LMFDB - Embedded newform 4925.2.a.q.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	\(q-1.27073 q^{2} +2.56743 q^{3} -0.385233 q^{4} -3.26253 q^{6} -0.840739 q^{7} +3.03100 q^{8} +3.59172 q^{9} -4.34010 q^{11} -0.989061 q^{12} +5.35753 q^{13} +1.06836 q^{14} -3.08113 q^{16} +3.54972 q^{17} -4.56412 q^{18} +1.48982 q^{19} -2.15854 q^{21} +5.51512 q^{22} +8.79752 q^{23} +7.78189 q^{24} -6.80800 q^{26} +1.51920 q^{27} +0.323881 q^{28} +0.249992 q^{29} -0.752380 q^{31} -2.14670 q^{32} -11.1429 q^{33} -4.51075 q^{34} -1.38365 q^{36} +4.94813 q^{37} -1.89316 q^{38} +13.7551 q^{39} +2.52610 q^{41} +2.74294 q^{42} +3.98446 q^{43} +1.67195 q^{44} -11.1793 q^{46} -2.10694 q^{47} -7.91060 q^{48} -6.29316 q^{49} +9.11368 q^{51} -2.06390 q^{52} -10.9059 q^{53} -1.93050 q^{54} -2.54828 q^{56} +3.82501 q^{57} -0.317674 q^{58} -7.97796 q^{59} -11.5832 q^{61} +0.956075 q^{62} -3.01970 q^{63} +8.89014 q^{64} +14.1597 q^{66} +0.0288247 q^{67} -1.36747 q^{68} +22.5871 q^{69} +11.3323 q^{71} +10.8865 q^{72} +12.2568 q^{73} -6.28775 q^{74} -0.573928 q^{76} +3.64889 q^{77} -17.4791 q^{78} -3.43029 q^{79} -6.87471 q^{81} -3.21001 q^{82} -0.952959 q^{83} +0.831543 q^{84} -5.06319 q^{86} +0.641838 q^{87} -13.1548 q^{88} -6.14398 q^{89} -4.50428 q^{91} -3.38910 q^{92} -1.93169 q^{93} +2.67736 q^{94} -5.51151 q^{96} +5.17024 q^{97} +7.99693 q^{98} -15.5884 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 2 q^{2} + q^{3} + 50 q^{4} + 8 q^{6} - 4 q^{7} + 50 q^{9} + 17 q^{11} - 2 q^{12} - 3 q^{13} + 14 q^{14} + 72 q^{16} - 10 q^{17} - 4 q^{18} + 54 q^{19} + 15 q^{21} - 11 q^{22} + 4 q^{23} + 28 q^{24}+ \cdots + 40 q^{99}+O(q^{100}) $ 37 * q + 2 * q^2 + q^3 + 50 * q^4 + 8 * q^6 - 4 * q^7 + 50 * q^9 + 17 * q^11 - 2 * q^12 - 3 * q^13 + 14 * q^14 + 72 * q^16 - 10 * q^17 - 4 * q^18 + 54 * q^19 + 15 * q^21 - 11 * q^22 + 4 * q^23 + 28 * q^24 + 19 * q^26 - 2 * q^27 + 3 * q^28 - 4 * q^29 + 58 * q^31 + 22 * q^32 - 7 * q^33 + 63 * q^34 + 66 * q^36 + 7 * q^37 + 12 * q^38 + 23 * q^39 + 17 * q^41 - 17 * q^42 + 11 * q^43 + 34 * q^44 + 41 * q^46 - 19 * q^47 + 10 * q^48 + 81 * q^49 + 39 * q^51 + 31 * q^52 - 14 * q^53 + 5 * q^54 + 21 * q^56 + 24 * q^57 - 15 * q^58 + 56 * q^59 + 40 * q^61 + 4 * q^62 + 5 * q^63 + 78 * q^64 + 22 * q^66 + 2 * q^67 - 3 * q^68 + 24 * q^69 + 31 * q^71 + 85 * q^72 + 4 * q^73 - 32 * q^74 + 137 * q^76 + 16 * q^77 - 113 * q^78 + 17 * q^79 + 85 * q^81 + 52 * q^82 - 18 * q^83 - 32 * q^84 + 35 * q^86 + 39 * q^87 - 63 * q^88 + 23 * q^89 + 152 * q^91 + 75 * q^92 + 8 * q^93 + 32 * q^94 + 72 * q^96 + 34 * q^97 - 40 * q^98 + 40 * 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 $

$2$	−1.27073	−0.898545	−0.449273	−	0.893395i	$-0.648317\pi$
$2$	−1.27073	−0.898545	−0.449273	+	0.893395i	$0.648317\pi$
$3$	2.56743	1.48231	0.741154	−	0.671335i	$-0.234279\pi$
$3$	2.56743	1.48231	0.741154	+	0.671335i	$0.234279\pi$
$4$	−0.385233	−0.192617
$5$	0	0
$5$	0	0
$6$	−3.26253	−1.33192
$7$	−0.840739	−0.317770	−0.158885	−	0.987297i	$-0.550790\pi$
$7$	−0.840739	−0.317770	−0.158885	+	0.987297i	$0.550790\pi$
$8$	3.03100	1.07162
$9$	3.59172	1.19724
$10$	0	0
$11$	−4.34010	−1.30859	−0.654295	−	0.756240i	$-0.727034\pi$
$11$	−4.34010	−1.30859	−0.654295	+	0.756240i	$0.727034\pi$
$12$	−0.989061	−0.285517
$13$	5.35753	1.48591	0.742956	−	0.669341i	$-0.233423\pi$
$13$	5.35753	1.48591	0.742956	+	0.669341i	$0.233423\pi$
$14$	1.06836	0.285530
$15$	0	0
$16$	−3.08113	−0.770282
$17$	3.54972	0.860934	0.430467	−	0.902606i	$-0.358349\pi$
$17$	3.54972	0.860934	0.430467	+	0.902606i	$0.358349\pi$
$18$	−4.56412	−1.07577
$19$	1.48982	0.341788	0.170894	−	0.985289i	$-0.445334\pi$
$19$	1.48982	0.341788	0.170894	+	0.985289i	$0.445334\pi$
$20$	0	0
$21$	−2.15854	−0.471033
$22$	5.51512	1.17583
$23$	8.79752	1.83441	0.917205	−	0.398415i	$-0.130440\pi$
$23$	8.79752	1.83441	0.917205	+	0.398415i	$0.130440\pi$
$24$	7.78189	1.58847
$25$	0	0
$26$	−6.80800	−1.33516
$27$	1.51920	0.292370
$28$	0.323881	0.0612077
$29$	0.249992	0.0464223	0.0232112	−	0.999731i	$-0.492611\pi$
$29$	0.249992	0.0464223	0.0232112	+	0.999731i	$0.492611\pi$
$30$	0	0
$31$	−0.752380	−0.135131	−0.0675657	−	0.997715i	$-0.521523\pi$
$31$	−0.752380	−0.135131	−0.0675657	+	0.997715i	$0.521523\pi$
$32$	−2.14670	−0.379487
$33$	−11.1429	−1.93973
$34$	−4.51075	−0.773588
$35$	0	0
$36$	−1.38365	−0.230608
$37$	4.94813	0.813467	0.406733	−	0.913547i	$-0.366668\pi$
$37$	4.94813	0.813467	0.406733	+	0.913547i	$0.366668\pi$
$38$	−1.89316	−0.307112
$39$	13.7551	2.20258
$40$	0	0
$41$	2.52610	0.394511	0.197256	−	0.980352i	$-0.436797\pi$
$41$	2.52610	0.394511	0.197256	+	0.980352i	$0.436797\pi$
$42$	2.74294	0.423244
$43$	3.98446	0.607624	0.303812	−	0.952732i	$-0.401740\pi$
$43$	3.98446	0.607624	0.303812	+	0.952732i	$0.401740\pi$
$44$	1.67195	0.252056
$45$	0	0
$46$	−11.1793	−1.64830
$47$	−2.10694	−0.307329	−0.153665	−	0.988123i	$-0.549108\pi$
$47$	−2.10694	−0.307329	−0.153665	+	0.988123i	$0.549108\pi$
$48$	−7.91060	−1.14180
$49$	−6.29316	−0.899022
$50$	0	0
$51$	9.11368	1.27617
$52$	−2.06390	−0.286211
$53$	−10.9059	−1.49804	−0.749018	−	0.662550i	$-0.769474\pi$
$53$	−10.9059	−1.49804	−0.749018	+	0.662550i	$0.769474\pi$
$54$	−1.93050	−0.262708
$55$	0	0
$56$	−2.54828	−0.340528
$57$	3.82501	0.506635
$58$	−0.317674	−0.0417126
$59$	−7.97796	−1.03864	−0.519321	−	0.854579i	$-0.673815\pi$
$59$	−7.97796	−1.03864	−0.519321	+	0.854579i	$0.673815\pi$
$60$	0	0
$61$	−11.5832	−1.48307	−0.741536	−	0.670914i	$-0.765902\pi$
$61$	−11.5832	−1.48307	−0.741536	+	0.670914i	$0.765902\pi$
$62$	0.956075	0.121422
$63$	−3.01970	−0.380446
$64$	8.89014	1.11127
$65$	0	0
$66$	14.1597	1.74294
$67$	0.0288247	0.00352150	0.00176075	−	0.999998i	$-0.499440\pi$
$67$	0.0288247	0.00352150	0.00176075	+	0.999998i	$0.499440\pi$
$68$	−1.36747	−0.165830
$69$	22.5871	2.71916
$70$	0	0
$71$	11.3323	1.34490	0.672448	−	0.740144i	$-0.265243\pi$
$71$	11.3323	1.34490	0.672448	+	0.740144i	$0.265243\pi$
$72$	10.8865	1.28299
$73$	12.2568	1.43455	0.717273	−	0.696792i	$-0.245390\pi$
$73$	12.2568	1.43455	0.717273	+	0.696792i	$0.245390\pi$
$74$	−6.28775	−0.730937
$75$	0	0
$76$	−0.573928	−0.0658340
$77$	3.64889	0.415830
$78$	−17.4791	−1.97912
$79$	−3.43029	−0.385938	−0.192969	−	0.981205i	$-0.561812\pi$
$79$	−3.43029	−0.385938	−0.192969	+	0.981205i	$0.561812\pi$
$80$	0	0
$81$	−6.87471	−0.763857
$82$	−3.21001	−0.354486
$83$	−0.952959	−0.104601	−0.0523004	−	0.998631i	$-0.516655\pi$
$83$	−0.952959	−0.104601	−0.0523004	+	0.998631i	$0.516655\pi$
$84$	0.831543	0.0907287
$85$	0	0
$86$	−5.06319	−0.545978
$87$	0.641838	0.0688123
$88$	−13.1548	−1.40231
$89$	−6.14398	−0.651261	−0.325631	−	0.945497i	$-0.605577\pi$
$89$	−6.14398	−0.651261	−0.325631	+	0.945497i	$0.605577\pi$
$90$	0	0
$91$	−4.50428	−0.472177
$92$	−3.38910	−0.353338
$93$	−1.93169	−0.200306
$94$	2.67736	0.276149
$95$	0	0
$96$	−5.51151	−0.562516
$97$	5.17024	0.524959	0.262479	−	0.964938i	$-0.415460\pi$
$97$	5.17024	0.524959	0.262479	+	0.964938i	$0.415460\pi$
$98$	7.99693	0.807812
$99$	−15.5884	−1.56670
$100$	0	0
$101$	9.56484	0.951737	0.475868	−	0.879516i	$-0.342134\pi$
$101$	9.56484	0.951737	0.475868	+	0.879516i	$0.342134\pi$
$102$	−11.5811	−1.14670
$103$	−7.55823	−0.744735	−0.372367	−	0.928085i	$-0.621454\pi$
$103$	−7.55823	−0.744735	−0.372367	+	0.928085i	$0.621454\pi$
$104$	16.2387	1.59233
$105$	0	0
$106$	13.8585	1.34605
$107$	5.89996	0.570371	0.285185	−	0.958472i	$-0.407945\pi$
$107$	5.89996	0.570371	0.285185	+	0.958472i	$0.407945\pi$
$108$	−0.585246	−0.0563154
$109$	15.0084	1.43755	0.718773	−	0.695245i	$-0.244704\pi$
$109$	15.0084	1.43755	0.718773	+	0.695245i	$0.244704\pi$
$110$	0	0
$111$	12.7040	1.20581
$112$	2.59043	0.244772
$113$	8.53901	0.803283	0.401641	−	0.915797i	$-0.368440\pi$
$113$	8.53901	0.803283	0.401641	+	0.915797i	$0.368440\pi$
$114$	−4.86057	−0.455234
$115$	0	0
$116$	−0.0963052	−0.00894172
$117$	19.2427	1.77899
$118$	10.1379	0.933266
$119$	−2.98439	−0.273579
$120$	0	0
$121$	7.83648	0.712407
$122$	14.7191	1.33261
$123$	6.48561	0.584787
$124$	0.289842	0.0260286
$125$	0	0
$126$	3.83724	0.341848
$127$	9.88262	0.876940	0.438470	−	0.898746i	$-0.355520\pi$
$127$	9.88262	0.876940	0.438470	+	0.898746i	$0.355520\pi$
$128$	−7.00361	−0.619038
$129$	10.2298	0.900687
$130$	0	0
$131$	6.11501	0.534270	0.267135	−	0.963659i	$-0.413923\pi$
$131$	6.11501	0.534270	0.267135	+	0.963659i	$0.413923\pi$
$132$	4.29263	0.373625
$133$	−1.25255	−0.108610
$134$	−0.0366285	−0.00316422
$135$	0	0
$136$	10.7592	0.922594
$137$	8.43937	0.721024	0.360512	−	0.932755i	$-0.382602\pi$
$137$	8.43937	0.721024	0.360512	+	0.932755i	$0.382602\pi$
$138$	−28.7022	−2.44329
$139$	10.1596	0.861729	0.430864	−	0.902417i	$-0.358209\pi$
$139$	10.1596	0.861729	0.430864	+	0.902417i	$0.358209\pi$
$140$	0	0
$141$	−5.40944	−0.455557
$142$	−14.4003	−1.20845
$143$	−23.2522	−1.94445
$144$	−11.0665	−0.922212
$145$	0	0
$146$	−15.5751	−1.28901
$147$	−16.1573	−1.33263
$148$	−1.90618	−0.156687
$149$	2.67438	0.219094	0.109547	−	0.993982i	$-0.465060\pi$
$149$	2.67438	0.219094	0.109547	+	0.993982i	$0.465060\pi$
$150$	0	0
$151$	3.11672	0.253635	0.126818	−	0.991926i	$-0.459524\pi$
$151$	3.11672	0.253635	0.126818	+	0.991926i	$0.459524\pi$
$152$	4.51564	0.366267
$153$	12.7496	1.03074
$154$	−4.63678	−0.373642
$155$	0	0
$156$	−5.29892	−0.424253
$157$	1.50182	0.119859	0.0599293	−	0.998203i	$-0.480912\pi$
$157$	1.50182	0.119859	0.0599293	+	0.998203i	$0.480912\pi$
$158$	4.35899	0.346783
$159$	−28.0001	−2.22055
$160$	0	0
$161$	−7.39642	−0.582920
$162$	8.73593	0.686360
$163$	1.19875	0.0938935	0.0469468	−	0.998897i	$-0.485051\pi$
$163$	1.19875	0.0938935	0.0469468	+	0.998897i	$0.485051\pi$
$164$	−0.973139	−0.0759894
$165$	0	0
$166$	1.21096	0.0939886
$167$	0.493224	0.0381668	0.0190834	−	0.999818i	$-0.493925\pi$
$167$	0.493224	0.0381668	0.0190834	+	0.999818i	$0.493925\pi$
$168$	−6.54254	−0.504768
$169$	15.7031	1.20793
$170$	0	0
$171$	5.35101	0.409202
$172$	−1.53495	−0.117039
$173$	−13.3902	−1.01804	−0.509020	−	0.860755i	$-0.669992\pi$
$173$	−13.3902	−1.01804	−0.509020	+	0.860755i	$0.669992\pi$
$174$	−0.815606	−0.0618309
$175$	0	0
$176$	13.3724	1.00798
$177$	−20.4829	−1.53959
$178$	7.80737	0.585187
$179$	−10.2304	−0.764653	−0.382327	−	0.924027i	$-0.624877\pi$
$179$	−10.2304	−0.764653	−0.382327	+	0.924027i	$0.624877\pi$
$180$	0	0
$181$	14.7223	1.09430	0.547149	−	0.837035i	$-0.315713\pi$
$181$	14.7223	1.09430	0.547149	+	0.837035i	$0.315713\pi$
$182$	5.72375	0.424273
$183$	−29.7390	−2.19837
$184$	26.6653	1.96579
$185$	0	0
$186$	2.45466	0.179984
$187$	−15.4061	−1.12661
$188$	0.811664	0.0591967
$189$	−1.27725	−0.0929063
$190$	0	0
$191$	13.7718	0.996490	0.498245	−	0.867036i	$-0.333978\pi$
$191$	13.7718	0.996490	0.498245	+	0.867036i	$0.333978\pi$
$192$	22.8249	1.64724
$193$	4.20388	0.302602	0.151301	−	0.988488i	$-0.451654\pi$
$193$	4.20388	0.302602	0.151301	+	0.988488i	$0.451654\pi$
$194$	−6.57001	−0.471699
$195$	0	0
$196$	2.42433	0.173167
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	19.8088	1.40775
$199$	25.7088	1.82245	0.911226	−	0.411908i	$-0.135137\pi$
$199$	25.7088	1.82245	0.911226	+	0.411908i	$0.135137\pi$
$200$	0	0
$201$	0.0740055	0.00521995
$202$	−12.1544	−0.855178
$203$	−0.210178	−0.0147516
$204$	−3.51089	−0.245812
$205$	0	0
$206$	9.60451	0.669178
$207$	31.5982	2.19623
$208$	−16.5072	−1.14457
$209$	−6.46596	−0.447260
$210$	0	0
$211$	12.1797	0.838485	0.419242	−	0.907874i	$-0.362296\pi$
$211$	12.1797	0.838485	0.419242	+	0.907874i	$0.362296\pi$
$212$	4.20130	0.288547
$213$	29.0949	1.99355
$214$	−7.49728	−0.512504
$215$	0	0
$216$	4.60469	0.313310
$217$	0.632555	0.0429406
$218$	−19.0717	−1.29170
$219$	31.4685	2.12644
$220$	0	0
$221$	19.0177	1.27927
$222$	−16.1434	−1.08347
$223$	−4.46872	−0.299247	−0.149624	−	0.988743i	$-0.547806\pi$
$223$	−4.46872	−0.299247	−0.149624	+	0.988743i	$0.547806\pi$
$224$	1.80482	0.120589
$225$	0	0
$226$	−10.8508	−0.721786
$227$	10.5115	0.697671	0.348836	−	0.937184i	$-0.386577\pi$
$227$	10.5115	0.697671	0.348836	+	0.937184i	$0.386577\pi$
$228$	−1.47352	−0.0975863
$229$	25.5028	1.68527	0.842635	−	0.538485i	$-0.181003\pi$
$229$	25.5028	1.68527	0.842635	+	0.538485i	$0.181003\pi$
$230$	0	0
$231$	9.36829	0.616389
$232$	0.757725	0.0497471
$233$	3.51329	0.230163	0.115082	−	0.993356i	$-0.463287\pi$
$233$	3.51329	0.230163	0.115082	+	0.993356i	$0.463287\pi$
$234$	−24.4524	−1.59850
$235$	0	0
$236$	3.07337	0.200060
$237$	−8.80705	−0.572080
$238$	3.79237	0.245823
$239$	29.5503	1.91145	0.955725	−	0.294260i	$-0.0950732\pi$
$239$	29.5503	1.91145	0.955725	+	0.294260i	$0.0950732\pi$
$240$	0	0
$241$	−29.2735	−1.88567	−0.942836	−	0.333257i	$-0.891852\pi$
$241$	−29.2735	−1.88567	−0.942836	+	0.333257i	$0.891852\pi$
$242$	−9.95808	−0.640130
$243$	−22.2080	−1.42464
$244$	4.46222	0.285664
$245$	0	0
$246$	−8.24149	−0.525458
$247$	7.98174	0.507866
$248$	−2.28046	−0.144809
$249$	−2.44666	−0.155051
$250$	0	0
$251$	−0.554757	−0.0350159	−0.0175080	−	0.999847i	$-0.505573\pi$
$251$	−0.554757	−0.0350159	−0.0175080	+	0.999847i	$0.505573\pi$
$252$	1.16329	0.0732803
$253$	−38.1821	−2.40049
$254$	−12.5582	−0.787971
$255$	0	0
$256$	−8.88055	−0.555034
$257$	6.20512	0.387065	0.193532	−	0.981094i	$-0.438006\pi$
$257$	6.20512	0.387065	0.193532	+	0.981094i	$0.438006\pi$
$258$	−12.9994	−0.809308
$259$	−4.16008	−0.258495
$260$	0	0
$261$	0.897901	0.0555787
$262$	−7.77055	−0.480066
$263$	29.1872	1.79976	0.899880	−	0.436138i	$-0.143654\pi$
$263$	29.1872	1.79976	0.899880	+	0.436138i	$0.143654\pi$
$264$	−33.7742	−2.07866
$265$	0	0
$266$	1.59166	0.0975908
$267$	−15.7743	−0.965370
$268$	−0.0111042	−0.000678299 0
$269$	−2.57989	−0.157299	−0.0786495	−	0.996902i	$-0.525061\pi$
$269$	−2.57989	−0.157299	−0.0786495	+	0.996902i	$0.525061\pi$
$270$	0	0
$271$	13.2844	0.806972	0.403486	−	0.914986i	$-0.367798\pi$
$271$	13.2844	0.806972	0.403486	+	0.914986i	$0.367798\pi$
$272$	−10.9371	−0.663162
$273$	−11.5645	−0.699913
$274$	−10.7242	−0.647872
$275$	0	0
$276$	−8.70129	−0.523756
$277$	−24.3551	−1.46336	−0.731679	−	0.681649i	$-0.761263\pi$
$277$	−24.3551	−1.46336	−0.731679	+	0.681649i	$0.761263\pi$
$278$	−12.9102	−0.774302
$279$	−2.70234	−0.161785
$280$	0	0
$281$	−28.9239	−1.72545	−0.862727	−	0.505669i	$-0.831246\pi$
$281$	−28.9239	−1.72545	−0.862727	+	0.505669i	$0.831246\pi$
$282$	6.87396	0.409338
$283$	21.0715	1.25257	0.626285	−	0.779594i	$-0.284574\pi$
$283$	21.0715	1.25257	0.626285	+	0.779594i	$0.284574\pi$
$284$	−4.36558	−0.259049
$285$	0	0
$286$	29.5474	1.74717
$287$	−2.12380	−0.125364
$288$	−7.71034	−0.454336
$289$	−4.39948	−0.258793
$290$	0	0
$291$	13.2743	0.778151
$292$	−4.72172	−0.276318
$293$	19.0068	1.11039	0.555195	−	0.831720i	$-0.312643\pi$
$293$	19.0068	1.11039	0.555195	+	0.831720i	$0.312643\pi$
$294$	20.5316	1.19743
$295$	0	0
$296$	14.9978	0.871727
$297$	−6.59348	−0.382593
$298$	−3.39843	−0.196866
$299$	47.1330	2.72577
$300$	0	0
$301$	−3.34989	−0.193085
$302$	−3.96053	−0.227903
$303$	24.5571	1.41077
$304$	−4.59032	−0.263273
$305$	0	0
$306$	−16.2014	−0.926170
$307$	15.4963	0.884418	0.442209	−	0.896912i	$-0.354195\pi$
$307$	15.4963	0.884418	0.442209	+	0.896912i	$0.354195\pi$
$308$	−1.40568	−0.0800958
$309$	−19.4053	−1.10393
$310$	0	0
$311$	−16.2558	−0.921782	−0.460891	−	0.887457i	$-0.652470\pi$
$311$	−16.2558	−0.921782	−0.460891	+	0.887457i	$0.652470\pi$
$312$	41.6917	2.36033
$313$	−29.7230	−1.68004	−0.840022	−	0.542552i	$-0.817458\pi$
$313$	−29.7230	−1.68004	−0.840022	+	0.542552i	$0.817458\pi$
$314$	−1.90842	−0.107698
$315$	0	0
$316$	1.32146	0.0743381
$317$	−16.8258	−0.945031	−0.472515	−	0.881322i	$-0.656654\pi$
$317$	−16.8258	−0.945031	−0.472515	+	0.881322i	$0.656654\pi$
$318$	35.5807	1.99527
$319$	−1.08499	−0.0607478
$320$	0	0
$321$	15.1478	0.845465
$322$	9.39889	0.523780
$323$	5.28844	0.294257
$324$	2.64837	0.147132
$325$	0	0
$326$	−1.52330	−0.0843676
$327$	38.5331	2.13089
$328$	7.65662	0.422766
$329$	1.77139	0.0976598
$330$	0	0
$331$	23.5179	1.29266	0.646330	−	0.763058i	$-0.276303\pi$
$331$	23.5179	1.29266	0.646330	+	0.763058i	$0.276303\pi$
$332$	0.367111	0.0201479
$333$	17.7723	0.973915
$334$	−0.626756	−0.0342946
$335$	0	0
$336$	6.65075	0.362828
$337$	4.98221	0.271398	0.135699	−	0.990750i	$-0.456672\pi$
$337$	4.98221	0.271398	0.135699	+	0.990750i	$0.456672\pi$
$338$	−19.9545	−1.08538
$339$	21.9234	1.19071
$340$	0	0
$341$	3.26540	0.176832
$342$	−6.79971	−0.367686
$343$	11.1761	0.603452
$344$	12.0769	0.651142
$345$	0	0
$346$	17.0154	0.914754
$347$	−35.9354	−1.92911	−0.964557	−	0.263874i	$-0.915000\pi$
$347$	−35.9354	−1.92911	−0.964557	+	0.263874i	$0.915000\pi$
$348$	−0.247257	−0.0132544
$349$	4.84440	0.259315	0.129657	−	0.991559i	$-0.458612\pi$
$349$	4.84440	0.259315	0.129657	+	0.991559i	$0.458612\pi$
$350$	0	0
$351$	8.13916	0.434436
$352$	9.31690	0.496592
$353$	29.6559	1.57842	0.789211	−	0.614122i	$-0.210490\pi$
$353$	29.6559	1.57842	0.789211	+	0.614122i	$0.210490\pi$
$354$	26.0283	1.38339
$355$	0	0
$356$	2.36687	0.125444
$357$	−7.66223	−0.405528
$358$	13.0001	0.687075
$359$	−33.3896	−1.76224	−0.881118	−	0.472896i	$-0.843209\pi$
$359$	−33.3896	−1.76224	−0.881118	+	0.472896i	$0.843209\pi$
$360$	0	0
$361$	−16.7804	−0.883181
$362$	−18.7081	−0.983276
$363$	20.1196	1.05601
$364$	1.73520	0.0909492
$365$	0	0
$366$	37.7904	1.97533
$367$	−36.2306	−1.89122	−0.945612	−	0.325296i	$-0.894536\pi$
$367$	−36.2306	−1.89122	−0.945612	+	0.325296i	$0.894536\pi$
$368$	−27.1063	−1.41301
$369$	9.07306	0.472324
$370$	0	0
$371$	9.16899	0.476030
$372$	0.744150	0.0385824
$373$	−12.7415	−0.659730	−0.329865	−	0.944028i	$-0.607003\pi$
$373$	−12.7415	−0.659730	−0.329865	+	0.944028i	$0.607003\pi$
$374$	19.5771	1.01231
$375$	0	0
$376$	−6.38614	−0.329340
$377$	1.33934	0.0689795
$378$	1.62305	0.0834805
$379$	14.1352	0.726078	0.363039	−	0.931774i	$-0.381739\pi$
$379$	14.1352	0.726078	0.363039	+	0.931774i	$0.381739\pi$
$380$	0	0
$381$	25.3730	1.29990
$382$	−17.5003	−0.895391
$383$	−13.6470	−0.697331	−0.348665	−	0.937247i	$-0.613365\pi$
$383$	−13.6470	−0.697331	−0.348665	+	0.937247i	$0.613365\pi$
$384$	−17.9813	−0.917605
$385$	0	0
$386$	−5.34202	−0.271902
$387$	14.3111	0.727472
$388$	−1.99175	−0.101116
$389$	−8.53466	−0.432725	−0.216362	−	0.976313i	$-0.569419\pi$
$389$	−8.53466	−0.432725	−0.216362	+	0.976313i	$0.569419\pi$
$390$	0	0
$391$	31.2288	1.57931
$392$	−19.0746	−0.963410
$393$	15.6999	0.791954
$394$	−1.27073	−0.0640187
$395$	0	0
$396$	6.00518	0.301772
$397$	10.1191	0.507865	0.253933	−	0.967222i	$-0.418276\pi$
$397$	10.1191	0.507865	0.253933	+	0.967222i	$0.418276\pi$
$398$	−32.6691	−1.63755
$399$	−3.21584	−0.160993
$400$	0	0
$401$	−15.9024	−0.794130	−0.397065	−	0.917790i	$-0.629971\pi$
$401$	−15.9024	−0.794130	−0.397065	+	0.917790i	$0.629971\pi$
$402$	−0.0940414	−0.00469036
$403$	−4.03090	−0.200793
$404$	−3.68469	−0.183320
$405$	0	0
$406$	0.267081	0.0132550
$407$	−21.4754	−1.06449
$408$	27.6235	1.36757
$409$	−34.8872	−1.72506	−0.862531	−	0.506005i	$-0.831122\pi$
$409$	−34.8872	−1.72506	−0.862531	+	0.506005i	$0.831122\pi$
$410$	0	0
$411$	21.6675	1.06878
$412$	2.91168	0.143448
$413$	6.70738	0.330049
$414$	−40.1530	−1.97341
$415$	0	0
$416$	−11.5010	−0.563883
$417$	26.0842	1.27735
$418$	8.21652	0.401883
$419$	−10.2677	−0.501611	−0.250805	−	0.968038i	$-0.580695\pi$
$419$	−10.2677	−0.501611	−0.250805	+	0.968038i	$0.580695\pi$
$420$	0	0
$421$	14.2806	0.695996	0.347998	−	0.937495i	$-0.386862\pi$
$421$	14.2806	0.695996	0.347998	+	0.937495i	$0.386862\pi$
$422$	−15.4772	−0.753417
$423$	−7.56754	−0.367947
$424$	−33.0557	−1.60532
$425$	0	0
$426$	−36.9719	−1.79130
$427$	9.73841	0.471275
$428$	−2.27286	−0.109863
$429$	−59.6985	−2.88227
$430$	0	0
$431$	26.0782	1.25614	0.628072	−	0.778155i	$-0.283844\pi$
$431$	26.0782	1.25614	0.628072	+	0.778155i	$0.283844\pi$
$432$	−4.68085	−0.225208
$433$	14.2583	0.685211	0.342605	−	0.939479i	$-0.388691\pi$
$433$	14.2583	0.685211	0.342605	+	0.939479i	$0.388691\pi$
$434$	−0.803810	−0.0385841
$435$	0	0
$436$	−5.78174	−0.276895
$437$	13.1067	0.626979
$438$	−39.9881	−1.91070
$439$	−13.0603	−0.623332	−0.311666	−	0.950192i	$-0.600887\pi$
$439$	−13.0603	−0.623332	−0.311666	+	0.950192i	$0.600887\pi$
$440$	0	0
$441$	−22.6033	−1.07635
$442$	−24.1665	−1.14948
$443$	−1.07809	−0.0512214	−0.0256107	−	0.999672i	$-0.508153\pi$
$443$	−1.07809	−0.0512214	−0.0256107	+	0.999672i	$0.508153\pi$
$444$	−4.89400	−0.232259
$445$	0	0
$446$	5.67856	0.268887
$447$	6.86629	0.324764
$448$	−7.47429	−0.353127
$449$	−31.9691	−1.50871	−0.754357	−	0.656465i	$-0.772051\pi$
$449$	−31.9691	−1.50871	−0.754357	+	0.656465i	$0.772051\pi$
$450$	0	0
$451$	−10.9635	−0.516253
$452$	−3.28951	−0.154726
$453$	8.00198	0.375966
$454$	−13.3573	−0.626889
$455$	0	0
$456$	11.5936	0.542920
$457$	3.25528	0.152275	0.0761377	−	0.997097i	$-0.475741\pi$
$457$	3.25528	0.152275	0.0761377	+	0.997097i	$0.475741\pi$
$458$	−32.4073	−1.51429
$459$	5.39274	0.251711
$460$	0	0
$461$	−2.57036	−0.119713	−0.0598567	−	0.998207i	$-0.519064\pi$
$461$	−2.57036	−0.119713	−0.0598567	+	0.998207i	$0.519064\pi$
$462$	−11.9046	−0.553853
$463$	35.8942	1.66814	0.834072	−	0.551655i	$-0.186003\pi$
$463$	35.8942	1.66814	0.834072	+	0.551655i	$0.186003\pi$
$464$	−0.770258	−0.0357583
$465$	0	0
$466$	−4.46446	−0.206812
$467$	−0.985633	−0.0456096	−0.0228048	−	0.999740i	$-0.507260\pi$
$467$	−0.985633	−0.0456096	−0.0228048	+	0.999740i	$0.507260\pi$
$468$	−7.41294	−0.342663
$469$	−0.0242341	−0.00111902
$470$	0	0
$471$	3.85583	0.177667
$472$	−24.1812	−1.11303
$473$	−17.2930	−0.795131
$474$	11.1914	0.514039
$475$	0	0
$476$	1.14969	0.0526958
$477$	−39.1708	−1.79351
$478$	−37.5506	−1.71752
$479$	6.66573	0.304565	0.152283	−	0.988337i	$-0.451338\pi$
$479$	6.66573	0.304565	0.152283	+	0.988337i	$0.451338\pi$
$480$	0	0
$481$	26.5097	1.20874
$482$	37.1989	1.69436
$483$	−18.9898	−0.864067
$484$	−3.01887	−0.137221
$485$	0	0
$486$	28.2204	1.28011
$487$	−35.7666	−1.62074	−0.810370	−	0.585919i	$-0.800734\pi$
$487$	−35.7666	−1.62074	−0.810370	+	0.585919i	$0.800734\pi$
$488$	−35.1085	−1.58929
$489$	3.07772	0.139179
$490$	0	0
$491$	−32.3477	−1.45983	−0.729916	−	0.683537i	$-0.760441\pi$
$491$	−32.3477	−1.45983	−0.729916	+	0.683537i	$0.760441\pi$
$492$	−2.49847	−0.112640
$493$	0.887402	0.0399666
$494$	−10.1427	−0.456341
$495$	0	0
$496$	2.31818	0.104089
$497$	−9.52751	−0.427367
$498$	3.10905	0.139320
$499$	10.1041	0.452323	0.226161	−	0.974090i	$-0.427382\pi$
$499$	10.1041	0.452323	0.226161	+	0.974090i	$0.427382\pi$
$500$	0	0
$501$	1.26632	0.0565750
$502$	0.704948	0.0314634
$503$	−3.84475	−0.171429	−0.0857146	−	0.996320i	$-0.527317\pi$
$503$	−3.84475	−0.171429	−0.0857146	+	0.996320i	$0.527317\pi$
$504$	−9.15271	−0.407694
$505$	0	0
$506$	48.5194	2.15695
$507$	40.3167	1.79053
$508$	−3.80711	−0.168913
$509$	−28.0430	−1.24298	−0.621491	−	0.783421i	$-0.713473\pi$
$509$	−28.0430	−1.24298	−0.621491	+	0.783421i	$0.713473\pi$
$510$	0	0
$511$	−10.3048	−0.455855
$512$	25.2921	1.11776
$513$	2.26333	0.0999285
$514$	−7.88506	−0.347795
$515$	0	0
$516$	−3.94087	−0.173487
$517$	9.14434	0.402168
$518$	5.28636	0.232269
$519$	−34.3785	−1.50905
$520$	0	0
$521$	22.7189	0.995333	0.497666	−	0.867369i	$-0.334190\pi$
$521$	22.7189	0.995333	0.497666	+	0.867369i	$0.334190\pi$
$522$	−1.14099	−0.0499400
$523$	15.5424	0.679622	0.339811	−	0.940494i	$-0.389637\pi$
$523$	15.5424	0.679622	0.339811	+	0.940494i	$0.389637\pi$
$524$	−2.35570	−0.102909
$525$	0	0
$526$	−37.0892	−1.61717
$527$	−2.67074	−0.116339
$528$	34.3328	1.49414
$529$	54.3964	2.36506
$530$	0	0
$531$	−28.6546	−1.24350
$532$	0.482523	0.0209200
$533$	13.5337	0.586209
$534$	20.0449	0.867429
$535$	0	0
$536$	0.0873676	0.00377371
$537$	−26.2658	−1.13345
$538$	3.27836	0.141340
$539$	27.3129	1.17645
$540$	0	0
$541$	−29.7765	−1.28019	−0.640097	−	0.768294i	$-0.721106\pi$
$541$	−29.7765	−1.28019	−0.640097	+	0.768294i	$0.721106\pi$
$542$	−16.8810	−0.725100
$543$	37.7985	1.62209
$544$	−7.62019	−0.326713
$545$	0	0
$546$	14.6954	0.628903
$547$	31.5735	1.34998	0.674992	−	0.737825i	$-0.264147\pi$
$547$	31.5735	1.34998	0.674992	+	0.737825i	$0.264147\pi$
$548$	−3.25113	−0.138881
$549$	−41.6034	−1.77559
$550$	0	0
$551$	0.372443	0.0158666
$552$	68.4614	2.91391
$553$	2.88398	0.122639
$554$	30.9489	1.31489
$555$	0	0
$556$	−3.91383	−0.165983
$557$	−0.211991	−0.00898235	−0.00449118	−	0.999990i	$-0.501430\pi$
$557$	−0.211991	−0.00898235	−0.00449118	+	0.999990i	$0.501430\pi$
$558$	3.43395	0.145371
$559$	21.3469	0.902876
$560$	0	0
$561$	−39.5543	−1.66998
$562$	36.7546	1.55040
$563$	20.5800	0.867346	0.433673	−	0.901070i	$-0.357217\pi$
$563$	20.5800	0.867346	0.433673	+	0.901070i	$0.357217\pi$
$564$	2.08389	0.0877478
$565$	0	0
$566$	−26.7763	−1.12549
$567$	5.77984	0.242730
$568$	34.3482	1.44122
$569$	−8.21040	−0.344198	−0.172099	−	0.985080i	$-0.555055\pi$
$569$	−8.21040	−0.344198	−0.172099	+	0.985080i	$0.555055\pi$
$570$	0	0
$571$	−30.4973	−1.27627	−0.638135	−	0.769924i	$-0.720294\pi$
$571$	−30.4973	−1.27627	−0.638135	+	0.769924i	$0.720294\pi$
$572$	8.95753	0.374533
$573$	35.3581	1.47711
$574$	2.69878	0.112645
$575$	0	0
$576$	31.9309	1.33045
$577$	−10.5889	−0.440820	−0.220410	−	0.975407i	$-0.570740\pi$
$577$	−10.5889	−0.440820	−0.220410	+	0.975407i	$0.570740\pi$
$578$	5.59057	0.232537
$579$	10.7932	0.448550
$580$	0	0
$581$	0.801190	0.0332390
$582$	−16.8681	−0.699204
$583$	47.3326	1.96031
$584$	37.1503	1.53729
$585$	0	0
$586$	−24.1526	−0.997736
$587$	−44.9239	−1.85421	−0.927104	−	0.374804i	$-0.877710\pi$
$587$	−44.9239	−1.85421	−0.927104	+	0.374804i	$0.877710\pi$
$588$	6.22432	0.256687
$589$	−1.12091	−0.0461863
$590$	0	0
$591$	2.56743	0.105610
$592$	−15.2458	−0.626599
$593$	9.24694	0.379726	0.189863	−	0.981811i	$-0.439196\pi$
$593$	9.24694	0.379726	0.189863	+	0.981811i	$0.439196\pi$
$594$	8.37857	0.343777
$595$	0	0
$596$	−1.03026	−0.0422011
$597$	66.0058	2.70144
$598$	−59.8935	−2.44923
$599$	−24.1790	−0.987926	−0.493963	−	0.869483i	$-0.664452\pi$
$599$	−24.1790	−0.987926	−0.493963	+	0.869483i	$0.664452\pi$
$600$	0	0
$601$	−24.2974	−0.991112	−0.495556	−	0.868576i	$-0.665036\pi$
$601$	−24.2974	−0.991112	−0.495556	+	0.868576i	$0.665036\pi$
$602$	4.25682	0.173495
$603$	0.103530	0.00421608
$604$	−1.20067	−0.0488544
$605$	0	0
$606$	−31.2055	−1.26764
$607$	8.52637	0.346075	0.173037	−	0.984915i	$-0.444642\pi$
$607$	8.52637	0.346075	0.173037	+	0.984915i	$0.444642\pi$
$608$	−3.19819	−0.129704
$609$	−0.539618	−0.0218664
$610$	0	0
$611$	−11.2880	−0.456664
$612$	−4.91157	−0.198538
$613$	−15.2646	−0.616532	−0.308266	−	0.951300i	$-0.599749\pi$
$613$	−15.2646	−0.616532	−0.308266	+	0.951300i	$0.599749\pi$
$614$	−19.6916	−0.794690
$615$	0	0
$616$	11.0598	0.445612
$617$	−29.5735	−1.19058	−0.595291	−	0.803510i	$-0.702963\pi$
$617$	−29.5735	−1.19058	−0.595291	+	0.803510i	$0.702963\pi$
$618$	24.6589	0.991928
$619$	21.5621	0.866654	0.433327	−	0.901237i	$-0.357339\pi$
$619$	21.5621	0.866654	0.433327	+	0.901237i	$0.357339\pi$
$620$	0	0
$621$	13.3652	0.536327
$622$	20.6568	0.828263
$623$	5.16549	0.206951
$624$	−42.3812	−1.69661
$625$	0	0
$626$	37.7701	1.50960
$627$	−16.6009	−0.662977
$628$	−0.578552	−0.0230868
$629$	17.5645	0.700341
$630$	0	0
$631$	−9.85441	−0.392298	−0.196149	−	0.980574i	$-0.562844\pi$
$631$	−9.85441	−0.392298	−0.196149	+	0.980574i	$0.562844\pi$
$632$	−10.3972	−0.413579
$633$	31.2706	1.24289
$634$	21.3811	0.849153
$635$	0	0
$636$	10.7866	0.427715
$637$	−33.7158	−1.33587
$638$	1.37874	0.0545846
$639$	40.7024	1.61016
$640$	0	0
$641$	−7.63907	−0.301725	−0.150863	−	0.988555i	$-0.548205\pi$
$641$	−7.63907	−0.301725	−0.150863	+	0.988555i	$0.548205\pi$
$642$	−19.2488	−0.759689
$643$	15.5644	0.613801	0.306901	−	0.951742i	$-0.400708\pi$
$643$	15.5644	0.613801	0.306901	+	0.951742i	$0.400708\pi$
$644$	2.84935	0.112280
$645$	0	0
$646$	−6.72020	−0.264403
$647$	41.8999	1.64725	0.823627	−	0.567132i	$-0.191947\pi$
$647$	41.8999	1.64725	0.823627	+	0.567132i	$0.191947\pi$
$648$	−20.8372	−0.818564
$649$	34.6251	1.35916
$650$	0	0
$651$	1.62404	0.0636513
$652$	−0.461799	−0.0180855
$653$	4.58237	0.179322	0.0896611	−	0.995972i	$-0.471422\pi$
$653$	4.58237	0.179322	0.0896611	+	0.995972i	$0.471422\pi$
$654$	−48.9654	−1.91470
$655$	0	0
$656$	−7.78325	−0.303885
$657$	44.0229	1.71750
$658$	−2.25097	−0.0877518
$659$	−2.20124	−0.0857483	−0.0428741	−	0.999080i	$-0.513651\pi$
$659$	−2.20124	−0.0857483	−0.0428741	+	0.999080i	$0.513651\pi$
$660$	0	0
$661$	31.1237	1.21057	0.605286	−	0.796008i	$-0.293059\pi$
$661$	31.1237	1.21057	0.605286	+	0.796008i	$0.293059\pi$
$662$	−29.8850	−1.16151
$663$	48.8268	1.89628
$664$	−2.88842	−0.112092
$665$	0	0
$666$	−22.5838	−0.875106
$667$	2.19931	0.0851576
$668$	−0.190006	−0.00735156
$669$	−11.4731	−0.443577
$670$	0	0
$671$	50.2721	1.94073
$672$	4.63374	0.178751
$673$	−16.0139	−0.617290	−0.308645	−	0.951177i	$-0.599875\pi$
$673$	−16.0139	−0.617290	−0.308645	+	0.951177i	$0.599875\pi$
$674$	−6.33107	−0.243864
$675$	0	0
$676$	−6.04936	−0.232668
$677$	12.4482	0.478424	0.239212	−	0.970967i	$-0.423111\pi$
$677$	12.4482	0.478424	0.239212	+	0.970967i	$0.423111\pi$
$678$	−27.8588	−1.06991
$679$	−4.34683	−0.166816
$680$	0	0
$681$	26.9875	1.03416
$682$	−4.14946	−0.158891
$683$	27.0020	1.03320	0.516601	−	0.856226i	$-0.327197\pi$
$683$	27.0020	1.03320	0.516601	+	0.856226i	$0.327197\pi$
$684$	−2.06139	−0.0788191
$685$	0	0
$686$	−14.2018	−0.542229
$687$	65.4767	2.49809
$688$	−12.2766	−0.468042
$689$	−58.4285	−2.22595
$690$	0	0
$691$	26.3961	1.00416	0.502078	−	0.864822i	$-0.332569\pi$
$691$	26.3961	1.00416	0.502078	+	0.864822i	$0.332569\pi$
$692$	5.15836	0.196091
$693$	13.1058	0.497848
$694$	45.6644	1.73340
$695$	0	0
$696$	1.94541	0.0737406
$697$	8.96697	0.339648
$698$	−6.15594	−0.233006
$699$	9.02014	0.341173
$700$	0	0
$701$	−28.2190	−1.06582	−0.532908	−	0.846173i	$-0.678901\pi$
$701$	−28.2190	−1.06582	−0.532908	+	0.846173i	$0.678901\pi$
$702$	−10.3427	−0.390360
$703$	7.37181	0.278033
$704$	−38.5841	−1.45419
$705$	0	0
$706$	−37.6847	−1.41828
$707$	−8.04153	−0.302433
$708$	7.89069	0.296550
$709$	2.62332	0.0985207	0.0492604	−	0.998786i	$-0.484314\pi$
$709$	2.62332	0.0985207	0.0492604	+	0.998786i	$0.484314\pi$
$710$	0	0
$711$	−12.3207	−0.462061
$712$	−18.6224	−0.697904
$713$	−6.61908	−0.247886
$714$	9.73666	0.364385
$715$	0	0
$716$	3.94108	0.147285
$717$	75.8685	2.83336
$718$	42.4293	1.58345
$719$	7.79924	0.290863	0.145431	−	0.989368i	$-0.453543\pi$
$719$	7.79924	0.290863	0.145431	+	0.989368i	$0.453543\pi$
$720$	0	0
$721$	6.35450	0.236654
$722$	21.3235	0.793578
$723$	−75.1578	−2.79515
$724$	−5.67151	−0.210780
$725$	0	0
$726$	−25.5667	−0.948870
$727$	−32.8349	−1.21778	−0.608889	−	0.793256i	$-0.708384\pi$
$727$	−32.8349	−1.21778	−0.608889	+	0.793256i	$0.708384\pi$
$728$	−13.6525	−0.505995
$729$	−36.3934	−1.34790
$730$	0	0
$731$	14.1437	0.523125
$732$	11.4564	0.423443
$733$	6.54261	0.241657	0.120828	−	0.992673i	$-0.461445\pi$
$733$	6.54261	0.241657	0.120828	+	0.992673i	$0.461445\pi$
$734$	46.0395	1.69935
$735$	0	0
$736$	−18.8856	−0.696134
$737$	−0.125102	−0.00460820
$738$	−11.5294	−0.424405
$739$	41.6103	1.53066	0.765330	−	0.643639i	$-0.222576\pi$
$739$	41.6103	1.53066	0.765330	+	0.643639i	$0.222576\pi$
$740$	0	0
$741$	20.4926	0.752815
$742$	−11.6514	−0.427735
$743$	9.77987	0.358789	0.179394	−	0.983777i	$-0.442586\pi$
$743$	9.77987	0.358789	0.179394	+	0.983777i	$0.442586\pi$
$744$	−5.85494	−0.214652
$745$	0	0
$746$	16.1911	0.592797
$747$	−3.42276	−0.125232
$748$	5.93496	0.217004
$749$	−4.96033	−0.181246
$750$	0	0
$751$	28.5826	1.04299	0.521496	−	0.853253i	$-0.325374\pi$
$751$	28.5826	1.04299	0.521496	+	0.853253i	$0.325374\pi$
$752$	6.49176	0.236730
$753$	−1.42430	−0.0519044
$754$	−1.70194	−0.0619812
$755$	0	0
$756$	0.492040	0.0178953
$757$	8.24184	0.299555	0.149777	−	0.988720i	$-0.452144\pi$
$757$	8.24184	0.299555	0.149777	+	0.988720i	$0.452144\pi$
$758$	−17.9621	−0.652414
$759$	−98.0301	−3.55827
$760$	0	0
$761$	42.3724	1.53600	0.767999	−	0.640451i	$-0.221253\pi$
$761$	42.3724	1.53600	0.767999	+	0.640451i	$0.221253\pi$
$762$	−32.2423	−1.16802
$763$	−12.6182	−0.456808
$764$	−5.30534	−0.191941
$765$	0	0
$766$	17.3418	0.626583
$767$	−42.7421	−1.54333
$768$	−22.8002	−0.822733
$769$	30.7290	1.10812	0.554058	−	0.832478i	$-0.313079\pi$
$769$	30.7290	1.10812	0.554058	+	0.832478i	$0.313079\pi$
$770$	0	0
$771$	15.9312	0.573750
$772$	−1.61948	−0.0582862
$773$	37.3634	1.34387	0.671934	−	0.740611i	$-0.265464\pi$
$773$	37.3634	1.34387	0.671934	+	0.740611i	$0.265464\pi$
$774$	−18.1856	−0.653667
$775$	0	0
$776$	15.6710	0.562556
$777$	−10.6807	−0.383169
$778$	10.8453	0.388823
$779$	3.76344	0.134839
$780$	0	0
$781$	−49.1833	−1.75992
$782$	−39.6835	−1.41908
$783$	0.379788	0.0135725
$784$	19.3900	0.692501
$785$	0	0
$786$	−19.9504	−0.711606
$787$	−22.5392	−0.803437	−0.401719	−	0.915763i	$-0.631587\pi$
$787$	−22.5392	−0.803437	−0.401719	+	0.915763i	$0.631587\pi$
$788$	−0.385233	−0.0137234
$789$	74.9362	2.66780
$790$	0	0
$791$	−7.17909	−0.255259
$792$	−47.2485	−1.67890
$793$	−62.0571	−2.20371
$794$	−12.8588	−0.456340
$795$	0	0
$796$	−9.90390	−0.351034
$797$	21.8779	0.774956	0.387478	−	0.921879i	$-0.373346\pi$
$797$	21.8779	0.774956	0.387478	+	0.921879i	$0.373346\pi$
$798$	4.08648	0.144660
$799$	−7.47906	−0.264590
$800$	0	0
$801$	−22.0675	−0.779716
$802$	20.2078	0.713562
$803$	−53.1956	−1.87723
$804$	−0.0285094	−0.00100545
$805$	0	0
$806$	5.12220	0.180422
$807$	−6.62371	−0.233166
$808$	28.9910	1.01990
$809$	−30.9092	−1.08671	−0.543355	−	0.839503i	$-0.682846\pi$
$809$	−30.9092	−1.08671	−0.543355	+	0.839503i	$0.682846\pi$
$810$	0	0
$811$	29.7736	1.04549	0.522746	−	0.852488i	$-0.324907\pi$
$811$	29.7736	1.04549	0.522746	+	0.852488i	$0.324907\pi$
$812$	0.0809676	0.00284141
$813$	34.1069	1.19618
$814$	27.2895	0.956496
$815$	0	0
$816$	−28.0804	−0.983011
$817$	5.93612	0.207679
$818$	44.3324	1.55005
$819$	−16.1781	−0.565309
$820$	0	0
$821$	3.56378	0.124377	0.0621884	−	0.998064i	$-0.480192\pi$
$821$	3.56378	0.124377	0.0621884	+	0.998064i	$0.480192\pi$
$822$	−27.5337	−0.960347
$823$	54.9352	1.91492	0.957461	−	0.288564i	$-0.0931777\pi$
$823$	54.9352	1.91492	0.957461	+	0.288564i	$0.0931777\pi$
$824$	−22.9090	−0.798073
$825$	0	0
$826$	−8.52330	−0.296564
$827$	55.6540	1.93528	0.967640	−	0.252335i	$-0.0811986\pi$
$827$	55.6540	1.93528	0.967640	+	0.252335i	$0.0811986\pi$
$828$	−12.1727	−0.423030
$829$	24.1406	0.838439	0.419219	−	0.907885i	$-0.362304\pi$
$829$	24.1406	0.838439	0.419219	+	0.907885i	$0.362304\pi$
$830$	0	0
$831$	−62.5302	−2.16915
$832$	47.6292	1.65125
$833$	−22.3390	−0.773999
$834$	−33.1461	−1.14775
$835$	0	0
$836$	2.49090	0.0861497
$837$	−1.14302	−0.0395084
$838$	13.0475	0.450720
$839$	−5.80565	−0.200433	−0.100217	−	0.994966i	$-0.531954\pi$
$839$	−5.80565	−0.200433	−0.100217	+	0.994966i	$0.531954\pi$
$840$	0	0
$841$	−28.9375	−0.997845
$842$	−18.1469	−0.625384
$843$	−74.2602	−2.55766
$844$	−4.69202	−0.161506
$845$	0	0
$846$	9.61634	0.330617
$847$	−6.58843	−0.226381
$848$	33.6024	1.15391
$849$	54.0997	1.85670
$850$	0	0
$851$	43.5313	1.49223
$852$	−11.2083	−0.383991
$853$	−1.99743	−0.0683908	−0.0341954	−	0.999415i	$-0.510887\pi$
$853$	−1.99743	−0.0683908	−0.0341954	+	0.999415i	$0.510887\pi$
$854$	−12.3749	−0.423462
$855$	0	0
$856$	17.8828	0.611220
$857$	−33.1164	−1.13123	−0.565617	−	0.824668i	$-0.691362\pi$
$857$	−33.1164	−1.13123	−0.565617	+	0.824668i	$0.691362\pi$
$858$	75.8610	2.58985
$859$	12.0670	0.411719	0.205860	−	0.978582i	$-0.434001\pi$
$859$	12.0670	0.411719	0.205860	+	0.978582i	$0.434001\pi$
$860$	0	0
$861$	−5.45270	−0.185828
$862$	−33.1385	−1.12870
$863$	−36.3710	−1.23808	−0.619041	−	0.785359i	$-0.712479\pi$
$863$	−36.3710	−1.23808	−0.619041	+	0.785359i	$0.712479\pi$
$864$	−3.26127	−0.110951
$865$	0	0
$866$	−18.1185	−0.615693
$867$	−11.2954	−0.383611
$868$	−0.243681	−0.00827108
$869$	14.8878	0.505035
$870$	0	0
$871$	0.154429	0.00523263
$872$	45.4905	1.54050
$873$	18.5701	0.628501
$874$	−16.6552	−0.563369
$875$	0	0
$876$	−12.1227	−0.409588
$877$	−17.3189	−0.584818	−0.292409	−	0.956293i	$-0.594457\pi$
$877$	−17.3189	−0.584818	−0.292409	+	0.956293i	$0.594457\pi$
$878$	16.5961	0.560092
$879$	48.7988	1.64594
$880$	0	0
$881$	−53.3439	−1.79720	−0.898600	−	0.438768i	$-0.855415\pi$
$881$	−53.3439	−1.79720	−0.898600	+	0.438768i	$0.855415\pi$
$882$	28.7227	0.967145
$883$	−46.1134	−1.55184	−0.775919	−	0.630832i	$-0.782714\pi$
$883$	−46.1134	−1.55184	−0.775919	+	0.630832i	$0.782714\pi$
$884$	−7.32626	−0.246409
$885$	0	0
$886$	1.36996	0.0460248
$887$	−34.7806	−1.16782	−0.583908	−	0.811820i	$-0.698477\pi$
$887$	−34.7806	−1.16782	−0.583908	+	0.811820i	$0.698477\pi$
$888$	38.5058	1.29217
$889$	−8.30871	−0.278665
$890$	0	0
$891$	29.8369	0.999575
$892$	1.72150	0.0576400
$893$	−3.13896	−0.105041
$894$	−8.72523	−0.291815
$895$	0	0
$896$	5.88821	0.196711
$897$	121.011	4.04043
$898$	40.6242	1.35565
$899$	−0.188089	−0.00627312
$900$	0	0
$901$	−38.7128	−1.28971
$902$	13.9318	0.463877
$903$	−8.60063	−0.286211
$904$	25.8817	0.860814
$905$	0	0
$906$	−10.1684	−0.337822
$907$	39.7663	1.32042	0.660209	−	0.751082i	$-0.270468\pi$
$907$	39.7663	1.32042	0.660209	+	0.751082i	$0.270468\pi$
$908$	−4.04937	−0.134383
$909$	34.3542	1.13946
$910$	0	0
$911$	−33.6230	−1.11398	−0.556991	−	0.830519i	$-0.688044\pi$
$911$	−33.6230	−1.11398	−0.556991	+	0.830519i	$0.688044\pi$
$912$	−11.7854	−0.390252
$913$	4.13594	0.136880
$914$	−4.13659	−0.136826
$915$	0	0
$916$	−9.82452	−0.324611
$917$	−5.14113	−0.169775
$918$	−6.85274	−0.226174
$919$	−49.8301	−1.64374	−0.821872	−	0.569673i	$-0.807070\pi$
$919$	−49.8301	−1.64374	−0.821872	+	0.569673i	$0.807070\pi$
$920$	0	0
$921$	39.7856	1.31098
$922$	3.26624	0.107568
$923$	60.7131	1.99840
$924$	−3.60898	−0.118727
$925$	0	0
$926$	−45.6120	−1.49890
$927$	−27.1470	−0.891626
$928$	−0.536658	−0.0176167
$929$	10.7631	0.353127	0.176564	−	0.984289i	$-0.443502\pi$
$929$	10.7631	0.353127	0.176564	+	0.984289i	$0.443502\pi$
$930$	0	0
$931$	−9.37566	−0.307275
$932$	−1.35344	−0.0443333
$933$	−41.7357	−1.36637
$934$	1.25248	0.0409823
$935$	0	0
$936$	58.3247	1.90640
$937$	10.8163	0.353354	0.176677	−	0.984269i	$-0.443465\pi$
$937$	10.8163	0.353354	0.176677	+	0.984269i	$0.443465\pi$
$938$	0.0307951	0.00100549
$939$	−76.3119	−2.49034
$940$	0	0
$941$	−56.1398	−1.83011	−0.915053	−	0.403333i	$-0.867852\pi$
$941$	−56.1398	−1.83011	−0.915053	+	0.403333i	$0.867852\pi$
$942$	−4.89974	−0.159642
$943$	22.2235	0.723696
$944$	24.5811	0.800047
$945$	0	0
$946$	21.9748	0.714461
$947$	11.8386	0.384704	0.192352	−	0.981326i	$-0.438388\pi$
$947$	11.8386	0.384704	0.192352	+	0.981326i	$0.438388\pi$
$948$	3.39277	0.110192
$949$	65.6660	2.13161
$950$	0	0
$951$	−43.1991	−1.40083
$952$	−9.04568	−0.293172
$953$	41.9706	1.35956	0.679781	−	0.733416i	$-0.262075\pi$
$953$	41.9706	1.35956	0.679781	+	0.733416i	$0.262075\pi$
$954$	49.7757	1.61155
$955$	0	0
$956$	−11.3838	−0.368177
$957$	−2.78564	−0.0900470
$958$	−8.47038	−0.273665
$959$	−7.09531	−0.229119
$960$	0	0
$961$	−30.4339	−0.981740
$962$	−33.6868	−1.08611
$963$	21.1910	0.682870
$964$	11.2771	0.363212
$965$	0	0
$966$	24.1310	0.776404
$967$	19.7955	0.636580	0.318290	−	0.947993i	$-0.396891\pi$
$967$	19.7955	0.636580	0.318290	+	0.947993i	$0.396891\pi$
$968$	23.7524	0.763430
$969$	13.5777	0.436179
$970$	0	0
$971$	−23.3780	−0.750235	−0.375118	−	0.926977i	$-0.622398\pi$
$971$	−23.3780	−0.750235	−0.375118	+	0.926977i	$0.622398\pi$
$972$	8.55525	0.274410
$973$	−8.54160	−0.273831
$974$	45.4499	1.45631
$975$	0	0
$976$	35.6892	1.14238
$977$	11.4712	0.366995	0.183498	−	0.983020i	$-0.441258\pi$
$977$	11.4712	0.366995	0.183498	+	0.983020i	$0.441258\pi$
$978$	−3.91096	−0.125059
$979$	26.6655	0.852234
$980$	0	0
$981$	53.9060	1.72109
$982$	41.1054	1.31173
$983$	−32.2131	−1.02744	−0.513719	−	0.857959i	$-0.671732\pi$
$983$	−32.2131	−1.02744	−0.513719	+	0.857959i	$0.671732\pi$
$984$	19.6579	0.626670
$985$	0	0
$986$	−1.12765	−0.0359118
$987$	4.54793	0.144762
$988$	−3.07483	−0.0978235
$989$	35.0534	1.11463
$990$	0	0
$991$	−12.0981	−0.384308	−0.192154	−	0.981365i	$-0.561547\pi$
$991$	−12.0981	−0.384308	−0.192154	+	0.981365i	$0.561547\pi$
$992$	1.61513	0.0512805
$993$	60.3806	1.91612
$994$	12.1069	0.384009
$995$	0	0
$996$	0.942534	0.0298653
$997$	−28.4299	−0.900383	−0.450192	−	0.892932i	$-0.648644\pi$
$997$	−28.4299	−0.900383	−0.450192	+	0.892932i	$0.648644\pi$
$998$	−12.8397	−0.406432
$999$	7.51719	0.237833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.q.1.12	yes	37
5.4	even	2		4925.2.a.p.1.26	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4925.2.a.p.1.26	✓	37	5.4	even	2
4925.2.a.q.1.12	yes	37	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.27073 q^{2} +2.56743 q^{3} -0.385233 q^{4} -3.26253 q^{6} -0.840739 q^{7} +3.03100 q^{8} +3.59172 q^{9} -4.34010 q^{11} -0.989061 q^{12} +5.35753 q^{13} +1.06836 q^{14} -3.08113 q^{16} +3.54972 q^{17} -4.56412 q^{18} +1.48982 q^{19} -2.15854 q^{21} +5.51512 q^{22} +8.79752 q^{23} +7.78189 q^{24} -6.80800 q^{26} +1.51920 q^{27} +0.323881 q^{28} +0.249992 q^{29} -0.752380 q^{31} -2.14670 q^{32} -11.1429 q^{33} -4.51075 q^{34} -1.38365 q^{36} +4.94813 q^{37} -1.89316 q^{38} +13.7551 q^{39} +2.52610 q^{41} +2.74294 q^{42} +3.98446 q^{43} +1.67195 q^{44} -11.1793 q^{46} -2.10694 q^{47} -7.91060 q^{48} -6.29316 q^{49} +9.11368 q^{51} -2.06390 q^{52} -10.9059 q^{53} -1.93050 q^{54} -2.54828 q^{56} +3.82501 q^{57} -0.317674 q^{58} -7.97796 q^{59} -11.5832 q^{61} +0.956075 q^{62} -3.01970 q^{63} +8.89014 q^{64} +14.1597 q^{66} +0.0288247 q^{67} -1.36747 q^{68} +22.5871 q^{69} +11.3323 q^{71} +10.8865 q^{72} +12.2568 q^{73} -6.28775 q^{74} -0.573928 q^{76} +3.64889 q^{77} -17.4791 q^{78} -3.43029 q^{79} -6.87471 q^{81} -3.21001 q^{82} -0.952959 q^{83} +0.831543 q^{84} -5.06319 q^{86} +0.641838 q^{87} -13.1548 q^{88} -6.14398 q^{89} -4.50428 q^{91} -3.38910 q^{92} -1.93169 q^{93} +2.67736 q^{94} -5.51151 q^{96} +5.17024 q^{97} +7.99693 q^{98} -15.5884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 2 q^{2} + q^{3} + 50 q^{4} + 8 q^{6} - 4 q^{7} + 50 q^{9} + 17 q^{11} - 2 q^{12} - 3 q^{13} + 14 q^{14} + 72 q^{16} - 10 q^{17} - 4 q^{18} + 54 q^{19} + 15 q^{21} - 11 q^{22} + 4 q^{23} + 28 q^{24}+ \cdots + 40 q^{99}+O(q^{100}) \) 37 * q + 2 * q^2 + q^3 + 50 * q^4 + 8 * q^6 - 4 * q^7 + 50 * q^9 + 17 * q^11 - 2 * q^12 - 3 * q^13 + 14 * q^14 + 72 * q^16 - 10 * q^17 - 4 * q^18 + 54 * q^19 + 15 * q^21 - 11 * q^22 + 4 * q^23 + 28 * q^24 + 19 * q^26 - 2 * q^27 + 3 * q^28 - 4 * q^29 + 58 * q^31 + 22 * q^32 - 7 * q^33 + 63 * q^34 + 66 * q^36 + 7 * q^37 + 12 * q^38 + 23 * q^39 + 17 * q^41 - 17 * q^42 + 11 * q^43 + 34 * q^44 + 41 * q^46 - 19 * q^47 + 10 * q^48 + 81 * q^49 + 39 * q^51 + 31 * q^52 - 14 * q^53 + 5 * q^54 + 21 * q^56 + 24 * q^57 - 15 * q^58 + 56 * q^59 + 40 * q^61 + 4 * q^62 + 5 * q^63 + 78 * q^64 + 22 * q^66 + 2 * q^67 - 3 * q^68 + 24 * q^69 + 31 * q^71 + 85 * q^72 + 4 * q^73 - 32 * q^74 + 137 * q^76 + 16 * q^77 - 113 * q^78 + 17 * q^79 + 85 * q^81 + 52 * q^82 - 18 * q^83 - 32 * q^84 + 35 * q^86 + 39 * q^87 - 63 * q^88 + 23 * q^89 + 152 * q^91 + 75 * q^92 + 8 * q^93 + 32 * q^94 + 72 * q^96 + 34 * q^97 - 40 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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