LMFDB - Embedded newform 6025.2.a.o.1.15

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			1.15
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)

$f(q)$	$=$	$q-0.607027 q^{2} -2.70729 q^{3} -1.63152 q^{4} +1.64340 q^{6} +4.13328 q^{7} +2.20443 q^{8} +4.32942 q^{9} +O(q^{10})$	\(q-0.607027 q^{2} -2.70729 q^{3} -1.63152 q^{4} +1.64340 q^{6} +4.13328 q^{7} +2.20443 q^{8} +4.32942 q^{9} -2.90950 q^{11} +4.41699 q^{12} -5.65016 q^{13} -2.50901 q^{14} +1.92489 q^{16} -5.58131 q^{17} -2.62808 q^{18} -4.71394 q^{19} -11.1900 q^{21} +1.76614 q^{22} -0.792647 q^{23} -5.96803 q^{24} +3.42980 q^{26} -3.59913 q^{27} -6.74352 q^{28} +6.62616 q^{29} -9.77777 q^{31} -5.57732 q^{32} +7.87686 q^{33} +3.38800 q^{34} -7.06353 q^{36} -0.377832 q^{37} +2.86149 q^{38} +15.2966 q^{39} -1.27451 q^{41} +6.79263 q^{42} +4.04101 q^{43} +4.74690 q^{44} +0.481158 q^{46} +8.11332 q^{47} -5.21123 q^{48} +10.0840 q^{49} +15.1102 q^{51} +9.21834 q^{52} -4.86345 q^{53} +2.18477 q^{54} +9.11153 q^{56} +12.7620 q^{57} -4.02226 q^{58} -7.15558 q^{59} -14.5913 q^{61} +5.93537 q^{62} +17.8947 q^{63} -0.464193 q^{64} -4.78146 q^{66} +5.68784 q^{67} +9.10600 q^{68} +2.14593 q^{69} -2.26738 q^{71} +9.54391 q^{72} +15.3964 q^{73} +0.229354 q^{74} +7.69089 q^{76} -12.0258 q^{77} -9.28546 q^{78} -10.4215 q^{79} -3.24438 q^{81} +0.773663 q^{82} -9.54507 q^{83} +18.2567 q^{84} -2.45300 q^{86} -17.9389 q^{87} -6.41378 q^{88} -2.36422 q^{89} -23.3537 q^{91} +1.29322 q^{92} +26.4713 q^{93} -4.92501 q^{94} +15.0994 q^{96} +12.9964 q^{97} -6.12126 q^{98} -12.5964 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.607027	−0.429233	−0.214616	−	0.976698i	$-0.568850\pi$
$2$	−0.607027	−0.429233	−0.214616	+	0.976698i	$0.568850\pi$
$3$	−2.70729	−1.56305	−0.781527	−	0.623871i	$-0.785559\pi$
$3$	−2.70729	−1.56305	−0.781527	+	0.623871i	$0.785559\pi$
$4$	−1.63152	−0.815759
$5$	0	0
$5$	0	0
$6$	1.64340	0.670915
$7$	4.13328	1.56223	0.781117	−	0.624385i	$-0.214650\pi$
$7$	4.13328	1.56223	0.781117	+	0.624385i	$0.214650\pi$
$8$	2.20443	0.779384
$9$	4.32942	1.44314
$10$	0	0
$11$	−2.90950	−0.877247	−0.438623	−	0.898671i	$-0.644534\pi$
$11$	−2.90950	−0.877247	−0.438623	+	0.898671i	$0.644534\pi$
$12$	4.41699	1.27508
$13$	−5.65016	−1.56707	−0.783536	−	0.621346i	$-0.786586\pi$
$13$	−5.65016	−1.56707	−0.783536	+	0.621346i	$0.786586\pi$
$14$	−2.50901	−0.670562
$15$	0	0
$16$	1.92489	0.481222
$17$	−5.58131	−1.35367	−0.676833	−	0.736137i	$-0.736648\pi$
$17$	−5.58131	−1.35367	−0.676833	+	0.736137i	$0.736648\pi$
$18$	−2.62808	−0.619443
$19$	−4.71394	−1.08145	−0.540727	−	0.841198i	$-0.681851\pi$
$19$	−4.71394	−1.08145	−0.540727	+	0.841198i	$0.681851\pi$
$20$	0	0
$21$	−11.1900	−2.44186
$22$	1.76614	0.376543
$23$	−0.792647	−0.165278	−0.0826392	−	0.996580i	$-0.526335\pi$
$23$	−0.792647	−0.165278	−0.0826392	+	0.996580i	$0.526335\pi$
$24$	−5.96803	−1.21822
$25$	0	0
$26$	3.42980	0.672639
$27$	−3.59913	−0.692653
$28$	−6.74352	−1.27441
$29$	6.62616	1.23045	0.615223	−	0.788353i	$-0.289066\pi$
$29$	6.62616	1.23045	0.615223	+	0.788353i	$0.289066\pi$
$30$	0	0
$31$	−9.77777	−1.75614	−0.878069	−	0.478533i	$-0.841169\pi$
$31$	−9.77777	−1.75614	−0.878069	+	0.478533i	$0.841169\pi$
$32$	−5.57732	−0.985940
$33$	7.87686	1.37118
$34$	3.38800	0.581038
$35$	0	0
$36$	−7.06353	−1.17725
$37$	−0.377832	−0.0621153	−0.0310576	−	0.999518i	$-0.509888\pi$
$37$	−0.377832	−0.0621153	−0.0310576	+	0.999518i	$0.509888\pi$
$38$	2.86149	0.464195
$39$	15.2966	2.44942
$40$	0	0
$41$	−1.27451	−0.199045	−0.0995226	−	0.995035i	$-0.531732\pi$
$41$	−1.27451	−0.199045	−0.0995226	+	0.995035i	$0.531732\pi$
$42$	6.79263	1.04813
$43$	4.04101	0.616249	0.308124	−	0.951346i	$-0.400299\pi$
$43$	4.04101	0.616249	0.308124	+	0.951346i	$0.400299\pi$
$44$	4.74690	0.715622
$45$	0	0
$46$	0.481158	0.0709429
$47$	8.11332	1.18345	0.591725	−	0.806140i	$-0.298447\pi$
$47$	8.11332	1.18345	0.591725	+	0.806140i	$0.298447\pi$
$48$	−5.21123	−0.752176
$49$	10.0840	1.44057
$50$	0	0
$51$	15.1102	2.11585
$52$	9.21834	1.27835
$53$	−4.86345	−0.668046	−0.334023	−	0.942565i	$-0.608406\pi$
$53$	−4.86345	−0.668046	−0.334023	+	0.942565i	$0.608406\pi$
$54$	2.18477	0.297309
$55$	0	0
$56$	9.11153	1.21758
$57$	12.7620	1.69037
$58$	−4.02226	−0.528148
$59$	−7.15558	−0.931577	−0.465789	−	0.884896i	$-0.654229\pi$
$59$	−7.15558	−0.931577	−0.465789	+	0.884896i	$0.654229\pi$
$60$	0	0
$61$	−14.5913	−1.86822	−0.934112	−	0.356979i	$-0.883807\pi$
$61$	−14.5913	−1.86822	−0.934112	+	0.356979i	$0.883807\pi$
$62$	5.93537	0.753793
$63$	17.8947	2.25452
$64$	−0.464193	−0.0580241
$65$	0	0
$66$	−4.78146	−0.588558
$67$	5.68784	0.694881	0.347440	−	0.937702i	$-0.387051\pi$
$67$	5.68784	0.694881	0.347440	+	0.937702i	$0.387051\pi$
$68$	9.10600	1.10427
$69$	2.14593	0.258339
$70$	0	0
$71$	−2.26738	−0.269088	−0.134544	−	0.990908i	$-0.542957\pi$
$71$	−2.26738	−0.269088	−0.134544	+	0.990908i	$0.542957\pi$
$72$	9.54391	1.12476
$73$	15.3964	1.80201	0.901007	−	0.433804i	$-0.142829\pi$
$73$	15.3964	1.80201	0.901007	+	0.433804i	$0.142829\pi$
$74$	0.229354	0.0266619
$75$	0	0
$76$	7.69089	0.882205
$77$	−12.0258	−1.37046
$78$	−9.28546	−1.05137
$79$	−10.4215	−1.17251	−0.586253	−	0.810128i	$-0.699397\pi$
$79$	−10.4215	−1.17251	−0.586253	+	0.810128i	$0.699397\pi$
$80$	0	0
$81$	−3.24438	−0.360486
$82$	0.773663	0.0854368
$83$	−9.54507	−1.04771	−0.523854	−	0.851808i	$-0.675506\pi$
$83$	−9.54507	−1.04771	−0.523854	+	0.851808i	$0.675506\pi$
$84$	18.2567	1.99197
$85$	0	0
$86$	−2.45300	−0.264514
$87$	−17.9389	−1.92326
$88$	−6.41378	−0.683712
$89$	−2.36422	−0.250607	−0.125303	−	0.992118i	$-0.539990\pi$
$89$	−2.36422	−0.250607	−0.125303	+	0.992118i	$0.539990\pi$
$90$	0	0
$91$	−23.3537	−2.44813
$92$	1.29322	0.134827
$93$	26.4713	2.74494
$94$	−4.92501	−0.507976
$95$	0	0
$96$	15.0994	1.54108
$97$	12.9964	1.31958	0.659791	−	0.751449i	$-0.270645\pi$
$97$	12.9964	1.31958	0.659791	+	0.751449i	$0.270645\pi$
$98$	−6.12126	−0.618341
$99$	−12.5964	−1.26599
$100$	0	0
$101$	−7.86033	−0.782132	−0.391066	−	0.920363i	$-0.627894\pi$
$101$	−7.86033	−0.782132	−0.391066	+	0.920363i	$0.627894\pi$
$102$	−9.17231	−0.908194
$103$	−4.21555	−0.415370	−0.207685	−	0.978196i	$-0.566593\pi$
$103$	−4.21555	−0.415370	−0.207685	+	0.978196i	$0.566593\pi$
$104$	−12.4554	−1.22135
$105$	0	0
$106$	2.95225	0.286747
$107$	−9.15004	−0.884568	−0.442284	−	0.896875i	$-0.645832\pi$
$107$	−9.15004	−0.884568	−0.442284	+	0.896875i	$0.645832\pi$
$108$	5.87205	0.565038
$109$	−7.83909	−0.750849	−0.375424	−	0.926853i	$-0.622503\pi$
$109$	−7.83909	−0.750849	−0.375424	+	0.926853i	$0.622503\pi$
$110$	0	0
$111$	1.02290	0.0970896
$112$	7.95610	0.751781
$113$	−4.71870	−0.443898	−0.221949	−	0.975058i	$-0.571242\pi$
$113$	−4.71870	−0.443898	−0.221949	+	0.975058i	$0.571242\pi$
$114$	−7.74689	−0.725563
$115$	0	0
$116$	−10.8107	−1.00375
$117$	−24.4619	−2.26151
$118$	4.34363	0.399864
$119$	−23.0691	−2.11474
$120$	0	0
$121$	−2.53482	−0.230438
$122$	8.85731	0.801903
$123$	3.45047	0.311119
$124$	15.9526	1.43259
$125$	0	0
$126$	−10.8626	−0.967715
$127$	4.72860	0.419595	0.209798	−	0.977745i	$-0.432719\pi$
$127$	4.72860	0.419595	0.209798	+	0.977745i	$0.432719\pi$
$128$	11.4364	1.01085
$129$	−10.9402	−0.963230
$130$	0	0
$131$	1.63558	0.142902	0.0714509	−	0.997444i	$-0.477237\pi$
$131$	1.63558	0.142902	0.0714509	+	0.997444i	$0.477237\pi$
$132$	−12.8512	−1.11856
$133$	−19.4841	−1.68948
$134$	−3.45267	−0.298266
$135$	0	0
$136$	−12.3036	−1.05502
$137$	−19.4929	−1.66539	−0.832693	−	0.553734i	$-0.813202\pi$
$137$	−19.4929	−1.66539	−0.832693	+	0.553734i	$0.813202\pi$
$138$	−1.30264	−0.110888
$139$	14.6691	1.24422	0.622108	−	0.782932i	$-0.286277\pi$
$139$	14.6691	1.24422	0.622108	+	0.782932i	$0.286277\pi$
$140$	0	0
$141$	−21.9651	−1.84980
$142$	1.37636	0.115501
$143$	16.4391	1.37471
$144$	8.33365	0.694471
$145$	0	0
$146$	−9.34604	−0.773484
$147$	−27.3003	−2.25169
$148$	0.616440	0.0506711
$149$	4.93337	0.404157	0.202079	−	0.979369i	$-0.435230\pi$
$149$	4.93337	0.404157	0.202079	+	0.979369i	$0.435230\pi$
$150$	0	0
$151$	−0.476715	−0.0387945	−0.0193972	−	0.999812i	$-0.506175\pi$
$151$	−0.476715	−0.0387945	−0.0193972	+	0.999812i	$0.506175\pi$
$152$	−10.3916	−0.842867
$153$	−24.1638	−1.95353
$154$	7.29997	0.588248
$155$	0	0
$156$	−24.9567	−1.99814
$157$	−2.27270	−0.181381	−0.0906904	−	0.995879i	$-0.528907\pi$
$157$	−2.27270	−0.181381	−0.0906904	+	0.995879i	$0.528907\pi$
$158$	6.32611	0.503278
$159$	13.1668	1.04419
$160$	0	0
$161$	−3.27623	−0.258203
$162$	1.96942	0.154733
$163$	−0.336507	−0.0263573	−0.0131786	−	0.999913i	$-0.504195\pi$
$163$	−0.336507	−0.0263573	−0.0131786	+	0.999913i	$0.504195\pi$
$164$	2.07939	0.162373
$165$	0	0
$166$	5.79412	0.449711
$167$	16.2691	1.25894	0.629470	−	0.777024i	$-0.283272\pi$
$167$	16.2691	1.25894	0.629470	+	0.777024i	$0.283272\pi$
$168$	−24.6675	−1.90314
$169$	18.9243	1.45572
$170$	0	0
$171$	−20.4087	−1.56069
$172$	−6.59299	−0.502710
$173$	5.73548	0.436061	0.218030	−	0.975942i	$-0.430037\pi$
$173$	5.73548	0.436061	0.218030	+	0.975942i	$0.430037\pi$
$174$	10.8894	0.825524
$175$	0	0
$176$	−5.60046	−0.422150
$177$	19.3722	1.45611
$178$	1.43514	0.107569
$179$	−5.72860	−0.428176	−0.214088	−	0.976814i	$-0.568678\pi$
$179$	−5.72860	−0.428176	−0.214088	+	0.976814i	$0.568678\pi$
$180$	0	0
$181$	−3.25620	−0.242031	−0.121016	−	0.992651i	$-0.538615\pi$
$181$	−3.25620	−0.242031	−0.121016	+	0.992651i	$0.538615\pi$
$182$	14.1763	1.05082
$183$	39.5029	2.92014
$184$	−1.74734	−0.128815
$185$	0	0
$186$	−16.0688	−1.17822
$187$	16.2388	1.18750
$188$	−13.2370	−0.965410
$189$	−14.8762	−1.08209
$190$	0	0
$191$	17.6801	1.27929	0.639644	−	0.768672i	$-0.279082\pi$
$191$	17.6801	1.27929	0.639644	+	0.768672i	$0.279082\pi$
$192$	1.25670	0.0906948
$193$	−3.60080	−0.259191	−0.129596	−	0.991567i	$-0.541368\pi$
$193$	−3.60080	−0.259191	−0.129596	+	0.991567i	$0.541368\pi$
$194$	−7.88915	−0.566408
$195$	0	0
$196$	−16.4522	−1.17516
$197$	11.0793	0.789368	0.394684	−	0.918817i	$-0.370854\pi$
$197$	11.0793	0.789368	0.394684	+	0.918817i	$0.370854\pi$
$198$	7.64638	0.543405
$199$	−9.77120	−0.692662	−0.346331	−	0.938112i	$-0.612573\pi$
$199$	−9.77120	−0.692662	−0.346331	+	0.938112i	$0.612573\pi$
$200$	0	0
$201$	−15.3986	−1.08614
$202$	4.77143	0.335717
$203$	27.3878	1.92224
$204$	−24.6526	−1.72603
$205$	0	0
$206$	2.55895	0.178290
$207$	−3.43170	−0.238520
$208$	−10.8759	−0.754110
$209$	13.7152	0.948701
$210$	0	0
$211$	−25.0539	−1.72478	−0.862390	−	0.506244i	$-0.831034\pi$
$211$	−25.0539	−1.72478	−0.862390	+	0.506244i	$0.831034\pi$
$212$	7.93481	0.544965
$213$	6.13845	0.420599
$214$	5.55432	0.379686
$215$	0	0
$216$	−7.93403	−0.539842
$217$	−40.4142	−2.74350
$218$	4.75854	0.322289
$219$	−41.6826	−2.81665
$220$	0	0
$221$	31.5353	2.12129
$222$	−0.620929	−0.0416740
$223$	−14.3359	−0.960000	−0.480000	−	0.877268i	$-0.659363\pi$
$223$	−14.3359	−0.960000	−0.480000	+	0.877268i	$0.659363\pi$
$224$	−23.0526	−1.54027
$225$	0	0
$226$	2.86438	0.190535
$227$	−10.7895	−0.716125	−0.358062	−	0.933698i	$-0.616562\pi$
$227$	−10.7895	−0.716125	−0.358062	+	0.933698i	$0.616562\pi$
$228$	−20.8215	−1.37893
$229$	−13.1400	−0.868315	−0.434158	−	0.900837i	$-0.642954\pi$
$229$	−13.1400	−0.868315	−0.434158	+	0.900837i	$0.642954\pi$
$230$	0	0
$231$	32.5573	2.14211
$232$	14.6069	0.958990
$233$	27.0571	1.77257	0.886285	−	0.463141i	$-0.153278\pi$
$233$	27.0571	1.77257	0.886285	+	0.463141i	$0.153278\pi$
$234$	14.8490	0.970713
$235$	0	0
$236$	11.6745	0.759943
$237$	28.2139	1.83269
$238$	14.0036	0.907717
$239$	5.07753	0.328438	0.164219	−	0.986424i	$-0.447490\pi$
$239$	5.07753	0.328438	0.164219	+	0.986424i	$0.447490\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	1.53871	0.0989118
$243$	19.5809	1.25611
$244$	23.8060	1.52402
$245$	0	0
$246$	−2.09453	−0.133542
$247$	26.6345	1.69472
$248$	−21.5544	−1.36871
$249$	25.8413	1.63762
$250$	0	0
$251$	16.5033	1.04168	0.520841	−	0.853654i	$-0.325619\pi$
$251$	16.5033	1.04168	0.520841	+	0.853654i	$0.325619\pi$
$252$	−29.1955	−1.83915
$253$	2.30621	0.144990
$254$	−2.87039	−0.180104
$255$	0	0
$256$	−6.01383	−0.375864
$257$	7.78649	0.485708	0.242854	−	0.970063i	$-0.421916\pi$
$257$	7.78649	0.485708	0.242854	+	0.970063i	$0.421916\pi$
$258$	6.64099	0.413450
$259$	−1.56169	−0.0970385
$260$	0	0
$261$	28.6874	1.77571
$262$	−0.992844	−0.0613381
$263$	7.41946	0.457504	0.228752	−	0.973485i	$-0.426536\pi$
$263$	7.41946	0.457504	0.228752	+	0.973485i	$0.426536\pi$
$264$	17.3640	1.06868
$265$	0	0
$266$	11.8273	0.725181
$267$	6.40063	0.391712
$268$	−9.27982	−0.566855
$269$	−31.5197	−1.92179	−0.960894	−	0.276918i	$-0.910687\pi$
$269$	−31.5197	−1.92179	−0.960894	+	0.276918i	$0.910687\pi$
$270$	0	0
$271$	7.61045	0.462302	0.231151	−	0.972918i	$-0.425751\pi$
$271$	7.61045	0.462302	0.231151	+	0.972918i	$0.425751\pi$
$272$	−10.7434	−0.651414
$273$	63.2252	3.82657
$274$	11.8327	0.714839
$275$	0	0
$276$	−3.50112	−0.210743
$277$	3.99387	0.239969	0.119984	−	0.992776i	$-0.461716\pi$
$277$	3.99387	0.239969	0.119984	+	0.992776i	$0.461716\pi$
$278$	−8.90453	−0.534058
$279$	−42.3321	−2.53435
$280$	0	0
$281$	13.5892	0.810664	0.405332	−	0.914170i	$-0.367156\pi$
$281$	13.5892	0.810664	0.405332	+	0.914170i	$0.367156\pi$
$282$	13.3334	0.793994
$283$	−25.2098	−1.49857	−0.749283	−	0.662250i	$-0.769602\pi$
$283$	−25.2098	−1.49857	−0.749283	+	0.662250i	$0.769602\pi$
$284$	3.69927	0.219511
$285$	0	0
$286$	−9.97900	−0.590070
$287$	−5.26791	−0.310955
$288$	−24.1466	−1.42285
$289$	14.1510	0.832411
$290$	0	0
$291$	−35.1849	−2.06258
$292$	−25.1195	−1.47001
$293$	−1.99294	−0.116429	−0.0582144	−	0.998304i	$-0.518541\pi$
$293$	−1.99294	−0.116429	−0.0582144	+	0.998304i	$0.518541\pi$
$294$	16.5720	0.966501
$295$	0	0
$296$	−0.832905	−0.0484116
$297$	10.4717	0.607627
$298$	−2.99469	−0.173478
$299$	4.47858	0.259003
$300$	0	0
$301$	16.7026	0.962724
$302$	0.289379	0.0166519
$303$	21.2802	1.22252
$304$	−9.07381	−0.520419
$305$	0	0
$306$	14.6681	0.838519
$307$	−13.6976	−0.781762	−0.390881	−	0.920441i	$-0.627830\pi$
$307$	−13.6976	−0.781762	−0.390881	+	0.920441i	$0.627830\pi$
$308$	19.6203	1.11797
$309$	11.4127	0.649246
$310$	0	0
$311$	22.7974	1.29272	0.646362	−	0.763031i	$-0.276289\pi$
$311$	22.7974	1.29272	0.646362	+	0.763031i	$0.276289\pi$
$312$	33.7203	1.90904
$313$	−10.5386	−0.595679	−0.297840	−	0.954616i	$-0.596266\pi$
$313$	−10.5386	−0.595679	−0.297840	+	0.954616i	$0.596266\pi$
$314$	1.37959	0.0778547
$315$	0	0
$316$	17.0028	0.956482
$317$	−30.9033	−1.73570	−0.867852	−	0.496823i	$-0.834500\pi$
$317$	−30.9033	−1.73570	−0.867852	+	0.496823i	$0.834500\pi$
$318$	−7.99259	−0.448202
$319$	−19.2788	−1.07940
$320$	0	0
$321$	24.7718	1.38263
$322$	1.98876	0.110829
$323$	26.3100	1.46393
$324$	5.29326	0.294070
$325$	0	0
$326$	0.204269	0.0113134
$327$	21.2227	1.17362
$328$	−2.80957	−0.155133
$329$	33.5346	1.84882
$330$	0	0
$331$	−34.6526	−1.90468	−0.952340	−	0.305038i	$-0.901331\pi$
$331$	−34.6526	−1.90468	−0.952340	+	0.305038i	$0.901331\pi$
$332$	15.5730	0.854677
$333$	−1.63580	−0.0896410
$334$	−9.87579	−0.540379
$335$	0	0
$336$	−21.5395	−1.17507
$337$	10.5022	0.572090	0.286045	−	0.958216i	$-0.407659\pi$
$337$	10.5022	0.572090	0.286045	+	0.958216i	$0.407659\pi$
$338$	−11.4876	−0.624841
$339$	12.7749	0.693836
$340$	0	0
$341$	28.4484	1.54057
$342$	12.3886	0.669899
$343$	12.7471	0.688276
$344$	8.90813	0.480294
$345$	0	0
$346$	−3.48159	−0.187172
$347$	28.4358	1.52651	0.763257	−	0.646095i	$-0.223599\pi$
$347$	28.4358	1.52651	0.763257	+	0.646095i	$0.223599\pi$
$348$	29.2677	1.56891
$349$	5.27666	0.282453	0.141227	−	0.989977i	$-0.454895\pi$
$349$	5.27666	0.282453	0.141227	+	0.989977i	$0.454895\pi$
$350$	0	0
$351$	20.3357	1.08544
$352$	16.2272	0.864912
$353$	−8.04481	−0.428182	−0.214091	−	0.976814i	$-0.568679\pi$
$353$	−8.04481	−0.428182	−0.214091	+	0.976814i	$0.568679\pi$
$354$	−11.7595	−0.625009
$355$	0	0
$356$	3.85727	0.204435
$357$	62.4548	3.30546
$358$	3.47741	0.183787
$359$	15.4762	0.816805	0.408402	−	0.912802i	$-0.366086\pi$
$359$	15.4762	0.816805	0.408402	+	0.912802i	$0.366086\pi$
$360$	0	0
$361$	3.22127	0.169541
$362$	1.97660	0.103888
$363$	6.86250	0.360188
$364$	38.1020	1.99709
$365$	0	0
$366$	−23.9793	−1.25342
$367$	2.08050	0.108601	0.0543005	−	0.998525i	$-0.482707\pi$
$367$	2.08050	0.108601	0.0543005	+	0.998525i	$0.482707\pi$
$368$	−1.52576	−0.0795356
$369$	−5.51790	−0.287250
$370$	0	0
$371$	−20.1020	−1.04364
$372$	−43.1883	−2.23921
$373$	28.9770	1.50037	0.750185	−	0.661228i	$-0.229964\pi$
$373$	28.9770	1.50037	0.750185	+	0.661228i	$0.229964\pi$
$374$	−9.85739	−0.509714
$375$	0	0
$376$	17.8853	0.922362
$377$	−37.4388	−1.92820
$378$	9.03026	0.464467
$379$	0.284443	0.0146108	0.00730542	−	0.999973i	$-0.497675\pi$
$379$	0.284443	0.0146108	0.00730542	+	0.999973i	$0.497675\pi$
$380$	0	0
$381$	−12.8017	−0.655851
$382$	−10.7323	−0.549112
$383$	−27.3223	−1.39611	−0.698053	−	0.716046i	$-0.745950\pi$
$383$	−27.3223	−1.39611	−0.698053	+	0.716046i	$0.745950\pi$
$384$	−30.9617	−1.58001
$385$	0	0
$386$	2.18578	0.111253
$387$	17.4952	0.889333
$388$	−21.2038	−1.07646
$389$	6.58807	0.334029	0.167014	−	0.985954i	$-0.446587\pi$
$389$	6.58807	0.334029	0.167014	+	0.985954i	$0.446587\pi$
$390$	0	0
$391$	4.42401	0.223732
$392$	22.2295	1.12276
$393$	−4.42800	−0.223363
$394$	−6.72544	−0.338823
$395$	0	0
$396$	20.5513	1.03274
$397$	20.9494	1.05142	0.525709	−	0.850665i	$-0.323800\pi$
$397$	20.9494	1.05142	0.525709	+	0.850665i	$0.323800\pi$
$398$	5.93138	0.297313
$399$	52.7490	2.64075
$400$	0	0
$401$	−27.1437	−1.35549	−0.677746	−	0.735296i	$-0.737043\pi$
$401$	−27.1437	−1.35549	−0.677746	+	0.735296i	$0.737043\pi$
$402$	9.34739	0.466206
$403$	55.2459	2.75200
$404$	12.8243	0.638031
$405$	0	0
$406$	−16.6251	−0.825090
$407$	1.09930	0.0544904
$408$	33.3094	1.64906
$409$	28.4542	1.40697	0.703484	−	0.710711i	$-0.251626\pi$
$409$	28.4542	1.40697	0.703484	+	0.710711i	$0.251626\pi$
$410$	0	0
$411$	52.7728	2.60309
$412$	6.87774	0.338842
$413$	−29.5760	−1.45534
$414$	2.08314	0.102381
$415$	0	0
$416$	31.5127	1.54504
$417$	−39.7135	−1.94478
$418$	−8.32550	−0.407214
$419$	−4.58121	−0.223807	−0.111903	−	0.993719i	$-0.535695\pi$
$419$	−4.58121	−0.223807	−0.111903	+	0.993719i	$0.535695\pi$
$420$	0	0
$421$	31.3226	1.52657	0.763285	−	0.646062i	$-0.223585\pi$
$421$	31.3226	1.52657	0.763285	+	0.646062i	$0.223585\pi$
$422$	15.2084	0.740333
$423$	35.1260	1.70788
$424$	−10.7211	−0.520664
$425$	0	0
$426$	−3.72620	−0.180535
$427$	−60.3099	−2.91860
$428$	14.9285	0.721594
$429$	−44.5055	−2.14875
$430$	0	0
$431$	1.74639	0.0841207	0.0420604	−	0.999115i	$-0.486608\pi$
$431$	1.74639	0.0841207	0.0420604	+	0.999115i	$0.486608\pi$
$432$	−6.92792	−0.333320
$433$	−35.2703	−1.69498	−0.847490	−	0.530811i	$-0.821887\pi$
$433$	−35.2703	−1.69498	−0.847490	+	0.530811i	$0.821887\pi$
$434$	24.5325	1.17760
$435$	0	0
$436$	12.7896	0.612511
$437$	3.73650	0.178741
$438$	25.3024	1.20900
$439$	24.1927	1.15465	0.577327	−	0.816513i	$-0.304096\pi$
$439$	24.1927	1.15465	0.577327	+	0.816513i	$0.304096\pi$
$440$	0	0
$441$	43.6579	2.07895
$442$	−19.1428	−0.910529
$443$	−26.9380	−1.27986	−0.639932	−	0.768432i	$-0.721037\pi$
$443$	−26.9380	−1.27986	−0.639932	+	0.768432i	$0.721037\pi$
$444$	−1.66888	−0.0792017
$445$	0	0
$446$	8.70226	0.412064
$447$	−13.3561	−0.631720
$448$	−1.91864	−0.0906471
$449$	34.5786	1.63187	0.815933	−	0.578146i	$-0.196224\pi$
$449$	34.5786	1.63187	0.815933	+	0.578146i	$0.196224\pi$
$450$	0	0
$451$	3.70819	0.174612
$452$	7.69864	0.362113
$453$	1.29060	0.0606379
$454$	6.54952	0.307384
$455$	0	0
$456$	28.1330	1.31745
$457$	27.1576	1.27038	0.635190	−	0.772356i	$-0.280922\pi$
$457$	27.1576	1.27038	0.635190	+	0.772356i	$0.280922\pi$
$458$	7.97633	0.372710
$459$	20.0878	0.937620
$460$	0	0
$461$	26.2514	1.22265	0.611325	−	0.791380i	$-0.290637\pi$
$461$	26.2514	1.22265	0.611325	+	0.791380i	$0.290637\pi$
$462$	−19.7631	−0.919464
$463$	15.0126	0.697695	0.348848	−	0.937180i	$-0.386573\pi$
$463$	15.0126	0.697695	0.348848	+	0.937180i	$0.386573\pi$
$464$	12.7546	0.592118
$465$	0	0
$466$	−16.4244	−0.760845
$467$	1.86328	0.0862221	0.0431111	−	0.999070i	$-0.486273\pi$
$467$	1.86328	0.0862221	0.0431111	+	0.999070i	$0.486273\pi$
$468$	39.9101	1.84484
$469$	23.5094	1.08557
$470$	0	0
$471$	6.15285	0.283508
$472$	−15.7740	−0.726056
$473$	−11.7573	−0.540602
$474$	−17.1266	−0.786651
$475$	0	0
$476$	37.6377	1.72512
$477$	−21.0559	−0.964085
$478$	−3.08220	−0.140976
$479$	−32.9252	−1.50439	−0.752195	−	0.658941i	$-0.771005\pi$
$479$	−32.9252	−1.50439	−0.752195	+	0.658941i	$0.771005\pi$
$480$	0	0
$481$	2.13481	0.0973391
$482$	0.607027	0.0276493
$483$	8.86972	0.403586
$484$	4.13561	0.187982
$485$	0	0
$486$	−11.8861	−0.539165
$487$	27.2081	1.23292	0.616459	−	0.787387i	$-0.288567\pi$
$487$	27.2081	1.23292	0.616459	+	0.787387i	$0.288567\pi$
$488$	−32.1655	−1.45606
$489$	0.911022	0.0411979
$490$	0	0
$491$	30.4162	1.37266	0.686332	−	0.727288i	$-0.259220\pi$
$491$	30.4162	1.37266	0.686332	+	0.727288i	$0.259220\pi$
$492$	−5.62951	−0.253798
$493$	−36.9826	−1.66561
$494$	−16.1679	−0.727428
$495$	0	0
$496$	−18.8211	−0.845093
$497$	−9.37170	−0.420378
$498$	−15.6864	−0.702922
$499$	14.7082	0.658430	0.329215	−	0.944255i	$-0.393216\pi$
$499$	14.7082	0.658430	0.329215	+	0.944255i	$0.393216\pi$
$500$	0	0
$501$	−44.0452	−1.96779
$502$	−10.0180	−0.447124
$503$	37.6554	1.67897	0.839486	−	0.543382i	$-0.182856\pi$
$503$	37.6554	1.67897	0.839486	+	0.543382i	$0.182856\pi$
$504$	39.4476	1.75714
$505$	0	0
$506$	−1.39993	−0.0622344
$507$	−51.2336	−2.27536
$508$	−7.71480	−0.342289
$509$	−0.613385	−0.0271878	−0.0135939	−	0.999908i	$-0.504327\pi$
$509$	−0.613385	−0.0271878	−0.0135939	+	0.999908i	$0.504327\pi$
$510$	0	0
$511$	63.6377	2.81517
$512$	−19.2223	−0.849512
$513$	16.9661	0.749071
$514$	−4.72661	−0.208482
$515$	0	0
$516$	17.8491	0.785764
$517$	−23.6057	−1.03818
$518$	0.947986	0.0416521
$519$	−15.5276	−0.681587
$520$	0	0
$521$	−43.8907	−1.92288	−0.961442	−	0.275007i	$-0.911320\pi$
$521$	−43.8907	−1.92288	−0.961442	+	0.275007i	$0.911320\pi$
$522$	−17.4140	−0.762192
$523$	7.89434	0.345195	0.172598	−	0.984992i	$-0.444784\pi$
$523$	7.89434	0.345195	0.172598	+	0.984992i	$0.444784\pi$
$524$	−2.66849	−0.116573
$525$	0	0
$526$	−4.50381	−0.196376
$527$	54.5727	2.37723
$528$	15.1621	0.659844
$529$	−22.3717	−0.972683
$530$	0	0
$531$	−30.9795	−1.34440
$532$	31.7886	1.37821
$533$	7.20119	0.311918
$534$	−3.88535	−0.168136
$535$	0	0
$536$	12.5384	0.541579
$537$	15.5090	0.669262
$538$	19.1333	0.824894
$539$	−29.3394	−1.26374
$540$	0	0
$541$	−9.58585	−0.412128	−0.206064	−	0.978539i	$-0.566065\pi$
$541$	−9.58585	−0.412128	−0.206064	+	0.978539i	$0.566065\pi$
$542$	−4.61975	−0.198435
$543$	8.81548	0.378308
$544$	31.1287	1.33463
$545$	0	0
$546$	−38.3794	−1.64249
$547$	17.4807	0.747421	0.373711	−	0.927545i	$-0.378085\pi$
$547$	17.4807	0.747421	0.373711	+	0.927545i	$0.378085\pi$
$548$	31.8029	1.35855
$549$	−63.1719	−2.69611
$550$	0	0
$551$	−31.2353	−1.33067
$552$	4.73054	0.201345
$553$	−43.0748	−1.83173
$554$	−2.42439	−0.103002
$555$	0	0
$556$	−23.9329	−1.01498
$557$	21.0814	0.893246	0.446623	−	0.894722i	$-0.352627\pi$
$557$	21.0814	0.893246	0.446623	+	0.894722i	$0.352627\pi$
$558$	25.6967	1.08783
$559$	−22.8324	−0.965706
$560$	0	0
$561$	−43.9632	−1.85613
$562$	−8.24901	−0.347964
$563$	27.4025	1.15488	0.577439	−	0.816434i	$-0.304052\pi$
$563$	27.4025	1.15488	0.577439	+	0.816434i	$0.304052\pi$
$564$	35.8365	1.50899
$565$	0	0
$566$	15.3030	0.643234
$567$	−13.4099	−0.563163
$568$	−4.99827	−0.209723
$569$	44.1332	1.85016	0.925081	−	0.379771i	$-0.123997\pi$
$569$	44.1332	1.85016	0.925081	+	0.379771i	$0.123997\pi$
$570$	0	0
$571$	45.9351	1.92233	0.961163	−	0.275981i	$-0.0890027\pi$
$571$	45.9351	1.92233	0.961163	+	0.275981i	$0.0890027\pi$
$572$	−26.8207	−1.12143
$573$	−47.8652	−1.99960
$574$	3.19777	0.133472
$575$	0	0
$576$	−2.00969	−0.0837369
$577$	−37.7574	−1.57186	−0.785932	−	0.618313i	$-0.787816\pi$
$577$	−37.7574	−1.57186	−0.785932	+	0.618313i	$0.787816\pi$
$578$	−8.59003	−0.357298
$579$	9.74840	0.405130
$580$	0	0
$581$	−39.4524	−1.63676
$582$	21.3582	0.885326
$583$	14.1502	0.586041
$584$	33.9403	1.40446
$585$	0	0
$586$	1.20977	0.0499751
$587$	−4.42260	−0.182540	−0.0912702	−	0.995826i	$-0.529093\pi$
$587$	−4.42260	−0.182540	−0.0912702	+	0.995826i	$0.529093\pi$
$588$	44.5410	1.83684
$589$	46.0918	1.89918
$590$	0	0
$591$	−29.9949	−1.23383
$592$	−0.727285	−0.0298912
$593$	−15.9604	−0.655415	−0.327707	−	0.944779i	$-0.606276\pi$
$593$	−15.9604	−0.655415	−0.327707	+	0.944779i	$0.606276\pi$
$594$	−6.35658	−0.260814
$595$	0	0
$596$	−8.04888	−0.329695
$597$	26.4535	1.08267
$598$	−2.71862	−0.111173
$599$	37.0618	1.51430	0.757152	−	0.653239i	$-0.226590\pi$
$599$	37.0618	1.51430	0.757152	+	0.653239i	$0.226590\pi$
$600$	0	0
$601$	−14.4504	−0.589443	−0.294721	−	0.955583i	$-0.595227\pi$
$601$	−14.4504	−0.589443	−0.294721	+	0.955583i	$0.595227\pi$
$602$	−10.1390	−0.413233
$603$	24.6251	1.00281
$604$	0.777769	0.0316470
$605$	0	0
$606$	−12.9177	−0.524744
$607$	36.3002	1.47338	0.736689	−	0.676232i	$-0.236388\pi$
$607$	36.3002	1.47338	0.736689	+	0.676232i	$0.236388\pi$
$608$	26.2912	1.06625
$609$	−74.1466	−3.00457
$610$	0	0
$611$	−45.8416	−1.85455
$612$	39.4237	1.59361
$613$	−18.2726	−0.738022	−0.369011	−	0.929425i	$-0.620303\pi$
$613$	−18.2726	−0.738022	−0.369011	+	0.929425i	$0.620303\pi$
$614$	8.31480	0.335558
$615$	0	0
$616$	−26.5100	−1.06812
$617$	17.7778	0.715707	0.357854	−	0.933778i	$-0.383509\pi$
$617$	17.7778	0.715707	0.357854	+	0.933778i	$0.383509\pi$
$618$	−6.92782	−0.278678
$619$	−24.4347	−0.982116	−0.491058	−	0.871127i	$-0.663390\pi$
$619$	−24.4347	−0.982116	−0.491058	+	0.871127i	$0.663390\pi$
$620$	0	0
$621$	2.85284	0.114481
$622$	−13.8387	−0.554880
$623$	−9.77198	−0.391506
$624$	29.4443	1.17871
$625$	0	0
$626$	6.39724	0.255685
$627$	−37.1311	−1.48287
$628$	3.70795	0.147963
$629$	2.10880	0.0840833
$630$	0	0
$631$	−44.7875	−1.78296	−0.891482	−	0.453056i	$-0.850334\pi$
$631$	−44.7875	−1.78296	−0.891482	+	0.453056i	$0.850334\pi$
$632$	−22.9734	−0.913832
$633$	67.8282	2.69593
$634$	18.7592	0.745021
$635$	0	0
$636$	−21.4818	−0.851810
$637$	−56.9762	−2.25748
$638$	11.7027	0.463316
$639$	−9.81643	−0.388332
$640$	0	0
$641$	−22.6242	−0.893604	−0.446802	−	0.894633i	$-0.647437\pi$
$641$	−22.6242	−0.893604	−0.446802	+	0.894633i	$0.647437\pi$
$642$	−15.0372	−0.593470
$643$	14.7151	0.580307	0.290154	−	0.956980i	$-0.406294\pi$
$643$	14.7151	0.580307	0.290154	+	0.956980i	$0.406294\pi$
$644$	5.34523	0.210632
$645$	0	0
$646$	−15.9709	−0.628365
$647$	7.76275	0.305185	0.152593	−	0.988289i	$-0.451238\pi$
$647$	7.76275	0.305185	0.152593	+	0.988289i	$0.451238\pi$
$648$	−7.15200	−0.280957
$649$	20.8191	0.817223
$650$	0	0
$651$	109.413	4.28824
$652$	0.549017	0.0215012
$653$	−13.6486	−0.534112	−0.267056	−	0.963681i	$-0.586051\pi$
$653$	−13.6486	−0.534112	−0.267056	+	0.963681i	$0.586051\pi$
$654$	−12.8827	−0.503755
$655$	0	0
$656$	−2.45329	−0.0957849
$657$	66.6576	2.60056
$658$	−20.3564	−0.793576
$659$	19.5281	0.760706	0.380353	−	0.924841i	$-0.375803\pi$
$659$	19.5281	0.760706	0.380353	+	0.924841i	$0.375803\pi$
$660$	0	0
$661$	25.0030	0.972502	0.486251	−	0.873819i	$-0.338364\pi$
$661$	25.0030	0.972502	0.486251	+	0.873819i	$0.338364\pi$
$662$	21.0351	0.817551
$663$	−85.3752	−3.31570
$664$	−21.0414	−0.816566
$665$	0	0
$666$	0.992972	0.0384769
$667$	−5.25221	−0.203366
$668$	−26.5433	−1.02699
$669$	38.8113	1.50053
$670$	0	0
$671$	42.4534	1.63889
$672$	62.4101	2.40752
$673$	−20.6533	−0.796126	−0.398063	−	0.917358i	$-0.630317\pi$
$673$	−20.6533	−0.796126	−0.398063	+	0.917358i	$0.630317\pi$
$674$	−6.37511	−0.245560
$675$	0	0
$676$	−30.8754	−1.18751
$677$	−12.8943	−0.495569	−0.247784	−	0.968815i	$-0.579702\pi$
$677$	−12.8943	−0.495569	−0.247784	+	0.968815i	$0.579702\pi$
$678$	−7.75470	−0.297817
$679$	53.7176	2.06149
$680$	0	0
$681$	29.2103	1.11934
$682$	−17.2689	−0.661262
$683$	30.7317	1.17592	0.587958	−	0.808891i	$-0.299932\pi$
$683$	30.7317	1.17592	0.587958	+	0.808891i	$0.299932\pi$
$684$	33.2971	1.27315
$685$	0	0
$686$	−7.73781	−0.295431
$687$	35.5738	1.35722
$688$	7.77850	0.296552
$689$	27.4793	1.04688
$690$	0	0
$691$	28.4059	1.08061	0.540306	−	0.841469i	$-0.318309\pi$
$691$	28.4059	1.08061	0.540306	+	0.841469i	$0.318309\pi$
$692$	−9.35755	−0.355721
$693$	−52.0646	−1.97777
$694$	−17.2613	−0.655230
$695$	0	0
$696$	−39.5451	−1.49895
$697$	7.11344	0.269441
$698$	−3.20308	−0.121238
$699$	−73.2514	−2.77062
$700$	0	0
$701$	31.3636	1.18459	0.592294	−	0.805722i	$-0.298223\pi$
$701$	31.3636	1.18459	0.592294	+	0.805722i	$0.298223\pi$
$702$	−12.3443	−0.465905
$703$	1.78108	0.0671747
$704$	1.35057	0.0509014
$705$	0	0
$706$	4.88342	0.183790
$707$	−32.4889	−1.22187
$708$	−31.6062	−1.18783
$709$	−27.0084	−1.01432	−0.507161	−	0.861851i	$-0.669305\pi$
$709$	−27.0084	−1.01432	−0.507161	+	0.861851i	$0.669305\pi$
$710$	0	0
$711$	−45.1189	−1.69209
$712$	−5.21175	−0.195319
$713$	7.75032	0.290252
$714$	−37.9117	−1.41881
$715$	0	0
$716$	9.34631	0.349288
$717$	−13.7463	−0.513367
$718$	−9.39449	−0.350599
$719$	39.3285	1.46670	0.733352	−	0.679849i	$-0.237955\pi$
$719$	39.3285	1.46670	0.733352	+	0.679849i	$0.237955\pi$
$720$	0	0
$721$	−17.4240	−0.648905
$722$	−1.95540	−0.0727724
$723$	2.70729	0.100685
$724$	5.31255	0.197439
$725$	0	0
$726$	−4.16572	−0.154605
$727$	−32.0357	−1.18814	−0.594069	−	0.804414i	$-0.702479\pi$
$727$	−32.0357	−1.18814	−0.594069	+	0.804414i	$0.702479\pi$
$728$	−51.4816	−1.90803
$729$	−43.2779	−1.60289
$730$	0	0
$731$	−22.5541	−0.834195
$732$	−64.4497	−2.38213
$733$	31.7332	1.17209	0.586046	−	0.810278i	$-0.300684\pi$
$733$	31.7332	1.17209	0.586046	+	0.810278i	$0.300684\pi$
$734$	−1.26292	−0.0466151
$735$	0	0
$736$	4.42085	0.162955
$737$	−16.5488	−0.609582
$738$	3.34951	0.123297
$739$	8.52453	0.313580	0.156790	−	0.987632i	$-0.449885\pi$
$739$	8.52453	0.313580	0.156790	+	0.987632i	$0.449885\pi$
$740$	0	0
$741$	−72.1074	−2.64893
$742$	12.2025	0.447966
$743$	22.1670	0.813228	0.406614	−	0.913600i	$-0.366709\pi$
$743$	22.1670	0.813228	0.406614	+	0.913600i	$0.366709\pi$
$744$	58.3540	2.13936
$745$	0	0
$746$	−17.5898	−0.644008
$747$	−41.3246	−1.51199
$748$	−26.4939	−0.968713
$749$	−37.8197	−1.38190
$750$	0	0
$751$	−17.9357	−0.654482	−0.327241	−	0.944941i	$-0.606119\pi$
$751$	−17.9357	−0.654482	−0.327241	+	0.944941i	$0.606119\pi$
$752$	15.6172	0.569502
$753$	−44.6793	−1.62820
$754$	22.7264	0.827646
$755$	0	0
$756$	24.2708	0.882721
$757$	12.0712	0.438734	0.219367	−	0.975642i	$-0.429601\pi$
$757$	12.0712	0.438734	0.219367	+	0.975642i	$0.429601\pi$
$758$	−0.172664	−0.00627145
$759$	−6.24357	−0.226627
$760$	0	0
$761$	41.6212	1.50877	0.754383	−	0.656435i	$-0.227936\pi$
$761$	41.6212	1.50877	0.754383	+	0.656435i	$0.227936\pi$
$762$	7.77098	0.281513
$763$	−32.4011	−1.17300
$764$	−28.8454	−1.04359
$765$	0	0
$766$	16.5854	0.599255
$767$	40.4302	1.45985
$768$	16.2812	0.587496
$769$	−16.1993	−0.584161	−0.292080	−	0.956394i	$-0.594347\pi$
$769$	−16.1993	−0.584161	−0.292080	+	0.956394i	$0.594347\pi$
$770$	0	0
$771$	−21.0803	−0.759188
$772$	5.87477	0.211437
$773$	−34.1932	−1.22984	−0.614922	−	0.788588i	$-0.710813\pi$
$773$	−34.1932	−1.22984	−0.614922	+	0.788588i	$0.710813\pi$
$774$	−10.6201	−0.381731
$775$	0	0
$776$	28.6496	1.02846
$777$	4.22794	0.151677
$778$	−3.99914	−0.143376
$779$	6.00798	0.215258
$780$	0	0
$781$	6.59693	0.236057
$782$	−2.68549	−0.0960330
$783$	−23.8484	−0.852272
$784$	19.4106	0.693235
$785$	0	0
$786$	2.68792	0.0958748
$787$	50.2675	1.79184	0.895922	−	0.444212i	$-0.146516\pi$
$787$	50.2675	1.79184	0.895922	+	0.444212i	$0.146516\pi$
$788$	−18.0761	−0.643934
$789$	−20.0866	−0.715103
$790$	0	0
$791$	−19.5037	−0.693471
$792$	−27.7680	−0.986692
$793$	82.4432	2.92764
$794$	−12.7168	−0.451303
$795$	0	0
$796$	15.9419	0.565046
$797$	23.2038	0.821921	0.410960	−	0.911653i	$-0.365193\pi$
$797$	23.2038	0.821921	0.410960	+	0.911653i	$0.365193\pi$
$798$	−32.0201	−1.13350
$799$	−45.2830	−1.60200
$800$	0	0
$801$	−10.2357	−0.361661
$802$	16.4770	0.581822
$803$	−44.7958	−1.58081
$804$	25.1232	0.886026
$805$	0	0
$806$	−33.5358	−1.18125
$807$	85.3329	3.00386
$808$	−17.3275	−0.609581
$809$	7.36051	0.258782	0.129391	−	0.991594i	$-0.458698\pi$
$809$	7.36051	0.258782	0.129391	+	0.991594i	$0.458698\pi$
$810$	0	0
$811$	−42.0564	−1.47680	−0.738401	−	0.674362i	$-0.764419\pi$
$811$	−42.0564	−1.47680	−0.738401	+	0.674362i	$0.764419\pi$
$812$	−44.6836	−1.56809
$813$	−20.6037	−0.722603
$814$	−0.667306	−0.0233891
$815$	0	0
$816$	29.0855	1.01820
$817$	−19.0491	−0.666444
$818$	−17.2725	−0.603917
$819$	−101.108	−3.53300
$820$	0	0
$821$	4.76222	0.166203	0.0831013	−	0.996541i	$-0.473517\pi$
$821$	4.76222	0.166203	0.0831013	+	0.996541i	$0.473517\pi$
$822$	−32.0345	−1.11733
$823$	−31.7139	−1.10548	−0.552738	−	0.833355i	$-0.686417\pi$
$823$	−31.7139	−1.10548	−0.552738	+	0.833355i	$0.686417\pi$
$824$	−9.29287	−0.323733
$825$	0	0
$826$	17.9534	0.624680
$827$	−27.5015	−0.956321	−0.478160	−	0.878273i	$-0.658696\pi$
$827$	−27.5015	−0.956321	−0.478160	+	0.878273i	$0.658696\pi$
$828$	5.59889	0.194575
$829$	4.10892	0.142709	0.0713543	−	0.997451i	$-0.477268\pi$
$829$	4.10892	0.142709	0.0713543	+	0.997451i	$0.477268\pi$
$830$	0	0
$831$	−10.8126	−0.375084
$832$	2.62276	0.0909279
$833$	−56.2819	−1.95005
$834$	24.1072	0.834762
$835$	0	0
$836$	−22.3766	−0.773911
$837$	35.1914	1.21639
$838$	2.78092	0.0960653
$839$	20.3076	0.701098	0.350549	−	0.936544i	$-0.385995\pi$
$839$	20.3076	0.701098	0.350549	+	0.936544i	$0.385995\pi$
$840$	0	0
$841$	14.9059	0.513998
$842$	−19.0137	−0.655254
$843$	−36.7899	−1.26711
$844$	40.8759	1.40701
$845$	0	0
$846$	−21.3224	−0.733080
$847$	−10.4771	−0.359999
$848$	−9.36160	−0.321479
$849$	68.2502	2.34234
$850$	0	0
$851$	0.299488	0.0102663
$852$	−10.0150	−0.343108
$853$	−43.2228	−1.47992	−0.739961	−	0.672650i	$-0.765156\pi$
$853$	−43.2228	−1.47992	−0.739961	+	0.672650i	$0.765156\pi$
$854$	36.6098	1.25276
$855$	0	0
$856$	−20.1706	−0.689418
$857$	1.39094	0.0475136	0.0237568	−	0.999718i	$-0.492437\pi$
$857$	1.39094	0.0475136	0.0237568	+	0.999718i	$0.492437\pi$
$858$	27.0160	0.922312
$859$	−31.1259	−1.06200	−0.531001	−	0.847371i	$-0.678184\pi$
$859$	−31.1259	−1.06200	−0.531001	+	0.847371i	$0.678184\pi$
$860$	0	0
$861$	14.2618	0.486040
$862$	−1.06011	−0.0361074
$863$	14.5679	0.495897	0.247949	−	0.968773i	$-0.420244\pi$
$863$	14.5679	0.495897	0.247949	+	0.968773i	$0.420244\pi$
$864$	20.0735	0.682914
$865$	0	0
$866$	21.4100	0.727541
$867$	−38.3108	−1.30110
$868$	65.9366	2.23803
$869$	30.3212	1.02858
$870$	0	0
$871$	−32.1372	−1.08893
$872$	−17.2807	−0.585199
$873$	56.2668	1.90434
$874$	−2.26815	−0.0767215
$875$	0	0
$876$	68.0059	2.29771
$877$	−19.5928	−0.661602	−0.330801	−	0.943701i	$-0.607319\pi$
$877$	−19.5928	−0.661602	−0.330801	+	0.943701i	$0.607319\pi$
$878$	−14.6856	−0.495616
$879$	5.39547	0.181985
$880$	0	0
$881$	−18.3933	−0.619685	−0.309843	−	0.950788i	$-0.600276\pi$
$881$	−18.3933	−0.619685	−0.309843	+	0.950788i	$0.600276\pi$
$882$	−26.5015	−0.892353
$883$	55.6562	1.87298	0.936490	−	0.350695i	$-0.114055\pi$
$883$	55.6562	1.87298	0.936490	+	0.350695i	$0.114055\pi$
$884$	−51.4504	−1.73046
$885$	0	0
$886$	16.3521	0.549359
$887$	9.15109	0.307263	0.153632	−	0.988128i	$-0.450903\pi$
$887$	9.15109	0.307263	0.153632	+	0.988128i	$0.450903\pi$
$888$	2.25492	0.0756700
$889$	19.5446	0.655506
$890$	0	0
$891$	9.43950	0.316235
$892$	23.3892	0.783129
$893$	−38.2458	−1.27985
$894$	8.10749	0.271155
$895$	0	0
$896$	47.2699	1.57918
$897$	−12.1248	−0.404836
$898$	−20.9902	−0.700451
$899$	−64.7890	−2.16083
$900$	0	0
$901$	27.1444	0.904311
$902$	−2.25097	−0.0749491
$903$	−45.2189	−1.50479
$904$	−10.4020	−0.345966
$905$	0	0
$906$	−0.783432	−0.0260278
$907$	45.0356	1.49538	0.747692	−	0.664046i	$-0.231162\pi$
$907$	45.0356	1.49538	0.747692	+	0.664046i	$0.231162\pi$
$908$	17.6033	0.584185
$909$	−34.0307	−1.12873
$910$	0	0
$911$	48.9405	1.62147	0.810735	−	0.585413i	$-0.199068\pi$
$911$	48.9405	1.62147	0.810735	+	0.585413i	$0.199068\pi$
$912$	24.5655	0.813443
$913$	27.7714	0.919098
$914$	−16.4854	−0.545289
$915$	0	0
$916$	21.4381	0.708336
$917$	6.76033	0.223246
$918$	−12.1939	−0.402458
$919$	−33.3371	−1.09969	−0.549844	−	0.835267i	$-0.685313\pi$
$919$	−33.3371	−1.09969	−0.549844	+	0.835267i	$0.685313\pi$
$920$	0	0
$921$	37.0833	1.22194
$922$	−15.9353	−0.524802
$923$	12.8110	0.421680
$924$	−53.1177	−1.74745
$925$	0	0
$926$	−9.11306	−0.299474
$927$	−18.2509	−0.599437
$928$	−36.9562	−1.21315
$929$	−29.0107	−0.951809	−0.475905	−	0.879497i	$-0.657879\pi$
$929$	−29.0107	−0.951809	−0.475905	+	0.879497i	$0.657879\pi$
$930$	0	0
$931$	−47.5354	−1.55791
$932$	−44.1442	−1.44599
$933$	−61.7193	−2.02060
$934$	−1.13106	−0.0370094
$935$	0	0
$936$	−53.9246	−1.76258
$937$	−43.6179	−1.42493	−0.712467	−	0.701706i	$-0.752422\pi$
$937$	−43.6179	−1.42493	−0.712467	+	0.701706i	$0.752422\pi$
$938$	−14.2709	−0.465960
$939$	28.5312	0.931079
$940$	0	0
$941$	27.3402	0.891265	0.445632	−	0.895216i	$-0.352979\pi$
$941$	27.3402	0.891265	0.445632	+	0.895216i	$0.352979\pi$
$942$	−3.73495	−0.121691
$943$	1.01024	0.0328979
$944$	−13.7737	−0.448295
$945$	0	0
$946$	7.13701	0.232044
$947$	34.3965	1.11774	0.558868	−	0.829257i	$-0.311236\pi$
$947$	34.3965	1.11774	0.558868	+	0.829257i	$0.311236\pi$
$948$	−46.0315	−1.49503
$949$	−86.9922	−2.82389
$950$	0	0
$951$	83.6643	2.71300
$952$	−50.8542	−1.64819
$953$	−60.1196	−1.94746	−0.973732	−	0.227699i	$-0.926880\pi$
$953$	−60.1196	−1.94746	−0.973732	+	0.227699i	$0.926880\pi$
$954$	12.7815	0.413817
$955$	0	0
$956$	−8.28408	−0.267926
$957$	52.1933	1.68717
$958$	19.9865	0.645734
$959$	−80.5694	−2.60172
$960$	0	0
$961$	64.6047	2.08402
$962$	−1.29589	−0.0417812
$963$	−39.6144	−1.27656
$964$	1.63152	0.0525477
$965$	0	0
$966$	−5.38416	−0.173232
$967$	7.14059	0.229626	0.114813	−	0.993387i	$-0.463373\pi$
$967$	7.14059	0.229626	0.114813	+	0.993387i	$0.463373\pi$
$968$	−5.58784	−0.179600
$969$	−71.2287	−2.28820
$970$	0	0
$971$	53.2415	1.70860	0.854300	−	0.519780i	$-0.173986\pi$
$971$	53.2415	1.70860	0.854300	+	0.519780i	$0.173986\pi$
$972$	−31.9465	−1.02469
$973$	60.6314	1.94375
$974$	−16.5161	−0.529209
$975$	0	0
$976$	−28.0866	−0.899031
$977$	56.6807	1.81338	0.906688	−	0.421801i	$-0.138602\pi$
$977$	56.6807	1.81338	0.906688	+	0.421801i	$0.138602\pi$
$978$	−0.553015	−0.0176835
$979$	6.87869	0.219844
$980$	0	0
$981$	−33.9387	−1.08358
$982$	−18.4635	−0.589193
$983$	45.1699	1.44070	0.720348	−	0.693613i	$-0.243982\pi$
$983$	45.1699	1.44070	0.720348	+	0.693613i	$0.243982\pi$
$984$	7.60632	0.242481
$985$	0	0
$986$	22.4494	0.714936
$987$	−90.7880	−2.88981
$988$	−43.4547	−1.38248
$989$	−3.20310	−0.101853
$990$	0	0
$991$	−34.3662	−1.09168	−0.545839	−	0.837890i	$-0.683789\pi$
$991$	−34.3662	−1.09168	−0.545839	+	0.837890i	$0.683789\pi$
$992$	54.5337	1.73145
$993$	93.8147	2.97712
$994$	5.68888	0.180440
$995$	0	0
$996$	−42.1605	−1.33591
$997$	20.9578	0.663741	0.331871	−	0.943325i	$-0.392320\pi$
$997$	20.9578	0.663741	0.331871	+	0.943325i	$0.392320\pi$
$998$	−8.92829	−0.282620
$999$	1.35987	0.0430243

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.15	yes	40
5.4	even	2		6025.2.a.l.1.26	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.26	✓	40	5.4	even	2
6025.2.a.o.1.15	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)

\(f(q)\)	\(=\)	\(q-0.607027 q^{2} -2.70729 q^{3} -1.63152 q^{4} +1.64340 q^{6} +4.13328 q^{7} +2.20443 q^{8} +4.32942 q^{9} +O(q^{10})\)	\(q-0.607027 q^{2} -2.70729 q^{3} -1.63152 q^{4} +1.64340 q^{6} +4.13328 q^{7} +2.20443 q^{8} +4.32942 q^{9} -2.90950 q^{11} +4.41699 q^{12} -5.65016 q^{13} -2.50901 q^{14} +1.92489 q^{16} -5.58131 q^{17} -2.62808 q^{18} -4.71394 q^{19} -11.1900 q^{21} +1.76614 q^{22} -0.792647 q^{23} -5.96803 q^{24} +3.42980 q^{26} -3.59913 q^{27} -6.74352 q^{28} +6.62616 q^{29} -9.77777 q^{31} -5.57732 q^{32} +7.87686 q^{33} +3.38800 q^{34} -7.06353 q^{36} -0.377832 q^{37} +2.86149 q^{38} +15.2966 q^{39} -1.27451 q^{41} +6.79263 q^{42} +4.04101 q^{43} +4.74690 q^{44} +0.481158 q^{46} +8.11332 q^{47} -5.21123 q^{48} +10.0840 q^{49} +15.1102 q^{51} +9.21834 q^{52} -4.86345 q^{53} +2.18477 q^{54} +9.11153 q^{56} +12.7620 q^{57} -4.02226 q^{58} -7.15558 q^{59} -14.5913 q^{61} +5.93537 q^{62} +17.8947 q^{63} -0.464193 q^{64} -4.78146 q^{66} +5.68784 q^{67} +9.10600 q^{68} +2.14593 q^{69} -2.26738 q^{71} +9.54391 q^{72} +15.3964 q^{73} +0.229354 q^{74} +7.69089 q^{76} -12.0258 q^{77} -9.28546 q^{78} -10.4215 q^{79} -3.24438 q^{81} +0.773663 q^{82} -9.54507 q^{83} +18.2567 q^{84} -2.45300 q^{86} -17.9389 q^{87} -6.41378 q^{88} -2.36422 q^{89} -23.3537 q^{91} +1.29322 q^{92} +26.4713 q^{93} -4.92501 q^{94} +15.0994 q^{96} +12.9964 q^{97} -6.12126 q^{98} -12.5964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 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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