LMFDB - Embedded newform 8281.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.86620 q^{2} -3.34889 q^{3} +1.48270 q^{4} -0.866198 q^{5} +6.24970 q^{6} +0.965392 q^{8} +8.21509 q^{9} +1.61650 q^{10} +3.86620 q^{11} -4.96539 q^{12} +2.90081 q^{15} -4.76700 q^{16} -3.34889 q^{17} -15.3310 q^{18} +5.38350 q^{19} -1.28431 q^{20} -7.21509 q^{22} -5.24970 q^{23} -3.23300 q^{24} -4.24970 q^{25} -17.4648 q^{27} +1.69779 q^{29} -5.41348 q^{30} -7.56399 q^{31} +6.96539 q^{32} -12.9475 q^{33} +6.24970 q^{34} +12.1805 q^{36} +4.83159 q^{37} -10.0467 q^{38} -0.836221 q^{40} +4.06922 q^{41} +4.03461 q^{43} +5.73240 q^{44} -7.11590 q^{45} +9.79698 q^{46} +3.65111 q^{47} +15.9642 q^{48} +7.93078 q^{50} +11.2151 q^{51} -0.215092 q^{53} +32.5928 q^{54} -3.34889 q^{55} -18.0288 q^{57} -3.16841 q^{58} +2.78491 q^{59} +4.30101 q^{60} +9.03461 q^{61} +14.1159 q^{62} -3.46479 q^{64} +24.1626 q^{66} +7.66318 q^{67} -4.96539 q^{68} +17.5807 q^{69} -4.90081 q^{71} +7.93078 q^{72} -15.5461 q^{73} -9.01671 q^{74} +14.2318 q^{75} +7.98210 q^{76} +9.43018 q^{79} +4.12917 q^{80} +33.8425 q^{81} -7.59396 q^{82} +4.09919 q^{83} +2.90081 q^{85} -7.52938 q^{86} -5.68571 q^{87} +3.73240 q^{88} +0.418110 q^{89} +13.2797 q^{90} -7.78371 q^{92} +25.3310 q^{93} -6.81369 q^{94} -4.66318 q^{95} -23.3264 q^{96} +7.11590 q^{97} +31.7612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} + 11 q^{9} + 14 q^{10} + 4 q^{11} - 18 q^{12} - 2 q^{15} + 4 q^{16} - 4 q^{17} - 8 q^{18} + 7 q^{19} + 16 q^{20} - 8 q^{22} + q^{23} - 28 q^{24}+ \cdots + 30 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 + 6 * q^8 + 11 * q^9 + 14 * q^10 + 4 * q^11 - 18 * q^12 - 2 * q^15 + 4 * q^16 - 4 * q^17 - 8 * q^18 + 7 * q^19 + 16 * q^20 - 8 * q^22 + q^23 - 28 * q^24 + 4 * q^25 - 22 * q^27 - 7 * q^29 - 24 * q^30 - 3 * q^31 + 24 * q^32 - 10 * q^33 + 2 * q^34 + 26 * q^36 + 10 * q^37 - 12 * q^38 + 22 * q^40 + 6 * q^41 + 9 * q^43 + 2 * q^44 + 3 * q^45 + 28 * q^46 + 17 * q^47 - 16 * q^48 + 30 * q^50 + 20 * q^51 + 13 * q^53 + 28 * q^54 - 4 * q^55 - 4 * q^57 - 14 * q^58 + 22 * q^59 - 42 * q^60 + 24 * q^61 + 18 * q^62 + 20 * q^64 + 30 * q^66 + 14 * q^67 - 18 * q^68 - 2 * q^69 - 4 * q^71 + 30 * q^72 + 5 * q^73 + 8 * q^74 - 6 * q^75 - 8 * q^76 + q^79 + 40 * q^80 + 15 * q^81 - 20 * q^82 + 23 * q^83 - 2 * q^85 - 6 * q^86 - 20 * q^87 - 4 * q^88 - 11 * q^89 + 40 * q^90 + 30 * q^92 + 38 * q^93 + 16 * q^94 - 5 * q^95 - 52 * q^96 - 3 * q^97 + 30 * q^99

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 $

$2$	−1.86620	−1.31960	−0.659801	−	0.751441i	$-0.729359\pi$
$2$	−1.86620	−1.31960	−0.659801	+	0.751441i	$0.729359\pi$
$3$	−3.34889	−1.93348	−0.966742	−	0.255752i	$-0.917677\pi$
$3$	−3.34889	−1.93348	−0.966742	+	0.255752i	$0.917677\pi$
$4$	1.48270	0.741348
$5$	−0.866198	−0.387376	−0.193688	−	0.981063i	$-0.562045\pi$
$5$	−0.866198	−0.387376	−0.193688	+	0.981063i	$0.562045\pi$
$6$	6.24970	2.55143
$7$	0	0
$7$	0	0
$8$	0.965392	0.341318
$9$	8.21509	2.73836
$10$	1.61650	0.511181
$11$	3.86620	1.16570	0.582851	−	0.812579i	$-0.301937\pi$
$11$	3.86620	1.16570	0.582851	+	0.812579i	$0.301937\pi$
$12$	−4.96539	−1.43339
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.90081	0.748985
$16$	−4.76700	−1.19175
$17$	−3.34889	−0.812226	−0.406113	−	0.913823i	$-0.633116\pi$
$17$	−3.34889	−0.812226	−0.406113	+	0.913823i	$0.633116\pi$
$18$	−15.3310	−3.61355
$19$	5.38350	1.23506	0.617530	−	0.786547i	$-0.288133\pi$
$19$	5.38350	1.23506	0.617530	+	0.786547i	$0.288133\pi$
$20$	−1.28431	−0.287180
$21$	0	0
$22$	−7.21509	−1.53826
$23$	−5.24970	−1.09464	−0.547319	−	0.836924i	$-0.684352\pi$
$23$	−5.24970	−1.09464	−0.547319	+	0.836924i	$0.684352\pi$
$24$	−3.23300	−0.659932
$25$	−4.24970	−0.849940
$26$	0	0
$27$	−17.4648	−3.36110
$28$	0	0
$29$	1.69779	0.315271	0.157636	−	0.987497i	$-0.449613\pi$
$29$	1.69779	0.315271	0.157636	+	0.987497i	$0.449613\pi$
$30$	−5.41348	−0.988362
$31$	−7.56399	−1.35853	−0.679266	−	0.733892i	$-0.737702\pi$
$31$	−7.56399	−1.35853	−0.679266	+	0.733892i	$0.737702\pi$
$32$	6.96539	1.23132
$33$	−12.9475	−2.25387
$34$	6.24970	1.07181
$35$	0	0
$36$	12.1805	2.03008
$37$	4.83159	0.794309	0.397154	−	0.917752i	$-0.369998\pi$
$37$	4.83159	0.794309	0.397154	+	0.917752i	$0.369998\pi$
$38$	−10.0467	−1.62979
$39$	0	0
$40$	−0.836221	−0.132218
$41$	4.06922	0.635505	0.317752	−	0.948174i	$-0.397072\pi$
$41$	4.06922	0.635505	0.317752	+	0.948174i	$0.397072\pi$
$42$	0	0
$43$	4.03461	0.615272	0.307636	−	0.951504i	$-0.400462\pi$
$43$	4.03461	0.615272	0.307636	+	0.951504i	$0.400462\pi$
$44$	5.73240	0.864191
$45$	−7.11590	−1.06078
$46$	9.79698	1.44449
$47$	3.65111	0.532569	0.266284	−	0.963895i	$-0.414204\pi$
$47$	3.65111	0.532569	0.266284	+	0.963895i	$0.414204\pi$
$48$	15.9642	2.30423
$49$	0	0
$50$	7.93078	1.12158
$51$	11.2151	1.57043
$52$	0	0
$53$	−0.215092	−0.0295452	−0.0147726	−	0.999891i	$-0.504702\pi$
$53$	−0.215092	−0.0295452	−0.0147726	+	0.999891i	$0.504702\pi$
$54$	32.5928	4.43531
$55$	−3.34889	−0.451565
$56$	0	0
$57$	−18.0288	−2.38797
$58$	−3.16841	−0.416033
$59$	2.78491	0.362564	0.181282	−	0.983431i	$-0.441975\pi$
$59$	2.78491	0.362564	0.181282	+	0.983431i	$0.441975\pi$
$60$	4.30101	0.555258
$61$	9.03461	1.15676	0.578382	−	0.815766i	$-0.303685\pi$
$61$	9.03461	1.15676	0.578382	+	0.815766i	$0.303685\pi$
$62$	14.1159	1.79272
$63$	0	0
$64$	−3.46479	−0.433099
$65$	0	0
$66$	24.1626	2.97421
$67$	7.66318	0.936206	0.468103	−	0.883674i	$-0.344938\pi$
$67$	7.66318	0.936206	0.468103	+	0.883674i	$0.344938\pi$
$68$	−4.96539	−0.602142
$69$	17.5807	2.11647
$70$	0	0
$71$	−4.90081	−0.581619	−0.290809	−	0.956781i	$-0.593925\pi$
$71$	−4.90081	−0.581619	−0.290809	+	0.956781i	$0.593925\pi$
$72$	7.93078	0.934652
$73$	−15.5461	−1.81953	−0.909766	−	0.415122i	$-0.863739\pi$
$73$	−15.5461	−1.81953	−0.909766	+	0.415122i	$0.863739\pi$
$74$	−9.01671	−1.04817
$75$	14.2318	1.64335
$76$	7.98210	0.915609
$77$	0	0
$78$	0	0
$79$	9.43018	1.06098	0.530489	−	0.847692i	$-0.322008\pi$
$79$	9.43018	1.06098	0.530489	+	0.847692i	$0.322008\pi$
$80$	4.12917	0.461655
$81$	33.8425	3.76027
$82$	−7.59396	−0.838613
$83$	4.09919	0.449945	0.224972	−	0.974365i	$-0.427771\pi$
$83$	4.09919	0.449945	0.224972	+	0.974365i	$0.427771\pi$
$84$	0	0
$85$	2.90081	0.314637
$86$	−7.52938	−0.811914
$87$	−5.68571	−0.609573
$88$	3.73240	0.397875
$89$	0.418110	0.0443196	0.0221598	−	0.999754i	$-0.492946\pi$
$89$	0.418110	0.0443196	0.0221598	+	0.999754i	$0.492946\pi$
$90$	13.2797	1.39980
$91$	0	0
$92$	−7.78371	−0.811508
$93$	25.3310	2.62670
$94$	−6.81369	−0.702778
$95$	−4.66318	−0.478432
$96$	−23.3264	−2.38074
$97$	7.11590	0.722510	0.361255	−	0.932467i	$-0.382348\pi$
$97$	7.11590	0.722510	0.361255	+	0.932467i	$0.382348\pi$
$98$	0	0
$99$	31.7612	3.19212
$100$	−6.30101	−0.630101
$101$	−14.1159	−1.40458	−0.702292	−	0.711889i	$-0.747840\pi$
$101$	−14.1159	−1.40458	−0.702292	+	0.711889i	$0.747840\pi$
$102$	−20.9296	−2.07234
$103$	−16.8604	−1.66130	−0.830651	−	0.556794i	$-0.812031\pi$
$103$	−16.8604	−1.66130	−0.830651	+	0.556794i	$0.812031\pi$
$104$	0	0
$105$	0	0
$106$	0.401405	0.0389879
$107$	10.1805	0.984185	0.492092	−	0.870543i	$-0.336232\pi$
$107$	10.1805	0.984185	0.492092	+	0.870543i	$0.336232\pi$
$108$	−25.8950	−2.49175
$109$	6.20302	0.594141	0.297071	−	0.954855i	$-0.403990\pi$
$109$	6.20302	0.594141	0.297071	+	0.954855i	$0.403990\pi$
$110$	6.24970	0.595886
$111$	−16.1805	−1.53578
$112$	0	0
$113$	10.2843	0.967466	0.483733	−	0.875216i	$-0.339281\pi$
$113$	10.2843	0.967466	0.483733	+	0.875216i	$0.339281\pi$
$114$	33.6453	3.15117
$115$	4.54728	0.424036
$116$	2.51730	0.233726
$117$	0	0
$118$	−5.19719	−0.478440
$119$	0	0
$120$	2.80041	0.255642
$121$	3.94749	0.358863
$122$	−16.8604	−1.52647
$123$	−13.6274	−1.22874
$124$	−11.2151	−1.00715
$125$	8.01207	0.716622
$126$	0	0
$127$	−1.91288	−0.169741	−0.0848704	−	0.996392i	$-0.527048\pi$
$127$	−1.91288	−0.169741	−0.0848704	+	0.996392i	$0.527048\pi$
$128$	−7.46479	−0.659801
$129$	−13.5115	−1.18962
$130$	0	0
$131$	10.1626	0.887909	0.443954	−	0.896049i	$-0.353575\pi$
$131$	10.1626	0.887909	0.443954	+	0.896049i	$0.353575\pi$
$132$	−19.1972	−1.67090
$133$	0	0
$134$	−14.3010	−1.23542
$135$	15.1280	1.30201
$136$	−3.23300	−0.277227
$137$	−7.79698	−0.666141	−0.333071	−	0.942902i	$-0.608085\pi$
$137$	−7.79698	−0.666141	−0.333071	+	0.942902i	$0.608085\pi$
$138$	−32.8091	−2.79289
$139$	−5.08129	−0.430989	−0.215495	−	0.976505i	$-0.569136\pi$
$139$	−5.08129	−0.430989	−0.215495	+	0.976505i	$0.569136\pi$
$140$	0	0
$141$	−12.2272	−1.02971
$142$	9.14588	0.767505
$143$	0	0
$144$	−39.1614	−3.26345
$145$	−1.47062	−0.122128
$146$	29.0121	2.40106
$147$	0	0
$148$	7.16378	0.588859
$149$	2.49477	0.204380	0.102190	−	0.994765i	$-0.467415\pi$
$149$	2.49477	0.204380	0.102190	+	0.994765i	$0.467415\pi$
$150$	−26.5594	−2.16856
$151$	3.26178	0.265439	0.132720	−	0.991154i	$-0.457629\pi$
$151$	3.26178	0.265439	0.132720	+	0.991154i	$0.457629\pi$
$152$	5.19719	0.421548
$153$	−27.5115	−2.22417
$154$	0	0
$155$	6.55191	0.526262
$156$	0	0
$157$	−0.720322	−0.0574880	−0.0287440	−	0.999587i	$-0.509151\pi$
$157$	−0.720322	−0.0574880	−0.0287440	+	0.999587i	$0.509151\pi$
$158$	−17.5986	−1.40007
$159$	0.720322	0.0571252
$160$	−6.03341	−0.476983
$161$	0	0
$162$	−63.1568	−4.96206
$163$	−1.30221	−0.101997	−0.0509985	−	0.998699i	$-0.516240\pi$
$163$	−1.30221	−0.101997	−0.0509985	+	0.998699i	$0.516240\pi$
$164$	6.03341	0.471130
$165$	11.2151	0.873094
$166$	−7.64991	−0.593748
$167$	−16.1505	−1.24976	−0.624882	−	0.780719i	$-0.714853\pi$
$167$	−16.1505	−1.24976	−0.624882	+	0.780719i	$0.714853\pi$
$168$	0	0
$169$	0	0
$170$	−5.41348	−0.415195
$171$	44.2260	3.38204
$172$	5.98210	0.456131
$173$	−15.8483	−1.20492	−0.602462	−	0.798148i	$-0.705813\pi$
$173$	−15.8483	−1.20492	−0.602462	+	0.798148i	$0.705813\pi$
$174$	10.6107	0.804393
$175$	0	0
$176$	−18.4302	−1.38923
$177$	−9.32636	−0.701012
$178$	−0.780277	−0.0584842
$179$	−20.4648	−1.52961	−0.764805	−	0.644262i	$-0.777165\pi$
$179$	−20.4648	−1.52961	−0.764805	+	0.644262i	$0.777165\pi$
$180$	−10.5507	−0.786404
$181$	−6.58189	−0.489228	−0.244614	−	0.969621i	$-0.578661\pi$
$181$	−6.58189	−0.489228	−0.244614	+	0.969621i	$0.578661\pi$
$182$	0	0
$183$	−30.2559	−2.23658
$184$	−5.06802	−0.373619
$185$	−4.18512	−0.307696
$186$	−47.2727	−3.46620
$187$	−12.9475	−0.946814
$188$	5.41348	0.394819
$189$	0	0
$190$	8.70242	0.631340
$191$	12.7491	0.922493	0.461246	−	0.887272i	$-0.347402\pi$
$191$	12.7491	0.922493	0.461246	+	0.887272i	$0.347402\pi$
$192$	11.6032	0.837391
$193$	−2.26760	−0.163226	−0.0816128	−	0.996664i	$-0.526007\pi$
$193$	−2.26760	−0.163226	−0.0816128	+	0.996664i	$0.526007\pi$
$194$	−13.2797	−0.953425
$195$	0	0
$196$	0	0
$197$	18.6978	1.33216	0.666081	−	0.745879i	$-0.267970\pi$
$197$	18.6978	1.33216	0.666081	+	0.745879i	$0.267970\pi$
$198$	−59.2727	−4.21232
$199$	19.9175	1.41191	0.705957	−	0.708254i	$-0.250517\pi$
$199$	19.9175	1.41191	0.705957	+	0.708254i	$0.250517\pi$
$200$	−4.10263	−0.290100
$201$	−25.6632	−1.81014
$202$	26.3431	1.85349
$203$	0	0
$204$	16.6286	1.16423
$205$	−3.52475	−0.246179
$206$	31.4648	2.19226
$207$	−43.1268	−2.99752
$208$	0	0
$209$	20.8137	1.43971
$210$	0	0
$211$	0.645277	0.0444227	0.0222114	−	0.999753i	$-0.492929\pi$
$211$	0.645277	0.0444227	0.0222114	+	0.999753i	$0.492929\pi$
$212$	−0.318917	−0.0219033
$213$	16.4123	1.12455
$214$	−18.9988	−1.29873
$215$	−3.49477	−0.238341
$216$	−16.8604	−1.14720
$217$	0	0
$218$	−11.5761	−0.784030
$219$	52.0622	3.51804
$220$	−4.96539	−0.334767
$221$	0	0
$222$	30.1960	2.02662
$223$	−5.83159	−0.390512	−0.195256	−	0.980752i	$-0.562554\pi$
$223$	−5.83159	−0.390512	−0.195256	+	0.980752i	$0.562554\pi$
$224$	0	0
$225$	−34.9117	−2.32745
$226$	−19.1926	−1.27667
$227$	15.7324	1.04420	0.522098	−	0.852886i	$-0.325150\pi$
$227$	15.7324	1.04420	0.522098	+	0.852886i	$0.325150\pi$
$228$	−26.7312	−1.77032
$229$	8.87827	0.586693	0.293346	−	0.956006i	$-0.405231\pi$
$229$	8.87827	0.586693	0.293346	+	0.956006i	$0.405231\pi$
$230$	−8.48613	−0.559559
$231$	0	0
$232$	1.63903	0.107608
$233$	−11.7912	−0.772464	−0.386232	−	0.922402i	$-0.626224\pi$
$233$	−11.7912	−0.772464	−0.386232	+	0.922402i	$0.626224\pi$
$234$	0	0
$235$	−3.16258	−0.206304
$236$	4.12917	0.268786
$237$	−31.5807	−2.05139
$238$	0	0
$239$	16.0692	1.03943	0.519716	−	0.854339i	$-0.326038\pi$
$239$	16.0692	1.03943	0.519716	+	0.854339i	$0.326038\pi$
$240$	−13.8282	−0.892604
$241$	−4.93541	−0.317918	−0.158959	−	0.987285i	$-0.550814\pi$
$241$	−4.93541	−0.317918	−0.158959	+	0.987285i	$0.550814\pi$
$242$	−7.36680	−0.473556
$243$	−60.9405	−3.90933
$244$	13.3956	0.857564
$245$	0	0
$246$	25.4314	1.62145
$247$	0	0
$248$	−7.30221	−0.463691
$249$	−13.7278	−0.869962
$250$	−14.9521	−0.945655
$251$	15.2439	0.962185	0.481092	−	0.876670i	$-0.340240\pi$
$251$	15.2439	0.962185	0.481092	+	0.876670i	$0.340240\pi$
$252$	0	0
$253$	−20.2964	−1.27602
$254$	3.56982	0.223990
$255$	−9.71449	−0.608345
$256$	20.8604	1.30377
$257$	−15.6165	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$257$	−15.6165	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$258$	25.2151	1.56982
$259$	0	0
$260$	0	0
$261$	13.9475	0.863328
$262$	−18.9654	−1.17169
$263$	17.0934	1.05402	0.527011	−	0.849858i	$-0.323313\pi$
$263$	17.0934	1.05402	0.527011	+	0.849858i	$0.323313\pi$
$264$	−12.4994	−0.769285
$265$	0.186313	0.0114451
$266$	0	0
$267$	−1.40021	−0.0856913
$268$	11.3622	0.694055
$269$	16.3368	0.996073	0.498037	−	0.867156i	$-0.334054\pi$
$269$	16.3368	0.996073	0.498037	+	0.867156i	$0.334054\pi$
$270$	−28.2318	−1.71813
$271$	12.4994	0.759285	0.379642	−	0.925133i	$-0.376047\pi$
$271$	12.4994	0.759285	0.379642	+	0.925133i	$0.376047\pi$
$272$	15.9642	0.967971
$273$	0	0
$274$	14.5507	0.879041
$275$	−16.4302	−0.990777
$276$	26.0668	1.56904
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	9.48270	0.568734
$279$	−62.1389	−3.72016
$280$	0	0
$281$	−0.831590	−0.0496085	−0.0248043	−	0.999692i	$-0.507896\pi$
$281$	−0.831590	−0.0496085	−0.0248043	+	0.999692i	$0.507896\pi$
$282$	22.8183	1.35881
$283$	11.0588	0.657375	0.328687	−	0.944439i	$-0.393394\pi$
$283$	11.0588	0.657375	0.328687	+	0.944439i	$0.393394\pi$
$284$	−7.26641	−0.431182
$285$	15.6165	0.925041
$286$	0	0
$287$	0	0
$288$	57.2213	3.37180
$289$	−5.78491	−0.340289
$290$	2.74447	0.161161
$291$	−23.8304	−1.39696
$292$	−23.0501	−1.34891
$293$	26.9175	1.57254	0.786269	−	0.617884i	$-0.212010\pi$
$293$	26.9175	1.57254	0.786269	+	0.617884i	$0.212010\pi$
$294$	0	0
$295$	−2.41228	−0.140448
$296$	4.66438	0.271111
$297$	−67.5224	−3.91804
$298$	−4.65574	−0.269700
$299$	0	0
$300$	21.1014	1.21829
$301$	0	0
$302$	−6.08712	−0.350274
$303$	47.2727	2.71574
$304$	−25.6632	−1.47188
$305$	−7.82576	−0.448102
$306$	51.3419	2.93502
$307$	15.1580	0.865110	0.432555	−	0.901608i	$-0.357612\pi$
$307$	15.1580	0.865110	0.432555	+	0.901608i	$0.357612\pi$
$308$	0	0
$309$	56.4636	3.21210
$310$	−12.2272	−0.694456
$311$	4.24507	0.240716	0.120358	−	0.992731i	$-0.461596\pi$
$311$	4.24507	0.240716	0.120358	+	0.992731i	$0.461596\pi$
$312$	0	0
$313$	17.9533	1.01478	0.507391	−	0.861716i	$-0.330610\pi$
$313$	17.9533	1.01478	0.507391	+	0.861716i	$0.330610\pi$
$314$	1.34426	0.0758612
$315$	0	0
$316$	13.9821	0.786554
$317$	15.2664	0.857447	0.428723	−	0.903436i	$-0.358963\pi$
$317$	15.2664	0.857447	0.428723	+	0.903436i	$0.358963\pi$
$318$	−1.34426	−0.0753826
$319$	6.56399	0.367513
$320$	3.00120	0.167772
$321$	−34.0934	−1.90291
$322$	0	0
$323$	−18.0288	−1.00315
$324$	50.1781	2.78767
$325$	0	0
$326$	2.43018	0.134595
$327$	−20.7733	−1.14876
$328$	3.92839	0.216909
$329$	0	0
$330$	−20.9296	−1.15214
$331$	−17.2664	−0.949047	−0.474524	−	0.880243i	$-0.657380\pi$
$331$	−17.2664	−0.949047	−0.474524	+	0.880243i	$0.657380\pi$
$332$	6.07786	0.333566
$333$	39.6920	2.17511
$334$	30.1400	1.64919
$335$	−6.63783	−0.362664
$336$	0	0
$337$	−25.5415	−1.39133	−0.695666	−	0.718366i	$-0.744891\pi$
$337$	−25.5415	−1.39133	−0.695666	+	0.718366i	$0.744891\pi$
$338$	0	0
$339$	−34.4411	−1.87058
$340$	4.30101	0.233255
$341$	−29.2439	−1.58364
$342$	−82.5344	−4.46295
$343$	0	0
$344$	3.89498	0.210003
$345$	−15.2284	−0.819868
$346$	29.5761	1.59002
$347$	−12.6286	−0.677937	−0.338969	−	0.940798i	$-0.610078\pi$
$347$	−12.6286	−0.677937	−0.338969	+	0.940798i	$0.610078\pi$
$348$	−8.43018	−0.451905
$349$	−35.6394	−1.90774	−0.953868	−	0.300226i	$-0.902938\pi$
$349$	−35.6394	−1.90774	−0.953868	+	0.300226i	$0.902938\pi$
$350$	0	0
$351$	0	0
$352$	26.9296	1.43535
$353$	−5.41348	−0.288130	−0.144065	−	0.989568i	$-0.546018\pi$
$353$	−5.41348	−0.288130	−0.144065	+	0.989568i	$0.546018\pi$
$354$	17.4048	0.925057
$355$	4.24507	0.225305
$356$	0.619931	0.0328563
$357$	0	0
$358$	38.1914	2.01848
$359$	18.8316	0.993893	0.496947	−	0.867781i	$-0.334454\pi$
$359$	18.8316	0.993893	0.496947	+	0.867781i	$0.334454\pi$
$360$	−6.86963	−0.362061
$361$	9.98210	0.525374
$362$	12.2831	0.645586
$363$	−13.2197	−0.693856
$364$	0	0
$365$	13.4660	0.704842
$366$	56.4636	2.95140
$367$	−23.1505	−1.20845	−0.604223	−	0.796815i	$-0.706517\pi$
$367$	−23.1505	−1.20845	−0.604223	+	0.796815i	$0.706517\pi$
$368$	25.0253	1.30454
$369$	33.4290	1.74024
$370$	7.81025	0.406036
$371$	0	0
$372$	37.5582	1.94730
$373$	−33.1793	−1.71796	−0.858979	−	0.512011i	$-0.828901\pi$
$373$	−33.1793	−1.71796	−0.858979	+	0.512011i	$0.828901\pi$
$374$	24.1626	1.24942
$375$	−26.8316	−1.38558
$376$	3.52475	0.181775
$377$	0	0
$378$	0	0
$379$	−37.7853	−1.94090	−0.970451	−	0.241299i	$-0.922427\pi$
$379$	−37.7853	−1.94090	−0.970451	+	0.241299i	$0.922427\pi$
$380$	−6.91408	−0.354685
$381$	6.40604	0.328191
$382$	−23.7924	−1.21732
$383$	−0.231798	−0.0118443	−0.00592215	−	0.999982i	$-0.501885\pi$
$383$	−0.231798	−0.0118443	−0.00592215	+	0.999982i	$0.501885\pi$
$384$	24.9988	1.27571
$385$	0	0
$386$	4.23180	0.215393
$387$	33.1447	1.68484
$388$	10.5507	0.535631
$389$	9.35352	0.474243	0.237121	−	0.971480i	$-0.423796\pi$
$389$	9.35352	0.474243	0.237121	+	0.971480i	$0.423796\pi$
$390$	0	0
$391$	17.5807	0.889094
$392$	0	0
$393$	−34.0334	−1.71676
$394$	−34.8938	−1.75792
$395$	−8.16841	−0.410997
$396$	47.0922	2.36647
$397$	−9.74447	−0.489061	−0.244530	−	0.969642i	$-0.578634\pi$
$397$	−9.74447	−0.489061	−0.244530	+	0.969642i	$0.578634\pi$
$398$	−37.1700	−1.86317
$399$	0	0
$400$	20.2583	1.01292
$401$	−9.73240	−0.486013	−0.243006	−	0.970025i	$-0.578134\pi$
$401$	−9.73240	−0.486013	−0.243006	+	0.970025i	$0.578134\pi$
$402$	47.8926	2.38866
$403$	0	0
$404$	−20.9296	−1.04129
$405$	−29.3143	−1.45664
$406$	0	0
$407$	18.6799	0.925928
$408$	10.8270	0.536014
$409$	12.3730	0.611808	0.305904	−	0.952062i	$-0.401041\pi$
$409$	12.3730	0.611808	0.305904	+	0.952062i	$0.401041\pi$
$410$	6.57788	0.324858
$411$	26.1113	1.28797
$412$	−24.9988	−1.23160
$413$	0	0
$414$	80.4831	3.95553
$415$	−3.55071	−0.174298
$416$	0	0
$417$	17.0167	0.833312
$418$	−38.8425	−1.89985
$419$	−21.1054	−1.03107	−0.515534	−	0.856869i	$-0.672406\pi$
$419$	−21.1054	−1.03107	−0.515534	+	0.856869i	$0.672406\pi$
$420$	0	0
$421$	−23.2618	−1.13371	−0.566855	−	0.823818i	$-0.691840\pi$
$421$	−23.2618	−1.13371	−0.566855	+	0.823818i	$0.691840\pi$
$422$	−1.20422	−0.0586203
$423$	29.9942	1.45837
$424$	−0.207649	−0.0100843
$425$	14.2318	0.690344
$426$	−30.6286	−1.48396
$427$	0	0
$428$	15.0946	0.729623
$429$	0	0
$430$	6.52193	0.314516
$431$	16.9895	0.818357	0.409179	−	0.912454i	$-0.365815\pi$
$431$	16.9895	0.818357	0.409179	+	0.912454i	$0.365815\pi$
$432$	83.2547	4.00560
$433$	30.8604	1.48305	0.741527	−	0.670923i	$-0.234102\pi$
$433$	30.8604	1.48305	0.741527	+	0.670923i	$0.234102\pi$
$434$	0	0
$435$	4.92496	0.236134
$436$	9.19719	0.440465
$437$	−28.2618	−1.35194
$438$	−97.1584	−4.64241
$439$	−19.1972	−0.916232	−0.458116	−	0.888892i	$-0.651476\pi$
$439$	−19.1972	−0.916232	−0.458116	+	0.888892i	$0.651476\pi$
$440$	−3.23300	−0.154127
$441$	0	0
$442$	0	0
$443$	−34.3777	−1.63333	−0.816666	−	0.577110i	$-0.804180\pi$
$443$	−34.3777	−1.63333	−0.816666	+	0.577110i	$0.804180\pi$
$444$	−23.9907	−1.13855
$445$	−0.362166	−0.0171683
$446$	10.8829	0.515320
$447$	−8.35472	−0.395165
$448$	0	0
$449$	−29.6274	−1.39820	−0.699101	−	0.715023i	$-0.746416\pi$
$449$	−29.6274	−1.39820	−0.699101	+	0.715023i	$0.746416\pi$
$450$	65.1521	3.07130
$451$	15.7324	0.740810
$452$	15.2485	0.717229
$453$	−10.9233	−0.513223
$454$	−29.3598	−1.37792
$455$	0	0
$456$	−17.4048	−0.815056
$457$	17.0392	0.797062	0.398531	−	0.917155i	$-0.369520\pi$
$457$	17.0392	0.797062	0.398531	+	0.917155i	$0.369520\pi$
$458$	−16.5686	−0.774201
$459$	58.4877	2.72997
$460$	6.74224	0.314358
$461$	15.1280	0.704580	0.352290	−	0.935891i	$-0.385403\pi$
$461$	15.1280	0.704580	0.352290	+	0.935891i	$0.385403\pi$
$462$	0	0
$463$	−26.1221	−1.21400	−0.607000	−	0.794702i	$-0.707627\pi$
$463$	−26.1221	−1.21400	−0.607000	+	0.794702i	$0.707627\pi$
$464$	−8.09337	−0.375725
$465$	−21.9417	−1.01752
$466$	22.0046	1.01934
$467$	−22.9187	−1.06055	−0.530276	−	0.847825i	$-0.677912\pi$
$467$	−22.9187	−1.06055	−0.530276	+	0.847825i	$0.677912\pi$
$468$	0	0
$469$	0	0
$470$	5.90200	0.272239
$471$	2.41228	0.111152
$472$	2.68853	0.123749
$473$	15.5986	0.717224
$474$	58.9358	2.70701
$475$	−22.8783	−1.04973
$476$	0	0
$477$	−1.76700	−0.0809056
$478$	−29.9883	−1.37163
$479$	35.1914	1.60793	0.803967	−	0.594674i	$-0.202719\pi$
$479$	35.1914	1.60793	0.803967	+	0.594674i	$0.202719\pi$
$480$	20.2053	0.922239
$481$	0	0
$482$	9.21046	0.419525
$483$	0	0
$484$	5.85293	0.266042
$485$	−6.16378	−0.279883
$486$	113.727	5.15876
$487$	28.3010	1.28244	0.641221	−	0.767357i	$-0.278428\pi$
$487$	28.3010	1.28244	0.641221	+	0.767357i	$0.278428\pi$
$488$	8.72194	0.394824
$489$	4.36097	0.197210
$490$	0	0
$491$	8.24970	0.372304	0.186152	−	0.982521i	$-0.440398\pi$
$491$	8.24970	0.372304	0.186152	+	0.982521i	$0.440398\pi$
$492$	−20.2053	−0.910923
$493$	−5.68571	−0.256072
$494$	0	0
$495$	−27.5115	−1.23655
$496$	36.0576	1.61903
$497$	0	0
$498$	25.6187	1.14800
$499$	16.7266	0.748784	0.374392	−	0.927271i	$-0.377851\pi$
$499$	16.7266	0.748784	0.374392	+	0.927271i	$0.377851\pi$
$500$	11.8795	0.531266
$501$	54.0863	2.41640
$502$	−28.4481	−1.26970
$503$	−21.2213	−0.946213	−0.473106	−	0.881005i	$-0.656867\pi$
$503$	−21.2213	−0.946213	−0.473106	+	0.881005i	$0.656867\pi$
$504$	0	0
$505$	12.2272	0.544102
$506$	37.8771	1.68384
$507$	0	0
$508$	−2.83622	−0.125837
$509$	−40.6048	−1.79978	−0.899889	−	0.436119i	$-0.856353\pi$
$509$	−40.6048	−1.79978	−0.899889	+	0.436119i	$0.856353\pi$
$510$	18.1292	0.802773
$511$	0	0
$512$	−24.0000	−1.06066
$513$	−94.0218	−4.15116
$514$	29.1435	1.28546
$515$	14.6044	0.643548
$516$	−20.0334	−0.881922
$517$	14.1159	0.620817
$518$	0	0
$519$	53.0743	2.32970
$520$	0	0
$521$	−3.46479	−0.151795	−0.0758977	−	0.997116i	$-0.524182\pi$
$521$	−3.46479	−0.151795	−0.0758977	+	0.997116i	$0.524182\pi$
$522$	−26.0288	−1.13925
$523$	−5.56982	−0.243551	−0.121776	−	0.992558i	$-0.538859\pi$
$523$	−5.56982	−0.243551	−0.121776	+	0.992558i	$0.538859\pi$
$524$	15.0680	0.658249
$525$	0	0
$526$	−31.8996	−1.39089
$527$	25.3310	1.10344
$528$	61.7207	2.68605
$529$	4.55936	0.198233
$530$	−0.347696	−0.0151030
$531$	22.8783	0.992832
$532$	0	0
$533$	0	0
$534$	2.61306	0.113078
$535$	−8.81832	−0.381249
$536$	7.39797	0.319544
$537$	68.5344	2.95748
$538$	−30.4877	−1.31442
$539$	0	0
$540$	22.4302	0.965241
$541$	−24.2364	−1.04201	−0.521003	−	0.853555i	$-0.674442\pi$
$541$	−24.2364	−1.04201	−0.521003	+	0.853555i	$0.674442\pi$
$542$	−23.3264	−1.00195
$543$	22.0421	0.945915
$544$	−23.3264	−1.00011
$545$	−5.37304	−0.230156
$546$	0	0
$547$	15.7733	0.674416	0.337208	−	0.941430i	$-0.390518\pi$
$547$	15.7733	0.674416	0.337208	+	0.941430i	$0.390518\pi$
$548$	−11.5606	−0.493842
$549$	74.2201	3.16764
$550$	30.6620	1.30743
$551$	9.14005	0.389379
$552$	16.9723	0.722387
$553$	0	0
$554$	5.59859	0.237861
$555$	14.0155	0.594925
$556$	−7.53401	−0.319513
$557$	17.2213	0.729692	0.364846	−	0.931068i	$-0.381122\pi$
$557$	17.2213	0.729692	0.364846	+	0.931068i	$0.381122\pi$
$558$	115.963	4.90912
$559$	0	0
$560$	0	0
$561$	43.3598	1.83065
$562$	1.55191	0.0654635
$563$	15.3598	0.647337	0.323669	−	0.946171i	$-0.395084\pi$
$563$	15.3598	0.647337	0.323669	+	0.946171i	$0.395084\pi$
$564$	−18.1292	−0.763376
$565$	−8.90825	−0.374773
$566$	−20.6378	−0.867473
$567$	0	0
$568$	−4.73120	−0.198517
$569$	23.7219	0.994475	0.497238	−	0.867614i	$-0.334348\pi$
$569$	23.7219	0.994475	0.497238	+	0.867614i	$0.334348\pi$
$570$	−29.1435	−1.22069
$571$	−27.3189	−1.14326	−0.571631	−	0.820511i	$-0.693689\pi$
$571$	−27.3189	−1.14326	−0.571631	+	0.820511i	$0.693689\pi$
$572$	0	0
$573$	−42.6954	−1.78363
$574$	0	0
$575$	22.3097	0.930377
$576$	−28.4636	−1.18598
$577$	−23.2664	−0.968593	−0.484297	−	0.874904i	$-0.660924\pi$
$577$	−23.2664	−0.968593	−0.484297	+	0.874904i	$0.660924\pi$
$578$	10.7958	0.449045
$579$	7.59396	0.315594
$580$	−2.18048	−0.0905397
$581$	0	0
$582$	44.4722	1.84343
$583$	−0.831590	−0.0344409
$584$	−15.0081	−0.621038
$585$	0	0
$586$	−50.2334	−2.07512
$587$	45.7266	1.88734	0.943669	−	0.330892i	$-0.107349\pi$
$587$	45.7266	1.88734	0.943669	+	0.330892i	$0.107349\pi$
$588$	0	0
$589$	−40.7207	−1.67787
$590$	4.50180	0.185336
$591$	−62.6169	−2.57572
$592$	−23.0322	−0.946618
$593$	29.3897	1.20689	0.603446	−	0.797404i	$-0.293794\pi$
$593$	29.3897	1.20689	0.603446	+	0.797404i	$0.293794\pi$
$594$	126.010	5.17026
$595$	0	0
$596$	3.69899	0.151516
$597$	−66.7016	−2.72992
$598$	0	0
$599$	22.0588	0.901296	0.450648	−	0.892702i	$-0.351193\pi$
$599$	22.0588	0.901296	0.450648	+	0.892702i	$0.351193\pi$
$600$	13.7393	0.560903
$601$	30.8604	1.25882	0.629410	−	0.777073i	$-0.283296\pi$
$601$	30.8604	1.25882	0.629410	+	0.777073i	$0.283296\pi$
$602$	0	0
$603$	62.9537	2.56367
$604$	4.83622	0.196783
$605$	−3.41931	−0.139015
$606$	−88.2201	−3.58370
$607$	36.4861	1.48093	0.740463	−	0.672097i	$-0.234606\pi$
$607$	36.4861	1.48093	0.740463	+	0.672097i	$0.234606\pi$
$608$	37.4982	1.52075
$609$	0	0
$610$	14.6044	0.591316
$611$	0	0
$612$	−40.7912	−1.64888
$613$	−28.9988	−1.17125	−0.585625	−	0.810582i	$-0.699151\pi$
$613$	−28.9988	−1.17125	−0.585625	+	0.810582i	$0.699151\pi$
$614$	−28.2877	−1.14160
$615$	11.8040	0.475984
$616$	0	0
$617$	41.6515	1.67683	0.838414	−	0.545035i	$-0.183483\pi$
$617$	41.6515	1.67683	0.838414	+	0.545035i	$0.183483\pi$
$618$	−105.372	−4.23869
$619$	−12.4994	−0.502393	−0.251197	−	0.967936i	$-0.580824\pi$
$619$	−12.4994	−0.502393	−0.251197	+	0.967936i	$0.580824\pi$
$620$	9.71449	0.390143
$621$	91.6849	3.67919
$622$	−7.92214	−0.317649
$623$	0	0
$624$	0	0
$625$	14.3085	0.572338
$626$	−33.5044	−1.33911
$627$	−69.7028	−2.78366
$628$	−1.06802	−0.0426186
$629$	−16.1805	−0.645158
$630$	0	0
$631$	35.5582	1.41555	0.707774	−	0.706439i	$-0.249700\pi$
$631$	35.5582	1.41555	0.707774	+	0.706439i	$0.249700\pi$
$632$	9.10382	0.362131
$633$	−2.16097	−0.0858907
$634$	−28.4901	−1.13149
$635$	1.65693	0.0657534
$636$	1.06802	0.0423497
$637$	0	0
$638$	−12.2497	−0.484970
$639$	−40.2606	−1.59268
$640$	6.46599	0.255591
$641$	1.36097	0.0537550	0.0268775	−	0.999639i	$-0.491444\pi$
$641$	1.36097	0.0537550	0.0268775	+	0.999639i	$0.491444\pi$
$642$	63.6250	2.51108
$643$	−12.1867	−0.480598	−0.240299	−	0.970699i	$-0.577245\pi$
$643$	−12.1867	−0.480598	−0.240299	+	0.970699i	$0.577245\pi$
$644$	0	0
$645$	11.7036	0.460829
$646$	33.6453	1.32376
$647$	−3.72152	−0.146308	−0.0731540	−	0.997321i	$-0.523306\pi$
$647$	−3.72152	−0.146308	−0.0731540	+	0.997321i	$0.523306\pi$
$648$	32.6712	1.28345
$649$	10.7670	0.422642
$650$	0	0
$651$	0	0
$652$	−1.93078	−0.0756153
$653$	−45.0588	−1.76329	−0.881643	−	0.471917i	$-0.843562\pi$
$653$	−45.0588	−1.76329	−0.881643	+	0.471917i	$0.843562\pi$
$654$	38.7670	1.51591
$655$	−8.80281	−0.343954
$656$	−19.3980	−0.757364
$657$	−127.713	−4.98254
$658$	0	0
$659$	5.37887	0.209531	0.104766	−	0.994497i	$-0.466591\pi$
$659$	5.37887	0.209531	0.104766	+	0.994497i	$0.466591\pi$
$660$	16.6286	0.647266
$661$	42.4936	1.65281	0.826404	−	0.563077i	$-0.190383\pi$
$661$	42.4936	1.65281	0.826404	+	0.563077i	$0.190383\pi$
$662$	32.2225	1.25236
$663$	0	0
$664$	3.95733	0.153574
$665$	0	0
$666$	−74.0731	−2.87027
$667$	−8.91288	−0.345108
$668$	−23.9463	−0.926510
$669$	19.5294	0.755049
$670$	12.3875	0.478571
$671$	34.9296	1.34844
$672$	0	0
$673$	−37.3765	−1.44076	−0.720379	−	0.693581i	$-0.756032\pi$
$673$	−37.3765	−1.44076	−0.720379	+	0.693581i	$0.756032\pi$
$674$	47.6654	1.83600
$675$	74.2201	2.85673
$676$	0	0
$677$	−43.4757	−1.67091	−0.835453	−	0.549562i	$-0.814795\pi$
$677$	−43.4757	−1.67091	−0.835453	+	0.549562i	$0.814795\pi$
$678$	64.2738	2.46842
$679$	0	0
$680$	2.80041	0.107391
$681$	−52.6861	−2.01894
$682$	54.5749	2.08978
$683$	−31.3956	−1.20132	−0.600659	−	0.799505i	$-0.705095\pi$
$683$	−31.3956	−1.20132	−0.600659	+	0.799505i	$0.705095\pi$
$684$	65.5737	2.50727
$685$	6.75373	0.258047
$686$	0	0
$687$	−29.7324	−1.13436
$688$	−19.2330	−0.733251
$689$	0	0
$690$	28.4191	1.08190
$691$	−13.4411	−0.511322	−0.255661	−	0.966766i	$-0.582293\pi$
$691$	−13.4411	−0.511322	−0.255661	+	0.966766i	$0.582293\pi$
$692$	−23.4982	−0.893268
$693$	0	0
$694$	23.5674	0.894607
$695$	4.40141	0.166955
$696$	−5.48894	−0.208058
$697$	−13.6274	−0.516174
$698$	66.5103	2.51745
$699$	39.4873	1.49355
$700$	0	0
$701$	−28.0346	−1.05885	−0.529426	−	0.848356i	$-0.677593\pi$
$701$	−28.0346	−1.05885	−0.529426	+	0.848356i	$0.677593\pi$
$702$	0	0
$703$	26.0109	0.981019
$704$	−13.3956	−0.504865
$705$	10.5912	0.398886
$706$	10.1026	0.380217
$707$	0	0
$708$	−13.8282	−0.519694
$709$	40.2847	1.51292	0.756462	−	0.654037i	$-0.226926\pi$
$709$	40.2847	1.51292	0.756462	+	0.654037i	$0.226926\pi$
$710$	−7.92214	−0.297313
$711$	77.4698	2.90535
$712$	0.403640	0.0151271
$713$	39.7087	1.48710
$714$	0	0
$715$	0	0
$716$	−30.3431	−1.13397
$717$	−53.8141	−2.00972
$718$	−35.1435	−1.31154
$719$	16.7445	0.624463	0.312232	−	0.950006i	$-0.398923\pi$
$719$	16.7445	0.624463	0.312232	+	0.950006i	$0.398923\pi$
$720$	33.9215	1.26418
$721$	0	0
$722$	−18.6286	−0.693284
$723$	16.5282	0.614690
$724$	−9.75894	−0.362688
$725$	−7.21509	−0.267962
$726$	24.6706	0.915613
$727$	51.4982	1.90996	0.954981	−	0.296666i	$-0.0958748\pi$
$727$	51.4982	1.90996	0.954981	+	0.296666i	$0.0958748\pi$
$728$	0	0
$729$	102.556	3.79836
$730$	−25.1302	−0.930111
$731$	−13.5115	−0.499740
$732$	−44.8604	−1.65809
$733$	−33.8316	−1.24960	−0.624799	−	0.780786i	$-0.714819\pi$
$733$	−33.8316	−1.24960	−0.624799	+	0.780786i	$0.714819\pi$
$734$	43.2034	1.59467
$735$	0	0
$736$	−36.5662	−1.34785
$737$	29.6274	1.09134
$738$	−62.3851	−2.29643
$739$	−20.6690	−0.760322	−0.380161	−	0.924920i	$-0.624131\pi$
$739$	−20.6690	−0.760322	−0.380161	+	0.924920i	$0.624131\pi$
$740$	−6.20525	−0.228110
$741$	0	0
$742$	0	0
$743$	29.6966	1.08946	0.544731	−	0.838611i	$-0.316632\pi$
$743$	29.6966	1.08946	0.544731	+	0.838611i	$0.316632\pi$
$744$	24.4543	0.896539
$745$	−2.16097	−0.0791717
$746$	61.9191	2.26702
$747$	33.6753	1.23211
$748$	−19.1972	−0.701919
$749$	0	0
$750$	50.0731	1.82841
$751$	−12.0230	−0.438724	−0.219362	−	0.975644i	$-0.570398\pi$
$751$	−12.0230	−0.438724	−0.219362	+	0.975644i	$0.570398\pi$
$752$	−17.4048	−0.634689
$753$	−51.0501	−1.86037
$754$	0	0
$755$	−2.82534	−0.102825
$756$	0	0
$757$	−30.2906	−1.10093	−0.550464	−	0.834859i	$-0.685549\pi$
$757$	−30.2906	−1.10093	−0.550464	+	0.834859i	$0.685549\pi$
$758$	70.5149	2.56122
$759$	67.9704	2.46717
$760$	−4.50180	−0.163297
$761$	−45.9584	−1.66599	−0.832995	−	0.553281i	$-0.813376\pi$
$761$	−45.9584	−1.66599	−0.832995	+	0.553281i	$0.813376\pi$
$762$	−11.9549	−0.433082
$763$	0	0
$764$	18.9030	0.683888
$765$	23.8304	0.861590
$766$	0.432580	0.0156298
$767$	0	0
$768$	−69.8592	−2.52083
$769$	4.03924	0.145659	0.0728293	−	0.997344i	$-0.476797\pi$
$769$	4.03924	0.145659	0.0728293	+	0.997344i	$0.476797\pi$
$770$	0	0
$771$	52.2980	1.88347
$772$	−3.36217	−0.121007
$773$	36.1175	1.29906	0.649528	−	0.760337i	$-0.274966\pi$
$773$	36.1175	1.29906	0.649528	+	0.760337i	$0.274966\pi$
$774$	−61.8545	−2.22332
$775$	32.1447	1.15467
$776$	6.86963	0.246605
$777$	0	0
$778$	−17.4555	−0.625811
$779$	21.9066	0.784887
$780$	0	0
$781$	−18.9475	−0.677994
$782$	−32.8091	−1.17325
$783$	−29.6515	−1.05966
$784$	0	0
$785$	0.623942	0.0222694
$786$	63.5131	2.26544
$787$	−25.8650	−0.921988	−0.460994	−	0.887403i	$-0.652507\pi$
$787$	−25.8650	−0.921988	−0.460994	+	0.887403i	$0.652507\pi$
$788$	27.7231	0.987596
$789$	−57.2439	−2.03794
$790$	15.2439	0.542353
$791$	0	0
$792$	30.6620	1.08953
$793$	0	0
$794$	18.1851	0.645366
$795$	−0.623942	−0.0221289
$796$	29.5316	1.04672
$797$	0.291753	0.0103344	0.00516720	−	0.999987i	$-0.498355\pi$
$797$	0.291753	0.0103344	0.00516720	+	0.999987i	$0.498355\pi$
$798$	0	0
$799$	−12.2272	−0.432566
$800$	−29.6008	−1.04655
$801$	3.43482	0.121363
$802$	18.1626	0.641343
$803$	−60.1042	−2.12103
$804$	−38.0507	−1.34194
$805$	0	0
$806$	0	0
$807$	−54.7103	−1.92589
$808$	−13.6274	−0.479409
$809$	−5.25595	−0.184789	−0.0923946	−	0.995722i	$-0.529452\pi$
$809$	−5.25595	−0.184789	−0.0923946	+	0.995722i	$0.529452\pi$
$810$	54.7063	1.92218
$811$	−37.8499	−1.32909	−0.664545	−	0.747248i	$-0.731375\pi$
$811$	−37.8499	−1.32909	−0.664545	+	0.747248i	$0.731375\pi$
$812$	0	0
$813$	−41.8592	−1.46807
$814$	−34.8604	−1.22186
$815$	1.12797	0.0395112
$816$	−53.4624	−1.87156
$817$	21.7203	0.759898
$818$	−23.0906	−0.807342
$819$	0	0
$820$	−5.22613	−0.182504
$821$	25.7207	0.897660	0.448830	−	0.893617i	$-0.351841\pi$
$821$	25.7207	0.897660	0.448830	+	0.893617i	$0.351841\pi$
$822$	−48.7288	−1.69961
$823$	−19.4827	−0.679124	−0.339562	−	0.940584i	$-0.610279\pi$
$823$	−19.4827	−0.679124	−0.339562	+	0.940584i	$0.610279\pi$
$824$	−16.2769	−0.567031
$825$	55.0230	1.91565
$826$	0	0
$827$	32.0934	1.11600	0.557998	−	0.829842i	$-0.311570\pi$
$827$	32.0934	1.11600	0.557998	+	0.829842i	$0.311570\pi$
$828$	−63.9439	−2.22220
$829$	3.89336	0.135222	0.0676110	−	0.997712i	$-0.478462\pi$
$829$	3.89336	0.135222	0.0676110	+	0.997712i	$0.478462\pi$
$830$	6.62634	0.230004
$831$	10.0467	0.348516
$832$	0	0
$833$	0	0
$834$	−31.7565	−1.09964
$835$	13.9895	0.484128
$836$	30.8604	1.06733
$837$	132.103	4.56616
$838$	39.3869	1.36060
$839$	41.8592	1.44514	0.722570	−	0.691298i	$-0.242961\pi$
$839$	41.8592	1.44514	0.722570	+	0.691298i	$0.242961\pi$
$840$	0	0
$841$	−26.1175	−0.900604
$842$	43.4111	1.49604
$843$	2.78491	0.0959173
$844$	0.956750	0.0329327
$845$	0	0
$846$	−55.9751	−1.92446
$847$	0	0
$848$	1.02535	0.0352106
$849$	−37.0346	−1.27102
$850$	−26.5594	−0.910978
$851$	−25.3644	−0.869480
$852$	24.3344	0.833684
$853$	1.70242	0.0582897	0.0291449	−	0.999575i	$-0.490722\pi$
$853$	1.70242	0.0582897	0.0291449	+	0.999575i	$0.490722\pi$
$854$	0	0
$855$	−38.3085	−1.31012
$856$	9.82816	0.335919
$857$	9.57908	0.327215	0.163608	−	0.986526i	$-0.447687\pi$
$857$	9.57908	0.327215	0.163608	+	0.986526i	$0.447687\pi$
$858$	0	0
$859$	−6.17424	−0.210662	−0.105331	−	0.994437i	$-0.533590\pi$
$859$	−6.17424	−0.210662	−0.105331	+	0.994437i	$0.533590\pi$
$860$	−5.18168	−0.176694
$861$	0	0
$862$	−31.7059	−1.07991
$863$	−24.8604	−0.846257	−0.423128	−	0.906070i	$-0.639068\pi$
$863$	−24.8604	−0.846257	−0.423128	+	0.906070i	$0.639068\pi$
$864$	−121.649	−4.13859
$865$	13.7278	0.466758
$866$	−57.5916	−1.95704
$867$	19.3730	0.657943
$868$	0	0
$869$	36.4590	1.23679
$870$	−9.19094	−0.311602
$871$	0	0
$872$	5.98834	0.202791
$873$	58.4578	1.97850
$874$	52.7421	1.78403
$875$	0	0
$876$	77.1924	2.60809
$877$	−25.6562	−0.866347	−0.433173	−	0.901311i	$-0.642606\pi$
$877$	−25.6562	−0.866347	−0.433173	+	0.901311i	$0.642606\pi$
$878$	35.8258	1.20906
$879$	−90.1439	−3.04048
$880$	15.9642	0.538153
$881$	−25.6632	−0.864615	−0.432307	−	0.901726i	$-0.642300\pi$
$881$	−25.6632	−0.864615	−0.432307	+	0.901726i	$0.642300\pi$
$882$	0	0
$883$	48.6682	1.63782	0.818908	−	0.573925i	$-0.194580\pi$
$883$	48.6682	1.63782	0.818908	+	0.573925i	$0.194580\pi$
$884$	0	0
$885$	8.07848	0.271555
$886$	64.1556	2.15535
$887$	17.5247	0.588423	0.294212	−	0.955740i	$-0.404943\pi$
$887$	17.5247	0.588423	0.294212	+	0.955740i	$0.404943\pi$
$888$	−15.6205	−0.524190
$889$	0	0
$890$	0.675874	0.0226554
$891$	130.842	4.38336
$892$	−8.64648	−0.289505
$893$	19.6557	0.657754
$894$	15.5916	0.521460
$895$	17.7266	0.592534
$896$	0	0
$897$	0	0
$898$	55.2906	1.84507
$899$	−12.8420	−0.428306
$900$	−51.7634	−1.72545
$901$	0.720322	0.0239974
$902$	−29.3598	−0.977573
$903$	0	0
$904$	9.92839	0.330213
$905$	5.70122	0.189515
$906$	20.3851	0.677250
$907$	57.3765	1.90515	0.952577	−	0.304297i	$-0.0984214\pi$
$907$	57.3765	1.90515	0.952577	+	0.304297i	$0.0984214\pi$
$908$	23.3264	0.774112
$909$	−115.963	−3.84626
$910$	0	0
$911$	7.87203	0.260812	0.130406	−	0.991461i	$-0.458372\pi$
$911$	7.87203	0.260812	0.130406	+	0.991461i	$0.458372\pi$
$912$	85.9433	2.84587
$913$	15.8483	0.524502
$914$	−31.7986	−1.05180
$915$	26.2076	0.866398
$916$	13.1638	0.434944
$917$	0	0
$918$	−109.150	−3.60248
$919$	−47.0230	−1.55114	−0.775572	−	0.631259i	$-0.782538\pi$
$919$	−47.0230	−1.55114	−0.775572	+	0.631259i	$0.782538\pi$
$920$	4.38991	0.144731
$921$	−50.7624	−1.67268
$922$	−28.2318	−0.929765
$923$	0	0
$924$	0	0
$925$	−20.5328	−0.675115
$926$	48.7491	1.60199
$927$	−138.509	−4.54925
$928$	11.8258	0.388200
$929$	28.2618	0.927239	0.463619	−	0.886034i	$-0.346551\pi$
$929$	28.2618	0.927239	0.463619	+	0.886034i	$0.346551\pi$
$930$	40.9475	1.34272
$931$	0	0
$932$	−17.4827	−0.572665
$933$	−14.2163	−0.465420
$934$	42.7709	1.39951
$935$	11.2151	0.366773
$936$	0	0
$937$	−8.83784	−0.288720	−0.144360	−	0.989525i	$-0.546112\pi$
$937$	−8.83784	−0.288720	−0.144360	+	0.989525i	$0.546112\pi$
$938$	0	0
$939$	−60.1238	−1.96206
$940$	−4.68915	−0.152943
$941$	−42.7853	−1.39476	−0.697381	−	0.716701i	$-0.745651\pi$
$941$	−42.7853	−1.39476	−0.697381	+	0.716701i	$0.745651\pi$
$942$	−4.50180	−0.146676
$943$	−21.3622	−0.695648
$944$	−13.2757	−0.432086
$945$	0	0
$946$	−29.1101	−0.946450
$947$	−29.2213	−0.949566	−0.474783	−	0.880103i	$-0.657473\pi$
$947$	−29.2213	−0.949566	−0.474783	+	0.880103i	$0.657473\pi$
$948$	−46.8246	−1.52079
$949$	0	0
$950$	42.6954	1.38522
$951$	−51.1256	−1.65786
$952$	0	0
$953$	3.66198	0.118623	0.0593116	−	0.998240i	$-0.481109\pi$
$953$	3.66198	0.118623	0.0593116	+	0.998240i	$0.481109\pi$
$954$	3.29758	0.106763
$955$	−11.0432	−0.357351
$956$	23.8258	0.770580
$957$	−21.9821	−0.710580
$958$	−65.6741	−2.12183
$959$	0	0
$960$	−10.0507	−0.324385
$961$	26.2139	0.845610
$962$	0	0
$963$	83.6336	2.69506
$964$	−7.31772	−0.235688
$965$	1.96419	0.0632296
$966$	0	0
$967$	30.0288	0.965660	0.482830	−	0.875714i	$-0.339609\pi$
$967$	30.0288	0.965660	0.482830	+	0.875714i	$0.339609\pi$
$968$	3.81087	0.122486
$969$	60.3765	1.93957
$970$	11.5028	0.369334
$971$	4.90544	0.157423	0.0787115	−	0.996897i	$-0.474919\pi$
$971$	4.90544	0.157423	0.0787115	+	0.996897i	$0.474919\pi$
$972$	−90.3562	−2.89818
$973$	0	0
$974$	−52.8153	−1.69231
$975$	0	0
$976$	−43.0680	−1.37857
$977$	−22.6332	−0.724100	−0.362050	−	0.932159i	$-0.617923\pi$
$977$	−22.6332	−0.724100	−0.362050	+	0.932159i	$0.617923\pi$
$978$	−8.13843	−0.260238
$979$	1.61650	0.0516635
$980$	0	0
$981$	50.9584	1.62698
$982$	−15.3956	−0.491293
$983$	57.4053	1.83094	0.915472	−	0.402382i	$-0.131818\pi$
$983$	57.4053	1.83094	0.915472	+	0.402382i	$0.131818\pi$
$984$	−13.1558	−0.419390
$985$	−16.1960	−0.516047
$986$	10.6107	0.337913
$987$	0	0
$988$	0	0
$989$	−21.1805	−0.673500
$990$	51.3419	1.63175
$991$	31.1793	0.990443	0.495221	−	0.868767i	$-0.335087\pi$
$991$	31.1793	0.990443	0.495221	+	0.868767i	$0.335087\pi$
$992$	−52.6861	−1.67279
$993$	57.8234	1.83497
$994$	0	0
$995$	−17.2525	−0.546941
$996$	−20.3541	−0.644944
$997$	−36.0576	−1.14195	−0.570977	−	0.820966i	$-0.693436\pi$
$997$	−36.0576	−1.14195	−0.570977	+	0.820966i	$0.693436\pi$
$998$	−31.2151	−0.988096
$999$	−84.3827	−2.66975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bh.1.1		3
7.6	odd	2		8281.2.a.bk.1.1		3
13.12	even	2		637.2.a.h.1.3	✓	3
39.38	odd	2		5733.2.a.be.1.1		3
91.12	odd	6		637.2.e.k.508.1		6
91.25	even	6		637.2.e.l.79.1		6
91.38	odd	6		637.2.e.k.79.1		6
91.51	even	6		637.2.e.l.508.1		6
91.90	odd	2		637.2.a.i.1.3	yes	3
273.272	even	2		5733.2.a.bd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.h.1.3	✓	3	13.12	even	2
637.2.a.i.1.3	yes	3	91.90	odd	2
637.2.e.k.79.1		6	91.38	odd	6
637.2.e.k.508.1		6	91.12	odd	6
637.2.e.l.79.1		6	91.25	even	6
637.2.e.l.508.1		6	91.51	even	6
5733.2.a.bd.1.1		3	273.272	even	2
5733.2.a.be.1.1		3	39.38	odd	2
8281.2.a.bh.1.1		3	1.1	even	1	trivial
8281.2.a.bk.1.1		3	7.6	odd	2

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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86620 q^{2} -3.34889 q^{3} +1.48270 q^{4} -0.866198 q^{5} +6.24970 q^{6} +0.965392 q^{8} +8.21509 q^{9} +1.61650 q^{10} +3.86620 q^{11} -4.96539 q^{12} +2.90081 q^{15} -4.76700 q^{16} -3.34889 q^{17} -15.3310 q^{18} +5.38350 q^{19} -1.28431 q^{20} -7.21509 q^{22} -5.24970 q^{23} -3.23300 q^{24} -4.24970 q^{25} -17.4648 q^{27} +1.69779 q^{29} -5.41348 q^{30} -7.56399 q^{31} +6.96539 q^{32} -12.9475 q^{33} +6.24970 q^{34} +12.1805 q^{36} +4.83159 q^{37} -10.0467 q^{38} -0.836221 q^{40} +4.06922 q^{41} +4.03461 q^{43} +5.73240 q^{44} -7.11590 q^{45} +9.79698 q^{46} +3.65111 q^{47} +15.9642 q^{48} +7.93078 q^{50} +11.2151 q^{51} -0.215092 q^{53} +32.5928 q^{54} -3.34889 q^{55} -18.0288 q^{57} -3.16841 q^{58} +2.78491 q^{59} +4.30101 q^{60} +9.03461 q^{61} +14.1159 q^{62} -3.46479 q^{64} +24.1626 q^{66} +7.66318 q^{67} -4.96539 q^{68} +17.5807 q^{69} -4.90081 q^{71} +7.93078 q^{72} -15.5461 q^{73} -9.01671 q^{74} +14.2318 q^{75} +7.98210 q^{76} +9.43018 q^{79} +4.12917 q^{80} +33.8425 q^{81} -7.59396 q^{82} +4.09919 q^{83} +2.90081 q^{85} -7.52938 q^{86} -5.68571 q^{87} +3.73240 q^{88} +0.418110 q^{89} +13.2797 q^{90} -7.78371 q^{92} +25.3310 q^{93} -6.81369 q^{94} -4.66318 q^{95} -23.3264 q^{96} +7.11590 q^{97} +31.7612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} + 11 q^{9} + 14 q^{10} + 4 q^{11} - 18 q^{12} - 2 q^{15} + 4 q^{16} - 4 q^{17} - 8 q^{18} + 7 q^{19} + 16 q^{20} - 8 q^{22} + q^{23} - 28 q^{24}+ \cdots + 30 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 + 6 * q^8 + 11 * q^9 + 14 * q^10 + 4 * q^11 - 18 * q^12 - 2 * q^15 + 4 * q^16 - 4 * q^17 - 8 * q^18 + 7 * q^19 + 16 * q^20 - 8 * q^22 + q^23 - 28 * q^24 + 4 * q^25 - 22 * q^27 - 7 * q^29 - 24 * q^30 - 3 * q^31 + 24 * q^32 - 10 * q^33 + 2 * q^34 + 26 * q^36 + 10 * q^37 - 12 * q^38 + 22 * q^40 + 6 * q^41 + 9 * q^43 + 2 * q^44 + 3 * q^45 + 28 * q^46 + 17 * q^47 - 16 * q^48 + 30 * q^50 + 20 * q^51 + 13 * q^53 + 28 * q^54 - 4 * q^55 - 4 * q^57 - 14 * q^58 + 22 * q^59 - 42 * q^60 + 24 * q^61 + 18 * q^62 + 20 * q^64 + 30 * q^66 + 14 * q^67 - 18 * q^68 - 2 * q^69 - 4 * q^71 + 30 * q^72 + 5 * q^73 + 8 * q^74 - 6 * q^75 - 8 * q^76 + q^79 + 40 * q^80 + 15 * q^81 - 20 * q^82 + 23 * q^83 - 2 * q^85 - 6 * q^86 - 20 * q^87 - 4 * q^88 - 11 * q^89 + 40 * q^90 + 30 * q^92 + 38 * q^93 + 16 * q^94 - 5 * q^95 - 52 * q^96 - 3 * q^97 + 30 * q^99

Introduction

L-functions

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