LMFDB - Embedded newform 6025.2.a.q.1.18

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:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-1.54552 q^{2} -0.0372080 q^{3} +0.388631 q^{4} +0.0575057 q^{6} +5.06287 q^{7} +2.49040 q^{8} -2.99862 q^{9} +O(q^{10})$	\(q-1.54552 q^{2} -0.0372080 q^{3} +0.388631 q^{4} +0.0575057 q^{6} +5.06287 q^{7} +2.49040 q^{8} -2.99862 q^{9} -4.06486 q^{11} -0.0144602 q^{12} +1.45831 q^{13} -7.82476 q^{14} -4.62623 q^{16} +5.44373 q^{17} +4.63442 q^{18} +6.33862 q^{19} -0.188379 q^{21} +6.28232 q^{22} +0.741121 q^{23} -0.0926629 q^{24} -2.25385 q^{26} +0.223196 q^{27} +1.96759 q^{28} +3.82347 q^{29} -1.28663 q^{31} +2.16912 q^{32} +0.151245 q^{33} -8.41340 q^{34} -1.16536 q^{36} -6.35768 q^{37} -9.79647 q^{38} -0.0542608 q^{39} +2.63308 q^{41} +0.291144 q^{42} -9.54698 q^{43} -1.57973 q^{44} -1.14542 q^{46} +13.0365 q^{47} +0.172133 q^{48} +18.6326 q^{49} -0.202550 q^{51} +0.566744 q^{52} +4.87907 q^{53} -0.344955 q^{54} +12.6086 q^{56} -0.235848 q^{57} -5.90925 q^{58} -9.05344 q^{59} -13.3675 q^{61} +1.98851 q^{62} -15.1816 q^{63} +5.90004 q^{64} -0.233752 q^{66} +11.0842 q^{67} +2.11560 q^{68} -0.0275756 q^{69} +15.8511 q^{71} -7.46776 q^{72} +2.45112 q^{73} +9.82593 q^{74} +2.46339 q^{76} -20.5798 q^{77} +0.0838611 q^{78} -0.508873 q^{79} +8.98754 q^{81} -4.06947 q^{82} +0.736734 q^{83} -0.0732100 q^{84} +14.7550 q^{86} -0.142264 q^{87} -10.1231 q^{88} -9.93931 q^{89} +7.38322 q^{91} +0.288023 q^{92} +0.0478728 q^{93} -20.1482 q^{94} -0.0807087 q^{96} -2.18329 q^{97} -28.7971 q^{98} +12.1889 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * 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 $

$2$	−1.54552	−1.09285	−0.546424	−	0.837509i	$-0.684011\pi$
$2$	−1.54552	−1.09285	−0.546424	+	0.837509i	$0.684011\pi$
$3$	−0.0372080	−0.0214820	−0.0107410	−	0.999942i	$-0.503419\pi$
$3$	−0.0372080	−0.0214820	−0.0107410	+	0.999942i	$0.503419\pi$
$4$	0.388631	0.194316
$5$	0	0
$5$	0	0
$6$	0.0575057	0.0234766
$7$	5.06287	1.91358	0.956792	−	0.290774i	$-0.0939129\pi$
$7$	5.06287	1.91358	0.956792	+	0.290774i	$0.0939129\pi$
$8$	2.49040	0.880490
$9$	−2.99862	−0.999539
$10$	0	0
$11$	−4.06486	−1.22560	−0.612800	−	0.790238i	$-0.709957\pi$
$11$	−4.06486	−1.22560	−0.612800	+	0.790238i	$0.709957\pi$
$12$	−0.0144602	−0.00417430
$13$	1.45831	0.404462	0.202231	−	0.979338i	$-0.435181\pi$
$13$	1.45831	0.404462	0.202231	+	0.979338i	$0.435181\pi$
$14$	−7.82476	−2.09125
$15$	0	0
$16$	−4.62623	−1.15656
$17$	5.44373	1.32030	0.660150	−	0.751134i	$-0.270493\pi$
$17$	5.44373	1.32030	0.660150	+	0.751134i	$0.270493\pi$
$18$	4.63442	1.09234
$19$	6.33862	1.45418	0.727090	−	0.686542i	$-0.240872\pi$
$19$	6.33862	1.45418	0.727090	+	0.686542i	$0.240872\pi$
$20$	0	0
$21$	−0.188379	−0.0411077
$22$	6.28232	1.33939
$23$	0.741121	0.154534	0.0772672	−	0.997010i	$-0.475381\pi$
$23$	0.741121	0.154534	0.0772672	+	0.997010i	$0.475381\pi$
$24$	−0.0926629	−0.0189147
$25$	0	0
$26$	−2.25385	−0.442015
$27$	0.223196	0.0429542
$28$	1.96759	0.371839
$29$	3.82347	0.710001	0.355001	−	0.934866i	$-0.384481\pi$
$29$	3.82347	0.710001	0.355001	+	0.934866i	$0.384481\pi$
$30$	0	0
$31$	−1.28663	−0.231085	−0.115542	−	0.993303i	$-0.536861\pi$
$31$	−1.28663	−0.231085	−0.115542	+	0.993303i	$0.536861\pi$
$32$	2.16912	0.383450
$33$	0.151245	0.0263284
$34$	−8.41340	−1.44289
$35$	0	0
$36$	−1.16536	−0.194226
$37$	−6.35768	−1.04520	−0.522598	−	0.852579i	$-0.675037\pi$
$37$	−6.35768	−1.04520	−0.522598	+	0.852579i	$0.675037\pi$
$38$	−9.79647	−1.58920
$39$	−0.0542608	−0.00868868
$40$	0	0
$41$	2.63308	0.411217	0.205609	−	0.978634i	$-0.434083\pi$
$41$	2.63308	0.411217	0.205609	+	0.978634i	$0.434083\pi$
$42$	0.291144	0.0449244
$43$	−9.54698	−1.45590	−0.727950	−	0.685630i	$-0.759527\pi$
$43$	−9.54698	−1.45590	−0.727950	+	0.685630i	$0.759527\pi$
$44$	−1.57973	−0.238153
$45$	0	0
$46$	−1.14542	−0.168882
$47$	13.0365	1.90157	0.950786	−	0.309848i	$-0.100278\pi$
$47$	13.0365	1.90157	0.950786	+	0.309848i	$0.100278\pi$
$48$	0.172133	0.0248452
$49$	18.6326	2.66180
$50$	0	0
$51$	−0.202550	−0.0283627
$52$	0.566744	0.0785933
$53$	4.87907	0.670192	0.335096	−	0.942184i	$-0.391231\pi$
$53$	4.87907	0.670192	0.335096	+	0.942184i	$0.391231\pi$
$54$	−0.344955	−0.0469424
$55$	0	0
$56$	12.6086	1.68489
$57$	−0.235848	−0.0312388
$58$	−5.90925	−0.775923
$59$	−9.05344	−1.17866	−0.589329	−	0.807893i	$-0.700608\pi$
$59$	−9.05344	−1.17866	−0.589329	+	0.807893i	$0.700608\pi$
$60$	0	0
$61$	−13.3675	−1.71153	−0.855765	−	0.517365i	$-0.826913\pi$
$61$	−13.3675	−1.71153	−0.855765	+	0.517365i	$0.826913\pi$
$62$	1.98851	0.252540
$63$	−15.1816	−1.91270
$64$	5.90004	0.737504
$65$	0	0
$66$	−0.233752	−0.0287729
$67$	11.0842	1.35415	0.677077	−	0.735912i	$-0.263246\pi$
$67$	11.0842	1.35415	0.677077	+	0.735912i	$0.263246\pi$
$68$	2.11560	0.256555
$69$	−0.0275756	−0.00331971
$70$	0	0
$71$	15.8511	1.88118	0.940589	−	0.339548i	$-0.110274\pi$
$71$	15.8511	1.88118	0.940589	+	0.339548i	$0.110274\pi$
$72$	−7.46776	−0.880084
$73$	2.45112	0.286882	0.143441	−	0.989659i	$-0.454183\pi$
$73$	2.45112	0.286882	0.143441	+	0.989659i	$0.454183\pi$
$74$	9.82593	1.14224
$75$	0	0
$76$	2.46339	0.282570
$77$	−20.5798	−2.34529
$78$	0.0838611	0.00949540
$79$	−0.508873	−0.0572527	−0.0286264	−	0.999590i	$-0.509113\pi$
$79$	−0.508873	−0.0572527	−0.0286264	+	0.999590i	$0.509113\pi$
$80$	0	0
$81$	8.98754	0.998616
$82$	−4.06947	−0.449398
$83$	0.736734	0.0808671	0.0404336	−	0.999182i	$-0.487126\pi$
$83$	0.736734	0.0808671	0.0404336	+	0.999182i	$0.487126\pi$
$84$	−0.0732100	−0.00798786
$85$	0	0
$86$	14.7550	1.59108
$87$	−0.142264	−0.0152523
$88$	−10.1231	−1.07913
$89$	−9.93931	−1.05356	−0.526782	−	0.850000i	$-0.676602\pi$
$89$	−9.93931	−1.05356	−0.526782	+	0.850000i	$0.676602\pi$
$90$	0	0
$91$	7.38322	0.773972
$92$	0.288023	0.0300284
$93$	0.0478728	0.00496417
$94$	−20.1482	−2.07813
$95$	0	0
$96$	−0.0807087	−0.00823729
$97$	−2.18329	−0.221680	−0.110840	−	0.993838i	$-0.535354\pi$
$97$	−2.18329	−0.221680	−0.110840	+	0.993838i	$0.535354\pi$
$98$	−28.7971	−2.90894
$99$	12.1889	1.22503
$100$	0	0
$101$	9.82968	0.978089	0.489045	−	0.872259i	$-0.337345\pi$
$101$	9.82968	0.978089	0.489045	+	0.872259i	$0.337345\pi$
$102$	0.313046	0.0309961
$103$	−5.51114	−0.543029	−0.271514	−	0.962434i	$-0.587524\pi$
$103$	−5.51114	−0.543029	−0.271514	+	0.962434i	$0.587524\pi$
$104$	3.63178	0.356125
$105$	0	0
$106$	−7.54070	−0.732417
$107$	−9.72635	−0.940282	−0.470141	−	0.882591i	$-0.655797\pi$
$107$	−9.72635	−0.940282	−0.470141	+	0.882591i	$0.655797\pi$
$108$	0.0867411	0.00834666
$109$	−1.44994	−0.138879	−0.0694395	−	0.997586i	$-0.522121\pi$
$109$	−1.44994	−0.138879	−0.0694395	+	0.997586i	$0.522121\pi$
$110$	0	0
$111$	0.236557	0.0224530
$112$	−23.4220	−2.21317
$113$	2.27639	0.214145	0.107072	−	0.994251i	$-0.465852\pi$
$113$	2.27639	0.214145	0.107072	+	0.994251i	$0.465852\pi$
$114$	0.364507	0.0341392
$115$	0	0
$116$	1.48592	0.137964
$117$	−4.37291	−0.404276
$118$	13.9923	1.28809
$119$	27.5609	2.52650
$120$	0	0
$121$	5.52306	0.502096
$122$	20.6597	1.87044
$123$	−0.0979715	−0.00883379
$124$	−0.500023	−0.0449034
$125$	0	0
$126$	23.4634	2.09029
$127$	4.47024	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$127$	4.47024	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$128$	−13.4569	−1.18943
$129$	0.355224	0.0312757
$130$	0	0
$131$	17.5461	1.53301	0.766505	−	0.642238i	$-0.221994\pi$
$131$	17.5461	1.53301	0.766505	+	0.642238i	$0.221994\pi$
$132$	0.0587786	0.00511602
$133$	32.0916	2.78269
$134$	−17.1309	−1.47988
$135$	0	0
$136$	13.5571	1.16251
$137$	14.7659	1.26153	0.630766	−	0.775973i	$-0.282741\pi$
$137$	14.7659	1.26153	0.630766	+	0.775973i	$0.282741\pi$
$138$	0.0426187	0.00362794
$139$	−14.5636	−1.23527	−0.617634	−	0.786466i	$-0.711909\pi$
$139$	−14.5636	−1.23527	−0.617634	+	0.786466i	$0.711909\pi$
$140$	0	0
$141$	−0.485063	−0.0408497
$142$	−24.4982	−2.05584
$143$	−5.92782	−0.495709
$144$	13.8723	1.15602
$145$	0	0
$146$	−3.78825	−0.313518
$147$	−0.693282	−0.0571809
$148$	−2.47079	−0.203098
$149$	10.5020	0.860359	0.430179	−	0.902743i	$-0.358450\pi$
$149$	10.5020	0.860359	0.430179	+	0.902743i	$0.358450\pi$
$150$	0	0
$151$	−10.7945	−0.878445	−0.439223	−	0.898378i	$-0.644746\pi$
$151$	−10.7945	−0.878445	−0.439223	+	0.898378i	$0.644746\pi$
$152$	15.7857	1.28039
$153$	−16.3237	−1.31969
$154$	31.8065	2.56304
$155$	0	0
$156$	−0.0210874	−0.00168834
$157$	20.3563	1.62461	0.812306	−	0.583232i	$-0.198212\pi$
$157$	20.3563	1.62461	0.812306	+	0.583232i	$0.198212\pi$
$158$	0.786474	0.0625685
$159$	−0.181540	−0.0143971
$160$	0	0
$161$	3.75219	0.295714
$162$	−13.8904	−1.09133
$163$	−4.36397	−0.341813	−0.170906	−	0.985287i	$-0.554670\pi$
$163$	−4.36397	−0.341813	−0.170906	+	0.985287i	$0.554670\pi$
$164$	1.02330	0.0799059
$165$	0	0
$166$	−1.13864	−0.0883754
$167$	−8.85394	−0.685138	−0.342569	−	0.939493i	$-0.611297\pi$
$167$	−8.85394	−0.685138	−0.342569	+	0.939493i	$0.611297\pi$
$168$	−0.469140	−0.0361949
$169$	−10.8733	−0.836410
$170$	0	0
$171$	−19.0071	−1.45351
$172$	−3.71025	−0.282904
$173$	2.38714	0.181491	0.0907454	−	0.995874i	$-0.471075\pi$
$173$	2.38714	0.181491	0.0907454	+	0.995874i	$0.471075\pi$
$174$	0.219872	0.0166684
$175$	0	0
$176$	18.8050	1.41748
$177$	0.336860	0.0253200
$178$	15.3614	1.15139
$179$	8.01142	0.598802	0.299401	−	0.954127i	$-0.403213\pi$
$179$	8.01142	0.598802	0.299401	+	0.954127i	$0.403213\pi$
$180$	0	0
$181$	−17.3646	−1.29070	−0.645350	−	0.763887i	$-0.723288\pi$
$181$	−17.3646	−1.29070	−0.645350	+	0.763887i	$0.723288\pi$
$182$	−11.4109	−0.845833
$183$	0.497377	0.0367672
$184$	1.84569	0.136066
$185$	0	0
$186$	−0.0739883	−0.00542508
$187$	−22.1280	−1.61816
$188$	5.06640	0.369505
$189$	1.13001	0.0821964
$190$	0	0
$191$	5.43854	0.393519	0.196759	−	0.980452i	$-0.436958\pi$
$191$	5.43854	0.393519	0.196759	+	0.980452i	$0.436958\pi$
$192$	−0.219529	−0.0158431
$193$	−7.83807	−0.564197	−0.282098	−	0.959385i	$-0.591030\pi$
$193$	−7.83807	−0.564197	−0.282098	+	0.959385i	$0.591030\pi$
$194$	3.37432	0.242262
$195$	0	0
$196$	7.24121	0.517229
$197$	−1.62401	−0.115706	−0.0578530	−	0.998325i	$-0.518425\pi$
$197$	−1.62401	−0.115706	−0.0578530	+	0.998325i	$0.518425\pi$
$198$	−18.8382	−1.33878
$199$	−6.48042	−0.459385	−0.229693	−	0.973263i	$-0.573772\pi$
$199$	−6.48042	−0.459385	−0.229693	+	0.973263i	$0.573772\pi$
$200$	0	0
$201$	−0.412422	−0.0290900
$202$	−15.1920	−1.06890
$203$	19.3577	1.35865
$204$	−0.0787174	−0.00551132
$205$	0	0
$206$	8.51758	0.593448
$207$	−2.22234	−0.154463
$208$	−6.74647	−0.467784
$209$	−25.7656	−1.78224
$210$	0	0
$211$	−14.4663	−0.995903	−0.497951	−	0.867205i	$-0.665914\pi$
$211$	−14.4663	−0.995903	−0.497951	+	0.867205i	$0.665914\pi$
$212$	1.89616	0.130229
$213$	−0.589787	−0.0404115
$214$	15.0323	1.02758
$215$	0	0
$216$	0.555849	0.0378207
$217$	−6.51401	−0.442200
$218$	2.24091	0.151774
$219$	−0.0912012	−0.00616281
$220$	0	0
$221$	7.93865	0.534011
$222$	−0.365603	−0.0245377
$223$	−29.2323	−1.95754	−0.978770	−	0.204963i	$-0.934293\pi$
$223$	−29.2323	−1.95754	−0.978770	+	0.204963i	$0.934293\pi$
$224$	10.9820	0.733764
$225$	0	0
$226$	−3.51820	−0.234027
$227$	27.8129	1.84600	0.923002	−	0.384795i	$-0.125728\pi$
$227$	27.8129	1.84600	0.923002	+	0.384795i	$0.125728\pi$
$228$	−0.0916577	−0.00607018
$229$	−13.4799	−0.890774	−0.445387	−	0.895338i	$-0.646934\pi$
$229$	−13.4799	−0.890774	−0.445387	+	0.895338i	$0.646934\pi$
$230$	0	0
$231$	0.765734	0.0503816
$232$	9.52199	0.625149
$233$	6.34697	0.415804	0.207902	−	0.978150i	$-0.433337\pi$
$233$	6.34697	0.415804	0.207902	+	0.978150i	$0.433337\pi$
$234$	6.75842	0.441811
$235$	0	0
$236$	−3.51845	−0.229031
$237$	0.0189342	0.00122991
$238$	−42.5959	−2.76108
$239$	3.66465	0.237047	0.118523	−	0.992951i	$-0.462184\pi$
$239$	3.66465	0.237047	0.118523	+	0.992951i	$0.462184\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−8.53599	−0.548714
$243$	−1.00400	−0.0644065
$244$	−5.19502	−0.332577
$245$	0	0
$246$	0.151417	0.00965399
$247$	9.24367	0.588161
$248$	−3.20422	−0.203468
$249$	−0.0274124	−0.00173719
$250$	0	0
$251$	5.50004	0.347159	0.173580	−	0.984820i	$-0.444467\pi$
$251$	5.50004	0.347159	0.173580	+	0.984820i	$0.444467\pi$
$252$	−5.90004	−0.371667
$253$	−3.01255	−0.189397
$254$	−6.90885	−0.433500
$255$	0	0
$256$	8.99778	0.562361
$257$	−8.43297	−0.526034	−0.263017	−	0.964791i	$-0.584718\pi$
$257$	−8.43297	−0.526034	−0.263017	+	0.964791i	$0.584718\pi$
$258$	−0.549005	−0.0341796
$259$	−32.1881	−2.00007
$260$	0	0
$261$	−11.4651	−0.709674
$262$	−27.1179	−1.67535
$263$	7.59351	0.468236	0.234118	−	0.972208i	$-0.424780\pi$
$263$	7.59351	0.468236	0.234118	+	0.972208i	$0.424780\pi$
$264$	0.376661	0.0231819
$265$	0	0
$266$	−49.5982	−3.04106
$267$	0.369822	0.0226327
$268$	4.30768	0.263133
$269$	10.4193	0.635278	0.317639	−	0.948212i	$-0.397110\pi$
$269$	10.4193	0.635278	0.317639	+	0.948212i	$0.397110\pi$
$270$	0	0
$271$	25.6834	1.56015	0.780076	−	0.625684i	$-0.215180\pi$
$271$	25.6834	1.56015	0.780076	+	0.625684i	$0.215180\pi$
$272$	−25.1839	−1.52700
$273$	−0.274715	−0.0166265
$274$	−22.8209	−1.37866
$275$	0	0
$276$	−0.0107167	−0.000645072 0
$277$	−7.18371	−0.431627	−0.215814	−	0.976435i	$-0.569240\pi$
$277$	−7.18371	−0.431627	−0.215814	+	0.976435i	$0.569240\pi$
$278$	22.5083	1.34996
$279$	3.85810	0.230978
$280$	0	0
$281$	−7.96121	−0.474926	−0.237463	−	0.971397i	$-0.576316\pi$
$281$	−7.96121	−0.474926	−0.237463	+	0.971397i	$0.576316\pi$
$282$	0.749674	0.0446424
$283$	1.82901	0.108723	0.0543617	−	0.998521i	$-0.482688\pi$
$283$	1.82901	0.108723	0.0543617	+	0.998521i	$0.482688\pi$
$284$	6.16022	0.365542
$285$	0	0
$286$	9.16156	0.541734
$287$	13.3309	0.786899
$288$	−6.50436	−0.383273
$289$	12.6342	0.743190
$290$	0	0
$291$	0.0812360	0.00476214
$292$	0.952581	0.0557456
$293$	−20.9132	−1.22176	−0.610880	−	0.791723i	$-0.709184\pi$
$293$	−20.9132	−1.22176	−0.610880	+	0.791723i	$0.709184\pi$
$294$	1.07148	0.0624900
$295$	0	0
$296$	−15.8332	−0.920285
$297$	−0.907262	−0.0526447
$298$	−16.2311	−0.940241
$299$	1.08078	0.0625033
$300$	0	0
$301$	−48.3351	−2.78599
$302$	16.6831	0.960007
$303$	−0.365743	−0.0210114
$304$	−29.3239	−1.68184
$305$	0	0
$306$	25.2285	1.44222
$307$	−23.3801	−1.33437	−0.667187	−	0.744890i	$-0.732502\pi$
$307$	−23.3801	−1.33437	−0.667187	+	0.744890i	$0.732502\pi$
$308$	−7.99796	−0.455726
$309$	0.205058	0.0116654
$310$	0	0
$311$	0.414526	0.0235056	0.0117528	−	0.999931i	$-0.496259\pi$
$311$	0.414526	0.0235056	0.0117528	+	0.999931i	$0.496259\pi$
$312$	−0.135131	−0.00765029
$313$	17.0916	0.966074	0.483037	−	0.875600i	$-0.339534\pi$
$313$	17.0916	0.966074	0.483037	+	0.875600i	$0.339534\pi$
$314$	−31.4611	−1.77545
$315$	0	0
$316$	−0.197764	−0.0111251
$317$	−9.54612	−0.536163	−0.268082	−	0.963396i	$-0.586390\pi$
$317$	−9.54612	−0.536163	−0.268082	+	0.963396i	$0.586390\pi$
$318$	0.280574	0.0157338
$319$	−15.5419	−0.870178
$320$	0	0
$321$	0.361898	0.0201992
$322$	−5.79909	−0.323171
$323$	34.5058	1.91995
$324$	3.49284	0.194047
$325$	0	0
$326$	6.74461	0.373549
$327$	0.0539494	0.00298341
$328$	6.55742	0.362073
$329$	66.0021	3.63882
$330$	0	0
$331$	7.80966	0.429258	0.214629	−	0.976696i	$-0.431146\pi$
$331$	7.80966	0.429258	0.214629	+	0.976696i	$0.431146\pi$
$332$	0.286318	0.0157137
$333$	19.0642	1.04471
$334$	13.6839	0.748751
$335$	0	0
$336$	0.871485	0.0475434
$337$	8.81006	0.479915	0.239957	−	0.970783i	$-0.422867\pi$
$337$	8.81006	0.479915	0.239957	+	0.970783i	$0.422867\pi$
$338$	16.8050	0.914069
$339$	−0.0846998	−0.00460026
$340$	0	0
$341$	5.22995	0.283218
$342$	29.3758	1.58846
$343$	58.8943	3.17999
$344$	−23.7758	−1.28191
$345$	0	0
$346$	−3.68937	−0.198342
$347$	−29.8188	−1.60076	−0.800379	−	0.599494i	$-0.795368\pi$
$347$	−29.8188	−1.60076	−0.800379	+	0.599494i	$0.795368\pi$
$348$	−0.0552881	−0.00296376
$349$	28.7714	1.54010	0.770049	−	0.637984i	$-0.220232\pi$
$349$	28.7714	1.54010	0.770049	+	0.637984i	$0.220232\pi$
$350$	0	0
$351$	0.325489	0.0173733
$352$	−8.81717	−0.469957
$353$	5.08285	0.270533	0.135267	−	0.990809i	$-0.456811\pi$
$353$	5.08285	0.270533	0.135267	+	0.990809i	$0.456811\pi$
$354$	−0.520624	−0.0276709
$355$	0	0
$356$	−3.86272	−0.204724
$357$	−1.02549	−0.0542744
$358$	−12.3818	−0.654399
$359$	−0.292937	−0.0154606	−0.00773030	−	0.999970i	$-0.502461\pi$
$359$	−0.292937	−0.0154606	−0.00773030	+	0.999970i	$0.502461\pi$
$360$	0	0
$361$	21.1782	1.11464
$362$	26.8373	1.41054
$363$	−0.205502	−0.0107861
$364$	2.86935	0.150395
$365$	0	0
$366$	−0.768706	−0.0401809
$367$	27.8233	1.45236	0.726181	−	0.687504i	$-0.241293\pi$
$367$	27.8233	1.45236	0.726181	+	0.687504i	$0.241293\pi$
$368$	−3.42859	−0.178728
$369$	−7.89558	−0.411028
$370$	0	0
$371$	24.7021	1.28247
$372$	0.0186048	0.000964616 0
$373$	−1.81941	−0.0942054	−0.0471027	−	0.998890i	$-0.514999\pi$
$373$	−1.81941	−0.0942054	−0.0471027	+	0.998890i	$0.514999\pi$
$374$	34.1992	1.76840
$375$	0	0
$376$	32.4662	1.67432
$377$	5.57581	0.287169
$378$	−1.74646	−0.0898281
$379$	30.5694	1.57024	0.785121	−	0.619342i	$-0.212600\pi$
$379$	30.5694	1.57024	0.785121	+	0.619342i	$0.212600\pi$
$380$	0	0
$381$	−0.166329	−0.00852128
$382$	−8.40537	−0.430056
$383$	5.76282	0.294466	0.147233	−	0.989102i	$-0.452963\pi$
$383$	5.76282	0.294466	0.147233	+	0.989102i	$0.452963\pi$
$384$	0.500703	0.0255514
$385$	0	0
$386$	12.1139	0.616581
$387$	28.6277	1.45523
$388$	−0.848496	−0.0430759
$389$	5.42392	0.275004	0.137502	−	0.990502i	$-0.456093\pi$
$389$	5.42392	0.275004	0.137502	+	0.990502i	$0.456093\pi$
$390$	0	0
$391$	4.03446	0.204032
$392$	46.4027	2.34369
$393$	−0.652856	−0.0329322
$394$	2.50994	0.126449
$395$	0	0
$396$	4.73700	0.238043
$397$	3.75181	0.188298	0.0941490	−	0.995558i	$-0.469987\pi$
$397$	3.75181	0.188298	0.0941490	+	0.995558i	$0.469987\pi$
$398$	10.0156	0.502038
$399$	−1.19406	−0.0597780
$400$	0	0
$401$	16.6551	0.831714	0.415857	−	0.909430i	$-0.363482\pi$
$401$	16.6551	0.831714	0.415857	+	0.909430i	$0.363482\pi$
$402$	0.637406	0.0317909
$403$	−1.87630	−0.0934650
$404$	3.82012	0.190058
$405$	0	0
$406$	−29.9178	−1.48479
$407$	25.8431	1.28099
$408$	−0.504432	−0.0249731
$409$	19.5346	0.965924	0.482962	−	0.875641i	$-0.339561\pi$
$409$	19.5346	0.965924	0.482962	+	0.875641i	$0.339561\pi$
$410$	0	0
$411$	−0.549408	−0.0271003
$412$	−2.14180	−0.105519
$413$	−45.8364	−2.25546
$414$	3.43466	0.168805
$415$	0	0
$416$	3.16325	0.155091
$417$	0.541882	0.0265361
$418$	39.8212	1.94772
$419$	14.9475	0.730233	0.365117	−	0.930962i	$-0.381029\pi$
$419$	14.9475	0.730233	0.365117	+	0.930962i	$0.381029\pi$
$420$	0	0
$421$	21.8853	1.06662	0.533312	−	0.845918i	$-0.320947\pi$
$421$	21.8853	1.06662	0.533312	+	0.845918i	$0.320947\pi$
$422$	22.3580	1.08837
$423$	−39.0915	−1.90069
$424$	12.1508	0.590097
$425$	0	0
$426$	0.911527	0.0441636
$427$	−67.6777	−3.27515
$428$	−3.77996	−0.182711
$429$	0.220562	0.0106488
$430$	0	0
$431$	1.99708	0.0961960	0.0480980	−	0.998843i	$-0.484684\pi$
$431$	1.99708	0.0961960	0.0480980	+	0.998843i	$0.484684\pi$
$432$	−1.03256	−0.0496790
$433$	20.8686	1.00288	0.501441	−	0.865192i	$-0.332803\pi$
$433$	20.8686	1.00288	0.501441	+	0.865192i	$0.332803\pi$
$434$	10.0675	0.483257
$435$	0	0
$436$	−0.563492	−0.0269864
$437$	4.69768	0.224721
$438$	0.140953	0.00673501
$439$	−5.43238	−0.259273	−0.129637	−	0.991562i	$-0.541381\pi$
$439$	−5.43238	−0.259273	−0.129637	+	0.991562i	$0.541381\pi$
$440$	0	0
$441$	−55.8720	−2.66057
$442$	−12.2693	−0.583593
$443$	−16.6946	−0.793185	−0.396592	−	0.917995i	$-0.629807\pi$
$443$	−16.6946	−0.793185	−0.396592	+	0.917995i	$0.629807\pi$
$444$	0.0919333	0.00436296
$445$	0	0
$446$	45.1791	2.13929
$447$	−0.390759	−0.0184823
$448$	29.8711	1.41128
$449$	−28.0472	−1.32363	−0.661815	−	0.749667i	$-0.730214\pi$
$449$	−28.0472	−1.32363	−0.661815	+	0.749667i	$0.730214\pi$
$450$	0	0
$451$	−10.7031	−0.503988
$452$	0.884675	0.0416116
$453$	0.401642	0.0188708
$454$	−42.9853	−2.01740
$455$	0	0
$456$	−0.587355	−0.0275054
$457$	22.3046	1.04336	0.521682	−	0.853140i	$-0.325305\pi$
$457$	22.3046	1.04336	0.521682	+	0.853140i	$0.325305\pi$
$458$	20.8334	0.973481
$459$	1.21502	0.0567124
$460$	0	0
$461$	25.5453	1.18976	0.594881	−	0.803814i	$-0.297199\pi$
$461$	25.5453	1.18976	0.594881	+	0.803814i	$0.297199\pi$
$462$	−1.18346	−0.0550594
$463$	37.5533	1.74525	0.872625	−	0.488391i	$-0.162416\pi$
$463$	37.5533	1.74525	0.872625	+	0.488391i	$0.162416\pi$
$464$	−17.6883	−0.821157
$465$	0	0
$466$	−9.80937	−0.454410
$467$	−21.3303	−0.987049	−0.493525	−	0.869732i	$-0.664292\pi$
$467$	−21.3303	−0.987049	−0.493525	+	0.869732i	$0.664292\pi$
$468$	−1.69945	−0.0785570
$469$	56.1180	2.59129
$470$	0	0
$471$	−0.757418	−0.0349000
$472$	−22.5467	−1.03780
$473$	38.8071	1.78435
$474$	−0.0292631	−0.00134410
$475$	0	0
$476$	10.7110	0.490939
$477$	−14.6305	−0.669882
$478$	−5.66380	−0.259056
$479$	2.49931	0.114197	0.0570983	−	0.998369i	$-0.481815\pi$
$479$	2.49931	0.114197	0.0570983	+	0.998369i	$0.481815\pi$
$480$	0	0
$481$	−9.27147	−0.422743
$482$	1.54552	0.0703965
$483$	−0.139612	−0.00635255
$484$	2.14643	0.0975651
$485$	0	0
$486$	1.55170	0.0703865
$487$	32.3933	1.46788	0.733941	−	0.679213i	$-0.237679\pi$
$487$	32.3933	1.46788	0.733941	+	0.679213i	$0.237679\pi$
$488$	−33.2904	−1.50699
$489$	0.162375	0.00734284
$490$	0	0
$491$	−25.7985	−1.16427	−0.582136	−	0.813091i	$-0.697783\pi$
$491$	−25.7985	−1.16427	−0.582136	+	0.813091i	$0.697783\pi$
$492$	−0.0380748	−0.00171654
$493$	20.8140	0.937414
$494$	−14.2863	−0.642770
$495$	0	0
$496$	5.95222	0.267263
$497$	80.2519	3.59979
$498$	0.0423664	0.00189848
$499$	−9.47263	−0.424053	−0.212027	−	0.977264i	$-0.568006\pi$
$499$	−9.47263	−0.424053	−0.212027	+	0.977264i	$0.568006\pi$
$500$	0	0
$501$	0.329437	0.0147182
$502$	−8.50041	−0.379392
$503$	7.11924	0.317431	0.158716	−	0.987324i	$-0.449265\pi$
$503$	7.11924	0.317431	0.158716	+	0.987324i	$0.449265\pi$
$504$	−37.8083	−1.68411
$505$	0	0
$506$	4.65595	0.206982
$507$	0.404575	0.0179678
$508$	1.73727	0.0770791
$509$	4.70389	0.208496	0.104248	−	0.994551i	$-0.466756\pi$
$509$	4.70389	0.208496	0.104248	+	0.994551i	$0.466756\pi$
$510$	0	0
$511$	12.4097	0.548972
$512$	13.0075	0.574855
$513$	1.41476	0.0624631
$514$	13.0333	0.574875
$515$	0	0
$516$	0.138051	0.00607736
$517$	−52.9916	−2.33057
$518$	49.7473	2.18577
$519$	−0.0888206	−0.00389879
$520$	0	0
$521$	23.0641	1.01046	0.505228	−	0.862986i	$-0.331408\pi$
$521$	23.0641	1.01046	0.505228	+	0.862986i	$0.331408\pi$
$522$	17.7196	0.775565
$523$	34.0179	1.48750	0.743750	−	0.668458i	$-0.233045\pi$
$523$	34.0179	1.48750	0.743750	+	0.668458i	$0.233045\pi$
$524$	6.81896	0.297888
$525$	0	0
$526$	−11.7359	−0.511711
$527$	−7.00405	−0.305101
$528$	−0.699695	−0.0304503
$529$	−22.4507	−0.976119
$530$	0	0
$531$	27.1478	1.17811
$532$	12.4718	0.540721
$533$	3.83984	0.166322
$534$	−0.571567	−0.0247341
$535$	0	0
$536$	27.6042	1.19232
$537$	−0.298089	−0.0128635
$538$	−16.1033	−0.694262
$539$	−75.7389	−3.26230
$540$	0	0
$541$	−8.98916	−0.386474	−0.193237	−	0.981152i	$-0.561899\pi$
$541$	−8.98916	−0.386474	−0.193237	+	0.981152i	$0.561899\pi$
$542$	−39.6941	−1.70501
$543$	0.646102	0.0277269
$544$	11.8081	0.506269
$545$	0	0
$546$	0.424577	0.0181702
$547$	−44.9348	−1.92127	−0.960637	−	0.277805i	$-0.910393\pi$
$547$	−44.9348	−1.92127	−0.960637	+	0.277805i	$0.910393\pi$
$548$	5.73847	0.245135
$549$	40.0839	1.71074
$550$	0	0
$551$	24.2356	1.03247
$552$	−0.0686744	−0.00292298
$553$	−2.57636	−0.109558
$554$	11.1026	0.471703
$555$	0	0
$556$	−5.65986	−0.240032
$557$	1.94116	0.0822495	0.0411247	−	0.999154i	$-0.486906\pi$
$557$	1.94116	0.0822495	0.0411247	+	0.999154i	$0.486906\pi$
$558$	−5.96276	−0.252424
$559$	−13.9224	−0.588857
$560$	0	0
$561$	0.823338	0.0347614
$562$	12.3042	0.519021
$563$	−37.7897	−1.59265	−0.796323	−	0.604872i	$-0.793224\pi$
$563$	−37.7897	−1.59265	−0.796323	+	0.604872i	$0.793224\pi$
$564$	−0.188510	−0.00793772
$565$	0	0
$566$	−2.82677	−0.118818
$567$	45.5027	1.91093
$568$	39.4756	1.65636
$569$	15.0702	0.631776	0.315888	−	0.948797i	$-0.397698\pi$
$569$	15.0702	0.631776	0.315888	+	0.948797i	$0.397698\pi$
$570$	0	0
$571$	27.5415	1.15258	0.576288	−	0.817247i	$-0.304501\pi$
$571$	27.5415	1.15258	0.576288	+	0.817247i	$0.304501\pi$
$572$	−2.30373	−0.0963240
$573$	−0.202357	−0.00845359
$574$	−20.6032	−0.859960
$575$	0	0
$576$	−17.6919	−0.737164
$577$	23.9489	0.997007	0.498503	−	0.866888i	$-0.333883\pi$
$577$	23.9489	0.997007	0.498503	+	0.866888i	$0.333883\pi$
$578$	−19.5264	−0.812193
$579$	0.291639	0.0121201
$580$	0	0
$581$	3.72999	0.154746
$582$	−0.125552	−0.00520429
$583$	−19.8327	−0.821387
$584$	6.10427	0.252597
$585$	0	0
$586$	32.3217	1.33520
$587$	15.6396	0.645516	0.322758	−	0.946481i	$-0.395390\pi$
$587$	15.6396	0.645516	0.322758	+	0.946481i	$0.395390\pi$
$588$	−0.269431	−0.0111111
$589$	−8.15544	−0.336039
$590$	0	0
$591$	0.0604262	0.00248560
$592$	29.4121	1.20883
$593$	24.5345	1.00751	0.503755	−	0.863847i	$-0.331952\pi$
$593$	24.5345	1.00751	0.503755	+	0.863847i	$0.331952\pi$
$594$	1.40219	0.0575326
$595$	0	0
$596$	4.08141	0.167181
$597$	0.241124	0.00986853
$598$	−1.67037	−0.0683066
$599$	28.9411	1.18250	0.591251	−	0.806488i	$-0.298634\pi$
$599$	28.9411	1.18250	0.591251	+	0.806488i	$0.298634\pi$
$600$	0	0
$601$	21.3208	0.869695	0.434848	−	0.900504i	$-0.356802\pi$
$601$	21.3208	0.869695	0.434848	+	0.900504i	$0.356802\pi$
$602$	74.7028	3.04466
$603$	−33.2374	−1.35353
$604$	−4.19508	−0.170696
$605$	0	0
$606$	0.565262	0.0229622
$607$	30.6729	1.24498	0.622488	−	0.782629i	$-0.286122\pi$
$607$	30.6729	1.24498	0.622488	+	0.782629i	$0.286122\pi$
$608$	13.7492	0.557606
$609$	−0.720263	−0.0291865
$610$	0	0
$611$	19.0113	0.769114
$612$	−6.34388	−0.256436
$613$	−8.71017	−0.351800	−0.175900	−	0.984408i	$-0.556284\pi$
$613$	−8.71017	−0.351800	−0.175900	+	0.984408i	$0.556284\pi$
$614$	36.1344	1.45827
$615$	0	0
$616$	−51.2520	−2.06500
$617$	−28.2534	−1.13744	−0.568719	−	0.822532i	$-0.692561\pi$
$617$	−28.2534	−1.13744	−0.568719	+	0.822532i	$0.692561\pi$
$618$	−0.316922	−0.0127485
$619$	−47.7047	−1.91741	−0.958706	−	0.284399i	$-0.908206\pi$
$619$	−47.7047	−1.91741	−0.958706	+	0.284399i	$0.908206\pi$
$620$	0	0
$621$	0.165416	0.00663790
$622$	−0.640658	−0.0256880
$623$	−50.3214	−2.01608
$624$	0.251023	0.0100489
$625$	0	0
$626$	−26.4154	−1.05577
$627$	0.958686	0.0382862
$628$	7.91110	0.315687
$629$	−34.6095	−1.37997
$630$	0	0
$631$	12.3819	0.492918	0.246459	−	0.969153i	$-0.420733\pi$
$631$	12.3819	0.492918	0.246459	+	0.969153i	$0.420733\pi$
$632$	−1.26730	−0.0504105
$633$	0.538263	0.0213940
$634$	14.7537	0.585945
$635$	0	0
$636$	−0.0705522	−0.00279758
$637$	27.1721	1.07660
$638$	24.0203	0.950972
$639$	−47.5313	−1.88031
$640$	0	0
$641$	19.1983	0.758287	0.379143	−	0.925338i	$-0.376219\pi$
$641$	19.1983	0.758287	0.379143	+	0.925338i	$0.376219\pi$
$642$	−0.559320	−0.0220746
$643$	22.0427	0.869279	0.434639	−	0.900605i	$-0.356876\pi$
$643$	22.0427	0.869279	0.434639	+	0.900605i	$0.356876\pi$
$644$	1.45822	0.0574619
$645$	0	0
$646$	−53.3294	−2.09822
$647$	−7.64953	−0.300734	−0.150367	−	0.988630i	$-0.548046\pi$
$647$	−7.64953	−0.300734	−0.150367	+	0.988630i	$0.548046\pi$
$648$	22.3826	0.879271
$649$	36.8009	1.44456
$650$	0	0
$651$	0.242373	0.00949936
$652$	−1.69598	−0.0664195
$653$	2.52759	0.0989121	0.0494560	−	0.998776i	$-0.484251\pi$
$653$	2.52759	0.0989121	0.0494560	+	0.998776i	$0.484251\pi$
$654$	−0.0833798	−0.00326041
$655$	0	0
$656$	−12.1812	−0.475596
$657$	−7.34996	−0.286749
$658$	−102.008	−3.97667
$659$	−0.00229940	−8.95719e−5 0	−4.47859e−5	−	1.00000i	$-0.500014\pi$
$659$	−0.00229940	−8.95719e−5 0	−4.47859e−5		1.00000i	$0.500014\pi$
$660$	0	0
$661$	33.3623	1.29764	0.648822	−	0.760941i	$-0.275262\pi$
$661$	33.3623	1.29764	0.648822	+	0.760941i	$0.275262\pi$
$662$	−12.0700	−0.469113
$663$	−0.295381	−0.0114717
$664$	1.83477	0.0712027
$665$	0	0
$666$	−29.4642	−1.14171
$667$	2.83366	0.109720
$668$	−3.44091	−0.133133
$669$	1.08768	0.0420520
$670$	0	0
$671$	54.3369	2.09765
$672$	−0.408617	−0.0157627
$673$	34.2751	1.32121	0.660603	−	0.750735i	$-0.270301\pi$
$673$	34.2751	1.32121	0.660603	+	0.750735i	$0.270301\pi$
$674$	−13.6161	−0.524473
$675$	0	0
$676$	−4.22572	−0.162528
$677$	5.55645	0.213552	0.106776	−	0.994283i	$-0.465947\pi$
$677$	5.55645	0.213552	0.106776	+	0.994283i	$0.465947\pi$
$678$	0.130905	0.00502739
$679$	−11.0537	−0.424203
$680$	0	0
$681$	−1.03486	−0.0396559
$682$	−8.08299	−0.309514
$683$	−24.3119	−0.930269	−0.465135	−	0.885240i	$-0.653994\pi$
$683$	−24.3119	−0.930269	−0.465135	+	0.885240i	$0.653994\pi$
$684$	−7.38675	−0.282439
$685$	0	0
$686$	−91.0223	−3.47525
$687$	0.501559	0.0191357
$688$	44.1665	1.68383
$689$	7.11519	0.271067
$690$	0	0
$691$	30.9510	1.17743	0.588716	−	0.808340i	$-0.299634\pi$
$691$	30.9510	1.17743	0.588716	+	0.808340i	$0.299634\pi$
$692$	0.927716	0.0352665
$693$	61.7110	2.34421
$694$	46.0856	1.74938
$695$	0	0
$696$	−0.354294	−0.0134295
$697$	14.3338	0.542930
$698$	−44.4668	−1.68309
$699$	−0.236158	−0.00893232
$700$	0	0
$701$	30.5142	1.15251	0.576253	−	0.817271i	$-0.304514\pi$
$701$	30.5142	1.15251	0.576253	+	0.817271i	$0.304514\pi$
$702$	−0.503050	−0.0189864
$703$	−40.2990	−1.51990
$704$	−23.9828	−0.903886
$705$	0	0
$706$	−7.85565	−0.295651
$707$	49.7663	1.87166
$708$	0.130914	0.00492007
$709$	22.5083	0.845315	0.422658	−	0.906289i	$-0.361097\pi$
$709$	22.5083	0.845315	0.422658	+	0.906289i	$0.361097\pi$
$710$	0	0
$711$	1.52592	0.0572263
$712$	−24.7529	−0.927653
$713$	−0.953545	−0.0357105
$714$	1.58491	0.0593137
$715$	0	0
$716$	3.11349	0.116356
$717$	−0.136354	−0.00509225
$718$	0.452739	0.0168961
$719$	22.1897	0.827535	0.413767	−	0.910383i	$-0.364213\pi$
$719$	22.1897	0.827535	0.413767	+	0.910383i	$0.364213\pi$
$720$	0	0
$721$	−27.9022	−1.03913
$722$	−32.7313	−1.21813
$723$	0.0372080	0.00138378
$724$	−6.74842	−0.250803
$725$	0	0
$726$	0.317607	0.0117875
$727$	−32.5076	−1.20564	−0.602821	−	0.797877i	$-0.705957\pi$
$727$	−32.5076	−1.20564	−0.602821	+	0.797877i	$0.705957\pi$
$728$	18.3872	0.681475
$729$	−26.9253	−0.997232
$730$	0	0
$731$	−51.9712	−1.92222
$732$	0.193296	0.00714443
$733$	−38.8209	−1.43388	−0.716940	−	0.697134i	$-0.754458\pi$
$733$	−38.8209	−1.43388	−0.716940	+	0.697134i	$0.754458\pi$
$734$	−43.0014	−1.58721
$735$	0	0
$736$	1.60758	0.0592562
$737$	−45.0558	−1.65965
$738$	12.2028	0.449191
$739$	35.3834	1.30160	0.650799	−	0.759250i	$-0.274434\pi$
$739$	35.3834	1.30160	0.650799	+	0.759250i	$0.274434\pi$
$740$	0	0
$741$	−0.343939	−0.0126349
$742$	−38.1775	−1.40154
$743$	−33.0087	−1.21097	−0.605485	−	0.795857i	$-0.707021\pi$
$743$	−33.0087	−1.21097	−0.605485	+	0.795857i	$0.707021\pi$
$744$	0.119222	0.00437091
$745$	0	0
$746$	2.81193	0.102952
$747$	−2.20918	−0.0808298
$748$	−8.59962	−0.314433
$749$	−49.2432	−1.79931
$750$	0	0
$751$	−23.7981	−0.868405	−0.434202	−	0.900815i	$-0.642970\pi$
$751$	−23.7981	−0.868405	−0.434202	+	0.900815i	$0.642970\pi$
$752$	−60.3099	−2.19928
$753$	−0.204645	−0.00745769
$754$	−8.61752	−0.313832
$755$	0	0
$756$	0.439158	0.0159720
$757$	1.12334	0.0408283	0.0204142	−	0.999792i	$-0.493502\pi$
$757$	1.12334	0.0408283	0.0204142	+	0.999792i	$0.493502\pi$
$758$	−47.2456	−1.71604
$759$	0.112091	0.00406864
$760$	0	0
$761$	−11.2193	−0.406698	−0.203349	−	0.979106i	$-0.565183\pi$
$761$	−11.2193	−0.406698	−0.203349	+	0.979106i	$0.565183\pi$
$762$	0.257064	0.00931246
$763$	−7.34085	−0.265757
$764$	2.11359	0.0764668
$765$	0	0
$766$	−8.90656	−0.321807
$767$	−13.2027	−0.476722
$768$	−0.334789	−0.0120807
$769$	11.4429	0.412643	0.206321	−	0.978484i	$-0.433851\pi$
$769$	11.4429	0.412643	0.206321	+	0.978484i	$0.433851\pi$
$770$	0	0
$771$	0.313774	0.0113003
$772$	−3.04612	−0.109632
$773$	6.53383	0.235006	0.117503	−	0.993073i	$-0.462511\pi$
$773$	6.53383	0.235006	0.117503	+	0.993073i	$0.462511\pi$
$774$	−44.2447	−1.59034
$775$	0	0
$776$	−5.43728	−0.195187
$777$	1.19765	0.0429656
$778$	−8.38277	−0.300537
$779$	16.6901	0.597984
$780$	0	0
$781$	−64.4324	−2.30557
$782$	−6.23534	−0.222975
$783$	0.853386	0.0304975
$784$	−86.1987	−3.07852
$785$	0	0
$786$	1.00900	0.0359899
$787$	9.15036	0.326175	0.163088	−	0.986612i	$-0.447855\pi$
$787$	9.15036	0.326175	0.163088	+	0.986612i	$0.447855\pi$
$788$	−0.631141	−0.0224835
$789$	−0.282539	−0.0100587
$790$	0	0
$791$	11.5250	0.409783
$792$	30.3554	1.07863
$793$	−19.4939	−0.692249
$794$	−5.79850	−0.205781
$795$	0	0
$796$	−2.51849	−0.0892657
$797$	−0.534788	−0.0189432	−0.00947159	−	0.999955i	$-0.503015\pi$
$797$	−0.534788	−0.0189432	−0.00947159	+	0.999955i	$0.503015\pi$
$798$	1.84545	0.0653282
$799$	70.9673	2.51064
$800$	0	0
$801$	29.8042	1.05308
$802$	−25.7407	−0.908936
$803$	−9.96344	−0.351602
$804$	−0.160280	−0.00565264
$805$	0	0
$806$	2.89986	0.102143
$807$	−0.387683	−0.0136471
$808$	24.4799	0.861198
$809$	18.1162	0.636932	0.318466	−	0.947934i	$-0.396832\pi$
$809$	18.1162	0.636932	0.318466	+	0.947934i	$0.396832\pi$
$810$	0	0
$811$	20.1490	0.707528	0.353764	−	0.935335i	$-0.384902\pi$
$811$	20.1490	0.707528	0.353764	+	0.935335i	$0.384902\pi$
$812$	7.52302	0.264006
$813$	−0.955626	−0.0335153
$814$	−39.9410	−1.39993
$815$	0	0
$816$	0.937044	0.0328031
$817$	−60.5147	−2.11714
$818$	−30.1911	−1.05561
$819$	−22.1394	−0.773615
$820$	0	0
$821$	−11.4019	−0.397930	−0.198965	−	0.980007i	$-0.563758\pi$
$821$	−11.4019	−0.397930	−0.198965	+	0.980007i	$0.563758\pi$
$822$	0.849121	0.0296165
$823$	−11.6081	−0.404632	−0.202316	−	0.979320i	$-0.564847\pi$
$823$	−11.6081	−0.404632	−0.202316	+	0.979320i	$0.564847\pi$
$824$	−13.7250	−0.478131
$825$	0	0
$826$	70.8410	2.46487
$827$	30.5124	1.06102	0.530509	−	0.847679i	$-0.322001\pi$
$827$	30.5124	1.06102	0.530509	+	0.847679i	$0.322001\pi$
$828$	−0.863669	−0.0300146
$829$	19.6882	0.683801	0.341900	−	0.939736i	$-0.388929\pi$
$829$	19.6882	0.683801	0.341900	+	0.939736i	$0.388929\pi$
$830$	0	0
$831$	0.267291	0.00927223
$832$	8.60408	0.298293
$833$	101.431	3.51437
$834$	−0.837489	−0.0289999
$835$	0	0
$836$	−10.0133	−0.346318
$837$	−0.287170	−0.00992606
$838$	−23.1017	−0.798033
$839$	40.9176	1.41263	0.706316	−	0.707897i	$-0.250356\pi$
$839$	40.9176	1.41263	0.706316	+	0.707897i	$0.250356\pi$
$840$	0	0
$841$	−14.3810	−0.495898
$842$	−33.8242	−1.16566
$843$	0.296221	0.0102024
$844$	−5.62206	−0.193519
$845$	0	0
$846$	60.4167	2.07717
$847$	27.9625	0.960803
$848$	−22.5717	−0.775115
$849$	−0.0680538	−0.00233560
$850$	0	0
$851$	−4.71181	−0.161519
$852$	−0.229210	−0.00785259
$853$	1.08928	0.0372962	0.0186481	−	0.999826i	$-0.494064\pi$
$853$	1.08928	0.0372962	0.0186481	+	0.999826i	$0.494064\pi$
$854$	104.597	3.57924
$855$	0	0
$856$	−24.2225	−0.827909
$857$	−14.2908	−0.488165	−0.244083	−	0.969754i	$-0.578487\pi$
$857$	−14.2908	−0.488165	−0.244083	+	0.969754i	$0.578487\pi$
$858$	−0.340883	−0.0116376
$859$	50.2506	1.71453	0.857264	−	0.514877i	$-0.172163\pi$
$859$	50.2506	1.71453	0.857264	+	0.514877i	$0.172163\pi$
$860$	0	0
$861$	−0.496017	−0.0169042
$862$	−3.08653	−0.105128
$863$	−38.9315	−1.32524	−0.662621	−	0.748955i	$-0.730556\pi$
$863$	−38.9315	−1.32524	−0.662621	+	0.748955i	$0.730556\pi$
$864$	0.484140	0.0164708
$865$	0	0
$866$	−32.2529	−1.09600
$867$	−0.470094	−0.0159652
$868$	−2.53155	−0.0859263
$869$	2.06850	0.0701690
$870$	0	0
$871$	16.1642	0.547704
$872$	−3.61093	−0.122282
$873$	6.54686	0.221578
$874$	−7.26036	−0.245585
$875$	0	0
$876$	−0.0354436	−0.00119753
$877$	−38.5745	−1.30257	−0.651284	−	0.758834i	$-0.725769\pi$
$877$	−38.5745	−1.30257	−0.651284	+	0.758834i	$0.725769\pi$
$878$	8.39585	0.283346
$879$	0.778137	0.0262459
$880$	0	0
$881$	20.5491	0.692316	0.346158	−	0.938176i	$-0.387486\pi$
$881$	20.5491	0.692316	0.346158	+	0.938176i	$0.387486\pi$
$882$	86.3513	2.90760
$883$	−48.2503	−1.62375	−0.811876	−	0.583830i	$-0.801553\pi$
$883$	−48.2503	−1.62375	−0.811876	+	0.583830i	$0.801553\pi$
$884$	3.08520	0.103767
$885$	0	0
$886$	25.8018	0.866830
$887$	30.0044	1.00745	0.503725	−	0.863864i	$-0.331963\pi$
$887$	30.0044	1.00745	0.503725	+	0.863864i	$0.331963\pi$
$888$	0.589121	0.0197696
$889$	22.6322	0.759061
$890$	0	0
$891$	−36.5331	−1.22390
$892$	−11.3606	−0.380380
$893$	82.6336	2.76523
$894$	0.603926	0.0201983
$895$	0	0
$896$	−68.1303	−2.27607
$897$	−0.0402138	−0.00134270
$898$	43.3475	1.44653
$899$	−4.91938	−0.164070
$900$	0	0
$901$	26.5603	0.884853
$902$	16.5418	0.550782
$903$	1.79845	0.0598487
$904$	5.66912	0.188552
$905$	0	0
$906$	−0.620746	−0.0206229
$907$	31.9502	1.06089	0.530444	−	0.847720i	$-0.322025\pi$
$907$	31.9502	1.06089	0.530444	+	0.847720i	$0.322025\pi$
$908$	10.8089	0.358707
$909$	−29.4754	−0.977638
$910$	0	0
$911$	−49.3553	−1.63521	−0.817607	−	0.575776i	$-0.804700\pi$
$911$	−49.3553	−1.63521	−0.817607	+	0.575776i	$0.804700\pi$
$912$	1.09108	0.0361294
$913$	−2.99472	−0.0991107
$914$	−34.4721	−1.14024
$915$	0	0
$916$	−5.23869	−0.173091
$917$	88.8336	2.93354
$918$	−1.87784	−0.0619780
$919$	26.1704	0.863281	0.431640	−	0.902046i	$-0.357935\pi$
$919$	26.1704	0.863281	0.431640	+	0.902046i	$0.357935\pi$
$920$	0	0
$921$	0.869928	0.0286651
$922$	−39.4807	−1.30023
$923$	23.1158	0.760865
$924$	0.297588	0.00978993
$925$	0	0
$926$	−58.0394	−1.90729
$927$	16.5258	0.542778
$928$	8.29358	0.272250
$929$	−27.1479	−0.890692	−0.445346	−	0.895359i	$-0.646919\pi$
$929$	−27.1479	−0.890692	−0.445346	+	0.895359i	$0.646919\pi$
$930$	0	0
$931$	118.105	3.87074
$932$	2.46663	0.0807971
$933$	−0.0154237	−0.000504948 0
$934$	32.9664	1.07869
$935$	0	0
$936$	−10.8903	−0.355961
$937$	−27.1488	−0.886912	−0.443456	−	0.896296i	$-0.646248\pi$
$937$	−27.1488	−0.886912	−0.443456	+	0.896296i	$0.646248\pi$
$938$	−86.7314	−2.83188
$939$	−0.635944	−0.0207533
$940$	0	0
$941$	−34.7773	−1.13371	−0.566854	−	0.823818i	$-0.691840\pi$
$941$	−34.7773	−1.13371	−0.566854	+	0.823818i	$0.691840\pi$
$942$	1.17060	0.0381404
$943$	1.95143	0.0635472
$944$	41.8833	1.36318
$945$	0	0
$946$	−59.9771	−1.95002
$947$	−32.2262	−1.04721	−0.523606	−	0.851961i	$-0.675413\pi$
$947$	−32.2262	−1.04721	−0.523606	+	0.851961i	$0.675413\pi$
$948$	0.00735840	0.000238990 0
$949$	3.57449	0.116033
$950$	0	0
$951$	0.355192	0.0115179
$952$	68.6377	2.22456
$953$	34.0016	1.10142	0.550710	−	0.834697i	$-0.314357\pi$
$953$	34.0016	1.10142	0.550710	+	0.834697i	$0.314357\pi$
$954$	22.6116	0.732079
$955$	0	0
$956$	1.42420	0.0460619
$957$	0.578282	0.0186932
$958$	−3.86274	−0.124799
$959$	74.7575	2.41405
$960$	0	0
$961$	−29.3446	−0.946600
$962$	14.3292	0.461993
$963$	29.1656	0.939848
$964$	−0.388631	−0.0125170
$965$	0	0
$966$	0.215773	0.00694237
$967$	−56.2583	−1.80914	−0.904572	−	0.426320i	$-0.859810\pi$
$967$	−56.2583	−1.80914	−0.904572	+	0.426320i	$0.859810\pi$
$968$	13.7546	0.442091
$969$	−1.28389	−0.0412445
$970$	0	0
$971$	1.55423	0.0498778	0.0249389	−	0.999689i	$-0.492061\pi$
$971$	1.55423	0.0498778	0.0249389	+	0.999689i	$0.492061\pi$
$972$	−0.390185	−0.0125152
$973$	−73.7335	−2.36379
$974$	−50.0646	−1.60417
$975$	0	0
$976$	61.8410	1.97948
$977$	56.0044	1.79174	0.895870	−	0.444315i	$-0.146553\pi$
$977$	56.0044	1.79174	0.895870	+	0.444315i	$0.146553\pi$
$978$	−0.250953	−0.00802460
$979$	40.4019	1.29125
$980$	0	0
$981$	4.34781	0.138815
$982$	39.8722	1.27237
$983$	45.2376	1.44286	0.721428	−	0.692489i	$-0.243486\pi$
$983$	45.2376	1.44286	0.721428	+	0.692489i	$0.243486\pi$
$984$	−0.243988	−0.00777807
$985$	0	0
$986$	−32.1684	−1.02445
$987$	−2.45581	−0.0781692
$988$	3.59238	0.114289
$989$	−7.07546	−0.224987
$990$	0	0
$991$	−20.5506	−0.652811	−0.326406	−	0.945230i	$-0.605838\pi$
$991$	−20.5506	−0.652811	−0.326406	+	0.945230i	$0.605838\pi$
$992$	−2.79085	−0.0886095
$993$	−0.290582	−0.00922134
$994$	−124.031	−3.93402
$995$	0	0
$996$	−0.0106533	−0.000337563 0
$997$	16.0234	0.507466	0.253733	−	0.967274i	$-0.418342\pi$
$997$	16.0234	0.507466	0.253733	+	0.967274i	$0.418342\pi$
$998$	14.6401	0.463426
$999$	−1.41901	−0.0448956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.18		66
5.2	odd	4		1205.2.b.d.724.18	✓	66
5.3	odd	4		1205.2.b.d.724.49	yes	66
5.4	even	2	inner	6025.2.a.q.1.49		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.18	✓	66	5.2	odd	4
1205.2.b.d.724.49	yes	66	5.3	odd	4
6025.2.a.q.1.18		66	1.1	even	1	trivial
6025.2.a.q.1.49		66	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.54552 q^{2} -0.0372080 q^{3} +0.388631 q^{4} +0.0575057 q^{6} +5.06287 q^{7} +2.49040 q^{8} -2.99862 q^{9} +O(q^{10})\)	\(q-1.54552 q^{2} -0.0372080 q^{3} +0.388631 q^{4} +0.0575057 q^{6} +5.06287 q^{7} +2.49040 q^{8} -2.99862 q^{9} -4.06486 q^{11} -0.0144602 q^{12} +1.45831 q^{13} -7.82476 q^{14} -4.62623 q^{16} +5.44373 q^{17} +4.63442 q^{18} +6.33862 q^{19} -0.188379 q^{21} +6.28232 q^{22} +0.741121 q^{23} -0.0926629 q^{24} -2.25385 q^{26} +0.223196 q^{27} +1.96759 q^{28} +3.82347 q^{29} -1.28663 q^{31} +2.16912 q^{32} +0.151245 q^{33} -8.41340 q^{34} -1.16536 q^{36} -6.35768 q^{37} -9.79647 q^{38} -0.0542608 q^{39} +2.63308 q^{41} +0.291144 q^{42} -9.54698 q^{43} -1.57973 q^{44} -1.14542 q^{46} +13.0365 q^{47} +0.172133 q^{48} +18.6326 q^{49} -0.202550 q^{51} +0.566744 q^{52} +4.87907 q^{53} -0.344955 q^{54} +12.6086 q^{56} -0.235848 q^{57} -5.90925 q^{58} -9.05344 q^{59} -13.3675 q^{61} +1.98851 q^{62} -15.1816 q^{63} +5.90004 q^{64} -0.233752 q^{66} +11.0842 q^{67} +2.11560 q^{68} -0.0275756 q^{69} +15.8511 q^{71} -7.46776 q^{72} +2.45112 q^{73} +9.82593 q^{74} +2.46339 q^{76} -20.5798 q^{77} +0.0838611 q^{78} -0.508873 q^{79} +8.98754 q^{81} -4.06947 q^{82} +0.736734 q^{83} -0.0732100 q^{84} +14.7550 q^{86} -0.142264 q^{87} -10.1231 q^{88} -9.93931 q^{89} +7.38322 q^{91} +0.288023 q^{92} +0.0478728 q^{93} -20.1482 q^{94} -0.0807087 q^{96} -2.18329 q^{97} -28.7971 q^{98} +12.1889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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