LMFDB - Embedded newform 6475.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6475,2,Mod(1,6475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6475.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6475 = 5^{2} \cdot 7 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.7031353088$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	6475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.84776 q^{2} +1.76537 q^{3} +1.41421 q^{4} +3.26197 q^{6} -1.00000 q^{7} -1.08239 q^{8} +0.116520 q^{9} -0.414214 q^{11} +2.49661 q^{12} -0.198912 q^{13} -1.84776 q^{14} -4.82843 q^{16} +2.08239 q^{17} +0.215301 q^{18} -5.26197 q^{19} -1.76537 q^{21} -0.765367 q^{22} -6.46088 q^{23} -1.91082 q^{24} -0.367542 q^{26} -5.09040 q^{27} -1.41421 q^{28} +4.14545 q^{29} +2.52395 q^{31} -6.75699 q^{32} -0.731239 q^{33} +3.84776 q^{34} +0.164784 q^{36} -1.00000 q^{37} -9.72286 q^{38} -0.351153 q^{39} +0.993212 q^{41} -3.26197 q^{42} -2.31703 q^{43} -0.585786 q^{44} -11.9382 q^{46} +1.90281 q^{47} -8.52395 q^{48} +1.00000 q^{49} +3.67619 q^{51} -0.281305 q^{52} -11.1519 q^{53} -9.40583 q^{54} +1.08239 q^{56} -9.28931 q^{57} +7.65980 q^{58} -4.55967 q^{59} -5.78470 q^{61} +4.66364 q^{62} -0.116520 q^{63} -2.82843 q^{64} -1.35115 q^{66} +9.59154 q^{67} +2.94495 q^{68} -11.4058 q^{69} -12.1712 q^{71} -0.126121 q^{72} +4.70231 q^{73} -1.84776 q^{74} -7.44155 q^{76} +0.414214 q^{77} -0.648847 q^{78} -2.65338 q^{79} -9.33598 q^{81} +1.83522 q^{82} -12.9764 q^{83} -2.49661 q^{84} -4.28130 q^{86} +7.31824 q^{87} +0.448342 q^{88} -8.99321 q^{89} +0.198912 q^{91} -9.13707 q^{92} +4.45569 q^{93} +3.51594 q^{94} -11.9286 q^{96} -2.04373 q^{97} +1.84776 q^{98} -0.0482642 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 8 q^{16} + 4 q^{17} - 8 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{24} - 8 q^{26} + 4 q^{27} + 4 q^{29} - 16 q^{31} + 4 q^{33} + 8 q^{34} - 8 q^{36} - 4 q^{37} - 8 q^{38}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 8 * q^16 + 4 * q^17 - 8 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^24 - 8 * q^26 + 4 * q^27 + 4 * q^29 - 16 * q^31 + 4 * q^33 + 8 * q^34 - 8 * q^36 - 4 * q^37 - 8 * q^38 - 4 * q^39 - 16 * q^41 - 8 * q^43 - 8 * q^44 - 16 * q^46 + 12 * q^47 - 8 * q^48 + 4 * q^49 - 4 * q^51 + 8 * q^52 - 8 * q^53 - 8 * q^54 - 8 * q^57 + 8 * q^58 - 24 * q^61 + 16 * q^62 - 8 * q^66 + 8 * q^67 - 16 * q^69 - 16 * q^71 + 16 * q^72 + 24 * q^73 - 8 * q^76 - 4 * q^77 - 12 * q^79 + 4 * q^81 + 16 * q^82 + 16 * q^83 - 8 * q^86 - 12 * q^87 - 16 * q^89 - 4 * q^91 - 16 * q^93 + 8 * q^94 - 32 * q^96 + 4 * q^97 + 8 * 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 $

$2$	1.84776	1.30656	0.653281	−	0.757115i	$-0.273392\pi$
$2$	1.84776	1.30656	0.653281	+	0.757115i	$0.273392\pi$
$3$	1.76537	1.01924	0.509618	−	0.860401i	$-0.329787\pi$
$3$	1.76537	1.01924	0.509618	+	0.860401i	$0.329787\pi$
$4$	1.41421	0.707107
$5$	0	0
$5$	0	0
$6$	3.26197	1.33169
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.08239	−0.382683
$9$	0.116520	0.0388401
$10$	0	0
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	2.49661	0.720708
$13$	−0.198912	−0.0551684	−0.0275842	−	0.999619i	$-0.508781\pi$
$13$	−0.198912	−0.0551684	−0.0275842	+	0.999619i	$0.508781\pi$
$14$	−1.84776	−0.493834
$15$	0	0
$16$	−4.82843	−1.20711
$17$	2.08239	0.505054	0.252527	−	0.967590i	$-0.418738\pi$
$17$	2.08239	0.505054	0.252527	+	0.967590i	$0.418738\pi$
$18$	0.215301	0.0507470
$19$	−5.26197	−1.20718	−0.603590	−	0.797295i	$-0.706263\pi$
$19$	−5.26197	−1.20718	−0.603590	+	0.797295i	$0.706263\pi$
$20$	0	0
$21$	−1.76537	−0.385235
$22$	−0.765367	−0.163177
$23$	−6.46088	−1.34719	−0.673594	−	0.739102i	$-0.735250\pi$
$23$	−6.46088	−1.34719	−0.673594	+	0.739102i	$0.735250\pi$
$24$	−1.91082	−0.390044
$25$	0	0
$26$	−0.367542	−0.0720809
$27$	−5.09040	−0.979648
$28$	−1.41421	−0.267261
$29$	4.14545	0.769791	0.384896	−	0.922960i	$-0.374237\pi$
$29$	4.14545	0.769791	0.384896	+	0.922960i	$0.374237\pi$
$30$	0	0
$31$	2.52395	0.453314	0.226657	−	0.973975i	$-0.427220\pi$
$31$	2.52395	0.453314	0.226657	+	0.973975i	$0.427220\pi$
$32$	−6.75699	−1.19448
$33$	−0.731239	−0.127292
$34$	3.84776	0.659885
$35$	0	0
$36$	0.164784	0.0274641
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−9.72286	−1.57726
$39$	−0.351153	−0.0562295
$40$	0	0
$41$	0.993212	0.155114	0.0775568	−	0.996988i	$-0.475288\pi$
$41$	0.993212	0.155114	0.0775568	+	0.996988i	$0.475288\pi$
$42$	−3.26197	−0.503333
$43$	−2.31703	−0.353343	−0.176672	−	0.984270i	$-0.556533\pi$
$43$	−2.31703	−0.353343	−0.176672	+	0.984270i	$0.556533\pi$
$44$	−0.585786	−0.0883106
$45$	0	0
$46$	−11.9382	−1.76019
$47$	1.90281	0.277554	0.138777	−	0.990324i	$-0.455683\pi$
$47$	1.90281	0.277554	0.138777	+	0.990324i	$0.455683\pi$
$48$	−8.52395	−1.23033
$49$	1.00000	0.142857
$50$	0	0
$51$	3.67619	0.514769
$52$	−0.281305	−0.0390099
$53$	−11.1519	−1.53183	−0.765913	−	0.642944i	$-0.777713\pi$
$53$	−11.1519	−1.53183	−0.765913	+	0.642944i	$0.777713\pi$
$54$	−9.40583	−1.27997
$55$	0	0
$56$	1.08239	0.144641
$57$	−9.28931	−1.23040
$58$	7.65980	1.00578
$59$	−4.55967	−0.593618	−0.296809	−	0.954937i	$-0.595922\pi$
$59$	−4.55967	−0.593618	−0.296809	+	0.954937i	$0.595922\pi$
$60$	0	0
$61$	−5.78470	−0.740655	−0.370327	−	0.928901i	$-0.620754\pi$
$61$	−5.78470	−0.740655	−0.370327	+	0.928901i	$0.620754\pi$
$62$	4.66364	0.592283
$63$	−0.116520	−0.0146802
$64$	−2.82843	−0.353553
$65$	0	0
$66$	−1.35115	−0.166315
$67$	9.59154	1.17179	0.585897	−	0.810386i	$-0.300742\pi$
$67$	9.59154	1.17179	0.585897	+	0.810386i	$0.300742\pi$
$68$	2.94495	0.357127
$69$	−11.4058	−1.37310
$70$	0	0
$71$	−12.1712	−1.44446	−0.722228	−	0.691655i	$-0.756882\pi$
$71$	−12.1712	−1.44446	−0.722228	+	0.691655i	$0.756882\pi$
$72$	−0.126121	−0.0148634
$73$	4.70231	0.550363	0.275182	−	0.961392i	$-0.411262\pi$
$73$	4.70231	0.550363	0.275182	+	0.961392i	$0.411262\pi$
$74$	−1.84776	−0.214798
$75$	0	0
$76$	−7.44155	−0.853605
$77$	0.414214	0.0472040
$78$	−0.648847	−0.0734674
$79$	−2.65338	−0.298529	−0.149264	−	0.988797i	$-0.547691\pi$
$79$	−2.65338	−0.298529	−0.149264	+	0.988797i	$0.547691\pi$
$80$	0	0
$81$	−9.33598	−1.03733
$82$	1.83522	0.202666
$83$	−12.9764	−1.42435	−0.712175	−	0.702002i	$-0.752290\pi$
$83$	−12.9764	−1.42435	−0.712175	+	0.702002i	$0.752290\pi$
$84$	−2.49661	−0.272402
$85$	0	0
$86$	−4.28130	−0.461665
$87$	7.31824	0.784598
$88$	0.448342	0.0477934
$89$	−8.99321	−0.953279	−0.476639	−	0.879099i	$-0.658145\pi$
$89$	−8.99321	−0.953279	−0.476639	+	0.879099i	$0.658145\pi$
$90$	0	0
$91$	0.198912	0.0208517
$92$	−9.13707	−0.952606
$93$	4.45569	0.462033
$94$	3.51594	0.362641
$95$	0	0
$96$	−11.9286	−1.21745
$97$	−2.04373	−0.207509	−0.103755	−	0.994603i	$-0.533086\pi$
$97$	−2.04373	−0.207509	−0.103755	+	0.994603i	$0.533086\pi$
$98$	1.84776	0.186652
$99$	−0.0482642	−0.00485074
$100$	0	0
$101$	9.39104	0.934443	0.467222	−	0.884140i	$-0.345255\pi$
$101$	9.39104	0.934443	0.467222	+	0.884140i	$0.345255\pi$
$102$	6.79271	0.672578
$103$	15.6678	1.54379	0.771897	−	0.635747i	$-0.219308\pi$
$103$	15.6678	1.54379	0.771897	+	0.635747i	$0.219308\pi$
$104$	0.215301	0.0211120
$105$	0	0
$106$	−20.6060	−2.00143
$107$	−15.7263	−1.52032	−0.760161	−	0.649735i	$-0.774880\pi$
$107$	−15.7263	−1.52032	−0.760161	+	0.649735i	$0.774880\pi$
$108$	−7.19891	−0.692716
$109$	−10.9411	−1.04797	−0.523984	−	0.851728i	$-0.675555\pi$
$109$	−10.9411	−1.04797	−0.523984	+	0.851728i	$0.675555\pi$
$110$	0	0
$111$	−1.76537	−0.167561
$112$	4.82843	0.456243
$113$	−4.03347	−0.379437	−0.189718	−	0.981839i	$-0.560757\pi$
$113$	−4.03347	−0.379437	−0.189718	+	0.981839i	$0.560757\pi$
$114$	−17.1644	−1.60759
$115$	0	0
$116$	5.86256	0.544325
$117$	−0.0231773	−0.00214274
$118$	−8.42516	−0.775599
$119$	−2.08239	−0.190893
$120$	0	0
$121$	−10.8284	−0.984402
$122$	−10.6887	−0.967712
$123$	1.75338	0.158097
$124$	3.56940	0.320541
$125$	0	0
$126$	−0.215301	−0.0191806
$127$	16.5905	1.47217	0.736083	−	0.676891i	$-0.236673\pi$
$127$	16.5905	1.47217	0.736083	+	0.676891i	$0.236673\pi$
$128$	8.28772	0.732538
$129$	−4.09040	−0.360140
$130$	0	0
$131$	−6.53874	−0.571292	−0.285646	−	0.958335i	$-0.592208\pi$
$131$	−6.53874	−0.571292	−0.285646	+	0.958335i	$0.592208\pi$
$132$	−1.03413	−0.0900093
$133$	5.26197	0.456271
$134$	17.7229	1.53102
$135$	0	0
$136$	−2.25397	−0.193276
$137$	18.6690	1.59500	0.797501	−	0.603317i	$-0.206155\pi$
$137$	18.6690	1.59500	0.797501	+	0.603317i	$0.206155\pi$
$138$	−21.0752	−1.79404
$139$	3.41543	0.289693	0.144847	−	0.989454i	$-0.453731\pi$
$139$	3.41543	0.289693	0.144847	+	0.989454i	$0.453731\pi$
$140$	0	0
$141$	3.35916	0.282892
$142$	−22.4894	−1.88727
$143$	0.0823922	0.00688998
$144$	−0.562609	−0.0468841
$145$	0	0
$146$	8.68873	0.719084
$147$	1.76537	0.145605
$148$	−1.41421	−0.116248
$149$	−13.2359	−1.08432	−0.542162	−	0.840274i	$-0.682394\pi$
$149$	−13.2359	−1.08432	−0.542162	+	0.840274i	$0.682394\pi$
$150$	0	0
$151$	0.680032	0.0553402	0.0276701	−	0.999617i	$-0.491191\pi$
$151$	0.680032	0.0553402	0.0276701	+	0.999617i	$0.491191\pi$
$152$	5.69552	0.461968
$153$	0.242641	0.0196163
$154$	0.765367	0.0616750
$155$	0	0
$156$	−0.496606	−0.0397603
$157$	9.35462	0.746580	0.373290	−	0.927715i	$-0.378230\pi$
$157$	9.35462	0.746580	0.373290	+	0.927715i	$0.378230\pi$
$158$	−4.90281	−0.390047
$159$	−19.6871	−1.56129
$160$	0	0
$161$	6.46088	0.509189
$162$	−17.2506	−1.35534
$163$	−22.2532	−1.74301	−0.871503	−	0.490390i	$-0.836854\pi$
$163$	−22.2532	−1.74301	−0.871503	+	0.490390i	$0.836854\pi$
$164$	1.40461	0.109682
$165$	0	0
$166$	−23.9774	−1.86100
$167$	17.9008	1.38521	0.692604	−	0.721318i	$-0.256463\pi$
$167$	17.9008	1.38521	0.692604	+	0.721318i	$0.256463\pi$
$168$	1.91082	0.147423
$169$	−12.9604	−0.996956
$170$	0	0
$171$	−0.613126	−0.0468869
$172$	−3.27677	−0.249851
$173$	11.5429	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$173$	11.5429	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$174$	13.5224	1.02513
$175$	0	0
$176$	2.00000	0.150756
$177$	−8.04948	−0.605036
$178$	−16.6173	−1.24552
$179$	20.1695	1.50754	0.753769	−	0.657140i	$-0.228234\pi$
$179$	20.1695	1.50754	0.753769	+	0.657140i	$0.228234\pi$
$180$	0	0
$181$	−5.05280	−0.375572	−0.187786	−	0.982210i	$-0.560131\pi$
$181$	−5.05280	−0.375572	−0.187786	+	0.982210i	$0.560131\pi$
$182$	0.367542	0.0272440
$183$	−10.2121	−0.754901
$184$	6.99321	0.515546
$185$	0	0
$186$	8.23304	0.603676
$187$	−0.862555	−0.0630763
$188$	2.69098	0.196260
$189$	5.09040	0.370272
$190$	0	0
$191$	6.51594	0.471477	0.235738	−	0.971817i	$-0.424249\pi$
$191$	6.51594	0.471477	0.235738	+	0.971817i	$0.424249\pi$
$192$	−4.99321	−0.360354
$193$	18.8538	1.35713	0.678563	−	0.734542i	$-0.262603\pi$
$193$	18.8538	1.35713	0.678563	+	0.734542i	$0.262603\pi$
$194$	−3.77632	−0.271124
$195$	0	0
$196$	1.41421	0.101015
$197$	7.69968	0.548580	0.274290	−	0.961647i	$-0.411557\pi$
$197$	7.69968	0.548580	0.274290	+	0.961647i	$0.411557\pi$
$198$	−0.0891807	−0.00633779
$199$	6.34731	0.449949	0.224974	−	0.974365i	$-0.427770\pi$
$199$	6.34731	0.449949	0.224974	+	0.974365i	$0.427770\pi$
$200$	0	0
$201$	16.9326	1.19433
$202$	17.3524	1.22091
$203$	−4.14545	−0.290954
$204$	5.19891	0.363997
$205$	0	0
$206$	28.9503	2.01706
$207$	−0.752823	−0.0523248
$208$	0.960434	0.0665941
$209$	2.17958	0.150765
$210$	0	0
$211$	24.1863	1.66505	0.832527	−	0.553984i	$-0.186893\pi$
$211$	24.1863	1.66505	0.832527	+	0.553984i	$0.186893\pi$
$212$	−15.7711	−1.08317
$213$	−21.4866	−1.47224
$214$	−29.0585	−1.98640
$215$	0	0
$216$	5.50981	0.374895
$217$	−2.52395	−0.171337
$218$	−20.2165	−1.36924
$219$	8.30130	0.560950
$220$	0	0
$221$	−0.414214	−0.0278630
$222$	−3.26197	−0.218929
$223$	3.12318	0.209143	0.104572	−	0.994517i	$-0.466653\pi$
$223$	3.12318	0.209143	0.104572	+	0.994517i	$0.466653\pi$
$224$	6.75699	0.451470
$225$	0	0
$226$	−7.45288	−0.495758
$227$	9.87125	0.655178	0.327589	−	0.944820i	$-0.393764\pi$
$227$	9.87125	0.655178	0.327589	+	0.944820i	$0.393764\pi$
$228$	−13.1371	−0.870024
$229$	−21.2475	−1.40407	−0.702036	−	0.712142i	$-0.747725\pi$
$229$	−21.2475	−1.40407	−0.702036	+	0.712142i	$0.747725\pi$
$230$	0	0
$231$	0.731239	0.0481120
$232$	−4.48701	−0.294586
$233$	−8.59661	−0.563182	−0.281591	−	0.959534i	$-0.590862\pi$
$233$	−8.59661	−0.563182	−0.281591	+	0.959534i	$0.590862\pi$
$234$	−0.0428261	−0.00279963
$235$	0	0
$236$	−6.44834	−0.419751
$237$	−4.68419	−0.304271
$238$	−3.84776	−0.249413
$239$	−0.680722	−0.0440323	−0.0220161	−	0.999758i	$-0.507009\pi$
$239$	−0.680722	−0.0440323	−0.0220161	+	0.999758i	$0.507009\pi$
$240$	0	0
$241$	−11.9626	−0.770576	−0.385288	−	0.922796i	$-0.625898\pi$
$241$	−11.9626	−0.770576	−0.385288	+	0.922796i	$0.625898\pi$
$242$	−20.0083	−1.28618
$243$	−1.21024	−0.0776367
$244$	−8.18080	−0.523722
$245$	0	0
$246$	3.23983	0.206564
$247$	1.04667	0.0665981
$248$	−2.73190	−0.173476
$249$	−22.9082	−1.45175
$250$	0	0
$251$	−7.94041	−0.501194	−0.250597	−	0.968091i	$-0.580627\pi$
$251$	−7.94041	−0.501194	−0.250597	+	0.968091i	$0.580627\pi$
$252$	−0.164784	−0.0103804
$253$	2.67619	0.168250
$254$	30.6552	1.92348
$255$	0	0
$256$	20.9706	1.31066
$257$	15.9246	0.993348	0.496674	−	0.867937i	$-0.334554\pi$
$257$	15.9246	0.993348	0.496674	+	0.867937i	$0.334554\pi$
$258$	−7.55807	−0.470545
$259$	1.00000	0.0621370
$260$	0	0
$261$	0.483029	0.0298987
$262$	−12.0820	−0.746430
$263$	1.54847	0.0954829	0.0477415	−	0.998860i	$-0.484798\pi$
$263$	1.54847	0.0954829	0.0477415	+	0.998860i	$0.484798\pi$
$264$	0.791487	0.0487127
$265$	0	0
$266$	9.72286	0.596147
$267$	−15.8763	−0.971615
$268$	13.5645	0.828583
$269$	−20.3003	−1.23773	−0.618864	−	0.785498i	$-0.712407\pi$
$269$	−20.3003	−1.23773	−0.618864	+	0.785498i	$0.712407\pi$
$270$	0	0
$271$	14.1076	0.856978	0.428489	−	0.903547i	$-0.359046\pi$
$271$	14.1076	0.856978	0.428489	+	0.903547i	$0.359046\pi$
$272$	−10.0547	−0.609654
$273$	0.351153	0.0212528
$274$	34.4959	2.08397
$275$	0	0
$276$	−16.1303	−0.970929
$277$	−12.7814	−0.767959	−0.383979	−	0.923342i	$-0.625447\pi$
$277$	−12.7814	−0.767959	−0.383979	+	0.923342i	$0.625447\pi$
$278$	6.31090	0.378502
$279$	0.294091	0.0176067
$280$	0	0
$281$	6.81273	0.406413	0.203207	−	0.979136i	$-0.434864\pi$
$281$	6.81273	0.406413	0.203207	+	0.979136i	$0.434864\pi$
$282$	6.20692	0.369617
$283$	−6.92440	−0.411613	−0.205806	−	0.978593i	$-0.565982\pi$
$283$	−6.92440	−0.411613	−0.205806	+	0.978593i	$0.565982\pi$
$284$	−17.2127	−1.02138
$285$	0	0
$286$	0.152241	0.00900220
$287$	−0.993212	−0.0586274
$288$	−0.787325	−0.0463936
$289$	−12.6636	−0.744920
$290$	0	0
$291$	−3.60793	−0.211501
$292$	6.65007	0.389166
$293$	−20.6134	−1.20425	−0.602125	−	0.798402i	$-0.705679\pi$
$293$	−20.6134	−1.20425	−0.602125	+	0.798402i	$0.705679\pi$
$294$	3.26197	0.190242
$295$	0	0
$296$	1.08239	0.0629128
$297$	2.10851	0.122348
$298$	−24.4567	−1.41674
$299$	1.28515	0.0743221
$300$	0	0
$301$	2.31703	0.133551
$302$	1.25653	0.0723054
$303$	16.5786	0.952417
$304$	25.4071	1.45719
$305$	0	0
$306$	0.448342	0.0256300
$307$	22.7655	1.29930	0.649648	−	0.760235i	$-0.274916\pi$
$307$	22.7655	1.29930	0.649648	+	0.760235i	$0.274916\pi$
$308$	0.585786	0.0333783
$309$	27.6594	1.57349
$310$	0	0
$311$	21.6519	1.22777	0.613885	−	0.789396i	$-0.289606\pi$
$311$	21.6519	1.22777	0.613885	+	0.789396i	$0.289606\pi$
$312$	0.380086	0.0215181
$313$	0.182897	0.0103380	0.00516898	−	0.999987i	$-0.498355\pi$
$313$	0.182897	0.0103380	0.00516898	+	0.999987i	$0.498355\pi$
$314$	17.2851	0.975454
$315$	0	0
$316$	−3.75245	−0.211092
$317$	11.7056	0.657455	0.328727	−	0.944425i	$-0.393380\pi$
$317$	11.7056	0.657455	0.328727	+	0.944425i	$0.393380\pi$
$318$	−36.3771	−2.03993
$319$	−1.71710	−0.0961393
$320$	0	0
$321$	−27.7627	−1.54957
$322$	11.9382	0.665288
$323$	−10.9575	−0.609691
$324$	−13.2031	−0.733504
$325$	0	0
$326$	−41.1186	−2.27735
$327$	−19.3151	−1.06813
$328$	−1.07504	−0.0593594
$329$	−1.90281	−0.104905
$330$	0	0
$331$	−15.5004	−0.851979	−0.425989	−	0.904728i	$-0.640074\pi$
$331$	−15.5004	−0.851979	−0.425989	+	0.904728i	$0.640074\pi$
$332$	−18.3515	−1.00717
$333$	−0.116520	−0.00638527
$334$	33.0764	1.80986
$335$	0	0
$336$	8.52395	0.465019
$337$	−13.4856	−0.734607	−0.367304	−	0.930101i	$-0.619719\pi$
$337$	−13.4856	−0.734607	−0.367304	+	0.930101i	$0.619719\pi$
$338$	−23.9478	−1.30259
$339$	−7.12055	−0.386735
$340$	0	0
$341$	−1.04545	−0.0566144
$342$	−1.13291	−0.0612607
$343$	−1.00000	−0.0539949
$344$	2.50793	0.135219
$345$	0	0
$346$	21.3285	1.14663
$347$	4.91270	0.263727	0.131864	−	0.991268i	$-0.457904\pi$
$347$	4.91270	0.263727	0.131864	+	0.991268i	$0.457904\pi$
$348$	10.3496	0.554795
$349$	−15.9495	−0.853756	−0.426878	−	0.904309i	$-0.640387\pi$
$349$	−15.9495	−0.853756	−0.426878	+	0.904309i	$0.640387\pi$
$350$	0	0
$351$	1.01254	0.0540456
$352$	2.79884	0.149178
$353$	−17.5545	−0.934330	−0.467165	−	0.884170i	$-0.654725\pi$
$353$	−17.5545	−0.934330	−0.467165	+	0.884170i	$0.654725\pi$
$354$	−14.8735	−0.790518
$355$	0	0
$356$	−12.7183	−0.674070
$357$	−3.67619	−0.194564
$358$	37.2683	1.96969
$359$	−16.8258	−0.888030	−0.444015	−	0.896019i	$-0.646446\pi$
$359$	−16.8258	−0.888030	−0.444015	+	0.896019i	$0.646446\pi$
$360$	0	0
$361$	8.68836	0.457282
$362$	−9.33636	−0.490708
$363$	−19.1161	−1.00334
$364$	0.281305	0.0147444
$365$	0	0
$366$	−18.8695	−0.986326
$367$	20.5248	1.07139	0.535693	−	0.844413i	$-0.320051\pi$
$367$	20.5248	1.07139	0.535693	+	0.844413i	$0.320051\pi$
$368$	31.1959	1.62620
$369$	0.115729	0.00602462
$370$	0	0
$371$	11.1519	0.578976
$372$	6.30130	0.326707
$373$	−22.6629	−1.17344	−0.586720	−	0.809790i	$-0.699581\pi$
$373$	−22.6629	−1.17344	−0.586720	+	0.809790i	$0.699581\pi$
$374$	−1.59379	−0.0824131
$375$	0	0
$376$	−2.05959	−0.106215
$377$	−0.824582	−0.0424681
$378$	9.40583	0.483784
$379$	23.1508	1.18918	0.594589	−	0.804030i	$-0.297315\pi$
$379$	23.1508	1.18918	0.594589	+	0.804030i	$0.297315\pi$
$380$	0	0
$381$	29.2883	1.50048
$382$	12.0399	0.616014
$383$	2.01920	0.103176	0.0515882	−	0.998668i	$-0.483572\pi$
$383$	2.01920	0.103176	0.0515882	+	0.998668i	$0.483572\pi$
$384$	14.6309	0.746628
$385$	0	0
$386$	34.8373	1.77317
$387$	−0.269980	−0.0137239
$388$	−2.89027	−0.146731
$389$	−26.0868	−1.32266	−0.661328	−	0.750097i	$-0.730007\pi$
$389$	−26.0868	−1.32266	−0.661328	+	0.750097i	$0.730007\pi$
$390$	0	0
$391$	−13.4541	−0.680403
$392$	−1.08239	−0.0546691
$393$	−11.5433	−0.582281
$394$	14.2272	0.716754
$395$	0	0
$396$	−0.0682559	−0.00342999
$397$	−5.01307	−0.251599	−0.125799	−	0.992056i	$-0.540150\pi$
$397$	−5.01307	−0.251599	−0.125799	+	0.992056i	$0.540150\pi$
$398$	11.7283	0.587886
$399$	9.28931	0.465047
$400$	0	0
$401$	31.9153	1.59377	0.796887	−	0.604128i	$-0.206479\pi$
$401$	31.9153	1.59377	0.796887	+	0.604128i	$0.206479\pi$
$402$	31.2873	1.56047
$403$	−0.502044	−0.0250086
$404$	13.2809	0.660751
$405$	0	0
$406$	−7.65980	−0.380149
$407$	0.414214	0.0205318
$408$	−3.97908	−0.196994
$409$	−26.9514	−1.33266	−0.666330	−	0.745657i	$-0.732136\pi$
$409$	−26.9514	−1.33266	−0.666330	+	0.745657i	$0.732136\pi$
$410$	0	0
$411$	32.9577	1.62568
$412$	22.1576	1.09163
$413$	4.55967	0.224366
$414$	−1.39104	−0.0683657
$415$	0	0
$416$	1.34405	0.0658974
$417$	6.02949	0.295265
$418$	4.02734	0.196984
$419$	20.1965	0.986665	0.493332	−	0.869841i	$-0.335779\pi$
$419$	20.1965	0.986665	0.493332	+	0.869841i	$0.335779\pi$
$420$	0	0
$421$	−9.26704	−0.451648	−0.225824	−	0.974168i	$-0.572507\pi$
$421$	−9.26704	−0.451648	−0.225824	+	0.974168i	$0.572507\pi$
$422$	44.6905	2.17550
$423$	0.221716	0.0107802
$424$	12.0707	0.586205
$425$	0	0
$426$	−39.7021	−1.92357
$427$	5.78470	0.279941
$428$	−22.2404	−1.07503
$429$	0.145452	0.00702251
$430$	0	0
$431$	15.4764	0.745471	0.372735	−	0.927938i	$-0.378420\pi$
$431$	15.4764	0.745471	0.372735	+	0.927938i	$0.378420\pi$
$432$	24.5786	1.18254
$433$	0.308987	0.0148490	0.00742448	−	0.999972i	$-0.497637\pi$
$433$	0.308987	0.0148490	0.00742448	+	0.999972i	$0.497637\pi$
$434$	−4.66364	−0.223862
$435$	0	0
$436$	−15.4731	−0.741025
$437$	33.9970	1.62630
$438$	15.3388	0.732916
$439$	−19.9847	−0.953818	−0.476909	−	0.878953i	$-0.658243\pi$
$439$	−19.9847	−0.953818	−0.476909	+	0.878953i	$0.658243\pi$
$440$	0	0
$441$	0.116520	0.00554858
$442$	−0.765367	−0.0364048
$443$	14.2788	0.678407	0.339203	−	0.940713i	$-0.389843\pi$
$443$	14.2788	0.678407	0.339203	+	0.940713i	$0.389843\pi$
$444$	−2.49661	−0.118484
$445$	0	0
$446$	5.77088	0.273259
$447$	−23.3661	−1.10518
$448$	2.82843	0.133631
$449$	37.6732	1.77791	0.888954	−	0.457997i	$-0.151433\pi$
$449$	37.6732	1.77791	0.888954	+	0.457997i	$0.151433\pi$
$450$	0	0
$451$	−0.411402	−0.0193721
$452$	−5.70419	−0.268302
$453$	1.20051	0.0564047
$454$	18.2397	0.856031
$455$	0	0
$456$	10.0547	0.470853
$457$	4.82426	0.225670	0.112835	−	0.993614i	$-0.464007\pi$
$457$	4.82426	0.225670	0.112835	+	0.993614i	$0.464007\pi$
$458$	−39.2602	−1.83451
$459$	−10.6002	−0.494775
$460$	0	0
$461$	−41.5553	−1.93542	−0.967712	−	0.252058i	$-0.918892\pi$
$461$	−41.5553	−1.93542	−0.967712	+	0.252058i	$0.918892\pi$
$462$	1.35115	0.0628613
$463$	16.0916	0.747841	0.373920	−	0.927461i	$-0.378013\pi$
$463$	16.0916	0.747841	0.373920	+	0.927461i	$0.378013\pi$
$464$	−20.0160	−0.929220
$465$	0	0
$466$	−15.8845	−0.735833
$467$	−2.25115	−0.104171	−0.0520855	−	0.998643i	$-0.516587\pi$
$467$	−2.25115	−0.104171	−0.0520855	+	0.998643i	$0.516587\pi$
$468$	−0.0327777	−0.00151515
$469$	−9.59154	−0.442896
$470$	0	0
$471$	16.5143	0.760941
$472$	4.93535	0.227168
$473$	0.959743	0.0441290
$474$	−8.65526	−0.397549
$475$	0	0
$476$	−2.94495	−0.134981
$477$	−1.29942	−0.0594962
$478$	−1.25781	−0.0575309
$479$	12.2252	0.558583	0.279292	−	0.960206i	$-0.409900\pi$
$479$	12.2252	0.558583	0.279292	+	0.960206i	$0.409900\pi$
$480$	0	0
$481$	0.198912	0.00906962
$482$	−22.1039	−1.00681
$483$	11.4058	0.518983
$484$	−15.3137	−0.696078
$485$	0	0
$486$	−2.23623	−0.101437
$487$	1.24052	0.0562133	0.0281066	−	0.999605i	$-0.491052\pi$
$487$	1.24052	0.0562133	0.0281066	+	0.999605i	$0.491052\pi$
$488$	6.26131	0.283436
$489$	−39.2851	−1.77653
$490$	0	0
$491$	−17.9023	−0.807920	−0.403960	−	0.914777i	$-0.632366\pi$
$491$	−17.9023	−0.807920	−0.403960	+	0.914777i	$0.632366\pi$
$492$	2.47966	0.111792
$493$	8.63246	0.388786
$494$	1.93400	0.0870146
$495$	0	0
$496$	−12.1867	−0.547198
$497$	12.1712	0.545953
$498$	−42.3288	−1.89680
$499$	35.0891	1.57080	0.785402	−	0.618985i	$-0.212456\pi$
$499$	35.0891	1.57080	0.785402	+	0.618985i	$0.212456\pi$
$500$	0	0
$501$	31.6016	1.41185
$502$	−14.6720	−0.654842
$503$	−6.95192	−0.309971	−0.154985	−	0.987917i	$-0.549533\pi$
$503$	−6.95192	−0.309971	−0.154985	+	0.987917i	$0.549533\pi$
$504$	0.126121	0.00561785
$505$	0	0
$506$	4.94495	0.219830
$507$	−22.8799	−1.01613
$508$	23.4625	1.04098
$509$	13.1674	0.583633	0.291816	−	0.956474i	$-0.405740\pi$
$509$	13.1674	0.583633	0.291816	+	0.956474i	$0.405740\pi$
$510$	0	0
$511$	−4.70231	−0.208018
$512$	22.1731	0.979922
$513$	26.7855	1.18261
$514$	29.4248	1.29787
$515$	0	0
$516$	−5.78470	−0.254657
$517$	−0.788170	−0.0346637
$518$	1.84776	0.0811859
$519$	20.3775	0.894471
$520$	0	0
$521$	−22.2689	−0.975620	−0.487810	−	0.872950i	$-0.662204\pi$
$521$	−22.2689	−0.975620	−0.487810	+	0.872950i	$0.662204\pi$
$522$	0.892521	0.0390646
$523$	10.8143	0.472877	0.236439	−	0.971646i	$-0.424020\pi$
$523$	10.8143	0.472877	0.236439	+	0.971646i	$0.424020\pi$
$524$	−9.24718	−0.403965
$525$	0	0
$526$	2.86120	0.124754
$527$	5.25584	0.228948
$528$	3.53073	0.153655
$529$	18.7430	0.814915
$530$	0	0
$531$	−0.531293	−0.0230562
$532$	7.44155	0.322632
$533$	−0.197562	−0.00855736
$534$	−29.3356	−1.26948
$535$	0	0
$536$	−10.3818	−0.448426
$537$	35.6065	1.53653
$538$	−37.5100	−1.61717
$539$	−0.414214	−0.0178414
$540$	0	0
$541$	−16.6011	−0.713739	−0.356869	−	0.934154i	$-0.616156\pi$
$541$	−16.6011	−0.713739	−0.356869	+	0.934154i	$0.616156\pi$
$542$	26.0675	1.11970
$543$	−8.92005	−0.382796
$544$	−14.0707	−0.603276
$545$	0	0
$546$	0.648847	0.0277681
$547$	−5.22528	−0.223417	−0.111708	−	0.993741i	$-0.535632\pi$
$547$	−5.22528	−0.223417	−0.111708	+	0.993741i	$0.535632\pi$
$548$	26.4020	1.12784
$549$	−0.674034	−0.0287671
$550$	0	0
$551$	−21.8133	−0.929276
$552$	12.3456	0.525463
$553$	2.65338	0.112833
$554$	−23.6169	−1.00339
$555$	0	0
$556$	4.83015	0.204844
$557$	−9.68125	−0.410208	−0.205104	−	0.978740i	$-0.565753\pi$
$557$	−9.68125	−0.410208	−0.205104	+	0.978740i	$0.565753\pi$
$558$	0.543408	0.0230043
$559$	0.460885	0.0194934
$560$	0	0
$561$	−1.52273	−0.0642895
$562$	12.5883	0.531005
$563$	5.46195	0.230194	0.115097	−	0.993354i	$-0.463282\pi$
$563$	5.46195	0.230194	0.115097	+	0.993354i	$0.463282\pi$
$564$	4.75057	0.200035
$565$	0	0
$566$	−12.7946	−0.537798
$567$	9.33598	0.392074
$568$	13.1740	0.552769
$569$	5.98005	0.250697	0.125348	−	0.992113i	$-0.459995\pi$
$569$	5.98005	0.250697	0.125348	+	0.992113i	$0.459995\pi$
$570$	0	0
$571$	−18.8658	−0.789510	−0.394755	−	0.918786i	$-0.629170\pi$
$571$	−18.8658	−0.789510	−0.394755	+	0.918786i	$0.629170\pi$
$572$	0.116520	0.00487195
$573$	11.5030	0.480546
$574$	−1.83522	−0.0766004
$575$	0	0
$576$	−0.329569	−0.0137320
$577$	24.1571	1.00567	0.502836	−	0.864382i	$-0.332290\pi$
$577$	24.1571	1.00567	0.502836	+	0.864382i	$0.332290\pi$
$578$	−23.3994	−0.973285
$579$	33.2839	1.38323
$580$	0	0
$581$	12.9764	0.538354
$582$	−6.66659	−0.276339
$583$	4.61925	0.191310
$584$	−5.08974	−0.210615
$585$	0	0
$586$	−38.0887	−1.57343
$587$	13.1407	0.542375	0.271187	−	0.962527i	$-0.412584\pi$
$587$	13.1407	0.542375	0.271187	+	0.962527i	$0.412584\pi$
$588$	2.49661	0.102958
$589$	−13.2809	−0.547231
$590$	0	0
$591$	13.5928	0.559131
$592$	4.82843	0.198447
$593$	−40.8998	−1.67955	−0.839776	−	0.542932i	$-0.817314\pi$
$593$	−40.8998	−1.67955	−0.839776	+	0.542932i	$0.817314\pi$
$594$	3.89602	0.159856
$595$	0	0
$596$	−18.7183	−0.766732
$597$	11.2053	0.458604
$598$	2.37465	0.0971066
$599$	−6.72830	−0.274911	−0.137455	−	0.990508i	$-0.543892\pi$
$599$	−6.72830	−0.274911	−0.137455	+	0.990508i	$0.543892\pi$
$600$	0	0
$601$	−43.7070	−1.78285	−0.891423	−	0.453171i	$-0.850293\pi$
$601$	−43.7070	−1.78285	−0.891423	+	0.453171i	$0.850293\pi$
$602$	4.28130	0.174493
$603$	1.11761	0.0455125
$604$	0.961710	0.0391314
$605$	0	0
$606$	30.6333	1.24439
$607$	−22.8071	−0.925713	−0.462856	−	0.886433i	$-0.653175\pi$
$607$	−22.8071	−0.925713	−0.462856	+	0.886433i	$0.653175\pi$
$608$	35.5551	1.44195
$609$	−7.31824	−0.296550
$610$	0	0
$611$	−0.378493	−0.0153122
$612$	0.343146	0.0138708
$613$	−5.12400	−0.206956	−0.103478	−	0.994632i	$-0.532997\pi$
$613$	−5.12400	−0.206956	−0.103478	+	0.994632i	$0.532997\pi$
$614$	42.0652	1.69761
$615$	0	0
$616$	−0.448342	−0.0180642
$617$	41.6263	1.67581	0.837905	−	0.545816i	$-0.183780\pi$
$617$	41.6263	1.67581	0.837905	+	0.545816i	$0.183780\pi$
$618$	51.1080	2.05586
$619$	−13.3462	−0.536429	−0.268215	−	0.963359i	$-0.586434\pi$
$619$	−13.3462	−0.536429	−0.268215	+	0.963359i	$0.586434\pi$
$620$	0	0
$621$	32.8885	1.31977
$622$	40.0076	1.60416
$623$	8.99321	0.360305
$624$	1.69552	0.0678750
$625$	0	0
$626$	0.337950	0.0135072
$627$	3.84776	0.153665
$628$	13.2294	0.527912
$629$	−2.08239	−0.0830304
$630$	0	0
$631$	−10.4477	−0.415916	−0.207958	−	0.978138i	$-0.566682\pi$
$631$	−10.4477	−0.415916	−0.207958	+	0.978138i	$0.566682\pi$
$632$	2.87200	0.114242
$633$	42.6977	1.69708
$634$	21.6292	0.859006
$635$	0	0
$636$	−27.8418	−1.10400
$637$	−0.198912	−0.00788119
$638$	−3.17279	−0.125612
$639$	−1.41819	−0.0561027
$640$	0	0
$641$	−3.55635	−0.140467	−0.0702337	−	0.997531i	$-0.522374\pi$
$641$	−3.55635	−0.140467	−0.0702337	+	0.997531i	$0.522374\pi$
$642$	−51.2989	−2.02460
$643$	−4.69915	−0.185316	−0.0926582	−	0.995698i	$-0.529536\pi$
$643$	−4.69915	−0.185316	−0.0926582	+	0.995698i	$0.529536\pi$
$644$	9.13707	0.360051
$645$	0	0
$646$	−20.2468	−0.796600
$647$	−22.2176	−0.873462	−0.436731	−	0.899592i	$-0.643864\pi$
$647$	−22.2176	−0.873462	−0.436731	+	0.899592i	$0.643864\pi$
$648$	10.1052	0.396970
$649$	1.88868	0.0741370
$650$	0	0
$651$	−4.45569	−0.174632
$652$	−31.4708	−1.23249
$653$	−4.07345	−0.159406	−0.0797032	−	0.996819i	$-0.525397\pi$
$653$	−4.07345	−0.159406	−0.0797032	+	0.996819i	$0.525397\pi$
$654$	−35.6896	−1.39557
$655$	0	0
$656$	−4.79565	−0.187239
$657$	0.547914	0.0213761
$658$	−3.51594	−0.137066
$659$	−34.8942	−1.35929	−0.679643	−	0.733543i	$-0.737865\pi$
$659$	−34.8942	−1.35929	−0.679643	+	0.733543i	$0.737865\pi$
$660$	0	0
$661$	12.5624	0.488623	0.244311	−	0.969697i	$-0.421438\pi$
$661$	12.5624	0.488623	0.244311	+	0.969697i	$0.421438\pi$
$662$	−28.6410	−1.11316
$663$	−0.731239	−0.0283990
$664$	14.0456	0.545075
$665$	0	0
$666$	−0.215301	−0.00834275
$667$	−26.7833	−1.03705
$668$	25.3156	0.979491
$669$	5.51355	0.213166
$670$	0	0
$671$	2.39610	0.0925004
$672$	11.9286	0.460154
$673$	44.0942	1.69971	0.849854	−	0.527019i	$-0.176690\pi$
$673$	44.0942	1.69971	0.849854	+	0.527019i	$0.176690\pi$
$674$	−24.9181	−0.959811
$675$	0	0
$676$	−18.3288	−0.704955
$677$	10.3304	0.397030	0.198515	−	0.980098i	$-0.436388\pi$
$677$	10.3304	0.397030	0.198515	+	0.980098i	$0.436388\pi$
$678$	−13.1571	−0.505294
$679$	2.04373	0.0784311
$680$	0	0
$681$	17.4264	0.667780
$682$	−1.93174	−0.0739703
$683$	5.89509	0.225569	0.112785	−	0.993619i	$-0.464023\pi$
$683$	5.89509	0.225569	0.112785	+	0.993619i	$0.464023\pi$
$684$	−0.867091	−0.0331541
$685$	0	0
$686$	−1.84776	−0.0705478
$687$	−37.5096	−1.43108
$688$	11.1876	0.426523
$689$	2.21824	0.0845084
$690$	0	0
$691$	3.46311	0.131743	0.0658714	−	0.997828i	$-0.479017\pi$
$691$	3.46311	0.131743	0.0658714	+	0.997828i	$0.479017\pi$
$692$	16.3241	0.620550
$693$	0.0482642	0.00183341
$694$	9.07748	0.344577
$695$	0	0
$696$	−7.92121	−0.300253
$697$	2.06826	0.0783408
$698$	−29.4708	−1.11549
$699$	−15.1762	−0.574015
$700$	0	0
$701$	−3.16375	−0.119493	−0.0597466	−	0.998214i	$-0.519029\pi$
$701$	−3.16375	−0.119493	−0.0597466	+	0.998214i	$0.519029\pi$
$702$	1.87094	0.0706139
$703$	5.26197	0.198459
$704$	1.17157	0.0441553
$705$	0	0
$706$	−32.4364	−1.22076
$707$	−9.39104	−0.353186
$708$	−11.3837	−0.427825
$709$	−26.6608	−1.00127	−0.500633	−	0.865660i	$-0.666899\pi$
$709$	−26.6608	−1.00127	−0.500633	+	0.865660i	$0.666899\pi$
$710$	0	0
$711$	−0.309173	−0.0115949
$712$	9.73418	0.364804
$713$	−16.3069	−0.610699
$714$	−6.79271	−0.254211
$715$	0	0
$716$	28.5239	1.06599
$717$	−1.20172	−0.0448792
$718$	−31.0900	−1.16027
$719$	−10.8419	−0.404336	−0.202168	−	0.979351i	$-0.564799\pi$
$719$	−10.8419	−0.404336	−0.202168	+	0.979351i	$0.564799\pi$
$720$	0	0
$721$	−15.6678	−0.583500
$722$	16.0540	0.597468
$723$	−21.1183	−0.785398
$724$	−7.14574	−0.265569
$725$	0	0
$726$	−35.3220	−1.31092
$727$	−23.9509	−0.888291	−0.444146	−	0.895955i	$-0.646493\pi$
$727$	−23.9509	−0.888291	−0.444146	+	0.895955i	$0.646493\pi$
$728$	−0.215301	−0.00797959
$729$	25.8714	0.958201
$730$	0	0
$731$	−4.82496	−0.178457
$732$	−14.4421	−0.533796
$733$	29.5011	1.08965	0.544823	−	0.838551i	$-0.316597\pi$
$733$	29.5011	1.08965	0.544823	+	0.838551i	$0.316597\pi$
$734$	37.9249	1.39983
$735$	0	0
$736$	43.6561	1.60919
$737$	−3.97295	−0.146345
$738$	0.213840	0.00787155
$739$	17.0411	0.626867	0.313434	−	0.949610i	$-0.398521\pi$
$739$	17.0411	0.626867	0.313434	+	0.949610i	$0.398521\pi$
$740$	0	0
$741$	1.84776	0.0678791
$742$	20.6060	0.756469
$743$	32.4643	1.19100	0.595499	−	0.803356i	$-0.296954\pi$
$743$	32.4643	1.19100	0.595499	+	0.803356i	$0.296954\pi$
$744$	−4.82280	−0.176813
$745$	0	0
$746$	−41.8756	−1.53317
$747$	−1.51202	−0.0553218
$748$	−1.21984	−0.0446017
$749$	15.7263	0.574628
$750$	0	0
$751$	47.8901	1.74753	0.873767	−	0.486344i	$-0.161670\pi$
$751$	47.8901	1.74753	0.873767	+	0.486344i	$0.161670\pi$
$752$	−9.18759	−0.335037
$753$	−14.0177	−0.510835
$754$	−1.52363	−0.0554873
$755$	0	0
$756$	7.19891	0.261822
$757$	−47.6767	−1.73284	−0.866419	−	0.499317i	$-0.833584\pi$
$757$	−47.6767	−1.73284	−0.866419	+	0.499317i	$0.833584\pi$
$758$	42.7772	1.55374
$759$	4.72445	0.171487
$760$	0	0
$761$	1.12550	0.0407995	0.0203997	−	0.999792i	$-0.493506\pi$
$761$	1.12550	0.0407995	0.0203997	+	0.999792i	$0.493506\pi$
$762$	54.1177	1.96048
$763$	10.9411	0.396094
$764$	9.21493	0.333384
$765$	0	0
$766$	3.73100	0.134806
$767$	0.906974	0.0327489
$768$	37.0207	1.33587
$769$	21.2849	0.767551	0.383776	−	0.923426i	$-0.374624\pi$
$769$	21.2849	0.767551	0.383776	+	0.923426i	$0.374624\pi$
$770$	0	0
$771$	28.1127	1.01246
$772$	26.6633	0.959633
$773$	−3.29021	−0.118341	−0.0591704	−	0.998248i	$-0.518846\pi$
$773$	−3.29021	−0.118341	−0.0591704	+	0.998248i	$0.518846\pi$
$774$	−0.498858	−0.0179311
$775$	0	0
$776$	2.21212	0.0794103
$777$	1.76537	0.0633322
$778$	−48.2022	−1.72813
$779$	−5.22625	−0.187250
$780$	0	0
$781$	5.04148	0.180398
$782$	−24.8599	−0.888989
$783$	−21.1020	−0.754124
$784$	−4.82843	−0.172444
$785$	0	0
$786$	−21.3292	−0.760787
$787$	−34.5376	−1.23113	−0.615567	−	0.788085i	$-0.711073\pi$
$787$	−34.5376	−1.23113	−0.615567	+	0.788085i	$0.711073\pi$
$788$	10.8890	0.387904
$789$	2.73362	0.0973195
$790$	0	0
$791$	4.03347	0.143414
$792$	0.0522408	0.00185630
$793$	1.15065	0.0408607
$794$	−9.26295	−0.328730
$795$	0	0
$796$	8.97645	0.318162
$797$	−30.6642	−1.08618	−0.543091	−	0.839674i	$-0.682746\pi$
$797$	−30.6642	−1.08618	−0.543091	+	0.839674i	$0.682746\pi$
$798$	17.1644	0.607614
$799$	3.96240	0.140180
$800$	0	0
$801$	−1.04789	−0.0370254
$802$	58.9718	2.08237
$803$	−1.94776	−0.0687349
$804$	23.9463	0.844521
$805$	0	0
$806$	−0.927656	−0.0326753
$807$	−35.8374	−1.26154
$808$	−10.1648	−0.357596
$809$	−18.5638	−0.652668	−0.326334	−	0.945254i	$-0.605813\pi$
$809$	−18.5638	−0.652668	−0.326334	+	0.945254i	$0.605813\pi$
$810$	0	0
$811$	30.0999	1.05695	0.528476	−	0.848948i	$-0.322764\pi$
$811$	30.0999	1.05695	0.528476	+	0.848948i	$0.322764\pi$
$812$	−5.86256	−0.205735
$813$	24.9051	0.873462
$814$	0.765367	0.0268261
$815$	0	0
$816$	−17.7502	−0.621381
$817$	12.1921	0.426548
$818$	−49.7996	−1.74120
$819$	0.0231773	0.000809880 0
$820$	0	0
$821$	14.8822	0.519392	0.259696	−	0.965690i	$-0.416378\pi$
$821$	14.8822	0.519392	0.259696	+	0.965690i	$0.416378\pi$
$822$	60.8978	2.12406
$823$	−6.60960	−0.230396	−0.115198	−	0.993343i	$-0.536750\pi$
$823$	−6.60960	−0.230396	−0.115198	+	0.993343i	$0.536750\pi$
$824$	−16.9587	−0.590785
$825$	0	0
$826$	8.42516	0.293149
$827$	5.44052	0.189185	0.0945927	−	0.995516i	$-0.469845\pi$
$827$	5.44052	0.189185	0.0945927	+	0.995516i	$0.469845\pi$
$828$	−1.06465	−0.0369993
$829$	5.98364	0.207820	0.103910	−	0.994587i	$-0.466865\pi$
$829$	5.98364	0.207820	0.103910	+	0.994587i	$0.466865\pi$
$830$	0	0
$831$	−22.5638	−0.782731
$832$	0.562609	0.0195050
$833$	2.08239	0.0721506
$834$	11.1410	0.385783
$835$	0	0
$836$	3.08239	0.106607
$837$	−12.8479	−0.444088
$838$	37.3183	1.28914
$839$	−47.5405	−1.64128	−0.820641	−	0.571445i	$-0.806383\pi$
$839$	−47.5405	−1.64128	−0.820641	+	0.571445i	$0.806383\pi$
$840$	0	0
$841$	−11.8152	−0.407422
$842$	−17.1233	−0.590106
$843$	12.0270	0.414231
$844$	34.2046	1.17737
$845$	0	0
$846$	0.409678	0.0140850
$847$	10.8284	0.372069
$848$	53.8460	1.84908
$849$	−12.2241	−0.419530
$850$	0	0
$851$	6.46088	0.221476
$852$	−30.3867	−1.04103
$853$	−25.8590	−0.885396	−0.442698	−	0.896671i	$-0.645979\pi$
$853$	−25.8590	−0.885396	−0.442698	+	0.896671i	$0.645979\pi$
$854$	10.6887	0.365761
$855$	0	0
$856$	17.0221	0.581802
$857$	−5.82423	−0.198952	−0.0994760	−	0.995040i	$-0.531717\pi$
$857$	−5.82423	−0.198952	−0.0994760	+	0.995040i	$0.531717\pi$
$858$	0.268761	0.00917535
$859$	−2.64641	−0.0902943	−0.0451471	−	0.998980i	$-0.514376\pi$
$859$	−2.64641	−0.0902943	−0.0451471	+	0.998980i	$0.514376\pi$
$860$	0	0
$861$	−1.75338	−0.0597551
$862$	28.5966	0.974004
$863$	−11.5147	−0.391966	−0.195983	−	0.980607i	$-0.562790\pi$
$863$	−11.5147	−0.391966	−0.195983	+	0.980607i	$0.562790\pi$
$864$	34.3958	1.17017
$865$	0	0
$866$	0.570934	0.0194011
$867$	−22.3560	−0.759249
$868$	−3.56940	−0.121153
$869$	1.09907	0.0372833
$870$	0	0
$871$	−1.90788	−0.0646459
$872$	11.8426	0.401040
$873$	−0.238136	−0.00805967
$874$	62.8183	2.12486
$875$	0	0
$876$	11.7398	0.396651
$877$	−30.1953	−1.01962	−0.509811	−	0.860287i	$-0.670285\pi$
$877$	−30.1953	−1.01962	−0.509811	+	0.860287i	$0.670285\pi$
$878$	−36.9269	−1.24622
$879$	−36.3903	−1.22741
$880$	0	0
$881$	−13.9519	−0.470051	−0.235026	−	0.971989i	$-0.575517\pi$
$881$	−13.9519	−0.470051	−0.235026	+	0.971989i	$0.575517\pi$
$882$	0.215301	0.00724957
$883$	24.5461	0.826043	0.413022	−	0.910721i	$-0.364473\pi$
$883$	24.5461	0.826043	0.413022	+	0.910721i	$0.364473\pi$
$884$	−0.585786	−0.0197021
$885$	0	0
$886$	26.3838	0.886381
$887$	−43.2240	−1.45132	−0.725660	−	0.688054i	$-0.758465\pi$
$887$	−43.2240	−1.45132	−0.725660	+	0.688054i	$0.758465\pi$
$888$	1.91082	0.0641229
$889$	−16.5905	−0.556427
$890$	0	0
$891$	3.86709	0.129552
$892$	4.41684	0.147887
$893$	−10.0125	−0.335057
$894$	−43.1750	−1.44399
$895$	0	0
$896$	−8.28772	−0.276873
$897$	2.26876	0.0757517
$898$	69.6110	2.32295
$899$	10.4629	0.348957
$900$	0	0
$901$	−23.2226	−0.773656
$902$	−0.760171	−0.0253109
$903$	4.09040	0.136120
$904$	4.36579	0.145204
$905$	0	0
$906$	2.21824	0.0736962
$907$	−32.0014	−1.06259	−0.531295	−	0.847187i	$-0.678294\pi$
$907$	−32.0014	−1.06259	−0.531295	+	0.847187i	$0.678294\pi$
$908$	13.9601	0.463281
$909$	1.09425	0.0362938
$910$	0	0
$911$	−2.75773	−0.0913678	−0.0456839	−	0.998956i	$-0.514547\pi$
$911$	−2.75773	−0.0913678	−0.0456839	+	0.998956i	$0.514547\pi$
$912$	44.8528	1.48522
$913$	5.37502	0.177887
$914$	8.91408	0.294852
$915$	0	0
$916$	−30.0484	−0.992829
$917$	6.53874	0.215928
$918$	−19.5866	−0.646455
$919$	21.2411	0.700679	0.350339	−	0.936623i	$-0.386066\pi$
$919$	21.2411	0.700679	0.350339	+	0.936623i	$0.386066\pi$
$920$	0	0
$921$	40.1895	1.32429
$922$	−76.7842	−2.52875
$923$	2.42100	0.0796883
$924$	1.03413	0.0340203
$925$	0	0
$926$	29.7334	0.977101
$927$	1.82562	0.0599611
$928$	−28.0108	−0.919498
$929$	3.64328	0.119532	0.0597660	−	0.998212i	$-0.480965\pi$
$929$	3.64328	0.119532	0.0597660	+	0.998212i	$0.480965\pi$
$930$	0	0
$931$	−5.26197	−0.172454
$932$	−12.1574	−0.398230
$933$	38.2236	1.25139
$934$	−4.15959	−0.136106
$935$	0	0
$936$	0.0250869	0.000819992 0
$937$	52.0124	1.69917	0.849585	−	0.527452i	$-0.176853\pi$
$937$	52.0124	1.69917	0.849585	+	0.527452i	$0.176853\pi$
$938$	−17.7229	−0.578672
$939$	0.322881	0.0105368
$940$	0	0
$941$	−52.6276	−1.71561	−0.857806	−	0.513973i	$-0.828173\pi$
$941$	−52.6276	−1.71561	−0.857806	+	0.513973i	$0.828173\pi$
$942$	30.5145	0.994217
$943$	−6.41703	−0.208967
$944$	22.0160	0.716560
$945$	0	0
$946$	1.77337	0.0576574
$947$	−37.8509	−1.22999	−0.614994	−	0.788532i	$-0.710842\pi$
$947$	−37.8509	−1.22999	−0.614994	+	0.788532i	$0.710842\pi$
$948$	−6.62445	−0.215152
$949$	−0.935347	−0.0303626
$950$	0	0
$951$	20.6648	0.670101
$952$	2.25397	0.0730514
$953$	61.0724	1.97833	0.989164	−	0.146812i	$-0.0469012\pi$
$953$	61.0724	1.97833	0.989164	+	0.146812i	$0.0469012\pi$
$954$	−2.40101	−0.0777356
$955$	0	0
$956$	−0.962686	−0.0311355
$957$	−3.03132	−0.0979885
$958$	22.5892	0.729824
$959$	−18.6690	−0.602854
$960$	0	0
$961$	−24.6297	−0.794506
$962$	0.367542	0.0118500
$963$	−1.83243	−0.0590494
$964$	−16.9176	−0.544879
$965$	0	0
$966$	21.0752	0.678084
$967$	−42.6201	−1.37057	−0.685286	−	0.728275i	$-0.740323\pi$
$967$	−42.6201	−1.37057	−0.685286	+	0.728275i	$0.740323\pi$
$968$	11.7206	0.376715
$969$	−19.3440	−0.621419
$970$	0	0
$971$	−31.3911	−1.00739	−0.503694	−	0.863882i	$-0.668026\pi$
$971$	−31.3911	−1.00739	−0.503694	+	0.863882i	$0.668026\pi$
$972$	−1.71153	−0.0548975
$973$	−3.41543	−0.109494
$974$	2.29218	0.0734462
$975$	0	0
$976$	27.9310	0.894049
$977$	−34.8934	−1.11634	−0.558170	−	0.829727i	$-0.688496\pi$
$977$	−34.8934	−1.11634	−0.558170	+	0.829727i	$0.688496\pi$
$978$	−72.5894	−2.32115
$979$	3.72511	0.119055
$980$	0	0
$981$	−1.27486	−0.0407031
$982$	−33.0791	−1.05560
$983$	−20.6622	−0.659023	−0.329511	−	0.944152i	$-0.606884\pi$
$983$	−20.6622	−0.659023	−0.329511	+	0.944152i	$0.606884\pi$
$984$	−1.89785	−0.0605012
$985$	0	0
$986$	15.9507	0.507974
$987$	−3.35916	−0.106923
$988$	1.48022	0.0470920
$989$	14.9700	0.476019
$990$	0	0
$991$	−1.62877	−0.0517395	−0.0258697	−	0.999665i	$-0.508236\pi$
$991$	−1.62877	−0.0517395	−0.0258697	+	0.999665i	$0.508236\pi$
$992$	−17.0543	−0.541473
$993$	−27.3639	−0.868367
$994$	22.4894	0.713322
$995$	0	0
$996$	−32.3971	−1.02654
$997$	41.2487	1.30636	0.653180	−	0.757203i	$-0.273435\pi$
$997$	41.2487	1.30636	0.653180	+	0.757203i	$0.273435\pi$
$998$	64.8362	2.05236
$999$	5.09040	0.161053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6475.2.a.o.1.4		4
5.4	even	2		1295.2.a.d.1.1	✓	4
35.34	odd	2		9065.2.a.g.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.a.d.1.1	✓	4	5.4	even	2
6475.2.a.o.1.4		4	1.1	even	1	trivial
9065.2.a.g.1.1		4	35.34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6475 = 5^{2} \cdot 7 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.84776 q^{2} +1.76537 q^{3} +1.41421 q^{4} +3.26197 q^{6} -1.00000 q^{7} -1.08239 q^{8} +0.116520 q^{9} -0.414214 q^{11} +2.49661 q^{12} -0.198912 q^{13} -1.84776 q^{14} -4.82843 q^{16} +2.08239 q^{17} +0.215301 q^{18} -5.26197 q^{19} -1.76537 q^{21} -0.765367 q^{22} -6.46088 q^{23} -1.91082 q^{24} -0.367542 q^{26} -5.09040 q^{27} -1.41421 q^{28} +4.14545 q^{29} +2.52395 q^{31} -6.75699 q^{32} -0.731239 q^{33} +3.84776 q^{34} +0.164784 q^{36} -1.00000 q^{37} -9.72286 q^{38} -0.351153 q^{39} +0.993212 q^{41} -3.26197 q^{42} -2.31703 q^{43} -0.585786 q^{44} -11.9382 q^{46} +1.90281 q^{47} -8.52395 q^{48} +1.00000 q^{49} +3.67619 q^{51} -0.281305 q^{52} -11.1519 q^{53} -9.40583 q^{54} +1.08239 q^{56} -9.28931 q^{57} +7.65980 q^{58} -4.55967 q^{59} -5.78470 q^{61} +4.66364 q^{62} -0.116520 q^{63} -2.82843 q^{64} -1.35115 q^{66} +9.59154 q^{67} +2.94495 q^{68} -11.4058 q^{69} -12.1712 q^{71} -0.126121 q^{72} +4.70231 q^{73} -1.84776 q^{74} -7.44155 q^{76} +0.414214 q^{77} -0.648847 q^{78} -2.65338 q^{79} -9.33598 q^{81} +1.83522 q^{82} -12.9764 q^{83} -2.49661 q^{84} -4.28130 q^{86} +7.31824 q^{87} +0.448342 q^{88} -8.99321 q^{89} +0.198912 q^{91} -9.13707 q^{92} +4.45569 q^{93} +3.51594 q^{94} -11.9286 q^{96} -2.04373 q^{97} +1.84776 q^{98} -0.0482642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 8 q^{16} + 4 q^{17} - 8 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{24} - 8 q^{26} + 4 q^{27} + 4 q^{29} - 16 q^{31} + 4 q^{33} + 8 q^{34} - 8 q^{36} - 4 q^{37} - 8 q^{38}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 8 * q^16 + 4 * q^17 - 8 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^24 - 8 * q^26 + 4 * q^27 + 4 * q^29 - 16 * q^31 + 4 * q^33 + 8 * q^34 - 8 * q^36 - 4 * q^37 - 8 * q^38 - 4 * q^39 - 16 * q^41 - 8 * q^43 - 8 * q^44 - 16 * q^46 + 12 * q^47 - 8 * q^48 + 4 * q^49 - 4 * q^51 + 8 * q^52 - 8 * q^53 - 8 * q^54 - 8 * q^57 + 8 * q^58 - 24 * q^61 + 16 * q^62 - 8 * q^66 + 8 * q^67 - 16 * q^69 - 16 * q^71 + 16 * q^72 + 24 * q^73 - 8 * q^76 - 4 * q^77 - 12 * q^79 + 4 * q^81 + 16 * q^82 + 16 * q^83 - 8 * q^86 - 12 * q^87 - 16 * q^89 - 4 * q^91 - 16 * q^93 + 8 * q^94 - 32 * q^96 + 4 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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