LMFDB - Embedded newform 4925.2.a.s.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] 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("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [49,5,22] 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:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.420000 q^{2} -2.34519 q^{3} -1.82360 q^{4} +0.984980 q^{6} -1.96792 q^{7} +1.60591 q^{8} +2.49993 q^{9} +0.262244 q^{11} +4.27670 q^{12} +2.92808 q^{13} +0.826524 q^{14} +2.97272 q^{16} +3.46686 q^{17} -1.04997 q^{18} -0.910577 q^{19} +4.61515 q^{21} -0.110142 q^{22} +7.34742 q^{23} -3.76617 q^{24} -1.22979 q^{26} +1.17275 q^{27} +3.58869 q^{28} -3.55449 q^{29} -0.825867 q^{31} -4.46036 q^{32} -0.615014 q^{33} -1.45608 q^{34} -4.55888 q^{36} -5.87223 q^{37} +0.382442 q^{38} -6.86692 q^{39} +1.30619 q^{41} -1.93836 q^{42} +2.19557 q^{43} -0.478229 q^{44} -3.08591 q^{46} +11.4591 q^{47} -6.97160 q^{48} -3.12730 q^{49} -8.13047 q^{51} -5.33965 q^{52} -5.05298 q^{53} -0.492555 q^{54} -3.16030 q^{56} +2.13548 q^{57} +1.49288 q^{58} +3.30214 q^{59} +0.697567 q^{61} +0.346864 q^{62} -4.91966 q^{63} -4.07209 q^{64} +0.258305 q^{66} +12.7645 q^{67} -6.32217 q^{68} -17.2311 q^{69} -6.26113 q^{71} +4.01467 q^{72} -0.840670 q^{73} +2.46633 q^{74} +1.66053 q^{76} -0.516075 q^{77} +2.88410 q^{78} -16.5084 q^{79} -10.2501 q^{81} -0.548601 q^{82} -15.9653 q^{83} -8.41618 q^{84} -0.922138 q^{86} +8.33596 q^{87} +0.421141 q^{88} +3.78639 q^{89} -5.76222 q^{91} -13.3988 q^{92} +1.93682 q^{93} -4.81283 q^{94} +10.4604 q^{96} -7.17625 q^{97} +1.31347 q^{98} +0.655593 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * 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 $

$2$	−0.420000	−0.296985	−0.148492	−	0.988914i	$-0.547442\pi$
$2$	−0.420000	−0.296985	−0.148492	+	0.988914i	$0.547442\pi$
$3$	−2.34519	−1.35400	−0.676999	−	0.735984i	$-0.736720\pi$
$3$	−2.34519	−1.35400	−0.676999	+	0.735984i	$0.736720\pi$
$4$	−1.82360	−0.911800
$5$	0	0
$5$	0	0
$6$	0.984980	0.402117
$7$	−1.96792	−0.743803	−0.371901	−	0.928272i	$-0.621294\pi$
$7$	−1.96792	−0.743803	−0.371901	+	0.928272i	$0.621294\pi$
$8$	1.60591	0.567775
$9$	2.49993	0.833311
$10$	0	0
$11$	0.262244	0.0790696	0.0395348	−	0.999218i	$-0.487412\pi$
$11$	0.262244	0.0790696	0.0395348	+	0.999218i	$0.487412\pi$
$12$	4.27670	1.23458
$13$	2.92808	0.812104	0.406052	−	0.913850i	$-0.366905\pi$
$13$	2.92808	0.812104	0.406052	+	0.913850i	$0.366905\pi$
$14$	0.826524	0.220898
$15$	0	0
$16$	2.97272	0.743180
$17$	3.46686	0.840838	0.420419	−	0.907330i	$-0.361883\pi$
$17$	3.46686	0.840838	0.420419	+	0.907330i	$0.361883\pi$
$18$	−1.04997	−0.247481
$19$	−0.910577	−0.208901	−0.104450	−	0.994530i	$-0.533308\pi$
$19$	−0.910577	−0.208901	−0.104450	+	0.994530i	$0.533308\pi$
$20$	0	0
$21$	4.61515	1.00711
$22$	−0.110142	−0.0234825
$23$	7.34742	1.53204	0.766021	−	0.642815i	$-0.222234\pi$
$23$	7.34742	1.53204	0.766021	+	0.642815i	$0.222234\pi$
$24$	−3.76617	−0.768767
$25$	0	0
$26$	−1.22979	−0.241182
$27$	1.17275	0.225696
$28$	3.58869	0.678199
$29$	−3.55449	−0.660051	−0.330026	−	0.943972i	$-0.607057\pi$
$29$	−3.55449	−0.660051	−0.330026	+	0.943972i	$0.607057\pi$
$30$	0	0
$31$	−0.825867	−0.148330	−0.0741651	−	0.997246i	$-0.523629\pi$
$31$	−0.825867	−0.148330	−0.0741651	+	0.997246i	$0.523629\pi$
$32$	−4.46036	−0.788488
$33$	−0.615014	−0.107060
$34$	−1.45608	−0.249716
$35$	0	0
$36$	−4.55888	−0.759814
$37$	−5.87223	−0.965388	−0.482694	−	0.875789i	$-0.660342\pi$
$37$	−5.87223	−0.965388	−0.482694	+	0.875789i	$0.660342\pi$
$38$	0.382442	0.0620403
$39$	−6.86692	−1.09959
$40$	0	0
$41$	1.30619	0.203993	0.101997	−	0.994785i	$-0.467477\pi$
$41$	1.30619	0.203993	0.101997	+	0.994785i	$0.467477\pi$
$42$	−1.93836	−0.299095
$43$	2.19557	0.334821	0.167411	−	0.985887i	$-0.446459\pi$
$43$	2.19557	0.334821	0.167411	+	0.985887i	$0.446459\pi$
$44$	−0.478229	−0.0720957
$45$	0	0
$46$	−3.08591	−0.454993
$47$	11.4591	1.67149	0.835743	−	0.549120i	$-0.185037\pi$
$47$	11.4591	1.67149	0.835743	+	0.549120i	$0.185037\pi$
$48$	−6.97160	−1.00626
$49$	−3.12730	−0.446758
$50$	0	0
$51$	−8.13047	−1.13849
$52$	−5.33965	−0.740476
$53$	−5.05298	−0.694081	−0.347040	−	0.937850i	$-0.612813\pi$
$53$	−5.05298	−0.694081	−0.347040	+	0.937850i	$0.612813\pi$
$54$	−0.492555	−0.0670282
$55$	0	0
$56$	−3.16030	−0.422313
$57$	2.13548	0.282851
$58$	1.49288	0.196025
$59$	3.30214	0.429902	0.214951	−	0.976625i	$-0.431041\pi$
$59$	3.30214	0.429902	0.214951	+	0.976625i	$0.431041\pi$
$60$	0	0
$61$	0.697567	0.0893143	0.0446572	−	0.999002i	$-0.485780\pi$
$61$	0.697567	0.0893143	0.0446572	+	0.999002i	$0.485780\pi$
$62$	0.346864	0.0440518
$63$	−4.91966	−0.619819
$64$	−4.07209	−0.509011
$65$	0	0
$66$	0.258305	0.0317952
$67$	12.7645	1.55943	0.779717	−	0.626132i	$-0.215363\pi$
$67$	12.7645	1.55943	0.779717	+	0.626132i	$0.215363\pi$
$68$	−6.32217	−0.766676
$69$	−17.2311	−2.07438
$70$	0	0
$71$	−6.26113	−0.743060	−0.371530	−	0.928421i	$-0.621167\pi$
$71$	−6.26113	−0.743060	−0.371530	+	0.928421i	$0.621167\pi$
$72$	4.01467	0.473134
$73$	−0.840670	−0.0983929	−0.0491965	−	0.998789i	$-0.515666\pi$
$73$	−0.840670	−0.0983929	−0.0491965	+	0.998789i	$0.515666\pi$
$74$	2.46633	0.286705
$75$	0	0
$76$	1.66053	0.190476
$77$	−0.516075	−0.0588122
$78$	2.88410	0.326560
$79$	−16.5084	−1.85734	−0.928672	−	0.370903i	$-0.879048\pi$
$79$	−16.5084	−1.85734	−0.928672	+	0.370903i	$0.879048\pi$
$80$	0	0
$81$	−10.2501	−1.13890
$82$	−0.548601	−0.0605829
$83$	−15.9653	−1.75241	−0.876207	−	0.481934i	$-0.839934\pi$
$83$	−15.9653	−1.75241	−0.876207	+	0.481934i	$0.839934\pi$
$84$	−8.41618	−0.918281
$85$	0	0
$86$	−0.922138	−0.0994367
$87$	8.33596	0.893709
$88$	0.421141	0.0448938
$89$	3.78639	0.401357	0.200678	−	0.979657i	$-0.435685\pi$
$89$	3.78639	0.401357	0.200678	+	0.979657i	$0.435685\pi$
$90$	0	0
$91$	−5.76222	−0.604045
$92$	−13.3988	−1.39692
$93$	1.93682	0.200839
$94$	−4.81283	−0.496406
$95$	0	0
$96$	10.4604	1.06761
$97$	−7.17625	−0.728638	−0.364319	−	0.931274i	$-0.618698\pi$
$97$	−7.17625	−0.728638	−0.364319	+	0.931274i	$0.618698\pi$
$98$	1.31347	0.132680
$99$	0.655593	0.0658896
$100$	0	0
$101$	5.69799	0.566971	0.283485	−	0.958977i	$-0.408509\pi$
$101$	5.69799	0.566971	0.283485	+	0.958977i	$0.408509\pi$
$102$	3.41479	0.338115
$103$	−0.871010	−0.0858231	−0.0429116	−	0.999079i	$-0.513663\pi$
$103$	−0.871010	−0.0858231	−0.0429116	+	0.999079i	$0.513663\pi$
$104$	4.70224	0.461092
$105$	0	0
$106$	2.12225	0.206131
$107$	0.969497	0.0937248	0.0468624	−	0.998901i	$-0.485078\pi$
$107$	0.969497	0.0937248	0.0468624	+	0.998901i	$0.485078\pi$
$108$	−2.13863	−0.205790
$109$	6.03685	0.578225	0.289113	−	0.957295i	$-0.406640\pi$
$109$	6.03685	0.578225	0.289113	+	0.957295i	$0.406640\pi$
$110$	0	0
$111$	13.7715	1.30713
$112$	−5.85006	−0.552779
$113$	3.66113	0.344411	0.172205	−	0.985061i	$-0.444911\pi$
$113$	3.66113	0.344411	0.172205	+	0.985061i	$0.444911\pi$
$114$	−0.896900	−0.0840024
$115$	0	0
$116$	6.48196	0.601835
$117$	7.32001	0.676735
$118$	−1.38690	−0.127674
$119$	−6.82250	−0.625417
$120$	0	0
$121$	−10.9312	−0.993748
$122$	−0.292978	−0.0265250
$123$	−3.06328	−0.276207
$124$	1.50605	0.135247
$125$	0	0
$126$	2.06626	0.184077
$127$	15.1110	1.34088	0.670442	−	0.741962i	$-0.266105\pi$
$127$	15.1110	1.34088	0.670442	+	0.741962i	$0.266105\pi$
$128$	10.6310	0.939656
$129$	−5.14903	−0.453347
$130$	0	0
$131$	1.19923	0.104778	0.0523888	−	0.998627i	$-0.483316\pi$
$131$	1.19923	0.104778	0.0523888	+	0.998627i	$0.483316\pi$
$132$	1.12154	0.0976174
$133$	1.79194	0.155381
$134$	−5.36109	−0.463128
$135$	0	0
$136$	5.56747	0.477407
$137$	16.0952	1.37510	0.687552	−	0.726135i	$-0.258685\pi$
$137$	16.0952	1.37510	0.687552	+	0.726135i	$0.258685\pi$
$138$	7.23706	0.616060
$139$	−4.77889	−0.405340	−0.202670	−	0.979247i	$-0.564962\pi$
$139$	−4.77889	−0.405340	−0.202670	+	0.979247i	$0.564962\pi$
$140$	0	0
$141$	−26.8739	−2.26319
$142$	2.62967	0.220677
$143$	0.767873	0.0642127
$144$	7.43160	0.619300
$145$	0	0
$146$	0.353081	0.0292212
$147$	7.33414	0.604909
$148$	10.7086	0.880241
$149$	22.8959	1.87571	0.937853	−	0.347033i	$-0.112811\pi$
$149$	22.8959	1.87571	0.937853	+	0.347033i	$0.112811\pi$
$150$	0	0
$151$	−7.97484	−0.648983	−0.324492	−	0.945889i	$-0.605193\pi$
$151$	−7.97484	−0.648983	−0.324492	+	0.945889i	$0.605193\pi$
$152$	−1.46230	−0.118609
$153$	8.66693	0.700680
$154$	0.216751	0.0174663
$155$	0	0
$156$	12.5225	1.00260
$157$	−22.5524	−1.79988	−0.899938	−	0.436018i	$-0.856389\pi$
$157$	−22.5524	−1.79988	−0.899938	+	0.436018i	$0.856389\pi$
$158$	6.93353	0.551602
$159$	11.8502	0.939784
$160$	0	0
$161$	−14.4591	−1.13954
$162$	4.30505	0.338237
$163$	20.6248	1.61546	0.807729	−	0.589554i	$-0.200696\pi$
$163$	20.6248	1.61546	0.807729	+	0.589554i	$0.200696\pi$
$164$	−2.38198	−0.186001
$165$	0	0
$166$	6.70540	0.520440
$167$	4.14740	0.320935	0.160468	−	0.987041i	$-0.448700\pi$
$167$	4.14740	0.320935	0.160468	+	0.987041i	$0.448700\pi$
$168$	7.41151	0.571810
$169$	−4.42634	−0.340487
$170$	0	0
$171$	−2.27638	−0.174079
$172$	−4.00384	−0.305290
$173$	−21.7783	−1.65577	−0.827886	−	0.560896i	$-0.810457\pi$
$173$	−21.7783	−1.65577	−0.827886	+	0.560896i	$0.810457\pi$
$174$	−3.50110	−0.265418
$175$	0	0
$176$	0.779578	0.0587629
$177$	−7.74415	−0.582086
$178$	−1.59028	−0.119197
$179$	11.5992	0.866962	0.433481	−	0.901163i	$-0.357285\pi$
$179$	11.5992	0.866962	0.433481	+	0.901163i	$0.357285\pi$
$180$	0	0
$181$	3.95030	0.293623	0.146812	−	0.989164i	$-0.453099\pi$
$181$	3.95030	0.293623	0.146812	+	0.989164i	$0.453099\pi$
$182$	2.42013	0.179392
$183$	−1.63593	−0.120931
$184$	11.7993	0.869856
$185$	0	0
$186$	−0.813463	−0.0596460
$187$	0.909165	0.0664847
$188$	−20.8969	−1.52406
$189$	−2.30788	−0.167873
$190$	0	0
$191$	8.53415	0.617509	0.308755	−	0.951142i	$-0.400088\pi$
$191$	8.53415	0.617509	0.308755	+	0.951142i	$0.400088\pi$
$192$	9.54984	0.689200
$193$	1.18802	0.0855158	0.0427579	−	0.999085i	$-0.486386\pi$
$193$	1.18802	0.0855158	0.0427579	+	0.999085i	$0.486386\pi$
$194$	3.01402	0.216394
$195$	0	0
$196$	5.70295	0.407354
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−0.275349	−0.0195682
$199$	−6.78813	−0.481198	−0.240599	−	0.970625i	$-0.577344\pi$
$199$	−6.78813	−0.481198	−0.240599	+	0.970625i	$0.577344\pi$
$200$	0	0
$201$	−29.9353	−2.11147
$202$	−2.39315	−0.168382
$203$	6.99493	0.490948
$204$	14.8267	1.03808
$205$	0	0
$206$	0.365824	0.0254881
$207$	18.3681	1.27667
$208$	8.70436	0.603539
$209$	−0.238794	−0.0165177
$210$	0	0
$211$	0.358423	0.0246749	0.0123374	−	0.999924i	$-0.496073\pi$
$211$	0.358423	0.0246749	0.0123374	+	0.999924i	$0.496073\pi$
$212$	9.21462	0.632863
$213$	14.6836	1.00610
$214$	−0.407188	−0.0278348
$215$	0	0
$216$	1.88333	0.128145
$217$	1.62524	0.110328
$218$	−2.53547	−0.171724
$219$	1.97153	0.133224
$220$	0	0
$221$	10.1513	0.682847
$222$	−5.78403	−0.388198
$223$	−8.87146	−0.594077	−0.297039	−	0.954866i	$-0.595999\pi$
$223$	−8.87146	−0.594077	−0.297039	+	0.954866i	$0.595999\pi$
$224$	8.77762	0.586479
$225$	0	0
$226$	−1.53768	−0.102285
$227$	13.5773	0.901158	0.450579	−	0.892736i	$-0.351217\pi$
$227$	13.5773	0.901158	0.450579	+	0.892736i	$0.351217\pi$
$228$	−3.89426	−0.257904
$229$	15.2717	1.00918	0.504592	−	0.863358i	$-0.331643\pi$
$229$	15.2717	1.00918	0.504592	+	0.863358i	$0.331643\pi$
$230$	0	0
$231$	1.21030	0.0796316
$232$	−5.70819	−0.374761
$233$	7.70343	0.504668	0.252334	−	0.967640i	$-0.418802\pi$
$233$	7.70343	0.504668	0.252334	+	0.967640i	$0.418802\pi$
$234$	−3.07440	−0.200980
$235$	0	0
$236$	−6.02178	−0.391984
$237$	38.7155	2.51484
$238$	2.86545	0.185739
$239$	−15.2106	−0.983891	−0.491945	−	0.870626i	$-0.663714\pi$
$239$	−15.2106	−0.983891	−0.491945	+	0.870626i	$0.663714\pi$
$240$	0	0
$241$	−3.27032	−0.210660	−0.105330	−	0.994437i	$-0.533590\pi$
$241$	−3.27032	−0.210660	−0.105330	+	0.994437i	$0.533590\pi$
$242$	4.59111	0.295128
$243$	20.5203	1.31638
$244$	−1.27208	−0.0814368
$245$	0	0
$246$	1.28658	0.0820291
$247$	−2.66624	−0.169649
$248$	−1.32627	−0.0842181
$249$	37.4416	2.37277
$250$	0	0
$251$	2.02590	0.127874	0.0639368	−	0.997954i	$-0.479634\pi$
$251$	2.02590	0.127874	0.0639368	+	0.997954i	$0.479634\pi$
$252$	8.97150	0.565151
$253$	1.92682	0.121138
$254$	−6.34661	−0.398222
$255$	0	0
$256$	3.67916	0.229948
$257$	−21.2352	−1.32461	−0.662307	−	0.749232i	$-0.730423\pi$
$257$	−21.2352	−1.32461	−0.662307	+	0.749232i	$0.730423\pi$
$258$	2.16259	0.134637
$259$	11.5560	0.718058
$260$	0	0
$261$	−8.88598	−0.550028
$262$	−0.503678	−0.0311173
$263$	12.3719	0.762886	0.381443	−	0.924392i	$-0.375427\pi$
$263$	12.3719	0.762886	0.381443	+	0.924392i	$0.375427\pi$
$264$	−0.987657	−0.0607861
$265$	0	0
$266$	−0.752614	−0.0461457
$267$	−8.87982	−0.543436
$268$	−23.2774	−1.42189
$269$	11.7818	0.718351	0.359176	−	0.933270i	$-0.383058\pi$
$269$	11.7818	0.718351	0.359176	+	0.933270i	$0.383058\pi$
$270$	0	0
$271$	−5.95586	−0.361793	−0.180896	−	0.983502i	$-0.557900\pi$
$271$	−5.95586	−0.361793	−0.180896	+	0.983502i	$0.557900\pi$
$272$	10.3060	0.624894
$273$	13.5135	0.817876
$274$	−6.75997	−0.408385
$275$	0	0
$276$	31.4227	1.89142
$277$	6.02860	0.362224	0.181112	−	0.983463i	$-0.442030\pi$
$277$	6.02860	0.362224	0.181112	+	0.983463i	$0.442030\pi$
$278$	2.00713	0.120380
$279$	−2.06461	−0.123605
$280$	0	0
$281$	−32.6704	−1.94895	−0.974477	−	0.224486i	$-0.927930\pi$
$281$	−32.6704	−1.94895	−0.974477	+	0.224486i	$0.927930\pi$
$282$	11.2870	0.672132
$283$	−0.300454	−0.0178601	−0.00893006	−	0.999960i	$-0.502843\pi$
$283$	−0.300454	−0.0178601	−0.00893006	+	0.999960i	$0.502843\pi$
$284$	11.4178	0.677522
$285$	0	0
$286$	−0.322506	−0.0190702
$287$	−2.57048	−0.151731
$288$	−11.1506	−0.657056
$289$	−4.98086	−0.292992
$290$	0	0
$291$	16.8297	0.986574
$292$	1.53305	0.0897147
$293$	5.15837	0.301355	0.150678	−	0.988583i	$-0.451854\pi$
$293$	5.15837	0.301355	0.150678	+	0.988583i	$0.451854\pi$
$294$	−3.08033	−0.179649
$295$	0	0
$296$	−9.43027	−0.548123
$297$	0.307547	0.0178457
$298$	−9.61627	−0.557056
$299$	21.5138	1.24418
$300$	0	0
$301$	−4.32070	−0.249041
$302$	3.34943	0.192738
$303$	−13.3629	−0.767678
$304$	−2.70689	−0.155251
$305$	0	0
$306$	−3.64011	−0.208091
$307$	7.48717	0.427315	0.213658	−	0.976909i	$-0.431462\pi$
$307$	7.48717	0.427315	0.213658	+	0.976909i	$0.431462\pi$
$308$	0.941114	0.0536250
$309$	2.04269	0.116204
$310$	0	0
$311$	3.16816	0.179650	0.0898248	−	0.995958i	$-0.471369\pi$
$311$	3.16816	0.179650	0.0898248	+	0.995958i	$0.471369\pi$
$312$	−11.0277	−0.624318
$313$	15.3688	0.868697	0.434349	−	0.900745i	$-0.356979\pi$
$313$	15.3688	0.868697	0.434349	+	0.900745i	$0.356979\pi$
$314$	9.47199	0.534535
$315$	0	0
$316$	30.1048	1.69353
$317$	20.4252	1.14719	0.573597	−	0.819138i	$-0.305548\pi$
$317$	20.4252	1.14719	0.573597	+	0.819138i	$0.305548\pi$
$318$	−4.97709	−0.279101
$319$	−0.932143	−0.0521900
$320$	0	0
$321$	−2.27366	−0.126903
$322$	6.07282	0.338425
$323$	−3.15684	−0.175652
$324$	18.6921	1.03845
$325$	0	0
$326$	−8.66240	−0.479766
$327$	−14.1576	−0.782916
$328$	2.09763	0.115822
$329$	−22.5506	−1.24326
$330$	0	0
$331$	24.7462	1.36017	0.680087	−	0.733132i	$-0.261942\pi$
$331$	24.7462	1.36017	0.680087	+	0.733132i	$0.261942\pi$
$332$	29.1142	1.59785
$333$	−14.6802	−0.804469
$334$	−1.74190	−0.0953128
$335$	0	0
$336$	13.7195	0.748462
$337$	17.0879	0.930838	0.465419	−	0.885090i	$-0.345904\pi$
$337$	17.0879	0.930838	0.465419	+	0.885090i	$0.345904\pi$
$338$	1.85906	0.101120
$339$	−8.58607	−0.466331
$340$	0	0
$341$	−0.216579	−0.0117284
$342$	0.956080	0.0516989
$343$	19.9297	1.07610
$344$	3.52589	0.190103
$345$	0	0
$346$	9.14687	0.491739
$347$	15.7104	0.843379	0.421690	−	0.906740i	$-0.361437\pi$
$347$	15.7104	0.843379	0.421690	+	0.906740i	$0.361437\pi$
$348$	−15.2015	−0.814884
$349$	−23.9876	−1.28402	−0.642012	−	0.766694i	$-0.721900\pi$
$349$	−23.9876	−1.28402	−0.642012	+	0.766694i	$0.721900\pi$
$350$	0	0
$351$	3.43391	0.183289
$352$	−1.16970	−0.0623454
$353$	−13.9456	−0.742247	−0.371123	−	0.928584i	$-0.621027\pi$
$353$	−13.9456	−0.742247	−0.371123	+	0.928584i	$0.621027\pi$
$354$	3.25254	0.172871
$355$	0	0
$356$	−6.90487	−0.365957
$357$	16.0001	0.846814
$358$	−4.87164	−0.257474
$359$	14.1822	0.748509	0.374255	−	0.927326i	$-0.377899\pi$
$359$	14.1822	0.748509	0.374255	+	0.927326i	$0.377899\pi$
$360$	0	0
$361$	−18.1709	−0.956361
$362$	−1.65912	−0.0872016
$363$	25.6358	1.34553
$364$	10.5080	0.550768
$365$	0	0
$366$	0.687090	0.0359148
$367$	17.0353	0.889234	0.444617	−	0.895721i	$-0.353340\pi$
$367$	17.0353	0.889234	0.444617	+	0.895721i	$0.353340\pi$
$368$	21.8418	1.13858
$369$	3.26540	0.169990
$370$	0	0
$371$	9.94385	0.516259
$372$	−3.53198	−0.183125
$373$	7.40378	0.383353	0.191677	−	0.981458i	$-0.438608\pi$
$373$	7.40378	0.383353	0.191677	+	0.981458i	$0.438608\pi$
$374$	−0.381849	−0.0197449
$375$	0	0
$376$	18.4023	0.949028
$377$	−10.4078	−0.536030
$378$	0.969307	0.0498558
$379$	27.5227	1.41375	0.706873	−	0.707340i	$-0.250105\pi$
$379$	27.5227	1.41375	0.706873	+	0.707340i	$0.250105\pi$
$380$	0	0
$381$	−35.4382	−1.81555
$382$	−3.58434	−0.183391
$383$	−11.0860	−0.566470	−0.283235	−	0.959051i	$-0.591408\pi$
$383$	−11.0860	−0.566470	−0.283235	+	0.959051i	$0.591408\pi$
$384$	−24.9318	−1.27229
$385$	0	0
$386$	−0.498969	−0.0253969
$387$	5.48878	0.279010
$388$	13.0866	0.664372
$389$	−17.3331	−0.878824	−0.439412	−	0.898286i	$-0.644813\pi$
$389$	−17.3331	−0.878824	−0.439412	+	0.898286i	$0.644813\pi$
$390$	0	0
$391$	25.4725	1.28820
$392$	−5.02217	−0.253658
$393$	−2.81244	−0.141869
$394$	0.420000	0.0211593
$395$	0	0
$396$	−1.19554	−0.0600782
$397$	−24.5591	−1.23259	−0.616293	−	0.787517i	$-0.711366\pi$
$397$	−24.5591	−1.23259	−0.616293	+	0.787517i	$0.711366\pi$
$398$	2.85101	0.142908
$399$	−4.20244	−0.210385
$400$	0	0
$401$	−23.2354	−1.16032	−0.580160	−	0.814503i	$-0.697010\pi$
$401$	−23.2354	−1.16032	−0.580160	+	0.814503i	$0.697010\pi$
$402$	12.5728	0.627074
$403$	−2.41821	−0.120459
$404$	−10.3909	−0.516964
$405$	0	0
$406$	−2.93787	−0.145804
$407$	−1.53996	−0.0763329
$408$	−13.0568	−0.646408
$409$	20.1188	0.994811	0.497406	−	0.867518i	$-0.334286\pi$
$409$	20.1188	0.994811	0.497406	+	0.867518i	$0.334286\pi$
$410$	0	0
$411$	−37.7463	−1.86189
$412$	1.58837	0.0782536
$413$	−6.49833	−0.319762
$414$	−7.71458	−0.379151
$415$	0	0
$416$	−13.0603	−0.640334
$417$	11.2074	0.548830
$418$	0.100293	0.00490550
$419$	24.7359	1.20843	0.604215	−	0.796822i	$-0.293487\pi$
$419$	24.7359	1.20843	0.604215	+	0.796822i	$0.293487\pi$
$420$	0	0
$421$	11.5880	0.564766	0.282383	−	0.959302i	$-0.408875\pi$
$421$	11.5880	0.564766	0.282383	+	0.959302i	$0.408875\pi$
$422$	−0.150537	−0.00732805
$423$	28.6471	1.39287
$424$	−8.11464	−0.394082
$425$	0	0
$426$	−6.16709	−0.298797
$427$	−1.37275	−0.0664322
$428$	−1.76797	−0.0854583
$429$	−1.80081	−0.0869439
$430$	0	0
$431$	33.7897	1.62759	0.813795	−	0.581152i	$-0.197398\pi$
$431$	33.7897	1.62759	0.813795	+	0.581152i	$0.197398\pi$
$432$	3.48626	0.167733
$433$	7.75068	0.372474	0.186237	−	0.982505i	$-0.440371\pi$
$433$	7.75068	0.372474	0.186237	+	0.982505i	$0.440371\pi$
$434$	−0.682599	−0.0327658
$435$	0	0
$436$	−11.0088	−0.527226
$437$	−6.69039	−0.320045
$438$	−0.828043	−0.0395654
$439$	−26.5377	−1.26657	−0.633287	−	0.773917i	$-0.718295\pi$
$439$	−26.5377	−1.26657	−0.633287	+	0.773917i	$0.718295\pi$
$440$	0	0
$441$	−7.81806	−0.372288
$442$	−4.26352	−0.202795
$443$	35.7523	1.69864	0.849321	−	0.527877i	$-0.177012\pi$
$443$	35.7523	1.69864	0.849321	+	0.527877i	$0.177012\pi$
$444$	−25.1137	−1.19184
$445$	0	0
$446$	3.72601	0.176432
$447$	−53.6953	−2.53970
$448$	8.01353	0.378604
$449$	20.5021	0.967554	0.483777	−	0.875191i	$-0.339265\pi$
$449$	20.5021	0.967554	0.483777	+	0.875191i	$0.339265\pi$
$450$	0	0
$451$	0.342542	0.0161297
$452$	−6.67645	−0.314034
$453$	18.7025	0.878722
$454$	−5.70247	−0.267630
$455$	0	0
$456$	3.42939	0.160596
$457$	−32.4132	−1.51623	−0.758113	−	0.652123i	$-0.773878\pi$
$457$	−32.4132	−1.51623	−0.758113	+	0.652123i	$0.773878\pi$
$458$	−6.41412	−0.299712
$459$	4.06577	0.189774
$460$	0	0
$461$	2.71036	0.126234	0.0631170	−	0.998006i	$-0.479896\pi$
$461$	2.71036	0.126234	0.0631170	+	0.998006i	$0.479896\pi$
$462$	−0.508324	−0.0236494
$463$	−4.15267	−0.192991	−0.0964955	−	0.995333i	$-0.530763\pi$
$463$	−4.15267	−0.192991	−0.0964955	+	0.995333i	$0.530763\pi$
$464$	−10.5665	−0.490537
$465$	0	0
$466$	−3.23544	−0.149879
$467$	2.12695	0.0984235	0.0492117	−	0.998788i	$-0.484329\pi$
$467$	2.12695	0.0984235	0.0492117	+	0.998788i	$0.484329\pi$
$468$	−13.3488	−0.617047
$469$	−25.1195	−1.15991
$470$	0	0
$471$	52.8897	2.43703
$472$	5.30294	0.244087
$473$	0.575775	0.0264742
$474$	−16.2605	−0.746869
$475$	0	0
$476$	12.4415	0.570256
$477$	−12.6321	−0.578385
$478$	6.38844	0.292200
$479$	35.4058	1.61773	0.808867	−	0.587992i	$-0.200081\pi$
$479$	35.4058	1.61773	0.808867	+	0.587992i	$0.200081\pi$
$480$	0	0
$481$	−17.1944	−0.783995
$482$	1.37353	0.0625628
$483$	33.9094	1.54293
$484$	19.9342	0.906100
$485$	0	0
$486$	−8.61851	−0.390944
$487$	−8.17380	−0.370390	−0.185195	−	0.982702i	$-0.559292\pi$
$487$	−8.17380	−0.370390	−0.185195	+	0.982702i	$0.559292\pi$
$488$	1.12023	0.0507105
$489$	−48.3691	−2.18733
$490$	0	0
$491$	3.31109	0.149427	0.0747137	−	0.997205i	$-0.476196\pi$
$491$	3.31109	0.149427	0.0747137	+	0.997205i	$0.476196\pi$
$492$	5.58620	0.251845
$493$	−12.3229	−0.554996
$494$	1.11982	0.0503831
$495$	0	0
$496$	−2.45507	−0.110236
$497$	12.3214	0.552690
$498$	−15.7255	−0.704675
$499$	−5.86660	−0.262625	−0.131313	−	0.991341i	$-0.541919\pi$
$499$	−5.86660	−0.262625	−0.131313	+	0.991341i	$0.541919\pi$
$500$	0	0
$501$	−9.72645	−0.434546
$502$	−0.850877	−0.0379765
$503$	4.94195	0.220351	0.110175	−	0.993912i	$-0.464859\pi$
$503$	4.94195	0.220351	0.110175	+	0.993912i	$0.464859\pi$
$504$	−7.90054	−0.351918
$505$	0	0
$506$	−0.809263	−0.0359761
$507$	10.3806	0.461019
$508$	−27.5564	−1.22262
$509$	15.5285	0.688287	0.344143	−	0.938917i	$-0.388169\pi$
$509$	15.5285	0.688287	0.344143	+	0.938917i	$0.388169\pi$
$510$	0	0
$511$	1.65437	0.0731849
$512$	−22.8072	−1.00795
$513$	−1.06788	−0.0471480
$514$	8.91877	0.393390
$515$	0	0
$516$	9.38978	0.413362
$517$	3.00509	0.132164
$518$	−4.85354	−0.213252
$519$	51.0743	2.24191
$520$	0	0
$521$	41.3705	1.81247	0.906237	−	0.422771i	$-0.138943\pi$
$521$	41.3705	1.81247	0.906237	+	0.422771i	$0.138943\pi$
$522$	3.73211	0.163350
$523$	−14.0581	−0.614720	−0.307360	−	0.951593i	$-0.599445\pi$
$523$	−14.0581	−0.614720	−0.307360	+	0.951593i	$0.599445\pi$
$524$	−2.18692	−0.0955362
$525$	0	0
$526$	−5.19621	−0.226565
$527$	−2.86317	−0.124722
$528$	−1.82826	−0.0795649
$529$	30.9845	1.34715
$530$	0	0
$531$	8.25512	0.358242
$532$	−3.26778	−0.141676
$533$	3.82465	0.165664
$534$	3.72952	0.161392
$535$	0	0
$536$	20.4987	0.885408
$537$	−27.2023	−1.17387
$538$	−4.94837	−0.213339
$539$	−0.820118	−0.0353250
$540$	0	0
$541$	3.81284	0.163927	0.0819635	−	0.996635i	$-0.473881\pi$
$541$	3.81284	0.163927	0.0819635	+	0.996635i	$0.473881\pi$
$542$	2.50146	0.107447
$543$	−9.26422	−0.397566
$544$	−15.4635	−0.662990
$545$	0	0
$546$	−5.67567	−0.242896
$547$	28.7668	1.22998	0.614989	−	0.788536i	$-0.289160\pi$
$547$	28.7668	1.22998	0.614989	+	0.788536i	$0.289160\pi$
$548$	−29.3512	−1.25382
$549$	1.74387	0.0744267
$550$	0	0
$551$	3.23663	0.137885
$552$	−27.6716	−1.17778
$553$	32.4872	1.38150
$554$	−2.53201	−0.107575
$555$	0	0
$556$	8.71478	0.369589
$557$	−31.6837	−1.34248	−0.671241	−	0.741239i	$-0.734238\pi$
$557$	−31.6837	−1.34248	−0.671241	+	0.741239i	$0.734238\pi$
$558$	0.867137	0.0367088
$559$	6.42881	0.271909
$560$	0	0
$561$	−2.13217	−0.0900202
$562$	13.7216	0.578809
$563$	−18.7588	−0.790590	−0.395295	−	0.918554i	$-0.629358\pi$
$563$	−18.7588	−0.790590	−0.395295	+	0.918554i	$0.629358\pi$
$564$	49.0072	2.06358
$565$	0	0
$566$	0.126190	0.00530418
$567$	20.1714	0.847119
$568$	−10.0548	−0.421891
$569$	−34.9409	−1.46480	−0.732400	−	0.680875i	$-0.761600\pi$
$569$	−34.9409	−1.46480	−0.732400	+	0.680875i	$0.761600\pi$
$570$	0	0
$571$	9.46910	0.396270	0.198135	−	0.980175i	$-0.436512\pi$
$571$	9.46910	0.396270	0.198135	+	0.980175i	$0.436512\pi$
$572$	−1.40029	−0.0585492
$573$	−20.0142	−0.836106
$574$	1.07960	0.0450617
$575$	0	0
$576$	−10.1800	−0.424165
$577$	20.1885	0.840457	0.420229	−	0.907418i	$-0.361950\pi$
$577$	20.1885	0.840457	0.420229	+	0.907418i	$0.361950\pi$
$578$	2.09196	0.0870141
$579$	−2.78614	−0.115788
$580$	0	0
$581$	31.4183	1.30345
$582$	−7.06846	−0.292997
$583$	−1.32512	−0.0548807
$584$	−1.35004	−0.0558651
$585$	0	0
$586$	−2.16651	−0.0894979
$587$	−14.2199	−0.586919	−0.293460	−	0.955971i	$-0.594807\pi$
$587$	−14.2199	−0.586919	−0.293460	+	0.955971i	$0.594807\pi$
$588$	−13.3745	−0.551556
$589$	0.752015	0.0309863
$590$	0	0
$591$	2.34519	0.0964684
$592$	−17.4565	−0.717457
$593$	16.1285	0.662316	0.331158	−	0.943575i	$-0.392561\pi$
$593$	16.1285	0.662316	0.331158	+	0.943575i	$0.392561\pi$
$594$	−0.129170	−0.00529989
$595$	0	0
$596$	−41.7530	−1.71027
$597$	15.9195	0.651541
$598$	−9.03580	−0.369501
$599$	15.7943	0.645338	0.322669	−	0.946512i	$-0.395420\pi$
$599$	15.7943	0.645338	0.322669	+	0.946512i	$0.395420\pi$
$600$	0	0
$601$	15.5607	0.634735	0.317367	−	0.948303i	$-0.397201\pi$
$601$	15.5607	0.634735	0.317367	+	0.948303i	$0.397201\pi$
$602$	1.81469	0.0739613
$603$	31.9105	1.29949
$604$	14.5429	0.591743
$605$	0	0
$606$	5.61241	0.227988
$607$	24.4933	0.994150	0.497075	−	0.867707i	$-0.334407\pi$
$607$	24.4933	0.994150	0.497075	+	0.867707i	$0.334407\pi$
$608$	4.06150	0.164716
$609$	−16.4045	−0.664743
$610$	0	0
$611$	33.5533	1.35742
$612$	−15.8050	−0.638880
$613$	−47.3208	−1.91127	−0.955635	−	0.294554i	$-0.904829\pi$
$613$	−47.3208	−1.91127	−0.955635	+	0.294554i	$0.904829\pi$
$614$	−3.14461	−0.126906
$615$	0	0
$616$	−0.828770	−0.0333921
$617$	7.63310	0.307297	0.153649	−	0.988126i	$-0.450898\pi$
$617$	7.63310	0.307297	0.153649	+	0.988126i	$0.450898\pi$
$618$	−0.857927	−0.0345109
$619$	−18.9280	−0.760779	−0.380389	−	0.924826i	$-0.624210\pi$
$619$	−18.9280	−0.760779	−0.380389	+	0.924826i	$0.624210\pi$
$620$	0	0
$621$	8.61669	0.345776
$622$	−1.33062	−0.0533532
$623$	−7.45130	−0.298530
$624$	−20.4134	−0.817191
$625$	0	0
$626$	−6.45490	−0.257990
$627$	0.560017	0.0223649
$628$	41.1265	1.64113
$629$	−20.3582	−0.811735
$630$	0	0
$631$	−19.3273	−0.769410	−0.384705	−	0.923040i	$-0.625697\pi$
$631$	−19.3273	−0.769410	−0.384705	+	0.923040i	$0.625697\pi$
$632$	−26.5111	−1.05455
$633$	−0.840571	−0.0334097
$634$	−8.57858	−0.340699
$635$	0	0
$636$	−21.6101	−0.856895
$637$	−9.15700	−0.362814
$638$	0.391500	0.0154996
$639$	−15.6524	−0.619200
$640$	0	0
$641$	41.3956	1.63503	0.817515	−	0.575908i	$-0.195351\pi$
$641$	41.3956	1.63503	0.817515	+	0.575908i	$0.195351\pi$
$642$	0.954935	0.0376883
$643$	−35.3492	−1.39404	−0.697018	−	0.717054i	$-0.745490\pi$
$643$	−35.3492	−1.39404	−0.697018	+	0.717054i	$0.745490\pi$
$644$	26.3676	1.03903
$645$	0	0
$646$	1.32587	0.0521658
$647$	46.1951	1.81612	0.908058	−	0.418845i	$-0.137565\pi$
$647$	46.1951	1.81612	0.908058	+	0.418845i	$0.137565\pi$
$648$	−16.4608	−0.646641
$649$	0.865966	0.0339921
$650$	0	0
$651$	−3.81150	−0.149384
$652$	−37.6114	−1.47298
$653$	−13.3163	−0.521107	−0.260554	−	0.965459i	$-0.583905\pi$
$653$	−13.3163	−0.521107	−0.260554	+	0.965459i	$0.583905\pi$
$654$	5.94618	0.232514
$655$	0	0
$656$	3.88295	0.151604
$657$	−2.10162	−0.0819920
$658$	9.47125	0.369228
$659$	−16.3368	−0.636392	−0.318196	−	0.948025i	$-0.603077\pi$
$659$	−16.3368	−0.636392	−0.318196	+	0.948025i	$0.603077\pi$
$660$	0	0
$661$	34.0019	1.32252	0.661261	−	0.750156i	$-0.270022\pi$
$661$	34.0019	1.32252	0.661261	+	0.750156i	$0.270022\pi$
$662$	−10.3934	−0.403951
$663$	−23.8067	−0.924574
$664$	−25.6388	−0.994977
$665$	0	0
$666$	6.16567	0.238915
$667$	−26.1163	−1.01123
$668$	−7.56319	−0.292629
$669$	20.8053	0.804379
$670$	0	0
$671$	0.182933	0.00706205
$672$	−20.5852	−0.794092
$673$	10.2926	0.396749	0.198375	−	0.980126i	$-0.436434\pi$
$673$	10.2926	0.396749	0.198375	+	0.980126i	$0.436434\pi$
$674$	−7.17692	−0.276445
$675$	0	0
$676$	8.07187	0.310457
$677$	13.6379	0.524147	0.262074	−	0.965048i	$-0.415594\pi$
$677$	13.6379	0.524147	0.262074	+	0.965048i	$0.415594\pi$
$678$	3.60615	0.138493
$679$	14.1223	0.541962
$680$	0	0
$681$	−31.8415	−1.22017
$682$	0.0909630	0.00348315
$683$	−47.5466	−1.81932	−0.909660	−	0.415354i	$-0.863658\pi$
$683$	−47.5466	−1.81932	−0.909660	+	0.415354i	$0.863658\pi$
$684$	4.15121	0.158726
$685$	0	0
$686$	−8.37046	−0.319586
$687$	−35.8152	−1.36643
$688$	6.52681	0.248832
$689$	−14.7955	−0.563666
$690$	0	0
$691$	45.0219	1.71271	0.856356	−	0.516386i	$-0.172723\pi$
$691$	45.0219	1.71271	0.856356	+	0.516386i	$0.172723\pi$
$692$	39.7149	1.50973
$693$	−1.29015	−0.0490089
$694$	−6.59837	−0.250471
$695$	0	0
$696$	13.3868	0.507425
$697$	4.52840	0.171525
$698$	10.0748	0.381335
$699$	−18.0660	−0.683320
$700$	0	0
$701$	−1.11005	−0.0419259	−0.0209630	−	0.999780i	$-0.506673\pi$
$701$	−1.11005	−0.0419259	−0.0209630	+	0.999780i	$0.506673\pi$
$702$	−1.44224	−0.0544339
$703$	5.34711	0.201670
$704$	−1.06788	−0.0402473
$705$	0	0
$706$	5.85713	0.220436
$707$	−11.2132	−0.421714
$708$	14.1222	0.530746
$709$	0.820090	0.0307991	0.0153996	−	0.999881i	$-0.495098\pi$
$709$	0.820090	0.0307991	0.0153996	+	0.999881i	$0.495098\pi$
$710$	0	0
$711$	−41.2700	−1.54775
$712$	6.08061	0.227880
$713$	−6.06799	−0.227248
$714$	−6.72003	−0.251491
$715$	0	0
$716$	−21.1522	−0.790496
$717$	35.6718	1.33219
$718$	−5.95653	−0.222296
$719$	−29.2278	−1.09001	−0.545007	−	0.838431i	$-0.683473\pi$
$719$	−29.2278	−1.09001	−0.545007	+	0.838431i	$0.683473\pi$
$720$	0	0
$721$	1.71407	0.0638355
$722$	7.63175	0.284024
$723$	7.66954	0.285233
$724$	−7.20377	−0.267726
$725$	0	0
$726$	−10.7670	−0.399603
$727$	8.61722	0.319595	0.159797	−	0.987150i	$-0.448916\pi$
$727$	8.61722	0.319595	0.159797	+	0.987150i	$0.448916\pi$
$728$	−9.25361	−0.342962
$729$	−17.3737	−0.643469
$730$	0	0
$731$	7.61174	0.281530
$732$	2.98328	0.110265
$733$	−14.6933	−0.542708	−0.271354	−	0.962480i	$-0.587471\pi$
$733$	−14.6933	−0.542708	−0.271354	+	0.962480i	$0.587471\pi$
$734$	−7.15481	−0.264089
$735$	0	0
$736$	−32.7721	−1.20800
$737$	3.34742	0.123304
$738$	−1.37147	−0.0504844
$739$	−34.7113	−1.27688	−0.638439	−	0.769673i	$-0.720419\pi$
$739$	−34.7113	−1.27688	−0.638439	+	0.769673i	$0.720419\pi$
$740$	0	0
$741$	6.25286	0.229704
$742$	−4.17641	−0.153321
$743$	−45.4630	−1.66788	−0.833938	−	0.551857i	$-0.813919\pi$
$743$	−45.4630	−1.66788	−0.833938	+	0.551857i	$0.813919\pi$
$744$	3.11036	0.114031
$745$	0	0
$746$	−3.10958	−0.113850
$747$	−39.9121	−1.46031
$748$	−1.65795	−0.0606208
$749$	−1.90789	−0.0697127
$750$	0	0
$751$	−20.2715	−0.739718	−0.369859	−	0.929088i	$-0.620594\pi$
$751$	−20.2715	−0.739718	−0.369859	+	0.929088i	$0.620594\pi$
$752$	34.0648	1.24221
$753$	−4.75113	−0.173141
$754$	4.37128	0.159193
$755$	0	0
$756$	4.20864	0.153067
$757$	−32.5233	−1.18208	−0.591040	−	0.806642i	$-0.701283\pi$
$757$	−32.5233	−1.18208	−0.591040	+	0.806642i	$0.701283\pi$
$758$	−11.5595	−0.419861
$759$	−4.51876	−0.164021
$760$	0	0
$761$	−25.5381	−0.925757	−0.462878	−	0.886422i	$-0.653183\pi$
$761$	−25.5381	−0.925757	−0.462878	+	0.886422i	$0.653183\pi$
$762$	14.8840	0.539192
$763$	−11.8800	−0.430085
$764$	−15.5629	−0.563045
$765$	0	0
$766$	4.65613	0.168233
$767$	9.66893	0.349125
$768$	−8.62835	−0.311349
$769$	20.9064	0.753903	0.376951	−	0.926233i	$-0.376972\pi$
$769$	20.9064	0.753903	0.376951	+	0.926233i	$0.376972\pi$
$770$	0	0
$771$	49.8006	1.79353
$772$	−2.16648	−0.0779733
$773$	31.9157	1.14793	0.573965	−	0.818880i	$-0.305405\pi$
$773$	31.9157	1.14793	0.573965	+	0.818880i	$0.305405\pi$
$774$	−2.30528	−0.0828617
$775$	0	0
$776$	−11.5244	−0.413702
$777$	−27.1012	−0.972249
$778$	7.27990	0.260997
$779$	−1.18939	−0.0426143
$780$	0	0
$781$	−1.64195	−0.0587534
$782$	−10.6984	−0.382575
$783$	−4.16853	−0.148971
$784$	−9.29660	−0.332021
$785$	0	0
$786$	1.18122	0.0421328
$787$	12.6006	0.449163	0.224582	−	0.974455i	$-0.427898\pi$
$787$	12.6006	0.449163	0.224582	+	0.974455i	$0.427898\pi$
$788$	1.82360	0.0649631
$789$	−29.0146	−1.03295
$790$	0	0
$791$	−7.20481	−0.256173
$792$	1.05282	0.0374105
$793$	2.04253	0.0725325
$794$	10.3148	0.366059
$795$	0	0
$796$	12.3788	0.438757
$797$	−16.9438	−0.600179	−0.300090	−	0.953911i	$-0.597017\pi$
$797$	−16.9438	−0.600179	−0.300090	+	0.953911i	$0.597017\pi$
$798$	1.76502	0.0624812
$799$	39.7272	1.40545
$800$	0	0
$801$	9.46573	0.334455
$802$	9.75885	0.344597
$803$	−0.220461	−0.00777989
$804$	54.5900	1.92524
$805$	0	0
$806$	1.01565	0.0357746
$807$	−27.6307	−0.972647
$808$	9.15046	0.321912
$809$	3.28050	0.115336	0.0576681	−	0.998336i	$-0.481633\pi$
$809$	3.28050	0.115336	0.0576681	+	0.998336i	$0.481633\pi$
$810$	0	0
$811$	18.5865	0.652662	0.326331	−	0.945256i	$-0.394188\pi$
$811$	18.5865	0.652662	0.326331	+	0.945256i	$0.394188\pi$
$812$	−12.7560	−0.447646
$813$	13.9676	0.489867
$814$	0.646781	0.0226697
$815$	0	0
$816$	−24.1696	−0.846105
$817$	−1.99923	−0.0699443
$818$	−8.44989	−0.295444
$819$	−14.4052	−0.503357
$820$	0	0
$821$	−34.5854	−1.20704	−0.603520	−	0.797348i	$-0.706236\pi$
$821$	−34.5854	−1.20704	−0.603520	+	0.797348i	$0.706236\pi$
$822$	15.8534	0.552952
$823$	32.6063	1.13658	0.568292	−	0.822827i	$-0.307604\pi$
$823$	32.6063	1.13658	0.568292	+	0.822827i	$0.307604\pi$
$824$	−1.39876	−0.0487282
$825$	0	0
$826$	2.72930	0.0949643
$827$	−37.5683	−1.30638	−0.653190	−	0.757194i	$-0.726570\pi$
$827$	−37.5683	−1.30638	−0.653190	+	0.757194i	$0.726570\pi$
$828$	−33.4960	−1.16407
$829$	−47.2387	−1.64067	−0.820334	−	0.571885i	$-0.806212\pi$
$829$	−47.2387	−1.64067	−0.820334	+	0.571885i	$0.806212\pi$
$830$	0	0
$831$	−14.1382	−0.490450
$832$	−11.9234	−0.413370
$833$	−10.8419	−0.375651
$834$	−4.70711	−0.162994
$835$	0	0
$836$	0.435464	0.0150608
$837$	−0.968536	−0.0334775
$838$	−10.3891	−0.358885
$839$	−24.5739	−0.848384	−0.424192	−	0.905572i	$-0.639442\pi$
$839$	−24.5739	−0.848384	−0.424192	+	0.905572i	$0.639442\pi$
$840$	0	0
$841$	−16.3656	−0.564332
$842$	−4.86697	−0.167727
$843$	76.6185	2.63888
$844$	−0.653620	−0.0224985
$845$	0	0
$846$	−12.0318	−0.413661
$847$	21.5117	0.739152
$848$	−15.0211	−0.515827
$849$	0.704622	0.0241826
$850$	0	0
$851$	−43.1457	−1.47902
$852$	−26.7770	−0.917364
$853$	−24.9986	−0.855936	−0.427968	−	0.903794i	$-0.640770\pi$
$853$	−24.9986	−0.855936	−0.427968	+	0.903794i	$0.640770\pi$
$854$	0.576556	0.0197293
$855$	0	0
$856$	1.55693	0.0532146
$857$	15.2814	0.522002	0.261001	−	0.965339i	$-0.415947\pi$
$857$	15.2814	0.522002	0.261001	+	0.965339i	$0.415947\pi$
$858$	0.756339	0.0258210
$859$	38.0633	1.29870	0.649351	−	0.760489i	$-0.275041\pi$
$859$	38.0633	1.29870	0.649351	+	0.760489i	$0.275041\pi$
$860$	0	0
$861$	6.02828	0.205443
$862$	−14.1916	−0.483369
$863$	32.8868	1.11948	0.559741	−	0.828668i	$-0.310901\pi$
$863$	32.8868	1.11948	0.559741	+	0.828668i	$0.310901\pi$
$864$	−5.23089	−0.177959
$865$	0	0
$866$	−3.25528	−0.110619
$867$	11.6811	0.396711
$868$	−2.96378	−0.100597
$869$	−4.32924	−0.146859
$870$	0	0
$871$	37.3756	1.26642
$872$	9.69464	0.328302
$873$	−17.9402	−0.607182
$874$	2.80996	0.0950483
$875$	0	0
$876$	−3.59529	−0.121474
$877$	56.8070	1.91824	0.959118	−	0.283007i	$-0.0913321\pi$
$877$	56.8070	1.91824	0.959118	+	0.283007i	$0.0913321\pi$
$878$	11.1458	0.376153
$879$	−12.0974	−0.408035
$880$	0	0
$881$	−16.2674	−0.548063	−0.274031	−	0.961721i	$-0.588357\pi$
$881$	−16.2674	−0.548063	−0.274031	+	0.961721i	$0.588357\pi$
$882$	3.28358	0.110564
$883$	53.4665	1.79929	0.899645	−	0.436622i	$-0.143825\pi$
$883$	53.4665	1.79929	0.899645	+	0.436622i	$0.143825\pi$
$884$	−18.5118	−0.622620
$885$	0	0
$886$	−15.0159	−0.504470
$887$	−23.9495	−0.804147	−0.402073	−	0.915608i	$-0.631710\pi$
$887$	−23.9495	−0.804147	−0.402073	+	0.915608i	$0.631710\pi$
$888$	22.1158	0.742158
$889$	−29.7372	−0.997353
$890$	0	0
$891$	−2.68804	−0.0900527
$892$	16.1780	0.541680
$893$	−10.4344	−0.349175
$894$	22.5520	0.754252
$895$	0	0
$896$	−20.9209	−0.698919
$897$	−50.4541	−1.68461
$898$	−8.61087	−0.287349
$899$	2.93553	0.0979055
$900$	0	0
$901$	−17.5180	−0.583609
$902$	−0.143868	−0.00479026
$903$	10.1329	0.337201
$904$	5.87946	0.195548
$905$	0	0
$906$	−7.85506	−0.260967
$907$	5.11615	0.169879	0.0849395	−	0.996386i	$-0.472930\pi$
$907$	5.11615	0.169879	0.0849395	+	0.996386i	$0.472930\pi$
$908$	−24.7596	−0.821676
$909$	14.2446	0.472463
$910$	0	0
$911$	−19.9395	−0.660624	−0.330312	−	0.943872i	$-0.607154\pi$
$911$	−19.9395	−0.660624	−0.330312	+	0.943872i	$0.607154\pi$
$912$	6.34818	0.210209
$913$	−4.18680	−0.138563
$914$	13.6135	0.450296
$915$	0	0
$916$	−27.8495	−0.920174
$917$	−2.35999	−0.0779338
$918$	−1.70762	−0.0563598
$919$	45.7518	1.50921	0.754607	−	0.656177i	$-0.227827\pi$
$919$	45.7518	1.50921	0.754607	+	0.656177i	$0.227827\pi$
$920$	0	0
$921$	−17.5589	−0.578584
$922$	−1.13835	−0.0374896
$923$	−18.3331	−0.603442
$924$	−2.20710	−0.0726081
$925$	0	0
$926$	1.74412	0.0573153
$927$	−2.17747	−0.0715174
$928$	15.8543	0.520443
$929$	16.6626	0.546681	0.273340	−	0.961917i	$-0.411871\pi$
$929$	16.6626	0.546681	0.273340	+	0.961917i	$0.411871\pi$
$930$	0	0
$931$	2.84765	0.0933280
$932$	−14.0480	−0.460157
$933$	−7.42994	−0.243245
$934$	−0.893318	−0.0292302
$935$	0	0
$936$	11.7553	0.384234
$937$	−28.4480	−0.929355	−0.464678	−	0.885480i	$-0.653830\pi$
$937$	−28.4480	−0.929355	−0.464678	+	0.885480i	$0.653830\pi$
$938$	10.5502	0.344476
$939$	−36.0429	−1.17621
$940$	0	0
$941$	43.6163	1.42185	0.710926	−	0.703267i	$-0.248276\pi$
$941$	43.6163	1.42185	0.710926	+	0.703267i	$0.248276\pi$
$942$	−22.2137	−0.723760
$943$	9.59716	0.312526
$944$	9.81632	0.319494
$945$	0	0
$946$	−0.241825	−0.00786242
$947$	−21.4876	−0.698253	−0.349127	−	0.937076i	$-0.613522\pi$
$947$	−21.4876	−0.698253	−0.349127	+	0.937076i	$0.613522\pi$
$948$	−70.6015	−2.29303
$949$	−2.46155	−0.0799053
$950$	0	0
$951$	−47.9011	−1.55330
$952$	−10.9563	−0.355096
$953$	−1.44451	−0.0467924	−0.0233962	−	0.999726i	$-0.507448\pi$
$953$	−1.44451	−0.0467924	−0.0233962	+	0.999726i	$0.507448\pi$
$954$	5.30549	0.171772
$955$	0	0
$956$	27.7380	0.897112
$957$	2.18606	0.0706652
$958$	−14.8704	−0.480442
$959$	−31.6740	−1.02281
$960$	0	0
$961$	−30.3179	−0.977998
$962$	7.22162	0.232834
$963$	2.42368	0.0781019
$964$	5.96376	0.192080
$965$	0	0
$966$	−14.2419	−0.458227
$967$	−51.3114	−1.65006	−0.825031	−	0.565087i	$-0.808843\pi$
$967$	−51.3114	−1.65006	−0.825031	+	0.565087i	$0.808843\pi$
$968$	−17.5546	−0.564225
$969$	7.40341	0.237832
$970$	0	0
$971$	−34.4880	−1.10677	−0.553387	−	0.832924i	$-0.686665\pi$
$971$	−34.4880	−1.10677	−0.553387	+	0.832924i	$0.686665\pi$
$972$	−37.4208	−1.20027
$973$	9.40445	0.301493
$974$	3.43299	0.110000
$975$	0	0
$976$	2.07367	0.0663766
$977$	−5.16874	−0.165363	−0.0826814	−	0.996576i	$-0.526348\pi$
$977$	−5.16874	−0.165363	−0.0826814	+	0.996576i	$0.526348\pi$
$978$	20.3150	0.649603
$979$	0.992960	0.0317351
$980$	0	0
$981$	15.0917	0.481842
$982$	−1.39066	−0.0443776
$983$	37.3896	1.19254	0.596271	−	0.802783i	$-0.296648\pi$
$983$	37.3896	1.19254	0.596271	+	0.802783i	$0.296648\pi$
$984$	−4.91935	−0.156823
$985$	0	0
$986$	5.17562	0.164825
$987$	52.8856	1.68337
$988$	4.86216	0.154686
$989$	16.1318	0.512960
$990$	0	0
$991$	14.6919	0.466705	0.233352	−	0.972392i	$-0.425030\pi$
$991$	14.6919	0.466705	0.233352	+	0.972392i	$0.425030\pi$
$992$	3.68367	0.116957
$993$	−58.0346	−1.84167
$994$	−5.17498	−0.164140
$995$	0	0
$996$	−68.2786	−2.16349
$997$	55.0907	1.74474	0.872371	−	0.488845i	$-0.162582\pi$
$997$	55.0907	1.74474	0.872371	+	0.488845i	$0.162582\pi$
$998$	2.46397	0.0779957
$999$	−6.88666	−0.217884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.s.1.20		49
5.2	odd	4		985.2.b.a.789.42	✓	98
5.3	odd	4		985.2.b.a.789.57	yes	98
5.4	even	2		4925.2.a.r.1.30		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.42	✓	98	5.2	odd	4
985.2.b.a.789.57	yes	98	5.3	odd	4
4925.2.a.r.1.30		49	5.4	even	2
4925.2.a.s.1.20		49	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.420000 q^{2} -2.34519 q^{3} -1.82360 q^{4} +0.984980 q^{6} -1.96792 q^{7} +1.60591 q^{8} +2.49993 q^{9} +0.262244 q^{11} +4.27670 q^{12} +2.92808 q^{13} +0.826524 q^{14} +2.97272 q^{16} +3.46686 q^{17} -1.04997 q^{18} -0.910577 q^{19} +4.61515 q^{21} -0.110142 q^{22} +7.34742 q^{23} -3.76617 q^{24} -1.22979 q^{26} +1.17275 q^{27} +3.58869 q^{28} -3.55449 q^{29} -0.825867 q^{31} -4.46036 q^{32} -0.615014 q^{33} -1.45608 q^{34} -4.55888 q^{36} -5.87223 q^{37} +0.382442 q^{38} -6.86692 q^{39} +1.30619 q^{41} -1.93836 q^{42} +2.19557 q^{43} -0.478229 q^{44} -3.08591 q^{46} +11.4591 q^{47} -6.97160 q^{48} -3.12730 q^{49} -8.13047 q^{51} -5.33965 q^{52} -5.05298 q^{53} -0.492555 q^{54} -3.16030 q^{56} +2.13548 q^{57} +1.49288 q^{58} +3.30214 q^{59} +0.697567 q^{61} +0.346864 q^{62} -4.91966 q^{63} -4.07209 q^{64} +0.258305 q^{66} +12.7645 q^{67} -6.32217 q^{68} -17.2311 q^{69} -6.26113 q^{71} +4.01467 q^{72} -0.840670 q^{73} +2.46633 q^{74} +1.66053 q^{76} -0.516075 q^{77} +2.88410 q^{78} -16.5084 q^{79} -10.2501 q^{81} -0.548601 q^{82} -15.9653 q^{83} -8.41618 q^{84} -0.922138 q^{86} +8.33596 q^{87} +0.421141 q^{88} +3.78639 q^{89} -5.76222 q^{91} -13.3988 q^{92} +1.93682 q^{93} -4.81283 q^{94} +10.4604 q^{96} -7.17625 q^{97} +1.31347 q^{98} +0.655593 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more