.13127	−0.163451	−0.0817254	+	0.996655i	$0.526043\pi$
$368$	4.40837	0.229802
$369$	−4.84322	−0.252128
$370$	11.2664	0.585710
$371$	55.5663	2.88486
$372$	6.91925	0.358746
$373$	19.7254	1.02134	0.510672	−	0.859776i	$-0.329397\pi$
$373$	19.7254	1.02134	0.510672	+	0.859776i	$0.329397\pi$
$374$	23.3049	1.20507
$375$	11.6447	0.601328
$376$	12.6201	0.650834
$377$	0	0
$378$	25.7217	1.32298
$379$	−24.1551	−1.24076	−0.620381	−	0.784301i	$-0.713022\pi$
$379$	−24.1551	−1.24076	−0.620381	+	0.784301i	$0.713022\pi$
$380$	−1.43823	−0.0737798
$381$	15.1926	0.778340
$382$	13.6185	0.696781
$383$	−33.8361	−1.72894	−0.864472	−	0.502681i	$-0.832347\pi$
$383$	−33.8361	−1.72894	−0.864472	+	0.502681i	$0.832347\pi$
$384$	−1.02081	−0.0520928
$385$	30.3188	1.54519
$386$	7.07674	0.360197
$387$	10.3426	0.525746
$388$	0.438413	0.0222571
$389$	−6.35061	−0.321989	−0.160994	−	0.986955i	$-0.551470\pi$
$389$	−6.35061	−0.321989	−0.160994	+	0.986955i	$0.551470\pi$
$390$	0	0
$391$	24.7683	1.25259
$392$	18.8289	0.951005
$393$	−4.61542	−0.232817
$394$	−6.55682	−0.330328
$395$	21.0326	1.05826
$396$	−8.12143	−0.408117
$397$	7.66708	0.384800	0.192400	−	0.981317i	$-0.438373\pi$
$397$	7.66708	0.384800	0.192400	+	0.981317i	$0.438373\pi$
$398$	−0.996412	−0.0499456
$399$	5.18796	0.259723
$400$	−2.93149	−0.146574
$401$	−33.0707	−1.65147	−0.825736	−	0.564057i	$-0.809240\pi$
$401$	−33.0707	−1.65147	−0.825736	+	0.564057i	$0.809240\pi$
$402$	12.3204	0.614484
$403$	0	0
$404$	−5.15113	−0.256278
$405$	1.01749	0.0505597
$406$	−23.0485	−1.14388
$407$	32.4925	1.61060
$408$	−5.73536	−0.283943
$409$	−3.88917	−0.192307	−0.0961536	−	0.995367i	$-0.530654\pi$
$409$	−3.88917	−0.192307	−0.0961536	+	0.995367i	$0.530654\pi$
$410$	3.55763	0.175699
$411$	6.74980	0.332943
$412$	12.6076	0.621134
$413$	−36.9245	−1.81694
$414$	−8.63139	−0.424210
$415$	−4.71360	−0.231382
$416$	0	0
$417$	12.0481	0.589998
$418$	−4.14791	−0.202881
$419$	23.7872	1.16208	0.581039	−	0.813876i	$-0.302646\pi$
$419$	23.7872	1.16208	0.581039	+	0.813876i	$0.302646\pi$
$420$	−7.46149	−0.364083
$421$	−7.73035	−0.376754	−0.188377	−	0.982097i	$-0.560323\pi$
$421$	−7.73035	−0.376754	−0.188377	+	0.982097i	$0.560323\pi$
$422$	−5.38932	−0.262348
$423$	−24.7097	−1.20143
$424$	10.9335	0.530976
$425$	−16.4704	−0.798934
$426$	10.4956	0.508515
$427$	−37.9950	−1.83871
$428$	−13.9445	−0.674032
$429$	0	0
$430$	−7.59727	−0.366373
$431$	29.6981	1.43051	0.715253	−	0.698865i	$-0.246311\pi$
$431$	29.6981	1.43051	0.715253	+	0.698865i	$0.246311\pi$
$432$	5.06111	0.243503
$433$	−17.2122	−0.827166	−0.413583	−	0.910466i	$-0.635723\pi$
$433$	−17.2122	−0.827166	−0.413583	+	0.910466i	$0.635723\pi$
$434$	−34.4484	−1.65358
$435$	6.65827	0.319240
$436$	1.44804	0.0693485
$437$	−4.40837	−0.210881
$438$	15.1440	0.723609
$439$	23.7553	1.13378	0.566890	−	0.823793i	$-0.308146\pi$
$439$	23.7553	1.13378	0.566890	+	0.823793i	$0.308146\pi$
$440$	5.96566	0.284402
$441$	−36.8662	−1.75553
$442$	0	0
$443$	1.59318	0.0756942	0.0378471	−	0.999284i	$-0.487950\pi$
$443$	1.59318	0.0756942	0.0378471	+	0.999284i	$0.487950\pi$
$444$	−7.99645	−0.379495
$445$	5.33855	0.253072
$446$	13.0297	0.616974
$447$	−22.1983	−1.04994
$448$	5.08222	0.240112
$449$	−9.76811	−0.460985	−0.230493	−	0.973074i	$-0.574034\pi$
$449$	−9.76811	−0.460985	−0.230493	+	0.973074i	$0.574034\pi$
$450$	5.73972	0.270573
$451$	10.2603	0.483140
$452$	−7.95897	−0.374358
$453$	23.5600	1.10694
$454$	−9.04800	−0.424643
$455$	0	0
$456$	1.02081	0.0478036
$457$	31.4307	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$457$	31.4307	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$458$	25.3521	1.18463
$459$	28.4357	1.32726
$460$	6.34026	0.295616
$461$	11.3518	0.528705	0.264352	−	0.964426i	$-0.414842\pi$
$461$	11.3518	0.528705	0.264352	+	0.964426i	$0.414842\pi$
$462$	−21.5192	−1.00116
$463$	38.5658	1.79230	0.896152	−	0.443747i	$-0.146351\pi$
$463$	38.5658	1.79230	0.896152	+	0.443747i	$0.146351\pi$
$464$	−4.53513	−0.210538
$465$	9.95149	0.461489
$466$	14.3951	0.666840
$467$	5.08392	0.235256	0.117628	−	0.993058i	$-0.462471\pi$
$467$	5.08392	0.235256	0.117628	+	0.993058i	$0.462471\pi$
$468$	0	0
$469$	−61.3386	−2.83235
$470$	18.1507	0.837230
$471$	12.1969	0.562003
$472$	−7.26543	−0.334418
$473$	−21.9108	−1.00746
$474$	−14.9282	−0.685673
$475$	2.93149	0.134506
$476$	28.5543	1.30878
$477$	−21.4073	−0.980171
$478$	11.8226	0.540751
$479$	−24.7236	−1.12965	−0.564825	−	0.825211i	$-0.691056\pi$
$479$	−24.7236	−1.12965	−0.564825	+	0.825211i	$0.691056\pi$
$480$	−1.46816	−0.0670119
$481$	0	0
$482$	2.59870	0.118368
$483$	−22.8704	−1.04064
$484$	6.20517	0.282053
$485$	0.630541	0.0286314
$486$	−15.9055	−0.721488
$487$	−14.3193	−0.648870	−0.324435	−	0.945908i	$-0.605174\pi$
$487$	−14.3193	−0.648870	−0.324435	+	0.945908i	$0.605174\pi$
$488$	−7.47606	−0.338426
$489$	3.17883	0.143752
$490$	27.0804	1.22337
$491$	30.1007	1.35843	0.679213	−	0.733941i	$-0.262321\pi$
$491$	30.1007	1.35843	0.679213	+	0.733941i	$0.262321\pi$
$492$	−2.52508	−0.113839
$493$	−25.4804	−1.14758
$494$	0	0
$495$	−11.6805	−0.525000
$496$	−6.77822	−0.304351
$497$	−52.2539	−2.34391
$498$	3.34554	0.149917
$499$	−22.7422	−1.01808	−0.509041	−	0.860742i	$-0.670000\pi$
$499$	−22.7422	−1.01808	−0.509041	+	0.860742i	$0.670000\pi$
$500$	−11.4073	−0.510151
$501$	6.08811	0.271997
$502$	−2.65611	−0.118548
$503$	−2.80773	−0.125191	−0.0625953	−	0.998039i	$-0.519938\pi$
$503$	−2.80773	−0.125191	−0.0625953	+	0.998039i	$0.519938\pi$
$504$	−9.95076	−0.443242
$505$	−7.40852	−0.329675
$506$	18.2855	0.812891
$507$	0	0
$508$	−14.8829	−0.660324
$509$	−0.787853	−0.0349210	−0.0174605	−	0.999848i	$-0.505558\pi$
$509$	−0.787853	−0.0349210	−0.0174605	+	0.999848i	$0.505558\pi$
$510$	−8.24879	−0.365262
$511$	−75.3966	−3.33535
$512$	1.00000	0.0441942
$513$	−5.06111	−0.223453
$514$	−3.58554	−0.158152
$515$	18.1327	0.799023
$516$	5.39227	0.237381
$517$	52.3472	2.30223
$518$	39.8114	1.74921
$519$	−2.67259	−0.117314
$520$	0	0
$521$	−15.7328	−0.689265	−0.344633	−	0.938738i	$-0.611997\pi$
$521$	−15.7328	−0.689265	−0.344633	+	0.938738i	$0.611997\pi$
$522$	8.87958	0.388648
$523$	−14.1402	−0.618307	−0.309154	−	0.951012i	$-0.600046\pi$
$523$	−14.1402	−0.618307	−0.309154	+	0.951012i	$0.600046\pi$
$524$	4.52135	0.197516
$525$	15.2084	0.663750
$526$	8.71634	0.380051
$527$	−38.0832	−1.65893
$528$	−4.23421	−0.184270
$529$	−3.56628	−0.155056
$530$	15.7249	0.683045
$531$	14.2254	0.617329
$532$	−5.08222	−0.220342
$533$	0	0
$534$	−3.78911	−0.163971
$535$	−20.0554	−0.867071
$536$	−12.0693	−0.521313
$537$	−20.2952	−0.875802
$538$	−8.38391	−0.361456
$539$	78.1008	3.36404
$540$	7.27905	0.313241
$541$	−5.20666	−0.223852	−0.111926	−	0.993717i	$-0.535702\pi$
$541$	−5.20666	−0.223852	−0.111926	+	0.993717i	$0.535702\pi$
$542$	15.6311	0.671413
$543$	−12.1086	−0.519628
$544$	5.61847	0.240890
$545$	2.08262	0.0892096
$546$	0	0
$547$	21.0461	0.899866	0.449933	−	0.893062i	$-0.351448\pi$
$547$	21.0461	0.899866	0.449933	+	0.893062i	$0.351448\pi$
$548$	−6.61223	−0.282461
$549$	14.6378	0.624726
$550$	−12.1595	−0.518485
$551$	4.53513	0.193203
$552$	−4.50009	−0.191536
$553$	74.3218	3.16049
$554$	−30.3006	−1.28735
$555$	−11.5008	−0.488180
$556$	−11.8025	−0.500539
$557$	2.68696	0.113850	0.0569251	−	0.998378i	$-0.481870\pi$
$557$	2.68696	0.113850	0.0569251	+	0.998378i	$0.481870\pi$
$558$	13.2715	0.561826
$559$	0	0
$560$	7.30941	0.308879
$561$	−23.7898	−1.00440
$562$	−2.49333	−0.105175
$563$	46.3519	1.95350	0.976749	−	0.214386i	$-0.0687751\pi$
$563$	46.3519	1.95350	0.976749	+	0.214386i	$0.0687751\pi$
$564$	−12.8827	−0.542460
$565$	−11.4469	−0.481573
$566$	−15.6041	−0.655888
$567$	3.59547	0.150996
$568$	−10.2817	−0.431411
$569$	−19.8369	−0.831605	−0.415803	−	0.909455i	$-0.636499\pi$
$569$	−19.8369	−0.831605	−0.415803	+	0.909455i	$0.636499\pi$
$570$	1.46816	0.0614943
$571$	10.0534	0.420720	0.210360	−	0.977624i	$-0.432536\pi$
$571$	10.0534	0.420720	0.210360	+	0.977624i	$0.432536\pi$
$572$	0	0
$573$	−13.9018	−0.580756
$574$	12.5714	0.524721
$575$	−12.9231	−0.538929
$576$	−1.95796	−0.0815815
$577$	3.65960	0.152351	0.0761755	−	0.997094i	$-0.475729\pi$
$577$	3.65960	0.152351	0.0761755	+	0.997094i	$0.475729\pi$
$578$	14.5671	0.605914
$579$	−7.22398	−0.300218
$580$	−6.52257	−0.270835
$581$	−16.6562	−0.691017
$582$	−0.447535	−0.0185509
$583$	45.3511	1.87825
$584$	−14.8354	−0.613892
$585$	0	0
$586$	−5.81536	−0.240230
$587$	20.6601	0.852734	0.426367	−	0.904550i	$-0.359793\pi$
$587$	20.6601	0.852734	0.426367	+	0.904550i	$0.359793\pi$
$588$	−19.2207	−0.792648
$589$	6.77822	0.279292
$590$	−10.4494	−0.430194
$591$	6.69324	0.275323
$592$	7.83347	0.321954
$593$	−24.5621	−1.00864	−0.504322	−	0.863516i	$-0.668257\pi$
$593$	−24.5621	−1.00864	−0.504322	+	0.863516i	$0.668257\pi$
$594$	20.9930	0.861355
$595$	41.0677	1.68361
$596$	21.7459	0.890746
$597$	1.01714	0.0416289
$598$	0	0
$599$	27.1719	1.11021	0.555107	−	0.831779i	$-0.312677\pi$
$599$	27.1719	1.11021	0.555107	+	0.831779i	$0.312677\pi$
$600$	2.99248	0.122167
$601$	37.5882	1.53326	0.766628	−	0.642092i	$-0.221933\pi$
$601$	37.5882	1.53326	0.766628	+	0.642092i	$0.221933\pi$
$602$	−26.8461	−1.09417
$603$	23.6311	0.962332
$604$	−23.0798	−0.939103
$605$	8.92448	0.362832
$606$	5.25830	0.213604
$607$	−16.8337	−0.683257	−0.341629	−	0.939835i	$-0.610978\pi$
$607$	−16.8337	−0.683257	−0.341629	+	0.939835i	$0.610978\pi$
$608$	−1.00000	−0.0405554
$609$	23.5280	0.953404
$610$	−10.7523	−0.435349
$611$	0	0
$612$	−11.0007	−0.444677
$613$	41.5389	1.67774	0.838870	−	0.544332i	$-0.183217\pi$
$613$	41.5389	1.67774	0.838870	+	0.544332i	$0.183217\pi$
$614$	8.78515	0.354540
$615$	−3.63165	−0.146442
$616$	21.0806	0.849361
$617$	−6.40408	−0.257819	−0.128909	−	0.991656i	$-0.541148\pi$
$617$	−6.40408	−0.257819	−0.128909	+	0.991656i	$0.541148\pi$
$618$	−12.8699	−0.517705
$619$	17.6383	0.708942	0.354471	−	0.935067i	$-0.384661\pi$
$619$	17.6383	0.708942	0.354471	+	0.935067i	$0.384661\pi$
$620$	−9.74867	−0.391516
$621$	22.3112	0.895319
$622$	−5.34058	−0.214138
$623$	18.8646	0.755794
$624$	0	0
$625$	−1.74897	−0.0699588
$626$	−17.5931	−0.703162
$627$	4.23421	0.169098
$628$	−11.9483	−0.476789
$629$	44.0121	1.75488
$630$	−14.3115	−0.570184
$631$	−12.2173	−0.486364	−0.243182	−	0.969981i	$-0.578191\pi$
$631$	−12.2173	−0.486364	−0.243182	+	0.969981i	$0.578191\pi$
$632$	14.6239	0.581708
$633$	5.50145	0.218663
$634$	1.77517	0.0705008
$635$	−21.4051	−0.849437
$636$	−11.1609	−0.442560
$637$	0	0
$638$	−18.8113	−0.744746
$639$	20.1311	0.796375
$640$	1.43823	0.0568512
$641$	10.0572	0.397236	0.198618	−	0.980077i	$-0.436355\pi$
$641$	10.0572	0.397236	0.198618	+	0.980077i	$0.436355\pi$
$642$	14.2346	0.561795
$643$	−5.41810	−0.213669	−0.106835	−	0.994277i	$-0.534072\pi$
$643$	−5.41810	−0.213669	−0.106835	+	0.994277i	$0.534072\pi$
$644$	22.4043	0.882853
$645$	7.75534	0.305366
$646$	−5.61847	−0.221056
$647$	−22.8223	−0.897238	−0.448619	−	0.893723i	$-0.648084\pi$
$647$	−22.8223	−0.897238	−0.448619	+	0.893723i	$0.648084\pi$
$648$	0.707461	0.0277917
$649$	−30.1364	−1.18296
$650$	0	0
$651$	35.1651	1.37823
$652$	−3.11405	−0.121955
$653$	22.3211	0.873494	0.436747	−	0.899584i	$-0.356130\pi$
$653$	22.3211	0.873494	0.436747	+	0.899584i	$0.356130\pi$
$654$	−1.47817	−0.0578009
$655$	6.50276	0.254084
$656$	2.47361	0.0965783
$657$	29.0470	1.13323
$658$	64.1383	2.50037
$659$	−34.7370	−1.35316	−0.676580	−	0.736369i	$-0.736539\pi$
$659$	−34.7370	−1.35316	−0.676580	+	0.736369i	$0.736539\pi$
$660$	−6.08978	−0.237044
$661$	27.9056	1.08540	0.542702	−	0.839925i	$-0.317401\pi$
$661$	27.9056	1.08540	0.542702	+	0.839925i	$0.317401\pi$
$662$	0.416611	0.0161920
$663$	0	0
$664$	−3.27736	−0.127186
$665$	−7.30941	−0.283447
$666$	−15.3376	−0.594320
$667$	−19.9925	−0.774113
$668$	−5.96403	−0.230755
$669$	−13.3008	−0.514238
$670$	−17.3584	−0.670614
$671$	−31.0101	−1.19713
$672$	−5.18796	−0.200130
$673$	−31.4086	−1.21071	−0.605356	−	0.795955i	$-0.706969\pi$
$673$	−31.4086	−1.21071	−0.605356	+	0.795955i	$0.706969\pi$
$674$	−0.675427	−0.0260165
$675$	−14.8366	−0.571060
$676$	0	0
$677$	−24.2838	−0.933300	−0.466650	−	0.884442i	$-0.654539\pi$
$677$	−24.2838	−0.933300	−0.466650	+	0.884442i	$0.654539\pi$
$678$	8.12456	0.312022
$679$	2.22811	0.0855071
$680$	8.08066	0.309879
$681$	9.23624	0.353934
$682$	−28.1155	−1.07660
$683$	−32.3164	−1.23655	−0.618276	−	0.785961i	$-0.712169\pi$
$683$	−32.3164	−1.23655	−0.618276	+	0.785961i	$0.712169\pi$
$684$	1.95796	0.0748643
$685$	−9.50993	−0.363356
$686$	60.1172	2.29529
$687$	−25.8796	−0.987368
$688$	−5.28236	−0.201388
$689$	0	0
$690$	−6.47218	−0.246392
$691$	−22.6282	−0.860816	−0.430408	−	0.902634i	$-0.641630\pi$
$691$	−22.6282	−0.860816	−0.430408	+	0.902634i	$0.641630\pi$
$692$	2.61812	0.0995260
$693$	−41.2749	−1.56790
$694$	−6.04837	−0.229593
$695$	−16.9748	−0.643891
$696$	4.62948	0.175480
$697$	13.8979	0.526420
$698$	11.0953	0.419962
$699$	−14.6946	−0.555800
$700$	−14.8984	−0.563108
$701$	−0.169862	−0.00641559	−0.00320780	−	0.999995i	$-0.501021\pi$
$701$	−0.169862	−0.00641559	−0.00320780	+	0.999995i	$0.501021\pi$
$702$	0	0
$703$	−7.83347	−0.295445
$704$	4.14791	0.156330
$705$	−18.5283	−0.697818
$706$	25.0786	0.943845
$707$	−26.1792	−0.984568
$708$	7.41659	0.278732
$709$	−45.2806	−1.70055	−0.850274	−	0.526341i	$-0.823564\pi$
$709$	−45.2806	−1.70055	−0.850274	+	0.526341i	$0.823564\pi$
$710$	−14.7875	−0.554965
$711$	−28.6330	−1.07382
$712$	3.71188	0.139109
$713$	−29.8809	−1.11905
$714$	−29.1483	−1.09085
$715$	0	0
$716$	19.8815	0.743008
$717$	−12.0685	−0.450708
$718$	18.2808	0.682235
$719$	33.0108	1.23109	0.615547	−	0.788100i	$-0.288935\pi$
$719$	33.0108	1.23109	0.615547	+	0.788100i	$0.288935\pi$
$720$	−2.81600	−0.104946
$721$	64.0747	2.38627
$722$	1.00000	0.0372161
$723$	−2.65277	−0.0986576
$724$	11.8618	0.440839
$725$	13.2947	0.493751
$726$	−6.33427	−0.235087
$727$	46.9722	1.74210	0.871052	−	0.491192i	$-0.163439\pi$
$727$	46.9722	1.74210	0.871052	+	0.491192i	$0.163439\pi$
$728$	0	0
$729$	14.1140	0.522742
$730$	−21.3367	−0.789707
$731$	−29.6788	−1.09771
$732$	7.63161	0.282072
$733$	−19.1888	−0.708753	−0.354377	−	0.935103i	$-0.615307\pi$
$733$	−19.1888	−0.708753	−0.354377	+	0.935103i	$0.615307\pi$
$734$	−3.13127	−0.115577
$735$	−27.6438	−1.01966
$736$	4.40837	0.162495
$737$	−50.0622	−1.84407
$738$	−4.84322	−0.178281
$739$	19.3084	0.710272	0.355136	−	0.934815i	$-0.384435\pi$
$739$	19.3084	0.710272	0.355136	+	0.934815i	$0.384435\pi$
$740$	11.2664	0.414160
$741$	0	0
$742$	55.5663	2.03990
$743$	−10.8424	−0.397769	−0.198885	−	0.980023i	$-0.563732\pi$
$743$	−10.8424	−0.397769	−0.198885	+	0.980023i	$0.563732\pi$
$744$	6.91925	0.253672
$745$	31.2757	1.14585
$746$	19.7254	0.722199
$747$	6.41692	0.234783
$748$	23.3049	0.852111
$749$	−70.8689	−2.58949
$750$	11.6447	0.425203
$751$	29.6995	1.08375	0.541874	−	0.840460i	$-0.317715\pi$
$751$	29.6995	1.08375	0.541874	+	0.840460i	$0.317715\pi$
$752$	12.6201	0.460209
$753$	2.71137	0.0988079
$754$	0	0
$755$	−33.1941	−1.20806
$756$	25.7217	0.935488
$757$	18.1428	0.659410	0.329705	−	0.944084i	$-0.393051\pi$
$757$	18.1428	0.659410	0.329705	+	0.944084i	$0.393051\pi$
$758$	−24.1551	−0.877351
$759$	−18.6660	−0.677532
$760$	−1.43823	−0.0521702
$761$	−29.5418	−1.07089	−0.535444	−	0.844570i	$-0.679856\pi$
$761$	−29.5418	−1.07089	−0.535444	+	0.844570i	$0.679856\pi$
$762$	15.1926	0.550369
$763$	7.35925	0.266423
$764$	13.6185	0.492699
$765$	−15.8216	−0.572031
$766$	−33.8361	−1.22255
$767$	0	0
$768$	−1.02081	−0.0368351
$769$	−45.2834	−1.63296	−0.816481	−	0.577373i	$-0.804078\pi$
$769$	−45.2834	−1.63296	−0.816481	+	0.577373i	$0.804078\pi$
$770$	30.3188	1.09261
$771$	3.66014	0.131817
$772$	7.07674	0.254698
$773$	−14.1614	−0.509351	−0.254676	−	0.967027i	$-0.581969\pi$
$773$	−14.1614	−0.509351	−0.254676	+	0.967027i	$0.581969\pi$
$774$	10.3426	0.371758
$775$	19.8703	0.713761
$776$	0.438413	0.0157381
$777$	−40.6397	−1.45794
$778$	−6.35061	−0.227680
$779$	−2.47361	−0.0886263
$780$	0	0
$781$	−42.6476	−1.52605
$782$	24.7683	0.885711
$783$	−22.9528	−0.820265
$784$	18.8289	0.672462
$785$	−17.1844	−0.613339
$786$	−4.61542	−0.164627
$787$	23.8690	0.850836	0.425418	−	0.904997i	$-0.360127\pi$
$787$	23.8690	0.850836	0.425418	+	0.904997i	$0.360127\pi$
$788$	−6.55682	−0.233577
$789$	−8.89769	−0.316766
$790$	21.0326	0.748306
$791$	−40.4492	−1.43821
$792$	−8.12143	−0.288582
$793$	0	0
$794$	7.66708	0.272095
$795$	−16.0520	−0.569307
$796$	−0.996412	−0.0353169
$797$	−22.2088	−0.786674	−0.393337	−	0.919394i	$-0.628679\pi$
$797$	−22.2088	−0.786674	−0.393337	+	0.919394i	$0.628679\pi$
$798$	5.18796	0.183652
$799$	70.9058	2.50847
$800$	−2.93149	−0.103644
$801$	−7.26770	−0.256792
$802$	−33.0707	−1.16777
$803$	−61.5358	−2.17155
$804$	12.3204	0.434506
$805$	32.2226	1.13570
$806$	0	0
$807$	8.55834	0.301268
$808$	−5.15113	−0.181216
$809$	41.4729	1.45811	0.729055	−	0.684455i	$-0.239960\pi$
$809$	41.4729	1.45811	0.729055	+	0.684455i	$0.239960\pi$
$810$	1.01749	0.0357511
$811$	26.5633	0.932764	0.466382	−	0.884583i	$-0.345557\pi$
$811$	26.5633	0.932764	0.466382	+	0.884583i	$0.345557\pi$
$812$	−23.0485	−0.808844
$813$	−15.9563	−0.559612
$814$	32.4925	1.13886
$815$	−4.47872	−0.156883
$816$	−5.73536	−0.200778
$817$	5.28236	0.184807
$818$	−3.88917	−0.135982
$819$	0	0
$820$	3.55763	0.124238
$821$	−36.5090	−1.27417	−0.637086	−	0.770793i	$-0.719860\pi$
$821$	−36.5090	−1.27417	−0.637086	+	0.770793i	$0.719860\pi$
$822$	6.74980	0.235426
$823$	−32.2001	−1.12242	−0.561212	−	0.827672i	$-0.689665\pi$
$823$	−32.2001	−1.12242	−0.561212	+	0.827672i	$0.689665\pi$
$824$	12.6076	0.439208
$825$	12.4125	0.432149
$826$	−36.9245	−1.28477
$827$	−29.1947	−1.01520	−0.507600	−	0.861593i	$-0.669467\pi$
$827$	−29.1947	−1.01520	−0.507600	+	0.861593i	$0.669467\pi$
$828$	−8.63139	−0.299962
$829$	−36.7277	−1.27561	−0.637803	−	0.770199i	$-0.720157\pi$
$829$	−36.7277	−1.27561	−0.637803	+	0.770199i	$0.720157\pi$
$830$	−4.71360	−0.163612
$831$	30.9310	1.07298
$832$	0	0
$833$	105.790	3.66540
$834$	12.0481	0.417191
$835$	−8.57766	−0.296842
$836$	−4.14791	−0.143459
$837$	−34.3053	−1.18577
$838$	23.7872	0.821713
$839$	−27.7791	−0.959041	−0.479521	−	0.877531i	$-0.659189\pi$
$839$	−27.7791	−0.959041	−0.479521	+	0.877531i	$0.659189\pi$
$840$	−7.46149	−0.257446
$841$	−8.43264	−0.290781
$842$	−7.73035	−0.266406
$843$	2.54520	0.0876615
$844$	−5.38932	−0.185508
$845$	0	0
$846$	−24.7097	−0.849536
$847$	31.5360	1.08359
$848$	10.9335	0.375457
$849$	15.9287	0.546673
$850$	−16.4704	−0.564932
$851$	34.5328	1.18377
$852$	10.4956	0.359574
$853$	−5.16988	−0.177013	−0.0885066	−	0.996076i	$-0.528209\pi$
$853$	−5.16988	−0.177013	−0.0885066	+	0.996076i	$0.528209\pi$
$854$	−37.9950	−1.30016
$855$	2.81600	0.0963051
$856$	−13.9445	−0.476612
$857$	−49.4942	−1.69069	−0.845345	−	0.534222i	$-0.820605\pi$
$857$	−49.4942	−1.69069	−0.845345	+	0.534222i	$0.820605\pi$
$858$	0	0
$859$	−33.7978	−1.15316	−0.576582	−	0.817039i	$-0.695614\pi$
$859$	−33.7978	−1.15316	−0.576582	+	0.817039i	$0.695614\pi$
$860$	−7.59727	−0.259065
$861$	−12.8330	−0.437347
$862$	29.6981	1.01152
$863$	18.3081	0.623216	0.311608	−	0.950211i	$-0.399132\pi$
$863$	18.3081	0.623216	0.311608	+	0.950211i	$0.399132\pi$
$864$	5.06111	0.172182
$865$	3.76547	0.128030
$866$	−17.2122	−0.584894
$867$	−14.8702	−0.505019
$868$	−34.4484	−1.16926
$869$	60.6586	2.05770
$870$	6.65827	0.225737
$871$	0	0
$872$	1.44804	0.0490368
$873$	−0.858394	−0.0290522
$874$	−4.40837	−0.149115
$875$	−57.9745	−1.95990
$876$	15.1440	0.511669
$877$	29.8355	1.00747	0.503736	−	0.863858i	$-0.331958\pi$
$877$	29.8355	1.00747	0.503736	+	0.863858i	$0.331958\pi$
$878$	23.7553	0.801704
$879$	5.93635	0.200228
$880$	5.96566	0.201103
$881$	22.0300	0.742211	0.371106	−	0.928591i	$-0.378979\pi$
$881$	22.0300	0.742211	0.371106	+	0.928591i	$0.378979\pi$
$882$	−36.8662	−1.24135
$883$	−14.3251	−0.482076	−0.241038	−	0.970516i	$-0.577488\pi$
$883$	−14.3251	−0.482076	−0.241038	+	0.970516i	$0.577488\pi$
$884$	0	0
$885$	10.6668	0.358560
$886$	1.59318	0.0535239
$887$	1.92774	0.0647273	0.0323636	−	0.999476i	$-0.489697\pi$
$887$	1.92774	0.0647273	0.0323636	+	0.999476i	$0.489697\pi$
$888$	−7.99645	−0.268343
$889$	−75.6383	−2.53683
$890$	5.33855	0.178949
$891$	2.93449	0.0983090
$892$	13.0297	0.436266
$893$	−12.6201	−0.422317
$894$	−22.1983	−0.742423
$895$	28.5943	0.955802
$896$	5.08222	0.169785
$897$	0	0
$898$	−9.76811	−0.325966
$899$	30.7401	1.02524
$900$	5.73972	0.191324
$901$	61.4293	2.04651
$902$	10.2603	0.341631
$903$	27.4047	0.911970
$904$	−7.95897	−0.264711
$905$	17.0600	0.567093
$906$	23.5600	0.782727
$907$	−23.4758	−0.779501	−0.389751	−	0.920920i	$-0.627439\pi$
$907$	−23.4758	−0.779501	−0.389751	+	0.920920i	$0.627439\pi$
$908$	−9.04800	−0.300268
$909$	10.0857	0.334521
$910$	0	0
$911$	−39.4312	−1.30641	−0.653207	−	0.757180i	$-0.726577\pi$
$911$	−39.4312	−1.30641	−0.653207	+	0.757180i	$0.726577\pi$
$912$	1.02081	0.0338022
$913$	−13.5942	−0.449902
$914$	31.4307	1.03963
$915$	10.9760	0.362856
$916$	25.3521	0.837658
$917$	22.9785	0.758816
$918$	28.4357	0.938517
$919$	34.0313	1.12259	0.561295	−	0.827616i	$-0.310303\pi$
$919$	34.0313	1.12259	0.561295	+	0.827616i	$0.310303\pi$
$920$	6.34026	0.209032
$921$	−8.96792	−0.295503
$922$	11.3518	0.373851
$923$	0	0
$924$	−21.5192	−0.707929
$925$	−22.9637	−0.755042
$926$	38.5658	1.26735
$927$	−24.6852	−0.810768
$928$	−4.53513	−0.148873
$929$	15.1615	0.497432	0.248716	−	0.968577i	$-0.419992\pi$
$929$	15.1615	0.497432	0.248716	+	0.968577i	$0.419992\pi$
$930$	9.95149	0.326322
$931$	−18.8289	−0.617093
$932$	14.3951	0.471527
$933$	5.45169	0.178480
$934$	5.08392	0.166351
$935$	33.5179	1.09615
$936$	0	0
$937$	−2.83152	−0.0925018	−0.0462509	−	0.998930i	$-0.514727\pi$
$937$	−2.83152	−0.0925018	−0.0462509	+	0.998930i	$0.514727\pi$
$938$	−61.3386	−2.00278
$939$	17.9591	0.586074
$940$	18.1507	0.592011
$941$	−41.2047	−1.34324	−0.671618	−	0.740898i	$-0.734400\pi$
$941$	−41.2047	−1.34324	−0.671618	+	0.740898i	$0.734400\pi$
$942$	12.1969	0.397396
$943$	10.9046	0.355102
$944$	−7.26543	−0.236470
$945$	36.9937	1.20341
$946$	−21.9108	−0.712381
$947$	−52.2902	−1.69920	−0.849601	−	0.527426i	$-0.823157\pi$
$947$	−52.2902	−1.69920	−0.849601	+	0.527426i	$0.823157\pi$
$948$	−14.9282	−0.484844
$949$	0	0
$950$	2.93149	0.0951099
$951$	−1.81210	−0.0587613
$952$	28.5543	0.925449
$953$	−22.7391	−0.736593	−0.368296	−	0.929708i	$-0.620059\pi$
$953$	−22.7391	−0.736593	−0.368296	+	0.929708i	$0.620059\pi$
$954$	−21.4073	−0.693085
$955$	19.5865	0.633805
$956$	11.8226	0.382369
$957$	19.2027	0.620734
$958$	−24.7236	−0.798783
$959$	−33.6048	−1.08516
$960$	−1.46816	−0.0473845
$961$	14.9443	0.482075
$962$	0	0
$963$	27.3027	0.879816
$964$	2.59870	0.0836986
$965$	10.1780	0.327642
$966$	−22.8704	−0.735844
$967$	24.2276	0.779108	0.389554	−	0.921004i	$-0.372629\pi$
$967$	24.2276	0.779108	0.389554	+	0.921004i	$0.372629\pi$
$968$	6.20517	0.199442
$969$	5.73536	0.184246
$970$	0.630541	0.0202454
$971$	−0.153852	−0.00493734	−0.00246867	−	0.999997i	$-0.500786\pi$
$971$	−0.153852	−0.00493734	−0.00246867	+	0.999997i	$0.500786\pi$
$972$	−15.9055	−0.510169
$973$	−59.9830	−1.92297
$974$	−14.3193	−0.458820
$975$	0	0
$976$	−7.47606	−0.239303
$977$	−33.9459	−1.08603	−0.543013	−	0.839724i	$-0.682716\pi$
$977$	−33.9459	−1.08603	−0.543013	+	0.839724i	$0.682716\pi$
$978$	3.17883	0.101648
$979$	15.3966	0.492076
$980$	27.0804	0.865052
$981$	−2.83520	−0.0905209
$982$	30.1007	0.960552
$983$	5.14137	0.163984	0.0819921	−	0.996633i	$-0.473872\pi$
$983$	5.14137	0.163984	0.0819921	+	0.996633i	$0.473872\pi$
$984$	−2.52508	−0.0804965
$985$	−9.43024	−0.300472
$986$	−25.4804	−0.811463
$987$	−65.4727	−2.08402
$988$	0	0
$989$	−23.2866	−0.740471
$990$	−11.6805	−0.371231
$991$	4.63484	0.147230	0.0736152	−	0.997287i	$-0.476546\pi$
$991$	4.63484	0.147230	0.0736152	+	0.997287i	$0.476546\pi$
$992$	−6.77822	−0.215209
$993$	−0.425279	−0.0134958
$994$	−52.2539	−1.65739
$995$	−1.43307	−0.0454315
$996$	3.34554	0.106008
$997$	22.1717	0.702185	0.351092	−	0.936341i	$-0.385810\pi$
$997$	22.1717	0.702185	0.351092	+	0.936341i	$0.385810\pi$
$998$	−22.7422	−0.719893
$999$	39.6461	1.25435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bq.1.7	yes	15
13.12	even	2		6422.2.a.bo.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.7	✓	15	13.12	even	2
6422.2.a.bq.1.7	yes	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.02081 q^{3} +1.00000 q^{4} +1.43823 q^{5} -1.02081 q^{6} +5.08222 q^{7} +1.00000 q^{8} -1.95796 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.02081 q^{3} +1.00000 q^{4} +1.43823 q^{5} -1.02081 q^{6} +5.08222 q^{7} +1.00000 q^{8} -1.95796 q^{9} +1.43823 q^{10} +4.14791 q^{11} -1.02081 q^{12} +5.08222 q^{14} -1.46816 q^{15} +1.00000 q^{16} +5.61847 q^{17} -1.95796 q^{18} -1.00000 q^{19} +1.43823 q^{20} -5.18796 q^{21} +4.14791 q^{22} +4.40837 q^{23} -1.02081 q^{24} -2.93149 q^{25} +5.06111 q^{27} +5.08222 q^{28} -4.53513 q^{29} -1.46816 q^{30} -6.77822 q^{31} +1.00000 q^{32} -4.23421 q^{33} +5.61847 q^{34} +7.30941 q^{35} -1.95796 q^{36} +7.83347 q^{37} -1.00000 q^{38} +1.43823 q^{40} +2.47361 q^{41} -5.18796 q^{42} -5.28236 q^{43} +4.14791 q^{44} -2.81600 q^{45} +4.40837 q^{46} +12.6201 q^{47} -1.02081 q^{48} +18.8289 q^{49} -2.93149 q^{50} -5.73536 q^{51} +10.9335 q^{53} +5.06111 q^{54} +5.96566 q^{55} +5.08222 q^{56} +1.02081 q^{57} -4.53513 q^{58} -7.26543 q^{59} -1.46816 q^{60} -7.47606 q^{61} -6.77822 q^{62} -9.95076 q^{63} +1.00000 q^{64} -4.23421 q^{66} -12.0693 q^{67} +5.61847 q^{68} -4.50009 q^{69} +7.30941 q^{70} -10.2817 q^{71} -1.95796 q^{72} -14.8354 q^{73} +7.83347 q^{74} +2.99248 q^{75} -1.00000 q^{76} +21.0806 q^{77} +14.6239 q^{79} +1.43823 q^{80} +0.707461 q^{81} +2.47361 q^{82} -3.27736 q^{83} -5.18796 q^{84} +8.08066 q^{85} -5.28236 q^{86} +4.62948 q^{87} +4.14791 q^{88} +3.71188 q^{89} -2.81600 q^{90} +4.40837 q^{92} +6.91925 q^{93} +12.6201 q^{94} -1.43823 q^{95} -1.02081 q^{96} +0.438413 q^{97} +18.8289 q^{98} -8.12143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9	\( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9} + q^{10} + 4 q^{11} + 18 q^{14} + 23 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 15 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} + 18 q^{28} - 20 q^{29} + 23 q^{30} + 30 q^{31} + 15 q^{32} + 36 q^{33} + 2 q^{34} + 32 q^{35} + 17 q^{36} + 35 q^{37} - 15 q^{38} + q^{40} + 15 q^{41} + 2 q^{42} + q^{43} + 4 q^{44} - 11 q^{45} + 17 q^{46} + 29 q^{49} + 8 q^{50} - q^{51} - q^{53} + 12 q^{54} - 6 q^{55} + 18 q^{56} - 20 q^{58} - 7 q^{59} + 23 q^{60} - 2 q^{61} + 30 q^{62} + 42 q^{63} + 15 q^{64} + 36 q^{66} + 34 q^{67} + 2 q^{68} - 12 q^{69} + 32 q^{70} + 4 q^{71} + 17 q^{72} + 12 q^{73} + 35 q^{74} + 31 q^{75} - 15 q^{76} - 20 q^{77} + 23 q^{79} + q^{80} + 7 q^{81} + 15 q^{82} - 3 q^{83} + 2 q^{84} + 46 q^{85} + q^{86} + 22 q^{87} + 4 q^{88} + 17 q^{89} - 11 q^{90} + 17 q^{92} + 60 q^{93} - q^{95} + 18 q^{97} + 29 q^{98} - 6 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9 + q^10 + 4 * q^11 + 18 * q^14 + 23 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 15 * q^19 + q^20 + 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 + 18 * q^28 - 20 * q^29 + 23 * q^30 + 30 * q^31 + 15 * q^32 + 36 * q^33 + 2 * q^34 + 32 * q^35 + 17 * q^36 + 35 * q^37 - 15 * q^38 + q^40 + 15 * q^41 + 2 * q^42 + q^43 + 4 * q^44 - 11 * q^45 + 17 * q^46 + 29 * q^49 + 8 * q^50 - q^51 - q^53 + 12 * q^54 - 6 * q^55 + 18 * q^56 - 20 * q^58 - 7 * q^59 + 23 * q^60 - 2 * q^61 + 30 * q^62 + 42 * q^63 + 15 * q^64 + 36 * q^66 + 34 * q^67 + 2 * q^68 - 12 * q^69 + 32 * q^70 + 4 * q^71 + 17 * q^72 + 12 * q^73 + 35 * q^74 + 31 * q^75 - 15 * q^76 - 20 * q^77 + 23 * q^79 + q^80 + 7 * q^81 + 15 * q^82 - 3 * q^83 + 2 * q^84 + 46 * q^85 + q^86 + 22 * q^87 + 4 * q^88 + 17 * q^89 - 11 * q^90 + 17 * q^92 + 60 * q^93 - q^95 + 18 * q^97 + 29 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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