LMFDB - Embedded newform 6027.2.a.ba.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 $ x^8 - 2x^7 - 8x^6 + 14x^5 + 18x^4 - 24x^3 - 10x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.117246$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.117246 q^{2} +1.00000 q^{3} -1.98625 q^{4} +0.562834 q^{5} -0.117246 q^{6} +0.467371 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.117246 q^{2} +1.00000 q^{3} -1.98625 q^{4} +0.562834 q^{5} -0.117246 q^{6} +0.467371 q^{8} +1.00000 q^{9} -0.0659898 q^{10} -3.66118 q^{11} -1.98625 q^{12} -2.93646 q^{13} +0.562834 q^{15} +3.91771 q^{16} +5.92232 q^{17} -0.117246 q^{18} -2.30522 q^{19} -1.11793 q^{20} +0.429257 q^{22} +2.06188 q^{23} +0.467371 q^{24} -4.68322 q^{25} +0.344288 q^{26} +1.00000 q^{27} +8.65134 q^{29} -0.0659898 q^{30} -1.07656 q^{31} -1.39408 q^{32} -3.66118 q^{33} -0.694366 q^{34} -1.98625 q^{36} -8.26262 q^{37} +0.270277 q^{38} -2.93646 q^{39} +0.263052 q^{40} -1.00000 q^{41} +0.218401 q^{43} +7.27203 q^{44} +0.562834 q^{45} -0.241746 q^{46} +6.06982 q^{47} +3.91771 q^{48} +0.549087 q^{50} +5.92232 q^{51} +5.83256 q^{52} +8.14009 q^{53} -0.117246 q^{54} -2.06063 q^{55} -2.30522 q^{57} -1.01433 q^{58} -13.9226 q^{59} -1.11793 q^{60} +2.36306 q^{61} +0.126222 q^{62} -7.67197 q^{64} -1.65274 q^{65} +0.429257 q^{66} -7.65437 q^{67} -11.7632 q^{68} +2.06188 q^{69} +2.62791 q^{71} +0.467371 q^{72} -12.0095 q^{73} +0.968756 q^{74} -4.68322 q^{75} +4.57874 q^{76} +0.344288 q^{78} +14.2162 q^{79} +2.20502 q^{80} +1.00000 q^{81} +0.117246 q^{82} -9.95382 q^{83} +3.33328 q^{85} -0.0256066 q^{86} +8.65134 q^{87} -1.71113 q^{88} -5.74023 q^{89} -0.0659898 q^{90} -4.09541 q^{92} -1.07656 q^{93} -0.711660 q^{94} -1.29745 q^{95} -1.39408 q^{96} +4.97851 q^{97} -3.66118 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9	$ 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 4 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{18} - 6 q^{19} - 4 q^{20} - 14 q^{22} - 12 q^{23} - 6 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 10 q^{31} - 4 q^{32} - 2 q^{33} - 4 q^{34} + 4 q^{36} - 20 q^{37} + 18 q^{38} - 4 q^{39} - 12 q^{40} - 8 q^{41} - 8 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 24 q^{47} - 22 q^{50} - 8 q^{51} + 30 q^{52} - 36 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 4 q^{60} + 22 q^{61} - 30 q^{62} - 24 q^{64} + 8 q^{65} - 14 q^{66} - 14 q^{67} - 38 q^{68} - 12 q^{69} - 10 q^{71} - 6 q^{72} + 12 q^{73} - 2 q^{74} - 4 q^{75} - 32 q^{76} - 4 q^{78} + 16 q^{79} + 14 q^{80} + 8 q^{81} + 2 q^{82} - 24 q^{83} - 44 q^{85} + 36 q^{86} - 4 q^{87} - 34 q^{88} - 2 q^{89} + 2 q^{90} - 48 q^{92} + 10 q^{93} + 34 q^{94} - 24 q^{95} - 4 q^{96} - 16 q^{97} - 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 - 4 * q^13 - 2 * q^15 - 8 * q^17 - 2 * q^18 - 6 * q^19 - 4 * q^20 - 14 * q^22 - 12 * q^23 - 6 * q^24 - 4 * q^25 - 4 * q^26 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 10 * q^31 - 4 * q^32 - 2 * q^33 - 4 * q^34 + 4 * q^36 - 20 * q^37 + 18 * q^38 - 4 * q^39 - 12 * q^40 - 8 * q^41 - 8 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 24 * q^47 - 22 * q^50 - 8 * q^51 + 30 * q^52 - 36 * q^53 - 2 * q^54 - 4 * q^55 - 6 * q^57 + 14 * q^58 - 10 * q^59 - 4 * q^60 + 22 * q^61 - 30 * q^62 - 24 * q^64 + 8 * q^65 - 14 * q^66 - 14 * q^67 - 38 * q^68 - 12 * q^69 - 10 * q^71 - 6 * q^72 + 12 * q^73 - 2 * q^74 - 4 * q^75 - 32 * q^76 - 4 * q^78 + 16 * q^79 + 14 * q^80 + 8 * q^81 + 2 * q^82 - 24 * q^83 - 44 * q^85 + 36 * q^86 - 4 * q^87 - 34 * q^88 - 2 * q^89 + 2 * q^90 - 48 * q^92 + 10 * q^93 + 34 * q^94 - 24 * q^95 - 4 * q^96 - 16 * q^97 - 2 * 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 $

$2$	−0.117246	−0.0829052	−0.0414526	−	0.999140i	$-0.513199\pi$
$2$	−0.117246	−0.0829052	−0.0414526	+	0.999140i	$0.513199\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.98625	−0.993127
$5$	0.562834	0.251707	0.125853	−	0.992049i	$-0.459833\pi$
$5$	0.562834	0.251707	0.125853	+	0.992049i	$0.459833\pi$
$6$	−0.117246	−0.0478653
$7$	0	0
$7$	0	0
$8$	0.467371	0.165241
$9$	1.00000	0.333333
$10$	−0.0659898	−0.0208678
$11$	−3.66118	−1.10389	−0.551944	−	0.833881i	$-0.686114\pi$
$11$	−3.66118	−1.10389	−0.551944	+	0.833881i	$0.686114\pi$
$12$	−1.98625	−0.573382
$13$	−2.93646	−0.814429	−0.407214	−	0.913333i	$-0.633500\pi$
$13$	−2.93646	−0.814429	−0.407214	+	0.913333i	$0.633500\pi$
$14$	0	0
$15$	0.562834	0.145323
$16$	3.91771	0.979427
$17$	5.92232	1.43637	0.718186	−	0.695851i	$-0.244973\pi$
$17$	5.92232	1.43637	0.718186	+	0.695851i	$0.244973\pi$
$18$	−0.117246	−0.0276351
$19$	−2.30522	−0.528853	−0.264427	−	0.964406i	$-0.585183\pi$
$19$	−2.30522	−0.528853	−0.264427	+	0.964406i	$0.585183\pi$
$20$	−1.11793	−0.249977
$21$	0	0
$22$	0.429257	0.0915180
$23$	2.06188	0.429931	0.214965	−	0.976622i	$-0.431036\pi$
$23$	2.06188	0.429931	0.214965	+	0.976622i	$0.431036\pi$
$24$	0.467371	0.0954017
$25$	−4.68322	−0.936644
$26$	0.344288	0.0675204
$27$	1.00000	0.192450
$28$	0	0
$29$	8.65134	1.60651	0.803257	−	0.595633i	$-0.203099\pi$
$29$	8.65134	1.60651	0.803257	+	0.595633i	$0.203099\pi$
$30$	−0.0659898	−0.0120480
$31$	−1.07656	−0.193356	−0.0966782	−	0.995316i	$-0.530822\pi$
$31$	−1.07656	−0.193356	−0.0966782	+	0.995316i	$0.530822\pi$
$32$	−1.39408	−0.246440
$33$	−3.66118	−0.637330
$34$	−0.694366	−0.119083
$35$	0	0
$36$	−1.98625	−0.331042
$37$	−8.26262	−1.35837	−0.679183	−	0.733969i	$-0.737666\pi$
$37$	−8.26262	−1.35837	−0.679183	+	0.733969i	$0.737666\pi$
$38$	0.270277	0.0438447
$39$	−2.93646	−0.470211
$40$	0.263052	0.0415922
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	0.218401	0.0333059	0.0166530	−	0.999861i	$-0.494699\pi$
$43$	0.218401	0.0333059	0.0166530	+	0.999861i	$0.494699\pi$
$44$	7.27203	1.09630
$45$	0.562834	0.0839023
$46$	−0.241746	−0.0356435
$47$	6.06982	0.885375	0.442687	−	0.896676i	$-0.354025\pi$
$47$	6.06982	0.885375	0.442687	+	0.896676i	$0.354025\pi$
$48$	3.91771	0.565473
$49$	0	0
$50$	0.549087	0.0776526
$51$	5.92232	0.829290
$52$	5.83256	0.808831
$53$	8.14009	1.11813	0.559064	−	0.829125i	$-0.311161\pi$
$53$	8.14009	1.11813	0.559064	+	0.829125i	$0.311161\pi$
$54$	−0.117246	−0.0159551
$55$	−2.06063	−0.277856
$56$	0	0
$57$	−2.30522	−0.305333
$58$	−1.01433	−0.133188
$59$	−13.9226	−1.81257	−0.906286	−	0.422665i	$-0.861095\pi$
$59$	−13.9226	−1.81257	−0.906286	+	0.422665i	$0.861095\pi$
$60$	−1.11793	−0.144324
$61$	2.36306	0.302559	0.151279	−	0.988491i	$-0.451661\pi$
$61$	2.36306	0.302559	0.151279	+	0.988491i	$0.451661\pi$
$62$	0.126222	0.0160303
$63$	0	0
$64$	−7.67197	−0.958996
$65$	−1.65274	−0.204997
$66$	0.429257	0.0528379
$67$	−7.65437	−0.935130	−0.467565	−	0.883959i	$-0.654869\pi$
$67$	−7.65437	−0.935130	−0.467565	+	0.883959i	$0.654869\pi$
$68$	−11.7632	−1.42650
$69$	2.06188	0.248221
$70$	0	0
$71$	2.62791	0.311876	0.155938	−	0.987767i	$-0.450160\pi$
$71$	2.62791	0.311876	0.155938	+	0.987767i	$0.450160\pi$
$72$	0.467371	0.0550802
$73$	−12.0095	−1.40561	−0.702804	−	0.711384i	$-0.748069\pi$
$73$	−12.0095	−1.40561	−0.702804	+	0.711384i	$0.748069\pi$
$74$	0.968756	0.112616
$75$	−4.68322	−0.540771
$76$	4.57874	0.525218
$77$	0	0
$78$	0.344288	0.0389829
$79$	14.2162	1.59945	0.799723	−	0.600369i	$-0.204980\pi$
$79$	14.2162	1.59945	0.799723	+	0.600369i	$0.204980\pi$
$80$	2.20502	0.246529
$81$	1.00000	0.111111
$82$	0.117246	0.0129476
$83$	−9.95382	−1.09257	−0.546287	−	0.837598i	$-0.683959\pi$
$83$	−9.95382	−1.09257	−0.546287	+	0.837598i	$0.683959\pi$
$84$	0	0
$85$	3.33328	0.361545
$86$	−0.0256066	−0.00276123
$87$	8.65134	0.927521
$88$	−1.71113	−0.182407
$89$	−5.74023	−0.608463	−0.304232	−	0.952598i	$-0.598400\pi$
$89$	−5.74023	−0.608463	−0.304232	+	0.952598i	$0.598400\pi$
$90$	−0.0659898	−0.00695593
$91$	0	0
$92$	−4.09541	−0.426976
$93$	−1.07656	−0.111634
$94$	−0.711660	−0.0734022
$95$	−1.29745	−0.133116
$96$	−1.39408	−0.142282
$97$	4.97851	0.505491	0.252746	−	0.967533i	$-0.418666\pi$
$97$	4.97851	0.505491	0.252746	+	0.967533i	$0.418666\pi$
$98$	0	0
$99$	−3.66118	−0.367962
$100$	9.30206	0.930206
$101$	−5.54806	−0.552053	−0.276027	−	0.961150i	$-0.589018\pi$
$101$	−5.54806	−0.552053	−0.276027	+	0.961150i	$0.589018\pi$
$102$	−0.694366	−0.0687525
$103$	18.0913	1.78259	0.891296	−	0.453423i	$-0.149797\pi$
$103$	18.0913	1.78259	0.891296	+	0.453423i	$0.149797\pi$
$104$	−1.37242	−0.134577
$105$	0	0
$106$	−0.954390	−0.0926986
$107$	8.71524	0.842534	0.421267	−	0.906937i	$-0.361585\pi$
$107$	8.71524	0.842534	0.421267	+	0.906937i	$0.361585\pi$
$108$	−1.98625	−0.191127
$109$	−5.65802	−0.541940	−0.270970	−	0.962588i	$-0.587344\pi$
$109$	−5.65802	−0.541940	−0.270970	+	0.962588i	$0.587344\pi$
$110$	0.241600	0.0230357
$111$	−8.26262	−0.784253
$112$	0	0
$113$	−13.7174	−1.29042	−0.645212	−	0.764004i	$-0.723231\pi$
$113$	−13.7174	−1.29042	−0.645212	+	0.764004i	$0.723231\pi$
$114$	0.270277	0.0253137
$115$	1.16049	0.108217
$116$	−17.1838	−1.59547
$117$	−2.93646	−0.271476
$118$	1.63237	0.150272
$119$	0	0
$120$	0.263052	0.0240133
$121$	2.40424	0.218567
$122$	−0.277058	−0.0250837
$123$	−1.00000	−0.0901670
$124$	2.13833	0.192027
$125$	−5.45004	−0.487466
$126$	0	0
$127$	−19.9726	−1.77229	−0.886143	−	0.463412i	$-0.846625\pi$
$127$	−19.9726	−1.77229	−0.886143	+	0.463412i	$0.846625\pi$
$128$	3.68766	0.325946
$129$	0.218401	0.0192292
$130$	0.193777	0.0169953
$131$	−8.14739	−0.711841	−0.355920	−	0.934516i	$-0.615833\pi$
$131$	−8.14739	−0.711841	−0.355920	+	0.934516i	$0.615833\pi$
$132$	7.27203	0.632949
$133$	0	0
$134$	0.897441	0.0775271
$135$	0.562834	0.0484410
$136$	2.76792	0.237347
$137$	−0.0216108	−0.00184634	−0.000923168	−	1.00000i	$-0.500294\pi$
$137$	−0.0216108	−0.00184634	−0.000923168		1.00000i	$0.500294\pi$
$138$	−0.241746	−0.0205788
$139$	−17.3931	−1.47527	−0.737633	−	0.675202i	$-0.764057\pi$
$139$	−17.3931	−1.47527	−0.737633	+	0.675202i	$0.764057\pi$
$140$	0	0
$141$	6.06982	0.511171
$142$	−0.308111	−0.0258561
$143$	10.7509	0.899038
$144$	3.91771	0.326476
$145$	4.86926	0.404370
$146$	1.40806	0.116532
$147$	0	0
$148$	16.4117	1.34903
$149$	−5.78212	−0.473690	−0.236845	−	0.971547i	$-0.576113\pi$
$149$	−5.78212	−0.473690	−0.236845	+	0.971547i	$0.576113\pi$
$150$	0.549087	0.0448328
$151$	−0.865254	−0.0704133	−0.0352067	−	0.999380i	$-0.511209\pi$
$151$	−0.865254	−0.0704133	−0.0352067	+	0.999380i	$0.511209\pi$
$152$	−1.07739	−0.0873880
$153$	5.92232	0.478791
$154$	0	0
$155$	−0.605926	−0.0486691
$156$	5.83256	0.466979
$157$	−6.15684	−0.491369	−0.245685	−	0.969350i	$-0.579013\pi$
$157$	−6.15684	−0.491369	−0.245685	+	0.969350i	$0.579013\pi$
$158$	−1.66679	−0.132602
$159$	8.14009	0.645551
$160$	−0.784633	−0.0620307
$161$	0	0
$162$	−0.117246	−0.00921169
$163$	−22.9653	−1.79878	−0.899392	−	0.437144i	$-0.855990\pi$
$163$	−22.9653	−1.79878	−0.899392	+	0.437144i	$0.855990\pi$
$164$	1.98625	0.155100
$165$	−2.06063	−0.160420
$166$	1.16704	0.0905801
$167$	19.0532	1.47438	0.737192	−	0.675683i	$-0.236151\pi$
$167$	19.0532	1.47438	0.737192	+	0.675683i	$0.236151\pi$
$168$	0	0
$169$	−4.37717	−0.336706
$170$	−0.390812	−0.0299739
$171$	−2.30522	−0.176284
$172$	−0.433801	−0.0330770
$173$	−16.7532	−1.27372	−0.636861	−	0.770978i	$-0.719768\pi$
$173$	−16.7532	−1.27372	−0.636861	+	0.770978i	$0.719768\pi$
$174$	−1.01433	−0.0768963
$175$	0	0
$176$	−14.3434	−1.08118
$177$	−13.9226	−1.04649
$178$	0.673017	0.0504448
$179$	−13.2695	−0.991807	−0.495904	−	0.868378i	$-0.665163\pi$
$179$	−13.2695	−0.991807	−0.495904	+	0.868378i	$0.665163\pi$
$180$	−1.11793	−0.0833256
$181$	−25.0581	−1.86256	−0.931279	−	0.364308i	$-0.881306\pi$
$181$	−25.0581	−1.86256	−0.931279	+	0.364308i	$0.881306\pi$
$182$	0	0
$183$	2.36306	0.174682
$184$	0.963661	0.0710420
$185$	−4.65048	−0.341910
$186$	0.126222	0.00925507
$187$	−21.6827	−1.58559
$188$	−12.0562	−0.879289
$189$	0	0
$190$	0.152121	0.0110360
$191$	18.9006	1.36760	0.683801	−	0.729668i	$-0.260326\pi$
$191$	18.9006	1.36760	0.683801	+	0.729668i	$0.260326\pi$
$192$	−7.67197	−0.553677
$193$	−6.25461	−0.450217	−0.225108	−	0.974334i	$-0.572274\pi$
$193$	−6.25461	−0.450217	−0.225108	+	0.974334i	$0.572274\pi$
$194$	−0.583709	−0.0419078
$195$	−1.65274	−0.118355
$196$	0	0
$197$	−18.6911	−1.33168	−0.665842	−	0.746092i	$-0.731928\pi$
$197$	−18.6911	−1.33168	−0.665842	+	0.746092i	$0.731928\pi$
$198$	0.429257	0.0305060
$199$	19.1614	1.35832	0.679158	−	0.733993i	$-0.262345\pi$
$199$	19.1614	1.35832	0.679158	+	0.733993i	$0.262345\pi$
$200$	−2.18880	−0.154772
$201$	−7.65437	−0.539897
$202$	0.650486	0.0457681
$203$	0	0
$204$	−11.7632	−0.823590
$205$	−0.562834	−0.0393100
$206$	−2.12113	−0.147786
$207$	2.06188	0.143310
$208$	−11.5042	−0.797674
$209$	8.43981	0.583794
$210$	0	0
$211$	−6.41632	−0.441718	−0.220859	−	0.975306i	$-0.570886\pi$
$211$	−6.41632	−0.441718	−0.220859	+	0.975306i	$0.570886\pi$
$212$	−16.1683	−1.11044
$213$	2.62791	0.180062
$214$	−1.02182	−0.0698505
$215$	0.122924	0.00838333
$216$	0.467371	0.0318006
$217$	0	0
$218$	0.663379	0.0449297
$219$	−12.0095	−0.811528
$220$	4.09294	0.275946
$221$	−17.3907	−1.16982
$222$	0.968756	0.0650187
$223$	−16.9305	−1.13375	−0.566875	−	0.823804i	$-0.691848\pi$
$223$	−16.9305	−1.13375	−0.566875	+	0.823804i	$0.691848\pi$
$224$	0	0
$225$	−4.68322	−0.312215
$226$	1.60830	0.106983
$227$	26.1387	1.73489	0.867444	−	0.497534i	$-0.165761\pi$
$227$	26.1387	1.73489	0.867444	+	0.497534i	$0.165761\pi$
$228$	4.57874	0.303235
$229$	27.9844	1.84926	0.924630	−	0.380867i	$-0.124374\pi$
$229$	27.9844	1.84926	0.924630	+	0.380867i	$0.124374\pi$
$230$	−0.136063	−0.00897171
$231$	0	0
$232$	4.04338	0.265461
$233$	13.6927	0.897037	0.448519	−	0.893774i	$-0.351952\pi$
$233$	13.6927	0.897037	0.448519	+	0.893774i	$0.351952\pi$
$234$	0.344288	0.0225068
$235$	3.41630	0.222855
$236$	27.6539	1.80011
$237$	14.2162	0.923441
$238$	0	0
$239$	18.4282	1.19202	0.596011	−	0.802977i	$-0.296752\pi$
$239$	18.4282	1.19202	0.596011	+	0.802977i	$0.296752\pi$
$240$	2.20502	0.142333
$241$	11.8493	0.763278	0.381639	−	0.924312i	$-0.375360\pi$
$241$	11.8493	0.763278	0.381639	+	0.924312i	$0.375360\pi$
$242$	−0.281886	−0.0181203
$243$	1.00000	0.0641500
$244$	−4.69363	−0.300479
$245$	0	0
$246$	0.117246	0.00747531
$247$	6.76919	0.430713
$248$	−0.503154	−0.0319503
$249$	−9.95382	−0.630798
$250$	0.638994	0.0404135
$251$	6.43236	0.406007	0.203004	−	0.979178i	$-0.434930\pi$
$251$	6.43236	0.406007	0.203004	+	0.979178i	$0.434930\pi$
$252$	0	0
$253$	−7.54890	−0.474595
$254$	2.34171	0.146932
$255$	3.33328	0.208738
$256$	14.9116	0.931974
$257$	−2.51802	−0.157070	−0.0785349	−	0.996911i	$-0.525024\pi$
$257$	−2.51802	−0.157070	−0.0785349	+	0.996911i	$0.525024\pi$
$258$	−0.0256066	−0.00159420
$259$	0	0
$260$	3.28276	0.203588
$261$	8.65134	0.535504
$262$	0.955246	0.0590153
$263$	−12.1289	−0.747901	−0.373950	−	0.927449i	$-0.621997\pi$
$263$	−12.1289	−0.747901	−0.373950	+	0.927449i	$0.621997\pi$
$264$	−1.71113	−0.105313
$265$	4.58152	0.281440
$266$	0	0
$267$	−5.74023	−0.351296
$268$	15.2035	0.928702
$269$	−21.6039	−1.31721	−0.658605	−	0.752489i	$-0.728853\pi$
$269$	−21.6039	−1.31721	−0.658605	+	0.752489i	$0.728853\pi$
$270$	−0.0659898	−0.00401601
$271$	5.58625	0.339340	0.169670	−	0.985501i	$-0.445730\pi$
$271$	5.58625	0.339340	0.169670	+	0.985501i	$0.445730\pi$
$272$	23.2019	1.40682
$273$	0	0
$274$	0.00253377	0.000153071 0
$275$	17.1461	1.03395
$276$	−4.09541	−0.246515
$277$	−1.62473	−0.0976205	−0.0488103	−	0.998808i	$-0.515543\pi$
$277$	−1.62473	−0.0976205	−0.0488103	+	0.998808i	$0.515543\pi$
$278$	2.03927	0.122307
$279$	−1.07656	−0.0644521
$280$	0	0
$281$	−29.4166	−1.75485	−0.877424	−	0.479716i	$-0.840740\pi$
$281$	−29.4166	−1.75485	−0.877424	+	0.479716i	$0.840740\pi$
$282$	−0.711660	−0.0423788
$283$	−11.5265	−0.685180	−0.342590	−	0.939485i	$-0.611304\pi$
$283$	−11.5265	−0.685180	−0.342590	+	0.939485i	$0.611304\pi$
$284$	−5.21970	−0.309732
$285$	−1.29745	−0.0768545
$286$	−1.26050	−0.0745349
$287$	0	0
$288$	−1.39408	−0.0821467
$289$	18.0738	1.06317
$290$	−0.570900	−0.0335244
$291$	4.97851	0.291845
$292$	23.8539	1.39595
$293$	15.5546	0.908707	0.454353	−	0.890822i	$-0.349870\pi$
$293$	15.5546	0.908707	0.454353	+	0.890822i	$0.349870\pi$
$294$	0	0
$295$	−7.83612	−0.456237
$296$	−3.86171	−0.224457
$297$	−3.66118	−0.212443
$298$	0.677929	0.0392714
$299$	−6.05463	−0.350148
$300$	9.30206	0.537055
$301$	0	0
$302$	0.101447	0.00583763
$303$	−5.54806	−0.318728
$304$	−9.03117	−0.517973
$305$	1.33001	0.0761561
$306$	−0.694366	−0.0396943
$307$	−14.8205	−0.845848	−0.422924	−	0.906165i	$-0.638996\pi$
$307$	−14.8205	−0.845848	−0.422924	+	0.906165i	$0.638996\pi$
$308$	0	0
$309$	18.0913	1.02918
$310$	0.0710422	0.00403492
$311$	0.653256	0.0370427	0.0185214	−	0.999828i	$-0.494104\pi$
$311$	0.653256	0.0370427	0.0185214	+	0.999828i	$0.494104\pi$
$312$	−1.37242	−0.0776979
$313$	21.2461	1.20090	0.600450	−	0.799662i	$-0.294988\pi$
$313$	21.2461	1.20090	0.600450	+	0.799662i	$0.294988\pi$
$314$	0.721862	0.0407371
$315$	0	0
$316$	−28.2370	−1.58845
$317$	−18.2323	−1.02403	−0.512015	−	0.858976i	$-0.671101\pi$
$317$	−18.2323	−1.02403	−0.512015	+	0.858976i	$0.671101\pi$
$318$	−0.954390	−0.0535195
$319$	−31.6741	−1.77341
$320$	−4.31804	−0.241386
$321$	8.71524	0.486437
$322$	0	0
$323$	−13.6522	−0.759630
$324$	−1.98625	−0.110347
$325$	13.7521	0.762830
$326$	2.69259	0.149128
$327$	−5.65802	−0.312889
$328$	−0.467371	−0.0258062
$329$	0	0
$330$	0.241600	0.0132997
$331$	−2.72663	−0.149869	−0.0749346	−	0.997188i	$-0.523875\pi$
$331$	−2.72663	−0.149869	−0.0749346	+	0.997188i	$0.523875\pi$
$332$	19.7708	1.08506
$333$	−8.26262	−0.452789
$334$	−2.23391	−0.122234
$335$	−4.30813	−0.235379
$336$	0	0
$337$	−18.1839	−0.990538	−0.495269	−	0.868740i	$-0.664931\pi$
$337$	−18.1839	−0.990538	−0.495269	+	0.868740i	$0.664931\pi$
$338$	0.513205	0.0279146
$339$	−13.7174	−0.745026
$340$	−6.62074	−0.359060
$341$	3.94149	0.213444
$342$	0.270277	0.0146149
$343$	0	0
$344$	0.102074	0.00550349
$345$	1.16049	0.0624789
$346$	1.96424	0.105598
$347$	−5.36054	−0.287769	−0.143884	−	0.989595i	$-0.545959\pi$
$347$	−5.36054	−0.287769	−0.143884	+	0.989595i	$0.545959\pi$
$348$	−17.1838	−0.921146
$349$	7.64412	0.409181	0.204590	−	0.978848i	$-0.434414\pi$
$349$	7.64412	0.409181	0.204590	+	0.978848i	$0.434414\pi$
$350$	0	0
$351$	−2.93646	−0.156737
$352$	5.10396	0.272042
$353$	−1.57458	−0.0838062	−0.0419031	−	0.999122i	$-0.513342\pi$
$353$	−1.57458	−0.0838062	−0.0419031	+	0.999122i	$0.513342\pi$
$354$	1.63237	0.0867594
$355$	1.47908	0.0785012
$356$	11.4016	0.604281
$357$	0	0
$358$	1.55579	0.0822260
$359$	−14.3704	−0.758439	−0.379219	−	0.925307i	$-0.623807\pi$
$359$	−14.3704	−0.758439	−0.379219	+	0.925307i	$0.623807\pi$
$360$	0.263052	0.0138641
$361$	−13.6860	−0.720314
$362$	2.93796	0.154416
$363$	2.40424	0.126190
$364$	0	0
$365$	−6.75936	−0.353801
$366$	−0.277058	−0.0144821
$367$	−20.7996	−1.08573	−0.542866	−	0.839819i	$-0.682661\pi$
$367$	−20.7996	−1.08573	−0.542866	+	0.839819i	$0.682661\pi$
$368$	8.07783	0.421086
$369$	−1.00000	−0.0520579
$370$	0.545248	0.0283461
$371$	0	0
$372$	2.13833	0.110867
$373$	−5.41255	−0.280251	−0.140126	−	0.990134i	$-0.544751\pi$
$373$	−5.41255	−0.280251	−0.140126	+	0.990134i	$0.544751\pi$
$374$	2.54220	0.131454
$375$	−5.45004	−0.281439
$376$	2.83686	0.146300
$377$	−25.4044	−1.30839
$378$	0	0
$379$	4.54878	0.233655	0.116828	−	0.993152i	$-0.462727\pi$
$379$	4.54878	0.233655	0.116828	+	0.993152i	$0.462727\pi$
$380$	2.57707	0.132201
$381$	−19.9726	−1.02323
$382$	−2.21602	−0.113381
$383$	−11.8594	−0.605986	−0.302993	−	0.952993i	$-0.597986\pi$
$383$	−11.8594	−0.605986	−0.302993	+	0.952993i	$0.597986\pi$
$384$	3.68766	0.188185
$385$	0	0
$386$	0.733326	0.0373253
$387$	0.218401	0.0111020
$388$	−9.88858	−0.502017
$389$	−18.0699	−0.916182	−0.458091	−	0.888905i	$-0.651467\pi$
$389$	−18.0699	−0.916182	−0.458091	+	0.888905i	$0.651467\pi$
$390$	0.193777	0.00981226
$391$	12.2111	0.617541
$392$	0	0
$393$	−8.14739	−0.410981
$394$	2.19145	0.110404
$395$	8.00135	0.402591
$396$	7.27203	0.365433
$397$	25.4105	1.27532	0.637658	−	0.770320i	$-0.279903\pi$
$397$	25.4105	1.27532	0.637658	+	0.770320i	$0.279903\pi$
$398$	−2.24659	−0.112611
$399$	0	0
$400$	−18.3475	−0.917375
$401$	−18.4488	−0.921287	−0.460644	−	0.887585i	$-0.652381\pi$
$401$	−18.4488	−0.921287	−0.460644	+	0.887585i	$0.652381\pi$
$402$	0.897441	0.0447603
$403$	3.16129	0.157475
$404$	11.0199	0.548259
$405$	0.562834	0.0279674
$406$	0	0
$407$	30.2509	1.49948
$408$	2.76792	0.137032
$409$	−32.1898	−1.59168	−0.795841	−	0.605506i	$-0.792971\pi$
$409$	−32.1898	−1.59168	−0.795841	+	0.605506i	$0.792971\pi$
$410$	0.0659898	0.00325900
$411$	−0.0216108	−0.00106598
$412$	−35.9340	−1.77034
$413$	0	0
$414$	−0.241746	−0.0118812
$415$	−5.60235	−0.275008
$416$	4.09366	0.200708
$417$	−17.3931	−0.851745
$418$	−0.989531	−0.0483996
$419$	33.7142	1.64704	0.823522	−	0.567284i	$-0.192006\pi$
$419$	33.7142	1.64704	0.823522	+	0.567284i	$0.192006\pi$
$420$	0	0
$421$	6.44523	0.314121	0.157061	−	0.987589i	$-0.449798\pi$
$421$	6.44523	0.314121	0.157061	+	0.987589i	$0.449798\pi$
$422$	0.752286	0.0366207
$423$	6.06982	0.295125
$424$	3.80444	0.184760
$425$	−27.7355	−1.34537
$426$	−0.308111	−0.0149280
$427$	0	0
$428$	−17.3107	−0.836743
$429$	10.7509	0.519060
$430$	−0.0144123	−0.000695021 0
$431$	−11.9443	−0.575337	−0.287668	−	0.957730i	$-0.592880\pi$
$431$	−11.9443	−0.575337	−0.287668	+	0.957730i	$0.592880\pi$
$432$	3.91771	0.188491
$433$	25.8659	1.24304	0.621519	−	0.783399i	$-0.286516\pi$
$433$	25.8659	1.24304	0.621519	+	0.783399i	$0.286516\pi$
$434$	0	0
$435$	4.86926	0.233463
$436$	11.2383	0.538215
$437$	−4.75307	−0.227370
$438$	1.40806	0.0672799
$439$	13.8716	0.662054	0.331027	−	0.943621i	$-0.392605\pi$
$439$	13.8716	0.662054	0.331027	+	0.943621i	$0.392605\pi$
$440$	−0.963081	−0.0459131
$441$	0	0
$442$	2.03898	0.0969844
$443$	−11.1418	−0.529363	−0.264682	−	0.964336i	$-0.585267\pi$
$443$	−11.1418	−0.529363	−0.264682	+	0.964336i	$0.585267\pi$
$444$	16.4117	0.778863
$445$	−3.23079	−0.153154
$446$	1.98503	0.0939938
$447$	−5.78212	−0.273485
$448$	0	0
$449$	−35.1171	−1.65728	−0.828640	−	0.559782i	$-0.810885\pi$
$449$	−35.1171	−1.65728	−0.828640	+	0.559782i	$0.810885\pi$
$450$	0.549087	0.0258842
$451$	3.66118	0.172398
$452$	27.2462	1.28155
$453$	−0.865254	−0.0406532
$454$	−3.06465	−0.143831
$455$	0	0
$456$	−1.07739	−0.0504535
$457$	−38.0301	−1.77897	−0.889486	−	0.456963i	$-0.848937\pi$
$457$	−38.0301	−1.77897	−0.889486	+	0.456963i	$0.848937\pi$
$458$	−3.28105	−0.153313
$459$	5.92232	0.276430
$460$	−2.30503	−0.107473
$461$	−5.33168	−0.248321	−0.124161	−	0.992262i	$-0.539624\pi$
$461$	−5.33168	−0.248321	−0.124161	+	0.992262i	$0.539624\pi$
$462$	0	0
$463$	7.44282	0.345897	0.172949	−	0.984931i	$-0.444671\pi$
$463$	7.44282	0.345897	0.172949	+	0.984931i	$0.444671\pi$
$464$	33.8934	1.57346
$465$	−0.605926	−0.0280991
$466$	−1.60541	−0.0743690
$467$	−10.6975	−0.495020	−0.247510	−	0.968885i	$-0.579612\pi$
$467$	−10.6975	−0.495020	−0.247510	+	0.968885i	$0.579612\pi$
$468$	5.83256	0.269610
$469$	0	0
$470$	−0.400546	−0.0184758
$471$	−6.15684	−0.283692
$472$	−6.50703	−0.299510
$473$	−0.799607	−0.0367660
$474$	−1.66679	−0.0765580
$475$	10.7958	0.495347
$476$	0	0
$477$	8.14009	0.372709
$478$	−2.16063	−0.0988248
$479$	−25.8527	−1.18124	−0.590621	−	0.806949i	$-0.701117\pi$
$479$	−25.8527	−1.18124	−0.590621	+	0.806949i	$0.701117\pi$
$480$	−0.784633	−0.0358134
$481$	24.2629	1.10629
$482$	−1.38927	−0.0632797
$483$	0	0
$484$	−4.77542	−0.217065
$485$	2.80207	0.127236
$486$	−0.117246	−0.00531837
$487$	35.0858	1.58989	0.794945	−	0.606682i	$-0.207500\pi$
$487$	35.0858	1.58989	0.794945	+	0.606682i	$0.207500\pi$
$488$	1.10442	0.0499950
$489$	−22.9653	−1.03853
$490$	0	0
$491$	7.96601	0.359501	0.179751	−	0.983712i	$-0.442471\pi$
$491$	7.96601	0.359501	0.179751	+	0.983712i	$0.442471\pi$
$492$	1.98625	0.0895472
$493$	51.2360	2.30755
$494$	−0.793658	−0.0357084
$495$	−2.06063	−0.0926186
$496$	−4.21766	−0.189379
$497$	0	0
$498$	1.16704	0.0522964
$499$	26.8060	1.20000	0.600000	−	0.800000i	$-0.295167\pi$
$499$	26.8060	1.20000	0.600000	+	0.800000i	$0.295167\pi$
$500$	10.8252	0.484116
$501$	19.0532	0.851236
$502$	−0.754167	−0.0336601
$503$	12.4925	0.557014	0.278507	−	0.960434i	$-0.410161\pi$
$503$	12.4925	0.557014	0.278507	+	0.960434i	$0.410161\pi$
$504$	0	0
$505$	−3.12264	−0.138956
$506$	0.885076	0.0393464
$507$	−4.37717	−0.194397
$508$	39.6707	1.76010
$509$	−15.1351	−0.670853	−0.335427	−	0.942066i	$-0.608880\pi$
$509$	−15.1351	−0.670853	−0.335427	+	0.942066i	$0.608880\pi$
$510$	−0.390812	−0.0173055
$511$	0	0
$512$	−9.12363	−0.403211
$513$	−2.30522	−0.101778
$514$	0.295227	0.0130219
$515$	10.1824	0.448690
$516$	−0.433801	−0.0190970
$517$	−22.2227	−0.977354
$518$	0	0
$519$	−16.7532	−0.735384
$520$	−0.772443	−0.0338739
$521$	−14.1385	−0.619419	−0.309709	−	0.950831i	$-0.600232\pi$
$521$	−14.1385	−0.619419	−0.309709	+	0.950831i	$0.600232\pi$
$522$	−1.01433	−0.0443961
$523$	35.3444	1.54550	0.772751	−	0.634710i	$-0.218880\pi$
$523$	35.3444	1.54550	0.772751	+	0.634710i	$0.218880\pi$
$524$	16.1828	0.706948
$525$	0	0
$526$	1.42206	0.0620048
$527$	−6.37575	−0.277732
$528$	−14.3434	−0.624218
$529$	−18.7487	−0.815159
$530$	−0.537163	−0.0233329
$531$	−13.9226	−0.604191
$532$	0	0
$533$	2.93646	0.127192
$534$	0.673017	0.0291243
$535$	4.90523	0.212072
$536$	−3.57743	−0.154521
$537$	−13.2695	−0.572620
$538$	2.53296	0.109204
$539$	0	0
$540$	−1.11793	−0.0481081
$541$	−28.1913	−1.21204	−0.606020	−	0.795449i	$-0.707235\pi$
$541$	−28.1913	−1.21204	−0.606020	+	0.795449i	$0.707235\pi$
$542$	−0.654963	−0.0281331
$543$	−25.0581	−1.07535
$544$	−8.25616	−0.353980
$545$	−3.18452	−0.136410
$546$	0	0
$547$	−22.9569	−0.981566	−0.490783	−	0.871282i	$-0.663289\pi$
$547$	−22.9569	−0.981566	−0.490783	+	0.871282i	$0.663289\pi$
$548$	0.0429245	0.00183365
$549$	2.36306	0.100853
$550$	−2.01031	−0.0857197
$551$	−19.9432	−0.849609
$552$	0.963661	0.0410161
$553$	0	0
$554$	0.190492	0.00809325
$555$	−4.65048	−0.197402
$556$	34.5472	1.46513
$557$	−15.1009	−0.639847	−0.319924	−	0.947443i	$-0.603657\pi$
$557$	−15.1009	−0.639847	−0.319924	+	0.947443i	$0.603657\pi$
$558$	0.126222	0.00534342
$559$	−0.641328	−0.0271253
$560$	0	0
$561$	−21.6827	−0.915443
$562$	3.44897	0.145486
$563$	−8.55488	−0.360545	−0.180273	−	0.983617i	$-0.557698\pi$
$563$	−8.55488	−0.360545	−0.180273	+	0.983617i	$0.557698\pi$
$564$	−12.0562	−0.507658
$565$	−7.72060	−0.324808
$566$	1.35143	0.0568050
$567$	0	0
$568$	1.22821	0.0515345
$569$	−10.1191	−0.424215	−0.212107	−	0.977246i	$-0.568033\pi$
$569$	−10.1191	−0.424215	−0.212107	+	0.977246i	$0.568033\pi$
$570$	0.152121	0.00637164
$571$	−34.1656	−1.42978	−0.714892	−	0.699235i	$-0.753524\pi$
$571$	−34.1656	−1.42978	−0.714892	+	0.699235i	$0.753524\pi$
$572$	−21.3541	−0.892858
$573$	18.9006	0.789586
$574$	0	0
$575$	−9.65622	−0.402692
$576$	−7.67197	−0.319665
$577$	7.84351	0.326530	0.163265	−	0.986582i	$-0.447798\pi$
$577$	7.84351	0.326530	0.163265	+	0.986582i	$0.447798\pi$
$578$	−2.11908	−0.0881420
$579$	−6.25461	−0.259933
$580$	−9.67159	−0.401591
$581$	0	0
$582$	−0.583709	−0.0241955
$583$	−29.8023	−1.23429
$584$	−5.61290	−0.232263
$585$	−1.65274	−0.0683324
$586$	−1.82370	−0.0753365
$587$	13.3814	0.552310	0.276155	−	0.961113i	$-0.410940\pi$
$587$	13.3814	0.552310	0.276155	+	0.961113i	$0.410940\pi$
$588$	0	0
$589$	2.48171	0.102257
$590$	0.918751	0.0378244
$591$	−18.6911	−0.768849
$592$	−32.3705	−1.33042
$593$	−13.7207	−0.563441	−0.281721	−	0.959496i	$-0.590905\pi$
$593$	−13.7207	−0.563441	−0.281721	+	0.959496i	$0.590905\pi$
$594$	0.429257	0.0176126
$595$	0	0
$596$	11.4848	0.470434
$597$	19.1614	0.784224
$598$	0.709879	0.0290291
$599$	13.6121	0.556174	0.278087	−	0.960556i	$-0.410300\pi$
$599$	13.6121	0.556174	0.278087	+	0.960556i	$0.410300\pi$
$600$	−2.18880	−0.0893574
$601$	17.9729	0.733128	0.366564	−	0.930393i	$-0.380534\pi$
$601$	17.9729	0.733128	0.366564	+	0.930393i	$0.380534\pi$
$602$	0	0
$603$	−7.65437	−0.311710
$604$	1.71861	0.0699294
$605$	1.35318	0.0550148
$606$	0.650486	0.0264242
$607$	15.5693	0.631940	0.315970	−	0.948769i	$-0.397670\pi$
$607$	15.5693	0.631940	0.315970	+	0.948769i	$0.397670\pi$
$608$	3.21365	0.130331
$609$	0	0
$610$	−0.155938	−0.00631373
$611$	−17.8238	−0.721075
$612$	−11.7632	−0.475500
$613$	−16.8753	−0.681587	−0.340794	−	0.940138i	$-0.610696\pi$
$613$	−16.8753	−0.681587	−0.340794	+	0.940138i	$0.610696\pi$
$614$	1.73763	0.0701252
$615$	−0.562834	−0.0226956
$616$	0	0
$617$	41.0892	1.65419	0.827093	−	0.562064i	$-0.189993\pi$
$617$	41.0892	1.65419	0.827093	+	0.562064i	$0.189993\pi$
$618$	−2.12113	−0.0853243
$619$	16.8195	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$619$	16.8195	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$620$	1.20352	0.0483346
$621$	2.06188	0.0827402
$622$	−0.0765914	−0.00307103
$623$	0	0
$624$	−11.5042	−0.460537
$625$	20.3486	0.813945
$626$	−2.49101	−0.0995609
$627$	8.43981	0.337054
$628$	12.2290	0.487992
$629$	−48.9338	−1.95112
$630$	0	0
$631$	−26.8489	−1.06884	−0.534418	−	0.845220i	$-0.679469\pi$
$631$	−26.8489	−1.06884	−0.534418	+	0.845220i	$0.679469\pi$
$632$	6.64423	0.264293
$633$	−6.41632	−0.255026
$634$	2.13766	0.0848974
$635$	−11.2413	−0.446096
$636$	−16.1683	−0.641114
$637$	0	0
$638$	3.71365	0.147025
$639$	2.62791	0.103959
$640$	2.07554	0.0820428
$641$	11.2690	0.445098	0.222549	−	0.974922i	$-0.428562\pi$
$641$	11.2690	0.445098	0.222549	+	0.974922i	$0.428562\pi$
$642$	−1.02182	−0.0403282
$643$	33.0356	1.30280	0.651399	−	0.758735i	$-0.274182\pi$
$643$	33.0356	1.30280	0.651399	+	0.758735i	$0.274182\pi$
$644$	0	0
$645$	0.122924	0.00484011
$646$	1.60066	0.0629773
$647$	−24.5228	−0.964089	−0.482045	−	0.876147i	$-0.660106\pi$
$647$	−24.5228	−0.964089	−0.482045	+	0.876147i	$0.660106\pi$
$648$	0.467371	0.0183601
$649$	50.9732	2.00087
$650$	−1.61237	−0.0632425
$651$	0	0
$652$	45.6150	1.78642
$653$	−3.68476	−0.144196	−0.0720979	−	0.997398i	$-0.522969\pi$
$653$	−3.68476	−0.144196	−0.0720979	+	0.997398i	$0.522969\pi$
$654$	0.663379	0.0259402
$655$	−4.58562	−0.179175
$656$	−3.91771	−0.152961
$657$	−12.0095	−0.468536
$658$	0	0
$659$	6.37310	0.248261	0.124130	−	0.992266i	$-0.460386\pi$
$659$	6.37310	0.248261	0.124130	+	0.992266i	$0.460386\pi$
$660$	4.09294	0.159318
$661$	15.0507	0.585406	0.292703	−	0.956203i	$-0.405445\pi$
$661$	15.0507	0.585406	0.292703	+	0.956203i	$0.405445\pi$
$662$	0.319686	0.0124249
$663$	−17.3907	−0.675398
$664$	−4.65213	−0.180538
$665$	0	0
$666$	0.968756	0.0375385
$667$	17.8380	0.690690
$668$	−37.8446	−1.46425
$669$	−16.9305	−0.654571
$670$	0.505110	0.0195141
$671$	−8.65158	−0.333991
$672$	0	0
$673$	−7.51222	−0.289575	−0.144787	−	0.989463i	$-0.546250\pi$
$673$	−7.51222	−0.289575	−0.144787	+	0.989463i	$0.546250\pi$
$674$	2.13198	0.0821208
$675$	−4.68322	−0.180257
$676$	8.69418	0.334391
$677$	29.4270	1.13097	0.565486	−	0.824758i	$-0.308689\pi$
$677$	29.4270	1.13097	0.565486	+	0.824758i	$0.308689\pi$
$678$	1.60830	0.0617665
$679$	0	0
$680$	1.55788	0.0597419
$681$	26.1387	1.00164
$682$	−0.462123	−0.0176956
$683$	−11.3180	−0.433071	−0.216535	−	0.976275i	$-0.569476\pi$
$683$	−11.3180	−0.433071	−0.216535	+	0.976275i	$0.569476\pi$
$684$	4.57874	0.175073
$685$	−0.0121633	−0.000464735 0
$686$	0	0
$687$	27.9844	1.06767
$688$	0.855634	0.0326207
$689$	−23.9031	−0.910635
$690$	−0.136063	−0.00517982
$691$	−38.0471	−1.44738	−0.723690	−	0.690126i	$-0.757555\pi$
$691$	−38.0471	−1.44738	−0.723690	+	0.690126i	$0.757555\pi$
$692$	33.2761	1.26497
$693$	0	0
$694$	0.628500	0.0238575
$695$	−9.78944	−0.371335
$696$	4.04338	0.153264
$697$	−5.92232	−0.224324
$698$	−0.896240	−0.0339232
$699$	13.6927	0.517905
$700$	0	0
$701$	46.1894	1.74455	0.872275	−	0.489016i	$-0.162644\pi$
$701$	46.1894	1.74455	0.872275	+	0.489016i	$0.162644\pi$
$702$	0.344288	0.0129943
$703$	19.0471	0.718376
$704$	28.0885	1.05862
$705$	3.41630	0.128665
$706$	0.184612	0.00694797
$707$	0	0
$708$	27.6539	1.03930
$709$	−12.5843	−0.472611	−0.236306	−	0.971679i	$-0.575937\pi$
$709$	−12.5843	−0.472611	−0.236306	+	0.971679i	$0.575937\pi$
$710$	−0.173415	−0.00650816
$711$	14.2162	0.533149
$712$	−2.68282	−0.100543
$713$	−2.21974	−0.0831299
$714$	0	0
$715$	6.05098	0.226294
$716$	26.3565	0.984990
$717$	18.4282	0.688214
$718$	1.68486	0.0628785
$719$	−35.1814	−1.31204	−0.656022	−	0.754742i	$-0.727762\pi$
$719$	−35.1814	−1.31204	−0.656022	+	0.754742i	$0.727762\pi$
$720$	2.20502	0.0821762
$721$	0	0
$722$	1.60462	0.0597178
$723$	11.8493	0.440679
$724$	49.7718	1.84976
$725$	−40.5161	−1.50473
$726$	−0.281886	−0.0104618
$727$	−9.45535	−0.350679	−0.175340	−	0.984508i	$-0.556102\pi$
$727$	−9.45535	−0.350679	−0.175340	+	0.984508i	$0.556102\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.792505	0.0293319
$731$	1.29344	0.0478397
$732$	−4.69363	−0.173482
$733$	−4.27363	−0.157850	−0.0789251	−	0.996881i	$-0.525149\pi$
$733$	−4.27363	−0.157850	−0.0789251	+	0.996881i	$0.525149\pi$
$734$	2.43867	0.0900128
$735$	0	0
$736$	−2.87441	−0.105952
$737$	28.0240	1.03228
$738$	0.117246	0.00431587
$739$	2.93000	0.107782	0.0538909	−	0.998547i	$-0.482838\pi$
$739$	2.93000	0.107782	0.0538909	+	0.998547i	$0.482838\pi$
$740$	9.23703	0.339560
$741$	6.76919	0.248672
$742$	0	0
$743$	31.8902	1.16994	0.584969	−	0.811055i	$-0.301106\pi$
$743$	31.8902	1.16994	0.584969	+	0.811055i	$0.301106\pi$
$744$	−0.503154	−0.0184465
$745$	−3.25437	−0.119231
$746$	0.634598	0.0232343
$747$	−9.95382	−0.364191
$748$	43.0673	1.57470
$749$	0	0
$750$	0.638994	0.0233327
$751$	45.3814	1.65599	0.827996	−	0.560734i	$-0.189481\pi$
$751$	45.3814	1.65599	0.827996	+	0.560734i	$0.189481\pi$
$752$	23.7798	0.867160
$753$	6.43236	0.234408
$754$	2.97855	0.108472
$755$	−0.486994	−0.0177235
$756$	0	0
$757$	48.7265	1.77100	0.885498	−	0.464643i	$-0.153817\pi$
$757$	48.7265	1.77100	0.885498	+	0.464643i	$0.153817\pi$
$758$	−0.533325	−0.0193712
$759$	−7.54890	−0.274008
$760$	−0.606392	−0.0219961
$761$	21.9603	0.796059	0.398030	−	0.917373i	$-0.369694\pi$
$761$	21.9603	0.796059	0.398030	+	0.917373i	$0.369694\pi$
$762$	2.34171	0.0848310
$763$	0	0
$764$	−37.5415	−1.35820
$765$	3.33328	0.120515
$766$	1.39046	0.0502394
$767$	40.8833	1.47621
$768$	14.9116	0.538075
$769$	43.9020	1.58315	0.791573	−	0.611075i	$-0.209263\pi$
$769$	43.9020	1.58315	0.791573	+	0.611075i	$0.209263\pi$
$770$	0	0
$771$	−2.51802	−0.0906843
$772$	12.4232	0.447122
$773$	−26.9298	−0.968598	−0.484299	−	0.874903i	$-0.660925\pi$
$773$	−26.9298	−0.968598	−0.484299	+	0.874903i	$0.660925\pi$
$774$	−0.0256066	−0.000920411 0
$775$	5.04178	0.181106
$776$	2.32681	0.0835276
$777$	0	0
$778$	2.11862	0.0759563
$779$	2.30522	0.0825930
$780$	3.28276	0.117542
$781$	−9.62125	−0.344276
$782$	−1.43170	−0.0511974
$783$	8.65134	0.309174
$784$	0	0
$785$	−3.46527	−0.123681
$786$	0.955246	0.0340725
$787$	33.3392	1.18841	0.594207	−	0.804312i	$-0.297466\pi$
$787$	33.3392	1.18841	0.594207	+	0.804312i	$0.297466\pi$
$788$	37.1252	1.32253
$789$	−12.1289	−0.431801
$790$	−0.938123	−0.0333769
$791$	0	0
$792$	−1.71113	−0.0608023
$793$	−6.93904	−0.246412
$794$	−2.97927	−0.105730
$795$	4.58152	0.162490
$796$	−38.0594	−1.34898
$797$	−45.5433	−1.61323	−0.806614	−	0.591078i	$-0.798702\pi$
$797$	−45.5433	−1.61323	−0.806614	+	0.591078i	$0.798702\pi$
$798$	0	0
$799$	35.9474	1.27173
$800$	6.52876	0.230827
$801$	−5.74023	−0.202821
$802$	2.16304	0.0763795
$803$	43.9690	1.55163
$804$	15.2035	0.536187
$805$	0	0
$806$	−0.370647	−0.0130555
$807$	−21.6039	−0.760492
$808$	−2.59300	−0.0912216
$809$	36.3103	1.27660	0.638301	−	0.769787i	$-0.279637\pi$
$809$	36.3103	1.27660	0.638301	+	0.769787i	$0.279637\pi$
$810$	−0.0659898	−0.00231864
$811$	−47.3078	−1.66120	−0.830600	−	0.556869i	$-0.812003\pi$
$811$	−47.3078	−1.66120	−0.830600	+	0.556869i	$0.812003\pi$
$812$	0	0
$813$	5.58625	0.195918
$814$	−3.54679	−0.124315
$815$	−12.9257	−0.452766
$816$	23.2019	0.812230
$817$	−0.503463	−0.0176139
$818$	3.77411	0.131959
$819$	0	0
$820$	1.11793	0.0390398
$821$	−50.6804	−1.76876	−0.884378	−	0.466771i	$-0.845417\pi$
$821$	−50.6804	−1.76876	−0.884378	+	0.466771i	$0.845417\pi$
$822$	0.00253377	8.83755e−5 0
$823$	−11.9786	−0.417549	−0.208775	−	0.977964i	$-0.566947\pi$
$823$	−11.9786	−0.417549	−0.208775	+	0.977964i	$0.566947\pi$
$824$	8.45536	0.294556
$825$	17.1461	0.596951
$826$	0	0
$827$	−7.15241	−0.248714	−0.124357	−	0.992238i	$-0.539687\pi$
$827$	−7.15241	−0.248714	−0.124357	+	0.992238i	$0.539687\pi$
$828$	−4.09541	−0.142325
$829$	−15.4782	−0.537581	−0.268790	−	0.963199i	$-0.586624\pi$
$829$	−15.4782	−0.537581	−0.268790	+	0.963199i	$0.586624\pi$
$830$	0.656851	0.0227996
$831$	−1.62473	−0.0563612
$832$	22.5285	0.781034
$833$	0	0
$834$	2.03927	0.0706141
$835$	10.7238	0.371112
$836$	−16.7636	−0.579781
$837$	−1.07656	−0.0372115
$838$	−3.95284	−0.136549
$839$	−19.3804	−0.669085	−0.334543	−	0.942381i	$-0.608582\pi$
$839$	−19.3804	−0.669085	−0.334543	+	0.942381i	$0.608582\pi$
$840$	0	0
$841$	45.8457	1.58088
$842$	−0.755675	−0.0260423
$843$	−29.4166	−1.01316
$844$	12.7444	0.438682
$845$	−2.46362	−0.0847511
$846$	−0.711660	−0.0244674
$847$	0	0
$848$	31.8905	1.09512
$849$	−11.5265	−0.395589
$850$	3.25187	0.111538
$851$	−17.0365	−0.584004
$852$	−5.21970	−0.178824
$853$	−23.8980	−0.818252	−0.409126	−	0.912478i	$-0.634166\pi$
$853$	−23.8980	−0.818252	−0.409126	+	0.912478i	$0.634166\pi$
$854$	0	0
$855$	−1.29745	−0.0443720
$856$	4.07325	0.139221
$857$	10.3691	0.354201	0.177101	−	0.984193i	$-0.443328\pi$
$857$	10.3691	0.354201	0.177101	+	0.984193i	$0.443328\pi$
$858$	−1.26050	−0.0430327
$859$	41.2320	1.40682	0.703409	−	0.710786i	$-0.251660\pi$
$859$	41.2320	1.40682	0.703409	+	0.710786i	$0.251660\pi$
$860$	−0.244158	−0.00832570
$861$	0	0
$862$	1.40042	0.0476984
$863$	39.1252	1.33184	0.665919	−	0.746024i	$-0.268040\pi$
$863$	39.1252	1.33184	0.665919	+	0.746024i	$0.268040\pi$
$864$	−1.39408	−0.0474274
$865$	−9.42927	−0.320605
$866$	−3.03267	−0.103054
$867$	18.0738	0.613819
$868$	0	0
$869$	−52.0480	−1.76561
$870$	−0.570900	−0.0193553
$871$	22.4768	0.761597
$872$	−2.64440	−0.0895505
$873$	4.97851	0.168497
$874$	0.557277	0.0188502
$875$	0	0
$876$	23.8539	0.805950
$877$	−22.5131	−0.760212	−0.380106	−	0.924943i	$-0.624113\pi$
$877$	−22.5131	−0.760212	−0.380106	+	0.924943i	$0.624113\pi$
$878$	−1.62638	−0.0548877
$879$	15.5546	0.524642
$880$	−8.07297	−0.272140
$881$	−5.87889	−0.198065	−0.0990324	−	0.995084i	$-0.531575\pi$
$881$	−5.87889	−0.198065	−0.0990324	+	0.995084i	$0.531575\pi$
$882$	0	0
$883$	15.3483	0.516512	0.258256	−	0.966077i	$-0.416852\pi$
$883$	15.3483	0.516512	0.258256	+	0.966077i	$0.416852\pi$
$884$	34.5423	1.16178
$885$	−7.83612	−0.263408
$886$	1.30633	0.0438870
$887$	2.26580	0.0760781	0.0380390	−	0.999276i	$-0.487889\pi$
$887$	2.26580	0.0760781	0.0380390	+	0.999276i	$0.487889\pi$
$888$	−3.86171	−0.129590
$889$	0	0
$890$	0.378797	0.0126973
$891$	−3.66118	−0.122654
$892$	33.6283	1.12596
$893$	−13.9923	−0.468233
$894$	0.677929	0.0226733
$895$	−7.46851	−0.249645
$896$	0	0
$897$	−6.05463	−0.202158
$898$	4.11733	0.137397
$899$	−9.31371	−0.310630
$900$	9.30206	0.310069
$901$	48.2082	1.60605
$902$	−0.429257	−0.0142927
$903$	0	0
$904$	−6.41111	−0.213230
$905$	−14.1036	−0.468818
$906$	0.101447	0.00337036
$907$	54.1223	1.79710	0.898551	−	0.438869i	$-0.144621\pi$
$907$	54.1223	1.79710	0.898551	+	0.438869i	$0.144621\pi$
$908$	−51.9182	−1.72296
$909$	−5.54806	−0.184018
$910$	0	0
$911$	−8.33108	−0.276021	−0.138011	−	0.990431i	$-0.544071\pi$
$911$	−8.33108	−0.276021	−0.138011	+	0.990431i	$0.544071\pi$
$912$	−9.03117	−0.299052
$913$	36.4427	1.20608
$914$	4.45886	0.147486
$915$	1.33001	0.0439687
$916$	−55.5841	−1.83655
$917$	0	0
$918$	−0.694366	−0.0229175
$919$	38.8312	1.28092	0.640461	−	0.767991i	$-0.278743\pi$
$919$	38.8312	1.28092	0.640461	+	0.767991i	$0.278743\pi$
$920$	0.542381	0.0178818
$921$	−14.8205	−0.488351
$922$	0.625117	0.0205871
$923$	−7.71677	−0.254001
$924$	0	0
$925$	38.6956	1.27231
$926$	−0.872639	−0.0286767
$927$	18.0913	0.594197
$928$	−12.0606	−0.395909
$929$	22.7591	0.746700	0.373350	−	0.927690i	$-0.378209\pi$
$929$	22.7591	0.746700	0.373350	+	0.927690i	$0.378209\pi$
$930$	0.0710422	0.00232956
$931$	0	0
$932$	−27.1971	−0.890871
$933$	0.653256	0.0213866
$934$	1.25423	0.0410398
$935$	−12.2037	−0.399105
$936$	−1.37242	−0.0448589
$937$	−6.77546	−0.221345	−0.110672	−	0.993857i	$-0.535300\pi$
$937$	−6.77546	−0.221345	−0.110672	+	0.993857i	$0.535300\pi$
$938$	0	0
$939$	21.2461	0.693340
$940$	−6.78564	−0.221323
$941$	49.2273	1.60476	0.802381	−	0.596812i	$-0.203566\pi$
$941$	49.2273	1.60476	0.802381	+	0.596812i	$0.203566\pi$
$942$	0.721862	0.0235195
$943$	−2.06188	−0.0671439
$944$	−54.5448	−1.77528
$945$	0	0
$946$	0.0937504	0.00304809
$947$	21.2077	0.689158	0.344579	−	0.938757i	$-0.388022\pi$
$947$	21.2077	0.689158	0.344579	+	0.938757i	$0.388022\pi$
$948$	−28.2370	−0.917094
$949$	35.2655	1.14477
$950$	−1.26576	−0.0410668
$951$	−18.2323	−0.591224
$952$	0	0
$953$	28.0076	0.907256	0.453628	−	0.891191i	$-0.350130\pi$
$953$	28.0076	0.907256	0.453628	+	0.891191i	$0.350130\pi$
$954$	−0.954390	−0.0308995
$955$	10.6379	0.344235
$956$	−36.6031	−1.18383
$957$	−31.6741	−1.02388
$958$	3.03112	0.0979310
$959$	0	0
$960$	−4.31804	−0.139364
$961$	−29.8410	−0.962613
$962$	−2.84472	−0.0917174
$963$	8.71524	0.280845
$964$	−23.5356	−0.758032
$965$	−3.52031	−0.113323
$966$	0	0
$967$	31.2198	1.00396	0.501981	−	0.864878i	$-0.332605\pi$
$967$	31.2198	1.00396	0.501981	+	0.864878i	$0.332605\pi$
$968$	1.12367	0.0361161
$969$	−13.6522	−0.438573
$970$	−0.328531	−0.0105485
$971$	24.3846	0.782540	0.391270	−	0.920276i	$-0.372036\pi$
$971$	24.3846	0.782540	0.391270	+	0.920276i	$0.372036\pi$
$972$	−1.98625	−0.0637091
$973$	0	0
$974$	−4.11366	−0.131810
$975$	13.7521	0.440420
$976$	9.25778	0.296334
$977$	9.38073	0.300116	0.150058	−	0.988677i	$-0.452054\pi$
$977$	9.38073	0.300116	0.150058	+	0.988677i	$0.452054\pi$
$978$	2.69259	0.0860994
$979$	21.0160	0.671675
$980$	0	0
$981$	−5.65802	−0.180647
$982$	−0.933980	−0.0298045
$983$	−1.13489	−0.0361975	−0.0180987	−	0.999836i	$-0.505761\pi$
$983$	−1.13489	−0.0361975	−0.0180987	+	0.999836i	$0.505761\pi$
$984$	−0.467371	−0.0148992
$985$	−10.5200	−0.335194
$986$	−6.00719	−0.191308
$987$	0	0
$988$	−13.4453	−0.427753
$989$	0.450317	0.0143192
$990$	0.241600	0.00767857
$991$	−4.95653	−0.157449	−0.0787246	−	0.996896i	$-0.525085\pi$
$991$	−4.95653	−0.157449	−0.0787246	+	0.996896i	$0.525085\pi$
$992$	1.50081	0.0476508
$993$	−2.72663	−0.0865270
$994$	0	0
$995$	10.7847	0.341897
$996$	19.7708	0.626462
$997$	−49.2547	−1.55991	−0.779957	−	0.625834i	$-0.784759\pi$
$997$	−49.2547	−1.55991	−0.779957	+	0.625834i	$0.784759\pi$
$998$	−3.14288	−0.0994862
$999$	−8.26262	−0.261418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.ba.1.5	yes	8
7.6	odd	2		6027.2.a.z.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.z.1.5	✓	8	7.6	odd	2
6027.2.a.ba.1.5	yes	8	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-0.117246 q^{2} +1.00000 q^{3} -1.98625 q^{4} +0.562834 q^{5} -0.117246 q^{6} +0.467371 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.117246 q^{2} +1.00000 q^{3} -1.98625 q^{4} +0.562834 q^{5} -0.117246 q^{6} +0.467371 q^{8} +1.00000 q^{9} -0.0659898 q^{10} -3.66118 q^{11} -1.98625 q^{12} -2.93646 q^{13} +0.562834 q^{15} +3.91771 q^{16} +5.92232 q^{17} -0.117246 q^{18} -2.30522 q^{19} -1.11793 q^{20} +0.429257 q^{22} +2.06188 q^{23} +0.467371 q^{24} -4.68322 q^{25} +0.344288 q^{26} +1.00000 q^{27} +8.65134 q^{29} -0.0659898 q^{30} -1.07656 q^{31} -1.39408 q^{32} -3.66118 q^{33} -0.694366 q^{34} -1.98625 q^{36} -8.26262 q^{37} +0.270277 q^{38} -2.93646 q^{39} +0.263052 q^{40} -1.00000 q^{41} +0.218401 q^{43} +7.27203 q^{44} +0.562834 q^{45} -0.241746 q^{46} +6.06982 q^{47} +3.91771 q^{48} +0.549087 q^{50} +5.92232 q^{51} +5.83256 q^{52} +8.14009 q^{53} -0.117246 q^{54} -2.06063 q^{55} -2.30522 q^{57} -1.01433 q^{58} -13.9226 q^{59} -1.11793 q^{60} +2.36306 q^{61} +0.126222 q^{62} -7.67197 q^{64} -1.65274 q^{65} +0.429257 q^{66} -7.65437 q^{67} -11.7632 q^{68} +2.06188 q^{69} +2.62791 q^{71} +0.467371 q^{72} -12.0095 q^{73} +0.968756 q^{74} -4.68322 q^{75} +4.57874 q^{76} +0.344288 q^{78} +14.2162 q^{79} +2.20502 q^{80} +1.00000 q^{81} +0.117246 q^{82} -9.95382 q^{83} +3.33328 q^{85} -0.0256066 q^{86} +8.65134 q^{87} -1.71113 q^{88} -5.74023 q^{89} -0.0659898 q^{90} -4.09541 q^{92} -1.07656 q^{93} -0.711660 q^{94} -1.29745 q^{95} -1.39408 q^{96} +4.97851 q^{97} -3.66118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 4 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{18} - 6 q^{19} - 4 q^{20} - 14 q^{22} - 12 q^{23} - 6 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 10 q^{31} - 4 q^{32} - 2 q^{33} - 4 q^{34} + 4 q^{36} - 20 q^{37} + 18 q^{38} - 4 q^{39} - 12 q^{40} - 8 q^{41} - 8 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 24 q^{47} - 22 q^{50} - 8 q^{51} + 30 q^{52} - 36 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 4 q^{60} + 22 q^{61} - 30 q^{62} - 24 q^{64} + 8 q^{65} - 14 q^{66} - 14 q^{67} - 38 q^{68} - 12 q^{69} - 10 q^{71} - 6 q^{72} + 12 q^{73} - 2 q^{74} - 4 q^{75} - 32 q^{76} - 4 q^{78} + 16 q^{79} + 14 q^{80} + 8 q^{81} + 2 q^{82} - 24 q^{83} - 44 q^{85} + 36 q^{86} - 4 q^{87} - 34 q^{88} - 2 q^{89} + 2 q^{90} - 48 q^{92} + 10 q^{93} + 34 q^{94} - 24 q^{95} - 4 q^{96} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 - 4 * q^13 - 2 * q^15 - 8 * q^17 - 2 * q^18 - 6 * q^19 - 4 * q^20 - 14 * q^22 - 12 * q^23 - 6 * q^24 - 4 * q^25 - 4 * q^26 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 10 * q^31 - 4 * q^32 - 2 * q^33 - 4 * q^34 + 4 * q^36 - 20 * q^37 + 18 * q^38 - 4 * q^39 - 12 * q^40 - 8 * q^41 - 8 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 24 * q^47 - 22 * q^50 - 8 * q^51 + 30 * q^52 - 36 * q^53 - 2 * q^54 - 4 * q^55 - 6 * q^57 + 14 * q^58 - 10 * q^59 - 4 * q^60 + 22 * q^61 - 30 * q^62 - 24 * q^64 + 8 * q^65 - 14 * q^66 - 14 * q^67 - 38 * q^68 - 12 * q^69 - 10 * q^71 - 6 * q^72 + 12 * q^73 - 2 * q^74 - 4 * q^75 - 32 * q^76 - 4 * q^78 + 16 * q^79 + 14 * q^80 + 8 * q^81 + 2 * q^82 - 24 * q^83 - 44 * q^85 + 36 * q^86 - 4 * q^87 - 34 * q^88 - 2 * q^89 + 2 * q^90 - 48 * q^92 + 10 * q^93 + 34 * q^94 - 24 * q^95 - 4 * q^96 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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