LMFDB - Embedded newform 8910.2.a.bt.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8910.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,-4,0,-6,4,0,-4,-4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.1467082010$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.456850$ of defining polynomial
Character	$\chi$	$=$	8910.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.39761 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +4.24814 q^{13} -1.39761 q^{14} +1.00000 q^{16} +4.19615 q^{17} -5.31463 q^{19} -1.00000 q^{20} -1.00000 q^{22} -7.83071 q^{23} +1.00000 q^{25} +4.24814 q^{26} -1.39761 q^{28} -3.98019 q^{29} +0.957065 q^{31} +1.00000 q^{32} +4.19615 q^{34} +1.39761 q^{35} +0.118474 q^{37} -5.31463 q^{38} -1.00000 q^{40} -1.35425 q^{41} +4.73205 q^{43} -1.00000 q^{44} -7.83071 q^{46} +11.0037 q^{47} -5.04668 q^{49} +1.00000 q^{50} +4.24814 q^{52} -9.72129 q^{53} +1.00000 q^{55} -1.39761 q^{56} -3.98019 q^{58} -1.17260 q^{59} -1.82740 q^{61} +0.957065 q^{62} +1.00000 q^{64} -4.24814 q^{65} -6.75186 q^{67} +4.19615 q^{68} +1.39761 q^{70} +1.24814 q^{71} +9.62594 q^{73} +0.118474 q^{74} -5.31463 q^{76} +1.39761 q^{77} +2.59377 q^{79} -1.00000 q^{80} -1.35425 q^{82} -13.5971 q^{83} -4.19615 q^{85} +4.73205 q^{86} -1.00000 q^{88} +0.431927 q^{89} -5.93725 q^{91} -7.83071 q^{92} +11.0037 q^{94} +5.31463 q^{95} -5.45334 q^{97} -5.04668 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{7} + 4 q^{8} - 4 q^{10} - 4 q^{11} + 6 q^{13} - 6 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{22} - 2 q^{23} + 4 q^{25} + 6 q^{26} - 6 q^{28} + 2 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^7 + 4 * q^8 - 4 * q^10 - 4 * q^11 + 6 * q^13 - 6 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^22 - 2 * q^23 + 4 * q^25 + 6 * q^26 - 6 * q^28 + 2 * q^29 - 10 * q^31 + 4 * q^32 - 4 * q^34 + 6 * q^35 - 4 * q^37 + 4 * q^38 - 4 * q^40 - 16 * q^41 + 12 * q^43 - 4 * q^44 - 2 * q^46 - 2 * q^47 + 12 * q^49 + 4 * q^50 + 6 * q^52 - 10 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^62 + 4 * q^64 - 6 * q^65 - 38 * q^67 - 4 * q^68 + 6 * q^70 - 6 * q^71 + 10 * q^73 - 4 * q^74 + 4 * q^76 + 6 * q^77 - 10 * q^79 - 4 * q^80 - 16 * q^82 - 12 * q^83 + 4 * q^85 + 12 * q^86 - 4 * q^88 - 4 * q^89 + 8 * q^91 - 2 * q^92 - 2 * q^94 - 4 * q^95 + 14 * q^97 + 12 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.39761	−0.528248	−0.264124	−	0.964489i	$-0.585083\pi$
$7$	−1.39761	−0.528248	−0.264124	+	0.964489i	$0.585083\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.24814	1.17822	0.589111	−	0.808052i	$-0.299478\pi$
$13$	4.24814	1.17822	0.589111	+	0.808052i	$0.299478\pi$
$14$	−1.39761	−0.373528
$15$	0	0
$16$	1.00000	0.250000
$17$	4.19615	1.01772	0.508858	−	0.860850i	$-0.330068\pi$
$17$	4.19615	1.01772	0.508858	+	0.860850i	$0.330068\pi$
$18$	0	0
$19$	−5.31463	−1.21926	−0.609629	−	0.792687i	$-0.708682\pi$
$19$	−5.31463	−1.21926	−0.609629	+	0.792687i	$0.708682\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−7.83071	−1.63282	−0.816408	−	0.577475i	$-0.804038\pi$
$23$	−7.83071	−1.63282	−0.816408	+	0.577475i	$0.804038\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.24814	0.833128
$27$	0	0
$28$	−1.39761	−0.264124
$29$	−3.98019	−0.739103	−0.369551	−	0.929210i	$-0.620489\pi$
$29$	−3.98019	−0.739103	−0.369551	+	0.929210i	$0.620489\pi$
$30$	0	0
$31$	0.957065	0.171894	0.0859470	−	0.996300i	$-0.472608\pi$
$31$	0.957065	0.171894	0.0859470	+	0.996300i	$0.472608\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.19615	0.719634
$35$	1.39761	0.236240
$36$	0	0
$37$	0.118474	0.0194770	0.00973851	−	0.999953i	$-0.496900\pi$
$37$	0.118474	0.0194770	0.00973851	+	0.999953i	$0.496900\pi$
$38$	−5.31463	−0.862146
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−1.35425	−0.211498	−0.105749	−	0.994393i	$-0.533724\pi$
$41$	−1.35425	−0.211498	−0.105749	+	0.994393i	$0.533724\pi$
$42$	0	0
$43$	4.73205	0.721631	0.360815	−	0.932637i	$-0.382498\pi$
$43$	4.73205	0.721631	0.360815	+	0.932637i	$0.382498\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−7.83071	−1.15458
$47$	11.0037	1.60506	0.802530	−	0.596611i	$-0.203487\pi$
$47$	11.0037	1.60506	0.802530	+	0.596611i	$0.203487\pi$
$48$	0	0
$49$	−5.04668	−0.720954
$50$	1.00000	0.141421
$51$	0	0
$52$	4.24814	0.589111
$53$	−9.72129	−1.33532	−0.667661	−	0.744465i	$-0.732704\pi$
$53$	−9.72129	−1.33532	−0.667661	+	0.744465i	$0.732704\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.39761	−0.186764
$57$	0	0
$58$	−3.98019	−0.522624
$59$	−1.17260	−0.152659	−0.0763297	−	0.997083i	$-0.524320\pi$
$59$	−1.17260	−0.152659	−0.0763297	+	0.997083i	$0.524320\pi$
$60$	0	0
$61$	−1.82740	−0.233975	−0.116987	−	0.993133i	$-0.537324\pi$
$61$	−1.82740	−0.233975	−0.116987	+	0.993133i	$0.537324\pi$
$62$	0.957065	0.121547
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.24814	−0.526917
$66$	0	0
$67$	−6.75186	−0.824871	−0.412436	−	0.910987i	$-0.635322\pi$
$67$	−6.75186	−0.824871	−0.412436	+	0.910987i	$0.635322\pi$
$68$	4.19615	0.508858
$69$	0	0
$70$	1.39761	0.167047
$71$	1.24814	0.148127	0.0740634	−	0.997254i	$-0.476403\pi$
$71$	1.24814	0.148127	0.0740634	+	0.997254i	$0.476403\pi$
$72$	0	0
$73$	9.62594	1.12663	0.563316	−	0.826242i	$-0.309526\pi$
$73$	9.62594	1.12663	0.563316	+	0.826242i	$0.309526\pi$
$74$	0.118474	0.0137723
$75$	0	0
$76$	−5.31463	−0.609629
$77$	1.39761	0.159273
$78$	0	0
$79$	2.59377	0.291821	0.145911	−	0.989298i	$-0.453389\pi$
$79$	2.59377	0.291821	0.145911	+	0.989298i	$0.453389\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−1.35425	−0.149552
$83$	−13.5971	−1.49247	−0.746237	−	0.665681i	$-0.768141\pi$
$83$	−13.5971	−1.49247	−0.746237	+	0.665681i	$0.768141\pi$
$84$	0	0
$85$	−4.19615	−0.455137
$86$	4.73205	0.510270
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0.431927	0.0457842	0.0228921	−	0.999738i	$-0.492713\pi$
$89$	0.431927	0.0457842	0.0228921	+	0.999738i	$0.492713\pi$
$90$	0	0
$91$	−5.93725	−0.622393
$92$	−7.83071	−0.816408
$93$	0	0
$94$	11.0037	1.13495
$95$	5.31463	0.545269
$96$	0	0
$97$	−5.45334	−0.553703	−0.276851	−	0.960913i	$-0.589291\pi$
$97$	−5.45334	−0.553703	−0.276851	+	0.960913i	$0.589291\pi$
$98$	−5.04668	−0.509791
$99$	0	0
$100$	1.00000	0.100000
$101$	−17.5508	−1.74637	−0.873186	−	0.487386i	$-0.837950\pi$
$101$	−17.5508	−1.74637	−0.873186	+	0.487386i	$0.837950\pi$
$102$	0	0
$103$	−17.5632	−1.73055	−0.865276	−	0.501295i	$-0.832857\pi$
$103$	−17.5632	−1.73055	−0.865276	+	0.501295i	$0.832857\pi$
$104$	4.24814	0.416564
$105$	0	0
$106$	−9.72129	−0.944215
$107$	−0.201461	−0.0194759	−0.00973797	−	0.999953i	$-0.503100\pi$
$107$	−0.201461	−0.0194759	−0.00973797	+	0.999953i	$0.503100\pi$
$108$	0	0
$109$	6.93725	0.664468	0.332234	−	0.943197i	$-0.392198\pi$
$109$	6.93725	0.664468	0.332234	+	0.943197i	$0.392198\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.39761	−0.132062
$113$	12.5198	1.17777	0.588883	−	0.808218i	$-0.299568\pi$
$113$	12.5198	1.17777	0.588883	+	0.808218i	$0.299568\pi$
$114$	0	0
$115$	7.83071	0.730218
$116$	−3.98019	−0.369551
$117$	0	0
$118$	−1.17260	−0.107946
$119$	−5.86460	−0.537607
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.82740	−0.165445
$123$	0	0
$124$	0.957065	0.0859470
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.0916079	−0.00812888	−0.00406444	−	0.999992i	$-0.501294\pi$
$127$	−0.0916079	−0.00812888	−0.00406444	+	0.999992i	$0.501294\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.24814	−0.372586
$131$	−14.2519	−1.24519	−0.622596	−	0.782543i	$-0.713922\pi$
$131$	−14.2519	−1.24519	−0.622596	+	0.782543i	$0.713922\pi$
$132$	0	0
$133$	7.42779	0.644071
$134$	−6.75186	−0.583272
$135$	0	0
$136$	4.19615	0.359817
$137$	20.3935	1.74233	0.871166	−	0.490988i	$-0.163364\pi$
$137$	20.3935	1.74233	0.871166	+	0.490988i	$0.163364\pi$
$138$	0	0
$139$	−4.90508	−0.416043	−0.208022	−	0.978124i	$-0.566702\pi$
$139$	−4.90508	−0.416043	−0.208022	+	0.978124i	$0.566702\pi$
$140$	1.39761	0.118120
$141$	0	0
$142$	1.24814	0.104741
$143$	−4.24814	−0.355247
$144$	0	0
$145$	3.98019	0.330537
$146$	9.62594	0.796648
$147$	0	0
$148$	0.118474	0.00973851
$149$	−23.2230	−1.90250	−0.951252	−	0.308415i	$-0.900201\pi$
$149$	−23.2230	−1.90250	−0.951252	+	0.308415i	$0.900201\pi$
$150$	0	0
$151$	−3.19572	−0.260064	−0.130032	−	0.991510i	$-0.541508\pi$
$151$	−3.19572	−0.260064	−0.130032	+	0.991510i	$0.541508\pi$
$152$	−5.31463	−0.431073
$153$	0	0
$154$	1.39761	0.112623
$155$	−0.957065	−0.0768733
$156$	0	0
$157$	3.74398	0.298803	0.149401	−	0.988777i	$-0.452265\pi$
$157$	3.74398	0.298803	0.149401	+	0.988777i	$0.452265\pi$
$158$	2.59377	0.206349
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	10.9443	0.862532
$162$	0	0
$163$	3.20734	0.251218	0.125609	−	0.992080i	$-0.459911\pi$
$163$	3.20734	0.251218	0.125609	+	0.992080i	$0.459911\pi$
$164$	−1.35425	−0.105749
$165$	0	0
$166$	−13.5971	−1.05534
$167$	−0.0722263	−0.00558904	−0.00279452	−	0.999996i	$-0.500890\pi$
$167$	−0.0722263	−0.00558904	−0.00279452	+	0.999996i	$0.500890\pi$
$168$	0	0
$169$	5.04668	0.388206
$170$	−4.19615	−0.321830
$171$	0	0
$172$	4.73205	0.360815
$173$	−25.4592	−1.93563	−0.967814	−	0.251665i	$-0.919022\pi$
$173$	−25.4592	−1.93563	−0.967814	+	0.251665i	$0.919022\pi$
$174$	0	0
$175$	−1.39761	−0.105650
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0.431927	0.0323743
$179$	13.2515	0.990460	0.495230	−	0.868762i	$-0.335084\pi$
$179$	13.2515	0.990460	0.495230	+	0.868762i	$0.335084\pi$
$180$	0	0
$181$	−3.51277	−0.261102	−0.130551	−	0.991442i	$-0.541675\pi$
$181$	−3.51277	−0.261102	−0.130551	+	0.991442i	$0.541675\pi$
$182$	−5.93725	−0.440099
$183$	0	0
$184$	−7.83071	−0.577288
$185$	−0.118474	−0.00871039
$186$	0	0
$187$	−4.19615	−0.306853
$188$	11.0037	0.802530
$189$	0	0
$190$	5.31463	0.385564
$191$	−26.5777	−1.92309	−0.961547	−	0.274639i	$-0.911442\pi$
$191$	−26.5777	−1.92309	−0.961547	+	0.274639i	$0.911442\pi$
$192$	0	0
$193$	−1.56159	−0.112406	−0.0562029	−	0.998419i	$-0.517899\pi$
$193$	−1.56159	−0.112406	−0.0562029	+	0.998419i	$0.517899\pi$
$194$	−5.45334	−0.391527
$195$	0	0
$196$	−5.04668	−0.360477
$197$	5.43022	0.386887	0.193443	−	0.981111i	$-0.438034\pi$
$197$	5.43022	0.386887	0.193443	+	0.981111i	$0.438034\pi$
$198$	0	0
$199$	8.23695	0.583902	0.291951	−	0.956433i	$-0.405696\pi$
$199$	8.23695	0.583902	0.291951	+	0.956433i	$0.405696\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−17.5508	−1.23487
$203$	5.56276	0.390430
$204$	0	0
$205$	1.35425	0.0945848
$206$	−17.5632	−1.22369
$207$	0	0
$208$	4.24814	0.294555
$209$	5.31463	0.367620
$210$	0	0
$211$	4.47315	0.307945	0.153972	−	0.988075i	$-0.450793\pi$
$211$	4.47315	0.307945	0.153972	+	0.988075i	$0.450793\pi$
$212$	−9.72129	−0.667661
$213$	0	0
$214$	−0.201461	−0.0137716
$215$	−4.73205	−0.322723
$216$	0	0
$217$	−1.33761	−0.0908027
$218$	6.93725	0.469850
$219$	0	0
$220$	1.00000	0.0674200
$221$	17.8258	1.19910
$222$	0	0
$223$	−11.3919	−0.762856	−0.381428	−	0.924398i	$-0.624568\pi$
$223$	−11.3919	−0.762856	−0.381428	+	0.924398i	$0.624568\pi$
$224$	−1.39761	−0.0933820
$225$	0	0
$226$	12.5198	0.832807
$227$	−21.6495	−1.43693	−0.718464	−	0.695564i	$-0.755154\pi$
$227$	−21.6495	−1.43693	−0.718464	+	0.695564i	$0.755154\pi$
$228$	0	0
$229$	4.43724	0.293221	0.146610	−	0.989194i	$-0.453164\pi$
$229$	4.43724	0.293221	0.146610	+	0.989194i	$0.453164\pi$
$230$	7.83071	0.516342
$231$	0	0
$232$	−3.98019	−0.261312
$233$	18.2642	1.19653	0.598265	−	0.801298i	$-0.295857\pi$
$233$	18.2642	1.19653	0.598265	+	0.801298i	$0.295857\pi$
$234$	0	0
$235$	−11.0037	−0.717805
$236$	−1.17260	−0.0763297
$237$	0	0
$238$	−5.86460	−0.380145
$239$	−11.1061	−0.718395	−0.359197	−	0.933262i	$-0.616949\pi$
$239$	−11.1061	−0.718395	−0.359197	+	0.933262i	$0.616949\pi$
$240$	0	0
$241$	15.2915	0.985012	0.492506	−	0.870309i	$-0.336081\pi$
$241$	15.2915	0.985012	0.492506	+	0.870309i	$0.336081\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−1.82740	−0.116987
$245$	5.04668	0.322420
$246$	0	0
$247$	−22.5773	−1.43656
$248$	0.957065	0.0607737
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−3.65437	−0.230662	−0.115331	−	0.993327i	$-0.536793\pi$
$251$	−3.65437	−0.230662	−0.115331	+	0.993327i	$0.536793\pi$
$252$	0	0
$253$	7.83071	0.492313
$254$	−0.0916079	−0.00574799
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.79148	−0.298885	−0.149442	−	0.988770i	$-0.547748\pi$
$257$	−4.79148	−0.298885	−0.149442	+	0.988770i	$0.547748\pi$
$258$	0	0
$259$	−0.165581	−0.0102887
$260$	−4.24814	−0.263458
$261$	0	0
$262$	−14.2519	−0.880484
$263$	7.77659	0.479525	0.239763	−	0.970832i	$-0.422930\pi$
$263$	7.77659	0.479525	0.239763	+	0.970832i	$0.422930\pi$
$264$	0	0
$265$	9.72129	0.597174
$266$	7.42779	0.455427
$267$	0	0
$268$	−6.75186	−0.412436
$269$	−9.21639	−0.561933	−0.280967	−	0.959718i	$-0.590655\pi$
$269$	−9.21639	−0.561933	−0.280967	+	0.959718i	$0.590655\pi$
$270$	0	0
$271$	11.1008	0.674326	0.337163	−	0.941446i	$-0.390533\pi$
$271$	11.1008	0.674326	0.337163	+	0.941446i	$0.390533\pi$
$272$	4.19615	0.254429
$273$	0	0
$274$	20.3935	1.23202
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−7.43481	−0.446714	−0.223357	−	0.974737i	$-0.571702\pi$
$277$	−7.43481	−0.446714	−0.223357	+	0.974737i	$0.571702\pi$
$278$	−4.90508	−0.294187
$279$	0	0
$280$	1.39761	0.0835234
$281$	−21.9518	−1.30953	−0.654766	−	0.755832i	$-0.727233\pi$
$281$	−21.9518	−1.30953	−0.654766	+	0.755832i	$0.727233\pi$
$282$	0	0
$283$	−23.7734	−1.41318	−0.706592	−	0.707622i	$-0.749768\pi$
$283$	−23.7734	−1.41318	−0.706592	+	0.707622i	$0.749768\pi$
$284$	1.24814	0.0740634
$285$	0	0
$286$	−4.24814	−0.251198
$287$	1.89272	0.111723
$288$	0	0
$289$	0.607695	0.0357468
$290$	3.98019	0.233725
$291$	0	0
$292$	9.62594	0.563316
$293$	17.9476	1.04851	0.524256	−	0.851561i	$-0.324344\pi$
$293$	17.9476	1.04851	0.524256	+	0.851561i	$0.324344\pi$
$294$	0	0
$295$	1.17260	0.0682714
$296$	0.118474	0.00688617
$297$	0	0
$298$	−23.2230	−1.34527
$299$	−33.2660	−1.92382
$300$	0	0
$301$	−6.61358	−0.381200
$302$	−3.19572	−0.183893
$303$	0	0
$304$	−5.31463	−0.304815
$305$	1.82740	0.104637
$306$	0	0
$307$	−8.45003	−0.482269	−0.241134	−	0.970492i	$-0.577519\pi$
$307$	−8.45003	−0.482269	−0.241134	+	0.970492i	$0.577519\pi$
$308$	1.39761	0.0796364
$309$	0	0
$310$	−0.957065	−0.0543576
$311$	−0.985927	−0.0559068	−0.0279534	−	0.999609i	$-0.508899\pi$
$311$	−0.985927	−0.0559068	−0.0279534	+	0.999609i	$0.508899\pi$
$312$	0	0
$313$	12.9794	0.733642	0.366821	−	0.930292i	$-0.380446\pi$
$313$	12.9794	0.733642	0.366821	+	0.930292i	$0.380446\pi$
$314$	3.74398	0.211285
$315$	0	0
$316$	2.59377	0.145911
$317$	−17.1603	−0.963817	−0.481908	−	0.876222i	$-0.660056\pi$
$317$	−17.1603	−0.963817	−0.481908	+	0.876222i	$0.660056\pi$
$318$	0	0
$319$	3.98019	0.222848
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	10.9443	0.609902
$323$	−22.3010	−1.24086
$324$	0	0
$325$	4.24814	0.235644
$326$	3.20734	0.177638
$327$	0	0
$328$	−1.35425	−0.0747759
$329$	−15.3790	−0.847870
$330$	0	0
$331$	−25.2915	−1.39015	−0.695073	−	0.718939i	$-0.744628\pi$
$331$	−25.2915	−1.39015	−0.695073	+	0.718939i	$0.744628\pi$
$332$	−13.5971	−0.746237
$333$	0	0
$334$	−0.0722263	−0.00395205
$335$	6.75186	0.368894
$336$	0	0
$337$	−9.25188	−0.503982	−0.251991	−	0.967730i	$-0.581085\pi$
$337$	−9.25188	−0.503982	−0.251991	+	0.967730i	$0.581085\pi$
$338$	5.04668	0.274503
$339$	0	0
$340$	−4.19615	−0.227568
$341$	−0.957065	−0.0518280
$342$	0	0
$343$	16.8366	0.909091
$344$	4.73205	0.255135
$345$	0	0
$346$	−25.4592	−1.36870
$347$	−15.5918	−0.837010	−0.418505	−	0.908214i	$-0.637446\pi$
$347$	−15.5918	−0.837010	−0.418505	+	0.908214i	$0.637446\pi$
$348$	0	0
$349$	3.31919	0.177672	0.0888361	−	0.996046i	$-0.471685\pi$
$349$	3.31919	0.177672	0.0888361	+	0.996046i	$0.471685\pi$
$350$	−1.39761	−0.0747056
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−22.4368	−1.19419	−0.597096	−	0.802170i	$-0.703679\pi$
$353$	−22.4368	−1.19419	−0.597096	+	0.802170i	$0.703679\pi$
$354$	0	0
$355$	−1.24814	−0.0662443
$356$	0.431927	0.0228921
$357$	0	0
$358$	13.2515	0.700361
$359$	−11.0016	−0.580642	−0.290321	−	0.956929i	$-0.593762\pi$
$359$	−11.0016	−0.580642	−0.290321	+	0.956929i	$0.593762\pi$
$360$	0	0
$361$	9.24525	0.486592
$362$	−3.51277	−0.184627
$363$	0	0
$364$	−5.93725	−0.311197
$365$	−9.62594	−0.503845
$366$	0	0
$367$	−26.2276	−1.36907	−0.684534	−	0.728981i	$-0.739994\pi$
$367$	−26.2276	−1.36907	−0.684534	+	0.728981i	$0.739994\pi$
$368$	−7.83071	−0.408204
$369$	0	0
$370$	−0.118474	−0.00615917
$371$	13.5866	0.705381
$372$	0	0
$373$	31.8890	1.65115	0.825575	−	0.564292i	$-0.190851\pi$
$373$	31.8890	1.65115	0.825575	+	0.564292i	$0.190851\pi$
$374$	−4.19615	−0.216978
$375$	0	0
$376$	11.0037	0.567475
$377$	−16.9084	−0.870826
$378$	0	0
$379$	−12.2601	−0.629758	−0.314879	−	0.949132i	$-0.601964\pi$
$379$	−12.2601	−0.629758	−0.314879	+	0.949132i	$0.601964\pi$
$380$	5.31463	0.272635
$381$	0	0
$382$	−26.5777	−1.35983
$383$	−21.1693	−1.08170	−0.540851	−	0.841118i	$-0.681898\pi$
$383$	−21.1693	−1.08170	−0.540851	+	0.841118i	$0.681898\pi$
$384$	0	0
$385$	−1.39761	−0.0712290
$386$	−1.56159	−0.0794829
$387$	0	0
$388$	−5.45334	−0.276851
$389$	23.7193	1.20262	0.601308	−	0.799017i	$-0.294646\pi$
$389$	23.7193	1.20262	0.601308	+	0.799017i	$0.294646\pi$
$390$	0	0
$391$	−32.8589	−1.66174
$392$	−5.04668	−0.254896
$393$	0	0
$394$	5.43022	0.273570
$395$	−2.59377	−0.130507
$396$	0	0
$397$	35.6587	1.78966	0.894829	−	0.446409i	$-0.147297\pi$
$397$	35.6587	1.78966	0.894829	+	0.446409i	$0.147297\pi$
$398$	8.23695	0.412881
$399$	0	0
$400$	1.00000	0.0500000
$401$	−14.1491	−0.706571	−0.353286	−	0.935515i	$-0.614936\pi$
$401$	−14.1491	−0.706571	−0.353286	+	0.935515i	$0.614936\pi$
$402$	0	0
$403$	4.06574	0.202529
$404$	−17.5508	−0.873186
$405$	0	0
$406$	5.56276	0.276075
$407$	−0.118474	−0.00587254
$408$	0	0
$409$	−33.2907	−1.64612	−0.823059	−	0.567956i	$-0.807734\pi$
$409$	−33.2907	−1.64612	−0.823059	+	0.567956i	$0.807734\pi$
$410$	1.35425	0.0668816
$411$	0	0
$412$	−17.5632	−0.865276
$413$	1.63884	0.0806420
$414$	0	0
$415$	13.5971	0.667454
$416$	4.24814	0.208282
$417$	0	0
$418$	5.31463	0.259947
$419$	−9.84104	−0.480766	−0.240383	−	0.970678i	$-0.577273\pi$
$419$	−9.84104	−0.480766	−0.240383	+	0.970678i	$0.577273\pi$
$420$	0	0
$421$	33.8811	1.65127	0.825633	−	0.564208i	$-0.190818\pi$
$421$	33.8811	1.65127	0.825633	+	0.564208i	$0.190818\pi$
$422$	4.47315	0.217750
$423$	0	0
$424$	−9.72129	−0.472108
$425$	4.19615	0.203543
$426$	0	0
$427$	2.55400	0.123597
$428$	−0.201461	−0.00973797
$429$	0	0
$430$	−4.73205	−0.228200
$431$	−38.7317	−1.86564	−0.932819	−	0.360345i	$-0.882659\pi$
$431$	−38.7317	−1.86564	−0.932819	+	0.360345i	$0.882659\pi$
$432$	0	0
$433$	−29.7601	−1.43018	−0.715089	−	0.699033i	$-0.753614\pi$
$433$	−29.7601	−1.43018	−0.715089	+	0.699033i	$0.753614\pi$
$434$	−1.33761	−0.0642072
$435$	0	0
$436$	6.93725	0.332234
$437$	41.6173	1.99083
$438$	0	0
$439$	24.6004	1.17411	0.587056	−	0.809546i	$-0.300287\pi$
$439$	24.6004	1.17411	0.587056	+	0.809546i	$0.300287\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	17.8258	0.847889
$443$	−13.4286	−0.638013	−0.319006	−	0.947753i	$-0.603349\pi$
$443$	−13.4286	−0.638013	−0.319006	+	0.947753i	$0.603349\pi$
$444$	0	0
$445$	−0.431927	−0.0204753
$446$	−11.3919	−0.539421
$447$	0	0
$448$	−1.39761	−0.0660310
$449$	−4.79597	−0.226336	−0.113168	−	0.993576i	$-0.536100\pi$
$449$	−4.79597	−0.226336	−0.113168	+	0.993576i	$0.536100\pi$
$450$	0	0
$451$	1.35425	0.0637691
$452$	12.5198	0.588883
$453$	0	0
$454$	−21.6495	−1.01606
$455$	5.93725	0.278343
$456$	0	0
$457$	16.1458	0.755267	0.377634	−	0.925955i	$-0.376738\pi$
$457$	16.1458	0.755267	0.377634	+	0.925955i	$0.376738\pi$
$458$	4.43724	0.207338
$459$	0	0
$460$	7.83071	0.365109
$461$	−18.6755	−0.869805	−0.434902	−	0.900478i	$-0.643217\pi$
$461$	−18.6755	−0.869805	−0.434902	+	0.900478i	$0.643217\pi$
$462$	0	0
$463$	−16.8696	−0.783996	−0.391998	−	0.919966i	$-0.628216\pi$
$463$	−16.8696	−0.783996	−0.391998	+	0.919966i	$0.628216\pi$
$464$	−3.98019	−0.184776
$465$	0	0
$466$	18.2642	0.846075
$467$	27.3270	1.26454	0.632271	−	0.774747i	$-0.282123\pi$
$467$	27.3270	1.26454	0.632271	+	0.774747i	$0.282123\pi$
$468$	0	0
$469$	9.43649	0.435737
$470$	−11.0037	−0.507565
$471$	0	0
$472$	−1.17260	−0.0539732
$473$	−4.73205	−0.217580
$474$	0	0
$475$	−5.31463	−0.243852
$476$	−5.86460	−0.268803
$477$	0	0
$478$	−11.1061	−0.507982
$479$	−5.62049	−0.256807	−0.128403	−	0.991722i	$-0.540985\pi$
$479$	−5.62049	−0.256807	−0.128403	+	0.991722i	$0.540985\pi$
$480$	0	0
$481$	0.503294	0.0229482
$482$	15.2915	0.696509
$483$	0	0
$484$	1.00000	0.0454545
$485$	5.45334	0.247623
$486$	0	0
$487$	31.5562	1.42995	0.714973	−	0.699152i	$-0.246439\pi$
$487$	31.5562	1.42995	0.714973	+	0.699152i	$0.246439\pi$
$488$	−1.82740	−0.0827226
$489$	0	0
$490$	5.04668	0.227986
$491$	3.65163	0.164796	0.0823979	−	0.996600i	$-0.473742\pi$
$491$	3.65163	0.164796	0.0823979	+	0.996600i	$0.473742\pi$
$492$	0	0
$493$	−16.7015	−0.752197
$494$	−22.5773	−1.01580
$495$	0	0
$496$	0.957065	0.0429735
$497$	−1.74441	−0.0782477
$498$	0	0
$499$	24.0632	1.07722	0.538609	−	0.842556i	$-0.318950\pi$
$499$	24.0632	1.07722	0.538609	+	0.842556i	$0.318950\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−3.65437	−0.163103
$503$	23.7953	1.06098	0.530489	−	0.847692i	$-0.322008\pi$
$503$	23.7953	1.06098	0.530489	+	0.847692i	$0.322008\pi$
$504$	0	0
$505$	17.5508	0.781002
$506$	7.83071	0.348118
$507$	0	0
$508$	−0.0916079	−0.00406444
$509$	−13.5889	−0.602317	−0.301158	−	0.953574i	$-0.597373\pi$
$509$	−13.5889	−0.602317	−0.301158	+	0.953574i	$0.597373\pi$
$510$	0	0
$511$	−13.4533	−0.595141
$512$	1.00000	0.0441942
$513$	0	0
$514$	−4.79148	−0.211343
$515$	17.5632	0.773927
$516$	0	0
$517$	−11.0037	−0.483944
$518$	−0.165581	−0.00727521
$519$	0	0
$520$	−4.24814	−0.186293
$521$	5.06980	0.222112	0.111056	−	0.993814i	$-0.464577\pi$
$521$	5.06980	0.222112	0.111056	+	0.993814i	$0.464577\pi$
$522$	0	0
$523$	−11.5112	−0.503350	−0.251675	−	0.967812i	$-0.580981\pi$
$523$	−11.5112	−0.503350	−0.251675	+	0.967812i	$0.580981\pi$
$524$	−14.2519	−0.622596
$525$	0	0
$526$	7.77659	0.339075
$527$	4.01599	0.174939
$528$	0	0
$529$	38.3201	1.66609
$530$	9.72129	0.422266
$531$	0	0
$532$	7.42779	0.322036
$533$	−5.75304	−0.249192
$534$	0	0
$535$	0.201461	0.00870991
$536$	−6.75186	−0.291636
$537$	0	0
$538$	−9.21639	−0.397347
$539$	5.04668	0.217376
$540$	0	0
$541$	−10.1801	−0.437676	−0.218838	−	0.975761i	$-0.570227\pi$
$541$	−10.1801	−0.437676	−0.218838	+	0.975761i	$0.570227\pi$
$542$	11.1008	0.476820
$543$	0	0
$544$	4.19615	0.179909
$545$	−6.93725	−0.297159
$546$	0	0
$547$	−30.2217	−1.29219	−0.646094	−	0.763258i	$-0.723599\pi$
$547$	−30.2217	−1.29219	−0.646094	+	0.763258i	$0.723599\pi$
$548$	20.3935	0.871166
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	21.1532	0.901157
$552$	0	0
$553$	−3.62508	−0.154154
$554$	−7.43481	−0.315875
$555$	0	0
$556$	−4.90508	−0.208022
$557$	−20.1850	−0.855264	−0.427632	−	0.903953i	$-0.640652\pi$
$557$	−20.1850	−0.855264	−0.427632	+	0.903953i	$0.640652\pi$
$558$	0	0
$559$	20.1024	0.850241
$560$	1.39761	0.0590599
$561$	0	0
$562$	−21.9518	−0.925979
$563$	19.6090	0.826421	0.413211	−	0.910635i	$-0.364407\pi$
$563$	19.6090	0.826421	0.413211	+	0.910635i	$0.364407\pi$
$564$	0	0
$565$	−12.5198	−0.526713
$566$	−23.7734	−0.999271
$567$	0	0
$568$	1.24814	0.0523707
$569$	10.4523	0.438184	0.219092	−	0.975704i	$-0.429691\pi$
$569$	10.4523	0.438184	0.219092	+	0.975704i	$0.429691\pi$
$570$	0	0
$571$	4.80973	0.201281	0.100640	−	0.994923i	$-0.467911\pi$
$571$	4.80973	0.201281	0.100640	+	0.994923i	$0.467911\pi$
$572$	−4.24814	−0.177624
$573$	0	0
$574$	1.89272	0.0790004
$575$	−7.83071	−0.326563
$576$	0	0
$577$	2.32164	0.0966513	0.0483257	−	0.998832i	$-0.484611\pi$
$577$	2.32164	0.0966513	0.0483257	+	0.998832i	$0.484611\pi$
$578$	0.607695	0.0252768
$579$	0	0
$580$	3.98019	0.165268
$581$	19.0035	0.788396
$582$	0	0
$583$	9.72129	0.402615
$584$	9.62594	0.398324
$585$	0	0
$586$	17.9476	0.741409
$587$	−28.8540	−1.19093	−0.595465	−	0.803381i	$-0.703032\pi$
$587$	−28.8540	−1.19093	−0.595465	+	0.803381i	$0.703032\pi$
$588$	0	0
$589$	−5.08644	−0.209583
$590$	1.17260	0.0482751
$591$	0	0
$592$	0.118474	0.00486925
$593$	20.0744	0.824357	0.412178	−	0.911103i	$-0.364768\pi$
$593$	20.0744	0.824357	0.412178	+	0.911103i	$0.364768\pi$
$594$	0	0
$595$	5.86460	0.240425
$596$	−23.2230	−0.951252
$597$	0	0
$598$	−33.2660	−1.36035
$599$	23.6324	0.965595	0.482797	−	0.875732i	$-0.339621\pi$
$599$	23.6324	0.965595	0.482797	+	0.875732i	$0.339621\pi$
$600$	0	0
$601$	29.1735	1.19001	0.595005	−	0.803722i	$-0.297150\pi$
$601$	29.1735	1.19001	0.595005	+	0.803722i	$0.297150\pi$
$602$	−6.61358	−0.269549
$603$	0	0
$604$	−3.19572	−0.130032
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−24.5869	−0.997951	−0.498976	−	0.866616i	$-0.666290\pi$
$607$	−24.5869	−0.997951	−0.498976	+	0.866616i	$0.666290\pi$
$608$	−5.31463	−0.215537
$609$	0	0
$610$	1.82740	0.0739893
$611$	46.7454	1.89112
$612$	0	0
$613$	14.2222	0.574428	0.287214	−	0.957866i	$-0.407271\pi$
$613$	14.2222	0.574428	0.287214	+	0.957866i	$0.407271\pi$
$614$	−8.45003	−0.341015
$615$	0	0
$616$	1.39761	0.0563114
$617$	−4.26221	−0.171590	−0.0857951	−	0.996313i	$-0.527343\pi$
$617$	−4.26221	−0.171590	−0.0857951	+	0.996313i	$0.527343\pi$
$618$	0	0
$619$	−8.52026	−0.342458	−0.171229	−	0.985231i	$-0.554774\pi$
$619$	−8.52026	−0.342458	−0.171229	+	0.985231i	$0.554774\pi$
$620$	−0.957065	−0.0384367
$621$	0	0
$622$	−0.985927	−0.0395321
$623$	−0.603667	−0.0241854
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.9794	0.518763
$627$	0	0
$628$	3.74398	0.149401
$629$	0.497135	0.0198221
$630$	0	0
$631$	30.4733	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$631$	30.4733	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$632$	2.59377	0.103174
$633$	0	0
$634$	−17.1603	−0.681521
$635$	0.0916079	0.00363535
$636$	0	0
$637$	−21.4390	−0.849443
$638$	3.98019	0.157577
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	6.84147	0.270222	0.135111	−	0.990830i	$-0.456861\pi$
$641$	6.84147	0.270222	0.135111	+	0.990830i	$0.456861\pi$
$642$	0	0
$643$	−9.01611	−0.355561	−0.177780	−	0.984070i	$-0.556892\pi$
$643$	−9.01611	−0.355561	−0.177780	+	0.984070i	$0.556892\pi$
$644$	10.9443	0.431266
$645$	0	0
$646$	−22.3010	−0.877420
$647$	40.6301	1.59733	0.798667	−	0.601773i	$-0.205539\pi$
$647$	40.6301	1.59733	0.798667	+	0.601773i	$0.205539\pi$
$648$	0	0
$649$	1.17260	0.0460285
$650$	4.24814	0.166626
$651$	0	0
$652$	3.20734	0.125609
$653$	−30.1317	−1.17915	−0.589573	−	0.807715i	$-0.700704\pi$
$653$	−30.1317	−1.17915	−0.589573	+	0.807715i	$0.700704\pi$
$654$	0	0
$655$	14.2519	0.556867
$656$	−1.35425	−0.0528745
$657$	0	0
$658$	−15.3790	−0.599535
$659$	50.6336	1.97240	0.986202	−	0.165547i	$-0.0529390\pi$
$659$	50.6336	1.97240	0.986202	+	0.165547i	$0.0529390\pi$
$660$	0	0
$661$	32.4104	1.26062	0.630309	−	0.776344i	$-0.282928\pi$
$661$	32.4104	1.26062	0.630309	+	0.776344i	$0.282928\pi$
$662$	−25.2915	−0.982982
$663$	0	0
$664$	−13.5971	−0.527669
$665$	−7.42779	−0.288037
$666$	0	0
$667$	31.1677	1.20682
$668$	−0.0722263	−0.00279452
$669$	0	0
$670$	6.75186	0.260847
$671$	1.82740	0.0705460
$672$	0	0
$673$	−0.149905	−0.00577840	−0.00288920	−	0.999996i	$-0.500920\pi$
$673$	−0.149905	−0.00577840	−0.00288920	+	0.999996i	$0.500920\pi$
$674$	−9.25188	−0.356369
$675$	0	0
$676$	5.04668	0.194103
$677$	20.0226	0.769531	0.384765	−	0.923014i	$-0.374282\pi$
$677$	20.0226	0.769531	0.384765	+	0.923014i	$0.374282\pi$
$678$	0	0
$679$	7.62166	0.292493
$680$	−4.19615	−0.160915
$681$	0	0
$682$	−0.957065	−0.0366479
$683$	41.7322	1.59684	0.798420	−	0.602101i	$-0.205670\pi$
$683$	41.7322	1.59684	0.798420	+	0.602101i	$0.205670\pi$
$684$	0	0
$685$	−20.3935	−0.779195
$686$	16.8366	0.642824
$687$	0	0
$688$	4.73205	0.180408
$689$	−41.2974	−1.57331
$690$	0	0
$691$	44.7638	1.70290	0.851448	−	0.524439i	$-0.175725\pi$
$691$	44.7638	1.70290	0.851448	+	0.524439i	$0.175725\pi$
$692$	−25.4592	−0.967814
$693$	0	0
$694$	−15.5918	−0.591855
$695$	4.90508	0.186060
$696$	0	0
$697$	−5.68263	−0.215245
$698$	3.31919	0.125633
$699$	0	0
$700$	−1.39761	−0.0528248
$701$	−32.1479	−1.21421	−0.607106	−	0.794621i	$-0.707670\pi$
$701$	−32.1479	−1.21421	−0.607106	+	0.794621i	$0.707670\pi$
$702$	0	0
$703$	−0.629645	−0.0237475
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−22.4368	−0.844422
$707$	24.5293	0.922518
$708$	0	0
$709$	33.9064	1.27338	0.636691	−	0.771119i	$-0.280303\pi$
$709$	33.9064	1.27338	0.636691	+	0.771119i	$0.280303\pi$
$710$	−1.24814	−0.0468418
$711$	0	0
$712$	0.431927	0.0161871
$713$	−7.49450	−0.280671
$714$	0	0
$715$	4.24814	0.158871
$716$	13.2515	0.495230
$717$	0	0
$718$	−11.0016	−0.410576
$719$	−2.38482	−0.0889388	−0.0444694	−	0.999011i	$-0.514160\pi$
$719$	−2.38482	−0.0889388	−0.0444694	+	0.999011i	$0.514160\pi$
$720$	0	0
$721$	24.5466	0.914161
$722$	9.24525	0.344073
$723$	0	0
$724$	−3.51277	−0.130551
$725$	−3.98019	−0.147821
$726$	0	0
$727$	5.58831	0.207259	0.103630	−	0.994616i	$-0.466954\pi$
$727$	5.58831	0.207259	0.103630	+	0.994616i	$0.466954\pi$
$728$	−5.93725	−0.220049
$729$	0	0
$730$	−9.62594	−0.356272
$731$	19.8564	0.734416
$732$	0	0
$733$	−43.7520	−1.61602	−0.808009	−	0.589170i	$-0.799455\pi$
$733$	−43.7520	−1.61602	−0.808009	+	0.589170i	$0.799455\pi$
$734$	−26.2276	−0.968078
$735$	0	0
$736$	−7.83071	−0.288644
$737$	6.75186	0.248708
$738$	0	0
$739$	13.3060	0.489469	0.244735	−	0.969590i	$-0.421299\pi$
$739$	13.3060	0.489469	0.244735	+	0.969590i	$0.421299\pi$
$740$	−0.118474	−0.00435519
$741$	0	0
$742$	13.5866	0.498780
$743$	−38.2515	−1.40331	−0.701655	−	0.712517i	$-0.747555\pi$
$743$	−38.2515	−1.40331	−0.701655	+	0.712517i	$0.747555\pi$
$744$	0	0
$745$	23.2230	0.850826
$746$	31.8890	1.16754
$747$	0	0
$748$	−4.19615	−0.153427
$749$	0.281564	0.0102881
$750$	0	0
$751$	49.4741	1.80533	0.902667	−	0.430339i	$-0.141606\pi$
$751$	49.4741	1.80533	0.902667	+	0.430339i	$0.141606\pi$
$752$	11.0037	0.401265
$753$	0	0
$754$	−16.9084	−0.615767
$755$	3.19572	0.116304
$756$	0	0
$757$	15.7122	0.571071	0.285536	−	0.958368i	$-0.407829\pi$
$757$	15.7122	0.571071	0.285536	+	0.958368i	$0.407829\pi$
$758$	−12.2601	−0.445306
$759$	0	0
$760$	5.31463	0.192782
$761$	25.8399	0.936696	0.468348	−	0.883544i	$-0.344849\pi$
$761$	25.8399	0.936696	0.468348	+	0.883544i	$0.344849\pi$
$762$	0	0
$763$	−9.69560	−0.351004
$764$	−26.5777	−0.961547
$765$	0	0
$766$	−21.1693	−0.764879
$767$	−4.98136	−0.179867
$768$	0	0
$769$	5.23503	0.188780	0.0943900	−	0.995535i	$-0.469910\pi$
$769$	5.23503	0.188780	0.0943900	+	0.995535i	$0.469910\pi$
$770$	−1.39761	−0.0503665
$771$	0	0
$772$	−1.56159	−0.0562029
$773$	−30.7776	−1.10699	−0.553497	−	0.832851i	$-0.686707\pi$
$773$	−30.7776	−1.10699	−0.553497	+	0.832851i	$0.686707\pi$
$774$	0	0
$775$	0.957065	0.0343788
$776$	−5.45334	−0.195764
$777$	0	0
$778$	23.7193	0.850378
$779$	7.19733	0.257871
$780$	0	0
$781$	−1.24814	−0.0446619
$782$	−32.8589	−1.17503
$783$	0	0
$784$	−5.04668	−0.180238
$785$	−3.74398	−0.133629
$786$	0	0
$787$	11.5879	0.413063	0.206532	−	0.978440i	$-0.433782\pi$
$787$	11.5879	0.413063	0.206532	+	0.978440i	$0.433782\pi$
$788$	5.43022	0.193443
$789$	0	0
$790$	−2.59377	−0.0922820
$791$	−17.4979	−0.622153
$792$	0	0
$793$	−7.76305	−0.275674
$794$	35.6587	1.26548
$795$	0	0
$796$	8.23695	0.291951
$797$	−34.8992	−1.23619	−0.618097	−	0.786102i	$-0.712096\pi$
$797$	−34.8992	−1.23619	−0.618097	+	0.786102i	$0.712096\pi$
$798$	0	0
$799$	46.1734	1.63350
$800$	1.00000	0.0353553
$801$	0	0
$802$	−14.1491	−0.499621
$803$	−9.62594	−0.339692
$804$	0	0
$805$	−10.9443	−0.385736
$806$	4.06574	0.143210
$807$	0	0
$808$	−17.5508	−0.617436
$809$	49.8791	1.75366	0.876829	−	0.480803i	$-0.159655\pi$
$809$	49.8791	1.75366	0.876829	+	0.480803i	$0.159655\pi$
$810$	0	0
$811$	11.0546	0.388178	0.194089	−	0.980984i	$-0.437825\pi$
$811$	11.0546	0.388178	0.194089	+	0.980984i	$0.437825\pi$
$812$	5.56276	0.195215
$813$	0	0
$814$	−0.118474	−0.00415251
$815$	−3.20734	−0.112348
$816$	0	0
$817$	−25.1491	−0.879855
$818$	−33.2907	−1.16398
$819$	0	0
$820$	1.35425	0.0472924
$821$	−34.4750	−1.20318	−0.601592	−	0.798803i	$-0.705467\pi$
$821$	−34.4750	−1.20318	−0.601592	+	0.798803i	$0.705467\pi$
$822$	0	0
$823$	−41.0471	−1.43081	−0.715407	−	0.698708i	$-0.753759\pi$
$823$	−41.0471	−1.43081	−0.715407	+	0.698708i	$0.753759\pi$
$824$	−17.5632	−0.611843
$825$	0	0
$826$	1.63884	0.0570225
$827$	11.2842	0.392389	0.196194	−	0.980565i	$-0.437142\pi$
$827$	11.2842	0.392389	0.196194	+	0.980565i	$0.437142\pi$
$828$	0	0
$829$	21.0688	0.731751	0.365875	−	0.930664i	$-0.380770\pi$
$829$	21.0688	0.731751	0.365875	+	0.930664i	$0.380770\pi$
$830$	13.5971	0.471961
$831$	0	0
$832$	4.24814	0.147278
$833$	−21.1766	−0.733727
$834$	0	0
$835$	0.0722263	0.00249949
$836$	5.31463	0.183810
$837$	0	0
$838$	−9.84104	−0.339953
$839$	39.6822	1.36998	0.684992	−	0.728551i	$-0.259806\pi$
$839$	39.6822	1.36998	0.684992	+	0.728551i	$0.259806\pi$
$840$	0	0
$841$	−13.1581	−0.453727
$842$	33.8811	1.16762
$843$	0	0
$844$	4.47315	0.153972
$845$	−5.04668	−0.173611
$846$	0	0
$847$	−1.39761	−0.0480226
$848$	−9.72129	−0.333830
$849$	0	0
$850$	4.19615	0.143927
$851$	−0.927737	−0.0318024
$852$	0	0
$853$	53.9572	1.84746	0.923730	−	0.383043i	$-0.125124\pi$
$853$	53.9572	1.84746	0.923730	+	0.383043i	$0.125124\pi$
$854$	2.55400	0.0873961
$855$	0	0
$856$	−0.201461	−0.00688579
$857$	32.2697	1.10231	0.551156	−	0.834402i	$-0.314187\pi$
$857$	32.2697	1.10231	0.551156	+	0.834402i	$0.314187\pi$
$858$	0	0
$859$	−16.3919	−0.559285	−0.279642	−	0.960104i	$-0.590216\pi$
$859$	−16.3919	−0.559285	−0.279642	+	0.960104i	$0.590216\pi$
$860$	−4.73205	−0.161362
$861$	0	0
$862$	−38.7317	−1.31921
$863$	−19.6176	−0.667792	−0.333896	−	0.942610i	$-0.608363\pi$
$863$	−19.6176	−0.667792	−0.333896	+	0.942610i	$0.608363\pi$
$864$	0	0
$865$	25.4592	0.865640
$866$	−29.7601	−1.01129
$867$	0	0
$868$	−1.33761	−0.0454013
$869$	−2.59377	−0.0879875
$870$	0	0
$871$	−28.6828	−0.971881
$872$	6.93725	0.234925
$873$	0	0
$874$	41.6173	1.40773
$875$	1.39761	0.0472480
$876$	0	0
$877$	0.508325	0.0171649	0.00858246	−	0.999963i	$-0.497268\pi$
$877$	0.508325	0.0171649	0.00858246	+	0.999963i	$0.497268\pi$
$878$	24.6004	0.830223
$879$	0	0
$880$	1.00000	0.0337100
$881$	−10.5839	−0.356579	−0.178290	−	0.983978i	$-0.557056\pi$
$881$	−10.5839	−0.356579	−0.178290	+	0.983978i	$0.557056\pi$
$882$	0	0
$883$	−18.1160	−0.609654	−0.304827	−	0.952408i	$-0.598599\pi$
$883$	−18.1160	−0.609654	−0.304827	+	0.952408i	$0.598599\pi$
$884$	17.8258	0.599548
$885$	0	0
$886$	−13.4286	−0.451143
$887$	−48.4251	−1.62596	−0.812978	−	0.582295i	$-0.802155\pi$
$887$	−48.4251	−1.62596	−0.812978	+	0.582295i	$0.802155\pi$
$888$	0	0
$889$	0.128032	0.00429407
$890$	−0.431927	−0.0144782
$891$	0	0
$892$	−11.3919	−0.381428
$893$	−58.4808	−1.95698
$894$	0	0
$895$	−13.2515	−0.442947
$896$	−1.39761	−0.0466910
$897$	0	0
$898$	−4.79597	−0.160044
$899$	−3.80930	−0.127047
$900$	0	0
$901$	−40.7920	−1.35898
$902$	1.35425	0.0450915
$903$	0	0
$904$	12.5198	0.416403
$905$	3.51277	0.116769
$906$	0	0
$907$	33.7338	1.12011	0.560056	−	0.828455i	$-0.310780\pi$
$907$	33.7338	1.12011	0.560056	+	0.828455i	$0.310780\pi$
$908$	−21.6495	−0.718464
$909$	0	0
$910$	5.93725	0.196818
$911$	−47.5081	−1.57401	−0.787006	−	0.616945i	$-0.788370\pi$
$911$	−47.5081	−1.57401	−0.787006	+	0.616945i	$0.788370\pi$
$912$	0	0
$913$	13.5971	0.449998
$914$	16.1458	0.534055
$915$	0	0
$916$	4.43724	0.146610
$917$	19.9186	0.657771
$918$	0	0
$919$	−32.1034	−1.05899	−0.529497	−	0.848312i	$-0.677619\pi$
$919$	−32.1034	−1.05899	−0.529497	+	0.848312i	$0.677619\pi$
$920$	7.83071	0.258171
$921$	0	0
$922$	−18.6755	−0.615045
$923$	5.30226	0.174526
$924$	0	0
$925$	0.118474	0.00389540
$926$	−16.8696	−0.554369
$927$	0	0
$928$	−3.98019	−0.130656
$929$	8.57941	0.281481	0.140741	−	0.990047i	$-0.455052\pi$
$929$	8.57941	0.281481	0.140741	+	0.990047i	$0.455052\pi$
$930$	0	0
$931$	26.8212	0.879029
$932$	18.2642	0.598265
$933$	0	0
$934$	27.3270	0.894166
$935$	4.19615	0.137229
$936$	0	0
$937$	−15.6862	−0.512444	−0.256222	−	0.966618i	$-0.582478\pi$
$937$	−15.6862	−0.512444	−0.256222	+	0.966618i	$0.582478\pi$
$938$	9.43649	0.308112
$939$	0	0
$940$	−11.0037	−0.358903
$941$	26.9355	0.878074	0.439037	−	0.898469i	$-0.355320\pi$
$941$	26.9355	0.878074	0.439037	+	0.898469i	$0.355320\pi$
$942$	0	0
$943$	10.6047	0.345338
$944$	−1.17260	−0.0381648
$945$	0	0
$946$	−4.73205	−0.153852
$947$	−23.7524	−0.771850	−0.385925	−	0.922530i	$-0.626118\pi$
$947$	−23.7524	−0.771850	−0.385925	+	0.922530i	$0.626118\pi$
$948$	0	0
$949$	40.8923	1.32742
$950$	−5.31463	−0.172429
$951$	0	0
$952$	−5.86460	−0.190073
$953$	−34.0434	−1.10277	−0.551386	−	0.834250i	$-0.685901\pi$
$953$	−34.0434	−1.10277	−0.551386	+	0.834250i	$0.685901\pi$
$954$	0	0
$955$	26.5777	0.860034
$956$	−11.1061	−0.359197
$957$	0	0
$958$	−5.62049	−0.181590
$959$	−28.5022	−0.920384
$960$	0	0
$961$	−30.0840	−0.970452
$962$	0.503294	0.0162269
$963$	0	0
$964$	15.2915	0.492506
$965$	1.56159	0.0502694
$966$	0	0
$967$	32.9097	1.05830	0.529152	−	0.848527i	$-0.322510\pi$
$967$	32.9097	1.05830	0.529152	+	0.848527i	$0.322510\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	5.45334	0.175096
$971$	30.8666	0.990557	0.495279	−	0.868734i	$-0.335066\pi$
$971$	30.8666	0.990557	0.495279	+	0.868734i	$0.335066\pi$
$972$	0	0
$973$	6.85540	0.219774
$974$	31.5562	1.01113
$975$	0	0
$976$	−1.82740	−0.0584937
$977$	−7.42448	−0.237530	−0.118765	−	0.992922i	$-0.537894\pi$
$977$	−7.42448	−0.237530	−0.118765	+	0.992922i	$0.537894\pi$
$978$	0	0
$979$	−0.431927	−0.0138044
$980$	5.04668	0.161210
$981$	0	0
$982$	3.65163	0.116528
$983$	−28.1581	−0.898104	−0.449052	−	0.893506i	$-0.648238\pi$
$983$	−28.1581	−0.898104	−0.449052	+	0.893506i	$0.648238\pi$
$984$	0	0
$985$	−5.43022	−0.173021
$986$	−16.7015	−0.531883
$987$	0	0
$988$	−22.5773	−0.718279
$989$	−37.0553	−1.17829
$990$	0	0
$991$	−46.5021	−1.47719	−0.738593	−	0.674151i	$-0.764510\pi$
$991$	−46.5021	−1.47719	−0.738593	+	0.674151i	$0.764510\pi$
$992$	0.957065	0.0303868
$993$	0	0
$994$	−1.74441	−0.0553295
$995$	−8.23695	−0.261129
$996$	0	0
$997$	40.0438	1.26820	0.634100	−	0.773251i	$-0.281371\pi$
$997$	40.0438	1.26820	0.634100	+	0.773251i	$0.281371\pi$
$998$	24.0632	0.761708
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8910.2.a.bt.1.3	yes	4
3.2	odd	2		8910.2.a.br.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8910.2.a.br.1.3	✓	4	3.2	odd	2
8910.2.a.bt.1.3	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8910.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.39761 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +4.24814 q^{13} -1.39761 q^{14} +1.00000 q^{16} +4.19615 q^{17} -5.31463 q^{19} -1.00000 q^{20} -1.00000 q^{22} -7.83071 q^{23} +1.00000 q^{25} +4.24814 q^{26} -1.39761 q^{28} -3.98019 q^{29} +0.957065 q^{31} +1.00000 q^{32} +4.19615 q^{34} +1.39761 q^{35} +0.118474 q^{37} -5.31463 q^{38} -1.00000 q^{40} -1.35425 q^{41} +4.73205 q^{43} -1.00000 q^{44} -7.83071 q^{46} +11.0037 q^{47} -5.04668 q^{49} +1.00000 q^{50} +4.24814 q^{52} -9.72129 q^{53} +1.00000 q^{55} -1.39761 q^{56} -3.98019 q^{58} -1.17260 q^{59} -1.82740 q^{61} +0.957065 q^{62} +1.00000 q^{64} -4.24814 q^{65} -6.75186 q^{67} +4.19615 q^{68} +1.39761 q^{70} +1.24814 q^{71} +9.62594 q^{73} +0.118474 q^{74} -5.31463 q^{76} +1.39761 q^{77} +2.59377 q^{79} -1.00000 q^{80} -1.35425 q^{82} -13.5971 q^{83} -4.19615 q^{85} +4.73205 q^{86} -1.00000 q^{88} +0.431927 q^{89} -5.93725 q^{91} -7.83071 q^{92} +11.0037 q^{94} +5.31463 q^{95} -5.45334 q^{97} -5.04668 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{7} + 4 q^{8} - 4 q^{10} - 4 q^{11} + 6 q^{13} - 6 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{22} - 2 q^{23} + 4 q^{25} + 6 q^{26} - 6 q^{28} + 2 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^7 + 4 * q^8 - 4 * q^10 - 4 * q^11 + 6 * q^13 - 6 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^22 - 2 * q^23 + 4 * q^25 + 6 * q^26 - 6 * q^28 + 2 * q^29 - 10 * q^31 + 4 * q^32 - 4 * q^34 + 6 * q^35 - 4 * q^37 + 4 * q^38 - 4 * q^40 - 16 * q^41 + 12 * q^43 - 4 * q^44 - 2 * q^46 - 2 * q^47 + 12 * q^49 + 4 * q^50 + 6 * q^52 - 10 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^62 + 4 * q^64 - 6 * q^65 - 38 * q^67 - 4 * q^68 + 6 * q^70 - 6 * q^71 + 10 * q^73 - 4 * q^74 + 4 * q^76 + 6 * q^77 - 10 * q^79 - 4 * q^80 - 16 * q^82 - 12 * q^83 + 4 * q^85 + 12 * q^86 - 4 * q^88 - 4 * q^89 + 8 * q^91 - 2 * q^92 - 2 * q^94 - 4 * q^95 + 14 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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