LMFDB - Embedded newform 8281.2.a.cw.1.5

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:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	\(q-1.76732 q^{2} +2.32802 q^{3} +1.12344 q^{4} +4.06760 q^{5} -4.11437 q^{6} +1.54917 q^{8} +2.41970 q^{9} -7.18877 q^{10} -2.90676 q^{11} +2.61538 q^{12} +9.46947 q^{15} -4.98476 q^{16} +3.37669 q^{17} -4.27639 q^{18} -1.18631 q^{19} +4.56969 q^{20} +5.13720 q^{22} -0.00746406 q^{23} +3.60651 q^{24} +11.5454 q^{25} -1.35096 q^{27} -8.00705 q^{29} -16.7356 q^{30} -0.679440 q^{31} +5.71135 q^{32} -6.76702 q^{33} -5.96771 q^{34} +2.71837 q^{36} -2.58986 q^{37} +2.09659 q^{38} +6.30142 q^{40} +10.5725 q^{41} +4.33374 q^{43} -3.26556 q^{44} +9.84236 q^{45} +0.0131914 q^{46} +11.4149 q^{47} -11.6047 q^{48} -20.4044 q^{50} +7.86103 q^{51} -6.09729 q^{53} +2.38759 q^{54} -11.8236 q^{55} -2.76175 q^{57} +14.1511 q^{58} +3.95630 q^{59} +10.6383 q^{60} -3.60341 q^{61} +1.20079 q^{62} -0.124274 q^{64} +11.9595 q^{66} +1.67759 q^{67} +3.79350 q^{68} -0.0173765 q^{69} +7.13225 q^{71} +3.74853 q^{72} +10.4944 q^{73} +4.57712 q^{74} +26.8779 q^{75} -1.33274 q^{76} +11.4421 q^{79} -20.2760 q^{80} -10.4042 q^{81} -18.6850 q^{82} +1.82071 q^{83} +13.7350 q^{85} -7.65912 q^{86} -18.6406 q^{87} -4.50308 q^{88} -3.60636 q^{89} -17.3946 q^{90} -0.00838539 q^{92} -1.58175 q^{93} -20.1738 q^{94} -4.82543 q^{95} +13.2962 q^{96} +10.1115 q^{97} -7.03349 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * 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 $

$2$	−1.76732	−1.24969	−0.624844	−	0.780750i	$-0.714837\pi$
$2$	−1.76732	−1.24969	−0.624844	+	0.780750i	$0.714837\pi$
$3$	2.32802	1.34409	0.672043	−	0.740512i	$-0.265417\pi$
$3$	2.32802	1.34409	0.672043	+	0.740512i	$0.265417\pi$
$4$	1.12344	0.561718
$5$	4.06760	1.81909	0.909543	−	0.415610i	$-0.136432\pi$
$5$	4.06760	1.81909	0.909543	+	0.415610i	$0.136432\pi$
$6$	−4.11437	−1.67969
$7$	0	0
$7$	0	0
$8$	1.54917	0.547716
$9$	2.41970	0.806565
$10$	−7.18877	−2.27329
$11$	−2.90676	−0.876423	−0.438211	−	0.898872i	$-0.644388\pi$
$11$	−2.90676	−0.876423	−0.438211	+	0.898872i	$0.644388\pi$
$12$	2.61538	0.754997
$13$	0	0
$13$	0	0
$14$	0	0
$15$	9.46947	2.44501
$16$	−4.98476	−1.24619
$17$	3.37669	0.818969	0.409484	−	0.912317i	$-0.365709\pi$
$17$	3.37669	0.818969	0.409484	+	0.912317i	$0.365709\pi$
$18$	−4.27639	−1.00795
$19$	−1.18631	−0.272158	−0.136079	−	0.990698i	$-0.543450\pi$
$19$	−1.18631	−0.272158	−0.136079	+	0.990698i	$0.543450\pi$
$20$	4.56969	1.02181
$21$	0	0
$22$	5.13720	1.09525
$23$	−0.00746406	−0.00155636	−0.000778182	−	1.00000i	$-0.500248\pi$
$23$	−0.00746406	−0.00155636	−0.000778182		1.00000i	$0.500248\pi$
$24$	3.60651	0.736177
$25$	11.5454	2.30907
$26$	0	0
$27$	−1.35096	−0.259993
$28$	0	0
$29$	−8.00705	−1.48687	−0.743436	−	0.668807i	$-0.766805\pi$
$29$	−8.00705	−1.48687	−0.743436	+	0.668807i	$0.766805\pi$
$30$	−16.7356	−3.05549
$31$	−0.679440	−0.122031	−0.0610155	−	0.998137i	$-0.519434\pi$
$31$	−0.679440	−0.122031	−0.0610155	+	0.998137i	$0.519434\pi$
$32$	5.71135	1.00963
$33$	−6.76702	−1.17799
$34$	−5.96771	−1.02345
$35$	0	0
$36$	2.71837	0.453062
$37$	−2.58986	−0.425770	−0.212885	−	0.977077i	$-0.568286\pi$
$37$	−2.58986	−0.425770	−0.212885	+	0.977077i	$0.568286\pi$
$38$	2.09659	0.340112
$39$	0	0
$40$	6.30142	0.996342
$41$	10.5725	1.65115	0.825573	−	0.564295i	$-0.190852\pi$
$41$	10.5725	1.65115	0.825573	+	0.564295i	$0.190852\pi$
$42$	0	0
$43$	4.33374	0.660888	0.330444	−	0.943826i	$-0.392801\pi$
$43$	4.33374	0.660888	0.330444	+	0.943826i	$0.392801\pi$
$44$	−3.26556	−0.492302
$45$	9.84236	1.46721
$46$	0.0131914	0.00194497
$47$	11.4149	1.66503	0.832517	−	0.553999i	$-0.186899\pi$
$47$	11.4149	1.66503	0.832517	+	0.553999i	$0.186899\pi$
$48$	−11.6047	−1.67499
$49$	0	0
$50$	−20.4044	−2.88562
$51$	7.86103	1.10076
$52$	0	0
$53$	−6.09729	−0.837527	−0.418763	−	0.908095i	$-0.637536\pi$
$53$	−6.09729	−0.837527	−0.418763	+	0.908095i	$0.637536\pi$
$54$	2.38759	0.324909
$55$	−11.8236	−1.59429
$56$	0	0
$57$	−2.76175	−0.365803
$58$	14.1511	1.85812
$59$	3.95630	0.515067	0.257533	−	0.966269i	$-0.417090\pi$
$59$	3.95630	0.515067	0.257533	+	0.966269i	$0.417090\pi$
$60$	10.6383	1.37340
$61$	−3.60341	−0.461370	−0.230685	−	0.973029i	$-0.574097\pi$
$61$	−3.60341	−0.461370	−0.230685	+	0.973029i	$0.574097\pi$
$62$	1.20079	0.152501
$63$	0	0
$64$	−0.124274	−0.0155343
$65$	0	0
$66$	11.9595	1.47211
$67$	1.67759	0.204951	0.102475	−	0.994736i	$-0.467324\pi$
$67$	1.67759	0.204951	0.102475	+	0.994736i	$0.467324\pi$
$68$	3.79350	0.460029
$69$	−0.0173765	−0.00209189
$70$	0	0
$71$	7.13225	0.846442	0.423221	−	0.906026i	$-0.360899\pi$
$71$	7.13225	0.846442	0.423221	+	0.906026i	$0.360899\pi$
$72$	3.74853	0.441769
$73$	10.4944	1.22828	0.614139	−	0.789198i	$-0.289503\pi$
$73$	10.4944	1.22828	0.614139	+	0.789198i	$0.289503\pi$
$74$	4.57712	0.532079
$75$	26.8779	3.10359
$76$	−1.33274	−0.152876
$77$	0	0
$78$	0	0
$79$	11.4421	1.28733	0.643667	−	0.765306i	$-0.277412\pi$
$79$	11.4421	1.28733	0.643667	+	0.765306i	$0.277412\pi$
$80$	−20.2760	−2.26693
$81$	−10.4042	−1.15602
$82$	−18.6850	−2.06342
$83$	1.82071	0.199849	0.0999246	−	0.994995i	$-0.468140\pi$
$83$	1.82071	0.199849	0.0999246	+	0.994995i	$0.468140\pi$
$84$	0	0
$85$	13.7350	1.48977
$86$	−7.65912	−0.825904
$87$	−18.6406	−1.99848
$88$	−4.50308	−0.480030
$89$	−3.60636	−0.382273	−0.191137	−	0.981563i	$-0.561217\pi$
$89$	−3.60636	−0.382273	−0.191137	+	0.981563i	$0.561217\pi$
$90$	−17.3946	−1.83356
$91$	0	0
$92$	−0.00838539	−0.000874237 0
$93$	−1.58175	−0.164020
$94$	−20.1738	−2.08077
$95$	−4.82543	−0.495078
$96$	13.2962	1.35703
$97$	10.1115	1.02667	0.513333	−	0.858189i	$-0.328411\pi$
$97$	10.1115	1.02667	0.513333	+	0.858189i	$0.328411\pi$
$98$	0	0
$99$	−7.03349	−0.706892
$100$	12.9705	1.29705
$101$	−5.16963	−0.514398	−0.257199	−	0.966359i	$-0.582800\pi$
$101$	−5.16963	−0.514398	−0.257199	+	0.966359i	$0.582800\pi$
$102$	−13.8930	−1.37561
$103$	3.80467	0.374886	0.187443	−	0.982276i	$-0.439980\pi$
$103$	3.80467	0.374886	0.187443	+	0.982276i	$0.439980\pi$
$104$	0	0
$105$	0	0
$106$	10.7759	1.04665
$107$	7.91981	0.765637	0.382818	−	0.923824i	$-0.374954\pi$
$107$	7.91981	0.765637	0.382818	+	0.923824i	$0.374954\pi$
$108$	−1.51772	−0.146042
$109$	2.33496	0.223648	0.111824	−	0.993728i	$-0.464331\pi$
$109$	2.33496	0.223648	0.111824	+	0.993728i	$0.464331\pi$
$110$	20.8961	1.99236
$111$	−6.02925	−0.572271
$112$	0	0
$113$	7.22596	0.679761	0.339880	−	0.940469i	$-0.389613\pi$
$113$	7.22596	0.679761	0.339880	+	0.940469i	$0.389613\pi$
$114$	4.88092	0.457140
$115$	−0.0303608	−0.00283116
$116$	−8.99541	−0.835202
$117$	0	0
$118$	−6.99207	−0.643672
$119$	0	0
$120$	14.6699	1.33917
$121$	−2.55072	−0.231883
$122$	6.36840	0.576568
$123$	24.6130	2.21928
$124$	−0.763307	−0.0685470
$125$	26.6240	2.38132
$126$	0	0
$127$	3.31802	0.294427	0.147213	−	0.989105i	$-0.452970\pi$
$127$	3.31802	0.294427	0.147213	+	0.989105i	$0.452970\pi$
$128$	−11.2031	−0.990220
$129$	10.0890	0.888291
$130$	0	0
$131$	−19.5592	−1.70889	−0.854446	−	0.519540i	$-0.826103\pi$
$131$	−19.5592	−1.70889	−0.854446	+	0.519540i	$0.826103\pi$
$132$	−7.60231	−0.661696
$133$	0	0
$134$	−2.96485	−0.256124
$135$	−5.49517	−0.472949
$136$	5.23109	0.448562
$137$	−8.19509	−0.700154	−0.350077	−	0.936721i	$-0.613845\pi$
$137$	−8.19509	−0.700154	−0.350077	+	0.936721i	$0.613845\pi$
$138$	0.0307099	0.00261420
$139$	3.12762	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$139$	3.12762	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$140$	0	0
$141$	26.5742	2.23795
$142$	−12.6050	−1.05779
$143$	0	0
$144$	−12.0616	−1.00513
$145$	−32.5695	−2.70475
$146$	−18.5470	−1.53496
$147$	0	0
$148$	−2.90954	−0.239163
$149$	17.7943	1.45777	0.728884	−	0.684637i	$-0.240039\pi$
$149$	17.7943	1.45777	0.728884	+	0.684637i	$0.240039\pi$
$150$	−47.5020	−3.87852
$151$	−3.50944	−0.285594	−0.142797	−	0.989752i	$-0.545610\pi$
$151$	−3.50944	−0.285594	−0.142797	+	0.989752i	$0.545610\pi$
$152$	−1.83780	−0.149065
$153$	8.17058	0.660552
$154$	0	0
$155$	−2.76369	−0.221985
$156$	0	0
$157$	−2.12637	−0.169703	−0.0848515	−	0.996394i	$-0.527042\pi$
$157$	−2.12637	−0.169703	−0.0848515	+	0.996394i	$0.527042\pi$
$158$	−20.2218	−1.60876
$159$	−14.1946	−1.12571
$160$	23.2315	1.83661
$161$	0	0
$162$	18.3875	1.44466
$163$	19.0172	1.48955	0.744773	−	0.667318i	$-0.232558\pi$
$163$	19.0172	1.48955	0.744773	+	0.667318i	$0.232558\pi$
$164$	11.8775	0.927478
$165$	−27.5255	−2.14286
$166$	−3.21779	−0.249749
$167$	20.5398	1.58942	0.794711	−	0.606988i	$-0.207623\pi$
$167$	20.5398	1.58942	0.794711	+	0.606988i	$0.207623\pi$
$168$	0	0
$169$	0	0
$170$	−24.2743	−1.86175
$171$	−2.87051	−0.219513
$172$	4.86867	0.371233
$173$	5.26797	0.400516	0.200258	−	0.979743i	$-0.435822\pi$
$173$	5.26797	0.400516	0.200258	+	0.979743i	$0.435822\pi$
$174$	32.9440	2.49748
$175$	0	0
$176$	14.4895	1.09219
$177$	9.21037	0.692294
$178$	6.37360	0.477722
$179$	8.76221	0.654918	0.327459	−	0.944865i	$-0.393808\pi$
$179$	8.76221	0.654918	0.327459	+	0.944865i	$0.393808\pi$
$180$	11.0573	0.824159
$181$	16.9098	1.25690	0.628449	−	0.777851i	$-0.283690\pi$
$181$	16.9098	1.25690	0.628449	+	0.777851i	$0.283690\pi$
$182$	0	0
$183$	−8.38883	−0.620120
$184$	−0.0115631	−0.000852445 0
$185$	−10.5345	−0.774513
$186$	2.79547	0.204974
$187$	−9.81526	−0.717763
$188$	12.8239	0.935279
$189$	0	0
$190$	8.52810	0.618693
$191$	−6.49288	−0.469809	−0.234904	−	0.972018i	$-0.575478\pi$
$191$	−6.49288	−0.469809	−0.234904	+	0.972018i	$0.575478\pi$
$192$	−0.289314	−0.0208794
$193$	7.52381	0.541576	0.270788	−	0.962639i	$-0.412716\pi$
$193$	7.52381	0.541576	0.270788	+	0.962639i	$0.412716\pi$
$194$	−17.8703	−1.28301
$195$	0	0
$196$	0	0
$197$	21.4727	1.52987	0.764933	−	0.644109i	$-0.222772\pi$
$197$	21.4727	1.52987	0.764933	+	0.644109i	$0.222772\pi$
$198$	12.4305	0.883394
$199$	−1.21021	−0.0857894	−0.0428947	−	0.999080i	$-0.513658\pi$
$199$	−1.21021	−0.0857894	−0.0428947	+	0.999080i	$0.513658\pi$
$200$	17.8858	1.26472
$201$	3.90548	0.275471
$202$	9.13641	0.642836
$203$	0	0
$204$	8.83136	0.618319
$205$	43.0047	3.00358
$206$	−6.72409	−0.468490
$207$	−0.0180608	−0.00125531
$208$	0	0
$209$	3.44832	0.238525
$210$	0	0
$211$	17.4646	1.20232	0.601158	−	0.799130i	$-0.294706\pi$
$211$	17.4646	1.20232	0.601158	+	0.799130i	$0.294706\pi$
$212$	−6.84991	−0.470454
$213$	16.6040	1.13769
$214$	−13.9969	−0.956806
$215$	17.6279	1.20221
$216$	−2.09287	−0.142402
$217$	0	0
$218$	−4.12663	−0.279490
$219$	24.4313	1.65091
$220$	−13.2830	−0.895540
$221$	0	0
$222$	10.6556	0.715160
$223$	12.5328	0.839258	0.419629	−	0.907696i	$-0.362160\pi$
$223$	12.5328	0.839258	0.419629	+	0.907696i	$0.362160\pi$
$224$	0	0
$225$	27.9363	1.86242
$226$	−12.7706	−0.849488
$227$	−23.1920	−1.53931	−0.769653	−	0.638463i	$-0.779571\pi$
$227$	−23.1920	−1.53931	−0.769653	+	0.638463i	$0.779571\pi$
$228$	−3.10265	−0.205478
$229$	7.61349	0.503113	0.251557	−	0.967843i	$-0.419057\pi$
$229$	7.61349	0.503113	0.251557	+	0.967843i	$0.419057\pi$
$230$	0.0536574	0.00353806
$231$	0	0
$232$	−12.4043	−0.814383
$233$	23.9588	1.56959	0.784797	−	0.619753i	$-0.212767\pi$
$233$	23.9588	1.56959	0.784797	+	0.619753i	$0.212767\pi$
$234$	0	0
$235$	46.4313	3.02884
$236$	4.44465	0.289322
$237$	26.6374	1.73029
$238$	0	0
$239$	15.2589	0.987015	0.493508	−	0.869741i	$-0.335715\pi$
$239$	15.2589	0.987015	0.493508	+	0.869741i	$0.335715\pi$
$240$	−47.2031	−3.04695
$241$	−10.2271	−0.658786	−0.329393	−	0.944193i	$-0.606844\pi$
$241$	−10.2271	−0.658786	−0.329393	+	0.944193i	$0.606844\pi$
$242$	4.50795	0.289782
$243$	−20.1682	−1.29379
$244$	−4.04820	−0.259160
$245$	0	0
$246$	−43.4992	−2.77341
$247$	0	0
$248$	−1.05257	−0.0668383
$249$	4.23866	0.268614
$250$	−47.0532	−2.97590
$251$	−13.0463	−0.823473	−0.411737	−	0.911303i	$-0.635078\pi$
$251$	−13.0463	−0.823473	−0.411737	+	0.911303i	$0.635078\pi$
$252$	0	0
$253$	0.0216963	0.00136403
$254$	−5.86402	−0.367941
$255$	31.9755	2.00238
$256$	20.0480	1.25300
$257$	−15.1619	−0.945776	−0.472888	−	0.881123i	$-0.656789\pi$
$257$	−15.1619	−0.945776	−0.472888	+	0.881123i	$0.656789\pi$
$258$	−17.8306	−1.11009
$259$	0	0
$260$	0	0
$261$	−19.3746	−1.19926
$262$	34.5674	2.13558
$263$	0.671716	0.0414198	0.0207099	−	0.999786i	$-0.493407\pi$
$263$	0.671716	0.0414198	0.0207099	+	0.999786i	$0.493407\pi$
$264$	−10.4833	−0.645202
$265$	−24.8013	−1.52353
$266$	0	0
$267$	−8.39569	−0.513808
$268$	1.88467	0.115124
$269$	−18.0372	−1.09975	−0.549874	−	0.835248i	$-0.685324\pi$
$269$	−18.0372	−1.09975	−0.549874	+	0.835248i	$0.685324\pi$
$270$	9.71175	0.591038
$271$	9.51258	0.577848	0.288924	−	0.957352i	$-0.406703\pi$
$271$	9.51258	0.577848	0.288924	+	0.957352i	$0.406703\pi$
$272$	−16.8320	−1.02059
$273$	0	0
$274$	14.4834	0.874974
$275$	−33.5597	−2.02372
$276$	−0.0195214	−0.00117505
$277$	3.48620	0.209466	0.104733	−	0.994500i	$-0.466601\pi$
$277$	3.48620	0.209466	0.104733	+	0.994500i	$0.466601\pi$
$278$	−5.52753	−0.331519
$279$	−1.64404	−0.0984260
$280$	0	0
$281$	−13.0611	−0.779160	−0.389580	−	0.920993i	$-0.627380\pi$
$281$	−13.0611	−0.779160	−0.389580	+	0.920993i	$0.627380\pi$
$282$	−46.9652	−2.79674
$283$	−7.36674	−0.437907	−0.218954	−	0.975735i	$-0.570264\pi$
$283$	−7.36674	−0.437907	−0.218954	+	0.975735i	$0.570264\pi$
$284$	8.01262	0.475462
$285$	−11.2337	−0.665428
$286$	0	0
$287$	0	0
$288$	13.8197	0.814335
$289$	−5.59793	−0.329290
$290$	57.5608	3.38009
$291$	23.5398	1.37993
$292$	11.7898	0.689946
$293$	−23.9996	−1.40207	−0.701035	−	0.713127i	$-0.747278\pi$
$293$	−23.9996	−1.40207	−0.701035	+	0.713127i	$0.747278\pi$
$294$	0	0
$295$	16.0927	0.936951
$296$	−4.01214	−0.233201
$297$	3.92693	0.227863
$298$	−31.4484	−1.82175
$299$	0	0
$300$	30.1956	1.74334
$301$	0	0
$302$	6.20232	0.356903
$303$	−12.0350	−0.691394
$304$	5.91347	0.339161
$305$	−14.6572	−0.839271
$306$	−14.4401	−0.825483
$307$	−0.871450	−0.0497363	−0.0248681	−	0.999691i	$-0.507917\pi$
$307$	−0.871450	−0.0497363	−0.0248681	+	0.999691i	$0.507917\pi$
$308$	0	0
$309$	8.85737	0.503878
$310$	4.88434	0.277412
$311$	−16.3842	−0.929061	−0.464530	−	0.885557i	$-0.653777\pi$
$311$	−16.3842	−0.929061	−0.464530	+	0.885557i	$0.653777\pi$
$312$	0	0
$313$	−17.2300	−0.973898	−0.486949	−	0.873430i	$-0.661890\pi$
$313$	−17.2300	−0.973898	−0.486949	+	0.873430i	$0.661890\pi$
$314$	3.75799	0.212076
$315$	0	0
$316$	12.8544	0.723118
$317$	9.03976	0.507724	0.253862	−	0.967240i	$-0.418299\pi$
$317$	9.03976	0.507724	0.253862	+	0.967240i	$0.418299\pi$
$318$	25.0865	1.40678
$319$	23.2746	1.30313
$320$	−0.505498	−0.0282582
$321$	18.4375	1.02908
$322$	0	0
$323$	−4.00580	−0.222889
$324$	−11.6884	−0.649356
$325$	0	0
$326$	−33.6096	−1.86147
$327$	5.43584	0.300603
$328$	16.3786	0.904358
$329$	0	0
$330$	48.6465	2.67790
$331$	−35.8880	−1.97258	−0.986291	−	0.165014i	$-0.947233\pi$
$331$	−35.8880	−1.97258	−0.986291	+	0.165014i	$0.947233\pi$
$332$	2.04545	0.112259
$333$	−6.26667	−0.343412
$334$	−36.3006	−1.98628
$335$	6.82378	0.372823
$336$	0	0
$337$	17.5864	0.957992	0.478996	−	0.877817i	$-0.341001\pi$
$337$	17.5864	0.957992	0.478996	+	0.877817i	$0.341001\pi$
$338$	0	0
$339$	16.8222	0.913656
$340$	15.4304	0.836833
$341$	1.97497	0.106951
$342$	5.07312	0.274323
$343$	0	0
$344$	6.71371	0.361979
$345$	−0.0706807	−0.00380532
$346$	−9.31022	−0.500520
$347$	20.0826	1.07809	0.539045	−	0.842277i	$-0.318785\pi$
$347$	20.0826	1.07809	0.539045	+	0.842277i	$0.318785\pi$
$348$	−20.9415	−1.12258
$349$	−27.3383	−1.46339	−0.731693	−	0.681635i	$-0.761269\pi$
$349$	−27.3383	−1.46339	−0.731693	+	0.681635i	$0.761269\pi$
$350$	0	0
$351$	0	0
$352$	−16.6015	−0.884865
$353$	−16.4540	−0.875759	−0.437879	−	0.899034i	$-0.644270\pi$
$353$	−16.4540	−0.875759	−0.437879	+	0.899034i	$0.644270\pi$
$354$	−16.2777	−0.865150
$355$	29.0111	1.53975
$356$	−4.05151	−0.214730
$357$	0	0
$358$	−15.4857	−0.818443
$359$	−22.1418	−1.16860	−0.584299	−	0.811539i	$-0.698630\pi$
$359$	−22.1418	−1.16860	−0.584299	+	0.811539i	$0.698630\pi$
$360$	15.2475	0.803615
$361$	−17.5927	−0.925930
$362$	−29.8851	−1.57073
$363$	−5.93813	−0.311671
$364$	0	0
$365$	42.6871	2.23434
$366$	14.8258	0.774956
$367$	10.5997	0.553300	0.276650	−	0.960971i	$-0.410776\pi$
$367$	10.5997	0.553300	0.276650	+	0.960971i	$0.410776\pi$
$368$	0.0372066	0.00193953
$369$	25.5822	1.33176
$370$	18.6179	0.967898
$371$	0	0
$372$	−1.77700	−0.0921330
$373$	15.7596	0.816003	0.408002	−	0.912981i	$-0.366226\pi$
$373$	15.7596	0.816003	0.408002	+	0.912981i	$0.366226\pi$
$374$	17.3467	0.896979
$375$	61.9812	3.20070
$376$	17.6837	0.911966
$377$	0	0
$378$	0	0
$379$	22.6327	1.16257	0.581283	−	0.813702i	$-0.302551\pi$
$379$	22.6327	1.16257	0.581283	+	0.813702i	$0.302551\pi$
$380$	−5.42106	−0.278094
$381$	7.72443	0.395735
$382$	11.4750	0.587114
$383$	−0.278328	−0.0142219	−0.00711095	−	0.999975i	$-0.502264\pi$
$383$	−0.278328	−0.0142219	−0.00711095	+	0.999975i	$0.502264\pi$
$384$	−26.0810	−1.33094
$385$	0	0
$386$	−13.2970	−0.676800
$387$	10.4863	0.533050
$388$	11.3596	0.576697
$389$	−31.9693	−1.62091	−0.810453	−	0.585804i	$-0.800779\pi$
$389$	−31.9693	−1.62091	−0.810453	+	0.585804i	$0.800779\pi$
$390$	0	0
$391$	−0.0252038	−0.00127461
$392$	0	0
$393$	−45.5342	−2.29690
$394$	−37.9492	−1.91185
$395$	46.5418	2.34177
$396$	−7.90167	−0.397074
$397$	−18.2200	−0.914438	−0.457219	−	0.889354i	$-0.651154\pi$
$397$	−18.2200	−0.914438	−0.457219	+	0.889354i	$0.651154\pi$
$398$	2.13883	0.107210
$399$	0	0
$400$	−57.5510	−2.87755
$401$	−20.7625	−1.03683	−0.518414	−	0.855130i	$-0.673477\pi$
$401$	−20.7625	−1.03683	−0.518414	+	0.855130i	$0.673477\pi$
$402$	−6.90225	−0.344253
$403$	0	0
$404$	−5.80775	−0.288946
$405$	−42.3200	−2.10290
$406$	0	0
$407$	7.52811	0.373155
$408$	12.1781	0.602906
$409$	16.5799	0.819822	0.409911	−	0.912126i	$-0.365560\pi$
$409$	16.5799	0.819822	0.409911	+	0.912126i	$0.365560\pi$
$410$	−76.0032	−3.75353
$411$	−19.0784	−0.941067
$412$	4.27430	0.210580
$413$	0	0
$414$	0.0319192	0.00156874
$415$	7.40593	0.363543
$416$	0	0
$417$	7.28118	0.356561
$418$	−6.09430	−0.298082
$419$	4.21062	0.205702	0.102851	−	0.994697i	$-0.467203\pi$
$419$	4.21062	0.205702	0.102851	+	0.994697i	$0.467203\pi$
$420$	0	0
$421$	−33.9650	−1.65535	−0.827677	−	0.561205i	$-0.810338\pi$
$421$	−33.9650	−1.65535	−0.827677	+	0.561205i	$0.810338\pi$
$422$	−30.8657	−1.50252
$423$	27.6206	1.34296
$424$	−9.44576	−0.458727
$425$	38.9852	1.89106
$426$	−29.3447	−1.42176
$427$	0	0
$428$	8.89739	0.430072
$429$	0	0
$430$	−31.1542	−1.50239
$431$	−25.3140	−1.21933	−0.609667	−	0.792657i	$-0.708697\pi$
$431$	−25.3140	−1.21933	−0.609667	+	0.792657i	$0.708697\pi$
$432$	6.73422	0.324000
$433$	2.37946	0.114350	0.0571749	−	0.998364i	$-0.481791\pi$
$433$	2.37946	0.114350	0.0571749	+	0.998364i	$0.481791\pi$
$434$	0	0
$435$	−75.8226	−3.63541
$436$	2.62317	0.125627
$437$	0.00885467	0.000423577 0
$438$	−43.1780	−2.06312
$439$	−16.8172	−0.802642	−0.401321	−	0.915938i	$-0.631449\pi$
$439$	−16.8172	−0.802642	−0.401321	+	0.915938i	$0.631449\pi$
$440$	−18.3167	−0.873217
$441$	0	0
$442$	0	0
$443$	−16.1250	−0.766123	−0.383061	−	0.923723i	$-0.625130\pi$
$443$	−16.1250	−0.766123	−0.383061	+	0.923723i	$0.625130\pi$
$444$	−6.77348	−0.321455
$445$	−14.6692	−0.695388
$446$	−22.1495	−1.04881
$447$	41.4256	1.95936
$448$	0	0
$449$	−40.5289	−1.91268	−0.956339	−	0.292261i	$-0.905592\pi$
$449$	−40.5289	−1.91268	−0.956339	+	0.292261i	$0.905592\pi$
$450$	−49.3725	−2.32744
$451$	−30.7317	−1.44710
$452$	8.11789	0.381834
$453$	−8.17006	−0.383863
$454$	40.9877	1.92365
$455$	0	0
$456$	−4.27844	−0.200356
$457$	11.6091	0.543051	0.271526	−	0.962431i	$-0.412472\pi$
$457$	11.6091	0.543051	0.271526	+	0.962431i	$0.412472\pi$
$458$	−13.4555	−0.628734
$459$	−4.56178	−0.212926
$460$	−0.0341084	−0.00159031
$461$	26.4282	1.23088	0.615441	−	0.788183i	$-0.288978\pi$
$461$	26.4282	1.23088	0.615441	+	0.788183i	$0.288978\pi$
$462$	0	0
$463$	15.7251	0.730810	0.365405	−	0.930849i	$-0.380931\pi$
$463$	15.7251	0.730810	0.365405	+	0.930849i	$0.380931\pi$
$464$	39.9133	1.85293
$465$	−6.43394	−0.298367
$466$	−42.3430	−1.96150
$467$	27.4256	1.26911	0.634553	−	0.772879i	$-0.281184\pi$
$467$	27.4256	1.26911	0.634553	+	0.772879i	$0.281184\pi$
$468$	0	0
$469$	0	0
$470$	−82.0591	−3.78510
$471$	−4.95025	−0.228095
$472$	6.12900	0.282110
$473$	−12.5971	−0.579218
$474$	−47.0769	−2.16232
$475$	−13.6964	−0.628433
$476$	0	0
$477$	−14.7536	−0.675520
$478$	−26.9674	−1.23346
$479$	−17.8460	−0.815406	−0.407703	−	0.913115i	$-0.633670\pi$
$479$	−17.8460	−0.815406	−0.407703	+	0.913115i	$0.633670\pi$
$480$	54.0834	2.46856
$481$	0	0
$482$	18.0746	0.823277
$483$	0	0
$484$	−2.86557	−0.130253
$485$	41.1295	1.86759
$486$	35.6438	1.61684
$487$	−5.66101	−0.256525	−0.128262	−	0.991740i	$-0.540940\pi$
$487$	−5.66101	−0.256525	−0.128262	+	0.991740i	$0.540940\pi$
$488$	−5.58231	−0.252699
$489$	44.2726	2.00208
$490$	0	0
$491$	40.2435	1.81616	0.908081	−	0.418795i	$-0.137547\pi$
$491$	40.2435	1.81616	0.908081	+	0.418795i	$0.137547\pi$
$492$	27.6511	1.24661
$493$	−27.0374	−1.21770
$494$	0	0
$495$	−28.6094	−1.28590
$496$	3.38685	0.152074
$497$	0	0
$498$	−7.49109	−0.335684
$499$	−14.0573	−0.629293	−0.314647	−	0.949209i	$-0.601886\pi$
$499$	−14.0573	−0.629293	−0.314647	+	0.949209i	$0.601886\pi$
$500$	29.9103	1.33763
$501$	47.8173	2.13632
$502$	23.0570	1.02908
$503$	−7.72836	−0.344591	−0.172295	−	0.985045i	$-0.555118\pi$
$503$	−7.72836	−0.344591	−0.172295	+	0.985045i	$0.555118\pi$
$504$	0	0
$505$	−21.0280	−0.935733
$506$	−0.0383443	−0.00170461
$507$	0	0
$508$	3.72758	0.165385
$509$	6.08600	0.269757	0.134879	−	0.990862i	$-0.456936\pi$
$509$	6.08600	0.269757	0.134879	+	0.990862i	$0.456936\pi$
$510$	−56.5111	−2.50235
$511$	0	0
$512$	−13.0252	−0.575637
$513$	1.60266	0.0707590
$514$	26.7961	1.18192
$515$	15.4759	0.681949
$516$	11.3344	0.498969
$517$	−33.1804	−1.45927
$518$	0	0
$519$	12.2640	0.538328
$520$	0	0
$521$	0.0347104	0.00152069	0.000760344	−	1.00000i	$-0.499758\pi$
$521$	0.0347104	0.00152069	0.000760344		1.00000i	$0.499758\pi$
$522$	34.2413	1.49870
$523$	17.5984	0.769523	0.384762	−	0.923016i	$-0.374284\pi$
$523$	17.5984	0.769523	0.384762	+	0.923016i	$0.374284\pi$
$524$	−21.9735	−0.959915
$525$	0	0
$526$	−1.18714	−0.0517618
$527$	−2.29426	−0.0999395
$528$	33.7320	1.46800
$529$	−22.9999	−0.999998
$530$	43.8320	1.90394
$531$	9.57305	0.415435
$532$	0	0
$533$	0	0
$534$	14.8379	0.642099
$535$	32.2146	1.39276
$536$	2.59888	0.112255
$537$	20.3986	0.880266
$538$	31.8776	1.37434
$539$	0	0
$540$	−6.17347	−0.265664
$541$	0.578216	0.0248595	0.0124297	−	0.999923i	$-0.496043\pi$
$541$	0.578216	0.0248595	0.0124297	+	0.999923i	$0.496043\pi$
$542$	−16.8118	−0.722129
$543$	39.3665	1.68938
$544$	19.2855	0.826858
$545$	9.49767	0.406836
$546$	0	0
$547$	−13.6270	−0.582650	−0.291325	−	0.956624i	$-0.594096\pi$
$547$	−13.6270	−0.582650	−0.291325	+	0.956624i	$0.594096\pi$
$548$	−9.20666	−0.393289
$549$	−8.71916	−0.372125
$550$	59.3108	2.52902
$551$	9.49883	0.404664
$552$	−0.0269192	−0.00114576
$553$	0	0
$554$	−6.16125	−0.261766
$555$	−24.5246	−1.04101
$556$	3.51368	0.149013
$557$	−6.46311	−0.273851	−0.136925	−	0.990581i	$-0.543722\pi$
$557$	−6.46311	−0.273851	−0.136925	+	0.990581i	$0.543722\pi$
$558$	2.90555	0.123002
$559$	0	0
$560$	0	0
$561$	−22.8502	−0.964734
$562$	23.0832	0.973706
$563$	−33.1023	−1.39509	−0.697547	−	0.716539i	$-0.745725\pi$
$563$	−33.1023	−1.39509	−0.697547	+	0.716539i	$0.745725\pi$
$564$	29.8544	1.25710
$565$	29.3923	1.23654
$566$	13.0194	0.547247
$567$	0	0
$568$	11.0491	0.463610
$569$	−44.1263	−1.84987	−0.924936	−	0.380123i	$-0.875882\pi$
$569$	−44.1263	−1.84987	−0.924936	+	0.380123i	$0.875882\pi$
$570$	19.8536	0.831576
$571$	−12.1407	−0.508074	−0.254037	−	0.967195i	$-0.581758\pi$
$571$	−12.1407	−0.508074	−0.254037	+	0.967195i	$0.581758\pi$
$572$	0	0
$573$	−15.1156	−0.631463
$574$	0	0
$575$	−0.0861753	−0.00359376
$576$	−0.300706	−0.0125294
$577$	−42.2854	−1.76036	−0.880181	−	0.474637i	$-0.842579\pi$
$577$	−42.2854	−1.76036	−0.880181	+	0.474637i	$0.842579\pi$
$578$	9.89337	0.411510
$579$	17.5156	0.727924
$580$	−36.5897	−1.51931
$581$	0	0
$582$	−41.6025	−1.72448
$583$	17.7234	0.734027
$584$	16.2577	0.672748
$585$	0	0
$586$	42.4150	1.75215
$587$	34.0219	1.40424	0.702118	−	0.712061i	$-0.252238\pi$
$587$	34.0219	1.40424	0.702118	+	0.712061i	$0.252238\pi$
$588$	0	0
$589$	0.806025	0.0332117
$590$	−28.4409	−1.17090
$591$	49.9890	2.05627
$592$	12.9098	0.530591
$593$	−9.28987	−0.381489	−0.190745	−	0.981640i	$-0.561090\pi$
$593$	−9.28987	−0.381489	−0.190745	+	0.981640i	$0.561090\pi$
$594$	−6.94015	−0.284758
$595$	0	0
$596$	19.9908	0.818854
$597$	−2.81739	−0.115308
$598$	0	0
$599$	14.5963	0.596389	0.298195	−	0.954505i	$-0.403616\pi$
$599$	14.5963	0.596389	0.298195	+	0.954505i	$0.403616\pi$
$600$	41.6386	1.69989
$601$	−4.28534	−0.174803	−0.0874013	−	0.996173i	$-0.527856\pi$
$601$	−4.28534	−0.174803	−0.0874013	+	0.996173i	$0.527856\pi$
$602$	0	0
$603$	4.05927	0.165306
$604$	−3.94263	−0.160423
$605$	−10.3753	−0.421816
$606$	21.2698	0.864026
$607$	−36.6206	−1.48638	−0.743192	−	0.669078i	$-0.766689\pi$
$607$	−36.6206	−1.48638	−0.743192	+	0.669078i	$0.766689\pi$
$608$	−6.77542	−0.274779
$609$	0	0
$610$	25.9041	1.04883
$611$	0	0
$612$	9.17911	0.371044
$613$	43.0967	1.74066	0.870329	−	0.492471i	$-0.163906\pi$
$613$	43.0967	1.74066	0.870329	+	0.492471i	$0.163906\pi$
$614$	1.54013	0.0621548
$615$	100.116	4.03706
$616$	0	0
$617$	29.5143	1.18820	0.594101	−	0.804390i	$-0.297508\pi$
$617$	29.5143	1.18820	0.594101	+	0.804390i	$0.297508\pi$
$618$	−15.6538	−0.629690
$619$	−27.8765	−1.12045	−0.560225	−	0.828340i	$-0.689286\pi$
$619$	−27.8765	−1.12045	−0.560225	+	0.828340i	$0.689286\pi$
$620$	−3.10483	−0.124693
$621$	0.0100836	0.000404643 0
$622$	28.9561	1.16103
$623$	0	0
$624$	0	0
$625$	50.5688	2.02275
$626$	30.4510	1.21707
$627$	8.02777	0.320598
$628$	−2.38884	−0.0953252
$629$	−8.74516	−0.348692
$630$	0	0
$631$	−30.0124	−1.19477	−0.597387	−	0.801953i	$-0.703794\pi$
$631$	−30.0124	−1.19477	−0.597387	+	0.801953i	$0.703794\pi$
$632$	17.7258	0.705093
$633$	40.6581	1.61602
$634$	−15.9762	−0.634496
$635$	13.4964	0.535587
$636$	−15.9467	−0.632330
$637$	0	0
$638$	−41.1338	−1.62850
$639$	17.2579	0.682711
$640$	−45.5696	−1.80130
$641$	−9.37664	−0.370355	−0.185177	−	0.982705i	$-0.559286\pi$
$641$	−9.37664	−0.370355	−0.185177	+	0.982705i	$0.559286\pi$
$642$	−32.5850	−1.28603
$643$	−1.78629	−0.0704444	−0.0352222	−	0.999380i	$-0.511214\pi$
$643$	−1.78629	−0.0704444	−0.0352222	+	0.999380i	$0.511214\pi$
$644$	0	0
$645$	41.0382	1.61588
$646$	7.07955	0.278541
$647$	31.6251	1.24331	0.621656	−	0.783290i	$-0.286460\pi$
$647$	31.6251	1.24331	0.621656	+	0.783290i	$0.286460\pi$
$648$	−16.1179	−0.633169
$649$	−11.5000	−0.451416
$650$	0	0
$651$	0	0
$652$	21.3646	0.836704
$653$	−10.8423	−0.424291	−0.212145	−	0.977238i	$-0.568045\pi$
$653$	−10.8423	−0.424291	−0.212145	+	0.977238i	$0.568045\pi$
$654$	−9.60689	−0.375659
$655$	−79.5588	−3.10862
$656$	−52.7014	−2.05764
$657$	25.3933	0.990687
$658$	0	0
$659$	−29.4418	−1.14689	−0.573446	−	0.819244i	$-0.694394\pi$
$659$	−29.4418	−1.14689	−0.573446	+	0.819244i	$0.694394\pi$
$660$	−30.9232	−1.20368
$661$	−18.7726	−0.730170	−0.365085	−	0.930974i	$-0.618960\pi$
$661$	−18.7726	−0.730170	−0.365085	+	0.930974i	$0.618960\pi$
$662$	63.4257	2.46511
$663$	0	0
$664$	2.82060	0.109461
$665$	0	0
$666$	11.0752	0.429157
$667$	0.0597651	0.00231411
$668$	23.0752	0.892806
$669$	29.1767	1.12803
$670$	−12.0598	−0.465912
$671$	10.4743	0.404355
$672$	0	0
$673$	−6.72821	−0.259353	−0.129677	−	0.991556i	$-0.541394\pi$
$673$	−6.72821	−0.259353	−0.129677	+	0.991556i	$0.541394\pi$
$674$	−31.0809	−1.19719
$675$	−15.5973	−0.600342
$676$	0	0
$677$	27.7129	1.06509	0.532547	−	0.846400i	$-0.321235\pi$
$677$	27.7129	1.06509	0.532547	+	0.846400i	$0.321235\pi$
$678$	−29.7303	−1.14178
$679$	0	0
$680$	21.2780	0.815973
$681$	−53.9915	−2.06896
$682$	−3.49042	−0.133655
$683$	4.60138	0.176067	0.0880335	−	0.996118i	$-0.471942\pi$
$683$	4.60138	0.176067	0.0880335	+	0.996118i	$0.471942\pi$
$684$	−3.22483	−0.123304
$685$	−33.3344	−1.27364
$686$	0	0
$687$	17.7244	0.676227
$688$	−21.6026	−0.823593
$689$	0	0
$690$	0.124916	0.00475546
$691$	−29.3380	−1.11607	−0.558036	−	0.829817i	$-0.688445\pi$
$691$	−29.3380	−1.11607	−0.558036	+	0.829817i	$0.688445\pi$
$692$	5.91823	0.224977
$693$	0	0
$694$	−35.4924	−1.34727
$695$	12.7219	0.482570
$696$	−28.8775	−1.09460
$697$	35.7001	1.35224
$698$	48.3156	1.82877
$699$	55.7767	2.10967
$700$	0	0
$701$	15.1336	0.571588	0.285794	−	0.958291i	$-0.407743\pi$
$701$	15.1336	0.571588	0.285794	+	0.958291i	$0.407743\pi$
$702$	0	0
$703$	3.07237	0.115877
$704$	0.361236	0.0136146
$705$	108.093	4.07102
$706$	29.0796	1.09442
$707$	0	0
$708$	10.3473	0.388874
$709$	−21.7826	−0.818062	−0.409031	−	0.912521i	$-0.634133\pi$
$709$	−21.7826	−0.818062	−0.409031	+	0.912521i	$0.634133\pi$
$710$	−51.2721	−1.92421
$711$	27.6863	1.03832
$712$	−5.58688	−0.209377
$713$	0.00507138	0.000189925 0
$714$	0	0
$715$	0	0
$716$	9.84377	0.367879
$717$	35.5231	1.32663
$718$	39.1317	1.46038
$719$	37.6514	1.40416	0.702079	−	0.712099i	$-0.252255\pi$
$719$	37.6514	1.40416	0.702079	+	0.712099i	$0.252255\pi$
$720$	−49.0618	−1.82843
$721$	0	0
$722$	31.0920	1.15712
$723$	−23.8090	−0.885465
$724$	18.9971	0.706021
$725$	−92.4444	−3.43330
$726$	10.4946	0.389491
$727$	−34.5858	−1.28272	−0.641358	−	0.767242i	$-0.721629\pi$
$727$	−34.5858	−1.28272	−0.641358	+	0.767242i	$0.721629\pi$
$728$	0	0
$729$	−15.7397	−0.582952
$730$	−75.4419	−2.79223
$731$	14.6337	0.541247
$732$	−9.42431	−0.348333
$733$	1.17210	0.0432924	0.0216462	−	0.999766i	$-0.493109\pi$
$733$	1.17210	0.0432924	0.0216462	+	0.999766i	$0.493109\pi$
$734$	−18.7331	−0.691452
$735$	0	0
$736$	−0.0426298	−0.00157136
$737$	−4.87637	−0.179623
$738$	−45.2121	−1.66428
$739$	−25.4831	−0.937412	−0.468706	−	0.883354i	$-0.655280\pi$
$739$	−25.4831	−0.937412	−0.468706	+	0.883354i	$0.655280\pi$
$740$	−11.8348	−0.435057
$741$	0	0
$742$	0	0
$743$	−18.0403	−0.661835	−0.330918	−	0.943660i	$-0.607358\pi$
$743$	−18.0403	−0.661835	−0.330918	+	0.943660i	$0.607358\pi$
$744$	−2.45041	−0.0898364
$745$	72.3802	2.65181
$746$	−27.8524	−1.01975
$747$	4.40557	0.161191
$748$	−11.0268	−0.403180
$749$	0	0
$750$	−109.541	−3.99987
$751$	−0.738214	−0.0269378	−0.0134689	−	0.999909i	$-0.504287\pi$
$751$	−0.738214	−0.0269378	−0.0134689	+	0.999909i	$0.504287\pi$
$752$	−56.9006	−2.07495
$753$	−30.3720	−1.10682
$754$	0	0
$755$	−14.2750	−0.519521
$756$	0	0
$757$	−15.5551	−0.565359	−0.282680	−	0.959214i	$-0.591223\pi$
$757$	−15.5551	−0.565359	−0.282680	+	0.959214i	$0.591223\pi$
$758$	−39.9994	−1.45284
$759$	0.0505094	0.00183338
$760$	−7.47543	−0.271162
$761$	−35.6931	−1.29388	−0.646938	−	0.762543i	$-0.723951\pi$
$761$	−35.6931	−1.29388	−0.646938	+	0.762543i	$0.723951\pi$
$762$	−13.6516	−0.494544
$763$	0	0
$764$	−7.29434	−0.263900
$765$	33.2346	1.20160
$766$	0.491896	0.0177729
$767$	0	0
$768$	46.6722	1.68414
$769$	30.8652	1.11303	0.556514	−	0.830838i	$-0.312139\pi$
$769$	30.8652	1.11303	0.556514	+	0.830838i	$0.312139\pi$
$770$	0	0
$771$	−35.2974	−1.27120
$772$	8.45252	0.304213
$773$	−26.7279	−0.961337	−0.480669	−	0.876902i	$-0.659606\pi$
$773$	−26.7279	−0.961337	−0.480669	+	0.876902i	$0.659606\pi$
$774$	−18.5327	−0.666145
$775$	−7.84438	−0.281779
$776$	15.6645	0.562321
$777$	0	0
$778$	56.5001	2.02563
$779$	−12.5422	−0.449372
$780$	0	0
$781$	−20.7318	−0.741841
$782$	0.0445434	0.00159287
$783$	10.8172	0.386576
$784$	0	0
$785$	−8.64924	−0.308704
$786$	80.4737	2.87040
$787$	23.9432	0.853485	0.426742	−	0.904373i	$-0.359661\pi$
$787$	23.9432	0.853485	0.426742	+	0.904373i	$0.359661\pi$
$788$	24.1232	0.859353
$789$	1.56377	0.0556717
$790$	−82.2544	−2.92648
$791$	0	0
$792$	−10.8961	−0.387176
$793$	0	0
$794$	32.2007	1.14276
$795$	−57.7381	−2.04776
$796$	−1.35959	−0.0481894
$797$	−41.3242	−1.46378	−0.731889	−	0.681424i	$-0.761361\pi$
$797$	−41.3242	−1.46378	−0.731889	+	0.681424i	$0.761361\pi$
$798$	0	0
$799$	38.5446	1.36361
$800$	65.9396	2.33132
$801$	−8.72629	−0.308328
$802$	36.6940	1.29571
$803$	−30.5048	−1.07649
$804$	4.38755	0.154737
$805$	0	0
$806$	0	0
$807$	−41.9910	−1.47815
$808$	−8.00866	−0.281744
$809$	−43.8260	−1.54084	−0.770421	−	0.637536i	$-0.779954\pi$
$809$	−43.8260	−1.54084	−0.770421	+	0.637536i	$0.779954\pi$
$810$	74.7931	2.62796
$811$	−37.9985	−1.33431	−0.667154	−	0.744920i	$-0.732488\pi$
$811$	−37.9985	−1.33431	−0.667154	+	0.744920i	$0.732488\pi$
$812$	0	0
$813$	22.1455	0.776677
$814$	−13.3046	−0.466326
$815$	77.3545	2.70961
$816$	−39.1854	−1.37176
$817$	−5.14115	−0.179866
$818$	−29.3020	−1.02452
$819$	0	0
$820$	48.3130	1.68716
$821$	−20.6672	−0.721291	−0.360645	−	0.932703i	$-0.617444\pi$
$821$	−20.6672	−0.721291	−0.360645	+	0.932703i	$0.617444\pi$
$822$	33.7177	1.17604
$823$	7.02888	0.245011	0.122506	−	0.992468i	$-0.460907\pi$
$823$	7.02888	0.245011	0.122506	+	0.992468i	$0.460907\pi$
$824$	5.89410	0.205331
$825$	−78.1277	−2.72006
$826$	0	0
$827$	47.5367	1.65301	0.826506	−	0.562928i	$-0.190325\pi$
$827$	47.5367	1.65301	0.826506	+	0.562928i	$0.190325\pi$
$828$	−0.0202901	−0.000705129 0
$829$	−37.0237	−1.28589	−0.642944	−	0.765913i	$-0.722287\pi$
$829$	−37.0237	−1.28589	−0.642944	+	0.765913i	$0.722287\pi$
$830$	−13.0887	−0.454315
$831$	8.11596	0.281540
$832$	0	0
$833$	0	0
$834$	−12.8682	−0.445590
$835$	83.5479	2.89129
$836$	3.87396	0.133984
$837$	0.917896	0.0317271
$838$	−7.44154	−0.257064
$839$	−3.89858	−0.134594	−0.0672970	−	0.997733i	$-0.521437\pi$
$839$	−3.89858	−0.134594	−0.0672970	+	0.997733i	$0.521437\pi$
$840$	0	0
$841$	35.1129	1.21079
$842$	60.0272	2.06867
$843$	−30.4065	−1.04726
$844$	19.6204	0.675362
$845$	0	0
$846$	−48.8146	−1.67828
$847$	0	0
$848$	30.3935	1.04372
$849$	−17.1500	−0.588585
$850$	−68.8995	−2.36323
$851$	0.0193309	0.000662653 0
$852$	18.6536	0.639061
$853$	−12.3795	−0.423868	−0.211934	−	0.977284i	$-0.567976\pi$
$853$	−12.3795	−0.423868	−0.211934	+	0.977284i	$0.567976\pi$
$854$	0	0
$855$	−11.6761	−0.399313
$856$	12.2692	0.419351
$857$	9.39835	0.321041	0.160521	−	0.987032i	$-0.448683\pi$
$857$	9.39835	0.321041	0.160521	+	0.987032i	$0.448683\pi$
$858$	0	0
$859$	−23.7071	−0.808875	−0.404438	−	0.914566i	$-0.632533\pi$
$859$	−23.7071	−0.808875	−0.404438	+	0.914566i	$0.632533\pi$
$860$	19.8038	0.675304
$861$	0	0
$862$	44.7381	1.52379
$863$	−18.5288	−0.630726	−0.315363	−	0.948971i	$-0.602126\pi$
$863$	−18.5288	−0.630726	−0.315363	+	0.948971i	$0.602126\pi$
$864$	−7.71580	−0.262497
$865$	21.4280	0.728574
$866$	−4.20528	−0.142901
$867$	−13.0321	−0.442594
$868$	0	0
$869$	−33.2594	−1.12825
$870$	134.003	4.54313
$871$	0	0
$872$	3.61726	0.122496
$873$	24.4667	0.828074
$874$	−0.0156491	−0.000529338 0
$875$	0	0
$876$	27.4469	0.927346
$877$	−7.19084	−0.242817	−0.121409	−	0.992603i	$-0.538741\pi$
$877$	−7.19084	−0.242817	−0.121409	+	0.992603i	$0.538741\pi$
$878$	29.7215	1.00305
$879$	−55.8716	−1.88450
$880$	58.9376	1.98679
$881$	32.4334	1.09271	0.546355	−	0.837554i	$-0.316015\pi$
$881$	32.4334	1.09271	0.546355	+	0.837554i	$0.316015\pi$
$882$	0	0
$883$	−37.8020	−1.27214	−0.636069	−	0.771632i	$-0.719441\pi$
$883$	−37.8020	−1.27214	−0.636069	+	0.771632i	$0.719441\pi$
$884$	0	0
$885$	37.4641	1.25934
$886$	28.4981	0.957414
$887$	44.9619	1.50967	0.754836	−	0.655913i	$-0.227716\pi$
$887$	44.9619	1.50967	0.754836	+	0.655913i	$0.227716\pi$
$888$	−9.34036	−0.313442
$889$	0	0
$890$	25.9253	0.869017
$891$	30.2424	1.01316
$892$	14.0798	0.471426
$893$	−13.5416	−0.453152
$894$	−73.2125	−2.44859
$895$	35.6412	1.19135
$896$	0	0
$897$	0	0
$898$	71.6277	2.39025
$899$	5.44031	0.181444
$900$	31.3846	1.04615
$901$	−20.5887	−0.685908
$902$	54.3130	1.80842
$903$	0	0
$904$	11.1943	0.372316
$905$	68.7824	2.28640
$906$	14.4392	0.479709
$907$	34.3089	1.13921	0.569604	−	0.821919i	$-0.307097\pi$
$907$	34.3089	1.13921	0.569604	+	0.821919i	$0.307097\pi$
$908$	−26.0547	−0.864655
$909$	−12.5089	−0.414895
$910$	0	0
$911$	59.7775	1.98052	0.990258	−	0.139242i	$-0.0444666\pi$
$911$	59.7775	1.98052	0.990258	+	0.139242i	$0.0444666\pi$
$912$	13.7667	0.455861
$913$	−5.29239	−0.175152
$914$	−20.5171	−0.678644
$915$	−34.1224	−1.12805
$916$	8.55326	0.282608
$917$	0	0
$918$	8.06215	0.266091
$919$	23.4747	0.774359	0.387180	−	0.922004i	$-0.373449\pi$
$919$	23.4747	0.774359	0.387180	+	0.922004i	$0.373449\pi$
$920$	−0.0470342	−0.00155067
$921$	−2.02876	−0.0668498
$922$	−46.7071	−1.53822
$923$	0	0
$924$	0	0
$925$	−29.9009	−0.983135
$926$	−27.7914	−0.913283
$927$	9.20615	0.302370
$928$	−45.7310	−1.50120
$929$	4.96893	0.163025	0.0815127	−	0.996672i	$-0.474025\pi$
$929$	4.96893	0.163025	0.0815127	+	0.996672i	$0.474025\pi$
$930$	11.3709	0.372865
$931$	0	0
$932$	26.9162	0.881669
$933$	−38.1427	−1.24874
$934$	−48.4700	−1.58599
$935$	−39.9245	−1.30567
$936$	0	0
$937$	22.1788	0.724550	0.362275	−	0.932071i	$-0.382000\pi$
$937$	22.1788	0.724550	0.362275	+	0.932071i	$0.382000\pi$
$938$	0	0
$939$	−40.1119	−1.30900
$940$	52.1625	1.70135
$941$	−15.1828	−0.494946	−0.247473	−	0.968895i	$-0.579600\pi$
$941$	−15.1828	−0.494946	−0.247473	+	0.968895i	$0.579600\pi$
$942$	8.74869	0.285048
$943$	−0.0789137	−0.00256978
$944$	−19.7212	−0.641872
$945$	0	0
$946$	22.2632	0.723841
$947$	−42.8921	−1.39380	−0.696902	−	0.717166i	$-0.745439\pi$
$947$	−42.8921	−1.39380	−0.696902	+	0.717166i	$0.745439\pi$
$948$	29.9254	0.971932
$949$	0	0
$950$	24.2059	0.785344
$951$	21.0448	0.682424
$952$	0	0
$953$	−8.91446	−0.288768	−0.144384	−	0.989522i	$-0.546120\pi$
$953$	−8.91446	−0.288768	−0.144384	+	0.989522i	$0.546120\pi$
$954$	26.0744	0.844189
$955$	−26.4105	−0.854622
$956$	17.1424	0.554424
$957$	54.1839	1.75152
$958$	31.5397	1.01900
$959$	0	0
$960$	−1.17681	−0.0379814
$961$	−30.5384	−0.985108
$962$	0	0
$963$	19.1635	0.617536
$964$	−11.4895	−0.370052
$965$	30.6039	0.985173
$966$	0	0
$967$	13.8676	0.445952	0.222976	−	0.974824i	$-0.428423\pi$
$967$	13.8676	0.445952	0.222976	+	0.974824i	$0.428423\pi$
$968$	−3.95151	−0.127006
$969$	−9.32560	−0.299581
$970$	−72.6892	−2.33391
$971$	49.9947	1.60441	0.802203	−	0.597051i	$-0.203661\pi$
$971$	49.9947	1.60441	0.802203	+	0.597051i	$0.203661\pi$
$972$	−22.6577	−0.726747
$973$	0	0
$974$	10.0048	0.320576
$975$	0	0
$976$	17.9622	0.574955
$977$	19.5969	0.626960	0.313480	−	0.949595i	$-0.398505\pi$
$977$	19.5969	0.626960	0.313480	+	0.949595i	$0.398505\pi$
$978$	−78.2440	−2.50197
$979$	10.4828	0.335033
$980$	0	0
$981$	5.64989	0.180387
$982$	−71.1232	−2.26963
$983$	43.2093	1.37816	0.689081	−	0.724684i	$-0.258014\pi$
$983$	43.2093	1.37816	0.689081	+	0.724684i	$0.258014\pi$
$984$	38.1298	1.21554
$985$	87.3424	2.78296
$986$	47.7838	1.52175
$987$	0	0
$988$	0	0
$989$	−0.0323473	−0.00102858
$990$	50.5621	1.60697
$991$	29.9101	0.950126	0.475063	−	0.879952i	$-0.342425\pi$
$991$	29.9101	0.950126	0.475063	+	0.879952i	$0.342425\pi$
$992$	−3.88052	−0.123206
$993$	−83.5481	−2.65132
$994$	0	0
$995$	−4.92264	−0.156058
$996$	4.76187	0.150885
$997$	−9.36306	−0.296531	−0.148266	−	0.988948i	$-0.547369\pi$
$997$	−9.36306	−0.296531	−0.148266	+	0.988948i	$0.547369\pi$
$998$	24.8439	0.786420
$999$	3.49880	0.110697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.5		24
7.3	odd	6		1183.2.e.k.170.20	✓	48
7.5	odd	6		1183.2.e.k.508.20	yes	48
7.6	odd	2		8281.2.a.cv.1.5		24
13.12	even	2		8281.2.a.ct.1.20		24
91.12	odd	6		1183.2.e.l.508.5	yes	48
91.38	odd	6		1183.2.e.l.170.5	yes	48
91.90	odd	2		8281.2.a.cu.1.20		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.20	✓	48	7.3	odd	6
1183.2.e.k.508.20	yes	48	7.5	odd	6
1183.2.e.l.170.5	yes	48	91.38	odd	6
1183.2.e.l.508.5	yes	48	91.12	odd	6
8281.2.a.ct.1.20		24	13.12	even	2
8281.2.a.cu.1.20		24	91.90	odd	2
8281.2.a.cv.1.5		24	7.6	odd	2
8281.2.a.cw.1.5		24	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.76732 q^{2} +2.32802 q^{3} +1.12344 q^{4} +4.06760 q^{5} -4.11437 q^{6} +1.54917 q^{8} +2.41970 q^{9} -7.18877 q^{10} -2.90676 q^{11} +2.61538 q^{12} +9.46947 q^{15} -4.98476 q^{16} +3.37669 q^{17} -4.27639 q^{18} -1.18631 q^{19} +4.56969 q^{20} +5.13720 q^{22} -0.00746406 q^{23} +3.60651 q^{24} +11.5454 q^{25} -1.35096 q^{27} -8.00705 q^{29} -16.7356 q^{30} -0.679440 q^{31} +5.71135 q^{32} -6.76702 q^{33} -5.96771 q^{34} +2.71837 q^{36} -2.58986 q^{37} +2.09659 q^{38} +6.30142 q^{40} +10.5725 q^{41} +4.33374 q^{43} -3.26556 q^{44} +9.84236 q^{45} +0.0131914 q^{46} +11.4149 q^{47} -11.6047 q^{48} -20.4044 q^{50} +7.86103 q^{51} -6.09729 q^{53} +2.38759 q^{54} -11.8236 q^{55} -2.76175 q^{57} +14.1511 q^{58} +3.95630 q^{59} +10.6383 q^{60} -3.60341 q^{61} +1.20079 q^{62} -0.124274 q^{64} +11.9595 q^{66} +1.67759 q^{67} +3.79350 q^{68} -0.0173765 q^{69} +7.13225 q^{71} +3.74853 q^{72} +10.4944 q^{73} +4.57712 q^{74} +26.8779 q^{75} -1.33274 q^{76} +11.4421 q^{79} -20.2760 q^{80} -10.4042 q^{81} -18.6850 q^{82} +1.82071 q^{83} +13.7350 q^{85} -7.65912 q^{86} -18.6406 q^{87} -4.50308 q^{88} -3.60636 q^{89} -17.3946 q^{90} -0.00838539 q^{92} -1.58175 q^{93} -20.1738 q^{94} -4.82543 q^{95} +13.2962 q^{96} +10.1115 q^{97} -7.03349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Introduction

L-functions

Modular forms

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