LMFDB - Embedded newform 961.4.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,4,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [64,16,0] 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:	$56.7008355155$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.39615 q^{2} +5.75704 q^{3} +21.1185 q^{4} -7.08279 q^{5} -31.0659 q^{6} -23.7309 q^{7} -70.7893 q^{8} +6.14347 q^{9} +38.2198 q^{10} -41.3653 q^{11} +121.580 q^{12} -33.0731 q^{13} +128.056 q^{14} -40.7759 q^{15} +213.042 q^{16} +86.1662 q^{17} -33.1511 q^{18} -120.883 q^{19} -149.578 q^{20} -136.620 q^{21} +223.214 q^{22} -42.0245 q^{23} -407.537 q^{24} -74.8341 q^{25} +178.468 q^{26} -120.072 q^{27} -501.161 q^{28} -76.0693 q^{29} +220.033 q^{30} -583.295 q^{32} -238.142 q^{33} -464.966 q^{34} +168.081 q^{35} +129.741 q^{36} -16.4933 q^{37} +652.305 q^{38} -190.403 q^{39} +501.386 q^{40} +59.5959 q^{41} +737.222 q^{42} +164.381 q^{43} -873.573 q^{44} -43.5129 q^{45} +226.771 q^{46} -357.886 q^{47} +1226.49 q^{48} +220.158 q^{49} +403.816 q^{50} +496.062 q^{51} -698.454 q^{52} +485.193 q^{53} +647.926 q^{54} +292.982 q^{55} +1679.90 q^{56} -695.930 q^{57} +410.482 q^{58} +193.333 q^{59} -861.125 q^{60} +97.3901 q^{61} -145.790 q^{63} +1443.21 q^{64} +234.250 q^{65} +1285.05 q^{66} -580.891 q^{67} +1819.70 q^{68} -241.937 q^{69} -906.993 q^{70} -149.443 q^{71} -434.892 q^{72} +150.915 q^{73} +89.0003 q^{74} -430.823 q^{75} -2552.87 q^{76} +981.638 q^{77} +1027.44 q^{78} +1058.77 q^{79} -1508.93 q^{80} -857.131 q^{81} -321.589 q^{82} +823.504 q^{83} -2885.20 q^{84} -610.297 q^{85} -887.027 q^{86} -437.934 q^{87} +2928.22 q^{88} +793.082 q^{89} +234.802 q^{90} +784.856 q^{91} -887.494 q^{92} +1931.21 q^{94} +856.192 q^{95} -3358.05 q^{96} -766.204 q^{97} -1188.00 q^{98} -254.127 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 16 q^{2} + 288 q^{4} + 80 q^{5} + 112 q^{7} + 24 q^{8} + 720 q^{9} + 136 q^{10} + 712 q^{14} + 1408 q^{16} + 432 q^{18} + 608 q^{19} + 1656 q^{20} + 2000 q^{25} + 2168 q^{28} + 448 q^{32} + 1056 q^{33}+ \cdots + 16312 q^{98}+O(q^{100}) $ 64 * q + 16 * q^2 + 288 * q^4 + 80 * q^5 + 112 * q^7 + 24 * q^8 + 720 * q^9 + 136 * q^10 + 712 * q^14 + 1408 * q^16 + 432 * q^18 + 608 * q^19 + 1656 * q^20 + 2000 * q^25 + 2168 * q^28 + 448 * q^32 + 1056 * q^33 + 1328 * q^35 + 4320 * q^36 + 3816 * q^38 + 1280 * q^39 + 1728 * q^40 + 1600 * q^41 + 1520 * q^45 + 4576 * q^47 + 4704 * q^49 + 5656 * q^50 + 3200 * q^51 + 7488 * q^56 + 6368 * q^59 + 1264 * q^63 + 11544 * q^64 + 1024 * q^66 + 5840 * q^67 + 6080 * q^69 - 6040 * q^70 + 5712 * q^71 - 2376 * q^72 - 40 * q^76 + 1600 * q^78 + 19544 * q^80 + 13184 * q^81 - 1336 * q^82 + 9184 * q^87 + 6120 * q^90 + 3136 * q^94 + 24048 * q^95 + 512 * q^97 + 16312 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−5.39615	−1.90783	−0.953914	−	0.300079i	$-0.902987\pi$
$2$	−5.39615	−1.90783	−0.953914	+	0.300079i	$0.902987\pi$
$3$	5.75704	1.10794	0.553971	−	0.832536i	$-0.313112\pi$
$3$	5.75704	1.10794	0.553971	+	0.832536i	$0.313112\pi$
$4$	21.1185	2.63981
$5$	−7.08279	−0.633504	−0.316752	−	0.948508i	$-0.602592\pi$
$5$	−7.08279	−0.633504	−0.316752	+	0.948508i	$0.602592\pi$
$6$	−31.0659	−2.11376
$7$	−23.7309	−1.28135	−0.640675	−	0.767812i	$-0.721345\pi$
$7$	−23.7309	−1.28135	−0.640675	+	0.767812i	$0.721345\pi$
$8$	−70.7893	−3.12848
$9$	6.14347	0.227536
$10$	38.2198	1.20862
$11$	−41.3653	−1.13383	−0.566914	−	0.823777i	$-0.691863\pi$
$11$	−41.3653	−1.13383	−0.566914	+	0.823777i	$0.691863\pi$
$12$	121.580	2.92476
$13$	−33.0731	−0.705602	−0.352801	−	0.935698i	$-0.614771\pi$
$13$	−33.0731	−0.705602	−0.352801	+	0.935698i	$0.614771\pi$
$14$	128.056	2.44460
$15$	−40.7759	−0.701886
$16$	213.042	3.32879
$17$	86.1662	1.22932	0.614658	−	0.788794i	$-0.289294\pi$
$17$	86.1662	1.22932	0.614658	+	0.788794i	$0.289294\pi$
$18$	−33.1511	−0.434100
$19$	−120.883	−1.45961	−0.729804	−	0.683656i	$-0.760389\pi$
$19$	−120.883	−1.45961	−0.729804	+	0.683656i	$0.760389\pi$
$20$	−149.578	−1.67233
$21$	−136.620	−1.41966
$22$	223.214	2.16315
$23$	−42.0245	−0.380988	−0.190494	−	0.981688i	$-0.561009\pi$
$23$	−42.0245	−0.380988	−0.190494	+	0.981688i	$0.561009\pi$
$24$	−407.537	−3.46617
$25$	−74.8341	−0.598673
$26$	178.468	1.34617
$27$	−120.072	−0.855846
$28$	−501.161	−3.38252
$29$	−76.0693	−0.487094	−0.243547	−	0.969889i	$-0.578311\pi$
$29$	−76.0693	−0.487094	−0.243547	+	0.969889i	$0.578311\pi$
$30$	220.033	1.33908
$31$	0	0
$31$	0	0
$32$	−583.295	−3.22228
$33$	−238.142	−1.25622
$34$	−464.966	−2.34532
$35$	168.081	0.811741
$36$	129.741	0.600652
$37$	−16.4933	−0.0732832	−0.0366416	−	0.999328i	$-0.511666\pi$
$37$	−16.4933	−0.0732832	−0.0366416	+	0.999328i	$0.511666\pi$
$38$	652.305	2.78468
$39$	−190.403	−0.781766
$40$	501.386	1.98190
$41$	59.5959	0.227008	0.113504	−	0.993538i	$-0.463793\pi$
$41$	59.5959	0.227008	0.113504	+	0.993538i	$0.463793\pi$
$42$	737.222	2.70847
$43$	164.381	0.582975	0.291488	−	0.956575i	$-0.405850\pi$
$43$	164.381	0.582975	0.291488	+	0.956575i	$0.405850\pi$
$44$	−873.573	−2.99309
$45$	−43.5129	−0.144145
$46$	226.771	0.726859
$47$	−357.886	−1.11070	−0.555351	−	0.831616i	$-0.687416\pi$
$47$	−357.886	−1.11070	−0.555351	+	0.831616i	$0.687416\pi$
$48$	1226.49	3.68810
$49$	220.158	0.641859
$50$	403.816	1.14216
$51$	496.062	1.36201
$52$	−698.454	−1.86266
$53$	485.193	1.25748	0.628739	−	0.777616i	$-0.283571\pi$
$53$	485.193	1.25748	0.628739	+	0.777616i	$0.283571\pi$
$54$	647.926	1.63281
$55$	292.982	0.718285
$56$	1679.90	4.00867
$57$	−695.930	−1.61716
$58$	410.482	0.929291
$59$	193.333	0.426607	0.213303	−	0.976986i	$-0.431578\pi$
$59$	193.333	0.426607	0.213303	+	0.976986i	$0.431578\pi$
$60$	−861.125	−1.85285
$61$	97.3901	0.204418	0.102209	−	0.994763i	$-0.467409\pi$
$61$	97.3901	0.204418	0.102209	+	0.994763i	$0.467409\pi$
$62$	0	0
$63$	−145.790	−0.291553
$64$	1443.21	2.81877
$65$	234.250	0.447002
$66$	1285.05	2.39665
$67$	−580.891	−1.05921	−0.529605	−	0.848244i	$-0.677660\pi$
$67$	−580.891	−1.05921	−0.529605	+	0.848244i	$0.677660\pi$
$68$	1819.70	3.24516
$69$	−241.937	−0.422112
$70$	−906.993	−1.54866
$71$	−149.443	−0.249797	−0.124898	−	0.992170i	$-0.539860\pi$
$71$	−149.443	−0.249797	−0.124898	+	0.992170i	$0.539860\pi$
$72$	−434.892	−0.711841
$73$	150.915	0.241962	0.120981	−	0.992655i	$-0.461396\pi$
$73$	150.915	0.241962	0.120981	+	0.992655i	$0.461396\pi$
$74$	89.0003	0.139812
$75$	−430.823	−0.663295
$76$	−2552.87	−3.85309
$77$	981.638	1.45283
$78$	1027.44	1.49148
$79$	1058.77	1.50786	0.753929	−	0.656955i	$-0.228156\pi$
$79$	1058.77	1.50786	0.753929	+	0.656955i	$0.228156\pi$
$80$	−1508.93	−2.10880
$81$	−857.131	−1.17576
$82$	−321.589	−0.433092
$83$	823.504	1.08905	0.544526	−	0.838744i	$-0.316710\pi$
$83$	823.504	1.08905	0.544526	+	0.838744i	$0.316710\pi$
$84$	−2885.20	−3.74764
$85$	−610.297	−0.778777
$86$	−887.027	−1.11222
$87$	−437.934	−0.539672
$88$	2928.22	3.54716
$89$	793.082	0.944568	0.472284	−	0.881446i	$-0.343430\pi$
$89$	793.082	0.944568	0.472284	+	0.881446i	$0.343430\pi$
$90$	234.802	0.275004
$91$	784.856	0.904124
$92$	−887.494	−1.00574
$93$	0	0
$94$	1931.21	2.11903
$95$	856.192	0.924668
$96$	−3358.05	−3.57010
$97$	−766.204	−0.802023	−0.401012	−	0.916073i	$-0.631341\pi$
$97$	−766.204	−0.802023	−0.401012	+	0.916073i	$0.631341\pi$
$98$	−1188.00	−1.22456
$99$	−254.127	−0.257987
$100$	−1580.38	−1.58038
$101$	477.953	0.470872	0.235436	−	0.971890i	$-0.424348\pi$
$101$	477.953	0.470872	0.235436	+	0.971890i	$0.424348\pi$
$102$	−2676.83	−2.59848
$103$	−1556.86	−1.48934	−0.744670	−	0.667433i	$-0.767393\pi$
$103$	−1556.86	−1.48934	−0.744670	+	0.667433i	$0.767393\pi$
$104$	2341.22	2.20746
$105$	967.650	0.899362
$106$	−2618.18	−2.39905
$107$	1580.73	1.42818	0.714090	−	0.700054i	$-0.246841\pi$
$107$	1580.73	1.42818	0.714090	+	0.700054i	$0.246841\pi$
$108$	−2535.73	−2.25927
$109$	−1875.65	−1.64820	−0.824102	−	0.566441i	$-0.808320\pi$
$109$	−1875.65	−1.64820	−0.824102	+	0.566441i	$0.808320\pi$
$110$	−1580.98	−1.37037
$111$	−94.9524	−0.0811935
$112$	−5055.70	−4.26534
$113$	−1727.53	−1.43816	−0.719081	−	0.694926i	$-0.755437\pi$
$113$	−1727.53	−1.43816	−0.719081	+	0.694926i	$0.755437\pi$
$114$	3755.35	3.08527
$115$	297.651	0.241357
$116$	−1606.47	−1.28583
$117$	−203.184	−0.160550
$118$	−1043.25	−0.813892
$119$	−2044.80	−1.57518
$120$	2886.50	2.19583
$121$	380.091	0.285568
$122$	−525.532	−0.389995
$123$	343.096	0.251512
$124$	0	0
$125$	1415.38	1.01277
$126$	786.707	0.556234
$127$	2672.64	1.86739	0.933696	−	0.358066i	$-0.116564\pi$
$127$	2672.64	1.86739	0.933696	+	0.358066i	$0.116564\pi$
$128$	−3121.42	−2.15545
$129$	946.350	0.645903
$130$	−1264.05	−0.852803
$131$	317.079	0.211476	0.105738	−	0.994394i	$-0.466280\pi$
$131$	317.079	0.211476	0.105738	+	0.994394i	$0.466280\pi$
$132$	−5029.19	−3.31617
$133$	2868.68	1.87027
$134$	3134.58	2.02079
$135$	850.443	0.542182
$136$	−6099.65	−3.84589
$137$	−1764.51	−1.10038	−0.550190	−	0.835040i	$-0.685445\pi$
$137$	−1764.51	−1.10038	−0.550190	+	0.835040i	$0.685445\pi$
$138$	1305.53	0.805318
$139$	−1100.36	−0.671446	−0.335723	−	0.941961i	$-0.608981\pi$
$139$	−1100.36	−0.671446	−0.335723	+	0.941961i	$0.608981\pi$
$140$	3549.62	2.14284
$141$	−2060.36	−1.23059
$142$	806.415	0.476570
$143$	1368.08	0.800032
$144$	1308.82	0.757419
$145$	538.783	0.308576
$146$	−814.360	−0.461622
$147$	1267.46	0.711143
$148$	−348.313	−0.193454
$149$	1262.60	0.694201	0.347101	−	0.937828i	$-0.387166\pi$
$149$	1262.60	0.694201	0.347101	+	0.937828i	$0.387166\pi$
$150$	2324.78	1.26545
$151$	187.681	0.101147	0.0505737	−	0.998720i	$-0.483895\pi$
$151$	187.681	0.101147	0.0505737	+	0.998720i	$0.483895\pi$
$152$	8557.26	4.56635
$153$	529.359	0.279713
$154$	−5297.07	−2.77175
$155$	0	0
$156$	−4021.02	−2.06371
$157$	−424.705	−0.215893	−0.107946	−	0.994157i	$-0.534427\pi$
$157$	−424.705	−0.215893	−0.107946	+	0.994157i	$0.534427\pi$
$158$	−5713.28	−2.87674
$159$	2793.27	1.39321
$160$	4131.35	2.04133
$161$	997.281	0.488179
$162$	4625.21	2.24315
$163$	46.9190	0.0225459	0.0112730	−	0.999936i	$-0.496412\pi$
$163$	46.9190	0.0225459	0.0112730	+	0.999936i	$0.496412\pi$
$164$	1258.58	0.599258
$165$	1686.71	0.795818
$166$	−4443.75	−2.07772
$167$	833.885	0.386395	0.193197	−	0.981160i	$-0.438114\pi$
$167$	833.885	0.386395	0.193197	+	0.981160i	$0.438114\pi$
$168$	9671.23	4.44138
$169$	−1103.17	−0.502126
$170$	3293.26	1.48577
$171$	−742.643	−0.332113
$172$	3471.49	1.53894
$173$	1692.08	0.743621	0.371810	−	0.928309i	$-0.378737\pi$
$173$	1692.08	0.743621	0.371810	+	0.928309i	$0.378737\pi$
$174$	2363.16	1.02960
$175$	1775.88	0.767109
$176$	−8812.57	−3.77427
$177$	1113.02	0.472655
$178$	−4279.59	−1.80207
$179$	2837.48	1.18482	0.592411	−	0.805636i	$-0.298176\pi$
$179$	2837.48	1.18482	0.592411	+	0.805636i	$0.298176\pi$
$180$	−918.927	−0.380515
$181$	−2853.50	−1.17182	−0.585908	−	0.810378i	$-0.699262\pi$
$181$	−2853.50	−1.17182	−0.585908	+	0.810378i	$0.699262\pi$
$182$	−4235.20	−1.72491
$183$	560.678	0.226484
$184$	2974.89	1.19191
$185$	116.818	0.0464252
$186$	0	0
$187$	−3564.29	−1.39383
$188$	−7558.01	−2.93204
$189$	2849.42	1.09664
$190$	−4620.14	−1.76411
$191$	4172.75	1.58078	0.790392	−	0.612601i	$-0.209877\pi$
$191$	4172.75	1.58078	0.790392	+	0.612601i	$0.209877\pi$
$192$	8308.61	3.12303
$193$	−3117.44	−1.16269	−0.581343	−	0.813659i	$-0.697473\pi$
$193$	−3117.44	−1.16269	−0.581343	+	0.813659i	$0.697473\pi$
$194$	4134.56	1.53012
$195$	1348.58	0.495252
$196$	4649.40	1.69439
$197$	1452.27	0.525228	0.262614	−	0.964901i	$-0.415415\pi$
$197$	1452.27	0.525228	0.262614	+	0.964901i	$0.415415\pi$
$198$	1371.31	0.492195
$199$	1347.87	0.480142	0.240071	−	0.970755i	$-0.422829\pi$
$199$	1347.87	0.480142	0.240071	+	0.970755i	$0.422829\pi$
$200$	5297.46	1.87293
$201$	−3344.21	−1.17354
$202$	−2579.11	−0.898344
$203$	1805.20	0.624138
$204$	10476.1	3.59545
$205$	−422.106	−0.143810
$206$	8401.06	2.84141
$207$	−258.176	−0.0866884
$208$	−7045.97	−2.34880
$209$	5000.38	1.65495
$210$	−5221.59	−1.71583
$211$	4040.68	1.31835	0.659176	−	0.751989i	$-0.270905\pi$
$211$	4040.68	1.31835	0.659176	+	0.751989i	$0.270905\pi$
$212$	10246.5	3.31950
$213$	−860.347	−0.276760
$214$	−8529.88	−2.72472
$215$	−1164.28	−0.369317
$216$	8499.80	2.67749
$217$	0	0
$218$	10121.3	3.14449
$219$	868.822	0.268080
$220$	6187.33	1.89614
$221$	−2849.78	−0.867408
$222$	512.378	0.154903
$223$	−4834.36	−1.45172	−0.725859	−	0.687844i	$-0.758557\pi$
$223$	−4834.36	−1.45172	−0.725859	+	0.687844i	$0.758557\pi$
$224$	13842.1	4.12887
$225$	−459.741	−0.136220
$226$	9322.02	2.74377
$227$	4353.76	1.27299	0.636496	−	0.771280i	$-0.280383\pi$
$227$	4353.76	1.27299	0.636496	+	0.771280i	$0.280383\pi$
$228$	−14697.0	−4.26900
$229$	4645.61	1.34057	0.670284	−	0.742104i	$-0.266172\pi$
$229$	4645.61	1.34057	0.670284	+	0.742104i	$0.266172\pi$
$230$	−1606.17	−0.460468
$231$	5651.33	1.60965
$232$	5384.90	1.52386
$233$	−4237.39	−1.19142	−0.595709	−	0.803200i	$-0.703129\pi$
$233$	−4237.39	−1.19142	−0.595709	+	0.803200i	$0.703129\pi$
$234$	1096.41	0.306302
$235$	2534.83	0.703635
$236$	4082.89	1.12616
$237$	6095.37	1.67062
$238$	11034.1	3.00518
$239$	2150.70	0.582080	0.291040	−	0.956711i	$-0.405999\pi$
$239$	2150.70	0.582080	0.291040	+	0.956711i	$0.405999\pi$
$240$	−8686.99	−2.33643
$241$	−3237.89	−0.865439	−0.432719	−	0.901529i	$-0.642446\pi$
$241$	−3237.89	−0.865439	−0.432719	+	0.901529i	$0.642446\pi$
$242$	−2051.03	−0.544815
$243$	−1692.60	−0.446832
$244$	2056.73	0.539626
$245$	−1559.33	−0.406620
$246$	−1851.40	−0.479841
$247$	3997.99	1.02990
$248$	0	0
$249$	4740.94	1.20661
$250$	−7637.62	−1.93218
$251$	−4082.78	−1.02670	−0.513352	−	0.858178i	$-0.671597\pi$
$251$	−4082.78	−1.02670	−0.513352	+	0.858178i	$0.671597\pi$
$252$	−3078.87	−0.769645
$253$	1738.36	0.431975
$254$	−14422.0	−3.56266
$255$	−3513.50	−0.862839
$256$	5298.00	1.29346
$257$	−6292.12	−1.52721	−0.763603	−	0.645686i	$-0.776571\pi$
$257$	−6292.12	−1.52721	−0.763603	+	0.645686i	$0.776571\pi$
$258$	−5106.65	−1.23227
$259$	391.401	0.0939014
$260$	4947.00	1.18000
$261$	−467.329	−0.110831
$262$	−1711.01	−0.403460
$263$	−5033.00	−1.18003	−0.590015	−	0.807392i	$-0.700878\pi$
$263$	−5033.00	−1.18003	−0.590015	+	0.807392i	$0.700878\pi$
$264$	16857.9	3.93005
$265$	−3436.52	−0.796618
$266$	−15479.8	−3.56815
$267$	4565.80	1.04653
$268$	−12267.5	−2.79612
$269$	−2734.30	−0.619751	−0.309875	−	0.950777i	$-0.600287\pi$
$269$	−2734.30	−0.619751	−0.309875	+	0.950777i	$0.600287\pi$
$270$	−4589.12	−1.03439
$271$	6227.53	1.39592	0.697962	−	0.716135i	$-0.254091\pi$
$271$	6227.53	1.39592	0.697962	+	0.716135i	$0.254091\pi$
$272$	18357.0	4.09213
$273$	4518.44	1.00172
$274$	9521.55	2.09933
$275$	3095.54	0.678792
$276$	−5109.33	−1.11430
$277$	−2211.10	−0.479610	−0.239805	−	0.970821i	$-0.577084\pi$
$277$	−2211.10	−0.479610	−0.239805	+	0.970821i	$0.577084\pi$
$278$	5937.69	1.28100
$279$	0	0
$280$	−11898.4	−2.53951
$281$	3732.62	0.792417	0.396209	−	0.918160i	$-0.370326\pi$
$281$	3732.62	0.792417	0.396209	+	0.918160i	$0.370326\pi$
$282$	11118.0	2.34776
$283$	−3425.35	−0.719491	−0.359746	−	0.933050i	$-0.617137\pi$
$283$	−3425.35	−0.719491	−0.359746	+	0.933050i	$0.617137\pi$
$284$	−3156.00	−0.659416
$285$	4929.13	1.02448
$286$	−7382.37	−1.52632
$287$	−1414.27	−0.290877
$288$	−3583.45	−0.733184
$289$	2511.61	0.511217
$290$	−2907.36	−0.588710
$291$	−4411.07	−0.888596
$292$	3187.09	0.638734
$293$	1184.49	0.236173	0.118087	−	0.993003i	$-0.462324\pi$
$293$	1184.49	0.236173	0.118087	+	0.993003i	$0.462324\pi$
$294$	−6839.39	−1.35674
$295$	−1369.34	−0.270257
$296$	1167.55	0.229265
$297$	4966.81	0.970382
$298$	−6813.17	−1.32442
$299$	1389.88	0.268826
$300$	−9098.32	−1.75097
$301$	−3900.93	−0.746995
$302$	−1012.76	−0.192972
$303$	2751.59	0.521699
$304$	−25753.3	−4.85872
$305$	−689.793	−0.129500
$306$	−2856.50	−0.533645
$307$	9535.30	1.77267	0.886333	−	0.463049i	$-0.153245\pi$
$307$	9535.30	1.77267	0.886333	+	0.463049i	$0.153245\pi$
$308$	20730.7	3.83520
$309$	−8962.90	−1.65010
$310$	0	0
$311$	−3975.15	−0.724792	−0.362396	−	0.932024i	$-0.618041\pi$
$311$	−3975.15	−0.724792	−0.362396	+	0.932024i	$0.618041\pi$
$312$	13478.5	2.44574
$313$	−201.867	−0.0364542	−0.0182271	−	0.999834i	$-0.505802\pi$
$313$	−201.867	−0.0364542	−0.0182271	+	0.999834i	$0.505802\pi$
$314$	2291.77	0.411886
$315$	1032.60	0.184700
$316$	22359.6	3.98046
$317$	−5974.09	−1.05848	−0.529240	−	0.848472i	$-0.677523\pi$
$317$	−5974.09	−1.05848	−0.529240	+	0.848472i	$0.677523\pi$
$318$	−15072.9	−2.65801
$319$	3146.63	0.552281
$320$	−10221.9	−1.78570
$321$	9100.34	1.58234
$322$	−5381.48	−0.931361
$323$	−10416.1	−1.79432
$324$	−18101.3	−3.10379
$325$	2475.00	0.422425
$326$	−253.182	−0.0430137
$327$	−10798.2	−1.82611
$328$	−4218.76	−0.710189
$329$	8492.97	1.42320
$330$	−9101.74	−1.51829
$331$	7864.23	1.30591	0.652957	−	0.757395i	$-0.273528\pi$
$331$	7864.23	1.30591	0.652957	+	0.757395i	$0.273528\pi$
$332$	17391.1	2.87489
$333$	−101.326	−0.0166746
$334$	−4499.77	−0.737175
$335$	4114.33	0.671014
$336$	−29105.8	−4.72575
$337$	−11124.6	−1.79821	−0.899104	−	0.437734i	$-0.855781\pi$
$337$	−11124.6	−1.79821	−0.899104	+	0.437734i	$0.855781\pi$
$338$	5952.87	0.957970
$339$	−9945.45	−1.59340
$340$	−12888.5	−2.05582
$341$	0	0
$342$	4007.42	0.633615
$343$	2915.16	0.458904
$344$	−11636.5	−1.82382
$345$	1713.59	0.267410
$346$	−9130.72	−1.41870
$347$	−11534.7	−1.78448	−0.892238	−	0.451565i	$-0.850866\pi$
$347$	−11534.7	−1.78448	−0.892238	+	0.451565i	$0.850866\pi$
$348$	−9248.49	−1.42463
$349$	7246.39	1.11143	0.555717	−	0.831372i	$-0.312444\pi$
$349$	7246.39	1.11143	0.555717	+	0.831372i	$0.312444\pi$
$350$	−9582.94	−1.46351
$351$	3971.15	0.603886
$352$	24128.2	3.65351
$353$	3233.13	0.487485	0.243742	−	0.969840i	$-0.421625\pi$
$353$	3233.13	0.487485	0.243742	+	0.969840i	$0.421625\pi$
$354$	−6006.05	−0.901745
$355$	1058.47	0.158247
$356$	16748.7	2.49348
$357$	−11772.0	−1.74521
$358$	−15311.5	−2.26044
$359$	−749.795	−0.110230	−0.0551151	−	0.998480i	$-0.517553\pi$
$359$	−749.795	−0.110230	−0.0551151	+	0.998480i	$0.517553\pi$
$360$	3080.25	0.450954
$361$	7753.80	1.13046
$362$	15397.9	2.23562
$363$	2188.20	0.316393
$364$	16575.0	2.38671
$365$	−1068.90	−0.153284
$366$	−3025.51	−0.432092
$367$	−4879.10	−0.693970	−0.346985	−	0.937871i	$-0.612795\pi$
$367$	−4879.10	−0.693970	−0.346985	+	0.937871i	$0.612795\pi$
$368$	−8953.00	−1.26823
$369$	366.126	0.0516524
$370$	−630.370	−0.0885713
$371$	−11514.1	−1.61127
$372$	0	0
$373$	−9921.33	−1.37723	−0.688615	−	0.725127i	$-0.741781\pi$
$373$	−9921.33	−1.37723	−0.688615	+	0.725127i	$0.741781\pi$
$374$	19233.5	2.65920
$375$	8148.41	1.12209
$376$	25334.5	3.47481
$377$	2515.85	0.343694
$378$	−15375.9	−2.09220
$379$	571.935	0.0775154	0.0387577	−	0.999249i	$-0.487660\pi$
$379$	571.935	0.0775154	0.0387577	+	0.999249i	$0.487660\pi$
$380$	18081.5	2.44095
$381$	15386.5	2.06896
$382$	−22516.8	−3.01587
$383$	11153.5	1.48803	0.744015	−	0.668163i	$-0.232919\pi$
$383$	11153.5	1.48803	0.744015	+	0.668163i	$0.232919\pi$
$384$	−17970.1	−2.38811
$385$	−6952.74	−0.920375
$386$	16822.2	2.21821
$387$	1009.87	0.132648
$388$	−16181.1	−2.11719
$389$	9020.99	1.17579	0.587895	−	0.808937i	$-0.299957\pi$
$389$	9020.99	1.17579	0.587895	+	0.808937i	$0.299957\pi$
$390$	−7277.17	−0.944856
$391$	−3621.09	−0.468354
$392$	−15584.8	−2.00804
$393$	1825.44	0.234303
$394$	−7836.67	−1.00205
$395$	−7499.04	−0.955235
$396$	−5366.77	−0.681036
$397$	6982.61	0.882739	0.441369	−	0.897326i	$-0.354493\pi$
$397$	6982.61	0.882739	0.441369	+	0.897326i	$0.354493\pi$
$398$	−7273.34	−0.916029
$399$	16515.1	2.07215
$400$	−15942.8	−1.99285
$401$	−5347.23	−0.665905	−0.332952	−	0.942944i	$-0.608045\pi$
$401$	−5347.23	−0.665905	−0.332952	+	0.942944i	$0.608045\pi$
$402$	18045.9	2.23892
$403$	0	0
$404$	10093.6	1.24301
$405$	6070.88	0.744851
$406$	−9741.12	−1.19075
$407$	682.250	0.0830906
$408$	−35115.9	−4.26102
$409$	11151.7	1.34821	0.674106	−	0.738635i	$-0.264529\pi$
$409$	11151.7	1.34821	0.674106	+	0.738635i	$0.264529\pi$
$410$	2277.75	0.274366
$411$	−10158.3	−1.21916
$412$	−32878.5	−3.93158
$413$	−4587.97	−0.546632
$414$	1393.16	0.165387
$415$	−5832.70	−0.689919
$416$	19291.4	2.27365
$417$	−6334.79	−0.743923
$418$	−26982.8	−3.15735
$419$	5908.73	0.688927	0.344463	−	0.938800i	$-0.388061\pi$
$419$	5908.73	0.688927	0.344463	+	0.938800i	$0.388061\pi$
$420$	20435.3	2.37414
$421$	1426.38	0.165125	0.0825623	−	0.996586i	$-0.473690\pi$
$421$	1426.38	0.165125	0.0825623	+	0.996586i	$0.473690\pi$
$422$	−21804.2	−2.51519
$423$	−2198.66	−0.252725
$424$	−34346.5	−3.93399
$425$	−6448.17	−0.735958
$426$	4642.56	0.528012
$427$	−2311.16	−0.261932
$428$	33382.7	3.77012
$429$	7876.09	0.886389
$430$	6282.63	0.704594
$431$	4552.60	0.508796	0.254398	−	0.967100i	$-0.418123\pi$
$431$	4552.60	0.508796	0.254398	+	0.967100i	$0.418123\pi$
$432$	−25580.4	−2.84893
$433$	10329.4	1.14641	0.573207	−	0.819410i	$-0.305699\pi$
$433$	10329.4	1.14641	0.573207	+	0.819410i	$0.305699\pi$
$434$	0	0
$435$	3101.79	0.341884
$436$	−39610.8	−4.35095
$437$	5080.07	0.556093
$438$	−4688.30	−0.511451
$439$	5633.91	0.612510	0.306255	−	0.951950i	$-0.400924\pi$
$439$	5633.91	0.612510	0.306255	+	0.951950i	$0.400924\pi$
$440$	−20740.0	−2.24714
$441$	1352.53	0.146046
$442$	15377.9	1.65487
$443$	6672.19	0.715587	0.357794	−	0.933801i	$-0.383529\pi$
$443$	6672.19	0.715587	0.357794	+	0.933801i	$0.383529\pi$
$444$	−2005.25	−0.214336
$445$	−5617.24	−0.598388
$446$	26087.0	2.76963
$447$	7268.82	0.769135
$448$	−34248.7	−3.61183
$449$	18057.8	1.89800	0.948999	−	0.315279i	$-0.102098\pi$
$449$	18057.8	1.89800	0.948999	+	0.315279i	$0.102098\pi$
$450$	2480.83	0.259884
$451$	−2465.21	−0.257388
$452$	−36482.8	−3.79648
$453$	1080.49	0.112065
$454$	−23493.6	−2.42865
$455$	−5558.97	−0.572766
$456$	49264.4	5.05925
$457$	4522.63	0.462932	0.231466	−	0.972843i	$-0.425648\pi$
$457$	4522.63	0.462932	0.231466	+	0.972843i	$0.425648\pi$
$458$	−25068.4	−2.55758
$459$	−10346.1	−1.05210
$460$	6285.93	0.637137
$461$	4244.74	0.428845	0.214422	−	0.976741i	$-0.431213\pi$
$461$	4244.74	0.428845	0.214422	+	0.976741i	$0.431213\pi$
$462$	−30495.4	−3.07094
$463$	11496.4	1.15396	0.576978	−	0.816760i	$-0.304232\pi$
$463$	11496.4	1.15396	0.576978	+	0.816760i	$0.304232\pi$
$464$	−16206.0	−1.62143
$465$	0	0
$466$	22865.6	2.27302
$467$	−3023.96	−0.299640	−0.149820	−	0.988713i	$-0.547869\pi$
$467$	−3023.96	−0.299640	−0.149820	+	0.988713i	$0.547869\pi$
$468$	−4290.93	−0.423821
$469$	13785.1	1.35722
$470$	−13678.3	−1.34241
$471$	−2445.04	−0.239196
$472$	−13685.9	−1.33463
$473$	−6799.69	−0.660994
$474$	−32891.6	−3.18726
$475$	9046.20	0.873827
$476$	−43183.2	−4.15819
$477$	2980.77	0.286122
$478$	−11605.5	−1.11051
$479$	−5338.13	−0.509197	−0.254598	−	0.967047i	$-0.581943\pi$
$479$	−5338.13	−0.509197	−0.254598	+	0.967047i	$0.581943\pi$
$480$	23784.4	2.26167
$481$	545.484	0.0517088
$482$	17472.1	1.65111
$483$	5741.39	0.540874
$484$	8026.95	0.753845
$485$	5426.87	0.508085
$486$	9133.52	0.852479
$487$	15840.9	1.47396	0.736980	−	0.675914i	$-0.236251\pi$
$487$	15840.9	1.47396	0.736980	+	0.675914i	$0.236251\pi$
$488$	−6894.18	−0.639518
$489$	270.115	0.0249796
$490$	8414.39	0.775762
$491$	−10028.3	−0.921737	−0.460869	−	0.887468i	$-0.652462\pi$
$491$	−10028.3	−0.921737	−0.460869	+	0.887468i	$0.652462\pi$
$492$	7245.67	0.663943
$493$	−6554.60	−0.598792
$494$	−21573.8	−1.96488
$495$	1799.93	0.163436
$496$	0	0
$497$	3546.41	0.320077
$498$	−25582.8	−2.30200
$499$	−1555.85	−0.139578	−0.0697891	−	0.997562i	$-0.522233\pi$
$499$	−1555.85	−0.139578	−0.0697891	+	0.997562i	$0.522233\pi$
$500$	29890.7	2.67351
$501$	4800.70	0.428103
$502$	22031.3	1.95878
$503$	14459.8	1.28177	0.640886	−	0.767636i	$-0.278567\pi$
$503$	14459.8	1.28177	0.640886	+	0.767636i	$0.278567\pi$
$504$	10320.4	0.912118
$505$	−3385.24	−0.298300
$506$	−9380.45	−0.824134
$507$	−6350.99	−0.556326
$508$	56442.2	4.92956
$509$	−7508.81	−0.653875	−0.326937	−	0.945046i	$-0.606017\pi$
$509$	−7508.81	−0.653875	−0.326937	+	0.945046i	$0.606017\pi$
$510$	18959.4	1.64615
$511$	−3581.35	−0.310038
$512$	−3617.45	−0.312247
$513$	14514.7	1.24920
$514$	33953.3	2.91365
$515$	11026.9	0.943503
$516$	19985.5	1.70506
$517$	14804.1	1.25935
$518$	−2112.06	−0.179148
$519$	9741.36	0.823889
$520$	−16582.4	−1.39843
$521$	4469.83	0.375867	0.187934	−	0.982182i	$-0.439821\pi$
$521$	4469.83	0.375867	0.187934	+	0.982182i	$0.439821\pi$
$522$	2521.78	0.211447
$523$	−17773.2	−1.48598	−0.742990	−	0.669302i	$-0.766593\pi$
$523$	−17773.2	−1.48598	−0.742990	+	0.669302i	$0.766593\pi$
$524$	6696.24	0.558257
$525$	10223.8	0.849913
$526$	27158.8	2.25129
$527$	0	0
$528$	−50734.3	−4.18168
$529$	−10400.9	−0.854848
$530$	18544.0	1.51981
$531$	1187.73	0.0970683
$532$	60582.1	4.93716
$533$	−1971.02	−0.160177
$534$	−24637.8	−1.99659
$535$	−11196.0	−0.904758
$536$	41120.9	3.31372
$537$	16335.5	1.31271
$538$	14754.7	1.18238
$539$	−9106.90	−0.727758
$540$	17960.1	1.43126
$541$	−16721.1	−1.32883	−0.664414	−	0.747365i	$-0.731319\pi$
$541$	−16721.1	−1.32883	−0.664414	+	0.747365i	$0.731319\pi$
$542$	−33604.7	−2.66318
$543$	−16427.7	−1.29830
$544$	−50260.3	−3.96120
$545$	13284.8	1.04414
$546$	−24382.2	−1.91110
$547$	7242.44	0.566114	0.283057	−	0.959103i	$-0.408651\pi$
$547$	7242.44	0.566114	0.283057	+	0.959103i	$0.408651\pi$
$548$	−37263.7	−2.90479
$549$	598.313	0.0465125
$550$	−16704.0	−1.29502
$551$	9195.51	0.710966
$552$	17126.5	1.32057
$553$	−25125.6	−1.93210
$554$	11931.4	0.915014
$555$	672.528	0.0514364
$556$	−23237.8	−1.77249
$557$	−15328.1	−1.16602	−0.583010	−	0.812465i	$-0.698125\pi$
$557$	−15328.1	−1.16602	−0.583010	+	0.812465i	$0.698125\pi$
$558$	0	0
$559$	−5436.60	−0.411348
$560$	35808.4	2.70211
$561$	−20519.8	−1.54429
$562$	−20141.8	−1.51180
$563$	−15754.6	−1.17935	−0.589677	−	0.807639i	$-0.700745\pi$
$563$	−15754.6	−1.17935	−0.589677	+	0.807639i	$0.700745\pi$
$564$	−43511.7	−3.24854
$565$	12235.7	0.911082
$566$	18483.7	1.37267
$567$	20340.5	1.50656
$568$	10578.9	0.781484
$569$	16647.8	1.22656	0.613279	−	0.789866i	$-0.289850\pi$
$569$	16647.8	1.22656	0.613279	+	0.789866i	$0.289850\pi$
$570$	−26598.3	−1.95453
$571$	−1702.00	−0.124740	−0.0623700	−	0.998053i	$-0.519866\pi$
$571$	−1702.00	−0.124740	−0.0623700	+	0.998053i	$0.519866\pi$
$572$	28891.8	2.11193
$573$	24022.7	1.75142
$574$	7631.61	0.554943
$575$	3144.87	0.228087
$576$	8866.31	0.641371
$577$	3313.59	0.239076	0.119538	−	0.992830i	$-0.461859\pi$
$577$	3313.59	0.239076	0.119538	+	0.992830i	$0.461859\pi$
$578$	−13553.0	−0.975315
$579$	−17947.2	−1.28819
$580$	11378.3	0.814581
$581$	−19542.5	−1.39546
$582$	23802.8	1.69529
$583$	−20070.2	−1.42577
$584$	−10683.2	−0.756973
$585$	1439.11	0.101709
$586$	−6391.70	−0.450578
$587$	11210.6	0.788261	0.394130	−	0.919055i	$-0.371046\pi$
$587$	11210.6	0.788261	0.394130	+	0.919055i	$0.371046\pi$
$588$	26766.7	1.87728
$589$	0	0
$590$	7389.15	0.515604
$591$	8360.77	0.581923
$592$	−3513.77	−0.243944
$593$	−639.536	−0.0442877	−0.0221438	−	0.999755i	$-0.507049\pi$
$593$	−639.536	−0.0442877	−0.0221438	+	0.999755i	$0.507049\pi$
$594$	−26801.7	−1.85132
$595$	14482.9	0.997886
$596$	26664.1	1.83256
$597$	7759.76	0.531970
$598$	−7500.01	−0.512873
$599$	−4674.58	−0.318862	−0.159431	−	0.987209i	$-0.550966\pi$
$599$	−4674.58	−0.318862	−0.159431	+	0.987209i	$0.550966\pi$
$600$	30497.6	2.07510
$601$	−9152.10	−0.621168	−0.310584	−	0.950546i	$-0.600525\pi$
$601$	−9152.10	−0.621168	−0.310584	+	0.950546i	$0.600525\pi$
$602$	21050.0	1.42514
$603$	−3568.69	−0.241009
$604$	3963.54	0.267010
$605$	−2692.11	−0.180909
$606$	−14848.0	−0.995313
$607$	1805.96	0.120760	0.0603802	−	0.998175i	$-0.480769\pi$
$607$	1805.96	0.120760	0.0603802	+	0.998175i	$0.480769\pi$
$608$	70510.6	4.70326
$609$	10392.6	0.691508
$610$	3722.23	0.247064
$611$	11836.4	0.783714
$612$	11179.3	0.738390
$613$	18355.4	1.20941	0.604706	−	0.796449i	$-0.293291\pi$
$613$	18355.4	1.20941	0.604706	+	0.796449i	$0.293291\pi$
$614$	−51453.9	−3.38194
$615$	−2430.08	−0.159334
$616$	−69489.5	−4.54515
$617$	−5051.36	−0.329595	−0.164797	−	0.986327i	$-0.552697\pi$
$617$	−5051.36	−0.329595	−0.164797	+	0.986327i	$0.552697\pi$
$618$	48365.2	3.14811
$619$	−20397.6	−1.32448	−0.662238	−	0.749294i	$-0.730393\pi$
$619$	−20397.6	−1.32448	−0.662238	+	0.749294i	$0.730393\pi$
$620$	0	0
$621$	5045.96	0.326067
$622$	21450.5	1.38278
$623$	−18820.6	−1.21032
$624$	−40563.9	−2.60233
$625$	−670.601	−0.0429184
$626$	1089.30	0.0695484
$627$	28787.4	1.83358
$628$	−8969.12	−0.569915
$629$	−1421.16	−0.0900882
$630$	−5572.08	−0.352376
$631$	−5659.57	−0.357058	−0.178529	−	0.983935i	$-0.557134\pi$
$631$	−5659.57	−0.357058	−0.178529	+	0.983935i	$0.557134\pi$
$632$	−74949.6	−4.71730
$633$	23262.4	1.46066
$634$	32237.1	2.01940
$635$	−18929.8	−1.18300
$636$	58989.7	3.67782
$637$	−7281.30	−0.452897
$638$	−16979.7	−1.05366
$639$	−918.096	−0.0568378
$640$	22108.4	1.36549
$641$	4840.00	0.298235	0.149117	−	0.988820i	$-0.452357\pi$
$641$	4840.00	0.298235	0.149117	+	0.988820i	$0.452357\pi$
$642$	−49106.8	−3.01884
$643$	−28156.2	−1.72686	−0.863430	−	0.504468i	$-0.831689\pi$
$643$	−28156.2	−1.72686	−0.863430	+	0.504468i	$0.831689\pi$
$644$	21061.1	1.28870
$645$	−6702.80	−0.409182
$646$	56206.7	3.42325
$647$	−15477.7	−0.940484	−0.470242	−	0.882538i	$-0.655833\pi$
$647$	−15477.7	−0.940484	−0.470242	+	0.882538i	$0.655833\pi$
$648$	60675.8	3.67835
$649$	−7997.28	−0.483699
$650$	−13355.5	−0.805914
$651$	0	0
$652$	990.859	0.0595169
$653$	−14428.5	−0.864674	−0.432337	−	0.901712i	$-0.642311\pi$
$653$	−14428.5	−0.864674	−0.432337	+	0.901712i	$0.642311\pi$
$654$	58268.5	3.48391
$655$	−2245.81	−0.133971
$656$	12696.5	0.755661
$657$	927.141	0.0550551
$658$	−45829.4	−2.71522
$659$	−10106.8	−0.597428	−0.298714	−	0.954343i	$-0.596558\pi$
$659$	−10106.8	−0.597428	−0.298714	+	0.954343i	$0.596558\pi$
$660$	35620.7	2.10081
$661$	−16173.1	−0.951683	−0.475841	−	0.879531i	$-0.657856\pi$
$661$	−16173.1	−0.951683	−0.475841	+	0.879531i	$0.657856\pi$
$662$	−42436.6	−2.49146
$663$	−16406.3	−0.961038
$664$	−58295.3	−3.40707
$665$	−20318.2	−1.18482
$666$	546.770	0.0318122
$667$	3196.78	0.185577
$668$	17610.4	1.02001
$669$	−27831.6	−1.60842
$670$	−22201.6	−1.28018
$671$	−4028.57	−0.231775
$672$	79689.7	4.57455
$673$	−23042.5	−1.31980	−0.659898	−	0.751355i	$-0.729400\pi$
$673$	−23042.5	−1.31980	−0.659898	+	0.751355i	$0.729400\pi$
$674$	60030.1	3.43067
$675$	8985.46	0.512371
$676$	−23297.3	−1.32552
$677$	−4151.60	−0.235685	−0.117843	−	0.993032i	$-0.537598\pi$
$677$	−4151.60	−0.235685	−0.117843	+	0.993032i	$0.537598\pi$
$678$	53667.2	3.03994
$679$	18182.8	1.02767
$680$	43202.5	2.43638
$681$	25064.8	1.41040
$682$	0	0
$683$	−16603.3	−0.930174	−0.465087	−	0.885265i	$-0.653977\pi$
$683$	−16603.3	−0.930174	−0.465087	+	0.885265i	$0.653977\pi$
$684$	−15683.5	−0.876716
$685$	12497.6	0.697095
$686$	−15730.7	−0.875510
$687$	26744.9	1.48527
$688$	35020.2	1.94060
$689$	−16046.8	−0.887280
$690$	−9246.78	−0.510172
$691$	−15504.2	−0.853556	−0.426778	−	0.904356i	$-0.640351\pi$
$691$	−15504.2	−0.853556	−0.426778	+	0.904356i	$0.640351\pi$
$692$	35734.1	1.96302
$693$	6030.67	0.330572
$694$	62242.8	3.40448
$695$	7793.59	0.425364
$696$	31001.0	1.68835
$697$	5135.16	0.279064
$698$	−39102.6	−2.12042
$699$	−24394.8	−1.32002
$700$	37504.0	2.02502
$701$	−22509.3	−1.21279	−0.606393	−	0.795165i	$-0.707384\pi$
$701$	−22509.3	−1.21279	−0.606393	+	0.795165i	$0.707384\pi$
$702$	−21428.9	−1.15211
$703$	1993.76	0.106965
$704$	−59698.8	−3.19600
$705$	14593.1	0.779587
$706$	−17446.5	−0.930038
$707$	−11342.3	−0.603353
$708$	23505.4	1.24772
$709$	13381.8	0.708835	0.354418	−	0.935087i	$-0.384679\pi$
$709$	13381.8	0.708835	0.354418	+	0.935087i	$0.384679\pi$
$710$	−5711.67	−0.301909
$711$	6504.52	0.343092
$712$	−56141.8	−2.95506
$713$	0	0
$714$	63523.6	3.32957
$715$	−9689.82	−0.506824
$716$	59923.2	3.12770
$717$	12381.6	0.644911
$718$	4046.01	0.210300
$719$	23400.2	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$719$	23400.2	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$720$	−9270.09	−0.479828
$721$	36945.8	1.90837
$722$	−41840.7	−2.15672
$723$	−18640.6	−0.958856
$724$	−60261.5	−3.09337
$725$	5692.58	0.291610
$726$	−11807.9	−0.603623
$727$	22095.5	1.12720	0.563601	−	0.826047i	$-0.309415\pi$
$727$	22095.5	1.12720	0.563601	+	0.826047i	$0.309415\pi$
$728$	−55559.4	−2.82853
$729$	13398.2	0.680699
$730$	5767.94	0.292440
$731$	14164.1	0.716660
$732$	11840.7	0.597874
$733$	−33656.5	−1.69595	−0.847974	−	0.530038i	$-0.822178\pi$
$733$	−33656.5	−1.69595	−0.847974	+	0.530038i	$0.822178\pi$
$734$	26328.4	1.32398
$735$	−8977.12	−0.450512
$736$	24512.7	1.22765
$737$	24028.8	1.20096
$738$	−1975.67	−0.0985440
$739$	12248.8	0.609716	0.304858	−	0.952398i	$-0.401391\pi$
$739$	12248.8	0.609716	0.304858	+	0.952398i	$0.401391\pi$
$740$	2467.03	0.122554
$741$	23016.6	1.14107
$742$	62131.8	3.07403
$743$	8392.84	0.414406	0.207203	−	0.978298i	$-0.433564\pi$
$743$	8392.84	0.414406	0.207203	+	0.978298i	$0.433564\pi$
$744$	0	0
$745$	−8942.71	−0.439779
$746$	53537.0	2.62752
$747$	5059.17	0.247798
$748$	−75272.5	−3.67946
$749$	−37512.3	−1.83000
$750$	−43970.1	−2.14075
$751$	−4400.49	−0.213817	−0.106908	−	0.994269i	$-0.534095\pi$
$751$	−4400.49	−0.213817	−0.106908	+	0.994269i	$0.534095\pi$
$752$	−76244.9	−3.69729
$753$	−23504.7	−1.13753
$754$	−13575.9	−0.655710
$755$	−1329.30	−0.0640773
$756$	60175.4	2.89492
$757$	7899.31	0.379267	0.189634	−	0.981855i	$-0.439270\pi$
$757$	7899.31	0.379267	0.189634	+	0.981855i	$0.439270\pi$
$758$	−3086.25	−0.147886
$759$	10007.8	0.478603
$760$	−60609.3	−2.89280
$761$	−8102.08	−0.385940	−0.192970	−	0.981205i	$-0.561812\pi$
$761$	−8102.08	−0.385940	−0.192970	+	0.981205i	$0.561812\pi$
$762$	−83028.0	−3.94723
$763$	44510.8	2.11193
$764$	88122.2	4.17297
$765$	−3749.34	−0.177200
$766$	−60185.8	−2.83891
$767$	−6394.11	−0.301014
$768$	30500.8	1.43308
$769$	22438.7	1.05222	0.526111	−	0.850416i	$-0.323649\pi$
$769$	22438.7	1.05222	0.526111	+	0.850416i	$0.323649\pi$
$770$	37518.1	1.75592
$771$	−36224.0	−1.69206
$772$	−65835.7	−3.06927
$773$	−42602.9	−1.98230	−0.991151	−	0.132742i	$-0.957622\pi$
$773$	−42602.9	−1.98230	−0.991151	+	0.132742i	$0.957622\pi$
$774$	−5449.43	−0.253069
$775$	0	0
$776$	54239.1	2.50911
$777$	2253.31	0.104037
$778$	−48678.6	−2.24321
$779$	−7204.16	−0.331343
$780$	28480.1	1.30737
$781$	6181.74	0.283227
$782$	19540.0	0.893540
$783$	9133.78	0.416877
$784$	46902.9	2.13661
$785$	3008.09	0.136769
$786$	−9850.34	−0.447010
$787$	−38342.8	−1.73669	−0.868343	−	0.495964i	$-0.834815\pi$
$787$	−38342.8	−1.73669	−0.868343	+	0.495964i	$0.834815\pi$
$788$	30669.7	1.38650
$789$	−28975.1	−1.30740
$790$	40466.0	1.82242
$791$	40995.9	1.84279
$792$	17989.5	0.807106
$793$	−3220.99	−0.144238
$794$	−37679.3	−1.68411
$795$	−19784.2	−0.882606
$796$	28465.1	1.26748
$797$	−10977.8	−0.487896	−0.243948	−	0.969788i	$-0.578443\pi$
$797$	−10977.8	−0.487896	−0.243948	+	0.969788i	$0.578443\pi$
$798$	−89117.9	−3.95331
$799$	−30837.7	−1.36540
$800$	43650.3	1.92909
$801$	4872.28	0.214923
$802$	28854.5	1.27043
$803$	−6242.64	−0.274344
$804$	−70624.7	−3.09793
$805$	−7063.53	−0.309263
$806$	0	0
$807$	−15741.4	−0.686648
$808$	−33834.0	−1.47311
$809$	−38491.5	−1.67279	−0.836396	−	0.548125i	$-0.815342\pi$
$809$	−38491.5	−1.67279	−0.836396	+	0.548125i	$0.815342\pi$
$810$	−32759.4	−1.42105
$811$	−3879.02	−0.167954	−0.0839770	−	0.996468i	$-0.526762\pi$
$811$	−3879.02	−0.167954	−0.0839770	+	0.996468i	$0.526762\pi$
$812$	38123.0	1.64760
$813$	35852.1	1.54660
$814$	−3681.53	−0.158523
$815$	−332.318	−0.0142829
$816$	105682.	4.53384
$817$	−19871.0	−0.850915
$818$	−60176.5	−2.57216
$819$	4821.74	0.205721
$820$	−8914.23	−0.379632
$821$	17753.7	0.754698	0.377349	−	0.926071i	$-0.376836\pi$
$821$	17753.7	0.754698	0.377349	+	0.926071i	$0.376836\pi$
$822$	54815.9	2.32594
$823$	4749.80	0.201176	0.100588	−	0.994928i	$-0.467928\pi$
$823$	4749.80	0.201176	0.100588	+	0.994928i	$0.467928\pi$
$824$	110209.	4.65937
$825$	17821.1	0.752063
$826$	24757.4	1.04288
$827$	−12659.1	−0.532286	−0.266143	−	0.963934i	$-0.585749\pi$
$827$	−12659.1	−0.532286	−0.266143	+	0.963934i	$0.585749\pi$
$828$	−5452.29	−0.228841
$829$	−12994.3	−0.544402	−0.272201	−	0.962240i	$-0.587752\pi$
$829$	−12994.3	−0.544402	−0.272201	+	0.962240i	$0.587752\pi$
$830$	31474.2	1.31625
$831$	−12729.4	−0.531380
$832$	−47731.4	−1.98893
$833$	18970.1	0.789048
$834$	34183.5	1.41928
$835$	−5906.23	−0.244783
$836$	105600.	4.36874
$837$	0	0
$838$	−31884.4	−1.31435
$839$	−2310.98	−0.0950940	−0.0475470	−	0.998869i	$-0.515140\pi$
$839$	−2310.98	−0.0950940	−0.0475470	+	0.998869i	$0.515140\pi$
$840$	−68499.3	−2.81363
$841$	−18602.5	−0.762740
$842$	−7696.96	−0.315029
$843$	21488.8	0.877953
$844$	85333.1	3.48020
$845$	7813.52	0.318099
$846$	11864.3	0.482156
$847$	−9019.92	−0.365913
$848$	103367.	4.18588
$849$	−19719.9	−0.797155
$850$	34795.3	1.40408
$851$	693.122	0.0279200
$852$	−18169.2	−0.730595
$853$	4487.38	0.180123	0.0900615	−	0.995936i	$-0.471294\pi$
$853$	4487.38	0.180123	0.0900615	+	0.995936i	$0.471294\pi$
$854$	12471.4	0.499721
$855$	5259.99	0.210395
$856$	−111899.	−4.46803
$857$	25875.8	1.03139	0.515694	−	0.856773i	$-0.327534\pi$
$857$	25875.8	1.03139	0.515694	+	0.856773i	$0.327534\pi$
$858$	−42500.6	−1.69108
$859$	28118.7	1.11688	0.558439	−	0.829545i	$-0.311400\pi$
$859$	28118.7	1.11688	0.558439	+	0.829545i	$0.311400\pi$
$860$	−24587.8	−0.974927
$861$	−8141.99	−0.322275
$862$	−24566.5	−0.970696
$863$	5867.55	0.231441	0.115721	−	0.993282i	$-0.463082\pi$
$863$	5867.55	0.231441	0.115721	+	0.993282i	$0.463082\pi$
$864$	70037.2	2.75777
$865$	−11984.6	−0.471087
$866$	−55738.9	−2.18716
$867$	14459.4	0.566399
$868$	0	0
$869$	−43796.3	−1.70965
$870$	−16737.8	−0.652256
$871$	19211.9	0.747382
$872$	132776.	5.15637
$873$	−4707.15	−0.182489
$874$	−27412.8	−1.06093
$875$	−33588.4	−1.29771
$876$	18348.2	0.707681
$877$	12360.4	0.475920	0.237960	−	0.971275i	$-0.423521\pi$
$877$	12360.4	0.475920	0.237960	+	0.971275i	$0.423521\pi$
$878$	−30401.4	−1.16856
$879$	6819.16	0.261666
$880$	62417.6	2.39102
$881$	47633.9	1.82160	0.910799	−	0.412850i	$-0.135467\pi$
$881$	47633.9	1.82160	0.910799	+	0.412850i	$0.135467\pi$
$882$	−7298.47	−0.278631
$883$	41600.3	1.58546	0.792730	−	0.609572i	$-0.208659\pi$
$883$	41600.3	1.58546	0.792730	+	0.609572i	$0.208659\pi$
$884$	−60183.1	−2.28979
$885$	−7883.31	−0.299429
$886$	−36004.2	−1.36522
$887$	−650.835	−0.0246369	−0.0123184	−	0.999924i	$-0.503921\pi$
$887$	−650.835	−0.0246369	−0.0123184	+	0.999924i	$0.503921\pi$
$888$	6721.62	0.254012
$889$	−63424.4	−2.39278
$890$	30311.5	1.14162
$891$	35455.5	1.33311
$892$	−102094.	−3.83226
$893$	43262.5	1.62119
$894$	−39223.6	−1.46738
$895$	−20097.3	−0.750589
$896$	74074.3	2.76188
$897$	8001.60	0.297843
$898$	−97442.8	−3.62106
$899$	0	0
$900$	−9709.03	−0.359594
$901$	41807.2	1.54584
$902$	13302.6	0.491052
$903$	−22457.8	−0.827628
$904$	122291.	4.49926
$905$	20210.7	0.742350
$906$	−5830.47	−0.213802
$907$	44773.7	1.63913	0.819563	−	0.572989i	$-0.194216\pi$
$907$	44773.7	1.63913	0.819563	+	0.572989i	$0.194216\pi$
$908$	91944.9	3.36046
$909$	2936.29	0.107140
$910$	29997.1	1.09274
$911$	5486.43	0.199532	0.0997659	−	0.995011i	$-0.468191\pi$
$911$	5486.43	0.199532	0.0997659	+	0.995011i	$0.468191\pi$
$912$	−148263.	−5.38319
$913$	−34064.5	−1.23480
$914$	−24404.8	−0.883194
$915$	−3971.17	−0.143478
$916$	98108.1	3.53885
$917$	−7524.59	−0.270975
$918$	55829.3	2.00723
$919$	−6448.19	−0.231454	−0.115727	−	0.993281i	$-0.536920\pi$
$919$	−6448.19	−0.231454	−0.115727	+	0.993281i	$0.536920\pi$
$920$	−21070.5	−0.755080
$921$	54895.1	1.96401
$922$	−22905.3	−0.818162
$923$	4942.53	0.176257
$924$	119347.	4.24918
$925$	1234.26	0.0438726
$926$	−62036.2	−2.20155
$927$	−9564.53	−0.338878
$928$	44370.8	1.56955
$929$	23954.4	0.845983	0.422992	−	0.906134i	$-0.360980\pi$
$929$	23954.4	0.845983	0.422992	+	0.906134i	$0.360980\pi$
$930$	0	0
$931$	−26613.4	−0.936863
$932$	−89487.1	−3.14512
$933$	−22885.1	−0.803028
$934$	16317.7	0.571662
$935$	25245.1	0.882999
$936$	14383.2	0.502276
$937$	3704.00	0.129140	0.0645700	−	0.997913i	$-0.479432\pi$
$937$	3704.00	0.129140	0.0645700	+	0.997913i	$0.479432\pi$
$938$	−74386.5	−2.58934
$939$	−1162.15	−0.0403892
$940$	53531.8	1.85746
$941$	19465.6	0.674347	0.337174	−	0.941442i	$-0.390529\pi$
$941$	19465.6	0.674347	0.337174	+	0.941442i	$0.390529\pi$
$942$	13193.8	0.456346
$943$	−2504.49	−0.0864872
$944$	41188.1	1.42008
$945$	−20181.8	−0.694725
$946$	36692.2	1.26106
$947$	−25974.1	−0.891285	−0.445642	−	0.895211i	$-0.647025\pi$
$947$	−25974.1	−0.891285	−0.445642	+	0.895211i	$0.647025\pi$
$948$	128725.	4.41012
$949$	−4991.22	−0.170729
$950$	−48814.7	−1.66711
$951$	−34393.1	−1.17274
$952$	144750.	4.92793
$953$	20517.9	0.697417	0.348709	−	0.937231i	$-0.386620\pi$
$953$	20517.9	0.697417	0.348709	+	0.937231i	$0.386620\pi$
$954$	−16084.7	−0.545871
$955$	−29554.7	−1.00143
$956$	45419.5	1.53658
$957$	18115.3	0.611895
$958$	28805.4	0.971460
$959$	41873.4	1.40997
$960$	−58848.1	−1.97845
$961$	0	0
$962$	−2943.51	−0.0986515
$963$	9711.19	0.324962
$964$	−68379.3	−2.28459
$965$	22080.2	0.736566
$966$	−30981.4	−1.03189
$967$	−21495.0	−0.714822	−0.357411	−	0.933947i	$-0.616340\pi$
$967$	−21495.0	−0.714822	−0.357411	+	0.933947i	$0.616340\pi$
$968$	−26906.4	−0.893393
$969$	−59965.6	−1.98800
$970$	−29284.2	−0.969339
$971$	21426.2	0.708136	0.354068	−	0.935220i	$-0.384798\pi$
$971$	21426.2	0.708136	0.354068	+	0.935220i	$0.384798\pi$
$972$	−35745.1	−1.17955
$973$	26112.5	0.860357
$974$	−85479.9	−2.81206
$975$	14248.6	0.468022
$976$	20748.2	0.680465
$977$	6050.25	0.198122	0.0990608	−	0.995081i	$-0.468416\pi$
$977$	6050.25	0.198122	0.0990608	+	0.995081i	$0.468416\pi$
$978$	−1457.58	−0.0476567
$979$	−32806.1	−1.07098
$980$	−32930.7	−1.07340
$981$	−11523.0	−0.375026
$982$	54114.5	1.75852
$983$	−39677.7	−1.28741	−0.643704	−	0.765275i	$-0.722603\pi$
$983$	−39677.7	−1.28741	−0.643704	+	0.765275i	$0.722603\pi$
$984$	−24287.5	−0.786848
$985$	−10286.1	−0.332734
$986$	35369.6	1.14239
$987$	48894.3	1.57682
$988$	84431.5	2.71875
$989$	−6908.05	−0.222106
$990$	−9712.68	−0.311807
$991$	−48098.8	−1.54178	−0.770892	−	0.636966i	$-0.780189\pi$
$991$	−48098.8	−1.54178	−0.770892	+	0.636966i	$0.780189\pi$
$992$	0	0
$993$	45274.7	1.44688
$994$	−19137.0	−0.610653
$995$	−9546.71	−0.304172
$996$	100121.	3.18521
$997$	4309.44	0.136892	0.0684459	−	0.997655i	$-0.478196\pi$
$997$	4309.44	0.136892	0.0684459	+	0.997655i	$0.478196\pi$
$998$	8395.62	0.266291
$999$	1980.38	0.0627191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.4.a.n.1.2	yes	64
31.30	odd	2	inner	961.4.a.n.1.1	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.4.a.n.1.1	✓	64	31.30	odd	2	inner
961.4.a.n.1.2	yes	64	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q-5.39615 q^{2} +5.75704 q^{3} +21.1185 q^{4} -7.08279 q^{5} -31.0659 q^{6} -23.7309 q^{7} -70.7893 q^{8} +6.14347 q^{9} +38.2198 q^{10} -41.3653 q^{11} +121.580 q^{12} -33.0731 q^{13} +128.056 q^{14} -40.7759 q^{15} +213.042 q^{16} +86.1662 q^{17} -33.1511 q^{18} -120.883 q^{19} -149.578 q^{20} -136.620 q^{21} +223.214 q^{22} -42.0245 q^{23} -407.537 q^{24} -74.8341 q^{25} +178.468 q^{26} -120.072 q^{27} -501.161 q^{28} -76.0693 q^{29} +220.033 q^{30} -583.295 q^{32} -238.142 q^{33} -464.966 q^{34} +168.081 q^{35} +129.741 q^{36} -16.4933 q^{37} +652.305 q^{38} -190.403 q^{39} +501.386 q^{40} +59.5959 q^{41} +737.222 q^{42} +164.381 q^{43} -873.573 q^{44} -43.5129 q^{45} +226.771 q^{46} -357.886 q^{47} +1226.49 q^{48} +220.158 q^{49} +403.816 q^{50} +496.062 q^{51} -698.454 q^{52} +485.193 q^{53} +647.926 q^{54} +292.982 q^{55} +1679.90 q^{56} -695.930 q^{57} +410.482 q^{58} +193.333 q^{59} -861.125 q^{60} +97.3901 q^{61} -145.790 q^{63} +1443.21 q^{64} +234.250 q^{65} +1285.05 q^{66} -580.891 q^{67} +1819.70 q^{68} -241.937 q^{69} -906.993 q^{70} -149.443 q^{71} -434.892 q^{72} +150.915 q^{73} +89.0003 q^{74} -430.823 q^{75} -2552.87 q^{76} +981.638 q^{77} +1027.44 q^{78} +1058.77 q^{79} -1508.93 q^{80} -857.131 q^{81} -321.589 q^{82} +823.504 q^{83} -2885.20 q^{84} -610.297 q^{85} -887.027 q^{86} -437.934 q^{87} +2928.22 q^{88} +793.082 q^{89} +234.802 q^{90} +784.856 q^{91} -887.494 q^{92} +1931.21 q^{94} +856.192 q^{95} -3358.05 q^{96} -766.204 q^{97} -1188.00 q^{98} -254.127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 16 q^{2} + 288 q^{4} + 80 q^{5} + 112 q^{7} + 24 q^{8} + 720 q^{9} + 136 q^{10} + 712 q^{14} + 1408 q^{16} + 432 q^{18} + 608 q^{19} + 1656 q^{20} + 2000 q^{25} + 2168 q^{28} + 448 q^{32} + 1056 q^{33}+ \cdots + 16312 q^{98}+O(q^{100}) \) 64 * q + 16 * q^2 + 288 * q^4 + 80 * q^5 + 112 * q^7 + 24 * q^8 + 720 * q^9 + 136 * q^10 + 712 * q^14 + 1408 * q^16 + 432 * q^18 + 608 * q^19 + 1656 * q^20 + 2000 * q^25 + 2168 * q^28 + 448 * q^32 + 1056 * q^33 + 1328 * q^35 + 4320 * q^36 + 3816 * q^38 + 1280 * q^39 + 1728 * q^40 + 1600 * q^41 + 1520 * q^45 + 4576 * q^47 + 4704 * q^49 + 5656 * q^50 + 3200 * q^51 + 7488 * q^56 + 6368 * q^59 + 1264 * q^63 + 11544 * q^64 + 1024 * q^66 + 5840 * q^67 + 6080 * q^69 - 6040 * q^70 + 5712 * q^71 - 2376 * q^72 - 40 * q^76 + 1600 * q^78 + 19544 * q^80 + 13184 * q^81 - 1336 * q^82 + 9184 * q^87 + 6120 * q^90 + 3136 * q^94 + 24048 * q^95 + 512 * q^97 + 16312 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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