LMFDB - Embedded newform 8649.2.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 $ x^16 - 24x^14 + 220x^12 - 992x^10 + 2366x^8 - 2944x^6 + 1688x^4 - 288*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 961)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.46287$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21652 q^{2} -0.520070 q^{4} -1.24784 q^{5} -1.96096 q^{7} +3.06572 q^{8} +O(q^{10})$	$q-1.21652 q^{2} -0.520070 q^{4} -1.24784 q^{5} -1.96096 q^{7} +3.06572 q^{8} +1.51803 q^{10} +1.16082 q^{11} -4.40139 q^{13} +2.38556 q^{14} -2.68939 q^{16} +1.37872 q^{17} -7.56437 q^{19} +0.648966 q^{20} -1.41217 q^{22} -8.73520 q^{23} -3.44289 q^{25} +5.35440 q^{26} +1.01984 q^{28} -6.85312 q^{29} -2.85975 q^{32} -1.67725 q^{34} +2.44697 q^{35} +7.41371 q^{37} +9.20224 q^{38} -3.82554 q^{40} -4.31003 q^{41} -5.62246 q^{43} -0.603710 q^{44} +10.6266 q^{46} +4.48355 q^{47} -3.15462 q^{49} +4.18836 q^{50} +2.28903 q^{52} -1.29761 q^{53} -1.44853 q^{55} -6.01178 q^{56} +8.33699 q^{58} -0.539311 q^{59} -7.98284 q^{61} +8.85772 q^{64} +5.49224 q^{65} -4.71720 q^{67} -0.717032 q^{68} -2.97680 q^{70} -2.92870 q^{71} -10.9567 q^{73} -9.01895 q^{74} +3.93400 q^{76} -2.27633 q^{77} +7.78004 q^{79} +3.35593 q^{80} +5.24326 q^{82} -2.80483 q^{83} -1.72043 q^{85} +6.83986 q^{86} +3.55877 q^{88} +16.8672 q^{89} +8.63097 q^{91} +4.54292 q^{92} -5.45434 q^{94} +9.43915 q^{95} +4.95119 q^{97} +3.83767 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7}+O(q^{10}) $ 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7	$ 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56} + 64 q^{59} - 16 q^{64} + 16 q^{67} + 88 q^{70} + 48 q^{71} + 40 q^{76} + 40 q^{80} + 88 q^{82} - 32 q^{94} + 48 q^{95} - 16 q^{98}+O(q^{100}) $ 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * q^98

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 $

$2$	−1.21652	−0.860212	−0.430106	−	0.902778i	$-0.641524\pi$
$2$	−1.21652	−0.860212	−0.430106	+	0.902778i	$0.641524\pi$
$3$	0	0
$3$	0	0
$4$	−0.520070	−0.260035
$5$	−1.24784	−0.558052	−0.279026	−	0.960284i	$-0.590012\pi$
$5$	−1.24784	−0.558052	−0.279026	+	0.960284i	$0.590012\pi$
$6$	0	0
$7$	−1.96096	−0.741175	−0.370587	−	0.928798i	$-0.620844\pi$
$7$	−1.96096	−0.741175	−0.370587	+	0.928798i	$0.620844\pi$
$8$	3.06572	1.08390
$9$	0	0
$10$	1.51803	0.480043
$11$	1.16082	0.350002	0.175001	−	0.984568i	$-0.444007\pi$
$11$	1.16082	0.350002	0.175001	+	0.984568i	$0.444007\pi$
$12$	0	0
$13$	−4.40139	−1.22073	−0.610363	−	0.792122i	$-0.708976\pi$
$13$	−4.40139	−1.22073	−0.610363	+	0.792122i	$0.708976\pi$
$14$	2.38556	0.637568
$15$	0	0
$16$	−2.68939	−0.672347
$17$	1.37872	0.334389	0.167195	−	0.985924i	$-0.446529\pi$
$17$	1.37872	0.334389	0.167195	+	0.985924i	$0.446529\pi$
$18$	0	0
$19$	−7.56437	−1.73539	−0.867693	−	0.497101i	$-0.834398\pi$
$19$	−7.56437	−1.73539	−0.867693	+	0.497101i	$0.834398\pi$
$20$	0.648966	0.145113
$21$	0	0
$22$	−1.41217	−0.301076
$23$	−8.73520	−1.82142	−0.910708	−	0.413051i	$-0.864463\pi$
$23$	−8.73520	−1.82142	−0.910708	+	0.413051i	$0.864463\pi$
$24$	0	0
$25$	−3.44289	−0.688578
$26$	5.35440	1.05008
$27$	0	0
$28$	1.01984	0.192731
$29$	−6.85312	−1.27259	−0.636296	−	0.771445i	$-0.719534\pi$
$29$	−6.85312	−1.27259	−0.636296	+	0.771445i	$0.719534\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−2.85975	−0.505536
$33$	0	0
$34$	−1.67725	−0.287646
$35$	2.44697	0.413614
$36$	0	0
$37$	7.41371	1.21881	0.609403	−	0.792861i	$-0.291409\pi$
$37$	7.41371	1.21881	0.609403	+	0.792861i	$0.291409\pi$
$38$	9.20224	1.49280
$39$	0	0
$40$	−3.82554	−0.604871
$41$	−4.31003	−0.673114	−0.336557	−	0.941663i	$-0.609262\pi$
$41$	−4.31003	−0.673114	−0.336557	+	0.941663i	$0.609262\pi$
$42$	0	0
$43$	−5.62246	−0.857417	−0.428709	−	0.903443i	$-0.641031\pi$
$43$	−5.62246	−0.857417	−0.428709	+	0.903443i	$0.641031\pi$
$44$	−0.603710	−0.0910127
$45$	0	0
$46$	10.6266	1.56680
$47$	4.48355	0.653993	0.326996	−	0.945026i	$-0.393963\pi$
$47$	4.48355	0.653993	0.326996	+	0.945026i	$0.393963\pi$
$48$	0	0
$49$	−3.15462	−0.450660
$50$	4.18836	0.592323
$51$	0	0
$52$	2.28903	0.317431
$53$	−1.29761	−0.178241	−0.0891205	−	0.996021i	$-0.528406\pi$
$53$	−1.29761	−0.178241	−0.0891205	+	0.996021i	$0.528406\pi$
$54$	0	0
$55$	−1.44853	−0.195319
$56$	−6.01178	−0.803357
$57$	0	0
$58$	8.33699	1.09470
$59$	−0.539311	−0.0702123	−0.0351062	−	0.999384i	$-0.511177\pi$
$59$	−0.539311	−0.0702123	−0.0351062	+	0.999384i	$0.511177\pi$
$60$	0	0
$61$	−7.98284	−1.02210	−0.511049	−	0.859551i	$-0.670743\pi$
$61$	−7.98284	−1.02210	−0.511049	+	0.859551i	$0.670743\pi$
$62$	0	0
$63$	0	0
$64$	8.85772	1.10722
$65$	5.49224	0.681229
$66$	0	0
$67$	−4.71720	−0.576297	−0.288149	−	0.957586i	$-0.593040\pi$
$67$	−4.71720	−0.576297	−0.288149	+	0.957586i	$0.593040\pi$
$68$	−0.717032	−0.0869529
$69$	0	0
$70$	−2.97680	−0.355796
$71$	−2.92870	−0.347573	−0.173787	−	0.984783i	$-0.555600\pi$
$71$	−2.92870	−0.347573	−0.173787	+	0.984783i	$0.555600\pi$
$72$	0	0
$73$	−10.9567	−1.28238	−0.641192	−	0.767380i	$-0.721560\pi$
$73$	−10.9567	−1.28238	−0.641192	+	0.767380i	$0.721560\pi$
$74$	−9.01895	−1.04843
$75$	0	0
$76$	3.93400	0.451261
$77$	−2.27633	−0.259412
$78$	0	0
$79$	7.78004	0.875322	0.437661	−	0.899140i	$-0.355807\pi$
$79$	7.78004	0.875322	0.437661	+	0.899140i	$0.355807\pi$
$80$	3.35593	0.375205
$81$	0	0
$82$	5.24326	0.579021
$83$	−2.80483	−0.307870	−0.153935	−	0.988081i	$-0.549195\pi$
$83$	−2.80483	−0.307870	−0.153935	+	0.988081i	$0.549195\pi$
$84$	0	0
$85$	−1.72043	−0.186607
$86$	6.83986	0.737561
$87$	0	0
$88$	3.55877	0.379366
$89$	16.8672	1.78792	0.893959	−	0.448149i	$-0.147917\pi$
$89$	16.8672	1.78792	0.893959	+	0.448149i	$0.147917\pi$
$90$	0	0
$91$	8.63097	0.904771
$92$	4.54292	0.473632
$93$	0	0
$94$	−5.45434	−0.562573
$95$	9.43915	0.968436
$96$	0	0
$97$	4.95119	0.502717	0.251358	−	0.967894i	$-0.419123\pi$
$97$	4.95119	0.502717	0.251358	+	0.967894i	$0.419123\pi$
$98$	3.83767	0.387663
$99$	0	0
$100$	1.79054	0.179054
$101$	−15.9564	−1.58772	−0.793859	−	0.608102i	$-0.791931\pi$
$101$	−15.9564	−1.58772	−0.793859	+	0.608102i	$0.791931\pi$
$102$	0	0
$103$	−11.8775	−1.17033	−0.585163	−	0.810916i	$-0.698969\pi$
$103$	−11.8775	−1.17033	−0.585163	+	0.810916i	$0.698969\pi$
$104$	−13.4935	−1.32314
$105$	0	0
$106$	1.57858	0.153325
$107$	6.83624	0.660885	0.330442	−	0.943826i	$-0.392802\pi$
$107$	6.83624	0.660885	0.330442	+	0.943826i	$0.392802\pi$
$108$	0	0
$109$	−4.66302	−0.446636	−0.223318	−	0.974746i	$-0.571689\pi$
$109$	−4.66302	−0.446636	−0.223318	+	0.974746i	$0.571689\pi$
$110$	1.76217	0.168016
$111$	0	0
$112$	5.27379	0.498326
$113$	6.54544	0.615743	0.307872	−	0.951428i	$-0.400383\pi$
$113$	6.54544	0.615743	0.307872	+	0.951428i	$0.400383\pi$
$114$	0	0
$115$	10.9002	1.01645
$116$	3.56410	0.330919
$117$	0	0
$118$	0.656085	0.0603975
$119$	−2.70362	−0.247841
$120$	0	0
$121$	−9.65249	−0.877499
$122$	9.71132	0.879221
$123$	0	0
$124$	0	0
$125$	10.5354	0.942315
$126$	0	0
$127$	−6.78442	−0.602020	−0.301010	−	0.953621i	$-0.597324\pi$
$127$	−6.78442	−0.602020	−0.301010	+	0.953621i	$0.597324\pi$
$128$	−5.05614	−0.446904
$129$	0	0
$130$	−6.68145	−0.586002
$131$	−3.34454	−0.292214	−0.146107	−	0.989269i	$-0.546674\pi$
$131$	−3.34454	−0.292214	−0.146107	+	0.989269i	$0.546674\pi$
$132$	0	0
$133$	14.8335	1.28622
$134$	5.73858	0.495738
$135$	0	0
$136$	4.22678	0.362444
$137$	−14.8814	−1.27140	−0.635702	−	0.771935i	$-0.719289\pi$
$137$	−14.8814	−1.27140	−0.635702	+	0.771935i	$0.719289\pi$
$138$	0	0
$139$	−17.1901	−1.45804	−0.729022	−	0.684490i	$-0.760025\pi$
$139$	−17.1901	−1.45804	−0.729022	+	0.684490i	$0.760025\pi$
$140$	−1.27260	−0.107554
$141$	0	0
$142$	3.56284	0.298987
$143$	−5.10924	−0.427256
$144$	0	0
$145$	8.55162	0.710173
$146$	13.3291	1.10312
$147$	0	0
$148$	−3.85565	−0.316932
$149$	−2.93570	−0.240502	−0.120251	−	0.992744i	$-0.538370\pi$
$149$	−2.93570	−0.240502	−0.120251	+	0.992744i	$0.538370\pi$
$150$	0	0
$151$	9.95326	0.809985	0.404992	−	0.914320i	$-0.367274\pi$
$151$	9.95326	0.809985	0.404992	+	0.914320i	$0.367274\pi$
$152$	−23.1903	−1.88098
$153$	0	0
$154$	2.76921	0.223150
$155$	0	0
$156$	0	0
$157$	−8.62837	−0.688619	−0.344309	−	0.938856i	$-0.611887\pi$
$157$	−8.62837	−0.688619	−0.344309	+	0.938856i	$0.611887\pi$
$158$	−9.46460	−0.752963
$159$	0	0
$160$	3.56851	0.282116
$161$	17.1294	1.34999
$162$	0	0
$163$	−8.74442	−0.684916	−0.342458	−	0.939533i	$-0.611259\pi$
$163$	−8.74442	−0.684916	−0.342458	+	0.939533i	$0.611259\pi$
$164$	2.24152	0.175033
$165$	0	0
$166$	3.41214	0.264833
$167$	10.5682	0.817792	0.408896	−	0.912581i	$-0.365914\pi$
$167$	10.5682	0.817792	0.408896	+	0.912581i	$0.365914\pi$
$168$	0	0
$169$	6.37224	0.490172
$170$	2.09294	0.160521
$171$	0	0
$172$	2.92407	0.222958
$173$	−15.5194	−1.17992	−0.589959	−	0.807434i	$-0.700856\pi$
$173$	−15.5194	−1.17992	−0.589959	+	0.807434i	$0.700856\pi$
$174$	0	0
$175$	6.75138	0.510356
$176$	−3.12191	−0.235322
$177$	0	0
$178$	−20.5193	−1.53799
$179$	−9.87783	−0.738304	−0.369152	−	0.929369i	$-0.620352\pi$
$179$	−9.87783	−0.738304	−0.369152	+	0.929369i	$0.620352\pi$
$180$	0	0
$181$	11.7736	0.875124	0.437562	−	0.899188i	$-0.355842\pi$
$181$	11.7736	0.875124	0.437562	+	0.899188i	$0.355842\pi$
$182$	−10.4998	−0.778295
$183$	0	0
$184$	−26.7797	−1.97423
$185$	−9.25114	−0.680157
$186$	0	0
$187$	1.60045	0.117037
$188$	−2.33176	−0.170061
$189$	0	0
$190$	−11.4829	−0.833060
$191$	−13.0470	−0.944049	−0.472025	−	0.881585i	$-0.656477\pi$
$191$	−13.0470	−0.944049	−0.472025	+	0.881585i	$0.656477\pi$
$192$	0	0
$193$	13.7402	0.989042	0.494521	−	0.869166i	$-0.335343\pi$
$193$	13.7402	0.989042	0.494521	+	0.869166i	$0.335343\pi$
$194$	−6.02324	−0.432443
$195$	0	0
$196$	1.64062	0.117187
$197$	−16.0248	−1.14172	−0.570860	−	0.821048i	$-0.693390\pi$
$197$	−16.0248	−1.14172	−0.570860	+	0.821048i	$0.693390\pi$
$198$	0	0
$199$	−15.4743	−1.09694	−0.548471	−	0.836169i	$-0.684790\pi$
$199$	−15.4743	−1.09694	−0.548471	+	0.836169i	$0.684790\pi$
$200$	−10.5549	−0.746348
$201$	0	0
$202$	19.4113	1.36577
$203$	13.4387	0.943214
$204$	0	0
$205$	5.37824	0.375633
$206$	14.4493	1.00673
$207$	0	0
$208$	11.8370	0.820751
$209$	−8.78090	−0.607388
$210$	0	0
$211$	−27.0029	−1.85895	−0.929477	−	0.368881i	$-0.879741\pi$
$211$	−27.0029	−1.85895	−0.929477	+	0.368881i	$0.879741\pi$
$212$	0.674850	0.0463489
$213$	0	0
$214$	−8.31645	−0.568501
$215$	7.01595	0.478484
$216$	0	0
$217$	0	0
$218$	5.67268	0.384202
$219$	0	0
$220$	0.753335	0.0507898
$221$	−6.06830	−0.408198
$222$	0	0
$223$	−7.96964	−0.533687	−0.266843	−	0.963740i	$-0.585981\pi$
$223$	−7.96964	−0.533687	−0.266843	+	0.963740i	$0.585981\pi$
$224$	5.60786	0.374691
$225$	0	0
$226$	−7.96268	−0.529670
$227$	12.8561	0.853287	0.426644	−	0.904420i	$-0.359696\pi$
$227$	12.8561	0.853287	0.426644	+	0.904420i	$0.359696\pi$
$228$	0	0
$229$	1.32400	0.0874922	0.0437461	−	0.999043i	$-0.486071\pi$
$229$	1.32400	0.0874922	0.0437461	+	0.999043i	$0.486071\pi$
$230$	−13.2603	−0.874359
$231$	0	0
$232$	−21.0098	−1.37936
$233$	−1.21897	−0.0798571	−0.0399286	−	0.999203i	$-0.512713\pi$
$233$	−1.21897	−0.0798571	−0.0399286	+	0.999203i	$0.512713\pi$
$234$	0	0
$235$	−5.59476	−0.364962
$236$	0.280480	0.0182577
$237$	0	0
$238$	3.28902	0.213196
$239$	22.3982	1.44882	0.724409	−	0.689370i	$-0.242113\pi$
$239$	22.3982	1.44882	0.724409	+	0.689370i	$0.242113\pi$
$240$	0	0
$241$	−0.748788	−0.0482337	−0.0241168	−	0.999709i	$-0.507677\pi$
$241$	−0.748788	−0.0482337	−0.0241168	+	0.999709i	$0.507677\pi$
$242$	11.7425	0.754835
$243$	0	0
$244$	4.15164	0.265781
$245$	3.93647	0.251492
$246$	0	0
$247$	33.2938	2.11843
$248$	0	0
$249$	0	0
$250$	−12.8166	−0.810591
$251$	−23.8240	−1.50376	−0.751880	−	0.659300i	$-0.770853\pi$
$251$	−23.8240	−1.50376	−0.751880	+	0.659300i	$0.770853\pi$
$252$	0	0
$253$	−10.1400	−0.637498
$254$	8.25341	0.517865
$255$	0	0
$256$	−11.5645	−0.722783
$257$	−16.6558	−1.03896	−0.519480	−	0.854482i	$-0.673874\pi$
$257$	−16.6558	−1.03896	−0.519480	+	0.854482i	$0.673874\pi$
$258$	0	0
$259$	−14.5380	−0.903348
$260$	−2.85635	−0.177143
$261$	0	0
$262$	4.06871	0.251366
$263$	15.3414	0.945988	0.472994	−	0.881066i	$-0.343173\pi$
$263$	15.3414	0.945988	0.472994	+	0.881066i	$0.343173\pi$
$264$	0	0
$265$	1.61922	0.0994678
$266$	−18.0453	−1.10643
$267$	0	0
$268$	2.45327	0.149857
$269$	1.91310	0.116644	0.0583218	−	0.998298i	$-0.481425\pi$
$269$	1.91310	0.116644	0.0583218	+	0.998298i	$0.481425\pi$
$270$	0	0
$271$	19.3875	1.17771	0.588854	−	0.808239i	$-0.299579\pi$
$271$	19.3875	1.17771	0.588854	+	0.808239i	$0.299579\pi$
$272$	−3.70792	−0.224826
$273$	0	0
$274$	18.1036	1.09368
$275$	−3.99659	−0.241003
$276$	0	0
$277$	−5.39483	−0.324144	−0.162072	−	0.986779i	$-0.551818\pi$
$277$	−5.39483	−0.324144	−0.162072	+	0.986779i	$0.551818\pi$
$278$	20.9122	1.25423
$279$	0	0
$280$	7.50175	0.448315
$281$	11.2977	0.673965	0.336983	−	0.941511i	$-0.390594\pi$
$281$	11.2977	0.673965	0.336983	+	0.941511i	$0.390594\pi$
$282$	0	0
$283$	3.92498	0.233316	0.116658	−	0.993172i	$-0.462782\pi$
$283$	3.92498	0.233316	0.116658	+	0.993172i	$0.462782\pi$
$284$	1.52313	0.0903811
$285$	0	0
$286$	6.21551	0.367531
$287$	8.45182	0.498895
$288$	0	0
$289$	−15.0991	−0.888184
$290$	−10.4032	−0.610900
$291$	0	0
$292$	5.69825	0.333465
$293$	10.6680	0.623230	0.311615	−	0.950208i	$-0.399130\pi$
$293$	10.6680	0.623230	0.311615	+	0.950208i	$0.399130\pi$
$294$	0	0
$295$	0.672976	0.0391822
$296$	22.7284	1.32106
$297$	0	0
$298$	3.57135	0.206883
$299$	38.4470	2.22345
$300$	0	0
$301$	11.0254	0.635496
$302$	−12.1084	−0.696759
$303$	0	0
$304$	20.3435	1.16678
$305$	9.96133	0.570384
$306$	0	0
$307$	11.9498	0.682011	0.341006	−	0.940061i	$-0.389232\pi$
$307$	11.9498	0.682011	0.341006	+	0.940061i	$0.389232\pi$
$308$	1.18385	0.0674563
$309$	0	0
$310$	0	0
$311$	21.8220	1.23741	0.618706	−	0.785623i	$-0.287657\pi$
$311$	21.8220	1.23741	0.618706	+	0.785623i	$0.287657\pi$
$312$	0	0
$313$	−5.41816	−0.306253	−0.153126	−	0.988207i	$-0.548934\pi$
$313$	−5.41816	−0.306253	−0.153126	+	0.988207i	$0.548934\pi$
$314$	10.4966	0.592358
$315$	0	0
$316$	−4.04616	−0.227614
$317$	11.5909	0.651008	0.325504	−	0.945541i	$-0.394466\pi$
$317$	11.5909	0.651008	0.325504	+	0.945541i	$0.394466\pi$
$318$	0	0
$319$	−7.95527	−0.445410
$320$	−11.0530	−0.617884
$321$	0	0
$322$	−20.8383	−1.16128
$323$	−10.4292	−0.580294
$324$	0	0
$325$	15.1535	0.840565
$326$	10.6378	0.589173
$327$	0	0
$328$	−13.2134	−0.729587
$329$	−8.79208	−0.484723
$330$	0	0
$331$	−6.43073	−0.353465	−0.176732	−	0.984259i	$-0.556553\pi$
$331$	−6.43073	−0.353465	−0.176732	+	0.984259i	$0.556553\pi$
$332$	1.45871	0.0800570
$333$	0	0
$334$	−12.8565	−0.703474
$335$	5.88632	0.321604
$336$	0	0
$337$	4.39764	0.239555	0.119777	−	0.992801i	$-0.461782\pi$
$337$	4.39764	0.239555	0.119777	+	0.992801i	$0.461782\pi$
$338$	−7.75198	−0.421652
$339$	0	0
$340$	0.894743	0.0485243
$341$	0	0
$342$	0	0
$343$	19.9128	1.07519
$344$	−17.2369	−0.929353
$345$	0	0
$346$	18.8797	1.01498
$347$	−14.0434	−0.753887	−0.376944	−	0.926236i	$-0.623025\pi$
$347$	−14.0434	−0.753887	−0.376944	+	0.926236i	$0.623025\pi$
$348$	0	0
$349$	−1.81526	−0.0971688	−0.0485844	−	0.998819i	$-0.515471\pi$
$349$	−1.81526	−0.0971688	−0.0485844	+	0.998819i	$0.515471\pi$
$350$	−8.21321	−0.439015
$351$	0	0
$352$	−3.31966	−0.176939
$353$	−16.5531	−0.881033	−0.440516	−	0.897745i	$-0.645205\pi$
$353$	−16.5531	−0.881033	−0.440516	+	0.897745i	$0.645205\pi$
$354$	0	0
$355$	3.65456	0.193964
$356$	−8.77211	−0.464921
$357$	0	0
$358$	12.0166	0.635098
$359$	9.60848	0.507116	0.253558	−	0.967320i	$-0.418399\pi$
$359$	9.60848	0.507116	0.253558	+	0.967320i	$0.418399\pi$
$360$	0	0
$361$	38.2197	2.01156
$362$	−14.3228	−0.752792
$363$	0	0
$364$	−4.48871	−0.235272
$365$	13.6722	0.715638
$366$	0	0
$367$	−13.3704	−0.697931	−0.348965	−	0.937136i	$-0.613467\pi$
$367$	−13.3704	−0.697931	−0.348965	+	0.937136i	$0.613467\pi$
$368$	23.4923	1.22462
$369$	0	0
$370$	11.2542	0.585080
$371$	2.54457	0.132108
$372$	0	0
$373$	1.93677	0.100282	0.0501412	−	0.998742i	$-0.484033\pi$
$373$	1.93677	0.100282	0.0501412	+	0.998742i	$0.484033\pi$
$374$	−1.94699	−0.100676
$375$	0	0
$376$	13.7453	0.708861
$377$	30.1633	1.55349
$378$	0	0
$379$	−7.23610	−0.371693	−0.185847	−	0.982579i	$-0.559503\pi$
$379$	−7.23610	−0.371693	−0.185847	+	0.982579i	$0.559503\pi$
$380$	−4.90902	−0.251827
$381$	0	0
$382$	15.8720	0.812083
$383$	−23.4390	−1.19768	−0.598839	−	0.800870i	$-0.704371\pi$
$383$	−23.4390	−1.19768	−0.598839	+	0.800870i	$0.704371\pi$
$384$	0	0
$385$	2.84051	0.144766
$386$	−16.7153	−0.850786
$387$	0	0
$388$	−2.57496	−0.130724
$389$	−12.5677	−0.637206	−0.318603	−	0.947888i	$-0.603214\pi$
$389$	−12.5677	−0.637206	−0.318603	+	0.947888i	$0.603214\pi$
$390$	0	0
$391$	−12.0434	−0.609062
$392$	−9.67120	−0.488469
$393$	0	0
$394$	19.4945	0.982121
$395$	−9.70826	−0.488476
$396$	0	0
$397$	19.7910	0.993284	0.496642	−	0.867955i	$-0.334566\pi$
$397$	19.7910	0.993284	0.496642	+	0.867955i	$0.334566\pi$
$398$	18.8248	0.943603
$399$	0	0
$400$	9.25926	0.462963
$401$	−14.6611	−0.732140	−0.366070	−	0.930587i	$-0.619297\pi$
$401$	−14.6611	−0.732140	−0.366070	+	0.930587i	$0.619297\pi$
$402$	0	0
$403$	0	0
$404$	8.29843	0.412862
$405$	0	0
$406$	−16.3485	−0.811364
$407$	8.60601	0.426584
$408$	0	0
$409$	24.4835	1.21063	0.605314	−	0.795987i	$-0.293047\pi$
$409$	24.4835	1.21063	0.605314	+	0.795987i	$0.293047\pi$
$410$	−6.54276	−0.323124
$411$	0	0
$412$	6.17714	0.304326
$413$	1.05757	0.0520396
$414$	0	0
$415$	3.49999	0.171808
$416$	12.5869	0.617122
$417$	0	0
$418$	10.6822	0.522482
$419$	27.9883	1.36732	0.683658	−	0.729802i	$-0.260388\pi$
$419$	27.9883	1.36732	0.683658	+	0.729802i	$0.260388\pi$
$420$	0	0
$421$	−18.9575	−0.923931	−0.461966	−	0.886898i	$-0.652856\pi$
$421$	−18.9575	−0.923931	−0.461966	+	0.886898i	$0.652856\pi$
$422$	32.8496	1.59909
$423$	0	0
$424$	−3.97813	−0.193195
$425$	−4.74679	−0.230253
$426$	0	0
$427$	15.6541	0.757553
$428$	−3.55532	−0.171853
$429$	0	0
$430$	−8.53507	−0.411598
$431$	11.0567	0.532580	0.266290	−	0.963893i	$-0.414202\pi$
$431$	11.0567	0.532580	0.266290	+	0.963893i	$0.414202\pi$
$432$	0	0
$433$	−23.4001	−1.12454	−0.562268	−	0.826955i	$-0.690071\pi$
$433$	−23.4001	−1.12454	−0.562268	+	0.826955i	$0.690071\pi$
$434$	0	0
$435$	0	0
$436$	2.42510	0.116141
$437$	66.0763	3.16086
$438$	0	0
$439$	−8.12280	−0.387680	−0.193840	−	0.981033i	$-0.562094\pi$
$439$	−8.12280	−0.387680	−0.193840	+	0.981033i	$0.562094\pi$
$440$	−4.44078	−0.211706
$441$	0	0
$442$	7.38223	0.351137
$443$	22.5347	1.07066	0.535329	−	0.844644i	$-0.320188\pi$
$443$	22.5347	1.07066	0.535329	+	0.844644i	$0.320188\pi$
$444$	0	0
$445$	−21.0476	−0.997752
$446$	9.69526	0.459084
$447$	0	0
$448$	−17.3697	−0.820640
$449$	−13.0648	−0.616566	−0.308283	−	0.951295i	$-0.599754\pi$
$449$	−13.0648	−0.616566	−0.308283	+	0.951295i	$0.599754\pi$
$450$	0	0
$451$	−5.00319	−0.235591
$452$	−3.40409	−0.160115
$453$	0	0
$454$	−15.6397	−0.734008
$455$	−10.7701	−0.504910
$456$	0	0
$457$	−31.3213	−1.46515	−0.732574	−	0.680687i	$-0.761681\pi$
$457$	−31.3213	−1.46515	−0.732574	+	0.680687i	$0.761681\pi$
$458$	−1.61067	−0.0752619
$459$	0	0
$460$	−5.66885	−0.264311
$461$	26.1846	1.21954	0.609770	−	0.792578i	$-0.291262\pi$
$461$	26.1846	1.21954	0.609770	+	0.792578i	$0.291262\pi$
$462$	0	0
$463$	9.47596	0.440385	0.220193	−	0.975456i	$-0.429331\pi$
$463$	9.47596	0.440385	0.220193	+	0.975456i	$0.429331\pi$
$464$	18.4307	0.855624
$465$	0	0
$466$	1.48290	0.0686941
$467$	24.7032	1.14313	0.571563	−	0.820558i	$-0.306337\pi$
$467$	24.7032	1.14313	0.571563	+	0.820558i	$0.306337\pi$
$468$	0	0
$469$	9.25025	0.427137
$470$	6.80616	0.313945
$471$	0	0
$472$	−1.65338	−0.0761030
$473$	−6.52669	−0.300097
$474$	0	0
$475$	26.0433	1.19495
$476$	1.40607	0.0644473
$477$	0	0
$478$	−27.2479	−1.24629
$479$	34.8764	1.59354	0.796772	−	0.604279i	$-0.206539\pi$
$479$	34.8764	1.59354	0.796772	+	0.604279i	$0.206539\pi$
$480$	0	0
$481$	−32.6306	−1.48783
$482$	0.910918	0.0414912
$483$	0	0
$484$	5.01997	0.228180
$485$	−6.17830	−0.280542
$486$	0	0
$487$	−18.8553	−0.854417	−0.427209	−	0.904153i	$-0.640503\pi$
$487$	−18.8553	−0.854417	−0.427209	+	0.904153i	$0.640503\pi$
$488$	−24.4732	−1.10785
$489$	0	0
$490$	−4.78881	−0.216336
$491$	22.7611	1.02720	0.513598	−	0.858031i	$-0.328312\pi$
$491$	22.7611	1.02720	0.513598	+	0.858031i	$0.328312\pi$
$492$	0	0
$493$	−9.44855	−0.425541
$494$	−40.5026	−1.82230
$495$	0	0
$496$	0	0
$497$	5.74308	0.257612
$498$	0	0
$499$	27.5840	1.23483	0.617414	−	0.786638i	$-0.288180\pi$
$499$	27.5840	1.23483	0.617414	+	0.786638i	$0.288180\pi$
$500$	−5.47914	−0.245035
$501$	0	0
$502$	28.9825	1.29355
$503$	35.5351	1.58443	0.792217	−	0.610240i	$-0.208927\pi$
$503$	35.5351	1.58443	0.792217	+	0.610240i	$0.208927\pi$
$504$	0	0
$505$	19.9110	0.886030
$506$	12.3356	0.548384
$507$	0	0
$508$	3.52837	0.156546
$509$	−40.1006	−1.77743	−0.888714	−	0.458463i	$-0.848400\pi$
$509$	−40.1006	−1.77743	−0.888714	+	0.458463i	$0.848400\pi$
$510$	0	0
$511$	21.4857	0.950471
$512$	24.1808	1.06865
$513$	0	0
$514$	20.2622	0.893727
$515$	14.8213	0.653103
$516$	0	0
$517$	5.20461	0.228899
$518$	17.6858	0.777071
$519$	0	0
$520$	16.8377	0.738382
$521$	−21.0751	−0.923319	−0.461659	−	0.887057i	$-0.652746\pi$
$521$	−21.0751	−0.923319	−0.461659	+	0.887057i	$0.652746\pi$
$522$	0	0
$523$	−9.34148	−0.408474	−0.204237	−	0.978921i	$-0.565471\pi$
$523$	−9.34148	−0.408474	−0.204237	+	0.978921i	$0.565471\pi$
$524$	1.73940	0.0759858
$525$	0	0
$526$	−18.6631	−0.813751
$527$	0	0
$528$	0	0
$529$	53.3038	2.31756
$530$	−1.96982	−0.0855634
$531$	0	0
$532$	−7.71443	−0.334463
$533$	18.9701	0.821688
$534$	0	0
$535$	−8.53056	−0.368808
$536$	−14.4616	−0.624647
$537$	0	0
$538$	−2.32733	−0.100338
$539$	−3.66196	−0.157732
$540$	0	0
$541$	11.6262	0.499851	0.249926	−	0.968265i	$-0.419594\pi$
$541$	11.6262	0.499851	0.249926	+	0.968265i	$0.419594\pi$
$542$	−23.5854	−1.01308
$543$	0	0
$544$	−3.94280	−0.169046
$545$	5.81872	0.249246
$546$	0	0
$547$	−0.509413	−0.0217809	−0.0108905	−	0.999941i	$-0.503467\pi$
$547$	−0.509413	−0.0217809	−0.0108905	+	0.999941i	$0.503467\pi$
$548$	7.73937	0.330610
$549$	0	0
$550$	4.86194	0.207314
$551$	51.8396	2.20844
$552$	0	0
$553$	−15.2564	−0.648767
$554$	6.56293	0.278832
$555$	0	0
$556$	8.94005	0.379143
$557$	3.59089	0.152151	0.0760754	−	0.997102i	$-0.475761\pi$
$557$	3.59089	0.152151	0.0760754	+	0.997102i	$0.475761\pi$
$558$	0	0
$559$	24.7467	1.04667
$560$	−6.58086	−0.278092
$561$	0	0
$562$	−13.7439	−0.579753
$563$	−30.8725	−1.30112	−0.650561	−	0.759454i	$-0.725466\pi$
$563$	−30.8725	−1.30112	−0.650561	+	0.759454i	$0.725466\pi$
$564$	0	0
$565$	−8.16768	−0.343617
$566$	−4.77484	−0.200701
$567$	0	0
$568$	−8.97860	−0.376733
$569$	4.80831	0.201575	0.100787	−	0.994908i	$-0.467864\pi$
$569$	4.80831	0.201575	0.100787	+	0.994908i	$0.467864\pi$
$570$	0	0
$571$	−32.5236	−1.36107	−0.680535	−	0.732716i	$-0.738253\pi$
$571$	−32.5236	−1.36107	−0.680535	+	0.732716i	$0.738253\pi$
$572$	2.65716	0.111102
$573$	0	0
$574$	−10.2818	−0.429156
$575$	30.0743	1.25419
$576$	0	0
$577$	−2.55258	−0.106265	−0.0531326	−	0.998587i	$-0.516921\pi$
$577$	−2.55258	−0.106265	−0.0531326	+	0.998587i	$0.516921\pi$
$578$	18.3684	0.764027
$579$	0	0
$580$	−4.44744	−0.184670
$581$	5.50017	0.228185
$582$	0	0
$583$	−1.50630	−0.0623846
$584$	−33.5902	−1.38997
$585$	0	0
$586$	−12.9779	−0.536110
$587$	17.8140	0.735263	0.367632	−	0.929971i	$-0.380169\pi$
$587$	17.8140	0.735263	0.367632	+	0.929971i	$0.380169\pi$
$588$	0	0
$589$	0	0
$590$	−0.818691	−0.0337050
$591$	0	0
$592$	−19.9383	−0.819460
$593$	17.9023	0.735159	0.367580	−	0.929992i	$-0.380187\pi$
$593$	17.9023	0.735159	0.367580	+	0.929992i	$0.380187\pi$
$594$	0	0
$595$	3.37370	0.138308
$596$	1.52677	0.0625389
$597$	0	0
$598$	−46.7717	−1.91264
$599$	−6.25118	−0.255416	−0.127708	−	0.991812i	$-0.540762\pi$
$599$	−6.25118	−0.255416	−0.127708	+	0.991812i	$0.540762\pi$
$600$	0	0
$601$	23.9451	0.976739	0.488370	−	0.872637i	$-0.337592\pi$
$601$	23.9451	0.976739	0.488370	+	0.872637i	$0.337592\pi$
$602$	−13.4127	−0.546661
$603$	0	0
$604$	−5.17639	−0.210624
$605$	12.0448	0.489690
$606$	0	0
$607$	1.78762	0.0725573	0.0362786	−	0.999342i	$-0.488450\pi$
$607$	1.78762	0.0725573	0.0362786	+	0.999342i	$0.488450\pi$
$608$	21.6322	0.877301
$609$	0	0
$610$	−12.1182	−0.490652
$611$	−19.7339	−0.798346
$612$	0	0
$613$	−9.12118	−0.368401	−0.184200	−	0.982889i	$-0.558970\pi$
$613$	−9.12118	−0.368401	−0.184200	+	0.982889i	$0.558970\pi$
$614$	−14.5372	−0.586674
$615$	0	0
$616$	−6.97861	−0.281176
$617$	−39.1622	−1.57661	−0.788306	−	0.615283i	$-0.789042\pi$
$617$	−39.1622	−1.57661	−0.788306	+	0.615283i	$0.789042\pi$
$618$	0	0
$619$	−13.7762	−0.553711	−0.276856	−	0.960912i	$-0.589292\pi$
$619$	−13.7762	−0.553711	−0.276856	+	0.960912i	$0.589292\pi$
$620$	0	0
$621$	0	0
$622$	−26.5470	−1.06444
$623$	−33.0759	−1.32516
$624$	0	0
$625$	4.06792	0.162717
$626$	6.59132	0.263442
$627$	0	0
$628$	4.48735	0.179065
$629$	10.2214	0.407556
$630$	0	0
$631$	6.21469	0.247403	0.123701	−	0.992319i	$-0.460523\pi$
$631$	6.21469	0.247403	0.123701	+	0.992319i	$0.460523\pi$
$632$	23.8515	0.948760
$633$	0	0
$634$	−14.1006	−0.560005
$635$	8.46589	0.335959
$636$	0	0
$637$	13.8847	0.550133
$638$	9.67777	0.383147
$639$	0	0
$640$	6.30927	0.249396
$641$	−18.2490	−0.720794	−0.360397	−	0.932799i	$-0.617359\pi$
$641$	−18.2490	−0.720794	−0.360397	+	0.932799i	$0.617359\pi$
$642$	0	0
$643$	27.5007	1.08452	0.542261	−	0.840210i	$-0.317568\pi$
$643$	27.5007	1.08452	0.542261	+	0.840210i	$0.317568\pi$
$644$	−8.90850	−0.351044
$645$	0	0
$646$	12.6873	0.499176
$647$	−35.4015	−1.39178	−0.695889	−	0.718149i	$-0.744990\pi$
$647$	−35.4015	−1.39178	−0.695889	+	0.718149i	$0.744990\pi$
$648$	0	0
$649$	−0.626045	−0.0245744
$650$	−18.4346	−0.723064
$651$	0	0
$652$	4.54771	0.178102
$653$	13.9080	0.544264	0.272132	−	0.962260i	$-0.412271\pi$
$653$	13.9080	0.544264	0.272132	+	0.962260i	$0.412271\pi$
$654$	0	0
$655$	4.17346	0.163071
$656$	11.5913	0.452566
$657$	0	0
$658$	10.6958	0.416965
$659$	44.2325	1.72306	0.861528	−	0.507711i	$-0.169508\pi$
$659$	44.2325	1.72306	0.861528	+	0.507711i	$0.169508\pi$
$660$	0	0
$661$	25.1378	0.977746	0.488873	−	0.872355i	$-0.337408\pi$
$661$	25.1378	0.977746	0.488873	+	0.872355i	$0.337408\pi$
$662$	7.82314	0.304055
$663$	0	0
$664$	−8.59883	−0.333699
$665$	−18.5098	−0.717780
$666$	0	0
$667$	59.8634	2.31792
$668$	−5.49620	−0.212654
$669$	0	0
$670$	−7.16085	−0.276648
$671$	−9.26668	−0.357736
$672$	0	0
$673$	−29.6891	−1.14443	−0.572214	−	0.820104i	$-0.693916\pi$
$673$	−29.6891	−1.14443	−0.572214	+	0.820104i	$0.693916\pi$
$674$	−5.34983	−0.206068
$675$	0	0
$676$	−3.31401	−0.127462
$677$	11.5868	0.445315	0.222658	−	0.974897i	$-0.428527\pi$
$677$	11.5868	0.445315	0.222658	+	0.974897i	$0.428527\pi$
$678$	0	0
$679$	−9.70910	−0.372601
$680$	−5.27436	−0.202263
$681$	0	0
$682$	0	0
$683$	−35.3501	−1.35264	−0.676318	−	0.736610i	$-0.736425\pi$
$683$	−35.3501	−1.35264	−0.676318	+	0.736610i	$0.736425\pi$
$684$	0	0
$685$	18.5697	0.709510
$686$	−24.2244	−0.924894
$687$	0	0
$688$	15.1210	0.576482
$689$	5.71130	0.217583
$690$	0	0
$691$	−18.6199	−0.708333	−0.354166	−	0.935182i	$-0.615235\pi$
$691$	−18.6199	−0.708333	−0.354166	+	0.935182i	$0.615235\pi$
$692$	8.07117	0.306820
$693$	0	0
$694$	17.0841	0.648503
$695$	21.4505	0.813665
$696$	0	0
$697$	−5.94234	−0.225082
$698$	2.20831	0.0835858
$699$	0	0
$700$	−3.51119	−0.132710
$701$	−11.6811	−0.441188	−0.220594	−	0.975366i	$-0.570800\pi$
$701$	−11.6811	−0.441188	−0.220594	+	0.975366i	$0.570800\pi$
$702$	0	0
$703$	−56.0800	−2.11510
$704$	10.2823	0.387527
$705$	0	0
$706$	20.1372	0.757875
$707$	31.2899	1.17678
$708$	0	0
$709$	3.77420	0.141743	0.0708715	−	0.997485i	$-0.477422\pi$
$709$	3.77420	0.141743	0.0708715	+	0.997485i	$0.477422\pi$
$710$	−4.44586	−0.166850
$711$	0	0
$712$	51.7101	1.93792
$713$	0	0
$714$	0	0
$715$	6.37553	0.238431
$716$	5.13716	0.191985
$717$	0	0
$718$	−11.6889	−0.436227
$719$	40.6508	1.51602	0.758009	−	0.652244i	$-0.226172\pi$
$719$	40.6508	1.51602	0.758009	+	0.652244i	$0.226172\pi$
$720$	0	0
$721$	23.2914	0.867416
$722$	−46.4952	−1.73037
$723$	0	0
$724$	−6.12309	−0.227563
$725$	23.5945	0.876279
$726$	0	0
$727$	11.5961	0.430075	0.215038	−	0.976606i	$-0.431013\pi$
$727$	11.5961	0.430075	0.215038	+	0.976606i	$0.431013\pi$
$728$	26.4602	0.980679
$729$	0	0
$730$	−16.6326	−0.615600
$731$	−7.75181	−0.286711
$732$	0	0
$733$	−30.1901	−1.11510	−0.557549	−	0.830144i	$-0.688258\pi$
$733$	−30.1901	−1.11510	−0.557549	+	0.830144i	$0.688258\pi$
$734$	16.2655	0.600369
$735$	0	0
$736$	24.9805	0.920792
$737$	−5.47583	−0.201705
$738$	0	0
$739$	14.8313	0.545579	0.272789	−	0.962074i	$-0.412054\pi$
$739$	14.8313	0.545579	0.272789	+	0.962074i	$0.412054\pi$
$740$	4.81124	0.176865
$741$	0	0
$742$	−3.09553	−0.113641
$743$	−39.7017	−1.45652	−0.728258	−	0.685304i	$-0.759670\pi$
$743$	−39.7017	−1.45652	−0.728258	+	0.685304i	$0.759670\pi$
$744$	0	0
$745$	3.66329	0.134213
$746$	−2.35613	−0.0862641
$747$	0	0
$748$	−0.832348	−0.0304337
$749$	−13.4056	−0.489831
$750$	0	0
$751$	0.176268	0.00643212	0.00321606	−	0.999995i	$-0.498976\pi$
$751$	0.176268	0.00643212	0.00321606	+	0.999995i	$0.498976\pi$
$752$	−12.0580	−0.439710
$753$	0	0
$754$	−36.6943	−1.33633
$755$	−12.4201	−0.452014
$756$	0	0
$757$	51.1490	1.85904	0.929522	−	0.368767i	$-0.120220\pi$
$757$	51.1490	1.85904	0.929522	+	0.368767i	$0.120220\pi$
$758$	8.80289	0.319735
$759$	0	0
$760$	28.9378	1.04969
$761$	33.2632	1.20579	0.602895	−	0.797821i	$-0.294014\pi$
$761$	33.2632	1.20579	0.602895	+	0.797821i	$0.294014\pi$
$762$	0	0
$763$	9.14401	0.331036
$764$	6.78536	0.245486
$765$	0	0
$766$	28.5141	1.03026
$767$	2.37372	0.0857100
$768$	0	0
$769$	1.00062	0.0360834	0.0180417	−	0.999837i	$-0.494257\pi$
$769$	1.00062	0.0360834	0.0180417	+	0.999837i	$0.494257\pi$
$770$	−3.45554	−0.124529
$771$	0	0
$772$	−7.14587	−0.257186
$773$	23.2344	0.835682	0.417841	−	0.908520i	$-0.362787\pi$
$773$	23.2344	0.835682	0.417841	+	0.908520i	$0.362787\pi$
$774$	0	0
$775$	0	0
$776$	15.1790	0.544894
$777$	0	0
$778$	15.2889	0.548132
$779$	32.6027	1.16811
$780$	0	0
$781$	−3.39971	−0.121651
$782$	14.6511	0.523923
$783$	0	0
$784$	8.48400	0.303000
$785$	10.7668	0.384285
$786$	0	0
$787$	−49.3296	−1.75841	−0.879205	−	0.476444i	$-0.841925\pi$
$787$	−49.3296	−1.75841	−0.879205	+	0.476444i	$0.841925\pi$
$788$	8.33401	0.296887
$789$	0	0
$790$	11.8103	0.420193
$791$	−12.8354	−0.456373
$792$	0	0
$793$	35.1356	1.24770
$794$	−24.0763	−0.854435
$795$	0	0
$796$	8.04771	0.285243
$797$	−5.67184	−0.200907	−0.100453	−	0.994942i	$-0.532029\pi$
$797$	−5.67184	−0.200907	−0.100453	+	0.994942i	$0.532029\pi$
$798$	0	0
$799$	6.18157	0.218688
$800$	9.84579	0.348101
$801$	0	0
$802$	17.8356	0.629796
$803$	−12.7188	−0.448837
$804$	0	0
$805$	−21.3748	−0.753363
$806$	0	0
$807$	0	0
$808$	−48.9178	−1.72092
$809$	−20.6208	−0.724989	−0.362494	−	0.931986i	$-0.618075\pi$
$809$	−20.6208	−0.724989	−0.362494	+	0.931986i	$0.618075\pi$
$810$	0	0
$811$	38.8250	1.36333	0.681664	−	0.731665i	$-0.261256\pi$
$811$	38.8250	1.36333	0.681664	+	0.731665i	$0.261256\pi$
$812$	−6.98908	−0.245269
$813$	0	0
$814$	−10.4694	−0.366953
$815$	10.9117	0.382219
$816$	0	0
$817$	42.5304	1.48795
$818$	−29.7847	−1.04140
$819$	0	0
$820$	−2.79706	−0.0976777
$821$	13.6237	0.475471	0.237735	−	0.971330i	$-0.423595\pi$
$821$	13.6237	0.475471	0.237735	+	0.971330i	$0.423595\pi$
$822$	0	0
$823$	−0.932186	−0.0324940	−0.0162470	−	0.999868i	$-0.505172\pi$
$823$	−0.932186	−0.0324940	−0.0162470	+	0.999868i	$0.505172\pi$
$824$	−36.4132	−1.26851
$825$	0	0
$826$	−1.28656	−0.0447651
$827$	−37.8830	−1.31732	−0.658660	−	0.752441i	$-0.728876\pi$
$827$	−37.8830	−1.31732	−0.658660	+	0.752441i	$0.728876\pi$
$828$	0	0
$829$	−47.3069	−1.64304	−0.821519	−	0.570182i	$-0.806873\pi$
$829$	−47.3069	−1.64304	−0.821519	+	0.570182i	$0.806873\pi$
$830$	−4.25782	−0.147791
$831$	0	0
$832$	−38.9863	−1.35161
$833$	−4.34935	−0.150696
$834$	0	0
$835$	−13.1875	−0.456371
$836$	4.56668	0.157942
$837$	0	0
$838$	−34.0484	−1.17618
$839$	−18.9947	−0.655771	−0.327885	−	0.944718i	$-0.606336\pi$
$839$	−18.9947	−0.655771	−0.327885	+	0.944718i	$0.606336\pi$
$840$	0	0
$841$	17.9653	0.619492
$842$	23.0622	0.794777
$843$	0	0
$844$	14.0434	0.483393
$845$	−7.95156	−0.273542
$846$	0	0
$847$	18.9282	0.650380
$848$	3.48979	0.119840
$849$	0	0
$850$	5.77458	0.198066
$851$	−64.7602	−2.21995
$852$	0	0
$853$	12.6147	0.431920	0.215960	−	0.976402i	$-0.430712\pi$
$853$	12.6147	0.431920	0.215960	+	0.976402i	$0.430712\pi$
$854$	−19.0435	−0.651657
$855$	0	0
$856$	20.9580	0.716331
$857$	−2.46194	−0.0840983	−0.0420492	−	0.999116i	$-0.513389\pi$
$857$	−2.46194	−0.0840983	−0.0420492	+	0.999116i	$0.513389\pi$
$858$	0	0
$859$	−45.2277	−1.54315	−0.771575	−	0.636139i	$-0.780531\pi$
$859$	−45.2277	−1.54315	−0.771575	+	0.636139i	$0.780531\pi$
$860$	−3.64878	−0.124422
$861$	0	0
$862$	−13.4507	−0.458132
$863$	−43.1363	−1.46838	−0.734188	−	0.678946i	$-0.762437\pi$
$863$	−43.1363	−1.46838	−0.734188	+	0.678946i	$0.762437\pi$
$864$	0	0
$865$	19.3658	0.658455
$866$	28.4668	0.967340
$867$	0	0
$868$	0	0
$869$	9.03125	0.306364
$870$	0	0
$871$	20.7622	0.703501
$872$	−14.2955	−0.484108
$873$	0	0
$874$	−80.3834	−2.71901
$875$	−20.6595	−0.698420
$876$	0	0
$877$	−8.21746	−0.277484	−0.138742	−	0.990329i	$-0.544306\pi$
$877$	−8.21746	−0.277484	−0.138742	+	0.990329i	$0.544306\pi$
$878$	9.88158	0.333487
$879$	0	0
$880$	3.89565	0.131322
$881$	−33.7928	−1.13851	−0.569255	−	0.822161i	$-0.692768\pi$
$881$	−33.7928	−1.13851	−0.569255	+	0.822161i	$0.692768\pi$
$882$	0	0
$883$	−41.9707	−1.41243	−0.706213	−	0.708000i	$-0.749598\pi$
$883$	−41.9707	−1.41243	−0.706213	+	0.708000i	$0.749598\pi$
$884$	3.15594	0.106146
$885$	0	0
$886$	−27.4141	−0.920993
$887$	22.2324	0.746492	0.373246	−	0.927732i	$-0.378245\pi$
$887$	22.2324	0.746492	0.373246	+	0.927732i	$0.378245\pi$
$888$	0	0
$889$	13.3040	0.446202
$890$	25.6049	0.858278
$891$	0	0
$892$	4.14477	0.138777
$893$	−33.9152	−1.13493
$894$	0	0
$895$	12.3260	0.412012
$896$	9.91491	0.331234
$897$	0	0
$898$	15.8936	0.530378
$899$	0	0
$900$	0	0
$901$	−1.78905	−0.0596019
$902$	6.08650	0.202658
$903$	0	0
$904$	20.0665	0.667402
$905$	−14.6916	−0.488365
$906$	0	0
$907$	32.6593	1.08443	0.542217	−	0.840239i	$-0.317585\pi$
$907$	32.6593	1.08443	0.542217	+	0.840239i	$0.317585\pi$
$908$	−6.68606	−0.221885
$909$	0	0
$910$	13.1021	0.434330
$911$	−55.6939	−1.84522	−0.922611	−	0.385732i	$-0.873949\pi$
$911$	−55.6939	−1.84522	−0.922611	+	0.385732i	$0.873949\pi$
$912$	0	0
$913$	−3.25591	−0.107755
$914$	38.1031	1.26034
$915$	0	0
$916$	−0.688571	−0.0227510
$917$	6.55852	0.216582
$918$	0	0
$919$	−11.9505	−0.394211	−0.197106	−	0.980382i	$-0.563154\pi$
$919$	−11.9505	−0.394211	−0.197106	+	0.980382i	$0.563154\pi$
$920$	33.4169	1.10172
$921$	0	0
$922$	−31.8542	−1.04906
$923$	12.8904	0.424291
$924$	0	0
$925$	−25.5246	−0.839243
$926$	−11.5277	−0.378825
$927$	0	0
$928$	19.5982	0.643342
$929$	45.2450	1.48444	0.742221	−	0.670156i	$-0.233773\pi$
$929$	45.2450	1.48444	0.742221	+	0.670156i	$0.233773\pi$
$930$	0	0
$931$	23.8627	0.782069
$932$	0.633948	0.0207656
$933$	0	0
$934$	−30.0520	−0.983331
$935$	−1.99712	−0.0653126
$936$	0	0
$937$	39.2884	1.28349	0.641747	−	0.766916i	$-0.278210\pi$
$937$	39.2884	1.28349	0.641747	+	0.766916i	$0.278210\pi$
$938$	−11.2531	−0.367428
$939$	0	0
$940$	2.90967	0.0949029
$941$	−22.1262	−0.721294	−0.360647	−	0.932702i	$-0.617444\pi$
$941$	−22.1262	−0.721294	−0.360647	+	0.932702i	$0.617444\pi$
$942$	0	0
$943$	37.6490	1.22602
$944$	1.45042	0.0472070
$945$	0	0
$946$	7.93987	0.258147
$947$	−29.2131	−0.949298	−0.474649	−	0.880175i	$-0.657425\pi$
$947$	−29.2131	−0.949298	−0.474649	+	0.880175i	$0.657425\pi$
$948$	0	0
$949$	48.2247	1.56544
$950$	−31.6823	−1.02791
$951$	0	0
$952$	−8.28857	−0.268634
$953$	31.3756	1.01635	0.508177	−	0.861253i	$-0.330319\pi$
$953$	31.3756	1.01635	0.508177	+	0.861253i	$0.330319\pi$
$954$	0	0
$955$	16.2806	0.526829
$956$	−11.6486	−0.376744
$957$	0	0
$958$	−42.4280	−1.37079
$959$	29.1819	0.942333
$960$	0	0
$961$	0	0
$962$	39.6959	1.27985
$963$	0	0
$964$	0.389422	0.0125424
$965$	−17.1456	−0.551937
$966$	0	0
$967$	2.13352	0.0686094	0.0343047	−	0.999411i	$-0.489078\pi$
$967$	2.13352	0.0686094	0.0343047	+	0.999411i	$0.489078\pi$
$968$	−29.5919	−0.951119
$969$	0	0
$970$	7.51605	0.241326
$971$	19.2043	0.616294	0.308147	−	0.951339i	$-0.400291\pi$
$971$	19.2043	0.616294	0.308147	+	0.951339i	$0.400291\pi$
$972$	0	0
$973$	33.7092	1.08067
$974$	22.9380	0.734980
$975$	0	0
$976$	21.4690	0.687205
$977$	−51.8885	−1.66006	−0.830030	−	0.557719i	$-0.811677\pi$
$977$	−51.8885	−1.66006	−0.830030	+	0.557719i	$0.811677\pi$
$978$	0	0
$979$	19.5798	0.625774
$980$	−2.04724	−0.0653967
$981$	0	0
$982$	−27.6894	−0.883606
$983$	−57.1145	−1.82167	−0.910835	−	0.412770i	$-0.864561\pi$
$983$	−57.1145	−1.82167	−0.910835	+	0.412770i	$0.864561\pi$
$984$	0	0
$985$	19.9964	0.637139
$986$	11.4944	0.366056
$987$	0	0
$988$	−17.3151	−0.550866
$989$	49.1133	1.56171
$990$	0	0
$991$	21.6046	0.686292	0.343146	−	0.939282i	$-0.388507\pi$
$991$	21.6046	0.686292	0.343146	+	0.939282i	$0.388507\pi$
$992$	0	0
$993$	0	0
$994$	−6.98659	−0.221601
$995$	19.3095	0.612151
$996$	0	0
$997$	−14.9162	−0.472399	−0.236200	−	0.971705i	$-0.575902\pi$
$997$	−14.9162	−0.472399	−0.236200	+	0.971705i	$0.575902\pi$
$998$	−33.5565	−1.06221
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bs.1.4		16
3.2	odd	2		961.2.a.l.1.14	yes	16
31.30	odd	2	inner	8649.2.a.bs.1.3		16
93.2	odd	10		961.2.d.s.531.14		64
93.5	odd	6		961.2.c.l.521.13		32
93.8	odd	10		961.2.d.s.374.3		64
93.11	even	30		961.2.g.w.338.4		128
93.14	odd	30		961.2.g.w.816.3		128
93.17	even	30		961.2.g.w.816.4		128
93.20	odd	30		961.2.g.w.338.3		128
93.23	even	10		961.2.d.s.374.4		64
93.26	even	6		961.2.c.l.521.14		32
93.29	even	10		961.2.d.s.531.13		64
93.35	odd	10		961.2.d.s.388.3		64
93.38	odd	30		961.2.g.w.235.4		128
93.41	odd	30		961.2.g.w.844.13		128
93.44	even	30		961.2.g.w.448.13		128
93.47	odd	10		961.2.d.s.628.14		64
93.50	odd	30		961.2.g.w.547.14		128
93.53	even	30		961.2.g.w.732.3		128
93.56	odd	6		961.2.c.l.439.13		32
93.59	odd	30		961.2.g.w.846.13		128
93.65	even	30		961.2.g.w.846.14		128
93.68	even	6		961.2.c.l.439.14		32
93.71	odd	30		961.2.g.w.732.4		128
93.74	even	30		961.2.g.w.547.13		128
93.77	even	10		961.2.d.s.628.13		64
93.80	odd	30		961.2.g.w.448.14		128
93.83	even	30		961.2.g.w.844.14		128
93.86	even	30		961.2.g.w.235.3		128
93.89	even	10		961.2.d.s.388.4		64
93.92	even	2		961.2.a.l.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.l.1.13	✓	16	93.92	even	2
961.2.a.l.1.14	yes	16	3.2	odd	2
961.2.c.l.439.13		32	93.56	odd	6
961.2.c.l.439.14		32	93.68	even	6
961.2.c.l.521.13		32	93.5	odd	6
961.2.c.l.521.14		32	93.26	even	6
961.2.d.s.374.3		64	93.8	odd	10
961.2.d.s.374.4		64	93.23	even	10
961.2.d.s.388.3		64	93.35	odd	10
961.2.d.s.388.4		64	93.89	even	10
961.2.d.s.531.13		64	93.29	even	10
961.2.d.s.531.14		64	93.2	odd	10
961.2.d.s.628.13		64	93.77	even	10
961.2.d.s.628.14		64	93.47	odd	10
961.2.g.w.235.3		128	93.86	even	30
961.2.g.w.235.4		128	93.38	odd	30
961.2.g.w.338.3		128	93.20	odd	30
961.2.g.w.338.4		128	93.11	even	30
961.2.g.w.448.13		128	93.44	even	30
961.2.g.w.448.14		128	93.80	odd	30
961.2.g.w.547.13		128	93.74	even	30
961.2.g.w.547.14		128	93.50	odd	30
961.2.g.w.732.3		128	93.53	even	30
961.2.g.w.732.4		128	93.71	odd	30
961.2.g.w.816.3		128	93.14	odd	30
961.2.g.w.816.4		128	93.17	even	30
961.2.g.w.844.13		128	93.41	odd	30
961.2.g.w.844.14		128	93.83	even	30
961.2.g.w.846.13		128	93.59	odd	30
961.2.g.w.846.14		128	93.65	even	30
8649.2.a.bs.1.3		16	31.30	odd	2	inner
8649.2.a.bs.1.4		16	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21652 q^{2} -0.520070 q^{4} -1.24784 q^{5} -1.96096 q^{7} +3.06572 q^{8} +O(q^{10})\)	\(q-1.21652 q^{2} -0.520070 q^{4} -1.24784 q^{5} -1.96096 q^{7} +3.06572 q^{8} +1.51803 q^{10} +1.16082 q^{11} -4.40139 q^{13} +2.38556 q^{14} -2.68939 q^{16} +1.37872 q^{17} -7.56437 q^{19} +0.648966 q^{20} -1.41217 q^{22} -8.73520 q^{23} -3.44289 q^{25} +5.35440 q^{26} +1.01984 q^{28} -6.85312 q^{29} -2.85975 q^{32} -1.67725 q^{34} +2.44697 q^{35} +7.41371 q^{37} +9.20224 q^{38} -3.82554 q^{40} -4.31003 q^{41} -5.62246 q^{43} -0.603710 q^{44} +10.6266 q^{46} +4.48355 q^{47} -3.15462 q^{49} +4.18836 q^{50} +2.28903 q^{52} -1.29761 q^{53} -1.44853 q^{55} -6.01178 q^{56} +8.33699 q^{58} -0.539311 q^{59} -7.98284 q^{61} +8.85772 q^{64} +5.49224 q^{65} -4.71720 q^{67} -0.717032 q^{68} -2.97680 q^{70} -2.92870 q^{71} -10.9567 q^{73} -9.01895 q^{74} +3.93400 q^{76} -2.27633 q^{77} +7.78004 q^{79} +3.35593 q^{80} +5.24326 q^{82} -2.80483 q^{83} -1.72043 q^{85} +6.83986 q^{86} +3.55877 q^{88} +16.8672 q^{89} +8.63097 q^{91} +4.54292 q^{92} -5.45434 q^{94} +9.43915 q^{95} +4.95119 q^{97} +3.83767 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7}+O(q^{10}) \) 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7	\( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56} + 64 q^{59} - 16 q^{64} + 16 q^{67} + 88 q^{70} + 48 q^{71} + 40 q^{76} + 40 q^{80} + 88 q^{82} - 32 q^{94} + 48 q^{95} - 16 q^{98}+O(q^{100}) \) 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * q^98

Introduction

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