LMFDB - Embedded newform 4477.2.a.t.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4477,2,Mod(1,4477)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4477.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4477, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4477 = 11^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4477.a (trivial)

Newform invariants

comment:select newform

sage:traces = [36,-8,-1,36,-1,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.7490249849$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 407)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4477.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.64401 q^{2} -0.209210 q^{3} +0.702767 q^{4} -1.61247 q^{5} +0.343944 q^{6} -3.77985 q^{7} +2.13266 q^{8} -2.95623 q^{9} +2.65091 q^{10} -0.147026 q^{12} -0.108499 q^{13} +6.21412 q^{14} +0.337345 q^{15} -4.91165 q^{16} +7.15834 q^{17} +4.86007 q^{18} -7.96186 q^{19} -1.13319 q^{20} +0.790785 q^{21} +6.93330 q^{23} -0.446175 q^{24} -2.39994 q^{25} +0.178373 q^{26} +1.24611 q^{27} -2.65636 q^{28} -1.42856 q^{29} -0.554599 q^{30} +2.12429 q^{31} +3.80948 q^{32} -11.7684 q^{34} +6.09490 q^{35} -2.07754 q^{36} -1.00000 q^{37} +13.0894 q^{38} +0.0226991 q^{39} -3.43885 q^{40} -10.5003 q^{41} -1.30006 q^{42} +6.55372 q^{43} +4.76683 q^{45} -11.3984 q^{46} +8.46332 q^{47} +1.02757 q^{48} +7.28730 q^{49} +3.94553 q^{50} -1.49760 q^{51} -0.0762495 q^{52} +5.26119 q^{53} -2.04861 q^{54} -8.06116 q^{56} +1.66570 q^{57} +2.34857 q^{58} +6.00752 q^{59} +0.237075 q^{60} -5.59592 q^{61} -3.49236 q^{62} +11.1741 q^{63} +3.56049 q^{64} +0.174951 q^{65} +4.55002 q^{67} +5.03065 q^{68} -1.45052 q^{69} -10.0201 q^{70} +7.19996 q^{71} -6.30464 q^{72} +6.56880 q^{73} +1.64401 q^{74} +0.502093 q^{75} -5.59534 q^{76} -0.0373175 q^{78} +2.50158 q^{79} +7.91989 q^{80} +8.60799 q^{81} +17.2626 q^{82} +4.32014 q^{83} +0.555738 q^{84} -11.5426 q^{85} -10.7744 q^{86} +0.298870 q^{87} -2.45641 q^{89} -7.83671 q^{90} +0.410110 q^{91} +4.87250 q^{92} -0.444424 q^{93} -13.9138 q^{94} +12.8383 q^{95} -0.796982 q^{96} -6.22646 q^{97} -11.9804 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 8 q^{2} - q^{3} + 36 q^{4} - q^{5} - 9 q^{6} - 19 q^{7} - 24 q^{8} + 31 q^{9} - 18 q^{10} - 23 q^{12} - 15 q^{13} + 13 q^{14} - 6 q^{15} + 40 q^{16} - 35 q^{17} - 11 q^{18} - 21 q^{19} + 10 q^{20}+ \cdots - 83 q^{98}+O(q^{100}) $ 36 * q - 8 * q^2 - q^3 + 36 * q^4 - q^5 - 9 * q^6 - 19 * q^7 - 24 * q^8 + 31 * q^9 - 18 * q^10 - 23 * q^12 - 15 * q^13 + 13 * q^14 - 6 * q^15 + 40 * q^16 - 35 * q^17 - 11 * q^18 - 21 * q^19 + 10 * q^20 - 22 * q^21 - 9 * q^23 - 33 * q^24 + 37 * q^25 + 3 * q^26 - 4 * q^27 - 56 * q^28 - 13 * q^29 + 12 * q^30 + 11 * q^31 - 61 * q^32 - 49 * q^34 - 61 * q^35 + 75 * q^36 - 36 * q^37 + 15 * q^38 - 16 * q^39 - 11 * q^40 - 45 * q^41 + 34 * q^42 - 38 * q^43 - 37 * q^45 - 28 * q^46 + 13 * q^47 - 46 * q^48 + 27 * q^49 - 8 * q^50 - 21 * q^51 - q^52 - 2 * q^53 - 41 * q^54 - 16 * q^56 - 47 * q^57 + 28 * q^58 - 8 * q^59 - 37 * q^60 - 5 * q^61 - 97 * q^62 - 56 * q^63 + 62 * q^64 - 36 * q^65 - 2 * q^67 - 105 * q^68 + 29 * q^69 - 24 * q^70 + 7 * q^71 - 22 * q^72 - 34 * q^73 + 8 * q^74 + 7 * q^75 - 10 * q^76 - 85 * q^78 - 22 * q^79 + 62 * q^80 - 56 * q^81 + 13 * q^82 - 166 * q^83 - 37 * q^84 - 43 * q^85 + 47 * q^86 - 26 * q^87 + 15 * q^89 - 29 * q^90 + 55 * q^91 + 19 * q^92 - 23 * q^93 - 13 * q^94 - 75 * q^95 - 14 * q^97 - 83 * q^98

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 $

$2$	−1.64401	−1.16249	−0.581245	−	0.813729i	$-0.697434\pi$
$2$	−1.64401	−1.16249	−0.581245	+	0.813729i	$0.697434\pi$
$3$	−0.209210	−0.120788	−0.0603938	−	0.998175i	$-0.519236\pi$
$3$	−0.209210	−0.120788	−0.0603938	+	0.998175i	$0.519236\pi$
$4$	0.702767	0.351384
$5$	−1.61247	−0.721118	−0.360559	−	0.932736i	$-0.617414\pi$
$5$	−1.61247	−0.721118	−0.360559	+	0.932736i	$0.617414\pi$
$6$	0.343944	0.140414
$7$	−3.77985	−1.42865	−0.714325	−	0.699814i	$-0.753266\pi$
$7$	−3.77985	−1.42865	−0.714325	+	0.699814i	$0.753266\pi$
$8$	2.13266	0.754010
$9$	−2.95623	−0.985410
$10$	2.65091	0.838293
$11$	0	0
$11$	0	0
$12$	−0.147026	−0.0424428
$13$	−0.108499	−0.0300922	−0.0150461	−	0.999887i	$-0.504789\pi$
$13$	−0.108499	−0.0300922	−0.0150461	+	0.999887i	$0.504789\pi$
$14$	6.21412	1.66079
$15$	0.337345	0.0871022
$16$	−4.91165	−1.22791
$17$	7.15834	1.73615	0.868076	−	0.496432i	$-0.165357\pi$
$17$	7.15834	1.73615	0.868076	+	0.496432i	$0.165357\pi$
$18$	4.86007	1.14553
$19$	−7.96186	−1.82658	−0.913288	−	0.407314i	$-0.866466\pi$
$19$	−7.96186	−1.82658	−0.913288	+	0.407314i	$0.866466\pi$
$20$	−1.13319	−0.253389
$21$	0.790785	0.172563
$22$	0	0
$23$	6.93330	1.44569	0.722847	−	0.691008i	$-0.242833\pi$
$23$	6.93330	1.44569	0.722847	+	0.691008i	$0.242833\pi$
$24$	−0.446175	−0.0910751
$25$	−2.39994	−0.479989
$26$	0.178373	0.0349819
$27$	1.24611	0.239813
$28$	−2.65636	−0.502005
$29$	−1.42856	−0.265278	−0.132639	−	0.991164i	$-0.542345\pi$
$29$	−1.42856	−0.265278	−0.132639	+	0.991164i	$0.542345\pi$
$30$	−0.554599	−0.101255
$31$	2.12429	0.381534	0.190767	−	0.981635i	$-0.438902\pi$
$31$	2.12429	0.381534	0.190767	+	0.981635i	$0.438902\pi$
$32$	3.80948	0.673427
$33$	0	0
$34$	−11.7684	−2.01826
$35$	6.09490	1.03023
$36$	−2.07754	−0.346257
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	13.0894	2.12338
$39$	0.0226991	0.00363476
$40$	−3.43885	−0.543730
$41$	−10.5003	−1.63987	−0.819934	−	0.572458i	$-0.805990\pi$
$41$	−10.5003	−1.63987	−0.819934	+	0.572458i	$0.805990\pi$
$42$	−1.30006	−0.200603
$43$	6.55372	0.999432	0.499716	−	0.866189i	$-0.333438\pi$
$43$	6.55372	0.999432	0.499716	+	0.866189i	$0.333438\pi$
$44$	0	0
$45$	4.76683	0.710597
$46$	−11.3984	−1.68060
$47$	8.46332	1.23450	0.617251	−	0.786766i	$-0.288246\pi$
$47$	8.46332	1.23450	0.617251	+	0.786766i	$0.288246\pi$
$48$	1.02757	0.148317
$49$	7.28730	1.04104
$50$	3.94553	0.557982
$51$	−1.49760	−0.209706
$52$	−0.0762495	−0.0105739
$53$	5.26119	0.722681	0.361340	−	0.932434i	$-0.382319\pi$
$53$	5.26119	0.722681	0.361340	+	0.932434i	$0.382319\pi$
$54$	−2.04861	−0.278780
$55$	0	0
$56$	−8.06116	−1.07722
$57$	1.66570	0.220628
$58$	2.34857	0.308383
$59$	6.00752	0.782113	0.391056	−	0.920367i	$-0.372110\pi$
$59$	6.00752	0.782113	0.391056	+	0.920367i	$0.372110\pi$
$60$	0.237075	0.0306063
$61$	−5.59592	−0.716485	−0.358242	−	0.933629i	$-0.616624\pi$
$61$	−5.59592	−0.716485	−0.358242	+	0.933629i	$0.616624\pi$
$62$	−3.49236	−0.443530
$63$	11.1741	1.40781
$64$	3.56049	0.445061
$65$	0.174951	0.0217000
$66$	0	0
$67$	4.55002	0.555873	0.277937	−	0.960599i	$-0.410349\pi$
$67$	4.55002	0.555873	0.277937	+	0.960599i	$0.410349\pi$
$68$	5.03065	0.610055
$69$	−1.45052	−0.174622
$70$	−10.0201	−1.19763
$71$	7.19996	0.854479	0.427239	−	0.904139i	$-0.359486\pi$
$71$	7.19996	0.854479	0.427239	+	0.904139i	$0.359486\pi$
$72$	−6.30464	−0.743009
$73$	6.56880	0.768819	0.384410	−	0.923163i	$-0.374405\pi$
$73$	6.56880	0.768819	0.384410	+	0.923163i	$0.374405\pi$
$74$	1.64401	0.191112
$75$	0.502093	0.0579767
$76$	−5.59534	−0.641829
$77$	0	0
$78$	−0.0373175	−0.00422538
$79$	2.50158	0.281450	0.140725	−	0.990049i	$-0.455057\pi$
$79$	2.50158	0.281450	0.140725	+	0.990049i	$0.455057\pi$
$80$	7.91989	0.885470
$81$	8.60799	0.956444
$82$	17.2626	1.90633
$83$	4.32014	0.474197	0.237098	−	0.971486i	$-0.423804\pi$
$83$	4.32014	0.474197	0.237098	+	0.971486i	$0.423804\pi$
$84$	0.555738	0.0606360
$85$	−11.5426	−1.25197
$86$	−10.7744	−1.16183
$87$	0.298870	0.0320423
$88$	0	0
$89$	−2.45641	−0.260379	−0.130189	−	0.991489i	$-0.541559\pi$
$89$	−2.45641	−0.260379	−0.130189	+	0.991489i	$0.541559\pi$
$90$	−7.83671	−0.826062
$91$	0.410110	0.0429912
$92$	4.87250	0.507993
$93$	−0.444424	−0.0460846
$94$	−13.9138	−1.43510
$95$	12.8383	1.31718
$96$	−0.796982	−0.0813417
$97$	−6.22646	−0.632201	−0.316101	−	0.948726i	$-0.602374\pi$
$97$	−6.22646	−0.632201	−0.316101	+	0.948726i	$0.602374\pi$
$98$	−11.9804	−1.21020
$99$	0	0
$100$	−1.68660	−0.168660
$101$	8.47257	0.843052	0.421526	−	0.906816i	$-0.361495\pi$
$101$	8.47257	0.843052	0.421526	+	0.906816i	$0.361495\pi$
$102$	2.46207	0.243781
$103$	−9.61381	−0.947277	−0.473638	−	0.880719i	$-0.657060\pi$
$103$	−9.61381	−0.947277	−0.473638	+	0.880719i	$0.657060\pi$
$104$	−0.231391	−0.0226898
$105$	−1.27512	−0.124439
$106$	−8.64945	−0.840109
$107$	−7.97682	−0.771148	−0.385574	−	0.922677i	$-0.625997\pi$
$107$	−7.97682	−0.771148	−0.385574	+	0.922677i	$0.625997\pi$
$108$	0.875722	0.0842664
$109$	−18.7754	−1.79835	−0.899177	−	0.437584i	$-0.855834\pi$
$109$	−18.7754	−1.79835	−0.899177	+	0.437584i	$0.855834\pi$
$110$	0	0
$111$	0.209210	0.0198574
$112$	18.5653	1.75426
$113$	19.9835	1.87989	0.939943	−	0.341332i	$-0.110878\pi$
$113$	19.9835	1.87989	0.939943	+	0.341332i	$0.110878\pi$
$114$	−2.73843	−0.256478
$115$	−11.1797	−1.04252
$116$	−1.00395	−0.0932142
$117$	0.320748	0.0296531
$118$	−9.87642	−0.909199
$119$	−27.0575	−2.48035
$120$	0.719444	0.0656759
$121$	0	0
$122$	9.19975	0.832906
$123$	2.19677	0.198076
$124$	1.49288	0.134065
$125$	11.9322	1.06725
$126$	−18.3704	−1.63656
$127$	3.97201	0.352459	0.176229	−	0.984349i	$-0.443610\pi$
$127$	3.97201	0.352459	0.176229	+	0.984349i	$0.443610\pi$
$128$	−13.4724	−1.19081
$129$	−1.37111	−0.120719
$130$	−0.287621	−0.0252260
$131$	10.4712	0.914869	0.457435	−	0.889243i	$-0.348768\pi$
$131$	10.4712	0.914869	0.457435	+	0.889243i	$0.348768\pi$
$132$	0	0
$133$	30.0947	2.60954
$134$	−7.48027	−0.646197
$135$	−2.00931	−0.172934
$136$	15.2663	1.30908
$137$	17.1817	1.46793	0.733965	−	0.679187i	$-0.237668\pi$
$137$	17.1817	1.46793	0.733965	+	0.679187i	$0.237668\pi$
$138$	2.38467	0.202996
$139$	−2.21550	−0.187917	−0.0939583	−	0.995576i	$-0.529952\pi$
$139$	−2.21550	−0.187917	−0.0939583	+	0.995576i	$0.529952\pi$
$140$	4.28330	0.362005
$141$	−1.77061	−0.149113
$142$	−11.8368	−0.993323
$143$	0	0
$144$	14.5200	1.21000
$145$	2.30351	0.191296
$146$	−10.7992	−0.893745
$147$	−1.52458	−0.125745
$148$	−0.702767	−0.0577671
$149$	−7.10223	−0.581838	−0.290919	−	0.956748i	$-0.593961\pi$
$149$	−7.10223	−0.581838	−0.290919	+	0.956748i	$0.593961\pi$
$150$	−0.825446	−0.0673974
$151$	−3.00728	−0.244729	−0.122365	−	0.992485i	$-0.539048\pi$
$151$	−3.00728	−0.244729	−0.122365	+	0.992485i	$0.539048\pi$
$152$	−16.9800	−1.37726
$153$	−21.1617	−1.71082
$154$	0	0
$155$	−3.42536	−0.275131
$156$	0.0159522	0.00127720
$157$	−3.02926	−0.241761	−0.120881	−	0.992667i	$-0.538572\pi$
$157$	−3.02926	−0.241761	−0.120881	+	0.992667i	$0.538572\pi$
$158$	−4.11262	−0.327182
$159$	−1.10070	−0.0872909
$160$	−6.14267	−0.485620
$161$	−26.2069	−2.06539
$162$	−14.1516	−1.11186
$163$	0.514392	0.0402903	0.0201451	−	0.999797i	$-0.493587\pi$
$163$	0.514392	0.0402903	0.0201451	+	0.999797i	$0.493587\pi$
$164$	−7.37925	−0.576223
$165$	0	0
$166$	−7.10235	−0.551249
$167$	−19.6192	−1.51818	−0.759088	−	0.650988i	$-0.774355\pi$
$167$	−19.6192	−1.51818	−0.759088	+	0.650988i	$0.774355\pi$
$168$	1.68648	0.130115
$169$	−12.9882	−0.999094
$170$	18.9761	1.45540
$171$	23.5371	1.79993
$172$	4.60574	0.351184
$173$	−2.04598	−0.155553	−0.0777766	−	0.996971i	$-0.524782\pi$
$173$	−2.04598	−0.155553	−0.0777766	+	0.996971i	$0.524782\pi$
$174$	−0.491346	−0.0372488
$175$	9.07144	0.685737
$176$	0	0
$177$	−1.25684	−0.0944696
$178$	4.03836	0.302688
$179$	−10.7635	−0.804504	−0.402252	−	0.915529i	$-0.631772\pi$
$179$	−10.7635	−0.804504	−0.402252	+	0.915529i	$0.631772\pi$
$180$	3.34997	0.249692
$181$	−16.6906	−1.24060	−0.620301	−	0.784364i	$-0.712990\pi$
$181$	−16.6906	−1.24060	−0.620301	+	0.784364i	$0.712990\pi$
$182$	−0.674225	−0.0499769
$183$	1.17073	0.0865425
$184$	14.7864	1.09007
$185$	1.61247	0.118551
$186$	0.730637	0.0535729
$187$	0	0
$188$	5.94775	0.433784
$189$	−4.71010	−0.342609
$190$	−21.1062	−1.53121
$191$	5.47013	0.395805	0.197902	−	0.980222i	$-0.436587\pi$
$191$	5.47013	0.395805	0.197902	+	0.980222i	$0.436587\pi$
$192$	−0.744891	−0.0537579
$193$	−1.15158	−0.0828928	−0.0414464	−	0.999141i	$-0.513197\pi$
$193$	−1.15158	−0.0828928	−0.0414464	+	0.999141i	$0.513197\pi$
$194$	10.2364	0.734928
$195$	−0.0366016	−0.00262109
$196$	5.12128	0.365806
$197$	1.08302	0.0771621	0.0385811	−	0.999255i	$-0.487716\pi$
$197$	1.08302	0.0771621	0.0385811	+	0.999255i	$0.487716\pi$
$198$	0	0
$199$	−12.0853	−0.856705	−0.428352	−	0.903612i	$-0.640906\pi$
$199$	−12.0853	−0.856705	−0.428352	+	0.903612i	$0.640906\pi$
$200$	−5.11827	−0.361916
$201$	−0.951911	−0.0671426
$202$	−13.9290	−0.980040
$203$	5.39976	0.378989
$204$	−1.05246	−0.0736872
$205$	16.9314	1.18254
$206$	15.8052	1.10120
$207$	−20.4964	−1.42460
$208$	0.532909	0.0369506
$209$	0	0
$210$	2.09630	0.144659
$211$	8.28912	0.570646	0.285323	−	0.958431i	$-0.407899\pi$
$211$	8.28912	0.570646	0.285323	+	0.958431i	$0.407899\pi$
$212$	3.69740	0.253938
$213$	−1.50631	−0.103210
$214$	13.1140	0.896452
$215$	−10.5677	−0.720708
$216$	2.65752	0.180822
$217$	−8.02952	−0.545079
$218$	30.8669	2.09057
$219$	−1.37426	−0.0928639
$220$	0	0
$221$	−0.776671	−0.0522446
$222$	−0.343944	−0.0230840
$223$	16.2534	1.08841	0.544204	−	0.838953i	$-0.316832\pi$
$223$	16.2534	1.08841	0.544204	+	0.838953i	$0.316832\pi$
$224$	−14.3993	−0.962092
$225$	7.09479	0.472986
$226$	−32.8530	−2.18535
$227$	−28.4779	−1.89014	−0.945071	−	0.326864i	$-0.894008\pi$
$227$	−28.4779	−1.89014	−0.945071	+	0.326864i	$0.894008\pi$
$228$	1.17060	0.0775251
$229$	17.7258	1.17135	0.585677	−	0.810545i	$-0.300829\pi$
$229$	17.7258	1.17135	0.585677	+	0.810545i	$0.300829\pi$
$230$	18.3796	1.21191
$231$	0	0
$232$	−3.04664	−0.200022
$233$	−16.2405	−1.06395	−0.531974	−	0.846761i	$-0.678549\pi$
$233$	−16.2405	−1.06395	−0.531974	+	0.846761i	$0.678549\pi$
$234$	−0.527312	−0.0344715
$235$	−13.6468	−0.890222
$236$	4.22189	0.274822
$237$	−0.523356	−0.0339956
$238$	44.4827	2.88339
$239$	2.17650	0.140786	0.0703931	−	0.997519i	$-0.477575\pi$
$239$	2.17650	0.140786	0.0703931	+	0.997519i	$0.477575\pi$
$240$	−1.65692	−0.106954
$241$	14.1945	0.914347	0.457174	−	0.889377i	$-0.348862\pi$
$241$	14.1945	0.914347	0.457174	+	0.889377i	$0.348862\pi$
$242$	0	0
$243$	−5.53920	−0.355340
$244$	−3.93263	−0.251761
$245$	−11.7505	−0.750715
$246$	−3.61151	−0.230261
$247$	0.863853	0.0549657
$248$	4.53040	0.287681
$249$	−0.903818	−0.0572771
$250$	−19.6166	−1.24066
$251$	−27.4378	−1.73186	−0.865930	−	0.500165i	$-0.833273\pi$
$251$	−27.4378	−1.73186	−0.865930	+	0.500165i	$0.833273\pi$
$252$	7.85281	0.494681
$253$	0	0
$254$	−6.53002	−0.409730
$255$	2.41483	0.151223
$256$	15.0278	0.939239
$257$	8.71562	0.543666	0.271833	−	0.962344i	$-0.412370\pi$
$257$	8.71562	0.543666	0.271833	+	0.962344i	$0.412370\pi$
$258$	2.25411	0.140335
$259$	3.77985	0.234869
$260$	0.122950	0.00762503
$261$	4.22316	0.261407
$262$	−17.2147	−1.06353
$263$	−3.71027	−0.228785	−0.114392	−	0.993436i	$-0.536492\pi$
$263$	−3.71027	−0.228785	−0.114392	+	0.993436i	$0.536492\pi$
$264$	0	0
$265$	−8.48351	−0.521138
$266$	−49.4760	−3.03357
$267$	0.513906	0.0314505
$268$	3.19760	0.195325
$269$	20.1896	1.23098	0.615491	−	0.788144i	$-0.288958\pi$
$269$	20.1896	1.23098	0.615491	+	0.788144i	$0.288958\pi$
$270$	3.30332	0.201034
$271$	−27.3489	−1.66133	−0.830663	−	0.556775i	$-0.812039\pi$
$271$	−27.3489	−1.66133	−0.830663	+	0.556775i	$0.812039\pi$
$272$	−35.1593	−2.13184
$273$	−0.0857993	−0.00519281
$274$	−28.2469	−1.70645
$275$	0	0
$276$	−1.01938	−0.0613593
$277$	−1.52345	−0.0915353	−0.0457676	−	0.998952i	$-0.514573\pi$
$277$	−1.52345	−0.0915353	−0.0457676	+	0.998952i	$0.514573\pi$
$278$	3.64231	0.218451
$279$	−6.27990	−0.375968
$280$	12.9984	0.776801
$281$	18.9532	1.13065	0.565327	−	0.824867i	$-0.308750\pi$
$281$	18.9532	1.13065	0.565327	+	0.824867i	$0.308750\pi$
$282$	2.91091	0.173342
$283$	26.7239	1.58857	0.794285	−	0.607546i	$-0.207846\pi$
$283$	26.7239	1.58857	0.794285	+	0.607546i	$0.207846\pi$
$284$	5.05990	0.300250
$285$	−2.68590	−0.159099
$286$	0	0
$287$	39.6895	2.34280
$288$	−11.2617	−0.663602
$289$	34.2418	2.01422
$290$	−3.78700	−0.222380
$291$	1.30264	0.0763621
$292$	4.61633	0.270151
$293$	5.03131	0.293932	0.146966	−	0.989142i	$-0.453049\pi$
$293$	5.03131	0.293932	0.146966	+	0.989142i	$0.453049\pi$
$294$	2.50642	0.146178
$295$	−9.68694	−0.563996
$296$	−2.13266	−0.123959
$297$	0	0
$298$	11.6761	0.676381
$299$	−0.752255	−0.0435041
$300$	0.352855	0.0203721
$301$	−24.7721	−1.42784
$302$	4.94400	0.284495
$303$	−1.77255	−0.101830
$304$	39.1059	2.24288
$305$	9.02325	0.516670
$306$	34.7900	1.98881
$307$	−4.76205	−0.271785	−0.135892	−	0.990724i	$-0.543390\pi$
$307$	−4.76205	−0.271785	−0.135892	+	0.990724i	$0.543390\pi$
$308$	0	0
$309$	2.01131	0.114419
$310$	5.63132	0.319837
$311$	−9.32853	−0.528972	−0.264486	−	0.964390i	$-0.585202\pi$
$311$	−9.32853	−0.528972	−0.264486	+	0.964390i	$0.585202\pi$
$312$	0.0484095	0.00274065
$313$	−14.1098	−0.797533	−0.398766	−	0.917053i	$-0.630562\pi$
$313$	−14.1098	−0.797533	−0.398766	+	0.917053i	$0.630562\pi$
$314$	4.98014	0.281045
$315$	−18.0179	−1.01520
$316$	1.75803	0.0988968
$317$	−19.3172	−1.08496	−0.542480	−	0.840069i	$-0.682515\pi$
$317$	−19.3172	−1.08496	−0.542480	+	0.840069i	$0.682515\pi$
$318$	1.80956	0.101475
$319$	0	0
$320$	−5.74117	−0.320941
$321$	1.66883	0.0931452
$322$	43.0844	2.40100
$323$	−56.9937	−3.17121
$324$	6.04942	0.336079
$325$	0.260391	0.0144439
$326$	−0.845665	−0.0468370
$327$	3.92800	0.217219
$328$	−22.3936	−1.23648
$329$	−31.9901	−1.76367
$330$	0	0
$331$	−15.4651	−0.850039	−0.425019	−	0.905184i	$-0.639733\pi$
$331$	−15.4651	−0.850039	−0.425019	+	0.905184i	$0.639733\pi$
$332$	3.03605	0.166625
$333$	2.95623	0.162000
$334$	32.2541	1.76486
$335$	−7.33676	−0.400850
$336$	−3.88406	−0.211893
$337$	29.2752	1.59472	0.797360	−	0.603504i	$-0.206229\pi$
$337$	29.2752	1.59472	0.797360	+	0.603504i	$0.206229\pi$
$338$	21.3528	1.16144
$339$	−4.18075	−0.227067
$340$	−8.11176	−0.439922
$341$	0	0
$342$	−38.6952	−2.09240
$343$	−1.08597	−0.0586367
$344$	13.9769	0.753582
$345$	2.33892	0.125923
$346$	3.36361	0.180829
$347$	−27.5904	−1.48113	−0.740565	−	0.671984i	$-0.765442\pi$
$347$	−27.5904	−1.48113	−0.740565	+	0.671984i	$0.765442\pi$
$348$	0.210036	0.0112591
$349$	−6.70826	−0.359085	−0.179542	−	0.983750i	$-0.557462\pi$
$349$	−6.70826	−0.359085	−0.179542	+	0.983750i	$0.557462\pi$
$350$	−14.9135	−0.797162
$351$	−0.135201	−0.00721650
$352$	0	0
$353$	−2.68824	−0.143080	−0.0715402	−	0.997438i	$-0.522791\pi$
$353$	−2.68824	−0.143080	−0.0715402	+	0.997438i	$0.522791\pi$
$354$	2.06625	0.109820
$355$	−11.6097	−0.616180
$356$	−1.72628	−0.0914929
$357$	5.66070	0.299596
$358$	17.6953	0.935228
$359$	−24.5827	−1.29743	−0.648714	−	0.761032i	$-0.724693\pi$
$359$	−24.5827	−1.29743	−0.648714	+	0.761032i	$0.724693\pi$
$360$	10.1660	0.535797
$361$	44.3913	2.33638
$362$	27.4395	1.44219
$363$	0	0
$364$	0.288212	0.0151064
$365$	−10.5920	−0.554409
$366$	−1.92468	−0.100605
$367$	−11.3782	−0.593935	−0.296967	−	0.954888i	$-0.595975\pi$
$367$	−11.3782	−0.593935	−0.296967	+	0.954888i	$0.595975\pi$
$368$	−34.0540	−1.77519
$369$	31.0413	1.61594
$370$	−2.65091	−0.137814
$371$	−19.8866	−1.03246
$372$	−0.312327	−0.0161934
$373$	−8.86333	−0.458926	−0.229463	−	0.973317i	$-0.573697\pi$
$373$	−8.86333	−0.458926	−0.229463	+	0.973317i	$0.573697\pi$
$374$	0	0
$375$	−2.49634	−0.128910
$376$	18.0494	0.930827
$377$	0.154997	0.00798278
$378$	7.74345	0.398280
$379$	−22.6288	−1.16236	−0.581181	−	0.813774i	$-0.697409\pi$
$379$	−22.6288	−1.16236	−0.581181	+	0.813774i	$0.697409\pi$
$380$	9.02231	0.462835
$381$	−0.830985	−0.0425727
$382$	−8.99295	−0.460119
$383$	27.9880	1.43012	0.715059	−	0.699064i	$-0.246400\pi$
$383$	27.9880	1.43012	0.715059	+	0.699064i	$0.246400\pi$
$384$	2.81857	0.143835
$385$	0	0
$386$	1.89322	0.0963621
$387$	−19.3743	−0.984851
$388$	−4.37575	−0.222145
$389$	34.4584	1.74711	0.873556	−	0.486723i	$-0.161808\pi$
$389$	34.4584	1.74711	0.873556	+	0.486723i	$0.161808\pi$
$390$	0.0601733	0.00304699
$391$	49.6309	2.50994
$392$	15.5414	0.784957
$393$	−2.19067	−0.110505
$394$	−1.78050	−0.0897002
$395$	−4.03372	−0.202958
$396$	0	0
$397$	1.42657	0.0715977	0.0357989	−	0.999359i	$-0.488602\pi$
$397$	1.42657	0.0715977	0.0357989	+	0.999359i	$0.488602\pi$
$398$	19.8684	0.995911
$399$	−6.29612	−0.315200
$400$	11.7877	0.589385
$401$	−37.4560	−1.87046	−0.935232	−	0.354035i	$-0.884809\pi$
$401$	−37.4560	−1.87046	−0.935232	+	0.354035i	$0.884809\pi$
$402$	1.56495	0.0780527
$403$	−0.230483	−0.0114812
$404$	5.95424	0.296235
$405$	−13.8801	−0.689709
$406$	−8.87726	−0.440571
$407$	0	0
$408$	−3.19387	−0.158120
$409$	27.2085	1.34538	0.672688	−	0.739927i	$-0.265140\pi$
$409$	27.2085	1.34538	0.672688	+	0.739927i	$0.265140\pi$
$410$	−27.8353	−1.37469
$411$	−3.59459	−0.177308
$412$	−6.75627	−0.332858
$413$	−22.7076	−1.11737
$414$	33.6964	1.65609
$415$	−6.96609	−0.341952
$416$	−0.413324	−0.0202649
$417$	0.463507	0.0226980
$418$	0	0
$419$	−33.0069	−1.61249	−0.806245	−	0.591581i	$-0.798504\pi$
$419$	−33.0069	−1.61249	−0.806245	+	0.591581i	$0.798504\pi$
$420$	−0.896110	−0.0437257
$421$	−3.38690	−0.165067	−0.0825336	−	0.996588i	$-0.526301\pi$
$421$	−3.38690	−0.165067	−0.0825336	+	0.996588i	$0.526301\pi$
$422$	−13.6274	−0.663371
$423$	−25.0195	−1.21649
$424$	11.2204	0.544909
$425$	−17.1796	−0.833333
$426$	2.47638	0.119981
$427$	21.1518	1.02361
$428$	−5.60585	−0.270969
$429$	0	0
$430$	17.3733	0.837817
$431$	23.5860	1.13610	0.568049	−	0.822994i	$-0.307698\pi$
$431$	23.5860	1.13610	0.568049	+	0.822994i	$0.307698\pi$
$432$	−6.12044	−0.294470
$433$	0.899198	0.0432127	0.0216064	−	0.999767i	$-0.493122\pi$
$433$	0.899198	0.0432127	0.0216064	+	0.999767i	$0.493122\pi$
$434$	13.2006	0.633649
$435$	−0.481919	−0.0231063
$436$	−13.1947	−0.631913
$437$	−55.2020	−2.64067
$438$	2.25930	0.107953
$439$	−28.2600	−1.34878	−0.674389	−	0.738376i	$-0.735593\pi$
$439$	−28.2600	−1.34878	−0.674389	+	0.738376i	$0.735593\pi$
$440$	0	0
$441$	−21.5430	−1.02585
$442$	1.27685	0.0607338
$443$	−7.45487	−0.354192	−0.177096	−	0.984194i	$-0.556670\pi$
$443$	−7.45487	−0.354192	−0.177096	+	0.984194i	$0.556670\pi$
$444$	0.147026	0.00697756
$445$	3.96088	0.187764
$446$	−26.7207	−1.26526
$447$	1.48586	0.0702788
$448$	−13.4581	−0.635837
$449$	−1.25450	−0.0592033	−0.0296017	−	0.999562i	$-0.509424\pi$
$449$	−1.25450	−0.0592033	−0.0296017	+	0.999562i	$0.509424\pi$
$450$	−11.6639	−0.549842
$451$	0	0
$452$	14.0437	0.660561
$453$	0.629155	0.0295603
$454$	46.8179	2.19727
$455$	−0.661289	−0.0310017
$456$	3.55239	0.166356
$457$	22.7608	1.06471	0.532353	−	0.846523i	$-0.321308\pi$
$457$	22.7608	1.06471	0.532353	+	0.846523i	$0.321308\pi$
$458$	−29.1414	−1.36169
$459$	8.92004	0.416352
$460$	−7.85675	−0.366323
$461$	−28.8239	−1.34246	−0.671231	−	0.741248i	$-0.734234\pi$
$461$	−28.8239	−1.34246	−0.671231	+	0.741248i	$0.734234\pi$
$462$	0	0
$463$	−17.8976	−0.831770	−0.415885	−	0.909417i	$-0.636528\pi$
$463$	−17.8976	−0.831770	−0.415885	+	0.909417i	$0.636528\pi$
$464$	7.01661	0.325738
$465$	0.716620	0.0332325
$466$	26.6995	1.23683
$467$	−2.26899	−0.104997	−0.0524983	−	0.998621i	$-0.516718\pi$
$467$	−2.26899	−0.104997	−0.0524983	+	0.998621i	$0.516718\pi$
$468$	0.225411	0.0104196
$469$	−17.1984	−0.794149
$470$	22.4355	1.03487
$471$	0.633753	0.0292018
$472$	12.8120	0.589721
$473$	0	0
$474$	0.860403	0.0395196
$475$	19.1080	0.876737
$476$	−19.0151	−0.871556
$477$	−15.5533	−0.712137
$478$	−3.57819	−0.163662
$479$	−35.6541	−1.62908	−0.814538	−	0.580110i	$-0.803009\pi$
$479$	−35.6541	−1.62908	−0.814538	+	0.580110i	$0.803009\pi$
$480$	1.28511	0.0586569
$481$	0.108499	0.00494712
$482$	−23.3359	−1.06292
$483$	5.48275	0.249474
$484$	0	0
$485$	10.0400	0.455892
$486$	9.10649	0.413079
$487$	4.01012	0.181716	0.0908579	−	0.995864i	$-0.471039\pi$
$487$	4.01012	0.181716	0.0908579	+	0.995864i	$0.471039\pi$
$488$	−11.9342	−0.540237
$489$	−0.107616	−0.00486657
$490$	19.3180	0.872699
$491$	10.5968	0.478228	0.239114	−	0.970991i	$-0.423143\pi$
$491$	10.5968	0.478228	0.239114	+	0.970991i	$0.423143\pi$
$492$	1.54382	0.0696006
$493$	−10.2261	−0.460562
$494$	−1.42018	−0.0638970
$495$	0	0
$496$	−10.4338	−0.468491
$497$	−27.2148	−1.22075
$498$	1.48588	0.0665841
$499$	4.42225	0.197967	0.0989836	−	0.995089i	$-0.468441\pi$
$499$	4.42225	0.197967	0.0989836	+	0.995089i	$0.468441\pi$
$500$	8.38555	0.375013
$501$	4.10453	0.183377
$502$	45.1080	2.01327
$503$	−9.92948	−0.442734	−0.221367	−	0.975191i	$-0.571052\pi$
$503$	−9.92948	−0.442734	−0.221367	+	0.975191i	$0.571052\pi$
$504$	23.8306	1.06150
$505$	−13.6618	−0.607940
$506$	0	0
$507$	2.71727	0.120678
$508$	2.79140	0.123848
$509$	−7.07542	−0.313613	−0.156806	−	0.987629i	$-0.550120\pi$
$509$	−7.07542	−0.313613	−0.156806	+	0.987629i	$0.550120\pi$
$510$	−3.97000	−0.175795
$511$	−24.8291	−1.09837
$512$	2.23896	0.0989491
$513$	−9.92132	−0.438037
$514$	−14.3286	−0.632006
$515$	15.5020	0.683098
$516$	−0.963568	−0.0424187
$517$	0	0
$518$	−6.21412	−0.273033
$519$	0.428041	0.0187889
$520$	0.373111	0.0163620
$521$	11.0745	0.485182	0.242591	−	0.970129i	$-0.422003\pi$
$521$	11.0745	0.485182	0.242591	+	0.970129i	$0.422003\pi$
$522$	−6.94292	−0.303883
$523$	−34.8624	−1.52443	−0.762213	−	0.647326i	$-0.775887\pi$
$523$	−34.8624	−1.52443	−0.762213	+	0.647326i	$0.775887\pi$
$524$	7.35879	0.321470
$525$	−1.89784	−0.0828285
$526$	6.09971	0.265960
$527$	15.2064	0.662401
$528$	0	0
$529$	25.0707	1.09003
$530$	13.9470	0.605818
$531$	−17.7596	−0.770702
$532$	21.1496	0.916950
$533$	1.13927	0.0493472
$534$	−0.844867	−0.0365610
$535$	12.8624	0.556089
$536$	9.70366	0.419134
$537$	2.25184	0.0971741
$538$	−33.1919	−1.43101
$539$	0	0
$540$	−1.41207	−0.0607660
$541$	−45.9086	−1.97377	−0.986883	−	0.161440i	$-0.948386\pi$
$541$	−45.9086	−1.97377	−0.986883	+	0.161440i	$0.948386\pi$
$542$	44.9618	1.93128
$543$	3.49185	0.149850
$544$	27.2695	1.16917
$545$	30.2747	1.29683
$546$	0.141055	0.00603659
$547$	8.59546	0.367515	0.183758	−	0.982972i	$-0.441174\pi$
$547$	8.59546	0.367515	0.183758	+	0.982972i	$0.441174\pi$
$548$	12.0747	0.515807
$549$	16.5428	0.706031
$550$	0	0
$551$	11.3740	0.484550
$552$	−3.09347	−0.131667
$553$	−9.45560	−0.402093
$554$	2.50457	0.106409
$555$	−0.337345	−0.0143195
$556$	−1.55698	−0.0660308
$557$	−7.97902	−0.338082	−0.169041	−	0.985609i	$-0.554067\pi$
$557$	−7.97902	−0.338082	−0.169041	+	0.985609i	$0.554067\pi$
$558$	10.3242	0.437059
$559$	−0.711071	−0.0300751
$560$	−29.9360	−1.26503
$561$	0	0
$562$	−31.1593	−1.31437
$563$	−12.6509	−0.533173	−0.266586	−	0.963811i	$-0.585896\pi$
$563$	−12.6509	−0.533173	−0.266586	+	0.963811i	$0.585896\pi$
$564$	−1.24433	−0.0523958
$565$	−32.2227	−1.35562
$566$	−43.9343	−1.84670
$567$	−32.5370	−1.36642
$568$	15.3551	0.644286
$569$	17.7233	0.743000	0.371500	−	0.928433i	$-0.378844\pi$
$569$	17.7233	0.743000	0.371500	+	0.928433i	$0.378844\pi$
$570$	4.41564	0.184951
$571$	41.1570	1.72237	0.861183	−	0.508296i	$-0.169724\pi$
$571$	41.1570	1.72237	0.861183	+	0.508296i	$0.169724\pi$
$572$	0	0
$573$	−1.14441	−0.0478083
$574$	−65.2500	−2.72348
$575$	−16.6395	−0.693917
$576$	−10.5256	−0.438568
$577$	−29.8053	−1.24081	−0.620406	−	0.784281i	$-0.713032\pi$
$577$	−29.8053	−1.24081	−0.620406	+	0.784281i	$0.713032\pi$
$578$	−56.2938	−2.34151
$579$	0.240923	0.0100124
$580$	1.61883	0.0672184
$581$	−16.3295	−0.677461
$582$	−2.14155	−0.0887702
$583$	0	0
$584$	14.0090	0.579698
$585$	−0.517196	−0.0213834
$586$	−8.27152	−0.341694
$587$	2.45373	0.101276	0.0506381	−	0.998717i	$-0.483874\pi$
$587$	2.45373	0.101276	0.0506381	+	0.998717i	$0.483874\pi$
$588$	−1.07142	−0.0441848
$589$	−16.9133	−0.696902
$590$	15.9254	0.655639
$591$	−0.226580	−0.00932024
$592$	4.91165	0.201868
$593$	10.1115	0.415230	0.207615	−	0.978211i	$-0.433430\pi$
$593$	10.1115	0.415230	0.207615	+	0.978211i	$0.433430\pi$
$594$	0	0
$595$	43.6293	1.78863
$596$	−4.99122	−0.204448
$597$	2.52837	0.103479
$598$	1.23671	0.0505730
$599$	−11.5565	−0.472187	−0.236094	−	0.971730i	$-0.575867\pi$
$599$	−11.5565	−0.472187	−0.236094	+	0.971730i	$0.575867\pi$
$600$	1.07080	0.0437151
$601$	−7.20310	−0.293821	−0.146910	−	0.989150i	$-0.546933\pi$
$601$	−7.20310	−0.293821	−0.146910	+	0.989150i	$0.546933\pi$
$602$	40.7256	1.65985
$603$	−13.4509	−0.547763
$604$	−2.11342	−0.0859939
$605$	0	0
$606$	2.91409	0.118377
$607$	20.4866	0.831526	0.415763	−	0.909473i	$-0.363515\pi$
$607$	20.4866	0.831526	0.415763	+	0.909473i	$0.363515\pi$
$608$	−30.3305	−1.23007
$609$	−1.12969	−0.0457772
$610$	−14.8343	−0.600624
$611$	−0.918261	−0.0371488
$612$	−14.8717	−0.601155
$613$	11.2723	0.455284	0.227642	−	0.973745i	$-0.426898\pi$
$613$	11.2723	0.455284	0.227642	+	0.973745i	$0.426898\pi$
$614$	7.82886	0.315947
$615$	−3.54222	−0.142836
$616$	0	0
$617$	−19.6982	−0.793021	−0.396511	−	0.918030i	$-0.629779\pi$
$617$	−19.6982	−0.793021	−0.396511	+	0.918030i	$0.629779\pi$
$618$	−3.30661	−0.133011
$619$	−12.2492	−0.492336	−0.246168	−	0.969227i	$-0.579172\pi$
$619$	−12.2492	−0.492336	−0.246168	+	0.969227i	$0.579172\pi$
$620$	−2.40723	−0.0966766
$621$	8.63963	0.346696
$622$	15.3362	0.614925
$623$	9.28487	0.371990
$624$	−0.111490	−0.00446317
$625$	−7.24054	−0.289622
$626$	23.1966	0.927124
$627$	0	0
$628$	−2.12887	−0.0849510
$629$	−7.15834	−0.285422
$630$	29.6216	1.18015
$631$	−2.08484	−0.0829963	−0.0414982	−	0.999139i	$-0.513213\pi$
$631$	−2.08484	−0.0829963	−0.0414982	+	0.999139i	$0.513213\pi$
$632$	5.33502	0.212216
$633$	−1.73417	−0.0689271
$634$	31.7576	1.26126
$635$	−6.40474	−0.254164
$636$	−0.773534	−0.0306726
$637$	−0.790664	−0.0313273
$638$	0	0
$639$	−21.2848	−0.842012
$640$	21.7239	0.858712
$641$	−19.8896	−0.785593	−0.392796	−	0.919625i	$-0.628492\pi$
$641$	−19.8896	−0.785593	−0.392796	+	0.919625i	$0.628492\pi$
$642$	−2.74358	−0.108280
$643$	20.3231	0.801464	0.400732	−	0.916195i	$-0.368756\pi$
$643$	20.3231	0.801464	0.400732	+	0.916195i	$0.368756\pi$
$644$	−18.4173	−0.725745
$645$	2.21086	0.0870527
$646$	93.6982	3.68651
$647$	−17.0321	−0.669603	−0.334801	−	0.942289i	$-0.608669\pi$
$647$	−17.0321	−0.669603	−0.334801	+	0.942289i	$0.608669\pi$
$648$	18.3580	0.721168
$649$	0	0
$650$	−0.428086	−0.0167909
$651$	1.67986	0.0658389
$652$	0.361498	0.0141573
$653$	48.1107	1.88272	0.941359	−	0.337407i	$-0.109550\pi$
$653$	48.1107	1.88272	0.941359	+	0.337407i	$0.109550\pi$
$654$	−6.45768	−0.252515
$655$	−16.8844	−0.659729
$656$	51.5737	2.01362
$657$	−19.4189	−0.757602
$658$	52.5921	2.05025
$659$	8.44549	0.328990	0.164495	−	0.986378i	$-0.447401\pi$
$659$	8.44549	0.328990	0.164495	+	0.986378i	$0.447401\pi$
$660$	0	0
$661$	7.41532	0.288423	0.144211	−	0.989547i	$-0.453936\pi$
$661$	7.41532	0.288423	0.144211	+	0.989547i	$0.453936\pi$
$662$	25.4248	0.988162
$663$	0.162488	0.00631050
$664$	9.21340	0.357549
$665$	−48.5267	−1.88179
$666$	−4.86007	−0.188324
$667$	−9.90466	−0.383510
$668$	−13.7877	−0.533462
$669$	−3.40038	−0.131466
$670$	12.0617	0.465984
$671$	0	0
$672$	3.01248	0.116209
$673$	−18.6214	−0.717800	−0.358900	−	0.933376i	$-0.616848\pi$
$673$	−18.6214	−0.717800	−0.358900	+	0.933376i	$0.616848\pi$
$674$	−48.1287	−1.85385
$675$	−2.99058	−0.115108
$676$	−9.12770	−0.351066
$677$	45.5655	1.75123	0.875613	−	0.483014i	$-0.160458\pi$
$677$	45.5655	1.75123	0.875613	+	0.483014i	$0.160458\pi$
$678$	6.87319	0.263963
$679$	23.5351	0.903195
$680$	−24.6165	−0.943998
$681$	5.95787	0.228306
$682$	0	0
$683$	−19.4950	−0.745954	−0.372977	−	0.927841i	$-0.621663\pi$
$683$	−19.4950	−0.745954	−0.372977	+	0.927841i	$0.621663\pi$
$684$	16.5411	0.632465
$685$	−27.7049	−1.05855
$686$	1.78534	0.0681646
$687$	−3.70842	−0.141485
$688$	−32.1896	−1.22722
$689$	−0.570833	−0.0217470
$690$	−3.84520	−0.146384
$691$	−33.3556	−1.26891	−0.634453	−	0.772962i	$-0.718774\pi$
$691$	−33.3556	−1.26891	−0.634453	+	0.772962i	$0.718774\pi$
$692$	−1.43785	−0.0546588
$693$	0	0
$694$	45.3589	1.72180
$695$	3.57243	0.135510
$696$	0.637390	0.0241602
$697$	−75.1645	−2.84706
$698$	11.0284	0.417433
$699$	3.39767	0.128512
$700$	6.37511	0.240957
$701$	10.4671	0.395338	0.197669	−	0.980269i	$-0.436663\pi$
$701$	10.4671	0.395338	0.197669	+	0.980269i	$0.436663\pi$
$702$	0.222272	0.00838911
$703$	7.96186	0.300287
$704$	0	0
$705$	2.85506	0.107528
$706$	4.41948	0.166330
$707$	−32.0251	−1.20443
$708$	−0.883263	−0.0331951
$709$	−38.7004	−1.45342	−0.726712	−	0.686942i	$-0.758953\pi$
$709$	−38.7004	−1.45342	−0.726712	+	0.686942i	$0.758953\pi$
$710$	19.0865	0.716303
$711$	−7.39524	−0.277343
$712$	−5.23869	−0.196328
$713$	14.7284	0.551582
$714$	−9.30625	−0.348278
$715$	0	0
$716$	−7.56425	−0.282689
$717$	−0.455346	−0.0170052
$718$	40.4143	1.50825
$719$	24.8190	0.925594	0.462797	−	0.886464i	$-0.346846\pi$
$719$	24.8190	0.925594	0.462797	+	0.886464i	$0.346846\pi$
$720$	−23.4130	−0.872552
$721$	36.3388	1.35333
$722$	−72.9797	−2.71602
$723$	−2.96963	−0.110442
$724$	−11.7296	−0.435928
$725$	3.42847	0.127330
$726$	0	0
$727$	−17.0392	−0.631948	−0.315974	−	0.948768i	$-0.602331\pi$
$727$	−17.0392	−0.631948	−0.315974	+	0.948768i	$0.602331\pi$
$728$	0.874626	0.0324158
$729$	−24.6651	−0.913523
$730$	17.4133	0.644496
$731$	46.9137	1.73517
$732$	0.822748	0.0304096
$733$	−15.3289	−0.566186	−0.283093	−	0.959092i	$-0.591361\pi$
$733$	−15.3289	−0.566186	−0.283093	+	0.959092i	$0.591361\pi$
$734$	18.7058	0.690443
$735$	2.45834	0.0906771
$736$	26.4123	0.973569
$737$	0	0
$738$	−51.0321	−1.87852
$739$	12.0612	0.443679	0.221840	−	0.975083i	$-0.428794\pi$
$739$	12.0612	0.443679	0.221840	+	0.975083i	$0.428794\pi$
$740$	1.13319	0.0416569
$741$	−0.180727	−0.00663917
$742$	32.6937	1.20022
$743$	−2.18957	−0.0803275	−0.0401638	−	0.999193i	$-0.512788\pi$
$743$	−2.18957	−0.0803275	−0.0401638	+	0.999193i	$0.512788\pi$
$744$	−0.947807	−0.0347483
$745$	11.4521	0.419574
$746$	14.5714	0.533497
$747$	−12.7713	−0.467278
$748$	0	0
$749$	30.1512	1.10170
$750$	4.10400	0.149857
$751$	28.3823	1.03568	0.517842	−	0.855476i	$-0.326736\pi$
$751$	28.3823	1.03568	0.517842	+	0.855476i	$0.326736\pi$
$752$	−41.5689	−1.51586
$753$	5.74028	0.209187
$754$	−0.254817	−0.00927990
$755$	4.84915	0.176479
$756$	−3.31010	−0.120387
$757$	−20.2955	−0.737654	−0.368827	−	0.929498i	$-0.620241\pi$
$757$	−20.2955	−0.737654	−0.368827	+	0.929498i	$0.620241\pi$
$758$	37.2019	1.35124
$759$	0	0
$760$	27.3797	0.993165
$761$	−14.6662	−0.531649	−0.265824	−	0.964021i	$-0.585644\pi$
$761$	−14.6662	−0.531649	−0.265824	+	0.964021i	$0.585644\pi$
$762$	1.36615	0.0494903
$763$	70.9682	2.56922
$764$	3.84423	0.139079
$765$	34.1226	1.23370
$766$	−46.0125	−1.66250
$767$	−0.651809	−0.0235355
$768$	−3.14398	−0.113449
$769$	20.8713	0.752638	0.376319	−	0.926490i	$-0.377190\pi$
$769$	20.8713	0.752638	0.376319	+	0.926490i	$0.377190\pi$
$770$	0	0
$771$	−1.82340	−0.0656681
$772$	−0.809296	−0.0291272
$773$	30.0811	1.08194	0.540972	−	0.841041i	$-0.318057\pi$
$773$	30.0811	1.08194	0.540972	+	0.841041i	$0.318057\pi$
$774$	31.8515	1.14488
$775$	−5.09818	−0.183132
$776$	−13.2789	−0.476686
$777$	−0.790785	−0.0283693
$778$	−56.6500	−2.03100
$779$	83.6018	2.99535
$780$	−0.0257224	−0.000921009 0
$781$	0	0
$782$	−81.5937	−2.91778
$783$	−1.78014	−0.0636170
$784$	−35.7927	−1.27831
$785$	4.88459	0.174339
$786$	3.60149	0.128461
$787$	−0.0882552	−0.00314596	−0.00157298	−	0.999999i	$-0.500501\pi$
$787$	−0.0882552	−0.00314596	−0.00157298	+	0.999999i	$0.500501\pi$
$788$	0.761113	0.0271135
$789$	0.776226	0.0276344
$790$	6.63147	0.235937
$791$	−75.5346	−2.68570
$792$	0	0
$793$	0.607151	0.0215606
$794$	−2.34530	−0.0832316
$795$	1.77484	0.0629470
$796$	−8.49316	−0.301032
$797$	−12.3335	−0.436874	−0.218437	−	0.975851i	$-0.570096\pi$
$797$	−12.3335	−0.436874	−0.218437	+	0.975851i	$0.570096\pi$
$798$	10.3509	0.366417
$799$	60.5833	2.14328
$800$	−9.14254	−0.323237
$801$	7.26171	0.256580
$802$	61.5780	2.17440
$803$	0	0
$804$	−0.668972	−0.0235928
$805$	42.2578	1.48939
$806$	0.378917	0.0133468
$807$	−4.22388	−0.148688
$808$	18.0691	0.635670
$809$	49.5901	1.74350	0.871748	−	0.489954i	$-0.162986\pi$
$809$	49.5901	1.74350	0.871748	+	0.489954i	$0.162986\pi$
$810$	22.8191	0.801780
$811$	−33.0041	−1.15893	−0.579465	−	0.814997i	$-0.696738\pi$
$811$	−33.0041	−1.15893	−0.579465	+	0.814997i	$0.696738\pi$
$812$	3.79478	0.133171
$813$	5.72167	0.200668
$814$	0	0
$815$	−0.829440	−0.0290540
$816$	7.35568	0.257500
$817$	−52.1798	−1.82554
$818$	−44.7311	−1.56399
$819$	−1.21238	−0.0423640
$820$	11.8988	0.415525
$821$	26.7892	0.934950	0.467475	−	0.884006i	$-0.345164\pi$
$821$	26.7892	0.934950	0.467475	+	0.884006i	$0.345164\pi$
$822$	5.90953	0.206119
$823$	−27.6409	−0.963502	−0.481751	−	0.876308i	$-0.659999\pi$
$823$	−27.6409	−0.963502	−0.481751	+	0.876308i	$0.659999\pi$
$824$	−20.5030	−0.714256
$825$	0	0
$826$	37.3315	1.29893
$827$	−19.4287	−0.675601	−0.337801	−	0.941218i	$-0.609683\pi$
$827$	−19.4287	−0.675601	−0.337801	+	0.941218i	$0.609683\pi$
$828$	−14.4042	−0.500582
$829$	20.6381	0.716792	0.358396	−	0.933570i	$-0.383324\pi$
$829$	20.6381	0.716792	0.358396	+	0.933570i	$0.383324\pi$
$830$	11.4523	0.397516
$831$	0.318722	0.0110563
$832$	−0.386309	−0.0133928
$833$	52.1650	1.80741
$834$	−0.762009	−0.0263862
$835$	31.6353	1.09478
$836$	0	0
$837$	2.64709	0.0914969
$838$	54.2636	1.87451
$839$	−4.24486	−0.146549	−0.0732745	−	0.997312i	$-0.523345\pi$
$839$	−4.24486	−0.146549	−0.0732745	+	0.997312i	$0.523345\pi$
$840$	−2.71939	−0.0938280
$841$	−26.9592	−0.929628
$842$	5.56809	0.191889
$843$	−3.96521	−0.136569
$844$	5.82532	0.200516
$845$	20.9431	0.720465
$846$	41.1324	1.41416
$847$	0	0
$848$	−25.8412	−0.887389
$849$	−5.59091	−0.191880
$850$	28.2434	0.968742
$851$	−6.93330	−0.237671
$852$	−1.05858	−0.0362665
$853$	−22.5069	−0.770622	−0.385311	−	0.922787i	$-0.625906\pi$
$853$	−22.5069	−0.770622	−0.385311	+	0.922787i	$0.625906\pi$
$854$	−34.7737	−1.18993
$855$	−37.9529	−1.29796
$856$	−17.0119	−0.581453
$857$	−6.83646	−0.233529	−0.116764	−	0.993160i	$-0.537252\pi$
$857$	−6.83646	−0.233529	−0.116764	+	0.993160i	$0.537252\pi$
$858$	0	0
$859$	43.7913	1.49414	0.747071	−	0.664745i	$-0.231460\pi$
$859$	43.7913	1.49414	0.747071	+	0.664745i	$0.231460\pi$
$860$	−7.42661	−0.253245
$861$	−8.30346	−0.282981
$862$	−38.7757	−1.32070
$863$	6.87387	0.233989	0.116995	−	0.993133i	$-0.462674\pi$
$863$	6.87387	0.233989	0.116995	+	0.993133i	$0.462674\pi$
$864$	4.74701	0.161497
$865$	3.29908	0.112172
$866$	−1.47829	−0.0502343
$867$	−7.16374	−0.243293
$868$	−5.64288	−0.191532
$869$	0	0
$870$	0.792280	0.0268608
$871$	−0.493672	−0.0167274
$872$	−40.0415	−1.35598
$873$	18.4069	0.622978
$874$	90.7526	3.06975
$875$	−45.1019	−1.52472
$876$	−0.965785	−0.0326309
$877$	19.8172	0.669179	0.334589	−	0.942364i	$-0.391402\pi$
$877$	19.8172	0.669179	0.334589	+	0.942364i	$0.391402\pi$
$878$	46.4598	1.56794
$879$	−1.05260	−0.0355034
$880$	0	0
$881$	−14.8825	−0.501404	−0.250702	−	0.968064i	$-0.580661\pi$
$881$	−14.8825	−0.501404	−0.250702	+	0.968064i	$0.580661\pi$
$882$	35.4168	1.19255
$883$	2.11112	0.0710450	0.0355225	−	0.999369i	$-0.488690\pi$
$883$	2.11112	0.0710450	0.0355225	+	0.999369i	$0.488690\pi$
$884$	−0.545819	−0.0183579
$885$	2.02661	0.0681237
$886$	12.2559	0.411744
$887$	−35.2332	−1.18302	−0.591508	−	0.806299i	$-0.701467\pi$
$887$	−35.2332	−1.18302	−0.591508	+	0.806299i	$0.701467\pi$
$888$	0.446175	0.0149727
$889$	−15.0136	−0.503540
$890$	−6.51173	−0.218274
$891$	0	0
$892$	11.4224	0.382449
$893$	−67.3838	−2.25491
$894$	−2.44277	−0.0816984
$895$	17.3558	0.580142
$896$	50.9238	1.70125
$897$	0.157380	0.00525475
$898$	2.06240	0.0688233
$899$	−3.03469	−0.101212
$900$	4.98599	0.166200
$901$	37.6614	1.25468
$902$	0	0
$903$	5.18258	0.172465
$904$	42.6180	1.41745
$905$	26.9131	0.894621
$906$	−1.03434	−0.0343635
$907$	−27.6638	−0.918561	−0.459280	−	0.888291i	$-0.651893\pi$
$907$	−27.6638	−0.918561	−0.459280	+	0.888291i	$0.651893\pi$
$908$	−20.0133	−0.664165
$909$	−25.0469	−0.830752
$910$	1.08717	0.0360392
$911$	41.3878	1.37124	0.685619	−	0.727960i	$-0.259531\pi$
$911$	41.3878	1.37124	0.685619	+	0.727960i	$0.259531\pi$
$912$	−8.18136	−0.270912
$913$	0	0
$914$	−37.4190	−1.23771
$915$	−1.88776	−0.0624074
$916$	12.4571	0.411594
$917$	−39.5795	−1.30703
$918$	−14.6646	−0.484005
$919$	32.0481	1.05717	0.528585	−	0.848881i	$-0.322723\pi$
$919$	32.0481	1.05717	0.528585	+	0.848881i	$0.322723\pi$
$920$	−23.8426	−0.786067
$921$	0.996271	0.0328283
$922$	47.3868	1.56060
$923$	−0.781188	−0.0257131
$924$	0	0
$925$	2.39994	0.0789097
$926$	29.4238	0.966925
$927$	28.4206	0.933456
$928$	−5.44208	−0.178645
$929$	40.6560	1.33388	0.666940	−	0.745111i	$-0.267604\pi$
$929$	40.6560	1.33388	0.666940	+	0.745111i	$0.267604\pi$
$930$	−1.17813	−0.0386324
$931$	−58.0205	−1.90155
$932$	−11.4133	−0.373854
$933$	1.95162	0.0638933
$934$	3.73025	0.122057
$935$	0	0
$936$	0.684047	0.0223588
$937$	−12.3221	−0.402545	−0.201272	−	0.979535i	$-0.564508\pi$
$937$	−12.3221	−0.402545	−0.201272	+	0.979535i	$0.564508\pi$
$938$	28.2744	0.923190
$939$	2.95191	0.0963321
$940$	−9.59056	−0.312809
$941$	42.3105	1.37928	0.689642	−	0.724151i	$-0.257768\pi$
$941$	42.3105	1.37928	0.689642	+	0.724151i	$0.257768\pi$
$942$	−1.04190	−0.0339468
$943$	−72.8016	−2.37075
$944$	−29.5069	−0.960367
$945$	7.59489	0.247062
$946$	0	0
$947$	−37.7783	−1.22763	−0.613816	−	0.789449i	$-0.710366\pi$
$947$	−37.7783	−1.22763	−0.613816	+	0.789449i	$0.710366\pi$
$948$	−0.367798	−0.0119455
$949$	−0.712707	−0.0231354
$950$	−31.4138	−1.01920
$951$	4.04135	0.131050
$952$	−57.7045	−1.87021
$953$	−1.74008	−0.0563668	−0.0281834	−	0.999603i	$-0.508972\pi$
$953$	−1.74008	−0.0563668	−0.0281834	+	0.999603i	$0.508972\pi$
$954$	25.5698	0.827852
$955$	−8.82041	−0.285422
$956$	1.52957	0.0494699
$957$	0	0
$958$	58.6156	1.89378
$959$	−64.9443	−2.09716
$960$	1.20111	0.0387658
$961$	−26.4874	−0.854432
$962$	−0.178373	−0.00575098
$963$	23.5813	0.759897
$964$	9.97542	0.321287
$965$	1.85689	0.0597755
$966$	−9.01370	−0.290011
$967$	42.1121	1.35423	0.677117	−	0.735876i	$-0.263229\pi$
$967$	42.1121	1.35423	0.677117	+	0.735876i	$0.263229\pi$
$968$	0	0
$969$	11.9237	0.383044
$970$	−16.5058	−0.529970
$971$	−22.1676	−0.711393	−0.355697	−	0.934602i	$-0.615756\pi$
$971$	−22.1676	−0.711393	−0.355697	+	0.934602i	$0.615756\pi$
$972$	−3.89277	−0.124861
$973$	8.37429	0.268467
$974$	−6.59267	−0.211243
$975$	−0.0544765	−0.00174465
$976$	27.4852	0.879781
$977$	31.2448	0.999609	0.499805	−	0.866138i	$-0.333405\pi$
$977$	31.2448	0.999609	0.499805	+	0.866138i	$0.333405\pi$
$978$	0.176922	0.00565734
$979$	0	0
$980$	−8.25790	−0.263789
$981$	55.5043	1.77212
$982$	−17.4213	−0.555936
$983$	34.2861	1.09356	0.546778	−	0.837277i	$-0.315854\pi$
$983$	34.2861	1.09356	0.546778	+	0.837277i	$0.315854\pi$
$984$	4.68496	0.149351
$985$	−1.74634	−0.0556430
$986$	16.8119	0.535399
$987$	6.69267	0.213030
$988$	0.607088	0.0193140
$989$	45.4389	1.44487
$990$	0	0
$991$	−40.5235	−1.28727	−0.643636	−	0.765331i	$-0.722575\pi$
$991$	−40.5235	−1.28727	−0.643636	+	0.765331i	$0.722575\pi$
$992$	8.09245	0.256935
$993$	3.23546	0.102674
$994$	44.7414	1.41911
$995$	19.4872	0.617785
$996$	−0.635173	−0.0201262
$997$	−15.6106	−0.494393	−0.247197	−	0.968965i	$-0.579509\pi$
$997$	−15.6106	−0.494393	−0.247197	+	0.968965i	$0.579509\pi$
$998$	−7.27022	−0.230135
$999$	−1.24611	−0.0394250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4477.2.a.t.1.11		36
11.5	even	5		407.2.h.b.223.13	✓	72
11.9	even	5		407.2.h.b.334.13	yes	72
11.10	odd	2		4477.2.a.u.1.26		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
407.2.h.b.223.13	✓	72	11.5	even	5
407.2.h.b.334.13	yes	72	11.9	even	5
4477.2.a.t.1.11		36	1.1	even	1	trivial
4477.2.a.u.1.26		36	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4477 = 11^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4477.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64401 q^{2} -0.209210 q^{3} +0.702767 q^{4} -1.61247 q^{5} +0.343944 q^{6} -3.77985 q^{7} +2.13266 q^{8} -2.95623 q^{9} +2.65091 q^{10} -0.147026 q^{12} -0.108499 q^{13} +6.21412 q^{14} +0.337345 q^{15} -4.91165 q^{16} +7.15834 q^{17} +4.86007 q^{18} -7.96186 q^{19} -1.13319 q^{20} +0.790785 q^{21} +6.93330 q^{23} -0.446175 q^{24} -2.39994 q^{25} +0.178373 q^{26} +1.24611 q^{27} -2.65636 q^{28} -1.42856 q^{29} -0.554599 q^{30} +2.12429 q^{31} +3.80948 q^{32} -11.7684 q^{34} +6.09490 q^{35} -2.07754 q^{36} -1.00000 q^{37} +13.0894 q^{38} +0.0226991 q^{39} -3.43885 q^{40} -10.5003 q^{41} -1.30006 q^{42} +6.55372 q^{43} +4.76683 q^{45} -11.3984 q^{46} +8.46332 q^{47} +1.02757 q^{48} +7.28730 q^{49} +3.94553 q^{50} -1.49760 q^{51} -0.0762495 q^{52} +5.26119 q^{53} -2.04861 q^{54} -8.06116 q^{56} +1.66570 q^{57} +2.34857 q^{58} +6.00752 q^{59} +0.237075 q^{60} -5.59592 q^{61} -3.49236 q^{62} +11.1741 q^{63} +3.56049 q^{64} +0.174951 q^{65} +4.55002 q^{67} +5.03065 q^{68} -1.45052 q^{69} -10.0201 q^{70} +7.19996 q^{71} -6.30464 q^{72} +6.56880 q^{73} +1.64401 q^{74} +0.502093 q^{75} -5.59534 q^{76} -0.0373175 q^{78} +2.50158 q^{79} +7.91989 q^{80} +8.60799 q^{81} +17.2626 q^{82} +4.32014 q^{83} +0.555738 q^{84} -11.5426 q^{85} -10.7744 q^{86} +0.298870 q^{87} -2.45641 q^{89} -7.83671 q^{90} +0.410110 q^{91} +4.87250 q^{92} -0.444424 q^{93} -13.9138 q^{94} +12.8383 q^{95} -0.796982 q^{96} -6.22646 q^{97} -11.9804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 8 q^{2} - q^{3} + 36 q^{4} - q^{5} - 9 q^{6} - 19 q^{7} - 24 q^{8} + 31 q^{9} - 18 q^{10} - 23 q^{12} - 15 q^{13} + 13 q^{14} - 6 q^{15} + 40 q^{16} - 35 q^{17} - 11 q^{18} - 21 q^{19} + 10 q^{20}+ \cdots - 83 q^{98}+O(q^{100}) \) 36 * q - 8 * q^2 - q^3 + 36 * q^4 - q^5 - 9 * q^6 - 19 * q^7 - 24 * q^8 + 31 * q^9 - 18 * q^10 - 23 * q^12 - 15 * q^13 + 13 * q^14 - 6 * q^15 + 40 * q^16 - 35 * q^17 - 11 * q^18 - 21 * q^19 + 10 * q^20 - 22 * q^21 - 9 * q^23 - 33 * q^24 + 37 * q^25 + 3 * q^26 - 4 * q^27 - 56 * q^28 - 13 * q^29 + 12 * q^30 + 11 * q^31 - 61 * q^32 - 49 * q^34 - 61 * q^35 + 75 * q^36 - 36 * q^37 + 15 * q^38 - 16 * q^39 - 11 * q^40 - 45 * q^41 + 34 * q^42 - 38 * q^43 - 37 * q^45 - 28 * q^46 + 13 * q^47 - 46 * q^48 + 27 * q^49 - 8 * q^50 - 21 * q^51 - q^52 - 2 * q^53 - 41 * q^54 - 16 * q^56 - 47 * q^57 + 28 * q^58 - 8 * q^59 - 37 * q^60 - 5 * q^61 - 97 * q^62 - 56 * q^63 + 62 * q^64 - 36 * q^65 - 2 * q^67 - 105 * q^68 + 29 * q^69 - 24 * q^70 + 7 * q^71 - 22 * q^72 - 34 * q^73 + 8 * q^74 + 7 * q^75 - 10 * q^76 - 85 * q^78 - 22 * q^79 + 62 * q^80 - 56 * q^81 + 13 * q^82 - 166 * q^83 - 37 * q^84 - 43 * q^85 + 47 * q^86 - 26 * q^87 + 15 * q^89 - 29 * q^90 + 55 * q^91 + 19 * q^92 - 23 * q^93 - 13 * q^94 - 75 * q^95 - 14 * q^97 - 83 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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