LMFDB - Embedded newform 4925.2.a.r.1.32

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:	$1$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
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)

$f(q)$	$=$	\(q+0.870537 q^{2} -1.92866 q^{3} -1.24217 q^{4} -1.67897 q^{6} -4.43166 q^{7} -2.82242 q^{8} +0.719722 q^{9} +1.70316 q^{11} +2.39571 q^{12} +0.123746 q^{13} -3.85792 q^{14} +0.0273066 q^{16} -1.14261 q^{17} +0.626545 q^{18} +8.45775 q^{19} +8.54715 q^{21} +1.48266 q^{22} -4.49103 q^{23} +5.44349 q^{24} +0.107725 q^{26} +4.39788 q^{27} +5.50485 q^{28} -3.89888 q^{29} +10.9216 q^{31} +5.66862 q^{32} -3.28481 q^{33} -0.994687 q^{34} -0.894014 q^{36} -3.64436 q^{37} +7.36279 q^{38} -0.238663 q^{39} +0.728472 q^{41} +7.44061 q^{42} -5.84768 q^{43} -2.11560 q^{44} -3.90961 q^{46} +11.6020 q^{47} -0.0526651 q^{48} +12.6396 q^{49} +2.20371 q^{51} -0.153713 q^{52} +12.1066 q^{53} +3.82851 q^{54} +12.5080 q^{56} -16.3121 q^{57} -3.39412 q^{58} -7.96100 q^{59} -12.8875 q^{61} +9.50769 q^{62} -3.18956 q^{63} +4.88013 q^{64} -2.85955 q^{66} -12.2474 q^{67} +1.41932 q^{68} +8.66166 q^{69} -10.6231 q^{71} -2.03136 q^{72} +6.59861 q^{73} -3.17255 q^{74} -10.5059 q^{76} -7.54781 q^{77} -0.207765 q^{78} +9.80240 q^{79} -10.6412 q^{81} +0.634162 q^{82} -2.94971 q^{83} -10.6170 q^{84} -5.09062 q^{86} +7.51960 q^{87} -4.80703 q^{88} +10.4622 q^{89} -0.548398 q^{91} +5.57860 q^{92} -21.0641 q^{93} +10.0999 q^{94} -10.9328 q^{96} -3.31388 q^{97} +11.0032 q^{98} +1.22580 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.870537	0.615563	0.307781	−	0.951457i	$-0.400414\pi$
$2$	0.870537	0.615563	0.307781	+	0.951457i	$0.400414\pi$
$3$	−1.92866	−1.11351	−0.556756	−	0.830676i	$-0.687954\pi$
$3$	−1.92866	−1.11351	−0.556756	+	0.830676i	$0.687954\pi$
$4$	−1.24217	−0.621083
$5$	0	0
$5$	0	0
$6$	−1.67897	−0.685436
$7$	−4.43166	−1.67501	−0.837505	−	0.546430i	$-0.815986\pi$
$7$	−4.43166	−1.67501	−0.837505	+	0.546430i	$0.815986\pi$
$8$	−2.82242	−0.997878
$9$	0.719722	0.239907
$10$	0	0
$11$	1.70316	0.513521	0.256761	−	0.966475i	$-0.417345\pi$
$11$	1.70316	0.513521	0.256761	+	0.966475i	$0.417345\pi$
$12$	2.39571	0.691583
$13$	0.123746	0.0343209	0.0171604	−	0.999853i	$-0.494537\pi$
$13$	0.123746	0.0343209	0.0171604	+	0.999853i	$0.494537\pi$
$14$	−3.85792	−1.03107
$15$	0	0
$16$	0.0273066	0.00682665
$17$	−1.14261	−0.277124	−0.138562	−	0.990354i	$-0.544248\pi$
$17$	−1.14261	−0.277124	−0.138562	+	0.990354i	$0.544248\pi$
$18$	0.626545	0.147678
$19$	8.45775	1.94034	0.970171	−	0.242422i	$-0.0779417\pi$
$19$	8.45775	1.94034	0.970171	+	0.242422i	$0.0779417\pi$
$20$	0	0
$21$	8.54715	1.86514
$22$	1.48266	0.316104
$23$	−4.49103	−0.936445	−0.468222	−	0.883611i	$-0.655105\pi$
$23$	−4.49103	−0.936445	−0.468222	+	0.883611i	$0.655105\pi$
$24$	5.44349	1.11115
$25$	0	0
$26$	0.107725	0.0211266
$27$	4.39788	0.846372
$28$	5.50485	1.04032
$29$	−3.89888	−0.724004	−0.362002	−	0.932177i	$-0.617907\pi$
$29$	−3.89888	−0.724004	−0.362002	+	0.932177i	$0.617907\pi$
$30$	0	0
$31$	10.9216	1.96158	0.980792	−	0.195056i	$-0.0624888\pi$
$31$	10.9216	1.96158	0.980792	+	0.195056i	$0.0624888\pi$
$32$	5.66862	1.00208
$33$	−3.28481	−0.571812
$34$	−0.994687	−0.170587
$35$	0	0
$36$	−0.894014	−0.149002
$37$	−3.64436	−0.599128	−0.299564	−	0.954076i	$-0.596841\pi$
$37$	−3.64436	−0.599128	−0.299564	+	0.954076i	$0.596841\pi$
$38$	7.36279	1.19440
$39$	−0.238663	−0.0382167
$40$	0	0
$41$	0.728472	0.113768	0.0568841	−	0.998381i	$-0.481883\pi$
$41$	0.728472	0.113768	0.0568841	+	0.998381i	$0.481883\pi$
$42$	7.44061	1.14811
$43$	−5.84768	−0.891763	−0.445881	−	0.895092i	$-0.647110\pi$
$43$	−5.84768	−0.891763	−0.445881	+	0.895092i	$0.647110\pi$
$44$	−2.11560	−0.318939
$45$	0	0
$46$	−3.90961	−0.576440
$47$	11.6020	1.69232	0.846160	−	0.532929i	$-0.178909\pi$
$47$	11.6020	1.69232	0.846160	+	0.532929i	$0.178909\pi$
$48$	−0.0526651	−0.00760155
$49$	12.6396	1.80566
$50$	0	0
$51$	2.20371	0.308581
$52$	−0.153713	−0.0213161
$53$	12.1066	1.66297	0.831485	−	0.555547i	$-0.187491\pi$
$53$	12.1066	1.66297	0.831485	+	0.555547i	$0.187491\pi$
$54$	3.82851	0.520995
$55$	0	0
$56$	12.5080	1.67145
$57$	−16.3121	−2.16059
$58$	−3.39412	−0.445669
$59$	−7.96100	−1.03643	−0.518217	−	0.855249i	$-0.673404\pi$
$59$	−7.96100	−1.03643	−0.518217	+	0.855249i	$0.673404\pi$
$60$	0	0
$61$	−12.8875	−1.65008	−0.825038	−	0.565077i	$-0.808846\pi$
$61$	−12.8875	−1.65008	−0.825038	+	0.565077i	$0.808846\pi$
$62$	9.50769	1.20748
$63$	−3.18956	−0.401847
$64$	4.88013	0.610016
$65$	0	0
$66$	−2.85955	−0.351986
$67$	−12.2474	−1.49626	−0.748129	−	0.663553i	$-0.769048\pi$
$67$	−12.2474	−1.49626	−0.748129	+	0.663553i	$0.769048\pi$
$68$	1.41932	0.172117
$69$	8.66166	1.04274
$70$	0	0
$71$	−10.6231	−1.26073	−0.630367	−	0.776297i	$-0.717096\pi$
$71$	−10.6231	−1.26073	−0.630367	+	0.776297i	$0.717096\pi$
$72$	−2.03136	−0.239398
$73$	6.59861	0.772309	0.386155	−	0.922434i	$-0.373803\pi$
$73$	6.59861	0.772309	0.386155	+	0.922434i	$0.373803\pi$
$74$	−3.17255	−0.368801
$75$	0	0
$76$	−10.5059	−1.20511
$77$	−7.54781	−0.860153
$78$	−0.207765	−0.0235248
$79$	9.80240	1.10286	0.551428	−	0.834222i	$-0.314083\pi$
$79$	9.80240	1.10286	0.551428	+	0.834222i	$0.314083\pi$
$80$	0	0
$81$	−10.6412	−1.18235
$82$	0.634162	0.0700315
$83$	−2.94971	−0.323773	−0.161887	−	0.986809i	$-0.551758\pi$
$83$	−2.94971	−0.323773	−0.161887	+	0.986809i	$0.551758\pi$
$84$	−10.6170	−1.15841
$85$	0	0
$86$	−5.09062	−0.548936
$87$	7.51960	0.806186
$88$	−4.80703	−0.512431
$89$	10.4622	1.10899	0.554496	−	0.832186i	$-0.312911\pi$
$89$	10.4622	1.10899	0.554496	+	0.832186i	$0.312911\pi$
$90$	0	0
$91$	−0.548398	−0.0574878
$92$	5.57860	0.581610
$93$	−21.0641	−2.18425
$94$	10.0999	1.04173
$95$	0	0
$96$	−10.9328	−1.11583
$97$	−3.31388	−0.336474	−0.168237	−	0.985747i	$-0.553807\pi$
$97$	−3.31388	−0.336474	−0.168237	+	0.985747i	$0.553807\pi$
$98$	11.0032	1.11149
$99$	1.22580	0.123198
$100$	0	0
$101$	12.2480	1.21872	0.609362	−	0.792892i	$-0.291426\pi$
$101$	12.2480	1.21872	0.609362	+	0.792892i	$0.291426\pi$
$102$	1.91841	0.189951
$103$	1.25200	0.123363	0.0616815	−	0.998096i	$-0.480354\pi$
$103$	1.25200	0.123363	0.0616815	+	0.998096i	$0.480354\pi$
$104$	−0.349263	−0.0342480
$105$	0	0
$106$	10.5392	1.02366
$107$	−10.1858	−0.984702	−0.492351	−	0.870397i	$-0.663862\pi$
$107$	−10.1858	−0.984702	−0.492351	+	0.870397i	$0.663862\pi$
$108$	−5.46289	−0.525667
$109$	−0.528072	−0.0505801	−0.0252900	−	0.999680i	$-0.508051\pi$
$109$	−0.528072	−0.0505801	−0.0252900	+	0.999680i	$0.508051\pi$
$110$	0	0
$111$	7.02872	0.667136
$112$	−0.121014	−0.0114347
$113$	3.75605	0.353339	0.176670	−	0.984270i	$-0.443468\pi$
$113$	3.75605	0.353339	0.176670	+	0.984270i	$0.443468\pi$
$114$	−14.2003	−1.32998
$115$	0	0
$116$	4.84305	0.449666
$117$	0.0890625	0.00823383
$118$	−6.93034	−0.637990
$119$	5.06367	0.464186
$120$	0	0
$121$	−8.09926	−0.736296
$122$	−11.2190	−1.01573
$123$	−1.40497	−0.126682
$124$	−13.5665	−1.21831
$125$	0	0
$126$	−2.77663	−0.247362
$127$	4.51202	0.400377	0.200188	−	0.979757i	$-0.435845\pi$
$127$	4.51202	0.400377	0.200188	+	0.979757i	$0.435845\pi$
$128$	−7.08891	−0.626577
$129$	11.2782	0.992988
$130$	0	0
$131$	−6.74933	−0.589692	−0.294846	−	0.955545i	$-0.595268\pi$
$131$	−6.74933	−0.589692	−0.294846	+	0.955545i	$0.595268\pi$
$132$	4.08028	0.355142
$133$	−37.4819	−3.25009
$134$	−10.6618	−0.921041
$135$	0	0
$136$	3.22494	0.276536
$137$	−14.4330	−1.23309	−0.616546	−	0.787319i	$-0.711468\pi$
$137$	−14.4330	−1.23309	−0.616546	+	0.787319i	$0.711468\pi$
$138$	7.54030	0.641873
$139$	−7.31604	−0.620538	−0.310269	−	0.950649i	$-0.600419\pi$
$139$	−7.31604	−0.620538	−0.310269	+	0.950649i	$0.600419\pi$
$140$	0	0
$141$	−22.3762	−1.88442
$142$	−9.24784	−0.776061
$143$	0.210758	0.0176245
$144$	0.0196532	0.00163776
$145$	0	0
$146$	5.74434	0.475405
$147$	−24.3774	−2.01062
$148$	4.52689	0.372108
$149$	4.10236	0.336079	0.168039	−	0.985780i	$-0.446256\pi$
$149$	4.10236	0.336079	0.168039	+	0.985780i	$0.446256\pi$
$150$	0	0
$151$	−13.3414	−1.08570	−0.542852	−	0.839828i	$-0.682656\pi$
$151$	−13.3414	−1.08570	−0.542852	+	0.839828i	$0.682656\pi$
$152$	−23.8714	−1.93622
$153$	−0.822364	−0.0664842
$154$	−6.57065	−0.529478
$155$	0	0
$156$	0.296459	0.0237357
$157$	15.6461	1.24870	0.624349	−	0.781145i	$-0.285364\pi$
$157$	15.6461	1.24870	0.624349	+	0.781145i	$0.285364\pi$
$158$	8.53335	0.678877
$159$	−23.3495	−1.85174
$160$	0	0
$161$	19.9027	1.56855
$162$	−9.26353	−0.727811
$163$	10.1824	0.797543	0.398772	−	0.917050i	$-0.369437\pi$
$163$	10.1824	0.797543	0.398772	+	0.917050i	$0.369437\pi$
$164$	−0.904883	−0.0706595
$165$	0	0
$166$	−2.56783	−0.199303
$167$	−11.4059	−0.882619	−0.441309	−	0.897355i	$-0.645486\pi$
$167$	−11.4059	−0.882619	−0.441309	+	0.897355i	$0.645486\pi$
$168$	−24.1237	−1.86118
$169$	−12.9847	−0.998822
$170$	0	0
$171$	6.08723	0.465502
$172$	7.26378	0.553858
$173$	24.8344	1.88812	0.944061	−	0.329771i	$-0.106972\pi$
$173$	24.8344	1.88812	0.944061	+	0.329771i	$0.106972\pi$
$174$	6.54609	0.496258
$175$	0	0
$176$	0.0465075	0.00350563
$177$	15.3540	1.15408
$178$	9.10775	0.682655
$179$	21.0550	1.57372	0.786861	−	0.617130i	$-0.211705\pi$
$179$	21.0550	1.57372	0.786861	+	0.617130i	$0.211705\pi$
$180$	0	0
$181$	−4.58253	−0.340617	−0.170308	−	0.985391i	$-0.554476\pi$
$181$	−4.58253	−0.340617	−0.170308	+	0.985391i	$0.554476\pi$
$182$	−0.477401	−0.0353873
$183$	24.8556	1.83738
$184$	12.6756	0.934457
$185$	0	0
$186$	−18.3371	−1.34454
$187$	−1.94605	−0.142309
$188$	−14.4116	−1.05107
$189$	−19.4899	−1.41768
$190$	0	0
$191$	−12.9687	−0.938385	−0.469193	−	0.883096i	$-0.655455\pi$
$191$	−12.9687	−0.938385	−0.469193	+	0.883096i	$0.655455\pi$
$192$	−9.41210	−0.679260
$193$	4.21268	0.303235	0.151618	−	0.988439i	$-0.451552\pi$
$193$	4.21268	0.303235	0.151618	+	0.988439i	$0.451552\pi$
$194$	−2.88486	−0.207121
$195$	0	0
$196$	−15.7005	−1.12146
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	1.06710	0.0758358
$199$	−9.85404	−0.698534	−0.349267	−	0.937023i	$-0.613569\pi$
$199$	−9.85404	−0.698534	−0.349267	+	0.937023i	$0.613569\pi$
$200$	0	0
$201$	23.6211	1.66610
$202$	10.6623	0.750200
$203$	17.2785	1.21271
$204$	−2.73737	−0.191654
$205$	0	0
$206$	1.08991	0.0759376
$207$	−3.23229	−0.224660
$208$	0.00337907	0.000234297 0
$209$	14.4049	0.996407
$210$	0	0
$211$	−9.42984	−0.649177	−0.324588	−	0.945855i	$-0.605226\pi$
$211$	−9.42984	−0.649177	−0.324588	+	0.945855i	$0.605226\pi$
$212$	−15.0384	−1.03284
$213$	20.4884	1.40384
$214$	−8.86714	−0.606145
$215$	0	0
$216$	−12.4127	−0.844576
$217$	−48.4010	−3.28567
$218$	−0.459706	−0.0311352
$219$	−12.7265	−0.859975
$220$	0	0
$221$	−0.141393	−0.00951115
$222$	6.11876	0.410664
$223$	−22.5192	−1.50800	−0.753999	−	0.656876i	$-0.771877\pi$
$223$	−22.5192	−1.50800	−0.753999	+	0.656876i	$0.771877\pi$
$224$	−25.1214	−1.67849
$225$	0	0
$226$	3.26978	0.217502
$227$	0.372248	0.0247070	0.0123535	−	0.999924i	$-0.496068\pi$
$227$	0.372248	0.0247070	0.0123535	+	0.999924i	$0.496068\pi$
$228$	20.2624	1.34191
$229$	16.0618	1.06139	0.530696	−	0.847562i	$-0.321930\pi$
$229$	16.0618	1.06139	0.530696	+	0.847562i	$0.321930\pi$
$230$	0	0
$231$	14.5571	0.957790
$232$	11.0043	0.722467
$233$	3.23653	0.212032	0.106016	−	0.994364i	$-0.466190\pi$
$233$	3.23653	0.212032	0.106016	+	0.994364i	$0.466190\pi$
$234$	0.0775322	0.00506844
$235$	0	0
$236$	9.88888	0.643711
$237$	−18.9055	−1.22804
$238$	4.40811	0.285736
$239$	−23.8140	−1.54040	−0.770200	−	0.637802i	$-0.779844\pi$
$239$	−23.8140	−1.54040	−0.770200	+	0.637802i	$0.779844\pi$
$240$	0	0
$241$	22.1911	1.42945	0.714727	−	0.699403i	$-0.246551\pi$
$241$	22.1911	1.42945	0.714727	+	0.699403i	$0.246551\pi$
$242$	−7.05070	−0.453236
$243$	7.32954	0.470190
$244$	16.0084	1.02483
$245$	0	0
$246$	−1.22308	−0.0779808
$247$	1.04661	0.0665942
$248$	−30.8255	−1.95742
$249$	5.68899	0.360525
$250$	0	0
$251$	−2.15167	−0.135812	−0.0679062	−	0.997692i	$-0.521632\pi$
$251$	−2.15167	−0.135812	−0.0679062	+	0.997692i	$0.521632\pi$
$252$	3.96196	0.249580
$253$	−7.64893	−0.480884
$254$	3.92788	0.246457
$255$	0	0
$256$	−15.9314	−0.995713
$257$	10.9521	0.683176	0.341588	−	0.939850i	$-0.389035\pi$
$257$	10.9521	0.683176	0.341588	+	0.939850i	$0.389035\pi$
$258$	9.81806	0.611246
$259$	16.1505	1.00355
$260$	0	0
$261$	−2.80611	−0.173694
$262$	−5.87554	−0.362992
$263$	−8.66154	−0.534093	−0.267047	−	0.963684i	$-0.586048\pi$
$263$	−8.66154	−0.534093	−0.267047	+	0.963684i	$0.586048\pi$
$264$	9.27112	0.570598
$265$	0	0
$266$	−32.6294	−2.00063
$267$	−20.1780	−1.23488
$268$	15.2133	0.929301
$269$	10.4447	0.636824	0.318412	−	0.947952i	$-0.396850\pi$
$269$	10.4447	0.636824	0.318412	+	0.947952i	$0.396850\pi$
$270$	0	0
$271$	4.55775	0.276864	0.138432	−	0.990372i	$-0.455794\pi$
$271$	4.55775	0.276864	0.138432	+	0.990372i	$0.455794\pi$
$272$	−0.0312009	−0.00189183
$273$	1.05767	0.0640133
$274$	−12.5644	−0.759045
$275$	0	0
$276$	−10.7592	−0.647629
$277$	−29.5789	−1.77722	−0.888612	−	0.458660i	$-0.848329\pi$
$277$	−29.5789	−1.77722	−0.888612	+	0.458660i	$0.848329\pi$
$278$	−6.36888	−0.381980
$279$	7.86055	0.470599
$280$	0	0
$281$	−5.02303	−0.299649	−0.149824	−	0.988713i	$-0.547871\pi$
$281$	−5.02303	−0.299649	−0.149824	+	0.988713i	$0.547871\pi$
$282$	−19.4793	−1.15998
$283$	−7.53723	−0.448042	−0.224021	−	0.974584i	$-0.571918\pi$
$283$	−7.53723	−0.448042	−0.224021	+	0.974584i	$0.571918\pi$
$284$	13.1957	0.783021
$285$	0	0
$286$	0.183473	0.0108490
$287$	−3.22834	−0.190563
$288$	4.07983	0.240406
$289$	−15.6944	−0.923202
$290$	0	0
$291$	6.39135	0.374668
$292$	−8.19657	−0.479668
$293$	−5.84987	−0.341753	−0.170877	−	0.985292i	$-0.554660\pi$
$293$	−5.84987	−0.341753	−0.170877	+	0.985292i	$0.554660\pi$
$294$	−21.2215	−1.23766
$295$	0	0
$296$	10.2859	0.597857
$297$	7.49028	0.434630
$298$	3.57126	0.206877
$299$	−0.555746	−0.0321396
$300$	0	0
$301$	25.9149	1.49371
$302$	−11.6141	−0.668319
$303$	−23.6222	−1.35706
$304$	0.230953	0.0132460
$305$	0	0
$306$	−0.715898	−0.0409252
$307$	22.8530	1.30429	0.652146	−	0.758094i	$-0.273869\pi$
$307$	22.8530	1.30429	0.652146	+	0.758094i	$0.273869\pi$
$308$	9.37563	0.534226
$309$	−2.41467	−0.137366
$310$	0	0
$311$	0.848574	0.0481182	0.0240591	−	0.999711i	$-0.492341\pi$
$311$	0.848574	0.0481182	0.0240591	+	0.999711i	$0.492341\pi$
$312$	0.673609	0.0381356
$313$	−24.2552	−1.37099	−0.685494	−	0.728078i	$-0.740414\pi$
$313$	−24.2552	−1.37099	−0.685494	+	0.728078i	$0.740414\pi$
$314$	13.6205	0.768652
$315$	0	0
$316$	−12.1762	−0.684965
$317$	9.50882	0.534069	0.267034	−	0.963687i	$-0.413956\pi$
$317$	9.50882	0.534069	0.267034	+	0.963687i	$0.413956\pi$
$318$	−20.3266	−1.13986
$319$	−6.64040	−0.371791
$320$	0	0
$321$	19.6450	1.09648
$322$	17.3260	0.965543
$323$	−9.66394	−0.537716
$324$	13.2181	0.734338
$325$	0	0
$326$	8.86411	0.490938
$327$	1.01847	0.0563215
$328$	−2.05606	−0.113527
$329$	−51.4159	−2.83465
$330$	0	0
$331$	2.55412	0.140387	0.0701936	−	0.997533i	$-0.477638\pi$
$331$	2.55412	0.140387	0.0701936	+	0.997533i	$0.477638\pi$
$332$	3.66403	0.201090
$333$	−2.62292	−0.143735
$334$	−9.92930	−0.543307
$335$	0	0
$336$	0.233394	0.0127327
$337$	−7.54624	−0.411070	−0.205535	−	0.978650i	$-0.565893\pi$
$337$	−7.54624	−0.411070	−0.205535	+	0.978650i	$0.565893\pi$
$338$	−11.3036	−0.614837
$339$	−7.24413	−0.393447
$340$	0	0
$341$	18.6013	1.00732
$342$	5.29916	0.286546
$343$	−24.9927	−1.34948
$344$	16.5046	0.889870
$345$	0	0
$346$	21.6192	1.16226
$347$	−13.6684	−0.733756	−0.366878	−	0.930269i	$-0.619573\pi$
$347$	−13.6684	−0.733756	−0.366878	+	0.930269i	$0.619573\pi$
$348$	−9.34059	−0.500708
$349$	30.5829	1.63706	0.818531	−	0.574462i	$-0.194789\pi$
$349$	30.5829	1.63706	0.818531	+	0.574462i	$0.194789\pi$
$350$	0	0
$351$	0.544218	0.0290482
$352$	9.65455	0.514589
$353$	−8.32483	−0.443086	−0.221543	−	0.975151i	$-0.571109\pi$
$353$	−8.32483	−0.443086	−0.221543	+	0.975151i	$0.571109\pi$
$354$	13.3663	0.710409
$355$	0	0
$356$	−12.9958	−0.688776
$357$	−9.76609	−0.516876
$358$	18.3291	0.968725
$359$	12.4077	0.654855	0.327428	−	0.944876i	$-0.393818\pi$
$359$	12.4077	0.654855	0.327428	+	0.944876i	$0.393818\pi$
$360$	0	0
$361$	52.5336	2.76493
$362$	−3.98926	−0.209671
$363$	15.6207	0.819874
$364$	0.681202	0.0357047
$365$	0	0
$366$	21.6377	1.13102
$367$	33.0594	1.72569	0.862843	−	0.505472i	$-0.168682\pi$
$367$	33.0594	1.72569	0.862843	+	0.505472i	$0.168682\pi$
$368$	−0.122635	−0.00639278
$369$	0.524298	0.0272938
$370$	0	0
$371$	−53.6523	−2.78549
$372$	26.1651	1.35660
$373$	−24.6346	−1.27553	−0.637765	−	0.770231i	$-0.720141\pi$
$373$	−24.6346	−1.27553	−0.637765	+	0.770231i	$0.720141\pi$
$374$	−1.69411	−0.0876003
$375$	0	0
$376$	−32.7457	−1.68873
$377$	−0.482469	−0.0248484
$378$	−16.9667	−0.872671
$379$	2.30362	0.118329	0.0591645	−	0.998248i	$-0.481156\pi$
$379$	2.30362	0.118329	0.0591645	+	0.998248i	$0.481156\pi$
$380$	0	0
$381$	−8.70214	−0.445824
$382$	−11.2898	−0.577635
$383$	−22.6666	−1.15821	−0.579104	−	0.815253i	$-0.696598\pi$
$383$	−22.6666	−1.15821	−0.579104	+	0.815253i	$0.696598\pi$
$384$	13.6721	0.697700
$385$	0	0
$386$	3.66729	0.186660
$387$	−4.20870	−0.213940
$388$	4.11639	0.208978
$389$	−10.8248	−0.548840	−0.274420	−	0.961610i	$-0.588486\pi$
$389$	−10.8248	−0.548840	−0.274420	+	0.961610i	$0.588486\pi$
$390$	0	0
$391$	5.13151	0.259512
$392$	−35.6743	−1.80182
$393$	13.0172	0.656629
$394$	0.870537	0.0438570
$395$	0	0
$396$	−1.52265	−0.0765159
$397$	−14.2272	−0.714040	−0.357020	−	0.934097i	$-0.616207\pi$
$397$	−14.2272	−0.714040	−0.357020	+	0.934097i	$0.616207\pi$
$398$	−8.57830	−0.429991
$399$	72.2897	3.61901
$400$	0	0
$401$	31.7277	1.58441	0.792203	−	0.610257i	$-0.208934\pi$
$401$	31.7277	1.58441	0.792203	+	0.610257i	$0.208934\pi$
$402$	20.5630	1.02559
$403$	1.35151	0.0673233
$404$	−15.2141	−0.756928
$405$	0	0
$406$	15.0416	0.746500
$407$	−6.20691	−0.307665
$408$	−6.21981	−0.307926
$409$	23.4878	1.16140	0.580698	−	0.814119i	$-0.302780\pi$
$409$	23.4878	1.16140	0.580698	+	0.814119i	$0.302780\pi$
$410$	0	0
$411$	27.8362	1.37306
$412$	−1.55519	−0.0766186
$413$	35.2804	1.73604
$414$	−2.81383	−0.138292
$415$	0	0
$416$	0.701467	0.0343923
$417$	14.1101	0.690976
$418$	12.5400	0.613351
$419$	18.4285	0.900292	0.450146	−	0.892955i	$-0.351372\pi$
$419$	18.4285	0.900292	0.450146	+	0.892955i	$0.351372\pi$
$420$	0	0
$421$	−12.9360	−0.630460	−0.315230	−	0.949015i	$-0.602082\pi$
$421$	−12.9360	−0.630460	−0.315230	+	0.949015i	$0.602082\pi$
$422$	−8.20902	−0.399609
$423$	8.35019	0.406000
$424$	−34.1700	−1.65944
$425$	0	0
$426$	17.8359	0.864153
$427$	57.1130	2.76389
$428$	12.6525	0.611581
$429$	−0.406481	−0.0196251
$430$	0	0
$431$	−17.2655	−0.831650	−0.415825	−	0.909445i	$-0.636507\pi$
$431$	−17.2655	−0.831650	−0.415825	+	0.909445i	$0.636507\pi$
$432$	0.120091	0.00577789
$433$	24.0953	1.15795	0.578973	−	0.815347i	$-0.303454\pi$
$433$	24.0953	1.15795	0.578973	+	0.815347i	$0.303454\pi$
$434$	−42.1348	−2.02254
$435$	0	0
$436$	0.655953	0.0314144
$437$	−37.9840	−1.81702
$438$	−11.0789	−0.529368
$439$	0.725758	0.0346386	0.0173193	−	0.999850i	$-0.494487\pi$
$439$	0.725758	0.0346386	0.0173193	+	0.999850i	$0.494487\pi$
$440$	0	0
$441$	9.09699	0.433190
$442$	−0.123088	−0.00585471
$443$	−25.6666	−1.21945	−0.609727	−	0.792611i	$-0.708721\pi$
$443$	−25.6666	−1.21945	−0.609727	+	0.792611i	$0.708721\pi$
$444$	−8.73083	−0.414347
$445$	0	0
$446$	−19.6038	−0.928266
$447$	−7.91205	−0.374227
$448$	−21.6271	−1.02178
$449$	−30.2263	−1.42647	−0.713235	−	0.700925i	$-0.752771\pi$
$449$	−30.2263	−1.42647	−0.713235	+	0.700925i	$0.752771\pi$
$450$	0	0
$451$	1.24070	0.0584224
$452$	−4.66563	−0.219453
$453$	25.7309	1.20894
$454$	0.324056	0.0152087
$455$	0	0
$456$	46.0397	2.15601
$457$	6.05210	0.283105	0.141553	−	0.989931i	$-0.454791\pi$
$457$	6.05210	0.283105	0.141553	+	0.989931i	$0.454791\pi$
$458$	13.9824	0.653354
$459$	−5.02507	−0.234550
$460$	0	0
$461$	−6.19623	−0.288587	−0.144293	−	0.989535i	$-0.546091\pi$
$461$	−6.19623	−0.288587	−0.144293	+	0.989535i	$0.546091\pi$
$462$	12.6725	0.589580
$463$	17.4621	0.811533	0.405767	−	0.913977i	$-0.367005\pi$
$463$	17.4621	0.811533	0.405767	+	0.913977i	$0.367005\pi$
$464$	−0.106465	−0.00494252
$465$	0	0
$466$	2.81752	0.130519
$467$	−26.8068	−1.24047	−0.620236	−	0.784416i	$-0.712963\pi$
$467$	−26.8068	−1.24047	−0.620236	+	0.784416i	$0.712963\pi$
$468$	−0.110630	−0.00511389
$469$	54.2763	2.50625
$470$	0	0
$471$	−30.1761	−1.39044
$472$	22.4693	1.03423
$473$	−9.95952	−0.457939
$474$	−16.4579	−0.755937
$475$	0	0
$476$	−6.28992	−0.288298
$477$	8.71339	0.398959
$478$	−20.7310	−0.948213
$479$	−17.4704	−0.798244	−0.399122	−	0.916898i	$-0.630685\pi$
$479$	−17.4704	−0.798244	−0.399122	+	0.916898i	$0.630685\pi$
$480$	0	0
$481$	−0.450973	−0.0205626
$482$	19.3182	0.879919
$483$	−38.3855	−1.74660
$484$	10.0606	0.457301
$485$	0	0
$486$	6.38064	0.289432
$487$	−13.3952	−0.606996	−0.303498	−	0.952832i	$-0.598155\pi$
$487$	−13.3952	−0.606996	−0.303498	+	0.952832i	$0.598155\pi$
$488$	36.3740	1.64657
$489$	−19.6383	−0.888074
$490$	0	0
$491$	2.76836	0.124934	0.0624672	−	0.998047i	$-0.480103\pi$
$491$	2.76836	0.124934	0.0624672	+	0.998047i	$0.480103\pi$
$492$	1.74521	0.0786802
$493$	4.45491	0.200639
$494$	0.911113	0.0409929
$495$	0	0
$496$	0.298233	0.0133911
$497$	47.0781	2.11174
$498$	4.95247	0.221926
$499$	−19.6857	−0.881253	−0.440627	−	0.897690i	$-0.645244\pi$
$499$	−19.6857	−0.881253	−0.440627	+	0.897690i	$0.645244\pi$
$500$	0	0
$501$	21.9982	0.982806
$502$	−1.87311	−0.0836010
$503$	−9.61308	−0.428626	−0.214313	−	0.976765i	$-0.568751\pi$
$503$	−9.61308	−0.428626	−0.214313	+	0.976765i	$0.568751\pi$
$504$	9.00230	0.400994
$505$	0	0
$506$	−6.65868	−0.296014
$507$	25.0430	1.11220
$508$	−5.60467	−0.248667
$509$	−27.0505	−1.19899	−0.599496	−	0.800377i	$-0.704632\pi$
$509$	−27.0505	−1.19899	−0.599496	+	0.800377i	$0.704632\pi$
$510$	0	0
$511$	−29.2428	−1.29363
$512$	0.308933	0.0136530
$513$	37.1962	1.64225
$514$	9.53425	0.420538
$515$	0	0
$516$	−14.0094	−0.616728
$517$	19.7600	0.869042
$518$	14.0596	0.617745
$519$	−47.8970	−2.10244
$520$	0	0
$521$	−22.9932	−1.00735	−0.503676	−	0.863893i	$-0.668019\pi$
$521$	−22.9932	−1.00735	−0.503676	+	0.863893i	$0.668019\pi$
$522$	−2.44282	−0.106919
$523$	−15.6933	−0.686219	−0.343110	−	0.939295i	$-0.611480\pi$
$523$	−15.6933	−0.686219	−0.343110	+	0.939295i	$0.611480\pi$
$524$	8.38379	0.366248
$525$	0	0
$526$	−7.54019	−0.328768
$527$	−12.4792	−0.543603
$528$	−0.0896970	−0.00390356
$529$	−2.83064	−0.123071
$530$	0	0
$531$	−5.72971	−0.248648
$532$	46.5587	2.01858
$533$	0.0901453	0.00390463
$534$	−17.5657	−0.760143
$535$	0	0
$536$	34.5674	1.49308
$537$	−40.6079	−1.75236
$538$	9.09249	0.392005
$539$	21.5272	0.927243
$540$	0	0
$541$	27.6492	1.18873	0.594366	−	0.804194i	$-0.297403\pi$
$541$	27.6492	1.18873	0.594366	+	0.804194i	$0.297403\pi$
$542$	3.96769	0.170427
$543$	8.83814	0.379281
$544$	−6.47704	−0.277701
$545$	0	0
$546$	0.920743	0.0394042
$547$	−27.9155	−1.19358	−0.596790	−	0.802397i	$-0.703558\pi$
$547$	−27.9155	−1.19358	−0.596790	+	0.802397i	$0.703558\pi$
$548$	17.9281	0.765852
$549$	−9.27542	−0.395865
$550$	0	0
$551$	−32.9758	−1.40481
$552$	−24.4469	−1.04053
$553$	−43.4409	−1.84729
$554$	−25.7495	−1.09399
$555$	0	0
$556$	9.08773	0.385406
$557$	9.60825	0.407115	0.203557	−	0.979063i	$-0.434750\pi$
$557$	9.60825	0.407115	0.203557	+	0.979063i	$0.434750\pi$
$558$	6.84289	0.289683
$559$	−0.723625	−0.0306061
$560$	0	0
$561$	3.75327	0.158463
$562$	−4.37273	−0.184452
$563$	22.1977	0.935520	0.467760	−	0.883855i	$-0.345061\pi$
$563$	22.1977	0.935520	0.467760	+	0.883855i	$0.345061\pi$
$564$	27.7950	1.17038
$565$	0	0
$566$	−6.56144	−0.275798
$567$	47.1580	1.98045
$568$	29.9830	1.25806
$569$	−12.3835	−0.519145	−0.259573	−	0.965724i	$-0.583582\pi$
$569$	−12.3835	−0.519145	−0.259573	+	0.965724i	$0.583582\pi$
$570$	0	0
$571$	−27.9621	−1.17018	−0.585090	−	0.810969i	$-0.698941\pi$
$571$	−27.9621	−1.17018	−0.585090	+	0.810969i	$0.698941\pi$
$572$	−0.261797	−0.0109463
$573$	25.0123	1.04490
$574$	−2.81039	−0.117303
$575$	0	0
$576$	3.51234	0.146347
$577$	34.3719	1.43092	0.715460	−	0.698654i	$-0.246217\pi$
$577$	34.3719	1.43092	0.715460	+	0.698654i	$0.246217\pi$
$578$	−13.6626	−0.568289
$579$	−8.12481	−0.337656
$580$	0	0
$581$	13.0721	0.542323
$582$	5.56391	0.230631
$583$	20.6194	0.853970
$584$	−18.6241	−0.770670
$585$	0	0
$586$	−5.09253	−0.210370
$587$	47.0526	1.94207	0.971033	−	0.238944i	$-0.0768013\pi$
$587$	47.0526	1.94207	0.971033	+	0.238944i	$0.0768013\pi$
$588$	30.2808	1.24876
$589$	92.3725	3.80614
$590$	0	0
$591$	−1.92866	−0.0793344
$592$	−0.0995150	−0.00409004
$593$	−10.0447	−0.412486	−0.206243	−	0.978501i	$-0.566124\pi$
$593$	−10.0447	−0.412486	−0.206243	+	0.978501i	$0.566124\pi$
$594$	6.52056	0.267542
$595$	0	0
$596$	−5.09581	−0.208733
$597$	19.0051	0.777826
$598$	−0.483797	−0.0197839
$599$	−29.0212	−1.18577	−0.592886	−	0.805286i	$-0.702012\pi$
$599$	−29.0212	−1.18577	−0.592886	+	0.805286i	$0.702012\pi$
$600$	0	0
$601$	−0.756067	−0.0308406	−0.0154203	−	0.999881i	$-0.504909\pi$
$601$	−0.756067	−0.0308406	−0.0154203	+	0.999881i	$0.504909\pi$
$602$	22.5599	0.919472
$603$	−8.81473	−0.358964
$604$	16.5722	0.674313
$605$	0	0
$606$	−20.5640	−0.835357
$607$	−0.980632	−0.0398026	−0.0199013	−	0.999802i	$-0.506335\pi$
$607$	−0.980632	−0.0398026	−0.0199013	+	0.999802i	$0.506335\pi$
$608$	47.9438	1.94438
$609$	−33.3243	−1.35037
$610$	0	0
$611$	1.43569	0.0580819
$612$	1.02151	0.0412922
$613$	−32.3831	−1.30794	−0.653971	−	0.756520i	$-0.726898\pi$
$613$	−32.3831	−1.30794	−0.653971	+	0.756520i	$0.726898\pi$
$614$	19.8944	0.802873
$615$	0	0
$616$	21.3031	0.858327
$617$	4.27331	0.172037	0.0860185	−	0.996294i	$-0.472586\pi$
$617$	4.27331	0.172037	0.0860185	+	0.996294i	$0.472586\pi$
$618$	−2.10206	−0.0845574
$619$	−23.6528	−0.950687	−0.475343	−	0.879800i	$-0.657676\pi$
$619$	−23.6528	−0.950687	−0.475343	+	0.879800i	$0.657676\pi$
$620$	0	0
$621$	−19.7510	−0.792580
$622$	0.738715	0.0296198
$623$	−46.3650	−1.85757
$624$	−0.00651708	−0.000260892 0
$625$	0	0
$626$	−21.1151	−0.843929
$627$	−27.7821	−1.10951
$628$	−19.4351	−0.775545
$629$	4.16409	0.166033
$630$	0	0
$631$	−2.75054	−0.109497	−0.0547487	−	0.998500i	$-0.517436\pi$
$631$	−2.75054	−0.109497	−0.0547487	+	0.998500i	$0.517436\pi$
$632$	−27.6665	−1.10052
$633$	18.1869	0.722866
$634$	8.27778	0.328753
$635$	0	0
$636$	29.0039	1.15008
$637$	1.56409	0.0619717
$638$	−5.78072	−0.228861
$639$	−7.64571	−0.302460
$640$	0	0
$641$	−11.3606	−0.448719	−0.224359	−	0.974506i	$-0.572029\pi$
$641$	−11.3606	−0.448719	−0.224359	+	0.974506i	$0.572029\pi$
$642$	17.1017	0.674950
$643$	−33.8558	−1.33514	−0.667572	−	0.744545i	$-0.732666\pi$
$643$	−33.8558	−1.33514	−0.667572	+	0.744545i	$0.732666\pi$
$644$	−24.7225	−0.974202
$645$	0	0
$646$	−8.41282	−0.330998
$647$	−37.0829	−1.45788	−0.728939	−	0.684579i	$-0.759986\pi$
$647$	−37.0829	−1.45788	−0.728939	+	0.684579i	$0.759986\pi$
$648$	30.0339	1.17984
$649$	−13.5588	−0.532231
$650$	0	0
$651$	93.3489	3.65863
$652$	−12.6482	−0.495340
$653$	−28.2327	−1.10483	−0.552415	−	0.833569i	$-0.686294\pi$
$653$	−28.2327	−1.10483	−0.552415	+	0.833569i	$0.686294\pi$
$654$	0.886615	0.0346694
$655$	0	0
$656$	0.0198921	0.000776656 0
$657$	4.74917	0.185283
$658$	−44.7594	−1.74490
$659$	32.3003	1.25824	0.629120	−	0.777308i	$-0.283415\pi$
$659$	32.3003	1.25824	0.629120	+	0.777308i	$0.283415\pi$
$660$	0	0
$661$	−11.4499	−0.445350	−0.222675	−	0.974893i	$-0.571479\pi$
$661$	−11.4499	−0.445350	−0.222675	+	0.974893i	$0.571479\pi$
$662$	2.22346	0.0864171
$663$	0.272700	0.0105908
$664$	8.32534	0.323086
$665$	0	0
$666$	−2.28335	−0.0884781
$667$	17.5100	0.677989
$668$	14.1681	0.548179
$669$	43.4318	1.67917
$670$	0	0
$671$	−21.9495	−0.847349
$672$	48.4506	1.86902
$673$	15.0187	0.578928	0.289464	−	0.957189i	$-0.406523\pi$
$673$	15.0187	0.578928	0.289464	+	0.957189i	$0.406523\pi$
$674$	−6.56928	−0.253039
$675$	0	0
$676$	16.1291	0.620351
$677$	−3.69635	−0.142062	−0.0710312	−	0.997474i	$-0.522629\pi$
$677$	−3.69635	−0.142062	−0.0710312	+	0.997474i	$0.522629\pi$
$678$	−6.30628	−0.242191
$679$	14.6860	0.563597
$680$	0	0
$681$	−0.717939	−0.0275115
$682$	16.1931	0.620065
$683$	9.96233	0.381198	0.190599	−	0.981668i	$-0.438957\pi$
$683$	9.96233	0.381198	0.190599	+	0.981668i	$0.438957\pi$
$684$	−7.56135	−0.289116
$685$	0	0
$686$	−21.7571	−0.830690
$687$	−30.9777	−1.18187
$688$	−0.159680	−0.00608775
$689$	1.49814	0.0570746
$690$	0	0
$691$	−24.9722	−0.949986	−0.474993	−	0.879990i	$-0.657549\pi$
$691$	−24.9722	−0.949986	−0.474993	+	0.879990i	$0.657549\pi$
$692$	−30.8484	−1.17268
$693$	−5.43233	−0.206357
$694$	−11.8988	−0.451673
$695$	0	0
$696$	−21.2235	−0.804475
$697$	−0.832362	−0.0315280
$698$	26.6235	1.00771
$699$	−6.24217	−0.236100
$700$	0	0
$701$	26.1263	0.986777	0.493388	−	0.869809i	$-0.335758\pi$
$701$	26.1263	0.986777	0.493388	+	0.869809i	$0.335758\pi$
$702$	0.473762	0.0178810
$703$	−30.8231	−1.16251
$704$	8.31163	0.313256
$705$	0	0
$706$	−7.24707	−0.272747
$707$	−54.2790	−2.04137
$708$	−19.0723	−0.716780
$709$	0.447942	0.0168228	0.00841142	−	0.999965i	$-0.497323\pi$
$709$	0.447942	0.0168228	0.00841142	+	0.999965i	$0.497323\pi$
$710$	0	0
$711$	7.05500	0.264583
$712$	−29.5288	−1.10664
$713$	−49.0494	−1.83692
$714$	−8.50174	−0.318170
$715$	0	0
$716$	−26.1538	−0.977412
$717$	45.9291	1.71525
$718$	10.8014	0.403104
$719$	−22.7145	−0.847107	−0.423553	−	0.905871i	$-0.639217\pi$
$719$	−22.7145	−0.847107	−0.423553	+	0.905871i	$0.639217\pi$
$720$	0	0
$721$	−5.54842	−0.206634
$722$	45.7324	1.70199
$723$	−42.7990	−1.59171
$724$	5.69226	0.211551
$725$	0	0
$726$	13.5984	0.504684
$727$	−25.7370	−0.954534	−0.477267	−	0.878758i	$-0.658373\pi$
$727$	−25.7370	−0.954534	−0.477267	+	0.878758i	$0.658373\pi$
$728$	1.54781	0.0573658
$729$	17.7873	0.658790
$730$	0	0
$731$	6.68164	0.247129
$732$	−30.8748	−1.14116
$733$	11.8624	0.438147	0.219074	−	0.975708i	$-0.429696\pi$
$733$	11.8624	0.438147	0.219074	+	0.975708i	$0.429696\pi$
$734$	28.7794	1.06227
$735$	0	0
$736$	−25.4580	−0.938393
$737$	−20.8593	−0.768361
$738$	0.456420	0.0168011
$739$	−38.0649	−1.40024	−0.700120	−	0.714026i	$-0.746870\pi$
$739$	−38.0649	−1.40024	−0.700120	+	0.714026i	$0.746870\pi$
$740$	0	0
$741$	−2.01855	−0.0741534
$742$	−46.7063	−1.71464
$743$	8.55413	0.313821	0.156910	−	0.987613i	$-0.449847\pi$
$743$	8.55413	0.313821	0.156910	+	0.987613i	$0.449847\pi$
$744$	59.4519	2.17961
$745$	0	0
$746$	−21.4453	−0.785169
$747$	−2.12297	−0.0776756
$748$	2.41732	0.0883859
$749$	45.1401	1.64938
$750$	0	0
$751$	−29.4966	−1.07635	−0.538174	−	0.842834i	$-0.680886\pi$
$751$	−29.4966	−1.07635	−0.538174	+	0.842834i	$0.680886\pi$
$752$	0.316810	0.0115529
$753$	4.14984	0.151229
$754$	−0.420007	−0.0152958
$755$	0	0
$756$	24.2097	0.880497
$757$	24.3569	0.885265	0.442633	−	0.896703i	$-0.354045\pi$
$757$	24.3569	0.885265	0.442633	+	0.896703i	$0.354045\pi$
$758$	2.00539	0.0728389
$759$	14.7522	0.535470
$760$	0	0
$761$	23.6562	0.857537	0.428768	−	0.903414i	$-0.358948\pi$
$761$	23.6562	0.857537	0.428768	+	0.903414i	$0.358948\pi$
$762$	−7.57553	−0.274433
$763$	2.34023	0.0847221
$764$	16.1093	0.582815
$765$	0	0
$766$	−19.7321	−0.712950
$767$	−0.985139	−0.0355713
$768$	30.7263	1.10874
$769$	4.60006	0.165883	0.0829413	−	0.996554i	$-0.473569\pi$
$769$	4.60006	0.165883	0.0829413	+	0.996554i	$0.473569\pi$
$770$	0	0
$771$	−21.1229	−0.760724
$772$	−5.23284	−0.188334
$773$	9.80949	0.352823	0.176411	−	0.984317i	$-0.443551\pi$
$773$	9.80949	0.352823	0.176411	+	0.984317i	$0.443551\pi$
$774$	−3.66383	−0.131694
$775$	0	0
$776$	9.35319	0.335760
$777$	−31.1489	−1.11746
$778$	−9.42340	−0.337845
$779$	6.16124	0.220749
$780$	0	0
$781$	−18.0929	−0.647414
$782$	4.46717	0.159746
$783$	−17.1468	−0.612776
$784$	0.345144	0.0123266
$785$	0	0
$786$	11.3319	0.404196
$787$	−14.8242	−0.528426	−0.264213	−	0.964464i	$-0.585112\pi$
$787$	−14.8242	−0.528426	−0.264213	+	0.964464i	$0.585112\pi$
$788$	−1.24217	−0.0442503
$789$	16.7051	0.594719
$790$	0	0
$791$	−16.6455	−0.591846
$792$	−3.45973	−0.122936
$793$	−1.59477	−0.0566321
$794$	−12.3853	−0.439536
$795$	0	0
$796$	12.2403	0.433848
$797$	8.53761	0.302418	0.151209	−	0.988502i	$-0.451683\pi$
$797$	8.53761	0.302418	0.151209	+	0.988502i	$0.451683\pi$
$798$	62.9309	2.22773
$799$	−13.2566	−0.468983
$800$	0	0
$801$	7.52989	0.266056
$802$	27.6201	0.975301
$803$	11.2385	0.396597
$804$	−29.3413	−1.03479
$805$	0	0
$806$	1.17654	0.0414417
$807$	−20.1442	−0.709111
$808$	−34.5691	−1.21614
$809$	17.6854	0.621786	0.310893	−	0.950445i	$-0.399372\pi$
$809$	17.6854	0.621786	0.310893	+	0.950445i	$0.399372\pi$
$810$	0	0
$811$	−45.0258	−1.58107	−0.790534	−	0.612418i	$-0.790197\pi$
$811$	−45.0258	−1.58107	−0.790534	+	0.612418i	$0.790197\pi$
$812$	−21.4628	−0.753195
$813$	−8.79035	−0.308291
$814$	−5.40334	−0.189387
$815$	0	0
$816$	0.0601759	0.00210658
$817$	−49.4582	−1.73032
$818$	20.4470	0.714911
$819$	−0.394695	−0.0137917
$820$	0	0
$821$	10.8106	0.377291	0.188646	−	0.982045i	$-0.439590\pi$
$821$	10.8106	0.377291	0.188646	+	0.982045i	$0.439590\pi$
$822$	24.2325	0.845205
$823$	16.3476	0.569841	0.284921	−	0.958551i	$-0.408033\pi$
$823$	16.3476	0.569841	0.284921	+	0.958551i	$0.408033\pi$
$824$	−3.53367	−0.123101
$825$	0	0
$826$	30.7129	1.06864
$827$	27.1543	0.944248	0.472124	−	0.881532i	$-0.343487\pi$
$827$	27.1543	0.944248	0.472124	+	0.881532i	$0.343487\pi$
$828$	4.01505	0.139532
$829$	14.3838	0.499572	0.249786	−	0.968301i	$-0.419640\pi$
$829$	14.3838	0.499572	0.249786	+	0.968301i	$0.419640\pi$
$830$	0	0
$831$	57.0476	1.97896
$832$	0.603895	0.0209363
$833$	−14.4422	−0.500391
$834$	12.2834	0.425339
$835$	0	0
$836$	−17.8933	−0.618851
$837$	48.0320	1.66023
$838$	16.0427	0.554186
$839$	−49.7122	−1.71626	−0.858128	−	0.513435i	$-0.828373\pi$
$839$	−49.7122	−1.71626	−0.858128	+	0.513435i	$0.828373\pi$
$840$	0	0
$841$	−13.7987	−0.475819
$842$	−11.2612	−0.388088
$843$	9.68770	0.333662
$844$	11.7134	0.403192
$845$	0	0
$846$	7.26915	0.249918
$847$	35.8931	1.23330
$848$	0.330590	0.0113525
$849$	14.5367	0.498900
$850$	0	0
$851$	16.3669	0.561051
$852$	−25.4500	−0.871903
$853$	0.189159	0.00647668	0.00323834	−	0.999995i	$-0.498969\pi$
$853$	0.189159	0.00647668	0.00323834	+	0.999995i	$0.498969\pi$
$854$	49.7190	1.70135
$855$	0	0
$856$	28.7487	0.982612
$857$	−12.1359	−0.414554	−0.207277	−	0.978282i	$-0.566460\pi$
$857$	−12.1359	−0.414554	−0.207277	+	0.978282i	$0.566460\pi$
$858$	−0.353857	−0.0120805
$859$	−8.97076	−0.306078	−0.153039	−	0.988220i	$-0.548906\pi$
$859$	−8.97076	−0.306078	−0.153039	+	0.988220i	$0.548906\pi$
$860$	0	0
$861$	6.22636	0.212194
$862$	−15.0303	−0.511933
$863$	48.5734	1.65346	0.826729	−	0.562600i	$-0.190199\pi$
$863$	48.5734	1.65346	0.826729	+	0.562600i	$0.190199\pi$
$864$	24.9299	0.848132
$865$	0	0
$866$	20.9758	0.712788
$867$	30.2692	1.02800
$868$	60.1220	2.04067
$869$	16.6950	0.566340
$870$	0	0
$871$	−1.51556	−0.0513529
$872$	1.49044	0.0504728
$873$	−2.38508	−0.0807226
$874$	−33.0665	−1.11849
$875$	0	0
$876$	15.8084	0.534116
$877$	33.1053	1.11789	0.558944	−	0.829206i	$-0.311207\pi$
$877$	33.1053	1.11789	0.558944	+	0.829206i	$0.311207\pi$
$878$	0.631799	0.0213222
$879$	11.2824	0.380546
$880$	0	0
$881$	34.8931	1.17558	0.587790	−	0.809014i	$-0.299998\pi$
$881$	34.8931	1.17558	0.587790	+	0.809014i	$0.299998\pi$
$882$	7.91927	0.266656
$883$	15.7765	0.530921	0.265461	−	0.964122i	$-0.414476\pi$
$883$	15.7765	0.530921	0.265461	+	0.964122i	$0.414476\pi$
$884$	0.175634	0.00590721
$885$	0	0
$886$	−22.3437	−0.750651
$887$	52.7863	1.77239	0.886196	−	0.463311i	$-0.153339\pi$
$887$	52.7863	1.77239	0.886196	+	0.463311i	$0.153339\pi$
$888$	−19.8380	−0.665720
$889$	−19.9957	−0.670635
$890$	0	0
$891$	−18.1236	−0.607163
$892$	27.9726	0.936591
$893$	98.1265	3.28368
$894$	−6.88773	−0.230360
$895$	0	0
$896$	31.4156	1.04952
$897$	1.07184	0.0357878
$898$	−26.3131	−0.878081
$899$	−42.5821	−1.42019
$900$	0	0
$901$	−13.8332	−0.460850
$902$	1.08008	0.0359627
$903$	−49.9810	−1.66326
$904$	−10.6012	−0.352589
$905$	0	0
$906$	22.3997	0.744181
$907$	−26.3884	−0.876212	−0.438106	−	0.898923i	$-0.644351\pi$
$907$	−26.3884	−0.876212	−0.438106	+	0.898923i	$0.644351\pi$
$908$	−0.462394	−0.0153451
$909$	8.81517	0.292381
$910$	0	0
$911$	54.8309	1.81663	0.908314	−	0.418289i	$-0.137370\pi$
$911$	54.8309	1.81663	0.908314	+	0.418289i	$0.137370\pi$
$912$	−0.445429	−0.0147496
$913$	−5.02383	−0.166264
$914$	5.26858	0.174269
$915$	0	0
$916$	−19.9514	−0.659213
$917$	29.9107	0.987740
$918$	−4.37451	−0.144380
$919$	−0.151198	−0.00498757	−0.00249378	−	0.999997i	$-0.500794\pi$
$919$	−0.151198	−0.00498757	−0.00249378	+	0.999997i	$0.500794\pi$
$920$	0	0
$921$	−44.0757	−1.45234
$922$	−5.39404	−0.177643
$923$	−1.31457	−0.0432695
$924$	−18.0824	−0.594867
$925$	0	0
$926$	15.2014	0.499549
$927$	0.901090	0.0295957
$928$	−22.1013	−0.725510
$929$	−53.5301	−1.75627	−0.878134	−	0.478416i	$-0.841211\pi$
$929$	−53.5301	−1.75627	−0.878134	+	0.478416i	$0.841211\pi$
$930$	0	0
$931$	106.903	3.50359
$932$	−4.02031	−0.131690
$933$	−1.63661	−0.0535802
$934$	−23.3363	−0.763587
$935$	0	0
$936$	−0.251372	−0.00821636
$937$	−35.9511	−1.17447	−0.587236	−	0.809416i	$-0.699784\pi$
$937$	−35.9511	−1.17447	−0.587236	+	0.809416i	$0.699784\pi$
$938$	47.2495	1.54275
$939$	46.7801	1.52661
$940$	0	0
$941$	11.7311	0.382423	0.191211	−	0.981549i	$-0.438758\pi$
$941$	11.7311	0.382423	0.191211	+	0.981549i	$0.438758\pi$
$942$	−26.2694	−0.855903
$943$	−3.27159	−0.106538
$944$	−0.217388	−0.00707537
$945$	0	0
$946$	−8.67012	−0.281890
$947$	−41.2862	−1.34162	−0.670810	−	0.741629i	$-0.734053\pi$
$947$	−41.2862	−1.34162	−0.670810	+	0.741629i	$0.734053\pi$
$948$	23.4837	0.762716
$949$	0.816550	0.0265063
$950$	0	0
$951$	−18.3393	−0.594691
$952$	−14.2918	−0.463201
$953$	25.0112	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$953$	25.0112	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$954$	7.58533	0.245584
$955$	0	0
$956$	29.5809	0.956716
$957$	12.8071	0.413994
$958$	−15.2086	−0.491369
$959$	63.9619	2.06544
$960$	0	0
$961$	88.2822	2.84781
$962$	−0.392589	−0.0126576
$963$	−7.33097	−0.236237
$964$	−27.5650	−0.887810
$965$	0	0
$966$	−33.4160	−1.07514
$967$	−9.45167	−0.303945	−0.151973	−	0.988385i	$-0.548563\pi$
$967$	−9.45167	−0.303945	−0.151973	+	0.988385i	$0.548563\pi$
$968$	22.8595	0.734733
$969$	18.6384	0.598753
$970$	0	0
$971$	−19.0888	−0.612589	−0.306295	−	0.951937i	$-0.599089\pi$
$971$	−19.0888	−0.612589	−0.306295	+	0.951937i	$0.599089\pi$
$972$	−9.10451	−0.292027
$973$	32.4222	1.03941
$974$	−11.6610	−0.373644
$975$	0	0
$976$	−0.351914	−0.0112645
$977$	−6.91399	−0.221198	−0.110599	−	0.993865i	$-0.535277\pi$
$977$	−6.91399	−0.221198	−0.110599	+	0.993865i	$0.535277\pi$
$978$	−17.0958	−0.546665
$979$	17.8188	0.569491
$980$	0	0
$981$	−0.380065	−0.0121345
$982$	2.40996	0.0769050
$983$	37.4654	1.19496	0.597481	−	0.801883i	$-0.296168\pi$
$983$	37.4654	1.19496	0.597481	+	0.801883i	$0.296168\pi$
$984$	3.96543	0.126413
$985$	0	0
$986$	3.87816	0.123506
$987$	99.1637	3.15642
$988$	−1.30006	−0.0413605
$989$	26.2621	0.835086
$990$	0	0
$991$	−29.5473	−0.938601	−0.469300	−	0.883039i	$-0.655494\pi$
$991$	−29.5473	−0.938601	−0.469300	+	0.883039i	$0.655494\pi$
$992$	61.9106	1.96566
$993$	−4.92603	−0.156323
$994$	40.9832	1.29991
$995$	0	0
$996$	−7.06667	−0.223916
$997$	10.9241	0.345970	0.172985	−	0.984924i	$-0.444659\pi$
$997$	10.9241	0.345970	0.172985	+	0.984924i	$0.444659\pi$
$998$	−17.1371	−0.542466
$999$	−16.0274	−0.507085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.r.1.32		49
5.2	odd	4		985.2.b.a.789.62	yes	98
5.3	odd	4		985.2.b.a.789.37	✓	98
5.4	even	2		4925.2.a.s.1.18		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.37	✓	98	5.3	odd	4
985.2.b.a.789.62	yes	98	5.2	odd	4
4925.2.a.r.1.32		49	1.1	even	1	trivial
4925.2.a.s.1.18		49	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.870537 q^{2} -1.92866 q^{3} -1.24217 q^{4} -1.67897 q^{6} -4.43166 q^{7} -2.82242 q^{8} +0.719722 q^{9} +1.70316 q^{11} +2.39571 q^{12} +0.123746 q^{13} -3.85792 q^{14} +0.0273066 q^{16} -1.14261 q^{17} +0.626545 q^{18} +8.45775 q^{19} +8.54715 q^{21} +1.48266 q^{22} -4.49103 q^{23} +5.44349 q^{24} +0.107725 q^{26} +4.39788 q^{27} +5.50485 q^{28} -3.89888 q^{29} +10.9216 q^{31} +5.66862 q^{32} -3.28481 q^{33} -0.994687 q^{34} -0.894014 q^{36} -3.64436 q^{37} +7.36279 q^{38} -0.238663 q^{39} +0.728472 q^{41} +7.44061 q^{42} -5.84768 q^{43} -2.11560 q^{44} -3.90961 q^{46} +11.6020 q^{47} -0.0526651 q^{48} +12.6396 q^{49} +2.20371 q^{51} -0.153713 q^{52} +12.1066 q^{53} +3.82851 q^{54} +12.5080 q^{56} -16.3121 q^{57} -3.39412 q^{58} -7.96100 q^{59} -12.8875 q^{61} +9.50769 q^{62} -3.18956 q^{63} +4.88013 q^{64} -2.85955 q^{66} -12.2474 q^{67} +1.41932 q^{68} +8.66166 q^{69} -10.6231 q^{71} -2.03136 q^{72} +6.59861 q^{73} -3.17255 q^{74} -10.5059 q^{76} -7.54781 q^{77} -0.207765 q^{78} +9.80240 q^{79} -10.6412 q^{81} +0.634162 q^{82} -2.94971 q^{83} -10.6170 q^{84} -5.09062 q^{86} +7.51960 q^{87} -4.80703 q^{88} +10.4622 q^{89} -0.548398 q^{91} +5.57860 q^{92} -21.0641 q^{93} +10.0999 q^{94} -10.9328 q^{96} -3.31388 q^{97} +11.0032 q^{98} +1.22580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 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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