LMFDB - Embedded newform 4925.2.a.s.1.16

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			1.16
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)

$f(q)$	$=$	\(q-1.18293 q^{2} +0.672647 q^{3} -0.600670 q^{4} -0.795696 q^{6} -3.32812 q^{7} +3.07642 q^{8} -2.54755 q^{9} -4.01450 q^{11} -0.404039 q^{12} +0.365360 q^{13} +3.93695 q^{14} -2.43785 q^{16} -5.21648 q^{17} +3.01358 q^{18} +2.16257 q^{19} -2.23865 q^{21} +4.74888 q^{22} -7.86503 q^{23} +2.06934 q^{24} -0.432196 q^{26} -3.73154 q^{27} +1.99911 q^{28} -7.24543 q^{29} -3.93707 q^{31} -3.26902 q^{32} -2.70034 q^{33} +6.17075 q^{34} +1.53023 q^{36} -3.21046 q^{37} -2.55817 q^{38} +0.245758 q^{39} +4.57854 q^{41} +2.64818 q^{42} +8.31813 q^{43} +2.41139 q^{44} +9.30380 q^{46} +3.03063 q^{47} -1.63982 q^{48} +4.07641 q^{49} -3.50885 q^{51} -0.219461 q^{52} -4.72562 q^{53} +4.41416 q^{54} -10.2387 q^{56} +1.45464 q^{57} +8.57085 q^{58} -5.78433 q^{59} -6.30248 q^{61} +4.65729 q^{62} +8.47855 q^{63} +8.74274 q^{64} +3.19432 q^{66} +11.0069 q^{67} +3.13339 q^{68} -5.29039 q^{69} -9.14673 q^{71} -7.83731 q^{72} +1.79241 q^{73} +3.79776 q^{74} -1.29899 q^{76} +13.3607 q^{77} -0.290716 q^{78} -14.4163 q^{79} +5.13263 q^{81} -5.41610 q^{82} -2.56116 q^{83} +1.34469 q^{84} -9.83978 q^{86} -4.87362 q^{87} -12.3503 q^{88} -13.7724 q^{89} -1.21596 q^{91} +4.72429 q^{92} -2.64826 q^{93} -3.58503 q^{94} -2.19890 q^{96} +4.69428 q^{97} -4.82212 q^{98} +10.2271 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.18293	−0.836460	−0.418230	−	0.908341i	$-0.637349\pi$
$2$	−1.18293	−0.836460	−0.418230	+	0.908341i	$0.637349\pi$
$3$	0.672647	0.388353	0.194177	−	0.980967i	$-0.437797\pi$
$3$	0.672647	0.388353	0.194177	+	0.980967i	$0.437797\pi$
$4$	−0.600670	−0.300335
$5$	0	0
$5$	0	0
$6$	−0.795696	−0.324842
$7$	−3.32812	−1.25791	−0.628956	−	0.777441i	$-0.716518\pi$
$7$	−3.32812	−1.25791	−0.628956	+	0.777441i	$0.716518\pi$
$8$	3.07642	1.08768
$9$	−2.54755	−0.849182
$10$	0	0
$11$	−4.01450	−1.21042	−0.605208	−	0.796067i	$-0.706910\pi$
$11$	−4.01450	−1.21042	−0.605208	+	0.796067i	$0.706910\pi$
$12$	−0.404039	−0.116636
$13$	0.365360	0.101333	0.0506663	−	0.998716i	$-0.483865\pi$
$13$	0.365360	0.101333	0.0506663	+	0.998716i	$0.483865\pi$
$14$	3.93695	1.05219
$15$	0	0
$16$	−2.43785	−0.609464
$17$	−5.21648	−1.26518	−0.632591	−	0.774486i	$-0.718009\pi$
$17$	−5.21648	−1.26518	−0.632591	+	0.774486i	$0.718009\pi$
$18$	3.01358	0.710306
$19$	2.16257	0.496127	0.248063	−	0.968744i	$-0.420206\pi$
$19$	2.16257	0.496127	0.248063	+	0.968744i	$0.420206\pi$
$20$	0	0
$21$	−2.23865	−0.488514
$22$	4.74888	1.01246
$23$	−7.86503	−1.63997	−0.819986	−	0.572384i	$-0.806019\pi$
$23$	−7.86503	−1.63997	−0.819986	+	0.572384i	$0.806019\pi$
$24$	2.06934	0.422403
$25$	0	0
$26$	−0.432196	−0.0847607
$27$	−3.73154	−0.718136
$28$	1.99911	0.377795
$29$	−7.24543	−1.34544	−0.672721	−	0.739896i	$-0.734875\pi$
$29$	−7.24543	−1.34544	−0.672721	+	0.739896i	$0.734875\pi$
$30$	0	0
$31$	−3.93707	−0.707119	−0.353560	−	0.935412i	$-0.615029\pi$
$31$	−3.93707	−0.707119	−0.353560	+	0.935412i	$0.615029\pi$
$32$	−3.26902	−0.577886
$33$	−2.70034	−0.470069
$34$	6.17075	1.05827
$35$	0	0
$36$	1.53023	0.255039
$37$	−3.21046	−0.527797	−0.263898	−	0.964551i	$-0.585008\pi$
$37$	−3.21046	−0.527797	−0.263898	+	0.964551i	$0.585008\pi$
$38$	−2.55817	−0.414990
$39$	0.245758	0.0393529
$40$	0	0
$41$	4.57854	0.715048	0.357524	−	0.933904i	$-0.383621\pi$
$41$	4.57854	0.715048	0.357524	+	0.933904i	$0.383621\pi$
$42$	2.64818	0.408623
$43$	8.31813	1.26850	0.634251	−	0.773127i	$-0.281308\pi$
$43$	8.31813	1.26850	0.634251	+	0.773127i	$0.281308\pi$
$44$	2.41139	0.363530
$45$	0	0
$46$	9.30380	1.37177
$47$	3.03063	0.442063	0.221031	−	0.975267i	$-0.429058\pi$
$47$	3.03063	0.442063	0.221031	+	0.975267i	$0.429058\pi$
$48$	−1.63982	−0.236687
$49$	4.07641	0.582345
$50$	0	0
$51$	−3.50885	−0.491338
$52$	−0.219461	−0.0304338
$53$	−4.72562	−0.649113	−0.324557	−	0.945866i	$-0.605215\pi$
$53$	−4.72562	−0.649113	−0.324557	+	0.945866i	$0.605215\pi$
$54$	4.41416	0.600691
$55$	0	0
$56$	−10.2387	−1.36820
$57$	1.45464	0.192672
$58$	8.57085	1.12541
$59$	−5.78433	−0.753055	−0.376528	−	0.926405i	$-0.622882\pi$
$59$	−5.78433	−0.753055	−0.376528	+	0.926405i	$0.622882\pi$
$60$	0	0
$61$	−6.30248	−0.806949	−0.403475	−	0.914991i	$-0.632198\pi$
$61$	−6.30248	−0.806949	−0.403475	+	0.914991i	$0.632198\pi$
$62$	4.65729	0.591477
$63$	8.47855	1.06820
$64$	8.74274	1.09284
$65$	0	0
$66$	3.19432	0.393194
$67$	11.0069	1.34470	0.672351	−	0.740232i	$-0.265285\pi$
$67$	11.0069	1.34470	0.672351	+	0.740232i	$0.265285\pi$
$68$	3.13339	0.379979
$69$	−5.29039	−0.636888
$70$	0	0
$71$	−9.14673	−1.08552	−0.542758	−	0.839889i	$-0.682620\pi$
$71$	−9.14673	−1.08552	−0.542758	+	0.839889i	$0.682620\pi$
$72$	−7.83731	−0.923636
$73$	1.79241	0.209786	0.104893	−	0.994484i	$-0.466550\pi$
$73$	1.79241	0.209786	0.104893	+	0.994484i	$0.466550\pi$
$74$	3.79776	0.441481
$75$	0	0
$76$	−1.29899	−0.149004
$77$	13.3607	1.52260
$78$	−0.290716	−0.0329171
$79$	−14.4163	−1.62196	−0.810982	−	0.585071i	$-0.801067\pi$
$79$	−14.4163	−1.62196	−0.810982	+	0.585071i	$0.801067\pi$
$80$	0	0
$81$	5.13263	0.570292
$82$	−5.41610	−0.598108
$83$	−2.56116	−0.281124	−0.140562	−	0.990072i	$-0.544891\pi$
$83$	−2.56116	−0.281124	−0.140562	+	0.990072i	$0.544891\pi$
$84$	1.34469	0.146718
$85$	0	0
$86$	−9.83978	−1.06105
$87$	−4.87362	−0.522507
$88$	−12.3503	−1.31654
$89$	−13.7724	−1.45987	−0.729937	−	0.683514i	$-0.760451\pi$
$89$	−13.7724	−1.45987	−0.729937	+	0.683514i	$0.760451\pi$
$90$	0	0
$91$	−1.21596	−0.127468
$92$	4.72429	0.492541
$93$	−2.64826	−0.274612
$94$	−3.58503	−0.369768
$95$	0	0
$96$	−2.19890	−0.224424
$97$	4.69428	0.476632	0.238316	−	0.971188i	$-0.423405\pi$
$97$	4.69428	0.476632	0.238316	+	0.971188i	$0.423405\pi$
$98$	−4.82212	−0.487108
$99$	10.2271	1.02786
$100$	0	0
$101$	−10.7288	−1.06756	−0.533779	−	0.845624i	$-0.679229\pi$
$101$	−10.7288	−1.06756	−0.533779	+	0.845624i	$0.679229\pi$
$102$	4.15074	0.410984
$103$	−5.27161	−0.519427	−0.259713	−	0.965686i	$-0.583628\pi$
$103$	−5.27161	−0.519427	−0.259713	+	0.965686i	$0.583628\pi$
$104$	1.12400	0.110217
$105$	0	0
$106$	5.59009	0.542957
$107$	−4.45979	−0.431144	−0.215572	−	0.976488i	$-0.569162\pi$
$107$	−4.45979	−0.431144	−0.215572	+	0.976488i	$0.569162\pi$
$108$	2.24143	0.215681
$109$	10.2412	0.980929	0.490465	−	0.871461i	$-0.336827\pi$
$109$	10.2412	0.980929	0.490465	+	0.871461i	$0.336827\pi$
$110$	0	0
$111$	−2.15951	−0.204971
$112$	8.11348	0.766652
$113$	10.5620	0.993593	0.496796	−	0.867867i	$-0.334510\pi$
$113$	10.5620	0.993593	0.496796	+	0.867867i	$0.334510\pi$
$114$	−1.72075	−0.161163
$115$	0	0
$116$	4.35211	0.404084
$117$	−0.930772	−0.0860499
$118$	6.84247	0.629900
$119$	17.3611	1.59149
$120$	0	0
$121$	5.11618	0.465107
$122$	7.45541	0.674981
$123$	3.07974	0.277691
$124$	2.36488	0.212373
$125$	0	0
$126$	−10.0296	−0.893504
$127$	13.5115	1.19895	0.599477	−	0.800392i	$-0.295375\pi$
$127$	13.5115	1.19895	0.599477	+	0.800392i	$0.295375\pi$
$128$	−3.80403	−0.336232
$129$	5.59517	0.492627
$130$	0	0
$131$	11.8689	1.03699	0.518495	−	0.855081i	$-0.326492\pi$
$131$	11.8689	1.03699	0.518495	+	0.855081i	$0.326492\pi$
$132$	1.62201	0.141178
$133$	−7.19729	−0.624084
$134$	−13.0204	−1.12479
$135$	0	0
$136$	−16.0481	−1.37611
$137$	18.3633	1.56888	0.784440	−	0.620205i	$-0.212950\pi$
$137$	18.3633	1.56888	0.784440	+	0.620205i	$0.212950\pi$
$138$	6.25818	0.532731
$139$	11.5674	0.981132	0.490566	−	0.871404i	$-0.336790\pi$
$139$	11.5674	0.981132	0.490566	+	0.871404i	$0.336790\pi$
$140$	0	0
$141$	2.03854	0.171676
$142$	10.8200	0.907991
$143$	−1.46674	−0.122655
$144$	6.21055	0.517546
$145$	0	0
$146$	−2.12030	−0.175477
$147$	2.74199	0.226155
$148$	1.92843	0.158516
$149$	−16.3235	−1.33727	−0.668637	−	0.743589i	$-0.733122\pi$
$149$	−16.3235	−1.33727	−0.668637	+	0.743589i	$0.733122\pi$
$150$	0	0
$151$	−15.6126	−1.27054	−0.635269	−	0.772291i	$-0.719111\pi$
$151$	−15.6126	−1.27054	−0.635269	+	0.772291i	$0.719111\pi$
$152$	6.65296	0.539626
$153$	13.2892	1.07437
$154$	−15.8049	−1.27359
$155$	0	0
$156$	−0.147620	−0.0118190
$157$	−14.6540	−1.16951	−0.584757	−	0.811208i	$-0.698810\pi$
$157$	−14.6540	−1.16951	−0.584757	+	0.811208i	$0.698810\pi$
$158$	17.0535	1.35671
$159$	−3.17867	−0.252085
$160$	0	0
$161$	26.1758	2.06294
$162$	−6.07155	−0.477026
$163$	−3.42042	−0.267908	−0.133954	−	0.990988i	$-0.542767\pi$
$163$	−3.42042	−0.267908	−0.133954	+	0.990988i	$0.542767\pi$
$164$	−2.75019	−0.214754
$165$	0	0
$166$	3.02968	0.235149
$167$	−0.265125	−0.0205160	−0.0102580	−	0.999947i	$-0.503265\pi$
$167$	−0.265125	−0.0205160	−0.0102580	+	0.999947i	$0.503265\pi$
$168$	−6.88703	−0.531346
$169$	−12.8665	−0.989732
$170$	0	0
$171$	−5.50924	−0.421302
$172$	−4.99645	−0.380976
$173$	11.3576	0.863500	0.431750	−	0.901993i	$-0.357896\pi$
$173$	11.3576	0.863500	0.431750	+	0.901993i	$0.357896\pi$
$174$	5.76516	0.437056
$175$	0	0
$176$	9.78676	0.737705
$177$	−3.89081	−0.292451
$178$	16.2918	1.22113
$179$	−10.6177	−0.793606	−0.396803	−	0.917904i	$-0.629880\pi$
$179$	−10.6177	−0.793606	−0.396803	+	0.917904i	$0.629880\pi$
$180$	0	0
$181$	18.6295	1.38472	0.692360	−	0.721552i	$-0.256571\pi$
$181$	18.6295	1.38472	0.692360	+	0.721552i	$0.256571\pi$
$182$	1.43840	0.106622
$183$	−4.23934	−0.313381
$184$	−24.1961	−1.78376
$185$	0	0
$186$	3.13272	0.229702
$187$	20.9415	1.53140
$188$	−1.82041	−0.132767
$189$	12.4190	0.903352
$190$	0	0
$191$	5.78207	0.418376	0.209188	−	0.977875i	$-0.432918\pi$
$191$	5.78207	0.418376	0.209188	+	0.977875i	$0.432918\pi$
$192$	5.88078	0.424409
$193$	0.502765	0.0361898	0.0180949	−	0.999836i	$-0.494240\pi$
$193$	0.502765	0.0361898	0.0180949	+	0.999836i	$0.494240\pi$
$194$	−5.55302	−0.398684
$195$	0	0
$196$	−2.44858	−0.174899
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−12.0980	−0.859766
$199$	14.3755	1.01905	0.509527	−	0.860455i	$-0.329820\pi$
$199$	14.3755	1.01905	0.509527	+	0.860455i	$0.329820\pi$
$200$	0	0
$201$	7.40374	0.522219
$202$	12.6915	0.892970
$203$	24.1137	1.69245
$204$	2.10766	0.147566
$205$	0	0
$206$	6.23596	0.434480
$207$	20.0365	1.39263
$208$	−0.890695	−0.0617586
$209$	−8.68162	−0.600520
$210$	0	0
$211$	−5.40187	−0.371880	−0.185940	−	0.982561i	$-0.559533\pi$
$211$	−5.40187	−0.371880	−0.185940	+	0.982561i	$0.559533\pi$
$212$	2.83854	0.194952
$213$	−6.15252	−0.421564
$214$	5.27563	0.360634
$215$	0	0
$216$	−11.4798	−0.781100
$217$	13.1031	0.889494
$218$	−12.1147	−0.820508
$219$	1.20566	0.0814709
$220$	0	0
$221$	−1.90589	−0.128204
$222$	2.55455	0.171450
$223$	−16.1807	−1.08354	−0.541771	−	0.840526i	$-0.682246\pi$
$223$	−16.1807	−1.08354	−0.541771	+	0.840526i	$0.682246\pi$
$224$	10.8797	0.726930
$225$	0	0
$226$	−12.4942	−0.831101
$227$	7.37366	0.489407	0.244704	−	0.969598i	$-0.421309\pi$
$227$	7.37366	0.489407	0.244704	+	0.969598i	$0.421309\pi$
$228$	−0.873762	−0.0578663
$229$	−19.8853	−1.31405	−0.657027	−	0.753867i	$-0.728186\pi$
$229$	−19.8853	−1.31405	−0.657027	+	0.753867i	$0.728186\pi$
$230$	0	0
$231$	8.98707	0.591306
$232$	−22.2900	−1.46341
$233$	−11.1333	−0.729369	−0.364684	−	0.931131i	$-0.618823\pi$
$233$	−11.1333	−0.729369	−0.364684	+	0.931131i	$0.618823\pi$
$234$	1.10104	0.0719772
$235$	0	0
$236$	3.47447	0.226169
$237$	−9.69710	−0.629895
$238$	−20.5370	−1.33122
$239$	−5.92300	−0.383127	−0.191563	−	0.981480i	$-0.561356\pi$
$239$	−5.92300	−0.383127	−0.191563	+	0.981480i	$0.561356\pi$
$240$	0	0
$241$	11.3665	0.732183	0.366091	−	0.930579i	$-0.380696\pi$
$241$	11.3665	0.732183	0.366091	+	0.930579i	$0.380696\pi$
$242$	−6.05209	−0.389043
$243$	14.6471	0.939610
$244$	3.78571	0.242355
$245$	0	0
$246$	−3.64313	−0.232277
$247$	0.790116	0.0502739
$248$	−12.1121	−0.769118
$249$	−1.72276	−0.109175
$250$	0	0
$251$	27.7984	1.75462	0.877311	−	0.479923i	$-0.159335\pi$
$251$	27.7984	1.75462	0.877311	+	0.479923i	$0.159335\pi$
$252$	−5.09281	−0.320817
$253$	31.5741	1.98505
$254$	−15.9832	−1.00288
$255$	0	0
$256$	−12.9856	−0.811597
$257$	12.6874	0.791418	0.395709	−	0.918376i	$-0.370499\pi$
$257$	12.6874	0.791418	0.395709	+	0.918376i	$0.370499\pi$
$258$	−6.61870	−0.412063
$259$	10.6848	0.663922
$260$	0	0
$261$	18.4581	1.14253
$262$	−14.0401	−0.867401
$263$	−29.2878	−1.80596	−0.902982	−	0.429678i	$-0.858627\pi$
$263$	−29.2878	−1.80596	−0.902982	+	0.429678i	$0.858627\pi$
$264$	−8.30737	−0.511283
$265$	0	0
$266$	8.51391	0.522021
$267$	−9.26398	−0.566947
$268$	−6.61149	−0.403861
$269$	−6.77642	−0.413166	−0.206583	−	0.978429i	$-0.566234\pi$
$269$	−6.77642	−0.413166	−0.206583	+	0.978429i	$0.566234\pi$
$270$	0	0
$271$	−24.4544	−1.48550	−0.742748	−	0.669571i	$-0.766478\pi$
$271$	−24.4544	−1.48550	−0.742748	+	0.669571i	$0.766478\pi$
$272$	12.7170	0.771083
$273$	−0.817915	−0.0495025
$274$	−21.7225	−1.31230
$275$	0	0
$276$	3.17778	0.191280
$277$	−1.64895	−0.0990758	−0.0495379	−	0.998772i	$-0.515775\pi$
$277$	−1.64895	−0.0990758	−0.0495379	+	0.998772i	$0.515775\pi$
$278$	−13.6834	−0.820677
$279$	10.0299	0.600473
$280$	0	0
$281$	−15.2800	−0.911530	−0.455765	−	0.890100i	$-0.650634\pi$
$281$	−15.2800	−0.911530	−0.455765	+	0.890100i	$0.650634\pi$
$282$	−2.41146	−0.143600
$283$	14.2874	0.849300	0.424650	−	0.905358i	$-0.360397\pi$
$283$	14.2874	0.849300	0.424650	+	0.905358i	$0.360397\pi$
$284$	5.49417	0.326019
$285$	0	0
$286$	1.73505	0.102596
$287$	−15.2379	−0.899467
$288$	8.32797	0.490730
$289$	10.2117	0.600688
$290$	0	0
$291$	3.15760	0.185102
$292$	−1.07665	−0.0630060
$293$	−9.81819	−0.573585	−0.286792	−	0.957993i	$-0.592589\pi$
$293$	−9.81819	−0.573585	−0.286792	+	0.957993i	$0.592589\pi$
$294$	−3.24359	−0.189170
$295$	0	0
$296$	−9.87672	−0.574073
$297$	14.9803	0.869243
$298$	19.3096	1.11858
$299$	−2.87357	−0.166183
$300$	0	0
$301$	−27.6838	−1.59567
$302$	18.4687	1.06275
$303$	−7.21672	−0.414590
$304$	−5.27202	−0.302371
$305$	0	0
$306$	−15.7203	−0.898668
$307$	18.4502	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$307$	18.4502	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$308$	−8.02540	−0.457290
$309$	−3.54593	−0.201721
$310$	0	0
$311$	34.6643	1.96563	0.982815	−	0.184593i	$-0.0590967\pi$
$311$	34.6643	1.96563	0.982815	+	0.184593i	$0.0590967\pi$
$312$	0.756056	0.0428032
$313$	−12.3956	−0.700641	−0.350320	−	0.936630i	$-0.613927\pi$
$313$	−12.3956	−0.700641	−0.350320	+	0.936630i	$0.613927\pi$
$314$	17.3347	0.978252
$315$	0	0
$316$	8.65946	0.487133
$317$	22.1635	1.24482	0.622412	−	0.782690i	$-0.286153\pi$
$317$	22.1635	1.24482	0.622412	+	0.782690i	$0.286153\pi$
$318$	3.76016	0.210859
$319$	29.0867	1.62854
$320$	0	0
$321$	−2.99986	−0.167436
$322$	−30.9642	−1.72557
$323$	−11.2810	−0.627691
$324$	−3.08302	−0.171279
$325$	0	0
$326$	4.04613	0.224095
$327$	6.88872	0.380947
$328$	14.0855	0.777741
$329$	−10.0863	−0.556076
$330$	0	0
$331$	−27.4133	−1.50677	−0.753386	−	0.657578i	$-0.771581\pi$
$331$	−27.4133	−1.50677	−0.753386	+	0.657578i	$0.771581\pi$
$332$	1.53841	0.0844314
$333$	8.17880	0.448195
$334$	0.313624	0.0171608
$335$	0	0
$336$	5.45751	0.297732
$337$	6.31336	0.343910	0.171955	−	0.985105i	$-0.444992\pi$
$337$	6.31336	0.343910	0.171955	+	0.985105i	$0.444992\pi$
$338$	15.2202	0.827871
$339$	7.10453	0.385865
$340$	0	0
$341$	15.8054	0.855908
$342$	6.51706	0.352402
$343$	9.73006	0.525374
$344$	25.5900	1.37972
$345$	0	0
$346$	−13.4352	−0.722283
$347$	14.9294	0.801451	0.400726	−	0.916198i	$-0.368758\pi$
$347$	14.9294	0.801451	0.400726	+	0.916198i	$0.368758\pi$
$348$	2.92744	0.156927
$349$	21.8095	1.16743	0.583717	−	0.811957i	$-0.301598\pi$
$349$	21.8095	1.16743	0.583717	+	0.811957i	$0.301598\pi$
$350$	0	0
$351$	−1.36336	−0.0727706
$352$	13.1235	0.699483
$353$	−9.32850	−0.496506	−0.248253	−	0.968695i	$-0.579856\pi$
$353$	−9.32850	−0.496506	−0.248253	+	0.968695i	$0.579856\pi$
$354$	4.60257	0.244624
$355$	0	0
$356$	8.27268	0.438451
$357$	11.6779	0.618060
$358$	12.5601	0.663819
$359$	14.1641	0.747554	0.373777	−	0.927519i	$-0.378063\pi$
$359$	14.1641	0.747554	0.373777	+	0.927519i	$0.378063\pi$
$360$	0	0
$361$	−14.3233	−0.753858
$362$	−22.0374	−1.15826
$363$	3.44138	0.180626
$364$	0.730393	0.0382830
$365$	0	0
$366$	5.01486	0.262131
$367$	−7.47147	−0.390008	−0.195004	−	0.980802i	$-0.562472\pi$
$367$	−7.47147	−0.390008	−0.195004	+	0.980802i	$0.562472\pi$
$368$	19.1738	0.999503
$369$	−11.6640	−0.607205
$370$	0	0
$371$	15.7274	0.816528
$372$	1.59073	0.0824756
$373$	28.3484	1.46782	0.733912	−	0.679245i	$-0.237693\pi$
$373$	28.3484	1.46782	0.733912	+	0.679245i	$0.237693\pi$
$374$	−24.7724	−1.28095
$375$	0	0
$376$	9.32348	0.480822
$377$	−2.64719	−0.136337
$378$	−14.6909	−0.755617
$379$	−24.3013	−1.24827	−0.624136	−	0.781316i	$-0.714549\pi$
$379$	−24.3013	−1.24827	−0.624136	+	0.781316i	$0.714549\pi$
$380$	0	0
$381$	9.08849	0.465617
$382$	−6.83981	−0.349955
$383$	0.942323	0.0481505	0.0240752	−	0.999710i	$-0.492336\pi$
$383$	0.942323	0.0481505	0.0240752	+	0.999710i	$0.492336\pi$
$384$	−2.55877	−0.130577
$385$	0	0
$386$	−0.594738	−0.0302713
$387$	−21.1908	−1.07719
$388$	−2.81972	−0.143149
$389$	4.36844	0.221489	0.110744	−	0.993849i	$-0.464677\pi$
$389$	4.36844	0.221489	0.110744	+	0.993849i	$0.464677\pi$
$390$	0	0
$391$	41.0278	2.07486
$392$	12.5407	0.633403
$393$	7.98358	0.402718
$394$	1.18293	0.0595953
$395$	0	0
$396$	−6.14312	−0.308703
$397$	−17.5141	−0.879009	−0.439504	−	0.898240i	$-0.644846\pi$
$397$	−17.5141	−0.879009	−0.439504	+	0.898240i	$0.644846\pi$
$398$	−17.0053	−0.852398
$399$	−4.84124	−0.242365
$400$	0	0
$401$	−35.5964	−1.77760	−0.888800	−	0.458294i	$-0.848461\pi$
$401$	−35.5964	−1.77760	−0.888800	+	0.458294i	$0.848461\pi$
$402$	−8.75812	−0.436815
$403$	−1.43845	−0.0716543
$404$	6.44449	0.320625
$405$	0	0
$406$	−28.5249	−1.41567
$407$	12.8884	0.638853
$408$	−10.7947	−0.534417
$409$	20.2533	1.00146	0.500730	−	0.865603i	$-0.333065\pi$
$409$	20.2533	1.00146	0.500730	+	0.865603i	$0.333065\pi$
$410$	0	0
$411$	12.3520	0.609279
$412$	3.16650	0.156002
$413$	19.2510	0.947278
$414$	−23.7019	−1.16488
$415$	0	0
$416$	−1.19437	−0.0585587
$417$	7.78076	0.381025
$418$	10.2698	0.502311
$419$	−12.5680	−0.613985	−0.306992	−	0.951712i	$-0.599323\pi$
$419$	−12.5680	−0.613985	−0.306992	+	0.951712i	$0.599323\pi$
$420$	0	0
$421$	−24.6205	−1.19993	−0.599964	−	0.800027i	$-0.704818\pi$
$421$	−24.6205	−1.19993	−0.599964	+	0.800027i	$0.704818\pi$
$422$	6.39005	0.311063
$423$	−7.72067	−0.375392
$424$	−14.5380	−0.706026
$425$	0	0
$426$	7.27802	0.352621
$427$	20.9754	1.01507
$428$	2.67886	0.129488
$429$	−0.986596	−0.0476333
$430$	0	0
$431$	25.8545	1.24537	0.622685	−	0.782473i	$-0.286042\pi$
$431$	25.8545	1.24537	0.622685	+	0.782473i	$0.286042\pi$
$432$	9.09696	0.437678
$433$	−14.5312	−0.698326	−0.349163	−	0.937062i	$-0.613534\pi$
$433$	−14.5312	−0.698326	−0.349163	+	0.937062i	$0.613534\pi$
$434$	−15.5000	−0.744026
$435$	0	0
$436$	−6.15159	−0.294608
$437$	−17.0087	−0.813634
$438$	−1.42621	−0.0681471
$439$	14.7262	0.702843	0.351421	−	0.936217i	$-0.385698\pi$
$439$	14.7262	0.702843	0.351421	+	0.936217i	$0.385698\pi$
$440$	0	0
$441$	−10.3848	−0.494516
$442$	2.25455	0.107238
$443$	16.8930	0.802611	0.401306	−	0.915944i	$-0.368557\pi$
$443$	16.8930	0.802611	0.401306	+	0.915944i	$0.368557\pi$
$444$	1.29715	0.0615601
$445$	0	0
$446$	19.1407	0.906340
$447$	−10.9800	−0.519335
$448$	−29.0969	−1.37470
$449$	40.2353	1.89882	0.949412	−	0.314034i	$-0.101681\pi$
$449$	40.2353	1.89882	0.949412	+	0.314034i	$0.101681\pi$
$450$	0	0
$451$	−18.3805	−0.865505
$452$	−6.34430	−0.298411
$453$	−10.5018	−0.493417
$454$	−8.72255	−0.409369
$455$	0	0
$456$	4.47509	0.209566
$457$	23.9328	1.11953	0.559764	−	0.828652i	$-0.310892\pi$
$457$	23.9328	1.11953	0.559764	+	0.828652i	$0.310892\pi$
$458$	23.5229	1.09915
$459$	19.4655	0.908573
$460$	0	0
$461$	−36.4943	−1.69971	−0.849855	−	0.527017i	$-0.823310\pi$
$461$	−36.4943	−1.69971	−0.849855	+	0.527017i	$0.823310\pi$
$462$	−10.6311	−0.494603
$463$	22.8528	1.06206	0.531030	−	0.847353i	$-0.321805\pi$
$463$	22.8528	1.06206	0.531030	+	0.847353i	$0.321805\pi$
$464$	17.6633	0.819998
$465$	0	0
$466$	13.1700	0.610087
$467$	−27.0948	−1.25380	−0.626898	−	0.779101i	$-0.715676\pi$
$467$	−27.0948	−1.25380	−0.626898	+	0.779101i	$0.715676\pi$
$468$	0.559087	0.0258438
$469$	−36.6322	−1.69152
$470$	0	0
$471$	−9.85695	−0.454184
$472$	−17.7950	−0.819081
$473$	−33.3931	−1.53542
$474$	11.4710	0.526881
$475$	0	0
$476$	−10.4283	−0.477980
$477$	12.0387	0.551215
$478$	7.00651	0.320470
$479$	−9.83837	−0.449527	−0.224763	−	0.974413i	$-0.572161\pi$
$479$	−9.83837	−0.449527	−0.224763	+	0.974413i	$0.572161\pi$
$480$	0	0
$481$	−1.17297	−0.0534830
$482$	−13.4458	−0.612441
$483$	17.6071	0.801150
$484$	−3.07313	−0.139688
$485$	0	0
$486$	−17.3265	−0.785946
$487$	21.7301	0.984685	0.492342	−	0.870402i	$-0.336141\pi$
$487$	21.7301	0.984685	0.492342	+	0.870402i	$0.336141\pi$
$488$	−19.3891	−0.877701
$489$	−2.30074	−0.104043
$490$	0	0
$491$	9.92067	0.447714	0.223857	−	0.974622i	$-0.428135\pi$
$491$	9.92067	0.447714	0.223857	+	0.974622i	$0.428135\pi$
$492$	−1.84991	−0.0834003
$493$	37.7957	1.70223
$494$	−0.934654	−0.0420521
$495$	0	0
$496$	9.59801	0.430963
$497$	30.4414	1.36549
$498$	2.03791	0.0913208
$499$	5.40678	0.242041	0.121020	−	0.992650i	$-0.461383\pi$
$499$	5.40678	0.242041	0.121020	+	0.992650i	$0.461383\pi$
$500$	0	0
$501$	−0.178335	−0.00796743
$502$	−32.8837	−1.46767
$503$	4.66981	0.208216	0.104108	−	0.994566i	$-0.466801\pi$
$503$	4.66981	0.208216	0.104108	+	0.994566i	$0.466801\pi$
$504$	26.0836	1.16185
$505$	0	0
$506$	−37.3501	−1.66041
$507$	−8.65462	−0.384365
$508$	−8.11597	−0.360088
$509$	−21.1778	−0.938690	−0.469345	−	0.883015i	$-0.655510\pi$
$509$	−21.1778	−0.938690	−0.469345	+	0.883015i	$0.655510\pi$
$510$	0	0
$511$	−5.96536	−0.263892
$512$	22.9691	1.01510
$513$	−8.06971	−0.356286
$514$	−15.0083	−0.661989
$515$	0	0
$516$	−3.36085	−0.147953
$517$	−12.1664	−0.535080
$518$	−12.6394	−0.555344
$519$	7.63964	0.335343
$520$	0	0
$521$	12.2704	0.537577	0.268789	−	0.963199i	$-0.413377\pi$
$521$	12.2704	0.537577	0.268789	+	0.963199i	$0.413377\pi$
$522$	−21.8346	−0.955676
$523$	7.41074	0.324049	0.162024	−	0.986787i	$-0.448198\pi$
$523$	7.41074	0.324049	0.162024	+	0.986787i	$0.448198\pi$
$524$	−7.12929	−0.311445
$525$	0	0
$526$	34.6455	1.51062
$527$	20.5377	0.894635
$528$	6.58304	0.286490
$529$	38.8587	1.68951
$530$	0	0
$531$	14.7358	0.639481
$532$	4.32320	0.187434
$533$	1.67282	0.0724577
$534$	10.9587	0.474228
$535$	0	0
$536$	33.8617	1.46260
$537$	−7.14198	−0.308199
$538$	8.01605	0.345596
$539$	−16.3647	−0.704879
$540$	0	0
$541$	−4.41764	−0.189929	−0.0949647	−	0.995481i	$-0.530274\pi$
$541$	−4.41764	−0.189929	−0.0949647	+	0.995481i	$0.530274\pi$
$542$	28.9279	1.24256
$543$	12.5311	0.537760
$544$	17.0528	0.731132
$545$	0	0
$546$	0.967538	0.0414068
$547$	−7.34172	−0.313909	−0.156955	−	0.987606i	$-0.550168\pi$
$547$	−7.34172	−0.313909	−0.156955	+	0.987606i	$0.550168\pi$
$548$	−11.0303	−0.471190
$549$	16.0558	0.685247
$550$	0	0
$551$	−15.6687	−0.667510
$552$	−16.2755	−0.692729
$553$	47.9793	2.04029
$554$	1.95060	0.0828729
$555$	0	0
$556$	−6.94818	−0.294668
$557$	3.51847	0.149082	0.0745412	−	0.997218i	$-0.476251\pi$
$557$	3.51847	0.149082	0.0745412	+	0.997218i	$0.476251\pi$
$558$	−11.8647	−0.502271
$559$	3.03911	0.128541
$560$	0	0
$561$	14.0863	0.594723
$562$	18.0752	0.762458
$563$	26.4829	1.11612	0.558060	−	0.829800i	$-0.311546\pi$
$563$	26.4829	1.11612	0.558060	+	0.829800i	$0.311546\pi$
$564$	−1.22449	−0.0515605
$565$	0	0
$566$	−16.9011	−0.710405
$567$	−17.0820	−0.717377
$568$	−28.1392	−1.18069
$569$	32.0339	1.34293	0.671466	−	0.741036i	$-0.265665\pi$
$569$	32.0339	1.34293	0.671466	+	0.741036i	$0.265665\pi$
$570$	0	0
$571$	−30.6558	−1.28290	−0.641452	−	0.767163i	$-0.721668\pi$
$571$	−30.6558	−1.28290	−0.641452	+	0.767163i	$0.721668\pi$
$572$	0.881025	0.0368375
$573$	3.88930	0.162478
$574$	18.0255	0.752368
$575$	0	0
$576$	−22.2725	−0.928022
$577$	−27.1211	−1.12907	−0.564533	−	0.825410i	$-0.690944\pi$
$577$	−27.1211	−1.12907	−0.564533	+	0.825410i	$0.690944\pi$
$578$	−12.0797	−0.502451
$579$	0.338184	0.0140544
$580$	0	0
$581$	8.52386	0.353629
$582$	−3.73522	−0.154830
$583$	18.9710	0.785697
$584$	5.51420	0.228179
$585$	0	0
$586$	11.6143	0.479780
$587$	−35.1593	−1.45118	−0.725589	−	0.688128i	$-0.758433\pi$
$587$	−35.1593	−1.45118	−0.725589	+	0.688128i	$0.758433\pi$
$588$	−1.64703	−0.0679224
$589$	−8.51418	−0.350821
$590$	0	0
$591$	−0.672647	−0.0276690
$592$	7.82664	0.321673
$593$	−40.0424	−1.64435	−0.822173	−	0.569238i	$-0.807238\pi$
$593$	−40.0424	−1.64435	−0.822173	+	0.569238i	$0.807238\pi$
$594$	−17.7206	−0.727087
$595$	0	0
$596$	9.80505	0.401630
$597$	9.66966	0.395753
$598$	3.39924	0.139005
$599$	6.08575	0.248657	0.124329	−	0.992241i	$-0.460322\pi$
$599$	6.08575	0.248657	0.124329	+	0.992241i	$0.460322\pi$
$600$	0	0
$601$	15.5884	0.635866	0.317933	−	0.948113i	$-0.397011\pi$
$601$	15.5884	0.635866	0.317933	+	0.948113i	$0.397011\pi$
$602$	32.7480	1.33471
$603$	−28.0405	−1.14190
$604$	9.37804	0.381587
$605$	0	0
$606$	8.53689	0.346788
$607$	43.0843	1.74874	0.874368	−	0.485263i	$-0.161276\pi$
$607$	43.0843	1.74874	0.874368	+	0.485263i	$0.161276\pi$
$608$	−7.06947	−0.286705
$609$	16.2200	0.657268
$610$	0	0
$611$	1.10727	0.0447954
$612$	−7.98244	−0.322671
$613$	9.83189	0.397106	0.198553	−	0.980090i	$-0.436376\pi$
$613$	9.83189	0.397106	0.198553	+	0.980090i	$0.436376\pi$
$614$	−21.8254	−0.880801
$615$	0	0
$616$	41.1032	1.65610
$617$	−40.8672	−1.64525	−0.822626	−	0.568583i	$-0.807492\pi$
$617$	−40.8672	−1.64525	−0.822626	+	0.568583i	$0.807492\pi$
$618$	4.19460	0.168732
$619$	−35.8404	−1.44055	−0.720274	−	0.693689i	$-0.755984\pi$
$619$	−35.8404	−1.44055	−0.720274	+	0.693689i	$0.755984\pi$
$620$	0	0
$621$	29.3487	1.17772
$622$	−41.0055	−1.64417
$623$	45.8363	1.83639
$624$	−0.599124	−0.0239841
$625$	0	0
$626$	14.6632	0.586058
$627$	−5.83966	−0.233214
$628$	8.80220	0.351246
$629$	16.7473	0.667759
$630$	0	0
$631$	−5.16698	−0.205694	−0.102847	−	0.994697i	$-0.532795\pi$
$631$	−5.16698	−0.205694	−0.102847	+	0.994697i	$0.532795\pi$
$632$	−44.3507	−1.76417
$633$	−3.63355	−0.144421
$634$	−26.2179	−1.04125
$635$	0	0
$636$	1.90933	0.0757100
$637$	1.48936	0.0590105
$638$	−34.4077	−1.36221
$639$	23.3017	0.921801
$640$	0	0
$641$	23.3893	0.923822	0.461911	−	0.886926i	$-0.347164\pi$
$641$	23.3893	0.923822	0.461911	+	0.886926i	$0.347164\pi$
$642$	3.54864	0.140053
$643$	48.9826	1.93168	0.965842	−	0.259130i	$-0.0834359\pi$
$643$	48.9826	1.93168	0.965842	+	0.259130i	$0.0834359\pi$
$644$	−15.7230	−0.619574
$645$	0	0
$646$	13.3447	0.525038
$647$	−44.2948	−1.74141	−0.870703	−	0.491809i	$-0.836336\pi$
$647$	−44.2948	−1.74141	−0.870703	+	0.491809i	$0.836336\pi$
$648$	15.7901	0.620294
$649$	23.2212	0.911510
$650$	0	0
$651$	8.81374	0.345438
$652$	2.05455	0.0804623
$653$	−20.8680	−0.816629	−0.408315	−	0.912841i	$-0.633883\pi$
$653$	−20.8680	−0.816629	−0.408315	+	0.912841i	$0.633883\pi$
$654$	−8.14889	−0.318647
$655$	0	0
$656$	−11.1618	−0.435796
$657$	−4.56624	−0.178146
$658$	11.9314	0.465135
$659$	−16.4987	−0.642697	−0.321349	−	0.946961i	$-0.604136\pi$
$659$	−16.4987	−0.642697	−0.321349	+	0.946961i	$0.604136\pi$
$660$	0	0
$661$	−35.2710	−1.37188	−0.685941	−	0.727658i	$-0.740609\pi$
$661$	−35.2710	−1.37188	−0.685941	+	0.727658i	$0.740609\pi$
$662$	32.4281	1.26035
$663$	−1.28199	−0.0497886
$664$	−7.87920	−0.305772
$665$	0	0
$666$	−9.67496	−0.374897
$667$	56.9855	2.20649
$668$	0.159252	0.00616166
$669$	−10.8839	−0.420797
$670$	0	0
$671$	25.3013	0.976745
$672$	7.31820	0.282306
$673$	−41.8981	−1.61505	−0.807527	−	0.589830i	$-0.799195\pi$
$673$	−41.8981	−1.61505	−0.807527	+	0.589830i	$0.799195\pi$
$674$	−7.46828	−0.287667
$675$	0	0
$676$	7.72853	0.297251
$677$	−16.8927	−0.649240	−0.324620	−	0.945845i	$-0.605236\pi$
$677$	−16.8927	−0.649240	−0.324620	+	0.945845i	$0.605236\pi$
$678$	−8.40418	−0.322760
$679$	−15.6232	−0.599562
$680$	0	0
$681$	4.95987	0.190063
$682$	−18.6967	−0.715933
$683$	34.9194	1.33615	0.668077	−	0.744093i	$-0.267118\pi$
$683$	34.9194	1.33615	0.668077	+	0.744093i	$0.267118\pi$
$684$	3.30923	0.126532
$685$	0	0
$686$	−11.5100	−0.439454
$687$	−13.3758	−0.510317
$688$	−20.2784	−0.773106
$689$	−1.72655	−0.0657764
$690$	0	0
$691$	37.2958	1.41880	0.709399	−	0.704807i	$-0.248966\pi$
$691$	37.2958	1.41880	0.709399	+	0.704807i	$0.248966\pi$
$692$	−6.82215	−0.259339
$693$	−34.0371	−1.29296
$694$	−17.6605	−0.670382
$695$	0	0
$696$	−14.9933	−0.568319
$697$	−23.8839	−0.904666
$698$	−25.7991	−0.976512
$699$	−7.48880	−0.283253
$700$	0	0
$701$	5.44329	0.205590	0.102795	−	0.994703i	$-0.467221\pi$
$701$	5.44329	0.205590	0.102795	+	0.994703i	$0.467221\pi$
$702$	1.61276	0.0608697
$703$	−6.94284	−0.261854
$704$	−35.0977	−1.32279
$705$	0	0
$706$	11.0350	0.415307
$707$	35.7069	1.34290
$708$	2.33709	0.0878334
$709$	−8.83117	−0.331661	−0.165831	−	0.986154i	$-0.553031\pi$
$709$	−8.83117	−0.331661	−0.165831	+	0.986154i	$0.553031\pi$
$710$	0	0
$711$	36.7263	1.37734
$712$	−42.3697	−1.58787
$713$	30.9652	1.15966
$714$	−13.8142	−0.516982
$715$	0	0
$716$	6.37775	0.238348
$717$	−3.98409	−0.148788
$718$	−16.7552	−0.625299
$719$	7.51607	0.280302	0.140151	−	0.990130i	$-0.455241\pi$
$719$	7.51607	0.280302	0.140151	+	0.990130i	$0.455241\pi$
$720$	0	0
$721$	17.5446	0.653394
$722$	16.9435	0.630572
$723$	7.64567	0.284345
$724$	−11.1902	−0.415880
$725$	0	0
$726$	−4.07092	−0.151086
$727$	−9.35179	−0.346839	−0.173419	−	0.984848i	$-0.555482\pi$
$727$	−9.35179	−0.346839	−0.173419	+	0.984848i	$0.555482\pi$
$728$	−3.74081	−0.138644
$729$	−5.54556	−0.205391
$730$	0	0
$731$	−43.3914	−1.60489
$732$	2.54645	0.0941194
$733$	−8.95136	−0.330626	−0.165313	−	0.986241i	$-0.552863\pi$
$733$	−8.95136	−0.330626	−0.165313	+	0.986241i	$0.552863\pi$
$734$	8.83825	0.326226
$735$	0	0
$736$	25.7109	0.947717
$737$	−44.1870	−1.62765
$738$	13.7978	0.507903
$739$	39.4888	1.45262	0.726309	−	0.687368i	$-0.241234\pi$
$739$	39.4888	1.45262	0.726309	+	0.687368i	$0.241234\pi$
$740$	0	0
$741$	0.531469	0.0195240
$742$	−18.6045	−0.682993
$743$	−12.2083	−0.447879	−0.223939	−	0.974603i	$-0.571892\pi$
$743$	−12.2083	−0.447879	−0.223939	+	0.974603i	$0.571892\pi$
$744$	−8.14716	−0.298689
$745$	0	0
$746$	−33.5342	−1.22778
$747$	6.52467	0.238725
$748$	−12.5790	−0.459932
$749$	14.8427	0.542341
$750$	0	0
$751$	31.3797	1.14506	0.572531	−	0.819883i	$-0.305962\pi$
$751$	31.3797	1.14506	0.572531	+	0.819883i	$0.305962\pi$
$752$	−7.38823	−0.269421
$753$	18.6985	0.681413
$754$	3.13145	0.114041
$755$	0	0
$756$	−7.45974	−0.271308
$757$	28.2134	1.02544	0.512718	−	0.858557i	$-0.328639\pi$
$757$	28.2134	1.02544	0.512718	+	0.858557i	$0.328639\pi$
$758$	28.7468	1.04413
$759$	21.2383	0.770900
$760$	0	0
$761$	35.6534	1.29243	0.646217	−	0.763154i	$-0.276350\pi$
$761$	35.6534	1.29243	0.646217	+	0.763154i	$0.276350\pi$
$762$	−10.7511	−0.389470
$763$	−34.0840	−1.23392
$764$	−3.47312	−0.125653
$765$	0	0
$766$	−1.11470	−0.0402759
$767$	−2.11336	−0.0763091
$768$	−8.73470	−0.315186
$769$	45.2644	1.63228	0.816139	−	0.577856i	$-0.196110\pi$
$769$	45.2644	1.63228	0.816139	+	0.577856i	$0.196110\pi$
$770$	0	0
$771$	8.53414	0.307350
$772$	−0.301996	−0.0108691
$773$	−24.4364	−0.878915	−0.439457	−	0.898263i	$-0.644829\pi$
$773$	−24.4364	−0.878915	−0.439457	+	0.898263i	$0.644829\pi$
$774$	25.0673	0.901026
$775$	0	0
$776$	14.4416	0.518422
$777$	7.18711	0.257836
$778$	−5.16757	−0.185266
$779$	9.90139	0.354754
$780$	0	0
$781$	36.7195	1.31393
$782$	−48.5331	−1.73554
$783$	27.0366	0.966210
$784$	−9.93770	−0.354918
$785$	0	0
$786$	−9.44404	−0.336858
$787$	−8.26001	−0.294437	−0.147219	−	0.989104i	$-0.547032\pi$
$787$	−8.26001	−0.294437	−0.147219	+	0.989104i	$0.547032\pi$
$788$	0.600670	0.0213980
$789$	−19.7004	−0.701352
$790$	0	0
$791$	−35.1518	−1.24985
$792$	31.4629	1.11798
$793$	−2.30267	−0.0817703
$794$	20.7180	0.735256
$795$	0	0
$796$	−8.63495	−0.306058
$797$	−6.32888	−0.224180	−0.112090	−	0.993698i	$-0.535755\pi$
$797$	−6.32888	−0.224180	−0.112090	+	0.993698i	$0.535755\pi$
$798$	5.72686	0.202729
$799$	−15.8092	−0.559290
$800$	0	0
$801$	35.0859	1.23970
$802$	42.1082	1.48689
$803$	−7.19562	−0.253928
$804$	−4.44720	−0.156841
$805$	0	0
$806$	1.70159	0.0599359
$807$	−4.55814	−0.160454
$808$	−33.0064	−1.16116
$809$	0.743242	0.0261310	0.0130655	−	0.999915i	$-0.495841\pi$
$809$	0.743242	0.0261310	0.0130655	+	0.999915i	$0.495841\pi$
$810$	0	0
$811$	−35.0526	−1.23086	−0.615431	−	0.788191i	$-0.711018\pi$
$811$	−35.0526	−1.23086	−0.615431	+	0.788191i	$0.711018\pi$
$812$	−14.4844	−0.508302
$813$	−16.4492	−0.576897
$814$	−15.2461	−0.534375
$815$	0	0
$816$	8.55407	0.299452
$817$	17.9885	0.629338
$818$	−23.9583	−0.837681
$819$	3.09772	0.108243
$820$	0	0
$821$	−29.8090	−1.04034	−0.520171	−	0.854062i	$-0.674132\pi$
$821$	−29.8090	−1.04034	−0.520171	+	0.854062i	$0.674132\pi$
$822$	−14.6116	−0.509638
$823$	20.0698	0.699591	0.349795	−	0.936826i	$-0.386251\pi$
$823$	20.0698	0.699591	0.349795	+	0.936826i	$0.386251\pi$
$824$	−16.2177	−0.564969
$825$	0	0
$826$	−22.7726	−0.792360
$827$	−18.9685	−0.659598	−0.329799	−	0.944051i	$-0.606981\pi$
$827$	−18.9685	−0.659598	−0.329799	+	0.944051i	$0.606981\pi$
$828$	−12.0353	−0.418257
$829$	20.7884	0.722011	0.361005	−	0.932564i	$-0.382434\pi$
$829$	20.7884	0.722011	0.361005	+	0.932564i	$0.382434\pi$
$830$	0	0
$831$	−1.10916	−0.0384764
$832$	3.19425	0.110741
$833$	−21.2645	−0.736772
$834$	−9.20412	−0.318712
$835$	0	0
$836$	5.21479	0.180357
$837$	14.6914	0.507807
$838$	14.8670	0.513573
$839$	52.9766	1.82896	0.914478	−	0.404636i	$-0.132602\pi$
$839$	52.9766	1.82896	0.914478	+	0.404636i	$0.132602\pi$
$840$	0	0
$841$	23.4962	0.810215
$842$	29.1243	1.00369
$843$	−10.2781	−0.353996
$844$	3.24474	0.111689
$845$	0	0
$846$	9.13303	0.314000
$847$	−17.0273	−0.585064
$848$	11.5204	0.395611
$849$	9.61040	0.329828
$850$	0	0
$851$	25.2504	0.865572
$852$	3.69564	0.126610
$853$	−23.3978	−0.801126	−0.400563	−	0.916269i	$-0.631185\pi$
$853$	−23.3978	−0.801126	−0.400563	+	0.916269i	$0.631185\pi$
$854$	−24.8125	−0.849067
$855$	0	0
$856$	−13.7202	−0.468946
$857$	−24.1325	−0.824349	−0.412175	−	0.911105i	$-0.635231\pi$
$857$	−24.1325	−0.824349	−0.412175	+	0.911105i	$0.635231\pi$
$858$	1.16708	0.0398434
$859$	4.08951	0.139532	0.0697661	−	0.997563i	$-0.477775\pi$
$859$	4.08951	0.139532	0.0697661	+	0.997563i	$0.477775\pi$
$860$	0	0
$861$	−10.2498	−0.349311
$862$	−30.5842	−1.04170
$863$	−44.4527	−1.51319	−0.756594	−	0.653885i	$-0.773138\pi$
$863$	−44.4527	−1.51319	−0.756594	+	0.653885i	$0.773138\pi$
$864$	12.1985	0.415001
$865$	0	0
$866$	17.1895	0.584122
$867$	6.86887	0.233279
$868$	−7.87062	−0.267146
$869$	57.8743	1.96325
$870$	0	0
$871$	4.02147	0.136262
$872$	31.5062	1.06694
$873$	−11.9589	−0.404747
$874$	20.1201	0.680572
$875$	0	0
$876$	−0.724203	−0.0244686
$877$	14.4263	0.487142	0.243571	−	0.969883i	$-0.421681\pi$
$877$	14.4263	0.487142	0.243571	+	0.969883i	$0.421681\pi$
$878$	−17.4201	−0.587900
$879$	−6.60418	−0.222753
$880$	0	0
$881$	−15.8712	−0.534715	−0.267357	−	0.963597i	$-0.586150\pi$
$881$	−15.8712	−0.534715	−0.267357	+	0.963597i	$0.586150\pi$
$882$	12.2846	0.413643
$883$	−4.89293	−0.164660	−0.0823301	−	0.996605i	$-0.526236\pi$
$883$	−4.89293	−0.164660	−0.0823301	+	0.996605i	$0.526236\pi$
$884$	1.14481	0.0385043
$885$	0	0
$886$	−19.9833	−0.671352
$887$	−26.0584	−0.874955	−0.437477	−	0.899229i	$-0.644128\pi$
$887$	−26.0584	−0.874955	−0.437477	+	0.899229i	$0.644128\pi$
$888$	−6.64355	−0.222943
$889$	−44.9680	−1.50818
$890$	0	0
$891$	−20.6049	−0.690290
$892$	9.71929	0.325426
$893$	6.55394	0.219319
$894$	12.9886	0.434402
$895$	0	0
$896$	12.6603	0.422951
$897$	−1.93290	−0.0645376
$898$	−47.5957	−1.58829
$899$	28.5258	0.951388
$900$	0	0
$901$	24.6511	0.821247
$902$	21.7429	0.723960
$903$	−18.6214	−0.619682
$904$	32.4932	1.08071
$905$	0	0
$906$	12.4229	0.412723
$907$	−57.2778	−1.90188	−0.950940	−	0.309376i	$-0.899880\pi$
$907$	−57.2778	−1.90188	−0.950940	+	0.309376i	$0.899880\pi$
$908$	−4.42914	−0.146986
$909$	27.3322	0.906551
$910$	0	0
$911$	−29.9179	−0.991225	−0.495613	−	0.868544i	$-0.665056\pi$
$911$	−29.9179	−0.991225	−0.495613	+	0.868544i	$0.665056\pi$
$912$	−3.54621	−0.117427
$913$	10.2818	0.340277
$914$	−28.3109	−0.936441
$915$	0	0
$916$	11.9445	0.394657
$917$	−39.5012	−1.30444
$918$	−23.0264	−0.759985
$919$	−41.7005	−1.37557	−0.687786	−	0.725913i	$-0.741417\pi$
$919$	−41.7005	−1.37557	−0.687786	+	0.725913i	$0.741417\pi$
$920$	0	0
$921$	12.4105	0.408940
$922$	43.1703	1.42174
$923$	−3.34185	−0.109998
$924$	−5.39826	−0.177590
$925$	0	0
$926$	−27.0333	−0.888371
$927$	13.4297	0.441088
$928$	23.6854	0.777512
$929$	−28.6930	−0.941387	−0.470694	−	0.882297i	$-0.655996\pi$
$929$	−28.6930	−0.941387	−0.470694	+	0.882297i	$0.655996\pi$
$930$	0	0
$931$	8.81551	0.288917
$932$	6.68746	0.219055
$933$	23.3168	0.763359
$934$	32.0513	1.04875
$935$	0	0
$936$	−2.86344	−0.0935945
$937$	−38.1640	−1.24676	−0.623382	−	0.781917i	$-0.714242\pi$
$937$	−38.1640	−1.24676	−0.623382	+	0.781917i	$0.714242\pi$
$938$	43.3334	1.41489
$939$	−8.33787	−0.272096
$940$	0	0
$941$	−51.7782	−1.68792	−0.843961	−	0.536404i	$-0.819782\pi$
$941$	−51.7782	−1.68792	−0.843961	+	0.536404i	$0.819782\pi$
$942$	11.6601	0.379907
$943$	−36.0103	−1.17266
$944$	14.1013	0.458960
$945$	0	0
$946$	39.5018	1.28431
$947$	1.35453	0.0440164	0.0220082	−	0.999758i	$-0.492994\pi$
$947$	1.35453	0.0440164	0.0220082	+	0.999758i	$0.492994\pi$
$948$	5.82476	0.189179
$949$	0.654875	0.0212581
$950$	0	0
$951$	14.9082	0.483431
$952$	53.4100	1.73103
$953$	−23.2905	−0.754452	−0.377226	−	0.926121i	$-0.623122\pi$
$953$	−23.2905	−0.754452	−0.377226	+	0.926121i	$0.623122\pi$
$954$	−14.2410	−0.461069
$955$	0	0
$956$	3.55777	0.115066
$957$	19.5651	0.632450
$958$	11.6381	0.376011
$959$	−61.1152	−1.97351
$960$	0	0
$961$	−15.4995	−0.499982
$962$	1.38755	0.0447364
$963$	11.3615	0.366120
$964$	−6.82754	−0.219900
$965$	0	0
$966$	−20.8280	−0.670130
$967$	−6.69207	−0.215203	−0.107601	−	0.994194i	$-0.534317\pi$
$967$	−6.69207	−0.215203	−0.107601	+	0.994194i	$0.534317\pi$
$968$	15.7395	0.505887
$969$	−7.58813	−0.243766
$970$	0	0
$971$	−41.7616	−1.34019	−0.670097	−	0.742274i	$-0.733748\pi$
$971$	−41.7616	−1.34019	−0.670097	+	0.742274i	$0.733748\pi$
$972$	−8.79806	−0.282198
$973$	−38.4976	−1.23418
$974$	−25.7052	−0.823649
$975$	0	0
$976$	15.3645	0.491806
$977$	18.8391	0.602715	0.301357	−	0.953511i	$-0.402560\pi$
$977$	18.8391	0.602715	0.301357	+	0.953511i	$0.402560\pi$
$978$	2.72162	0.0870278
$979$	55.2893	1.76705
$980$	0	0
$981$	−26.0899	−0.832987
$982$	−11.7355	−0.374494
$983$	41.7654	1.33211	0.666055	−	0.745903i	$-0.267982\pi$
$983$	41.7654	1.33211	0.666055	+	0.745903i	$0.267982\pi$
$984$	9.47457	0.302038
$985$	0	0
$986$	−44.7097	−1.42385
$987$	−6.78453	−0.215954
$988$	−0.474599	−0.0150990
$989$	−65.4223	−2.08031
$990$	0	0
$991$	−31.3931	−0.997235	−0.498617	−	0.866822i	$-0.666159\pi$
$991$	−31.3931	−0.997235	−0.498617	+	0.866822i	$0.666159\pi$
$992$	12.8704	0.408634
$993$	−18.4395	−0.585160
$994$	−36.0102	−1.14217
$995$	0	0
$996$	1.03481	0.0327892
$997$	−12.4169	−0.393247	−0.196623	−	0.980479i	$-0.562998\pi$
$997$	−12.4169	−0.393247	−0.196623	+	0.980479i	$0.562998\pi$
$998$	−6.39586	−0.202457
$999$	11.9800	0.379029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.s.1.16		49
5.2	odd	4		985.2.b.a.789.33	✓	98
5.3	odd	4		985.2.b.a.789.66	yes	98
5.4	even	2		4925.2.a.r.1.34		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.33	✓	98	5.2	odd	4
985.2.b.a.789.66	yes	98	5.3	odd	4
4925.2.a.r.1.34		49	5.4	even	2
4925.2.a.s.1.16		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)

\(f(q)\)	\(=\)	\(q-1.18293 q^{2} +0.672647 q^{3} -0.600670 q^{4} -0.795696 q^{6} -3.32812 q^{7} +3.07642 q^{8} -2.54755 q^{9} -4.01450 q^{11} -0.404039 q^{12} +0.365360 q^{13} +3.93695 q^{14} -2.43785 q^{16} -5.21648 q^{17} +3.01358 q^{18} +2.16257 q^{19} -2.23865 q^{21} +4.74888 q^{22} -7.86503 q^{23} +2.06934 q^{24} -0.432196 q^{26} -3.73154 q^{27} +1.99911 q^{28} -7.24543 q^{29} -3.93707 q^{31} -3.26902 q^{32} -2.70034 q^{33} +6.17075 q^{34} +1.53023 q^{36} -3.21046 q^{37} -2.55817 q^{38} +0.245758 q^{39} +4.57854 q^{41} +2.64818 q^{42} +8.31813 q^{43} +2.41139 q^{44} +9.30380 q^{46} +3.03063 q^{47} -1.63982 q^{48} +4.07641 q^{49} -3.50885 q^{51} -0.219461 q^{52} -4.72562 q^{53} +4.41416 q^{54} -10.2387 q^{56} +1.45464 q^{57} +8.57085 q^{58} -5.78433 q^{59} -6.30248 q^{61} +4.65729 q^{62} +8.47855 q^{63} +8.74274 q^{64} +3.19432 q^{66} +11.0069 q^{67} +3.13339 q^{68} -5.29039 q^{69} -9.14673 q^{71} -7.83731 q^{72} +1.79241 q^{73} +3.79776 q^{74} -1.29899 q^{76} +13.3607 q^{77} -0.290716 q^{78} -14.4163 q^{79} +5.13263 q^{81} -5.41610 q^{82} -2.56116 q^{83} +1.34469 q^{84} -9.83978 q^{86} -4.87362 q^{87} -12.3503 q^{88} -13.7724 q^{89} -1.21596 q^{91} +4.72429 q^{92} -2.64826 q^{93} -3.58503 q^{94} -2.19890 q^{96} +4.69428 q^{97} -4.82212 q^{98} +10.2271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 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

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