LMFDB - Embedded newform 6025.2.a.l.1.6

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:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.49478 q^{2} +2.89336 q^{3} +4.22391 q^{4} -7.21828 q^{6} +0.185663 q^{7} -5.54817 q^{8} +5.37152 q^{9} +O(q^{10})$	\(q-2.49478 q^{2} +2.89336 q^{3} +4.22391 q^{4} -7.21828 q^{6} +0.185663 q^{7} -5.54817 q^{8} +5.37152 q^{9} +5.68574 q^{11} +12.2213 q^{12} -6.29322 q^{13} -0.463189 q^{14} +5.39362 q^{16} -1.03345 q^{17} -13.4007 q^{18} -0.418306 q^{19} +0.537190 q^{21} -14.1846 q^{22} -5.21433 q^{23} -16.0528 q^{24} +15.7002 q^{26} +6.86165 q^{27} +0.784226 q^{28} -7.29566 q^{29} -8.47532 q^{31} -2.35955 q^{32} +16.4509 q^{33} +2.57823 q^{34} +22.6888 q^{36} -3.20366 q^{37} +1.04358 q^{38} -18.2085 q^{39} +9.21106 q^{41} -1.34017 q^{42} +1.97405 q^{43} +24.0161 q^{44} +13.0086 q^{46} -5.03668 q^{47} +15.6057 q^{48} -6.96553 q^{49} -2.99015 q^{51} -26.5820 q^{52} -9.52766 q^{53} -17.1183 q^{54} -1.03009 q^{56} -1.21031 q^{57} +18.2010 q^{58} -4.69247 q^{59} -7.67472 q^{61} +21.1440 q^{62} +0.997294 q^{63} -4.90070 q^{64} -41.0413 q^{66} -10.5482 q^{67} -4.36521 q^{68} -15.0869 q^{69} +4.48385 q^{71} -29.8021 q^{72} +8.34246 q^{73} +7.99242 q^{74} -1.76689 q^{76} +1.05563 q^{77} +45.4262 q^{78} +5.40340 q^{79} +3.73864 q^{81} -22.9795 q^{82} -8.45058 q^{83} +2.26905 q^{84} -4.92482 q^{86} -21.1089 q^{87} -31.5454 q^{88} -17.3526 q^{89} -1.16842 q^{91} -22.0249 q^{92} -24.5221 q^{93} +12.5654 q^{94} -6.82702 q^{96} -1.97553 q^{97} +17.3774 q^{98} +30.5410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.49478	−1.76407	−0.882037	−	0.471180i	$-0.843828\pi$
$2$	−2.49478	−1.76407	−0.882037	+	0.471180i	$0.843828\pi$
$3$	2.89336	1.67048	0.835240	−	0.549885i	$-0.185328\pi$
$3$	2.89336	1.67048	0.835240	+	0.549885i	$0.185328\pi$
$4$	4.22391	2.11196
$5$	0	0
$5$	0	0
$6$	−7.21828	−2.94685
$7$	0.185663	0.0701742	0.0350871	−	0.999384i	$-0.488829\pi$
$7$	0.185663	0.0701742	0.0350871	+	0.999384i	$0.488829\pi$
$8$	−5.54817	−1.96157
$9$	5.37152	1.79051
$10$	0	0
$11$	5.68574	1.71431	0.857157	−	0.515055i	$-0.172228\pi$
$11$	5.68574	1.71431	0.857157	+	0.515055i	$0.172228\pi$
$12$	12.2213	3.52798
$13$	−6.29322	−1.74542	−0.872712	−	0.488235i	$-0.837641\pi$
$13$	−6.29322	−1.74542	−0.872712	+	0.488235i	$0.837641\pi$
$14$	−0.463189	−0.123792
$15$	0	0
$16$	5.39362	1.34841
$17$	−1.03345	−0.250649	−0.125324	−	0.992116i	$-0.539997\pi$
$17$	−1.03345	−0.250649	−0.125324	+	0.992116i	$0.539997\pi$
$18$	−13.4007	−3.15858
$19$	−0.418306	−0.0959659	−0.0479830	−	0.998848i	$-0.515279\pi$
$19$	−0.418306	−0.0959659	−0.0479830	+	0.998848i	$0.515279\pi$
$20$	0	0
$21$	0.537190	0.117225
$22$	−14.1846	−3.02418
$23$	−5.21433	−1.08726	−0.543631	−	0.839324i	$-0.682951\pi$
$23$	−5.21433	−1.08726	−0.543631	+	0.839324i	$0.682951\pi$
$24$	−16.0528	−3.27677
$25$	0	0
$26$	15.7002	3.07906
$27$	6.86165	1.32052
$28$	0.784226	0.148205
$29$	−7.29566	−1.35477	−0.677385	−	0.735629i	$-0.736887\pi$
$29$	−7.29566	−1.35477	−0.677385	+	0.735629i	$0.736887\pi$
$30$	0	0
$31$	−8.47532	−1.52221	−0.761106	−	0.648627i	$-0.775343\pi$
$31$	−8.47532	−1.52221	−0.761106	+	0.648627i	$0.775343\pi$
$32$	−2.35955	−0.417113
$33$	16.4509	2.86373
$34$	2.57823	0.442163
$35$	0	0
$36$	22.6888	3.78147
$37$	−3.20366	−0.526678	−0.263339	−	0.964703i	$-0.584824\pi$
$37$	−3.20366	−0.526678	−0.263339	+	0.964703i	$0.584824\pi$
$38$	1.04358	0.169291
$39$	−18.2085	−2.91570
$40$	0	0
$41$	9.21106	1.43853	0.719263	−	0.694738i	$-0.244480\pi$
$41$	9.21106	1.43853	0.719263	+	0.694738i	$0.244480\pi$
$42$	−1.34017	−0.206793
$43$	1.97405	0.301040	0.150520	−	0.988607i	$-0.451905\pi$
$43$	1.97405	0.301040	0.150520	+	0.988607i	$0.451905\pi$
$44$	24.0161	3.62056
$45$	0	0
$46$	13.0086	1.91801
$47$	−5.03668	−0.734675	−0.367338	−	0.930088i	$-0.619731\pi$
$47$	−5.03668	−0.734675	−0.367338	+	0.930088i	$0.619731\pi$
$48$	15.6057	2.25249
$49$	−6.96553	−0.995076
$50$	0	0
$51$	−2.99015	−0.418704
$52$	−26.5820	−3.68626
$53$	−9.52766	−1.30872	−0.654362	−	0.756181i	$-0.727063\pi$
$53$	−9.52766	−1.30872	−0.654362	+	0.756181i	$0.727063\pi$
$54$	−17.1183	−2.32950
$55$	0	0
$56$	−1.03009	−0.137652
$57$	−1.21031	−0.160309
$58$	18.2010	2.38991
$59$	−4.69247	−0.610907	−0.305454	−	0.952207i	$-0.598808\pi$
$59$	−4.69247	−0.610907	−0.305454	+	0.952207i	$0.598808\pi$
$60$	0	0
$61$	−7.67472	−0.982647	−0.491323	−	0.870977i	$-0.663487\pi$
$61$	−7.67472	−0.982647	−0.491323	+	0.870977i	$0.663487\pi$
$62$	21.1440	2.68530
$63$	0.997294	0.125647
$64$	−4.90070	−0.612587
$65$	0	0
$66$	−41.0413	−5.05183
$67$	−10.5482	−1.28866	−0.644332	−	0.764746i	$-0.722864\pi$
$67$	−10.5482	−1.28866	−0.644332	+	0.764746i	$0.722864\pi$
$68$	−4.36521	−0.529360
$69$	−15.0869	−1.81625
$70$	0	0
$71$	4.48385	0.532135	0.266068	−	0.963954i	$-0.414276\pi$
$71$	4.48385	0.532135	0.266068	+	0.963954i	$0.414276\pi$
$72$	−29.8021	−3.51221
$73$	8.34246	0.976411	0.488206	−	0.872729i	$-0.337652\pi$
$73$	8.34246	0.976411	0.488206	+	0.872729i	$0.337652\pi$
$74$	7.99242	0.929099
$75$	0	0
$76$	−1.76689	−0.202676
$77$	1.05563	0.120301
$78$	45.4262	5.14351
$79$	5.40340	0.607931	0.303965	−	0.952683i	$-0.401689\pi$
$79$	5.40340	0.607931	0.303965	+	0.952683i	$0.401689\pi$
$80$	0	0
$81$	3.73864	0.415405
$82$	−22.9795	−2.53767
$83$	−8.45058	−0.927571	−0.463786	−	0.885947i	$-0.653509\pi$
$83$	−8.45058	−0.927571	−0.463786	+	0.885947i	$0.653509\pi$
$84$	2.26905	0.247573
$85$	0	0
$86$	−4.92482	−0.531056
$87$	−21.1089	−2.26312
$88$	−31.5454	−3.36276
$89$	−17.3526	−1.83937	−0.919687	−	0.392651i	$-0.871558\pi$
$89$	−17.3526	−1.83937	−0.919687	+	0.392651i	$0.871558\pi$
$90$	0	0
$91$	−1.16842	−0.122484
$92$	−22.0249	−2.29625
$93$	−24.5221	−2.54283
$94$	12.5654	1.29602
$95$	0	0
$96$	−6.82702	−0.696779
$97$	−1.97553	−0.200585	−0.100292	−	0.994958i	$-0.531978\pi$
$97$	−1.97553	−0.200585	−0.100292	+	0.994958i	$0.531978\pi$
$98$	17.3774	1.75539
$99$	30.5410	3.06949
$100$	0	0
$101$	−14.6910	−1.46181	−0.730903	−	0.682481i	$-0.760901\pi$
$101$	−14.6910	−1.46181	−0.730903	+	0.682481i	$0.760901\pi$
$102$	7.45975	0.738625
$103$	−0.847926	−0.0835486	−0.0417743	−	0.999127i	$-0.513301\pi$
$103$	−0.847926	−0.0835486	−0.0417743	+	0.999127i	$0.513301\pi$
$104$	34.9158	3.42378
$105$	0	0
$106$	23.7694	2.30869
$107$	−4.00627	−0.387301	−0.193650	−	0.981071i	$-0.562033\pi$
$107$	−4.00627	−0.387301	−0.193650	+	0.981071i	$0.562033\pi$
$108$	28.9830	2.78889
$109$	19.4578	1.86372	0.931859	−	0.362821i	$-0.118186\pi$
$109$	19.4578	1.86372	0.931859	+	0.362821i	$0.118186\pi$
$110$	0	0
$111$	−9.26933	−0.879806
$112$	1.00140	0.0946233
$113$	14.8998	1.40165	0.700827	−	0.713331i	$-0.252814\pi$
$113$	14.8998	1.40165	0.700827	+	0.713331i	$0.252814\pi$
$114$	3.01945	0.282797
$115$	0	0
$116$	−30.8162	−2.86121
$117$	−33.8041	−3.12519
$118$	11.7067	1.07769
$119$	−0.191874	−0.0175891
$120$	0	0
$121$	21.3276	1.93887
$122$	19.1467	1.73346
$123$	26.6509	2.40303
$124$	−35.7990	−3.21485
$125$	0	0
$126$	−2.48803	−0.221651
$127$	1.55006	0.137546	0.0687728	−	0.997632i	$-0.478092\pi$
$127$	1.55006	0.137546	0.0687728	+	0.997632i	$0.478092\pi$
$128$	16.9453	1.49776
$129$	5.71163	0.502881
$130$	0	0
$131$	4.43530	0.387514	0.193757	−	0.981050i	$-0.437933\pi$
$131$	4.43530	0.387514	0.193757	+	0.981050i	$0.437933\pi$
$132$	69.4871	6.04807
$133$	−0.0776641	−0.00673433
$134$	26.3153	2.27330
$135$	0	0
$136$	5.73377	0.491667
$137$	−1.31307	−0.112183	−0.0560917	−	0.998426i	$-0.517864\pi$
$137$	−1.31307	−0.112183	−0.0560917	+	0.998426i	$0.517864\pi$
$138$	37.6385	3.20400
$139$	11.1133	0.942619	0.471309	−	0.881968i	$-0.343781\pi$
$139$	11.1133	0.942619	0.471309	+	0.881968i	$0.343781\pi$
$140$	0	0
$141$	−14.5729	−1.22726
$142$	−11.1862	−0.938726
$143$	−35.7816	−2.99221
$144$	28.9719	2.41433
$145$	0	0
$146$	−20.8126	−1.72246
$147$	−20.1538	−1.66225
$148$	−13.5320	−1.11232
$149$	−14.6188	−1.19762	−0.598808	−	0.800893i	$-0.704359\pi$
$149$	−14.6188	−1.19762	−0.598808	+	0.800893i	$0.704359\pi$
$150$	0	0
$151$	17.0702	1.38915	0.694575	−	0.719420i	$-0.255592\pi$
$151$	17.0702	1.38915	0.694575	+	0.719420i	$0.255592\pi$
$152$	2.32083	0.188244
$153$	−5.55120	−0.448788
$154$	−2.63357	−0.212219
$155$	0	0
$156$	−76.9112	−6.15783
$157$	−0.889033	−0.0709525	−0.0354763	−	0.999371i	$-0.511295\pi$
$157$	−0.889033	−0.0709525	−0.0354763	+	0.999371i	$0.511295\pi$
$158$	−13.4803	−1.07243
$159$	−27.5669	−2.18620
$160$	0	0
$161$	−0.968110	−0.0762978
$162$	−9.32708	−0.732805
$163$	−7.63877	−0.598315	−0.299157	−	0.954204i	$-0.596706\pi$
$163$	−7.63877	−0.598315	−0.299157	+	0.954204i	$0.596706\pi$
$164$	38.9067	3.03810
$165$	0	0
$166$	21.0823	1.63630
$167$	−18.0674	−1.39810	−0.699048	−	0.715075i	$-0.746393\pi$
$167$	−18.0674	−1.39810	−0.699048	+	0.715075i	$0.746393\pi$
$168$	−2.98042	−0.229945
$169$	26.6046	2.04651
$170$	0	0
$171$	−2.24694	−0.171828
$172$	8.33822	0.635783
$173$	3.75907	0.285797	0.142898	−	0.989737i	$-0.454358\pi$
$173$	3.75907	0.285797	0.142898	+	0.989737i	$0.454358\pi$
$174$	52.6621	3.99230
$175$	0	0
$176$	30.6667	2.31159
$177$	−13.5770	−1.02051
$178$	43.2909	3.24479
$179$	−19.6464	−1.46844	−0.734222	−	0.678910i	$-0.762453\pi$
$179$	−19.6464	−1.46844	−0.734222	+	0.678910i	$0.762453\pi$
$180$	0	0
$181$	22.5984	1.67972	0.839862	−	0.542800i	$-0.182636\pi$
$181$	22.5984	1.67972	0.839862	+	0.542800i	$0.182636\pi$
$182$	2.91495	0.216070
$183$	−22.2057	−1.64149
$184$	28.9300	2.13275
$185$	0	0
$186$	61.1773	4.48573
$187$	−5.87594	−0.429691
$188$	−21.2745	−1.55160
$189$	1.27396	0.0926667
$190$	0	0
$191$	2.32721	0.168391	0.0841955	−	0.996449i	$-0.473168\pi$
$191$	2.32721	0.168391	0.0841955	+	0.996449i	$0.473168\pi$
$192$	−14.1795	−1.02332
$193$	−4.63276	−0.333474	−0.166737	−	0.986001i	$-0.553323\pi$
$193$	−4.63276	−0.333474	−0.166737	+	0.986001i	$0.553323\pi$
$194$	4.92851	0.353846
$195$	0	0
$196$	−29.4218	−2.10156
$197$	−7.43950	−0.530042	−0.265021	−	0.964243i	$-0.585379\pi$
$197$	−7.43950	−0.530042	−0.265021	+	0.964243i	$0.585379\pi$
$198$	−76.1931	−5.41481
$199$	−8.68045	−0.615341	−0.307670	−	0.951493i	$-0.599549\pi$
$199$	−8.68045	−0.615341	−0.307670	+	0.951493i	$0.599549\pi$
$200$	0	0
$201$	−30.5196	−2.15269
$202$	36.6507	2.57874
$203$	−1.35454	−0.0950698
$204$	−12.6301	−0.884285
$205$	0	0
$206$	2.11539	0.147386
$207$	−28.0089	−1.94675
$208$	−33.9432	−2.35354
$209$	−2.37838	−0.164516
$210$	0	0
$211$	26.4916	1.82376	0.911878	−	0.410462i	$-0.134633\pi$
$211$	26.4916	1.82376	0.911878	+	0.410462i	$0.134633\pi$
$212$	−40.2440	−2.76397
$213$	12.9734	0.888922
$214$	9.99475	0.683227
$215$	0	0
$216$	−38.0696	−2.59031
$217$	−1.57356	−0.106820
$218$	−48.5428	−3.28774
$219$	24.1377	1.63108
$220$	0	0
$221$	6.50374	0.437489
$222$	23.1249	1.55204
$223$	9.68831	0.648777	0.324388	−	0.945924i	$-0.394841\pi$
$223$	9.68831	0.648777	0.324388	+	0.945924i	$0.394841\pi$
$224$	−0.438082	−0.0292706
$225$	0	0
$226$	−37.1717	−2.47262
$227$	0.983505	0.0652775	0.0326387	−	0.999467i	$-0.489609\pi$
$227$	0.983505	0.0652775	0.0326387	+	0.999467i	$0.489609\pi$
$228$	−5.11224	−0.338566
$229$	11.1507	0.736856	0.368428	−	0.929656i	$-0.379896\pi$
$229$	11.1507	0.736856	0.368428	+	0.929656i	$0.379896\pi$
$230$	0	0
$231$	3.05432	0.200960
$232$	40.4776	2.65748
$233$	−25.5430	−1.67338	−0.836689	−	0.547679i	$-0.815512\pi$
$233$	−25.5430	−1.67338	−0.836689	+	0.547679i	$0.815512\pi$
$234$	84.3337	5.51307
$235$	0	0
$236$	−19.8206	−1.29021
$237$	15.6340	1.01554
$238$	0.478683	0.0310284
$239$	−13.9348	−0.901369	−0.450684	−	0.892683i	$-0.648820\pi$
$239$	−13.9348	−0.901369	−0.450684	+	0.892683i	$0.648820\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−53.2076	−3.42032
$243$	−9.76771	−0.626599
$244$	−32.4173	−2.07531
$245$	0	0
$246$	−66.4880	−4.23912
$247$	2.63249	0.167501
$248$	47.0225	2.98593
$249$	−24.4505	−1.54949
$250$	0	0
$251$	17.1415	1.08196	0.540980	−	0.841035i	$-0.318053\pi$
$251$	17.1415	1.08196	0.540980	+	0.841035i	$0.318053\pi$
$252$	4.21248	0.265362
$253$	−29.6473	−1.86391
$254$	−3.86706	−0.242641
$255$	0	0
$256$	−32.4732	−2.02958
$257$	17.4542	1.08876	0.544381	−	0.838838i	$-0.316764\pi$
$257$	17.4542	1.08876	0.544381	+	0.838838i	$0.316764\pi$
$258$	−14.2493	−0.887120
$259$	−0.594802	−0.0369592
$260$	0	0
$261$	−39.1887	−2.42572
$262$	−11.0651	−0.683604
$263$	17.6137	1.08611	0.543054	−	0.839698i	$-0.317268\pi$
$263$	17.6137	1.08611	0.543054	+	0.839698i	$0.317268\pi$
$264$	−91.2723	−5.61742
$265$	0	0
$266$	0.193755	0.0118799
$267$	−50.2074	−3.07264
$268$	−44.5545	−2.72160
$269$	−15.3977	−0.938816	−0.469408	−	0.882981i	$-0.655533\pi$
$269$	−15.3977	−0.938816	−0.469408	+	0.882981i	$0.655533\pi$
$270$	0	0
$271$	0.810481	0.0492332	0.0246166	−	0.999697i	$-0.492164\pi$
$271$	0.810481	0.0492332	0.0246166	+	0.999697i	$0.492164\pi$
$272$	−5.57405	−0.337976
$273$	−3.38066	−0.204607
$274$	3.27583	0.197900
$275$	0	0
$276$	−63.7259	−3.83585
$277$	4.65891	0.279927	0.139963	−	0.990157i	$-0.455302\pi$
$277$	4.65891	0.279927	0.139963	+	0.990157i	$0.455302\pi$
$278$	−27.7252	−1.66285
$279$	−45.5253	−2.72553
$280$	0	0
$281$	8.17956	0.487952	0.243976	−	0.969781i	$-0.421548\pi$
$281$	8.17956	0.487952	0.243976	+	0.969781i	$0.421548\pi$
$282$	36.3562	2.16498
$283$	0.277625	0.0165031	0.00825156	−	0.999966i	$-0.497373\pi$
$283$	0.277625	0.0165031	0.00825156	+	0.999966i	$0.497373\pi$
$284$	18.9394	1.12385
$285$	0	0
$286$	89.2671	5.27847
$287$	1.71016	0.100947
$288$	−12.6744	−0.746843
$289$	−15.9320	−0.937175
$290$	0	0
$291$	−5.71592	−0.335073
$292$	35.2378	2.06214
$293$	23.0356	1.34575	0.672877	−	0.739755i	$-0.265059\pi$
$293$	23.0356	1.34575	0.672877	+	0.739755i	$0.265059\pi$
$294$	50.2792	2.93234
$295$	0	0
$296$	17.7744	1.03312
$297$	39.0135	2.26379
$298$	36.4706	2.11268
$299$	32.8149	1.89773
$300$	0	0
$301$	0.366509	0.0211252
$302$	−42.5862	−2.45056
$303$	−42.5062	−2.44192
$304$	−2.25618	−0.129401
$305$	0	0
$306$	13.8490	0.791696
$307$	15.0541	0.859184	0.429592	−	0.903023i	$-0.358657\pi$
$307$	15.0541	0.859184	0.429592	+	0.903023i	$0.358657\pi$
$308$	4.45890	0.254070
$309$	−2.45335	−0.139566
$310$	0	0
$311$	14.6021	0.828008	0.414004	−	0.910275i	$-0.364130\pi$
$311$	14.6021	0.828008	0.414004	+	0.910275i	$0.364130\pi$
$312$	101.024	5.71936
$313$	0.512501	0.0289682	0.0144841	−	0.999895i	$-0.495389\pi$
$313$	0.512501	0.0289682	0.0144841	+	0.999895i	$0.495389\pi$
$314$	2.21794	0.125166
$315$	0	0
$316$	22.8235	1.28392
$317$	26.5332	1.49025	0.745126	−	0.666924i	$-0.232389\pi$
$317$	26.5332	1.49025	0.745126	+	0.666924i	$0.232389\pi$
$318$	68.7733	3.85662
$319$	−41.4812	−2.32250
$320$	0	0
$321$	−11.5916	−0.646978
$322$	2.41522	0.134595
$323$	0.432299	0.0240538
$324$	15.7917	0.877317
$325$	0	0
$326$	19.0570	1.05547
$327$	56.2983	3.11331
$328$	−51.1045	−2.82178
$329$	−0.935127	−0.0515552
$330$	0	0
$331$	2.75216	0.151273	0.0756363	−	0.997135i	$-0.475901\pi$
$331$	2.75216	0.151273	0.0756363	+	0.997135i	$0.475901\pi$
$332$	−35.6945	−1.95899
$333$	−17.2085	−0.943020
$334$	45.0741	2.46635
$335$	0	0
$336$	2.89740	0.158066
$337$	−16.9213	−0.921760	−0.460880	−	0.887462i	$-0.652466\pi$
$337$	−16.9213	−0.921760	−0.460880	+	0.887462i	$0.652466\pi$
$338$	−66.3725	−3.61019
$339$	43.1104	2.34144
$340$	0	0
$341$	−48.1884	−2.60955
$342$	5.60561	0.303117
$343$	−2.59289	−0.140003
$344$	−10.9524	−0.590512
$345$	0	0
$346$	−9.37804	−0.504167
$347$	5.42329	0.291137	0.145569	−	0.989348i	$-0.453499\pi$
$347$	5.42329	0.291137	0.145569	+	0.989348i	$0.453499\pi$
$348$	−89.1624	−4.77960
$349$	−31.3968	−1.68063	−0.840315	−	0.542099i	$-0.817630\pi$
$349$	−31.3968	−1.68063	−0.840315	+	0.542099i	$0.817630\pi$
$350$	0	0
$351$	−43.1818	−2.30488
$352$	−13.4158	−0.715063
$353$	8.09809	0.431018	0.215509	−	0.976502i	$-0.430859\pi$
$353$	8.09809	0.431018	0.215509	+	0.976502i	$0.430859\pi$
$354$	33.8715	1.80025
$355$	0	0
$356$	−73.2960	−3.88468
$357$	−0.555160	−0.0293822
$358$	49.0135	2.59044
$359$	−2.33900	−0.123448	−0.0617238	−	0.998093i	$-0.519660\pi$
$359$	−2.33900	−0.123448	−0.0617238	+	0.998093i	$0.519660\pi$
$360$	0	0
$361$	−18.8250	−0.990791
$362$	−56.3779	−2.96316
$363$	61.7084	3.23885
$364$	−4.93530	−0.258680
$365$	0	0
$366$	55.3983	2.89571
$367$	15.0927	0.787831	0.393915	−	0.919147i	$-0.371120\pi$
$367$	15.0927	0.787831	0.393915	+	0.919147i	$0.371120\pi$
$368$	−28.1241	−1.46607
$369$	49.4774	2.57569
$370$	0	0
$371$	−1.76894	−0.0918387
$372$	−103.579	−5.37034
$373$	29.8160	1.54381	0.771907	−	0.635736i	$-0.219303\pi$
$373$	29.8160	1.54381	0.771907	+	0.635736i	$0.219303\pi$
$374$	14.6592	0.758007
$375$	0	0
$376$	27.9444	1.44112
$377$	45.9131	2.36465
$378$	−3.17824	−0.163471
$379$	−35.2629	−1.81133	−0.905667	−	0.423991i	$-0.860629\pi$
$379$	−35.2629	−1.81133	−0.905667	+	0.423991i	$0.860629\pi$
$380$	0	0
$381$	4.48488	0.229767
$382$	−5.80587	−0.297054
$383$	−3.42500	−0.175010	−0.0875048	−	0.996164i	$-0.527889\pi$
$383$	−3.42500	−0.175010	−0.0875048	+	0.996164i	$0.527889\pi$
$384$	49.0287	2.50198
$385$	0	0
$386$	11.5577	0.588272
$387$	10.6036	0.539013
$388$	−8.34447	−0.423626
$389$	23.9111	1.21234	0.606170	−	0.795335i	$-0.292705\pi$
$389$	23.9111	1.21234	0.606170	+	0.795335i	$0.292705\pi$
$390$	0	0
$391$	5.38876	0.272521
$392$	38.6459	1.95192
$393$	12.8329	0.647335
$394$	18.5599	0.935034
$395$	0	0
$396$	129.003	6.48263
$397$	−10.0405	−0.503918	−0.251959	−	0.967738i	$-0.581075\pi$
$397$	−10.0405	−0.503918	−0.251959	+	0.967738i	$0.581075\pi$
$398$	21.6558	1.08551
$399$	−0.224710	−0.0112496
$400$	0	0
$401$	−3.15774	−0.157690	−0.0788451	−	0.996887i	$-0.525123\pi$
$401$	−3.15774	−0.157690	−0.0788451	+	0.996887i	$0.525123\pi$
$402$	76.1396	3.79750
$403$	53.3370	2.65691
$404$	−62.0534	−3.08727
$405$	0	0
$406$	3.37927	0.167710
$407$	−18.2152	−0.902892
$408$	16.5898	0.821319
$409$	28.3002	1.39936	0.699678	−	0.714458i	$-0.253327\pi$
$409$	28.3002	1.39936	0.699678	+	0.714458i	$0.253327\pi$
$410$	0	0
$411$	−3.79919	−0.187400
$412$	−3.58157	−0.176451
$413$	−0.871219	−0.0428699
$414$	69.8759	3.43421
$415$	0	0
$416$	14.8491	0.728039
$417$	32.1548	1.57463
$418$	5.93352	0.290218
$419$	17.4315	0.851585	0.425793	−	0.904821i	$-0.359995\pi$
$419$	17.4315	0.851585	0.425793	+	0.904821i	$0.359995\pi$
$420$	0	0
$421$	−9.56051	−0.465951	−0.232975	−	0.972483i	$-0.574846\pi$
$421$	−9.56051	−0.465951	−0.232975	+	0.972483i	$0.574846\pi$
$422$	−66.0906	−3.21724
$423$	−27.0546	−1.31544
$424$	52.8611	2.56716
$425$	0	0
$426$	−32.3657	−1.56812
$427$	−1.42491	−0.0689564
$428$	−16.9221	−0.817962
$429$	−103.529	−4.99842
$430$	0	0
$431$	25.9248	1.24875	0.624376	−	0.781124i	$-0.285353\pi$
$431$	25.9248	1.24875	0.624376	+	0.781124i	$0.285353\pi$
$432$	37.0091	1.78060
$433$	23.1374	1.11191	0.555956	−	0.831212i	$-0.312352\pi$
$433$	23.1374	1.11191	0.555956	+	0.831212i	$0.312352\pi$
$434$	3.92567	0.188438
$435$	0	0
$436$	82.1880	3.93609
$437$	2.18118	0.104340
$438$	−60.2182	−2.87734
$439$	9.00021	0.429556	0.214778	−	0.976663i	$-0.431097\pi$
$439$	9.00021	0.429556	0.214778	+	0.976663i	$0.431097\pi$
$440$	0	0
$441$	−37.4155	−1.78169
$442$	−16.2254	−0.771762
$443$	29.7663	1.41424	0.707120	−	0.707093i	$-0.249994\pi$
$443$	29.7663	1.41424	0.707120	+	0.707093i	$0.249994\pi$
$444$	−39.1529	−1.85811
$445$	0	0
$446$	−24.1702	−1.14449
$447$	−42.2973	−2.00059
$448$	−0.909880	−0.0429878
$449$	−24.9856	−1.17914	−0.589571	−	0.807716i	$-0.700703\pi$
$449$	−24.9856	−1.17914	−0.589571	+	0.807716i	$0.700703\pi$
$450$	0	0
$451$	52.3717	2.46609
$452$	62.9354	2.96023
$453$	49.3901	2.32055
$454$	−2.45363	−0.115154
$455$	0	0
$456$	6.71500	0.314459
$457$	−28.0775	−1.31341	−0.656705	−	0.754147i	$-0.728050\pi$
$457$	−28.0775	−1.31341	−0.656705	+	0.754147i	$0.728050\pi$
$458$	−27.8184	−1.29987
$459$	−7.09118	−0.330988
$460$	0	0
$461$	23.7429	1.10581	0.552907	−	0.833243i	$-0.313518\pi$
$461$	23.7429	1.10581	0.552907	+	0.833243i	$0.313518\pi$
$462$	−7.61986	−0.354508
$463$	−29.6836	−1.37952	−0.689758	−	0.724040i	$-0.742283\pi$
$463$	−29.6836	−1.37952	−0.689758	+	0.724040i	$0.742283\pi$
$464$	−39.3500	−1.82678
$465$	0	0
$466$	63.7241	2.95196
$467$	6.88153	0.318439	0.159220	−	0.987243i	$-0.449102\pi$
$467$	6.88153	0.318439	0.159220	+	0.987243i	$0.449102\pi$
$468$	−142.786	−6.60027
$469$	−1.95841	−0.0904309
$470$	0	0
$471$	−2.57229	−0.118525
$472$	26.0346	1.19834
$473$	11.2239	0.516077
$474$	−39.0033	−1.79148
$475$	0	0
$476$	−0.810460	−0.0371474
$477$	−51.1780	−2.34328
$478$	34.7643	1.59008
$479$	−17.5164	−0.800342	−0.400171	−	0.916440i	$-0.631049\pi$
$479$	−17.5164	−0.800342	−0.400171	+	0.916440i	$0.631049\pi$
$480$	0	0
$481$	20.1613	0.919277
$482$	2.49478	0.113634
$483$	−2.80109	−0.127454
$484$	90.0860	4.09482
$485$	0	0
$486$	24.3683	1.10537
$487$	31.7119	1.43700	0.718502	−	0.695525i	$-0.244828\pi$
$487$	31.7119	1.43700	0.718502	+	0.695525i	$0.244828\pi$
$488$	42.5806	1.92753
$489$	−22.1017	−0.999473
$490$	0	0
$491$	3.28239	0.148132	0.0740661	−	0.997253i	$-0.476402\pi$
$491$	3.28239	0.148132	0.0740661	+	0.997253i	$0.476402\pi$
$492$	112.571	5.07510
$493$	7.53971	0.339571
$494$	−6.56747	−0.295485
$495$	0	0
$496$	−45.7127	−2.05256
$497$	0.832487	0.0373422
$498$	60.9986	2.73341
$499$	17.7880	0.796298	0.398149	−	0.917321i	$-0.369653\pi$
$499$	17.7880	0.796298	0.398149	+	0.917321i	$0.369653\pi$
$500$	0	0
$501$	−52.2754	−2.33549
$502$	−42.7642	−1.90866
$503$	−25.7592	−1.14855	−0.574274	−	0.818663i	$-0.694716\pi$
$503$	−25.7592	−1.14855	−0.574274	+	0.818663i	$0.694716\pi$
$504$	−5.53316	−0.246466
$505$	0	0
$506$	73.9634	3.28808
$507$	76.9765	3.41865
$508$	6.54732	0.290490
$509$	22.7063	1.00644	0.503219	−	0.864159i	$-0.332149\pi$
$509$	22.7063	1.00644	0.503219	+	0.864159i	$0.332149\pi$
$510$	0	0
$511$	1.54889	0.0685188
$512$	47.1230	2.08256
$513$	−2.87027	−0.126725
$514$	−43.5443	−1.92066
$515$	0	0
$516$	24.1254	1.06206
$517$	−28.6372	−1.25946
$518$	1.48390	0.0651988
$519$	10.8763	0.477418
$520$	0	0
$521$	0.581106	0.0254587	0.0127294	−	0.999919i	$-0.495948\pi$
$521$	0.581106	0.0254587	0.0127294	+	0.999919i	$0.495948\pi$
$522$	97.7672	4.27915
$523$	17.7376	0.775611	0.387805	−	0.921741i	$-0.373233\pi$
$523$	17.7376	0.775611	0.387805	+	0.921741i	$0.373233\pi$
$524$	18.7343	0.818414
$525$	0	0
$526$	−43.9423	−1.91598
$527$	8.75884	0.381541
$528$	88.7298	3.86147
$529$	4.18923	0.182141
$530$	0	0
$531$	−25.2057	−1.09383
$532$	−0.328046	−0.0142226
$533$	−57.9672	−2.51084
$534$	125.256	5.42036
$535$	0	0
$536$	58.5230	2.52781
$537$	−56.8441	−2.45301
$538$	38.4139	1.65614
$539$	−39.6042	−1.70587
$540$	0	0
$541$	−20.4487	−0.879156	−0.439578	−	0.898204i	$-0.644872\pi$
$541$	−20.4487	−0.879156	−0.439578	+	0.898204i	$0.644872\pi$
$542$	−2.02197	−0.0868510
$543$	65.3852	2.80595
$544$	2.43848	0.104549
$545$	0	0
$546$	8.43398	0.360941
$547$	23.9802	1.02532	0.512661	−	0.858591i	$-0.328660\pi$
$547$	23.9802	1.02532	0.512661	+	0.858591i	$0.328660\pi$
$548$	−5.54631	−0.236927
$549$	−41.2249	−1.75943
$550$	0	0
$551$	3.05182	0.130012
$552$	83.7048	3.56271
$553$	1.00321	0.0426610
$554$	−11.6229	−0.493812
$555$	0	0
$556$	46.9417	1.99077
$557$	−36.8504	−1.56140	−0.780702	−	0.624904i	$-0.785138\pi$
$557$	−36.8504	−1.56140	−0.780702	+	0.624904i	$0.785138\pi$
$558$	113.576	4.80804
$559$	−12.4231	−0.525442
$560$	0	0
$561$	−17.0012	−0.717790
$562$	−20.4062	−0.860783
$563$	1.50424	0.0633962	0.0316981	−	0.999497i	$-0.489908\pi$
$563$	1.50424	0.0633962	0.0316981	+	0.999497i	$0.489908\pi$
$564$	−61.5547	−2.59192
$565$	0	0
$566$	−0.692614	−0.0291127
$567$	0.694129	0.0291507
$568$	−24.8772	−1.04382
$569$	−5.48494	−0.229941	−0.114970	−	0.993369i	$-0.536677\pi$
$569$	−5.48494	−0.229941	−0.114970	+	0.993369i	$0.536677\pi$
$570$	0	0
$571$	42.5997	1.78274	0.891371	−	0.453275i	$-0.149745\pi$
$571$	42.5997	1.78274	0.891371	+	0.453275i	$0.149745\pi$
$572$	−151.138	−6.31941
$573$	6.73345	0.281294
$574$	−4.26646	−0.178079
$575$	0	0
$576$	−26.3242	−1.09684
$577$	−25.0588	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$577$	−25.0588	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$578$	39.7467	1.65325
$579$	−13.4042	−0.557061
$580$	0	0
$581$	−1.56896	−0.0650915
$582$	14.2599	0.591094
$583$	−54.1718	−2.24357
$584$	−46.2854	−1.91530
$585$	0	0
$586$	−57.4686	−2.37401
$587$	−5.07185	−0.209338	−0.104669	−	0.994507i	$-0.533378\pi$
$587$	−5.07185	−0.209338	−0.104669	+	0.994507i	$0.533378\pi$
$588$	−85.1278	−3.51061
$589$	3.54528	0.146081
$590$	0	0
$591$	−21.5251	−0.885426
$592$	−17.2793	−0.710176
$593$	−24.4002	−1.00199	−0.500997	−	0.865449i	$-0.667033\pi$
$593$	−24.4002	−1.00199	−0.500997	+	0.865449i	$0.667033\pi$
$594$	−97.3300	−3.99350
$595$	0	0
$596$	−61.7484	−2.52931
$597$	−25.1156	−1.02791
$598$	−81.8659	−3.34774
$599$	−5.96971	−0.243916	−0.121958	−	0.992535i	$-0.538917\pi$
$599$	−5.96971	−0.243916	−0.121958	+	0.992535i	$0.538917\pi$
$600$	0	0
$601$	−17.8938	−0.729903	−0.364951	−	0.931027i	$-0.618914\pi$
$601$	−17.8938	−0.729903	−0.364951	+	0.931027i	$0.618914\pi$
$602$	−0.914358	−0.0372664
$603$	−56.6597	−2.30736
$604$	72.1029	2.93382
$605$	0	0
$606$	106.044	4.30773
$607$	20.3436	0.825721	0.412861	−	0.910794i	$-0.364530\pi$
$607$	20.3436	0.825721	0.412861	+	0.910794i	$0.364530\pi$
$608$	0.987013	0.0400287
$609$	−3.91916	−0.158812
$610$	0	0
$611$	31.6969	1.28232
$612$	−23.4478	−0.947822
$613$	−3.20174	−0.129317	−0.0646584	−	0.997907i	$-0.520596\pi$
$613$	−3.20174	−0.129317	−0.0646584	+	0.997907i	$0.520596\pi$
$614$	−37.5567	−1.51566
$615$	0	0
$616$	−5.85683	−0.235979
$617$	−11.6174	−0.467700	−0.233850	−	0.972273i	$-0.575132\pi$
$617$	−11.6174	−0.467700	−0.233850	+	0.972273i	$0.575132\pi$
$618$	6.12057	0.246205
$619$	0.811254	0.0326071	0.0163035	−	0.999867i	$-0.494810\pi$
$619$	0.811254	0.0326071	0.0163035	+	0.999867i	$0.494810\pi$
$620$	0	0
$621$	−35.7789	−1.43576
$622$	−36.4289	−1.46067
$623$	−3.22175	−0.129077
$624$	−98.2099	−3.93154
$625$	0	0
$626$	−1.27857	−0.0511021
$627$	−6.88150	−0.274820
$628$	−3.75520	−0.149849
$629$	3.31083	0.132011
$630$	0	0
$631$	1.43679	0.0571979	0.0285989	−	0.999591i	$-0.490895\pi$
$631$	1.43679	0.0571979	0.0285989	+	0.999591i	$0.490895\pi$
$632$	−29.9790	−1.19250
$633$	76.6496	3.04655
$634$	−66.1944	−2.62891
$635$	0	0
$636$	−116.440	−4.61716
$637$	43.8356	1.73683
$638$	103.486	4.09706
$639$	24.0851	0.952792
$640$	0	0
$641$	−39.6765	−1.56713	−0.783564	−	0.621311i	$-0.786600\pi$
$641$	−39.6765	−1.56713	−0.783564	+	0.621311i	$0.786600\pi$
$642$	28.9184	1.14132
$643$	29.0418	1.14530	0.572648	−	0.819801i	$-0.305916\pi$
$643$	29.0418	1.14530	0.572648	+	0.819801i	$0.305916\pi$
$644$	−4.08921	−0.161138
$645$	0	0
$646$	−1.07849	−0.0424326
$647$	40.5107	1.59264	0.796319	−	0.604877i	$-0.206778\pi$
$647$	40.5107	1.59264	0.796319	+	0.604877i	$0.206778\pi$
$648$	−20.7426	−0.814848
$649$	−26.6801	−1.04729
$650$	0	0
$651$	−4.55286	−0.178441
$652$	−32.2655	−1.26361
$653$	−41.3086	−1.61653	−0.808264	−	0.588820i	$-0.799593\pi$
$653$	−41.3086	−1.61653	−0.808264	+	0.588820i	$0.799593\pi$
$654$	−140.452	−5.49210
$655$	0	0
$656$	49.6810	1.93972
$657$	44.8117	1.74827
$658$	2.33293	0.0909472
$659$	−25.9584	−1.01120	−0.505598	−	0.862769i	$-0.668728\pi$
$659$	−25.9584	−1.01120	−0.505598	+	0.862769i	$0.668728\pi$
$660$	0	0
$661$	−34.8994	−1.35743	−0.678714	−	0.734403i	$-0.737462\pi$
$661$	−34.8994	−1.35743	−0.678714	+	0.734403i	$0.737462\pi$
$662$	−6.86603	−0.266856
$663$	18.8176	0.730816
$664$	46.8852	1.81950
$665$	0	0
$666$	42.9314	1.66356
$667$	38.0420	1.47299
$668$	−76.3151	−2.95272
$669$	28.0317	1.08377
$670$	0	0
$671$	−43.6364	−1.68457
$672$	−1.26753	−0.0488959
$673$	−18.3836	−0.708635	−0.354317	−	0.935125i	$-0.615287\pi$
$673$	−18.3836	−0.708635	−0.354317	+	0.935125i	$0.615287\pi$
$674$	42.2148	1.62605
$675$	0	0
$676$	112.375	4.32213
$677$	3.24098	0.124561	0.0622805	−	0.998059i	$-0.480163\pi$
$677$	3.24098	0.124561	0.0622805	+	0.998059i	$0.480163\pi$
$678$	−107.551	−4.13047
$679$	−0.366784	−0.0140759
$680$	0	0
$681$	2.84563	0.109045
$682$	120.219	4.60344
$683$	−46.4986	−1.77922	−0.889610	−	0.456720i	$-0.849024\pi$
$683$	−46.4986	−1.77922	−0.889610	+	0.456720i	$0.849024\pi$
$684$	−9.49087	−0.362892
$685$	0	0
$686$	6.46868	0.246975
$687$	32.2628	1.23090
$688$	10.6473	0.405924
$689$	59.9596	2.28428
$690$	0	0
$691$	−51.2423	−1.94935	−0.974674	−	0.223631i	$-0.928209\pi$
$691$	−51.2423	−1.94935	−0.974674	+	0.223631i	$0.928209\pi$
$692$	15.8780	0.603591
$693$	5.67035	0.215399
$694$	−13.5299	−0.513588
$695$	0	0
$696$	117.116	4.43927
$697$	−9.51919	−0.360565
$698$	78.3279	2.96476
$699$	−73.9050	−2.79534
$700$	0	0
$701$	13.9036	0.525131	0.262566	−	0.964914i	$-0.415431\pi$
$701$	13.9036	0.525131	0.262566	+	0.964914i	$0.415431\pi$
$702$	107.729	4.06597
$703$	1.34011	0.0505432
$704$	−27.8641	−1.05017
$705$	0	0
$706$	−20.2029	−0.760347
$707$	−2.72758	−0.102581
$708$	−57.3480	−2.15527
$709$	−15.6439	−0.587518	−0.293759	−	0.955879i	$-0.594906\pi$
$709$	−15.6439	−0.587518	−0.293759	+	0.955879i	$0.594906\pi$
$710$	0	0
$711$	29.0245	1.08850
$712$	96.2753	3.60807
$713$	44.1931	1.65505
$714$	1.38500	0.0518324
$715$	0	0
$716$	−82.9848	−3.10129
$717$	−40.3184	−1.50572
$718$	5.83528	0.217771
$719$	−11.8195	−0.440793	−0.220396	−	0.975410i	$-0.570735\pi$
$719$	−11.8195	−0.440793	−0.220396	+	0.975410i	$0.570735\pi$
$720$	0	0
$721$	−0.157429	−0.00586295
$722$	46.9642	1.74783
$723$	−2.89336	−0.107605
$724$	95.4536	3.54751
$725$	0	0
$726$	−153.949	−5.71357
$727$	−1.35855	−0.0503860	−0.0251930	−	0.999683i	$-0.508020\pi$
$727$	−1.35855	−0.0503860	−0.0251930	+	0.999683i	$0.508020\pi$
$728$	6.48259	0.240261
$729$	−39.4774	−1.46213
$730$	0	0
$731$	−2.04009	−0.0754553
$732$	−93.7949	−3.46676
$733$	−13.0708	−0.482782	−0.241391	−	0.970428i	$-0.577604\pi$
$733$	−13.0708	−0.482782	−0.241391	+	0.970428i	$0.577604\pi$
$734$	−37.6528	−1.38979
$735$	0	0
$736$	12.3035	0.453512
$737$	−59.9741	−2.20917
$738$	−123.435	−4.54371
$739$	51.1683	1.88226	0.941128	−	0.338051i	$-0.109768\pi$
$739$	51.1683	1.88226	0.941128	+	0.338051i	$0.109768\pi$
$740$	0	0
$741$	7.61673	0.279808
$742$	4.41311	0.162010
$743$	−21.9825	−0.806459	−0.403230	−	0.915099i	$-0.632112\pi$
$743$	−21.9825	−0.806459	−0.403230	+	0.915099i	$0.632112\pi$
$744$	136.053	4.98794
$745$	0	0
$746$	−74.3842	−2.72340
$747$	−45.3924	−1.66082
$748$	−24.8195	−0.907489
$749$	−0.743817	−0.0271785
$750$	0	0
$751$	−34.0636	−1.24300	−0.621498	−	0.783415i	$-0.713476\pi$
$751$	−34.0636	−1.24300	−0.621498	+	0.783415i	$0.713476\pi$
$752$	−27.1660	−0.990640
$753$	49.5964	1.80739
$754$	−114.543	−4.17141
$755$	0	0
$756$	5.38108	0.195708
$757$	−3.17141	−0.115267	−0.0576333	−	0.998338i	$-0.518355\pi$
$757$	−3.17141	−0.115267	−0.0576333	+	0.998338i	$0.518355\pi$
$758$	87.9731	3.19533
$759$	−85.7803	−3.11363
$760$	0	0
$761$	−34.6867	−1.25739	−0.628696	−	0.777652i	$-0.716411\pi$
$761$	−34.6867	−1.25739	−0.628696	+	0.777652i	$0.716411\pi$
$762$	−11.1888	−0.405326
$763$	3.61260	0.130785
$764$	9.82994	0.355635
$765$	0	0
$766$	8.54462	0.308730
$767$	29.5307	1.06629
$768$	−93.9567	−3.39037
$769$	14.0179	0.505500	0.252750	−	0.967532i	$-0.418665\pi$
$769$	14.0179	0.505500	0.252750	+	0.967532i	$0.418665\pi$
$770$	0	0
$771$	50.5012	1.81876
$772$	−19.5684	−0.704282
$773$	2.67336	0.0961541	0.0480770	−	0.998844i	$-0.484691\pi$
$773$	2.67336	0.0961541	0.0480770	+	0.998844i	$0.484691\pi$
$774$	−26.4537	−0.950860
$775$	0	0
$776$	10.9606	0.393462
$777$	−1.72098	−0.0617396
$778$	−59.6528	−2.13866
$779$	−3.85304	−0.138049
$780$	0	0
$781$	25.4940	0.912248
$782$	−13.4438	−0.480748
$783$	−50.0602	−1.78901
$784$	−37.5694	−1.34177
$785$	0	0
$786$	−32.0153	−1.14195
$787$	−2.91072	−0.103756	−0.0518780	−	0.998653i	$-0.516521\pi$
$787$	−2.91072	−0.103756	−0.0518780	+	0.998653i	$0.516521\pi$
$788$	−31.4238	−1.11943
$789$	50.9628	1.81432
$790$	0	0
$791$	2.76635	0.0983599
$792$	−169.447	−6.02103
$793$	48.2986	1.71514
$794$	25.0488	0.888949
$795$	0	0
$796$	−36.6655	−1.29957
$797$	7.04533	0.249558	0.124779	−	0.992185i	$-0.460178\pi$
$797$	7.04533	0.249558	0.124779	+	0.992185i	$0.460178\pi$
$798$	0.560601	0.0198451
$799$	5.20517	0.184146
$800$	0	0
$801$	−93.2099	−3.29341
$802$	7.87786	0.278177
$803$	47.4330	1.67388
$804$	−128.912	−4.54638
$805$	0	0
$806$	−133.064	−4.68698
$807$	−44.5511	−1.56827
$808$	81.5081	2.86744
$809$	−15.4823	−0.544328	−0.272164	−	0.962251i	$-0.587739\pi$
$809$	−15.4823	−0.544328	−0.272164	+	0.962251i	$0.587739\pi$
$810$	0	0
$811$	9.94876	0.349348	0.174674	−	0.984626i	$-0.444113\pi$
$811$	9.94876	0.349348	0.174674	+	0.984626i	$0.444113\pi$
$812$	−5.72144	−0.200783
$813$	2.34501	0.0822431
$814$	45.4428	1.59277
$815$	0	0
$816$	−16.1277	−0.564583
$817$	−0.825757	−0.0288896
$818$	−70.6027	−2.46857
$819$	−6.27619	−0.219308
$820$	0	0
$821$	−47.3254	−1.65167	−0.825835	−	0.563912i	$-0.809296\pi$
$821$	−47.3254	−1.65167	−0.825835	+	0.563912i	$0.809296\pi$
$822$	9.47814	0.330588
$823$	−53.1209	−1.85168	−0.925840	−	0.377916i	$-0.876641\pi$
$823$	−53.1209	−1.85168	−0.925840	+	0.377916i	$0.876641\pi$
$824$	4.70444	0.163887
$825$	0	0
$826$	2.17350	0.0756257
$827$	−23.7875	−0.827172	−0.413586	−	0.910465i	$-0.635724\pi$
$827$	−23.7875	−0.827172	−0.413586	+	0.910465i	$0.635724\pi$
$828$	−118.307	−4.11145
$829$	0.761676	0.0264541	0.0132271	−	0.999913i	$-0.495790\pi$
$829$	0.761676	0.0264541	0.0132271	+	0.999913i	$0.495790\pi$
$830$	0	0
$831$	13.4799	0.467612
$832$	30.8412	1.06922
$833$	7.19854	0.249415
$834$	−80.2190	−2.77776
$835$	0	0
$836$	−10.0461	−0.347450
$837$	−58.1546	−2.01012
$838$	−43.4877	−1.50226
$839$	−2.25531	−0.0778620	−0.0389310	−	0.999242i	$-0.512395\pi$
$839$	−2.25531	−0.0778620	−0.0389310	+	0.999242i	$0.512395\pi$
$840$	0	0
$841$	24.2266	0.835400
$842$	23.8514	0.821972
$843$	23.6664	0.815114
$844$	111.898	3.85169
$845$	0	0
$846$	67.4952	2.32053
$847$	3.95976	0.136059
$848$	−51.3886	−1.76469
$849$	0.803269	0.0275681
$850$	0	0
$851$	16.7049	0.572638
$852$	54.7985	1.87737
$853$	0.831787	0.0284798	0.0142399	−	0.999899i	$-0.495467\pi$
$853$	0.831787	0.0284798	0.0142399	+	0.999899i	$0.495467\pi$
$854$	3.55484	0.121644
$855$	0	0
$856$	22.2275	0.759719
$857$	9.74845	0.333001	0.166500	−	0.986041i	$-0.446753\pi$
$857$	9.74845	0.333001	0.166500	+	0.986041i	$0.446753\pi$
$858$	258.282	8.81759
$859$	−53.4365	−1.82323	−0.911615	−	0.411044i	$-0.865164\pi$
$859$	−53.4365	−1.82323	−0.911615	+	0.411044i	$0.865164\pi$
$860$	0	0
$861$	4.94809	0.168631
$862$	−64.6766	−2.20289
$863$	36.0244	1.22628	0.613142	−	0.789972i	$-0.289905\pi$
$863$	36.0244	1.22628	0.613142	+	0.789972i	$0.289905\pi$
$864$	−16.1904	−0.550808
$865$	0	0
$866$	−57.7227	−1.96150
$867$	−46.0969	−1.56553
$868$	−6.64657	−0.225599
$869$	30.7223	1.04218
$870$	0	0
$871$	66.3819	2.24926
$872$	−107.955	−3.65582
$873$	−10.6116	−0.359148
$874$	−5.44157	−0.184064
$875$	0	0
$876$	101.956	3.44476
$877$	54.3827	1.83637	0.918186	−	0.396149i	$-0.129654\pi$
$877$	54.3827	1.83637	0.918186	+	0.396149i	$0.129654\pi$
$878$	−22.4535	−0.757769
$879$	66.6502	2.24805
$880$	0	0
$881$	−34.0343	−1.14665	−0.573323	−	0.819329i	$-0.694346\pi$
$881$	−34.0343	−1.14665	−0.573323	+	0.819329i	$0.694346\pi$
$882$	93.3432	3.14303
$883$	13.4105	0.451300	0.225650	−	0.974208i	$-0.427549\pi$
$883$	13.4105	0.451300	0.225650	+	0.974208i	$0.427549\pi$
$884$	27.4712	0.923957
$885$	0	0
$886$	−74.2603	−2.49482
$887$	8.00619	0.268822	0.134411	−	0.990926i	$-0.457086\pi$
$887$	8.00619	0.268822	0.134411	+	0.990926i	$0.457086\pi$
$888$	51.4278	1.72580
$889$	0.287789	0.00965215
$890$	0	0
$891$	21.2569	0.712135
$892$	40.9226	1.37019
$893$	2.10687	0.0705038
$894$	105.522	3.52920
$895$	0	0
$896$	3.14611	0.105104
$897$	94.9452	3.17013
$898$	62.3335	2.08009
$899$	61.8330	2.06225
$900$	0	0
$901$	9.84638	0.328030
$902$	−130.656	−4.35036
$903$	1.06044	0.0352893
$904$	−82.6666	−2.74945
$905$	0	0
$906$	−123.217	−4.09362
$907$	−35.3910	−1.17514	−0.587570	−	0.809174i	$-0.699915\pi$
$907$	−35.3910	−1.17514	−0.587570	+	0.809174i	$0.699915\pi$
$908$	4.15424	0.137863
$909$	−78.9128	−2.61737
$910$	0	0
$911$	−19.2396	−0.637436	−0.318718	−	0.947850i	$-0.603252\pi$
$911$	−19.2396	−0.637436	−0.318718	+	0.947850i	$0.603252\pi$
$912$	−6.52795	−0.216162
$913$	−48.0478	−1.59015
$914$	70.0472	2.31695
$915$	0	0
$916$	47.0994	1.55621
$917$	0.823474	0.0271935
$918$	17.6909	0.583887
$919$	8.25754	0.272391	0.136195	−	0.990682i	$-0.456512\pi$
$919$	8.25754	0.272391	0.136195	+	0.990682i	$0.456512\pi$
$920$	0	0
$921$	43.5569	1.43525
$922$	−59.2331	−1.95074
$923$	−28.2179	−0.928802
$924$	12.9012	0.424418
$925$	0	0
$926$	74.0541	2.43357
$927$	−4.55465	−0.149594
$928$	17.2145	0.565092
$929$	10.0776	0.330635	0.165317	−	0.986240i	$-0.447135\pi$
$929$	10.0776	0.330635	0.165317	+	0.986240i	$0.447135\pi$
$930$	0	0
$931$	2.91372	0.0954934
$932$	−107.891	−3.53410
$933$	42.2490	1.38317
$934$	−17.1679	−0.561751
$935$	0	0
$936$	187.551	6.13030
$937$	43.5184	1.42168	0.710842	−	0.703352i	$-0.248314\pi$
$937$	43.5184	1.42168	0.710842	+	0.703352i	$0.248314\pi$
$938$	4.88579	0.159527
$939$	1.48285	0.0483909
$940$	0	0
$941$	44.8465	1.46195	0.730976	−	0.682403i	$-0.239065\pi$
$941$	44.8465	1.46195	0.730976	+	0.682403i	$0.239065\pi$
$942$	6.41729	0.209087
$943$	−48.0295	−1.56406
$944$	−25.3094	−0.823751
$945$	0	0
$946$	−28.0012	−0.910398
$947$	−28.9174	−0.939691	−0.469845	−	0.882749i	$-0.655690\pi$
$947$	−28.9174	−0.939691	−0.469845	+	0.882749i	$0.655690\pi$
$948$	66.0366	2.14477
$949$	−52.5009	−1.70425
$950$	0	0
$951$	76.7699	2.48944
$952$	1.06455	0.0345023
$953$	−57.7111	−1.86945	−0.934724	−	0.355376i	$-0.884353\pi$
$953$	−57.7111	−1.86945	−0.934724	+	0.355376i	$0.884353\pi$
$954$	127.678	4.13372
$955$	0	0
$956$	−58.8595	−1.90365
$957$	−120.020	−3.87969
$958$	43.6994	1.41186
$959$	−0.243790	−0.00787238
$960$	0	0
$961$	40.8311	1.31713
$962$	−50.2980	−1.62167
$963$	−21.5197	−0.693464
$964$	−4.22391	−0.136043
$965$	0	0
$966$	6.98809	0.224838
$967$	21.1595	0.680444	0.340222	−	0.940345i	$-0.389498\pi$
$967$	21.1595	0.680444	0.340222	+	0.940345i	$0.389498\pi$
$968$	−118.329	−3.80325
$969$	1.25080	0.0401813
$970$	0	0
$971$	−24.9393	−0.800341	−0.400170	−	0.916441i	$-0.631049\pi$
$971$	−24.9393	−0.800341	−0.400170	+	0.916441i	$0.631049\pi$
$972$	−41.2580	−1.32335
$973$	2.06334	0.0661475
$974$	−79.1141	−2.53498
$975$	0	0
$976$	−41.3945	−1.32501
$977$	18.8420	0.602810	0.301405	−	0.953496i	$-0.402544\pi$
$977$	18.8420	0.602810	0.301405	+	0.953496i	$0.402544\pi$
$978$	55.1388	1.76314
$979$	−98.6625	−3.15327
$980$	0	0
$981$	104.518	3.33700
$982$	−8.18883	−0.261316
$983$	45.1396	1.43973	0.719865	−	0.694114i	$-0.244204\pi$
$983$	45.1396	1.43973	0.719865	+	0.694114i	$0.244204\pi$
$984$	−147.864	−4.71372
$985$	0	0
$986$	−18.8099	−0.599029
$987$	−2.70566	−0.0861220
$988$	11.1194	0.353756
$989$	−10.2933	−0.327309
$990$	0	0
$991$	43.8047	1.39150	0.695751	−	0.718283i	$-0.255072\pi$
$991$	43.8047	1.39150	0.695751	+	0.718283i	$0.255072\pi$
$992$	19.9979	0.634935
$993$	7.96299	0.252698
$994$	−2.07687	−0.0658743
$995$	0	0
$996$	−103.277	−3.27246
$997$	−32.9897	−1.04479	−0.522397	−	0.852703i	$-0.674962\pi$
$997$	−32.9897	−1.04479	−0.522397	+	0.852703i	$0.674962\pi$
$998$	−44.3770	−1.40473
$999$	−21.9824	−0.695492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.6	✓	40
5.4	even	2		6025.2.a.o.1.35	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.6	✓	40	1.1	even	1	trivial
6025.2.a.o.1.35	yes	40	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49478 q^{2} +2.89336 q^{3} +4.22391 q^{4} -7.21828 q^{6} +0.185663 q^{7} -5.54817 q^{8} +5.37152 q^{9} +O(q^{10})\)	\(q-2.49478 q^{2} +2.89336 q^{3} +4.22391 q^{4} -7.21828 q^{6} +0.185663 q^{7} -5.54817 q^{8} +5.37152 q^{9} +5.68574 q^{11} +12.2213 q^{12} -6.29322 q^{13} -0.463189 q^{14} +5.39362 q^{16} -1.03345 q^{17} -13.4007 q^{18} -0.418306 q^{19} +0.537190 q^{21} -14.1846 q^{22} -5.21433 q^{23} -16.0528 q^{24} +15.7002 q^{26} +6.86165 q^{27} +0.784226 q^{28} -7.29566 q^{29} -8.47532 q^{31} -2.35955 q^{32} +16.4509 q^{33} +2.57823 q^{34} +22.6888 q^{36} -3.20366 q^{37} +1.04358 q^{38} -18.2085 q^{39} +9.21106 q^{41} -1.34017 q^{42} +1.97405 q^{43} +24.0161 q^{44} +13.0086 q^{46} -5.03668 q^{47} +15.6057 q^{48} -6.96553 q^{49} -2.99015 q^{51} -26.5820 q^{52} -9.52766 q^{53} -17.1183 q^{54} -1.03009 q^{56} -1.21031 q^{57} +18.2010 q^{58} -4.69247 q^{59} -7.67472 q^{61} +21.1440 q^{62} +0.997294 q^{63} -4.90070 q^{64} -41.0413 q^{66} -10.5482 q^{67} -4.36521 q^{68} -15.0869 q^{69} +4.48385 q^{71} -29.8021 q^{72} +8.34246 q^{73} +7.99242 q^{74} -1.76689 q^{76} +1.05563 q^{77} +45.4262 q^{78} +5.40340 q^{79} +3.73864 q^{81} -22.9795 q^{82} -8.45058 q^{83} +2.26905 q^{84} -4.92482 q^{86} -21.1089 q^{87} -31.5454 q^{88} -17.3526 q^{89} -1.16842 q^{91} -22.0249 q^{92} -24.5221 q^{93} +12.5654 q^{94} -6.82702 q^{96} -1.97553 q^{97} +17.3774 q^{98} +30.5410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 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

Varieties

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Database

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