LMFDB - Embedded newform 6025.2.a.o.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.625313 q^{2} +0.793037 q^{3} -1.60898 q^{4} -0.495897 q^{6} -2.86756 q^{7} +2.25675 q^{8} -2.37109 q^{9} +O(q^{10})$	\(q-0.625313 q^{2} +0.793037 q^{3} -1.60898 q^{4} -0.495897 q^{6} -2.86756 q^{7} +2.25675 q^{8} -2.37109 q^{9} -0.757164 q^{11} -1.27598 q^{12} +1.76574 q^{13} +1.79312 q^{14} +1.80679 q^{16} +2.04856 q^{17} +1.48268 q^{18} +2.54566 q^{19} -2.27408 q^{21} +0.473465 q^{22} -5.36607 q^{23} +1.78968 q^{24} -1.10414 q^{26} -4.25948 q^{27} +4.61385 q^{28} -2.67212 q^{29} -2.95864 q^{31} -5.64330 q^{32} -0.600459 q^{33} -1.28099 q^{34} +3.81505 q^{36} -4.94182 q^{37} -1.59183 q^{38} +1.40030 q^{39} +3.42561 q^{41} +1.42201 q^{42} +0.190976 q^{43} +1.21826 q^{44} +3.35548 q^{46} -5.06976 q^{47} +1.43285 q^{48} +1.22288 q^{49} +1.62458 q^{51} -2.84105 q^{52} +11.7819 q^{53} +2.66351 q^{54} -6.47135 q^{56} +2.01880 q^{57} +1.67091 q^{58} +6.50824 q^{59} -5.93708 q^{61} +1.85008 q^{62} +6.79924 q^{63} -0.0847519 q^{64} +0.375475 q^{66} -7.77917 q^{67} -3.29610 q^{68} -4.25549 q^{69} -4.06070 q^{71} -5.35095 q^{72} +8.15988 q^{73} +3.09019 q^{74} -4.09592 q^{76} +2.17121 q^{77} -0.875627 q^{78} -3.37973 q^{79} +3.73535 q^{81} -2.14208 q^{82} +1.25005 q^{83} +3.65896 q^{84} -0.119420 q^{86} -2.11909 q^{87} -1.70873 q^{88} -11.5389 q^{89} -5.06337 q^{91} +8.63392 q^{92} -2.34631 q^{93} +3.17019 q^{94} -4.47535 q^{96} -8.31745 q^{97} -0.764686 q^{98} +1.79531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−0.625313	−0.442163	−0.221082	−	0.975255i	$-0.570959\pi$
$2$	−0.625313	−0.442163	−0.221082	+	0.975255i	$0.570959\pi$
$3$	0.793037	0.457860	0.228930	−	0.973443i	$-0.426477\pi$
$3$	0.793037	0.457860	0.228930	+	0.973443i	$0.426477\pi$
$4$	−1.60898	−0.804492
$5$	0	0
$5$	0	0
$6$	−0.495897	−0.202449
$7$	−2.86756	−1.08383	−0.541917	−	0.840432i	$-0.682301\pi$
$7$	−2.86756	−1.08383	−0.541917	+	0.840432i	$0.682301\pi$
$8$	2.25675	0.797880
$9$	−2.37109	−0.790364
$10$	0	0
$11$	−0.757164	−0.228294	−0.114147	−	0.993464i	$-0.536413\pi$
$11$	−0.757164	−0.228294	−0.114147	+	0.993464i	$0.536413\pi$
$12$	−1.27598	−0.368345
$13$	1.76574	0.489729	0.244865	−	0.969557i	$-0.421256\pi$
$13$	1.76574	0.489729	0.244865	+	0.969557i	$0.421256\pi$
$14$	1.79312	0.479232
$15$	0	0
$16$	1.80679	0.451698
$17$	2.04856	0.496848	0.248424	−	0.968651i	$-0.420087\pi$
$17$	2.04856	0.496848	0.248424	+	0.968651i	$0.420087\pi$
$18$	1.48268	0.349470
$19$	2.54566	0.584014	0.292007	−	0.956416i	$-0.405677\pi$
$19$	2.54566	0.584014	0.292007	+	0.956416i	$0.405677\pi$
$20$	0	0
$21$	−2.27408	−0.496245
$22$	0.473465	0.100943
$23$	−5.36607	−1.11890	−0.559452	−	0.828863i	$-0.688988\pi$
$23$	−5.36607	−1.11890	−0.559452	+	0.828863i	$0.688988\pi$
$24$	1.78968	0.365318
$25$	0	0
$26$	−1.10414	−0.216540
$27$	−4.25948	−0.819737
$28$	4.61385	0.871936
$29$	−2.67212	−0.496200	−0.248100	−	0.968734i	$-0.579806\pi$
$29$	−2.67212	−0.496200	−0.248100	+	0.968734i	$0.579806\pi$
$30$	0	0
$31$	−2.95864	−0.531387	−0.265694	−	0.964058i	$-0.585601\pi$
$31$	−2.95864	−0.531387	−0.265694	+	0.964058i	$0.585601\pi$
$32$	−5.64330	−0.997604
$33$	−0.600459	−0.104527
$34$	−1.28099	−0.219688
$35$	0	0
$36$	3.81505	0.635841
$37$	−4.94182	−0.812430	−0.406215	−	0.913778i	$-0.633152\pi$
$37$	−4.94182	−0.812430	−0.406215	+	0.913778i	$0.633152\pi$
$38$	−1.59183	−0.258230
$39$	1.40030	0.224228
$40$	0	0
$41$	3.42561	0.534991	0.267496	−	0.963559i	$-0.413804\pi$
$41$	3.42561	0.534991	0.267496	+	0.963559i	$0.413804\pi$
$42$	1.42201	0.219421
$43$	0.190976	0.0291235	0.0145618	−	0.999894i	$-0.495365\pi$
$43$	0.190976	0.0291235	0.0145618	+	0.999894i	$0.495365\pi$
$44$	1.21826	0.183660
$45$	0	0
$46$	3.35548	0.494738
$47$	−5.06976	−0.739501	−0.369750	−	0.929131i	$-0.620557\pi$
$47$	−5.06976	−0.739501	−0.369750	+	0.929131i	$0.620557\pi$
$48$	1.43285	0.206815
$49$	1.22288	0.174698
$50$	0	0
$51$	1.62458	0.227487
$52$	−2.84105	−0.393983
$53$	11.7819	1.61837	0.809184	−	0.587555i	$-0.199910\pi$
$53$	11.7819	1.61837	0.809184	+	0.587555i	$0.199910\pi$
$54$	2.66351	0.362457
$55$	0	0
$56$	−6.47135	−0.864770
$57$	2.01880	0.267397
$58$	1.67091	0.219401
$59$	6.50824	0.847301	0.423650	−	0.905826i	$-0.360748\pi$
$59$	6.50824	0.847301	0.423650	+	0.905826i	$0.360748\pi$
$60$	0	0
$61$	−5.93708	−0.760165	−0.380083	−	0.924953i	$-0.624104\pi$
$61$	−5.93708	−0.760165	−0.380083	+	0.924953i	$0.624104\pi$
$62$	1.85008	0.234960
$63$	6.79924	0.856624
$64$	−0.0847519	−0.0105940
$65$	0	0
$66$	0.375475	0.0462178
$67$	−7.77917	−0.950377	−0.475188	−	0.879884i	$-0.657620\pi$
$67$	−7.77917	−0.950377	−0.475188	+	0.879884i	$0.657620\pi$
$68$	−3.29610	−0.399710
$69$	−4.25549	−0.512301
$70$	0	0
$71$	−4.06070	−0.481917	−0.240958	−	0.970535i	$-0.577462\pi$
$71$	−4.06070	−0.481917	−0.240958	+	0.970535i	$0.577462\pi$
$72$	−5.35095	−0.630616
$73$	8.15988	0.955041	0.477521	−	0.878621i	$-0.341536\pi$
$73$	8.15988	0.955041	0.477521	+	0.878621i	$0.341536\pi$
$74$	3.09019	0.359227
$75$	0	0
$76$	−4.09592	−0.469834
$77$	2.17121	0.247432
$78$	−0.875627	−0.0991452
$79$	−3.37973	−0.380249	−0.190125	−	0.981760i	$-0.560889\pi$
$79$	−3.37973	−0.380249	−0.190125	+	0.981760i	$0.560889\pi$
$80$	0	0
$81$	3.73535	0.415039
$82$	−2.14208	−0.236553
$83$	1.25005	0.137211	0.0686053	−	0.997644i	$-0.478145\pi$
$83$	1.25005	0.137211	0.0686053	+	0.997644i	$0.478145\pi$
$84$	3.65896	0.399225
$85$	0	0
$86$	−0.119420	−0.0128774
$87$	−2.11909	−0.227190
$88$	−1.70873	−0.182151
$89$	−11.5389	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$89$	−11.5389	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$90$	0	0
$91$	−5.06337	−0.530786
$92$	8.63392	0.900148
$93$	−2.34631	−0.243301
$94$	3.17019	0.326980
$95$	0	0
$96$	−4.47535	−0.456763
$97$	−8.31745	−0.844509	−0.422255	−	0.906477i	$-0.638761\pi$
$97$	−8.31745	−0.844509	−0.422255	+	0.906477i	$0.638761\pi$
$98$	−0.764686	−0.0772450
$99$	1.79531	0.180435
$100$	0	0
$101$	−10.2364	−1.01856	−0.509279	−	0.860601i	$-0.670088\pi$
$101$	−10.2364	−1.01856	−0.509279	+	0.860601i	$0.670088\pi$
$102$	−1.01587	−0.100586
$103$	9.20454	0.906951	0.453475	−	0.891269i	$-0.350184\pi$
$103$	9.20454	0.906951	0.453475	+	0.891269i	$0.350184\pi$
$104$	3.98483	0.390745
$105$	0	0
$106$	−7.36738	−0.715583
$107$	1.11056	0.107362	0.0536810	−	0.998558i	$-0.482905\pi$
$107$	1.11056	0.107362	0.0536810	+	0.998558i	$0.482905\pi$
$108$	6.85342	0.659471
$109$	1.83082	0.175360	0.0876801	−	0.996149i	$-0.472055\pi$
$109$	1.83082	0.175360	0.0876801	+	0.996149i	$0.472055\pi$
$110$	0	0
$111$	−3.91905	−0.371979
$112$	−5.18108	−0.489566
$113$	−11.2627	−1.05950	−0.529751	−	0.848153i	$-0.677715\pi$
$113$	−11.2627	−1.05950	−0.529751	+	0.848153i	$0.677715\pi$
$114$	−1.26238	−0.118233
$115$	0	0
$116$	4.29939	0.399189
$117$	−4.18674	−0.387064
$118$	−4.06969	−0.374645
$119$	−5.87436	−0.538502
$120$	0	0
$121$	−10.4267	−0.947882
$122$	3.71254	0.336117
$123$	2.71664	0.244951
$124$	4.76040	0.427497
$125$	0	0
$126$	−4.25166	−0.378768
$127$	18.9382	1.68049	0.840246	−	0.542206i	$-0.182411\pi$
$127$	18.9382	1.68049	0.840246	+	0.542206i	$0.182411\pi$
$128$	11.3396	1.00229
$129$	0.151451	0.0133345
$130$	0	0
$131$	16.9241	1.47867	0.739334	−	0.673339i	$-0.235140\pi$
$131$	16.9241	1.47867	0.739334	+	0.673339i	$0.235140\pi$
$132$	0.966129	0.0840907
$133$	−7.29982	−0.632975
$134$	4.86442	0.420222
$135$	0	0
$136$	4.62308	0.396425
$137$	−0.751925	−0.0642413	−0.0321206	−	0.999484i	$-0.510226\pi$
$137$	−0.751925	−0.0642413	−0.0321206	+	0.999484i	$0.510226\pi$
$138$	2.66102	0.226521
$139$	13.1368	1.11424	0.557122	−	0.830430i	$-0.311905\pi$
$139$	13.1368	1.11424	0.557122	+	0.830430i	$0.311905\pi$
$140$	0	0
$141$	−4.02051	−0.338588
$142$	2.53921	0.213086
$143$	−1.33696	−0.111802
$144$	−4.28407	−0.357006
$145$	0	0
$146$	−5.10248	−0.422284
$147$	0.969793	0.0799872
$148$	7.95130	0.653593
$149$	−9.39049	−0.769299	−0.384650	−	0.923063i	$-0.625678\pi$
$149$	−9.39049	−0.769299	−0.384650	+	0.923063i	$0.625678\pi$
$150$	0	0
$151$	15.9063	1.29444	0.647218	−	0.762305i	$-0.275932\pi$
$151$	15.9063	1.29444	0.647218	+	0.762305i	$0.275932\pi$
$152$	5.74490	0.465973
$153$	−4.85732	−0.392691
$154$	−1.35769	−0.109406
$155$	0	0
$156$	−2.25306	−0.180389
$157$	19.2289	1.53463	0.767317	−	0.641268i	$-0.221591\pi$
$157$	19.2289	1.53463	0.767317	+	0.641268i	$0.221591\pi$
$158$	2.11339	0.168132
$159$	9.34348	0.740986
$160$	0	0
$161$	15.3875	1.21271
$162$	−2.33577	−0.183515
$163$	−3.89500	−0.305080	−0.152540	−	0.988297i	$-0.548745\pi$
$163$	−3.89500	−0.305080	−0.152540	+	0.988297i	$0.548745\pi$
$164$	−5.51176	−0.430396
$165$	0	0
$166$	−0.781672	−0.0606695
$167$	5.00184	0.387054	0.193527	−	0.981095i	$-0.438007\pi$
$167$	5.00184	0.387054	0.193527	+	0.981095i	$0.438007\pi$
$168$	−5.13202	−0.395944
$169$	−9.88215	−0.760165
$170$	0	0
$171$	−6.03599	−0.461584
$172$	−0.307277	−0.0234296
$173$	11.8670	0.902233	0.451117	−	0.892465i	$-0.351026\pi$
$173$	11.8670	0.902233	0.451117	+	0.892465i	$0.351026\pi$
$174$	1.32510	0.100455
$175$	0	0
$176$	−1.36804	−0.103120
$177$	5.16128	0.387945
$178$	7.21541	0.540818
$179$	24.7281	1.84827	0.924133	−	0.382071i	$-0.124789\pi$
$179$	24.7281	1.84827	0.924133	+	0.382071i	$0.124789\pi$
$180$	0	0
$181$	−3.24617	−0.241286	−0.120643	−	0.992696i	$-0.538496\pi$
$181$	−3.24617	−0.241286	−0.120643	+	0.992696i	$0.538496\pi$
$182$	3.16619	0.234694
$183$	−4.70832	−0.348049
$184$	−12.1099	−0.892751
$185$	0	0
$186$	1.46718	0.107579
$187$	−1.55109	−0.113427
$188$	8.15716	0.594922
$189$	12.2143	0.888459
$190$	0	0
$191$	1.65925	0.120059	0.0600295	−	0.998197i	$-0.480881\pi$
$191$	1.65925	0.120059	0.0600295	+	0.998197i	$0.480881\pi$
$192$	−0.0672114	−0.00485057
$193$	13.3835	0.963368	0.481684	−	0.876345i	$-0.340025\pi$
$193$	13.3835	0.963368	0.481684	+	0.876345i	$0.340025\pi$
$194$	5.20101	0.373411
$195$	0	0
$196$	−1.96760	−0.140543
$197$	27.3776	1.95057	0.975286	−	0.220946i	$-0.0709145\pi$
$197$	27.3776	1.95057	0.975286	+	0.220946i	$0.0709145\pi$
$198$	−1.12263	−0.0797817
$199$	17.6734	1.25283	0.626417	−	0.779488i	$-0.284521\pi$
$199$	17.6734	1.25283	0.626417	+	0.779488i	$0.284521\pi$
$200$	0	0
$201$	−6.16917	−0.435140
$202$	6.40095	0.450369
$203$	7.66245	0.537799
$204$	−2.61393	−0.183011
$205$	0	0
$206$	−5.75573	−0.401020
$207$	12.7234	0.884341
$208$	3.19033	0.221210
$209$	−1.92748	−0.133327
$210$	0	0
$211$	−9.32184	−0.641742	−0.320871	−	0.947123i	$-0.603976\pi$
$211$	−9.32184	−0.641742	−0.320871	+	0.947123i	$0.603976\pi$
$212$	−18.9569	−1.30196
$213$	−3.22029	−0.220650
$214$	−0.694449	−0.0474715
$215$	0	0
$216$	−9.61255	−0.654051
$217$	8.48407	0.575936
$218$	−1.14483	−0.0775379
$219$	6.47109	0.437275
$220$	0	0
$221$	3.61723	0.243321
$222$	2.45063	0.164476
$223$	19.3803	1.29780	0.648899	−	0.760874i	$-0.275230\pi$
$223$	19.3803	1.29780	0.648899	+	0.760874i	$0.275230\pi$
$224$	16.1825	1.08124
$225$	0	0
$226$	7.04270	0.468473
$227$	19.0142	1.26202	0.631008	−	0.775776i	$-0.282642\pi$
$227$	19.0142	1.26202	0.631008	+	0.775776i	$0.282642\pi$
$228$	−3.24822	−0.215118
$229$	−3.02508	−0.199903	−0.0999514	−	0.994992i	$-0.531869\pi$
$229$	−3.02508	−0.199903	−0.0999514	+	0.994992i	$0.531869\pi$
$230$	0	0
$231$	1.72185	0.113289
$232$	−6.03029	−0.395908
$233$	11.1804	0.732451	0.366225	−	0.930526i	$-0.380650\pi$
$233$	11.1804	0.732451	0.366225	+	0.930526i	$0.380650\pi$
$234$	2.61803	0.171146
$235$	0	0
$236$	−10.4717	−0.681646
$237$	−2.68025	−0.174101
$238$	3.67332	0.238106
$239$	−15.7989	−1.02195	−0.510974	−	0.859596i	$-0.670715\pi$
$239$	−15.7989	−1.02195	−0.510974	+	0.859596i	$0.670715\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	6.51996	0.419119
$243$	15.7407	1.00977
$244$	9.55266	0.611546
$245$	0	0
$246$	−1.69875	−0.108308
$247$	4.49498	0.286009
$248$	−6.67690	−0.423983
$249$	0.991335	0.0628233
$250$	0	0
$251$	−21.4394	−1.35324	−0.676620	−	0.736332i	$-0.736556\pi$
$251$	−21.4394	−1.35324	−0.676620	+	0.736332i	$0.736556\pi$
$252$	−10.9399	−0.689147
$253$	4.06300	0.255438
$254$	−11.8423	−0.743052
$255$	0	0
$256$	−6.92130	−0.432581
$257$	−9.85853	−0.614958	−0.307479	−	0.951555i	$-0.599485\pi$
$257$	−9.85853	−0.614958	−0.307479	+	0.951555i	$0.599485\pi$
$258$	−0.0947043	−0.00589603
$259$	14.1709	0.880540
$260$	0	0
$261$	6.33584	0.392179
$262$	−10.5829	−0.653813
$263$	6.03135	0.371909	0.185954	−	0.982558i	$-0.440462\pi$
$263$	6.03135	0.371909	0.185954	+	0.982558i	$0.440462\pi$
$264$	−1.35508	−0.0833996
$265$	0	0
$266$	4.56468	0.279878
$267$	−9.15076	−0.560017
$268$	12.5166	0.764570
$269$	−21.2569	−1.29605	−0.648027	−	0.761617i	$-0.724406\pi$
$269$	−21.2569	−1.29605	−0.648027	+	0.761617i	$0.724406\pi$
$270$	0	0
$271$	12.9350	0.785743	0.392871	−	0.919593i	$-0.371482\pi$
$271$	12.9350	0.785743	0.392871	+	0.919593i	$0.371482\pi$
$272$	3.70132	0.224426
$273$	−4.01544	−0.243026
$274$	0.470189	0.0284051
$275$	0	0
$276$	6.84702	0.412142
$277$	6.91123	0.415256	0.207628	−	0.978208i	$-0.433426\pi$
$277$	6.91123	0.415256	0.207628	+	0.978208i	$0.433426\pi$
$278$	−8.21459	−0.492678
$279$	7.01520	0.419989
$280$	0	0
$281$	−5.05945	−0.301821	−0.150911	−	0.988547i	$-0.548221\pi$
$281$	−5.05945	−0.301821	−0.150911	+	0.988547i	$0.548221\pi$
$282$	2.51408	0.149711
$283$	−22.9620	−1.36495	−0.682475	−	0.730909i	$-0.739097\pi$
$283$	−22.9620	−1.36495	−0.682475	+	0.730909i	$0.739097\pi$
$284$	6.53360	0.387698
$285$	0	0
$286$	0.836018	0.0494348
$287$	−9.82314	−0.579842
$288$	13.3808	0.788471
$289$	−12.8034	−0.753142
$290$	0	0
$291$	−6.59605	−0.386667
$292$	−13.1291	−0.768323
$293$	21.4383	1.25244	0.626218	−	0.779648i	$-0.284602\pi$
$293$	21.4383	1.25244	0.626218	+	0.779648i	$0.284602\pi$
$294$	−0.606425	−0.0353674
$295$	0	0
$296$	−11.1524	−0.648222
$297$	3.22512	0.187141
$298$	5.87200	0.340156
$299$	−9.47511	−0.547960
$300$	0	0
$301$	−0.547634	−0.0315651
$302$	−9.94643	−0.572353
$303$	−8.11783	−0.466357
$304$	4.59948	0.263798
$305$	0	0
$306$	3.03735	0.173634
$307$	−8.02382	−0.457944	−0.228972	−	0.973433i	$-0.573536\pi$
$307$	−8.02382	−0.457944	−0.228972	+	0.973433i	$0.573536\pi$
$308$	−3.49344	−0.199057
$309$	7.29955	0.415257
$310$	0	0
$311$	10.7664	0.610504	0.305252	−	0.952272i	$-0.401259\pi$
$311$	10.7664	0.610504	0.305252	+	0.952272i	$0.401259\pi$
$312$	3.16012	0.178907
$313$	−4.51142	−0.255001	−0.127500	−	0.991839i	$-0.540695\pi$
$313$	−4.51142	−0.255001	−0.127500	+	0.991839i	$0.540695\pi$
$314$	−12.0241	−0.678559
$315$	0	0
$316$	5.43793	0.305907
$317$	26.8455	1.50779	0.753897	−	0.656993i	$-0.228172\pi$
$317$	26.8455	1.50779	0.753897	+	0.656993i	$0.228172\pi$
$318$	−5.84261	−0.327637
$319$	2.02323	0.113279
$320$	0	0
$321$	0.880716	0.0491568
$322$	−9.62202	−0.536214
$323$	5.21493	0.290166
$324$	−6.01012	−0.333895
$325$	0	0
$326$	2.43560	0.134895
$327$	1.45190	0.0802905
$328$	7.73074	0.426859
$329$	14.5378	0.801496
$330$	0	0
$331$	12.1662	0.668716	0.334358	−	0.942446i	$-0.391480\pi$
$331$	12.1662	0.668716	0.334358	+	0.942446i	$0.391480\pi$
$332$	−2.01131	−0.110385
$333$	11.7175	0.642115
$334$	−3.12772	−0.171141
$335$	0	0
$336$	−4.10879	−0.224153
$337$	−20.2272	−1.10185	−0.550923	−	0.834556i	$-0.685724\pi$
$337$	−20.2272	−1.10185	−0.550923	+	0.834556i	$0.685724\pi$
$338$	6.17944	0.336117
$339$	−8.93171	−0.485104
$340$	0	0
$341$	2.24017	0.121312
$342$	3.77438	0.204095
$343$	16.5662	0.894491
$344$	0.430984	0.0232371
$345$	0	0
$346$	−7.42061	−0.398935
$347$	7.93211	0.425818	0.212909	−	0.977072i	$-0.431706\pi$
$347$	7.93211	0.425818	0.212909	+	0.977072i	$0.431706\pi$
$348$	3.40958	0.182773
$349$	13.3361	0.713866	0.356933	−	0.934130i	$-0.383822\pi$
$349$	13.3361	0.713866	0.356933	+	0.934130i	$0.383822\pi$
$350$	0	0
$351$	−7.52114	−0.401449
$352$	4.27291	0.227747
$353$	−13.0021	−0.692032	−0.346016	−	0.938229i	$-0.612466\pi$
$353$	−13.0021	−0.692032	−0.346016	+	0.938229i	$0.612466\pi$
$354$	−3.22742	−0.171535
$355$	0	0
$356$	18.5659	0.983988
$357$	−4.65859	−0.246558
$358$	−15.4628	−0.817235
$359$	0.0595483	0.00314284	0.00157142	−	0.999999i	$-0.499500\pi$
$359$	0.0595483	0.00314284	0.00157142	+	0.999999i	$0.499500\pi$
$360$	0	0
$361$	−12.5196	−0.658928
$362$	2.02987	0.106688
$363$	−8.26876	−0.433998
$364$	8.14688	0.427012
$365$	0	0
$366$	2.94418	0.153895
$367$	5.04003	0.263087	0.131544	−	0.991310i	$-0.458007\pi$
$367$	5.04003	0.263087	0.131544	+	0.991310i	$0.458007\pi$
$368$	−9.69538	−0.505407
$369$	−8.12245	−0.422838
$370$	0	0
$371$	−33.7853	−1.75404
$372$	3.77517	0.195734
$373$	23.6008	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$373$	23.6008	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$374$	0.969920	0.0501534
$375$	0	0
$376$	−11.4412	−0.590033
$377$	−4.71828	−0.243004
$378$	−7.63776	−0.392844
$379$	16.1059	0.827303	0.413652	−	0.910435i	$-0.364253\pi$
$379$	16.1059	0.827303	0.413652	+	0.910435i	$0.364253\pi$
$380$	0	0
$381$	15.0187	0.769430
$382$	−1.03755	−0.0530857
$383$	11.3050	0.577657	0.288828	−	0.957381i	$-0.406734\pi$
$383$	11.3050	0.577657	0.288828	+	0.957381i	$0.406734\pi$
$384$	8.99273	0.458908
$385$	0	0
$386$	−8.36890	−0.425966
$387$	−0.452821	−0.0230182
$388$	13.3826	0.679400
$389$	−23.3791	−1.18537	−0.592684	−	0.805435i	$-0.701932\pi$
$389$	−23.3791	−1.18537	−0.592684	+	0.805435i	$0.701932\pi$
$390$	0	0
$391$	−10.9927	−0.555925
$392$	2.75974	0.139388
$393$	13.4215	0.677023
$394$	−17.1196	−0.862472
$395$	0	0
$396$	−2.88862	−0.145158
$397$	14.0294	0.704114	0.352057	−	0.935979i	$-0.385482\pi$
$397$	14.0294	0.704114	0.352057	+	0.935979i	$0.385482\pi$
$398$	−11.0514	−0.553957
$399$	−5.78903	−0.289814
$400$	0	0
$401$	−8.76024	−0.437465	−0.218733	−	0.975785i	$-0.570192\pi$
$401$	−8.76024	−0.437465	−0.218733	+	0.975785i	$0.570192\pi$
$402$	3.85767	0.192403
$403$	−5.22420	−0.260236
$404$	16.4702	0.819421
$405$	0	0
$406$	−4.79144	−0.237795
$407$	3.74177	0.185472
$408$	3.66627	0.181507
$409$	19.8589	0.981958	0.490979	−	0.871171i	$-0.336639\pi$
$409$	19.8589	0.981958	0.490979	+	0.871171i	$0.336639\pi$
$410$	0	0
$411$	−0.596304	−0.0294135
$412$	−14.8100	−0.729634
$413$	−18.6628	−0.918334
$414$	−7.95614	−0.391023
$415$	0	0
$416$	−9.96463	−0.488556
$417$	10.4179	0.510169
$418$	1.20528	0.0589521
$419$	−31.3944	−1.53372	−0.766860	−	0.641815i	$-0.778182\pi$
$419$	−31.3944	−1.53372	−0.766860	+	0.641815i	$0.778182\pi$
$420$	0	0
$421$	0.305158	0.0148725	0.00743625	−	0.999972i	$-0.497633\pi$
$421$	0.305158	0.0148725	0.00743625	+	0.999972i	$0.497633\pi$
$422$	5.82907	0.283755
$423$	12.0209	0.584475
$424$	26.5887	1.29126
$425$	0	0
$426$	2.01369	0.0975636
$427$	17.0249	0.823893
$428$	−1.78687	−0.0863718
$429$	−1.06026	−0.0511897
$430$	0	0
$431$	18.4107	0.886813	0.443406	−	0.896321i	$-0.353770\pi$
$431$	18.4107	0.886813	0.443406	+	0.896321i	$0.353770\pi$
$432$	−7.69599	−0.370273
$433$	22.7381	1.09272	0.546362	−	0.837549i	$-0.316012\pi$
$433$	22.7381	1.09272	0.546362	+	0.837549i	$0.316012\pi$
$434$	−5.30520	−0.254658
$435$	0	0
$436$	−2.94575	−0.141076
$437$	−13.6602	−0.653455
$438$	−4.04646	−0.193347
$439$	1.59341	0.0760494	0.0380247	−	0.999277i	$-0.487893\pi$
$439$	1.59341	0.0760494	0.0380247	+	0.999277i	$0.487893\pi$
$440$	0	0
$441$	−2.89957	−0.138075
$442$	−2.26190	−0.107588
$443$	−27.2527	−1.29482	−0.647408	−	0.762144i	$-0.724147\pi$
$443$	−27.2527	−1.29482	−0.647408	+	0.762144i	$0.724147\pi$
$444$	6.30568	0.299254
$445$	0	0
$446$	−12.1187	−0.573839
$447$	−7.44701	−0.352232
$448$	0.243031	0.0114821
$449$	−15.1097	−0.713070	−0.356535	−	0.934282i	$-0.616042\pi$
$449$	−15.1097	−0.713070	−0.356535	+	0.934282i	$0.616042\pi$
$450$	0	0
$451$	−2.59375	−0.122135
$452$	18.1214	0.852361
$453$	12.6143	0.592671
$454$	−11.8898	−0.558017
$455$	0	0
$456$	4.55592	0.213351
$457$	26.1957	1.22538	0.612691	−	0.790323i	$-0.290087\pi$
$457$	26.1957	1.22538	0.612691	+	0.790323i	$0.290087\pi$
$458$	1.89162	0.0883897
$459$	−8.72579	−0.407285
$460$	0	0
$461$	7.48120	0.348434	0.174217	−	0.984707i	$-0.444261\pi$
$461$	7.48120	0.348434	0.174217	+	0.984707i	$0.444261\pi$
$462$	−1.07670	−0.0500925
$463$	−15.4334	−0.717249	−0.358624	−	0.933482i	$-0.616754\pi$
$463$	−15.4334	−0.717249	−0.358624	+	0.933482i	$0.616754\pi$
$464$	−4.82797	−0.224133
$465$	0	0
$466$	−6.99124	−0.323863
$467$	22.6484	1.04804	0.524020	−	0.851706i	$-0.324432\pi$
$467$	22.6484	1.04804	0.524020	+	0.851706i	$0.324432\pi$
$468$	6.73640	0.311390
$469$	22.3072	1.03005
$470$	0	0
$471$	15.2492	0.702648
$472$	14.6874	0.676045
$473$	−0.144600	−0.00664871
$474$	1.67600	0.0769811
$475$	0	0
$476$	9.45174	0.433220
$477$	−27.9360	−1.27910
$478$	9.87928	0.451868
$479$	−16.7447	−0.765085	−0.382542	−	0.923938i	$-0.624951\pi$
$479$	−16.7447	−0.765085	−0.382542	+	0.923938i	$0.624951\pi$
$480$	0	0
$481$	−8.72598	−0.397871
$482$	0.625313	0.0284822
$483$	12.2029	0.555250
$484$	16.7764	0.762563
$485$	0	0
$486$	−9.84287	−0.446482
$487$	16.4770	0.746642	0.373321	−	0.927702i	$-0.378219\pi$
$487$	16.4770	0.746642	0.373321	+	0.927702i	$0.378219\pi$
$488$	−13.3985	−0.606521
$489$	−3.08888	−0.139684
$490$	0	0
$491$	−6.23529	−0.281395	−0.140697	−	0.990053i	$-0.544934\pi$
$491$	−6.23529	−0.281395	−0.140697	+	0.990053i	$0.544934\pi$
$492$	−4.37103	−0.197061
$493$	−5.47399	−0.246536
$494$	−2.81077	−0.126463
$495$	0	0
$496$	−5.34565	−0.240027
$497$	11.6443	0.522318
$498$	−0.619895	−0.0277782
$499$	31.5321	1.41157	0.705786	−	0.708425i	$-0.250594\pi$
$499$	31.5321	1.41157	0.705786	+	0.708425i	$0.250594\pi$
$500$	0	0
$501$	3.96665	0.177217
$502$	13.4063	0.598353
$503$	37.3580	1.66571	0.832856	−	0.553490i	$-0.186704\pi$
$503$	37.3580	1.66571	0.832856	+	0.553490i	$0.186704\pi$
$504$	15.3442	0.683483
$505$	0	0
$506$	−2.54065	−0.112945
$507$	−7.83691	−0.348049
$508$	−30.4712	−1.35194
$509$	3.28509	0.145609	0.0728046	−	0.997346i	$-0.476805\pi$
$509$	3.28509	0.145609	0.0728046	+	0.997346i	$0.476805\pi$
$510$	0	0
$511$	−23.3989	−1.03511
$512$	−18.3512	−0.811017
$513$	−10.8432	−0.478738
$514$	6.16467	0.271912
$515$	0	0
$516$	−0.243682	−0.0107275
$517$	3.83864	0.168823
$518$	−8.86128	−0.389342
$519$	9.41099	0.413097
$520$	0	0
$521$	34.8257	1.52574	0.762870	−	0.646552i	$-0.223790\pi$
$521$	34.8257	1.52574	0.762870	+	0.646552i	$0.223790\pi$
$522$	−3.96189	−0.173407
$523$	−14.6036	−0.638571	−0.319285	−	0.947659i	$-0.603443\pi$
$523$	−14.6036	−0.638571	−0.319285	+	0.947659i	$0.603443\pi$
$524$	−27.2306	−1.18958
$525$	0	0
$526$	−3.77148	−0.164444
$527$	−6.06094	−0.264019
$528$	−1.08491	−0.0472144
$529$	5.79473	0.251945
$530$	0	0
$531$	−15.4316	−0.669676
$532$	11.7453	0.509223
$533$	6.04876	0.262001
$534$	5.72209	0.247619
$535$	0	0
$536$	−17.5556	−0.758287
$537$	19.6103	0.846247
$538$	13.2922	0.573068
$539$	−0.925924	−0.0398824
$540$	0	0
$541$	39.0731	1.67989	0.839943	−	0.542675i	$-0.182588\pi$
$541$	39.0731	1.67989	0.839943	+	0.542675i	$0.182588\pi$
$542$	−8.08840	−0.347427
$543$	−2.57434	−0.110475
$544$	−11.5606	−0.495658
$545$	0	0
$546$	2.51091	0.107457
$547$	21.4939	0.919013	0.459507	−	0.888174i	$-0.348026\pi$
$547$	21.4939	0.919013	0.459507	+	0.888174i	$0.348026\pi$
$548$	1.20983	0.0516816
$549$	14.0774	0.600807
$550$	0	0
$551$	−6.80230	−0.289788
$552$	−9.60357	−0.408755
$553$	9.69157	0.412127
$554$	−4.32169	−0.183611
$555$	0	0
$556$	−21.1368	−0.896401
$557$	−24.3053	−1.02985	−0.514923	−	0.857236i	$-0.672180\pi$
$557$	−24.3053	−1.02985	−0.514923	+	0.857236i	$0.672180\pi$
$558$	−4.38670	−0.185704
$559$	0.337214	0.0142626
$560$	0	0
$561$	−1.23008	−0.0519338
$562$	3.16374	0.133454
$563$	12.9287	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$563$	12.9287	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$564$	6.46893	0.272391
$565$	0	0
$566$	14.3585	0.603531
$567$	−10.7113	−0.449834
$568$	−9.16397	−0.384512
$569$	−24.6551	−1.03360	−0.516798	−	0.856108i	$-0.672876\pi$
$569$	−24.6551	−1.03360	−0.516798	+	0.856108i	$0.672876\pi$
$570$	0	0
$571$	39.9053	1.66999	0.834993	−	0.550260i	$-0.185471\pi$
$571$	39.9053	1.66999	0.834993	+	0.550260i	$0.185471\pi$
$572$	2.15114	0.0899438
$573$	1.31585	0.0549703
$574$	6.14254	0.256385
$575$	0	0
$576$	0.200955	0.00837311
$577$	−29.6113	−1.23273	−0.616367	−	0.787459i	$-0.711396\pi$
$577$	−29.6113	−1.23273	−0.616367	+	0.787459i	$0.711396\pi$
$578$	8.00614	0.333012
$579$	10.6136	0.441088
$580$	0	0
$581$	−3.58459	−0.148714
$582$	4.12460	0.170970
$583$	−8.92083	−0.369463
$584$	18.4148	0.762008
$585$	0	0
$586$	−13.4056	−0.553781
$587$	6.58354	0.271732	0.135866	−	0.990727i	$-0.456618\pi$
$587$	6.58354	0.271732	0.135866	+	0.990727i	$0.456618\pi$
$588$	−1.56038	−0.0643490
$589$	−7.53168	−0.310338
$590$	0	0
$591$	21.7114	0.893089
$592$	−8.92884	−0.366973
$593$	37.9415	1.55807	0.779036	−	0.626979i	$-0.215709\pi$
$593$	37.9415	1.55807	0.779036	+	0.626979i	$0.215709\pi$
$594$	−2.01671	−0.0827467
$595$	0	0
$596$	15.1091	0.618895
$597$	14.0157	0.573623
$598$	5.92491	0.242288
$599$	−1.45269	−0.0593552	−0.0296776	−	0.999560i	$-0.509448\pi$
$599$	−1.45269	−0.0593552	−0.0296776	+	0.999560i	$0.509448\pi$
$600$	0	0
$601$	−20.0197	−0.816619	−0.408309	−	0.912844i	$-0.633882\pi$
$601$	−20.0197	−0.816619	−0.408309	+	0.912844i	$0.633882\pi$
$602$	0.342443	0.0139569
$603$	18.4451	0.751144
$604$	−25.5930	−1.04136
$605$	0	0
$606$	5.07619	0.206206
$607$	−2.87043	−0.116507	−0.0582537	−	0.998302i	$-0.518553\pi$
$607$	−2.87043	−0.116507	−0.0582537	+	0.998302i	$0.518553\pi$
$608$	−14.3659	−0.582615
$609$	6.07661	0.246237
$610$	0	0
$611$	−8.95190	−0.362155
$612$	7.81535	0.315917
$613$	8.57298	0.346259	0.173130	−	0.984899i	$-0.444612\pi$
$613$	8.57298	0.346259	0.173130	+	0.984899i	$0.444612\pi$
$614$	5.01740	0.202486
$615$	0	0
$616$	4.89987	0.197421
$617$	−11.9300	−0.480285	−0.240143	−	0.970738i	$-0.577194\pi$
$617$	−11.9300	−0.480285	−0.240143	+	0.970738i	$0.577194\pi$
$618$	−4.56451	−0.183611
$619$	13.7845	0.554047	0.277024	−	0.960863i	$-0.410652\pi$
$619$	13.7845	0.554047	0.277024	+	0.960863i	$0.410652\pi$
$620$	0	0
$621$	22.8567	0.917206
$622$	−6.73234	−0.269942
$623$	33.0884	1.32566
$624$	2.53005	0.101283
$625$	0	0
$626$	2.82105	0.112752
$627$	−1.52856	−0.0610450
$628$	−30.9390	−1.23460
$629$	−10.1236	−0.403654
$630$	0	0
$631$	−25.8071	−1.02736	−0.513682	−	0.857981i	$-0.671719\pi$
$631$	−25.8071	−1.02736	−0.513682	+	0.857981i	$0.671719\pi$
$632$	−7.62719	−0.303393
$633$	−7.39257	−0.293828
$634$	−16.7869	−0.666691
$635$	0	0
$636$	−15.0335	−0.596117
$637$	2.15930	0.0855546
$638$	−1.26515	−0.0500879
$639$	9.62830	0.380890
$640$	0	0
$641$	11.9184	0.470747	0.235373	−	0.971905i	$-0.424369\pi$
$641$	11.9184	0.470747	0.235373	+	0.971905i	$0.424369\pi$
$642$	−0.550724	−0.0217353
$643$	−8.86773	−0.349709	−0.174855	−	0.984594i	$-0.555946\pi$
$643$	−8.86773	−0.349709	−0.174855	+	0.984594i	$0.555946\pi$
$644$	−24.7583	−0.975612
$645$	0	0
$646$	−3.26096	−0.128301
$647$	4.42965	0.174148	0.0870738	−	0.996202i	$-0.472248\pi$
$647$	4.42965	0.174148	0.0870738	+	0.996202i	$0.472248\pi$
$648$	8.42974	0.331151
$649$	−4.92781	−0.193433
$650$	0	0
$651$	6.72818	0.263698
$652$	6.26699	0.245434
$653$	−21.4705	−0.840207	−0.420104	−	0.907476i	$-0.638006\pi$
$653$	−21.4705	−0.840207	−0.420104	+	0.907476i	$0.638006\pi$
$654$	−0.907896	−0.0355015
$655$	0	0
$656$	6.18937	0.241654
$657$	−19.3478	−0.754830
$658$	−9.09070	−0.354392
$659$	10.9678	0.427245	0.213623	−	0.976916i	$-0.431474\pi$
$659$	10.9678	0.427245	0.213623	+	0.976916i	$0.431474\pi$
$660$	0	0
$661$	−21.1702	−0.823426	−0.411713	−	0.911314i	$-0.635069\pi$
$661$	−21.1702	−0.823426	−0.411713	+	0.911314i	$0.635069\pi$
$662$	−7.60771	−0.295682
$663$	2.86860	0.111407
$664$	2.82104	0.109478
$665$	0	0
$666$	−7.32711	−0.283920
$667$	14.3388	0.555200
$668$	−8.04788	−0.311382
$669$	15.3693	0.594210
$670$	0	0
$671$	4.49534	0.173541
$672$	12.8333	0.495056
$673$	−30.4269	−1.17287	−0.586435	−	0.809996i	$-0.699469\pi$
$673$	−30.4269	−1.17287	−0.586435	+	0.809996i	$0.699469\pi$
$674$	12.6483	0.487196
$675$	0	0
$676$	15.9002	0.611547
$677$	19.4229	0.746483	0.373242	−	0.927734i	$-0.378246\pi$
$677$	19.4229	0.746483	0.373242	+	0.927734i	$0.378246\pi$
$678$	5.58512	0.214495
$679$	23.8508	0.915308
$680$	0	0
$681$	15.0790	0.577827
$682$	−1.40081	−0.0536398
$683$	−9.88104	−0.378088	−0.189044	−	0.981969i	$-0.560539\pi$
$683$	−9.88104	−0.378088	−0.189044	+	0.981969i	$0.560539\pi$
$684$	9.71180	0.371340
$685$	0	0
$686$	−10.3591	−0.395511
$687$	−2.39900	−0.0915275
$688$	0.345054	0.0131550
$689$	20.8038	0.792562
$690$	0	0
$691$	−28.7583	−1.09402	−0.547008	−	0.837128i	$-0.684233\pi$
$691$	−28.7583	−1.09402	−0.547008	+	0.837128i	$0.684233\pi$
$692$	−19.0938	−0.725839
$693$	−5.14814	−0.195562
$694$	−4.96005	−0.188281
$695$	0	0
$696$	−4.78225	−0.181271
$697$	7.01757	0.265809
$698$	−8.33926	−0.315646
$699$	8.86646	0.335360
$700$	0	0
$701$	1.68992	0.0638273	0.0319136	−	0.999491i	$-0.489840\pi$
$701$	1.68992	0.0638273	0.0319136	+	0.999491i	$0.489840\pi$
$702$	4.70307	0.177506
$703$	−12.5802	−0.474470
$704$	0.0641711	0.00241854
$705$	0	0
$706$	8.13039	0.305991
$707$	29.3534	1.10395
$708$	−8.30441	−0.312099
$709$	−24.3142	−0.913140	−0.456570	−	0.889687i	$-0.650922\pi$
$709$	−24.3142	−0.913140	−0.456570	+	0.889687i	$0.650922\pi$
$710$	0	0
$711$	8.01365	0.300535
$712$	−26.0403	−0.975902
$713$	15.8763	0.594571
$714$	2.91308	0.109019
$715$	0	0
$716$	−39.7871	−1.48691
$717$	−12.5291	−0.467909
$718$	−0.0372364	−0.00138965
$719$	10.6971	0.398933	0.199466	−	0.979905i	$-0.436079\pi$
$719$	10.6971	0.398933	0.199466	+	0.979905i	$0.436079\pi$
$720$	0	0
$721$	−26.3946	−0.982985
$722$	7.82869	0.291354
$723$	−0.793037	−0.0294934
$724$	5.22304	0.194113
$725$	0	0
$726$	5.17057	0.191898
$727$	23.2181	0.861111	0.430555	−	0.902564i	$-0.358318\pi$
$727$	23.2181	0.861111	0.430555	+	0.902564i	$0.358318\pi$
$728$	−11.4267	−0.423503
$729$	1.27691	0.0472928
$730$	0	0
$731$	0.391225	0.0144700
$732$	7.57561	0.280003
$733$	−0.854398	−0.0315579	−0.0157789	−	0.999876i	$-0.505023\pi$
$733$	−0.854398	−0.0315579	−0.0157789	+	0.999876i	$0.505023\pi$
$734$	−3.15160	−0.116328
$735$	0	0
$736$	30.2824	1.11622
$737$	5.89011	0.216965
$738$	5.07907	0.186963
$739$	−0.158737	−0.00583923	−0.00291962	−	0.999996i	$-0.500929\pi$
$739$	−0.158737	−0.00583923	−0.00291962	+	0.999996i	$0.500929\pi$
$740$	0	0
$741$	3.56469	0.130952
$742$	21.1264	0.775574
$743$	−18.5875	−0.681911	−0.340955	−	0.940079i	$-0.610751\pi$
$743$	−18.5875	−0.681911	−0.340955	+	0.940079i	$0.610751\pi$
$744$	−5.29503	−0.194125
$745$	0	0
$746$	−14.7579	−0.540324
$747$	−2.96398	−0.108446
$748$	2.49568	0.0912513
$749$	−3.18460	−0.116363
$750$	0	0
$751$	0.268190	0.00978641	0.00489320	−	0.999988i	$-0.498442\pi$
$751$	0.268190	0.00978641	0.00489320	+	0.999988i	$0.498442\pi$
$752$	−9.16001	−0.334031
$753$	−17.0022	−0.619595
$754$	2.95040	0.107447
$755$	0	0
$756$	−19.6526	−0.714758
$757$	15.8431	0.575826	0.287913	−	0.957656i	$-0.407039\pi$
$757$	15.8431	0.575826	0.287913	+	0.957656i	$0.407039\pi$
$758$	−10.0712	−0.365803
$759$	3.22211	0.116955
$760$	0	0
$761$	43.2172	1.56662	0.783312	−	0.621629i	$-0.213529\pi$
$761$	43.2172	1.56662	0.783312	+	0.621629i	$0.213529\pi$
$762$	−9.39138	−0.340214
$763$	−5.24997	−0.190062
$764$	−2.66970	−0.0965865
$765$	0	0
$766$	−7.06915	−0.255419
$767$	11.4919	0.414948
$768$	−5.48885	−0.198062
$769$	−37.9380	−1.36808	−0.684040	−	0.729444i	$-0.739779\pi$
$769$	−37.9380	−1.36808	−0.684040	+	0.729444i	$0.739779\pi$
$770$	0	0
$771$	−7.81818	−0.281565
$772$	−21.5339	−0.775021
$773$	−15.3418	−0.551806	−0.275903	−	0.961185i	$-0.588977\pi$
$773$	−15.3418	−0.551806	−0.275903	+	0.961185i	$0.588977\pi$
$774$	0.283155	0.0101778
$775$	0	0
$776$	−18.7704	−0.673817
$777$	11.2381	0.403164
$778$	14.6193	0.524126
$779$	8.72044	0.312442
$780$	0	0
$781$	3.07462	0.110018
$782$	6.87389	0.245810
$783$	11.3818	0.406753
$784$	2.20950	0.0789107
$785$	0	0
$786$	−8.39262	−0.299355
$787$	−7.53787	−0.268696	−0.134348	−	0.990934i	$-0.542894\pi$
$787$	−7.53787	−0.268696	−0.134348	+	0.990934i	$0.542894\pi$
$788$	−44.0501	−1.56922
$789$	4.78308	0.170282
$790$	0	0
$791$	32.2963	1.14833
$792$	4.05155	0.143965
$793$	−10.4834	−0.372275
$794$	−8.77276	−0.311334
$795$	0	0
$796$	−28.4362	−1.00789
$797$	−9.72342	−0.344421	−0.172211	−	0.985060i	$-0.555091\pi$
$797$	−9.72342	−0.344421	−0.172211	+	0.985060i	$0.555091\pi$
$798$	3.61996	0.128145
$799$	−10.3857	−0.367420
$800$	0	0
$801$	27.3597	0.966708
$802$	5.47789	0.193431
$803$	−6.17836	−0.218030
$804$	9.92609	0.350066
$805$	0	0
$806$	3.26676	0.115067
$807$	−16.8575	−0.593412
$808$	−23.1009	−0.812687
$809$	−31.2197	−1.09763	−0.548813	−	0.835945i	$-0.684920\pi$
$809$	−31.2197	−1.09763	−0.548813	+	0.835945i	$0.684920\pi$
$810$	0	0
$811$	39.9636	1.40331	0.701656	−	0.712516i	$-0.252444\pi$
$811$	39.9636	1.40331	0.701656	+	0.712516i	$0.252444\pi$
$812$	−12.3288	−0.432655
$813$	10.2579	0.359760
$814$	−2.33978	−0.0820091
$815$	0	0
$816$	2.93528	0.102756
$817$	0.486159	0.0170085
$818$	−12.4180	−0.434186
$819$	12.0057	0.419514
$820$	0	0
$821$	21.3540	0.745260	0.372630	−	0.927980i	$-0.378456\pi$
$821$	21.3540	0.745260	0.372630	+	0.927980i	$0.378456\pi$
$822$	0.372877	0.0130056
$823$	7.50218	0.261510	0.130755	−	0.991415i	$-0.458260\pi$
$823$	7.50218	0.261510	0.130755	+	0.991415i	$0.458260\pi$
$824$	20.7723	0.723638
$825$	0	0
$826$	11.6701	0.406054
$827$	−7.28944	−0.253479	−0.126739	−	0.991936i	$-0.540451\pi$
$827$	−7.28944	−0.253479	−0.126739	+	0.991936i	$0.540451\pi$
$828$	−20.4718	−0.711445
$829$	20.3562	0.707001	0.353500	−	0.935434i	$-0.384991\pi$
$829$	20.3562	0.707001	0.353500	+	0.935434i	$0.384991\pi$
$830$	0	0
$831$	5.48087	0.190129
$832$	−0.149650	−0.00518819
$833$	2.50515	0.0867983
$834$	−6.51448	−0.225578
$835$	0	0
$836$	3.10128	0.107260
$837$	12.6023	0.435598
$838$	19.6314	0.678154
$839$	18.1861	0.627854	0.313927	−	0.949447i	$-0.398355\pi$
$839$	18.1861	0.627854	0.313927	+	0.949447i	$0.398355\pi$
$840$	0	0
$841$	−21.8598	−0.753786
$842$	−0.190819	−0.00657607
$843$	−4.01233	−0.138192
$844$	14.9987	0.516276
$845$	0	0
$846$	−7.51681	−0.258433
$847$	29.8992	1.02735
$848$	21.2874	0.731014
$849$	−18.2097	−0.624957
$850$	0	0
$851$	26.5181	0.909030
$852$	5.18139	0.177511
$853$	−22.2490	−0.761791	−0.380895	−	0.924618i	$-0.624384\pi$
$853$	−22.2490	−0.761791	−0.380895	+	0.924618i	$0.624384\pi$
$854$	−10.6459	−0.364295
$855$	0	0
$856$	2.50625	0.0856620
$857$	7.12658	0.243439	0.121720	−	0.992565i	$-0.461159\pi$
$857$	7.12658	0.243439	0.121720	+	0.992565i	$0.461159\pi$
$858$	0.662993	0.0226342
$859$	45.3016	1.54567	0.772835	−	0.634607i	$-0.218838\pi$
$859$	45.3016	1.54567	0.772835	+	0.634607i	$0.218838\pi$
$860$	0	0
$861$	−7.79012	−0.265487
$862$	−11.5125	−0.392116
$863$	−27.0597	−0.921123	−0.460562	−	0.887628i	$-0.652352\pi$
$863$	−27.0597	−0.921123	−0.460562	+	0.887628i	$0.652352\pi$
$864$	24.0375	0.817773
$865$	0	0
$866$	−14.2185	−0.483163
$867$	−10.1536	−0.344834
$868$	−13.6507	−0.463336
$869$	2.55901	0.0868084
$870$	0	0
$871$	−13.7360	−0.465427
$872$	4.13168	0.139916
$873$	19.7214	0.667469
$874$	8.54190	0.288934
$875$	0	0
$876$	−10.4119	−0.351784
$877$	−14.5387	−0.490937	−0.245469	−	0.969405i	$-0.578942\pi$
$877$	−14.5387	−0.490937	−0.245469	+	0.969405i	$0.578942\pi$
$878$	−0.996382	−0.0336263
$879$	17.0013	0.573441
$880$	0	0
$881$	−3.99212	−0.134498	−0.0672490	−	0.997736i	$-0.521422\pi$
$881$	−3.99212	−0.134498	−0.0672490	+	0.997736i	$0.521422\pi$
$882$	1.81314	0.0610516
$883$	−17.9751	−0.604909	−0.302455	−	0.953164i	$-0.597806\pi$
$883$	−17.9751	−0.604909	−0.302455	+	0.953164i	$0.597806\pi$
$884$	−5.82006	−0.195750
$885$	0	0
$886$	17.0415	0.572520
$887$	6.30618	0.211741	0.105870	−	0.994380i	$-0.466237\pi$
$887$	6.30618	0.211741	0.105870	+	0.994380i	$0.466237\pi$
$888$	−8.84429	−0.296795
$889$	−54.3063	−1.82137
$890$	0	0
$891$	−2.82827	−0.0947507
$892$	−31.1825	−1.04407
$893$	−12.9059	−0.431879
$894$	4.65672	0.155744
$895$	0	0
$896$	−32.5170	−1.08632
$897$	−7.51411	−0.250889
$898$	9.44829	0.315293
$899$	7.90584	0.263674
$900$	0	0
$901$	24.1359	0.804084
$902$	1.62191	0.0540036
$903$	−0.434294	−0.0144524
$904$	−25.4170	−0.845356
$905$	0	0
$906$	−7.88789	−0.262057
$907$	−41.3860	−1.37420	−0.687100	−	0.726563i	$-0.741116\pi$
$907$	−41.3860	−1.37420	−0.687100	+	0.726563i	$0.741116\pi$
$908$	−30.5935	−1.01528
$909$	24.2714	0.805032
$910$	0	0
$911$	−11.9921	−0.397317	−0.198659	−	0.980069i	$-0.563658\pi$
$911$	−11.9921	−0.397317	−0.198659	+	0.980069i	$0.563658\pi$
$912$	3.64756	0.120783
$913$	−0.946492	−0.0313243
$914$	−16.3805	−0.541819
$915$	0	0
$916$	4.86730	0.160820
$917$	−48.5309	−1.60263
$918$	5.45635	0.180086
$919$	−29.1882	−0.962829	−0.481415	−	0.876493i	$-0.659877\pi$
$919$	−29.1882	−0.962829	−0.481415	+	0.876493i	$0.659877\pi$
$920$	0	0
$921$	−6.36319	−0.209674
$922$	−4.67809	−0.154065
$923$	−7.17016	−0.236009
$924$	−2.77043	−0.0911404
$925$	0	0
$926$	9.65069	0.317141
$927$	−21.8248	−0.716821
$928$	15.0796	0.495011
$929$	−8.95966	−0.293957	−0.146978	−	0.989140i	$-0.546955\pi$
$929$	−8.95966	−0.293957	−0.146978	+	0.989140i	$0.546955\pi$
$930$	0	0
$931$	3.11305	0.102026
$932$	−17.9890	−0.589251
$933$	8.53812	0.279525
$934$	−14.1623	−0.463405
$935$	0	0
$936$	−9.44841	−0.308831
$937$	−11.1076	−0.362869	−0.181435	−	0.983403i	$-0.558074\pi$
$937$	−11.1076	−0.362869	−0.181435	+	0.983403i	$0.558074\pi$
$938$	−13.9490	−0.455451
$939$	−3.57773	−0.116755
$940$	0	0
$941$	13.7607	0.448586	0.224293	−	0.974522i	$-0.427993\pi$
$941$	13.7607	0.448586	0.224293	+	0.974522i	$0.427993\pi$
$942$	−9.53556	−0.310685
$943$	−18.3821	−0.598603
$944$	11.7590	0.382724
$945$	0	0
$946$	0.0904203	0.00293982
$947$	2.41505	0.0784787	0.0392394	−	0.999230i	$-0.487507\pi$
$947$	2.41505	0.0784787	0.0392394	+	0.999230i	$0.487507\pi$
$948$	4.31248	0.140063
$949$	14.4083	0.467712
$950$	0	0
$951$	21.2895	0.690359
$952$	−13.2569	−0.429660
$953$	5.50480	0.178318	0.0891590	−	0.996017i	$-0.471582\pi$
$953$	5.50480	0.178318	0.0891590	+	0.996017i	$0.471582\pi$
$954$	17.4687	0.565571
$955$	0	0
$956$	25.4202	0.822148
$957$	1.60450	0.0518661
$958$	10.4707	0.338293
$959$	2.15619	0.0696269
$960$	0	0
$961$	−22.2465	−0.717628
$962$	5.45648	0.175924
$963$	−2.63324	−0.0848550
$964$	1.60898	0.0518219
$965$	0	0
$966$	−7.63062	−0.245511
$967$	−3.20039	−0.102918	−0.0514588	−	0.998675i	$-0.516387\pi$
$967$	−3.20039	−0.102918	−0.0514588	+	0.998675i	$0.516387\pi$
$968$	−23.5304	−0.756296
$969$	4.13563	0.132856
$970$	0	0
$971$	19.4594	0.624482	0.312241	−	0.950003i	$-0.398920\pi$
$971$	19.4594	0.624482	0.312241	+	0.950003i	$0.398920\pi$
$972$	−25.3265	−0.812349
$973$	−37.6704	−1.20766
$974$	−10.3033	−0.330138
$975$	0	0
$976$	−10.7271	−0.343365
$977$	−11.8951	−0.380557	−0.190278	−	0.981730i	$-0.560939\pi$
$977$	−11.8951	−0.380557	−0.190278	+	0.981730i	$0.560939\pi$
$978$	1.93152	0.0617631
$979$	8.73682	0.279230
$980$	0	0
$981$	−4.34103	−0.138598
$982$	3.89901	0.124422
$983$	18.2074	0.580726	0.290363	−	0.956917i	$-0.406224\pi$
$983$	18.2074	0.580726	0.290363	+	0.956917i	$0.406224\pi$
$984$	6.13076	0.195442
$985$	0	0
$986$	3.42296	0.109009
$987$	11.5290	0.366973
$988$	−7.23235	−0.230092
$989$	−1.02479	−0.0325864
$990$	0	0
$991$	−26.7572	−0.849972	−0.424986	−	0.905200i	$-0.639721\pi$
$991$	−26.7572	−0.849972	−0.424986	+	0.905200i	$0.639721\pi$
$992$	16.6965	0.530114
$993$	9.64827	0.306179
$994$	−7.28133	−0.230950
$995$	0	0
$996$	−1.59504	−0.0505408
$997$	3.53725	0.112026	0.0560129	−	0.998430i	$-0.482161\pi$
$997$	3.53725	0.112026	0.0560129	+	0.998430i	$0.482161\pi$
$998$	−19.7175	−0.624145
$999$	21.0496	0.665978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.14	yes	40
5.4	even	2		6025.2.a.l.1.27	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.27	✓	40	5.4	even	2
6025.2.a.o.1.14	yes	40	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.625313 q^{2} +0.793037 q^{3} -1.60898 q^{4} -0.495897 q^{6} -2.86756 q^{7} +2.25675 q^{8} -2.37109 q^{9} +O(q^{10})\)	\(q-0.625313 q^{2} +0.793037 q^{3} -1.60898 q^{4} -0.495897 q^{6} -2.86756 q^{7} +2.25675 q^{8} -2.37109 q^{9} -0.757164 q^{11} -1.27598 q^{12} +1.76574 q^{13} +1.79312 q^{14} +1.80679 q^{16} +2.04856 q^{17} +1.48268 q^{18} +2.54566 q^{19} -2.27408 q^{21} +0.473465 q^{22} -5.36607 q^{23} +1.78968 q^{24} -1.10414 q^{26} -4.25948 q^{27} +4.61385 q^{28} -2.67212 q^{29} -2.95864 q^{31} -5.64330 q^{32} -0.600459 q^{33} -1.28099 q^{34} +3.81505 q^{36} -4.94182 q^{37} -1.59183 q^{38} +1.40030 q^{39} +3.42561 q^{41} +1.42201 q^{42} +0.190976 q^{43} +1.21826 q^{44} +3.35548 q^{46} -5.06976 q^{47} +1.43285 q^{48} +1.22288 q^{49} +1.62458 q^{51} -2.84105 q^{52} +11.7819 q^{53} +2.66351 q^{54} -6.47135 q^{56} +2.01880 q^{57} +1.67091 q^{58} +6.50824 q^{59} -5.93708 q^{61} +1.85008 q^{62} +6.79924 q^{63} -0.0847519 q^{64} +0.375475 q^{66} -7.77917 q^{67} -3.29610 q^{68} -4.25549 q^{69} -4.06070 q^{71} -5.35095 q^{72} +8.15988 q^{73} +3.09019 q^{74} -4.09592 q^{76} +2.17121 q^{77} -0.875627 q^{78} -3.37973 q^{79} +3.73535 q^{81} -2.14208 q^{82} +1.25005 q^{83} +3.65896 q^{84} -0.119420 q^{86} -2.11909 q^{87} -1.70873 q^{88} -11.5389 q^{89} -5.06337 q^{91} +8.63392 q^{92} -2.34631 q^{93} +3.17019 q^{94} -4.47535 q^{96} -8.31745 q^{97} -0.764686 q^{98} +1.79531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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