LMFDB - Embedded newform 841.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [841,2,Mod(1,841)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("841.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$841 = 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	841.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.12.32268092290502656.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 15x^{10} + 78x^{8} - 169x^{6} + 148x^{4} - 36x^{2} + 1$ x^12 - 15x^10 + 78x^8 - 169x^6 + 148x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16310$ of defining polynomial
Character	$\chi$	$=$	841.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60244 q^{2} -0.439339 q^{3} +4.77269 q^{4} +2.58042 q^{5} +1.14335 q^{6} +0.0751311 q^{7} -7.21577 q^{8} -2.80698 q^{9} -6.71540 q^{10} +3.77836 q^{11} -2.09683 q^{12} +0.880995 q^{13} -0.195524 q^{14} -1.13368 q^{15} +9.23322 q^{16} +3.94108 q^{17} +7.30500 q^{18} +0.713617 q^{19} +12.3156 q^{20} -0.0330080 q^{21} -9.83294 q^{22} +1.17764 q^{23} +3.17017 q^{24} +1.65858 q^{25} -2.29274 q^{26} +2.55123 q^{27} +0.358578 q^{28} +2.95033 q^{30} +5.15087 q^{31} -9.59735 q^{32} -1.65998 q^{33} -10.2564 q^{34} +0.193870 q^{35} -13.3969 q^{36} -3.08751 q^{37} -1.85714 q^{38} -0.387055 q^{39} -18.6197 q^{40} +6.67122 q^{41} +0.0859013 q^{42} -8.31364 q^{43} +18.0329 q^{44} -7.24320 q^{45} -3.06474 q^{46} -10.5920 q^{47} -4.05651 q^{48} -6.99436 q^{49} -4.31637 q^{50} -1.73147 q^{51} +4.20472 q^{52} +5.55608 q^{53} -6.63943 q^{54} +9.74976 q^{55} -0.542129 q^{56} -0.313519 q^{57} +9.91885 q^{59} -5.41071 q^{60} +3.56734 q^{61} -13.4048 q^{62} -0.210892 q^{63} +6.51010 q^{64} +2.27334 q^{65} +4.31999 q^{66} +4.93956 q^{67} +18.8096 q^{68} -0.517383 q^{69} -0.504535 q^{70} +4.90681 q^{71} +20.2545 q^{72} -8.90215 q^{73} +8.03505 q^{74} -0.728681 q^{75} +3.40587 q^{76} +0.283872 q^{77} +1.00729 q^{78} +13.1895 q^{79} +23.8256 q^{80} +7.30009 q^{81} -17.3615 q^{82} +16.9864 q^{83} -0.157537 q^{84} +10.1697 q^{85} +21.6357 q^{86} -27.2637 q^{88} +6.79655 q^{89} +18.8500 q^{90} +0.0661901 q^{91} +5.62052 q^{92} -2.26298 q^{93} +27.5652 q^{94} +1.84143 q^{95} +4.21649 q^{96} -11.4171 q^{97} +18.2024 q^{98} -10.6058 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 8 q^{4} + 8 q^{5} + 24 q^{6} + 10 q^{7} + 10 q^{9} + 12 q^{13} + 16 q^{16} + 24 q^{20} - 38 q^{22} + 30 q^{23} + 10 q^{24} - 8 q^{25} - 12 q^{28} + 2 q^{30} - 4 q^{33} + 6 q^{34} + 44 q^{35} + 16 q^{36}+ \cdots - 58 q^{96}+O(q^{100})$ 12 * q + 8 * q^4 + 8 * q^5 + 24 * q^6 + 10 * q^7 + 10 * q^9 + 12 * q^13 + 16 * q^16 + 24 * q^20 - 38 * q^22 + 30 * q^23 + 10 * q^24 - 8 * q^25 - 12 * q^28 + 2 * q^30 - 4 * q^33 + 6 * q^34 + 44 * q^35 + 16 * q^36 + 6 * q^42 - 12 * q^45 + 6 * q^49 + 8 * q^51 - 6 * q^52 + 32 * q^53 + 32 * q^54 + 14 * q^57 + 44 * q^59 - 16 * q^62 + 34 * q^63 - 2 * q^64 + 8 * q^65 + 30 * q^67 + 70 * q^71 + 56 * q^74 - 4 * q^78 + 34 * q^80 + 8 * q^81 - 62 * q^82 + 82 * q^83 + 44 * q^86 - 66 * q^88 - 32 * q^91 - 22 * q^92 - 12 * q^93 + 10 * q^94 - 58 * q^96

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.60244	−1.84020	−0.920101	−	0.391680i	$-0.871894\pi$
$2$	−2.60244	−1.84020	−0.920101	+	0.391680i	$0.871894\pi$
$3$	−0.439339	−0.253652	−0.126826	−	0.991925i	$-0.540479\pi$
$3$	−0.439339	−0.253652	−0.126826	+	0.991925i	$0.540479\pi$
$4$	4.77269	2.38635
$5$	2.58042	1.15400	0.577000	−	0.816744i	$-0.304223\pi$
$5$	2.58042	1.15400	0.577000	+	0.816744i	$0.304223\pi$
$6$	1.14335	0.466772
$7$	0.0751311	0.0283969	0.0141984	−	0.999899i	$-0.495480\pi$
$7$	0.0751311	0.0283969	0.0141984	+	0.999899i	$0.495480\pi$
$8$	−7.21577	−2.55116
$9$	−2.80698	−0.935660
$10$	−6.71540	−2.12359
$11$	3.77836	1.13922	0.569609	−	0.821916i	$-0.307095\pi$
$11$	3.77836	1.13922	0.569609	+	0.821916i	$0.307095\pi$
$12$	−2.09683	−0.605302
$13$	0.880995	0.244344	0.122172	−	0.992509i	$-0.461014\pi$
$13$	0.880995	0.244344	0.122172	+	0.992509i	$0.461014\pi$
$14$	−0.195524	−0.0522560
$15$	−1.13368	−0.292715
$16$	9.23322	2.30830
$17$	3.94108	0.955854	0.477927	−	0.878400i	$-0.341388\pi$
$17$	3.94108	0.955854	0.477927	+	0.878400i	$0.341388\pi$
$18$	7.30500	1.72181
$19$	0.713617	0.163715	0.0818574	−	0.996644i	$-0.473915\pi$
$19$	0.713617	0.163715	0.0818574	+	0.996644i	$0.473915\pi$
$20$	12.3156	2.75385
$21$	−0.0330080	−0.00720294
$22$	−9.83294	−2.09639
$23$	1.17764	0.245555	0.122778	−	0.992434i	$-0.460820\pi$
$23$	1.17764	0.245555	0.122778	+	0.992434i	$0.460820\pi$
$24$	3.17017	0.647108
$25$	1.65858	0.331717
$26$	−2.29274	−0.449643
$27$	2.55123	0.490985
$28$	0.358578	0.0677648
$29$	0	0
$29$	0	0
$30$	2.95033	0.538655
$31$	5.15087	0.925124	0.462562	−	0.886587i	$-0.346930\pi$
$31$	5.15087	0.925124	0.462562	+	0.886587i	$0.346930\pi$
$32$	−9.59735	−1.69659
$33$	−1.65998	−0.288965
$34$	−10.2564	−1.75896
$35$	0.193870	0.0327700
$36$	−13.3969	−2.23281
$37$	−3.08751	−0.507583	−0.253791	−	0.967259i	$-0.581678\pi$
$37$	−3.08751	−0.507583	−0.253791	+	0.967259i	$0.581678\pi$
$38$	−1.85714	−0.301269
$39$	−0.387055	−0.0619785
$40$	−18.6197	−2.94404
$41$	6.67122	1.04187	0.520935	−	0.853596i	$-0.325583\pi$
$41$	6.67122	1.04187	0.520935	+	0.853596i	$0.325583\pi$
$42$	0.0859013	0.0132549
$43$	−8.31364	−1.26782	−0.633909	−	0.773408i	$-0.718551\pi$
$43$	−8.31364	−1.26782	−0.633909	+	0.773408i	$0.718551\pi$
$44$	18.0329	2.71857
$45$	−7.24320	−1.07975
$46$	−3.06474	−0.451871
$47$	−10.5920	−1.54501	−0.772504	−	0.635010i	$-0.780996\pi$
$47$	−10.5920	−1.54501	−0.772504	+	0.635010i	$0.780996\pi$
$48$	−4.05651	−0.585507
$49$	−6.99436	−0.999194
$50$	−4.31637	−0.610427
$51$	−1.73147	−0.242454
$52$	4.20472	0.583090
$53$	5.55608	0.763186	0.381593	−	0.924331i	$-0.375376\pi$
$53$	5.55608	0.763186	0.381593	+	0.924331i	$0.375376\pi$
$54$	−6.63943	−0.903512
$55$	9.74976	1.31466
$56$	−0.542129	−0.0724450
$57$	−0.313519	−0.0415267
$58$	0	0
$59$	9.91885	1.29132	0.645662	−	0.763623i	$-0.276582\pi$
$59$	9.91885	1.29132	0.645662	+	0.763623i	$0.276582\pi$
$60$	−5.41071	−0.698519
$61$	3.56734	0.456751	0.228375	−	0.973573i	$-0.426659\pi$
$61$	3.56734	0.456751	0.228375	+	0.973573i	$0.426659\pi$
$62$	−13.4048	−1.70242
$63$	−0.210892	−0.0265698
$64$	6.51010	0.813763
$65$	2.27334	0.281973
$66$	4.31999	0.531754
$67$	4.93956	0.603464	0.301732	−	0.953393i	$-0.402435\pi$
$67$	4.93956	0.603464	0.301732	+	0.953393i	$0.402435\pi$
$68$	18.8096	2.28100
$69$	−0.517383	−0.0622856
$70$	−0.504535	−0.0603035
$71$	4.90681	0.582331	0.291166	−	0.956673i	$-0.405957\pi$
$71$	4.90681	0.582331	0.291166	+	0.956673i	$0.405957\pi$
$72$	20.2545	2.38702
$73$	−8.90215	−1.04192	−0.520959	−	0.853582i	$-0.674426\pi$
$73$	−8.90215	−1.04192	−0.520959	+	0.853582i	$0.674426\pi$
$74$	8.03505	0.934056
$75$	−0.728681	−0.0841408
$76$	3.40587	0.390680
$77$	0.283872	0.0323502
$78$	1.00729	0.114053
$79$	13.1895	1.48394	0.741969	−	0.670435i	$-0.233892\pi$
$79$	13.1895	1.48394	0.741969	+	0.670435i	$0.233892\pi$
$80$	23.8256	2.66378
$81$	7.30009	0.811121
$82$	−17.3615	−1.91725
$83$	16.9864	1.86450	0.932249	−	0.361818i	$-0.117844\pi$
$83$	16.9864	1.86450	0.932249	+	0.361818i	$0.117844\pi$
$84$	−0.157537	−0.0171887
$85$	10.1697	1.10306
$86$	21.6357	2.33304
$87$	0	0
$88$	−27.2637	−2.90632
$89$	6.79655	0.720433	0.360216	−	0.932869i	$-0.382703\pi$
$89$	6.79655	0.720433	0.360216	+	0.932869i	$0.382703\pi$
$90$	18.8500	1.98696
$91$	0.0661901	0.00693861
$92$	5.62052	0.585980
$93$	−2.26298	−0.234660
$94$	27.5652	2.84313
$95$	1.84143	0.188927
$96$	4.21649	0.430344
$97$	−11.4171	−1.15923	−0.579615	−	0.814890i	$-0.696797\pi$
$97$	−11.4171	−1.15923	−0.579615	+	0.814890i	$0.696797\pi$
$98$	18.2024	1.83872
$99$	−10.6058	−1.06592
$100$	7.91592	0.791592
$101$	9.43826	0.939142	0.469571	−	0.882895i	$-0.344409\pi$
$101$	9.43826	0.939142	0.469571	+	0.882895i	$0.344409\pi$
$102$	4.50605	0.446165
$103$	−9.18317	−0.904845	−0.452422	−	0.891804i	$-0.649440\pi$
$103$	−9.18317	−0.904845	−0.452422	+	0.891804i	$0.649440\pi$
$104$	−6.35706	−0.623361
$105$	−0.0851746	−0.00831219
$106$	−14.4594	−1.40442
$107$	−4.86820	−0.470627	−0.235313	−	0.971920i	$-0.575612\pi$
$107$	−4.86820	−0.470627	−0.235313	+	0.971920i	$0.575612\pi$
$108$	12.1762	1.17166
$109$	4.33423	0.415144	0.207572	−	0.978220i	$-0.433444\pi$
$109$	4.33423	0.415144	0.207572	+	0.978220i	$0.433444\pi$
$110$	−25.3732	−2.41924
$111$	1.35646	0.128750
$112$	0.693702	0.0655486
$113$	4.52309	0.425496	0.212748	−	0.977107i	$-0.431759\pi$
$113$	4.52309	0.425496	0.212748	+	0.977107i	$0.431759\pi$
$114$	0.815916	0.0764175
$115$	3.03881	0.283371
$116$	0	0
$117$	−2.47294	−0.228623
$118$	−25.8132	−2.37630
$119$	0.296098	0.0271433
$120$	8.18037	0.746762
$121$	3.27597	0.297816
$122$	−9.28378	−0.840514
$123$	−2.93093	−0.264273
$124$	24.5835	2.20767
$125$	−8.62227	−0.771199
$126$	0.548833	0.0488939
$127$	5.83762	0.518005	0.259003	−	0.965877i	$-0.416606\pi$
$127$	5.83762	0.518005	0.259003	+	0.965877i	$0.416606\pi$
$128$	2.25255	0.199099
$129$	3.65250	0.321585
$130$	−5.91623	−0.518888
$131$	−6.62441	−0.578777	−0.289389	−	0.957212i	$-0.593452\pi$
$131$	−6.62441	−0.578777	−0.289389	+	0.957212i	$0.593452\pi$
$132$	−7.92257	−0.689571
$133$	0.0536148	0.00464899
$134$	−12.8549	−1.11050
$135$	6.58326	0.566597
$136$	−28.4380	−2.43853
$137$	1.55881	0.133178	0.0665890	−	0.997780i	$-0.478788\pi$
$137$	1.55881	0.133178	0.0665890	+	0.997780i	$0.478788\pi$
$138$	1.34646	0.114618
$139$	10.6416	0.902607	0.451303	−	0.892371i	$-0.350959\pi$
$139$	10.6416	0.902607	0.451303	+	0.892371i	$0.350959\pi$
$140$	0.925282	0.0782006
$141$	4.65349	0.391895
$142$	−12.7697	−1.07161
$143$	3.32871	0.278361
$144$	−25.9175	−2.15979
$145$	0	0
$146$	23.1673	1.91734
$147$	3.07289	0.253448
$148$	−14.7357	−1.21127
$149$	5.43253	0.445050	0.222525	−	0.974927i	$-0.428570\pi$
$149$	5.43253	0.445050	0.222525	+	0.974927i	$0.428570\pi$
$150$	1.89635	0.154836
$151$	17.8418	1.45194	0.725971	−	0.687725i	$-0.241390\pi$
$151$	17.8418	1.45194	0.725971	+	0.687725i	$0.241390\pi$
$152$	−5.14929	−0.417663
$153$	−11.0626	−0.894354
$154$	−0.738760	−0.0595310
$155$	13.2914	1.06759
$156$	−1.84730	−0.147902
$157$	9.46360	0.755278	0.377639	−	0.925953i	$-0.376736\pi$
$157$	9.46360	0.755278	0.377639	+	0.925953i	$0.376736\pi$
$158$	−34.3249	−2.73075
$159$	−2.44100	−0.193584
$160$	−24.7652	−1.95786
$161$	0.0884774	0.00697300
$162$	−18.9980	−1.49263
$163$	10.4578	0.819116	0.409558	−	0.912284i	$-0.365683\pi$
$163$	10.4578	0.819116	0.409558	+	0.912284i	$0.365683\pi$
$164$	31.8397	2.48626
$165$	−4.28345	−0.333466
$166$	−44.2060	−3.43105
$167$	0.170436	0.0131888	0.00659438	−	0.999978i	$-0.497901\pi$
$167$	0.170436	0.0131888	0.00659438	+	0.999978i	$0.497901\pi$
$168$	0.238178	0.0183758
$169$	−12.2238	−0.940296
$170$	−26.4659	−2.02985
$171$	−2.00311	−0.153182
$172$	−39.6785	−3.02545
$173$	−12.0694	−0.917616	−0.458808	−	0.888535i	$-0.651723\pi$
$173$	−12.0694	−0.917616	−0.458808	+	0.888535i	$0.651723\pi$
$174$	0	0
$175$	0.124611	0.00941973
$176$	34.8864	2.62966
$177$	−4.35773	−0.327547
$178$	−17.6876	−1.32574
$179$	−1.67483	−0.125183	−0.0625913	−	0.998039i	$-0.519936\pi$
$179$	−1.67483	−0.125183	−0.0625913	+	0.998039i	$0.519936\pi$
$180$	−34.5696	−2.57666
$181$	−15.3626	−1.14190	−0.570948	−	0.820986i	$-0.693424\pi$
$181$	−15.3626	−1.14190	−0.570948	+	0.820986i	$0.693424\pi$
$182$	−0.172256	−0.0127685
$183$	−1.56727	−0.115856
$184$	−8.49758	−0.626450
$185$	−7.96707	−0.585751
$186$	5.88926	0.431822
$187$	14.8908	1.08892
$188$	−50.5526	−3.68693
$189$	0.191677	0.0139424
$190$	−4.79222	−0.347664
$191$	20.4783	1.48176	0.740878	−	0.671640i	$-0.234410\pi$
$191$	20.4783	1.48176	0.740878	+	0.671640i	$0.234410\pi$
$192$	−2.86014	−0.206413
$193$	−3.13441	−0.225620	−0.112810	−	0.993617i	$-0.535985\pi$
$193$	−3.13441	−0.225620	−0.112810	+	0.993617i	$0.535985\pi$
$194$	29.7123	2.13322
$195$	−0.998767	−0.0715232
$196$	−33.3819	−2.38442
$197$	2.84040	0.202370	0.101185	−	0.994868i	$-0.467737\pi$
$197$	2.84040	0.202370	0.101185	+	0.994868i	$0.467737\pi$
$198$	27.6009	1.96151
$199$	−19.1189	−1.35530	−0.677652	−	0.735383i	$-0.737002\pi$
$199$	−19.1189	−1.35530	−0.677652	+	0.735383i	$0.737002\pi$
$200$	−11.9680	−0.846263
$201$	−2.17014	−0.153070
$202$	−24.5625	−1.72821
$203$	0	0
$204$	−8.26378	−0.578580
$205$	17.2146	1.20232
$206$	23.8987	1.66510
$207$	−3.30562	−0.229756
$208$	8.13442	0.564021
$209$	2.69630	0.186507
$210$	0.221662	0.0152961
$211$	17.8936	1.23184	0.615922	−	0.787807i	$-0.288784\pi$
$211$	17.8936	1.23184	0.615922	+	0.787807i	$0.288784\pi$
$212$	26.5174	1.82123
$213$	−2.15575	−0.147710
$214$	12.6692	0.866049
$215$	−21.4527	−1.46306
$216$	−18.4091	−1.25258
$217$	0.386991	0.0262706
$218$	−11.2796	−0.763949
$219$	3.91106	0.264285
$220$	46.5326	3.13723
$221$	3.47208	0.233557
$222$	−3.53011	−0.236925
$223$	17.7308	1.18735	0.593673	−	0.804707i	$-0.297677\pi$
$223$	17.7308	1.18735	0.593673	+	0.804707i	$0.297677\pi$
$224$	−0.721060	−0.0481778
$225$	−4.65562	−0.310374
$226$	−11.7711	−0.783000
$227$	22.2116	1.47424	0.737119	−	0.675763i	$-0.236186\pi$
$227$	22.2116	1.47424	0.737119	+	0.675763i	$0.236186\pi$
$228$	−1.49633	−0.0990970
$229$	−9.96812	−0.658712	−0.329356	−	0.944206i	$-0.606832\pi$
$229$	−9.96812	−0.658712	−0.329356	+	0.944206i	$0.606832\pi$
$230$	−7.90832	−0.521459
$231$	−0.124716	−0.00820571
$232$	0	0
$233$	−20.0765	−1.31525	−0.657627	−	0.753344i	$-0.728440\pi$
$233$	−20.0765	−1.31525	−0.657627	+	0.753344i	$0.728440\pi$
$234$	6.43567	0.420713
$235$	−27.3320	−1.78294
$236$	47.3396	3.08155
$237$	−5.79467	−0.376404
$238$	−0.770577	−0.0499491
$239$	−1.18085	−0.0763830	−0.0381915	−	0.999270i	$-0.512160\pi$
$239$	−1.18085	−0.0763830	−0.0381915	+	0.999270i	$0.512160\pi$
$240$	−10.4675	−0.675675
$241$	11.8311	0.762110	0.381055	−	0.924552i	$-0.375561\pi$
$241$	11.8311	0.762110	0.381055	+	0.924552i	$0.375561\pi$
$242$	−8.52552	−0.548041
$243$	−10.8609	−0.696728
$244$	17.0258	1.08997
$245$	−18.0484	−1.15307
$246$	7.62756	0.486316
$247$	0.628693	0.0400028
$248$	−37.1675	−2.36014
$249$	−7.46278	−0.472934
$250$	22.4389	1.41916
$251$	−7.66497	−0.483809	−0.241904	−	0.970300i	$-0.577772\pi$
$251$	−7.66497	−0.483809	−0.241904	+	0.970300i	$0.577772\pi$
$252$	−1.00652	−0.0634049
$253$	4.44955	0.279741
$254$	−15.1921	−0.953235
$255$	−4.46793	−0.279793
$256$	−18.8823	−1.18015
$257$	11.3728	0.709415	0.354707	−	0.934977i	$-0.384580\pi$
$257$	11.3728	0.709415	0.354707	+	0.934977i	$0.384580\pi$
$258$	−9.50542	−0.591782
$259$	−0.231968	−0.0144138
$260$	10.8500	0.672886
$261$	0	0
$262$	17.2396	1.06507
$263$	−16.7464	−1.03263	−0.516315	−	0.856399i	$-0.672697\pi$
$263$	−16.7464	−1.03263	−0.516315	+	0.856399i	$0.672697\pi$
$264$	11.9780	0.737196
$265$	14.3370	0.880717
$266$	−0.139529	−0.00855509
$267$	−2.98599	−0.182739
$268$	23.5750	1.44007
$269$	−21.9337	−1.33732	−0.668660	−	0.743568i	$-0.733132\pi$
$269$	−21.9337	−1.33732	−0.668660	+	0.743568i	$0.733132\pi$
$270$	−17.1325	−1.04265
$271$	−6.54431	−0.397539	−0.198769	−	0.980046i	$-0.563694\pi$
$271$	−6.54431	−0.397539	−0.198769	+	0.980046i	$0.563694\pi$
$272$	36.3889	2.20640
$273$	−0.0290799	−0.00176000
$274$	−4.05670	−0.245074
$275$	6.26672	0.377898
$276$	−2.46931	−0.148635
$277$	21.5277	1.29347	0.646736	−	0.762714i	$-0.276134\pi$
$277$	21.5277	1.29347	0.646736	+	0.762714i	$0.276134\pi$
$278$	−27.6941	−1.66098
$279$	−14.4584	−0.865602
$280$	−1.39892	−0.0836015
$281$	−8.69715	−0.518828	−0.259414	−	0.965766i	$-0.583529\pi$
$281$	−8.69715	−0.518828	−0.259414	+	0.965766i	$0.583529\pi$
$282$	−12.1104	−0.721166
$283$	21.9887	1.30709	0.653545	−	0.756888i	$-0.273281\pi$
$283$	21.9887	1.30709	0.653545	+	0.756888i	$0.273281\pi$
$284$	23.4187	1.38964
$285$	−0.809013	−0.0479218
$286$	−8.66278	−0.512241
$287$	0.501216	0.0295859
$288$	26.9396	1.58743
$289$	−1.46785	−0.0863441
$290$	0	0
$291$	5.01597	0.294042
$292$	−42.4872	−2.48638
$293$	−2.22609	−0.130050	−0.0650249	−	0.997884i	$-0.520713\pi$
$293$	−2.22609	−0.130050	−0.0650249	+	0.997884i	$0.520713\pi$
$294$	−7.99702	−0.466395
$295$	25.5948	1.49019
$296$	22.2787	1.29493
$297$	9.63946	0.559338
$298$	−14.1378	−0.818982
$299$	1.03750	0.0599999
$300$	−3.47777	−0.200789
$301$	−0.624613	−0.0360021
$302$	−46.4321	−2.67187
$303$	−4.14659	−0.238216
$304$	6.58898	0.377904
$305$	9.20524	0.527090
$306$	28.7896	1.64579
$307$	−13.8760	−0.791945	−0.395972	−	0.918262i	$-0.629592\pi$
$307$	−13.8760	−0.791945	−0.395972	+	0.918262i	$0.629592\pi$
$308$	1.35483	0.0771988
$309$	4.03452	0.229516
$310$	−34.5901	−1.96459
$311$	−1.41217	−0.0800768	−0.0400384	−	0.999198i	$-0.512748\pi$
$311$	−1.41217	−0.0800768	−0.0400384	+	0.999198i	$0.512748\pi$
$312$	2.79290	0.158117
$313$	−28.7487	−1.62497	−0.812487	−	0.582980i	$-0.801887\pi$
$313$	−28.7487	−1.62497	−0.812487	+	0.582980i	$0.801887\pi$
$314$	−24.6285	−1.38986
$315$	−0.544190	−0.0306616
$316$	62.9496	3.54119
$317$	−22.3203	−1.25363	−0.626815	−	0.779168i	$-0.715642\pi$
$317$	−22.3203	−1.25363	−0.626815	+	0.779168i	$0.715642\pi$
$318$	6.35255	0.356234
$319$	0	0
$320$	16.7988	0.939083
$321$	2.13879	0.119376
$322$	−0.230257	−0.0128317
$323$	2.81242	0.156487
$324$	34.8411	1.93562
$325$	1.46121	0.0810531
$326$	−27.2157	−1.50734
$327$	−1.90420	−0.105302
$328$	−48.1380	−2.65798
$329$	−0.795792	−0.0438734
$330$	11.1474	0.613645
$331$	9.34951	0.513895	0.256948	−	0.966425i	$-0.417283\pi$
$331$	9.34951	0.513895	0.256948	+	0.966425i	$0.417283\pi$
$332$	81.0708	4.44934
$333$	8.66657	0.474925
$334$	−0.443550	−0.0242700
$335$	12.7462	0.696397
$336$	−0.304770	−0.0166266
$337$	−15.4311	−0.840583	−0.420292	−	0.907389i	$-0.638072\pi$
$337$	−15.4311	−0.840583	−0.420292	+	0.907389i	$0.638072\pi$
$338$	31.8118	1.73034
$339$	−1.98717	−0.107928
$340$	48.5367	2.63227
$341$	19.4618	1.05392
$342$	5.21297	0.281885
$343$	−1.05141	−0.0567709
$344$	59.9893	3.23441
$345$	−1.33507	−0.0718776
$346$	31.4098	1.68860
$347$	−9.39939	−0.504586	−0.252293	−	0.967651i	$-0.581185\pi$
$347$	−9.39939	−0.504586	−0.252293	+	0.967651i	$0.581185\pi$
$348$	0	0
$349$	−24.5072	−1.31184	−0.655921	−	0.754830i	$-0.727720\pi$
$349$	−24.5072	−1.31184	−0.655921	+	0.754830i	$0.727720\pi$
$350$	−0.324293	−0.0173342
$351$	2.24762	0.119969
$352$	−36.2622	−1.93278
$353$	−24.1475	−1.28524	−0.642622	−	0.766183i	$-0.722153\pi$
$353$	−24.1475	−1.28524	−0.642622	+	0.766183i	$0.722153\pi$
$354$	11.3407	0.602753
$355$	12.6616	0.672011
$356$	32.4378	1.71920
$357$	−0.130087	−0.00688495
$358$	4.35864	0.230361
$359$	−15.2452	−0.804609	−0.402304	−	0.915506i	$-0.631791\pi$
$359$	−15.2452	−0.804609	−0.402304	+	0.915506i	$0.631791\pi$
$360$	52.2653	2.75462
$361$	−18.4908	−0.973197
$362$	39.9803	2.10132
$363$	−1.43926	−0.0755417
$364$	0.315905	0.0165579
$365$	−22.9713	−1.20237
$366$	4.07872	0.213198
$367$	−12.8020	−0.668261	−0.334130	−	0.942527i	$-0.608443\pi$
$367$	−12.8020	−0.668261	−0.334130	+	0.942527i	$0.608443\pi$
$368$	10.8734	0.566816
$369$	−18.7260	−0.974837
$370$	20.7338	1.07790
$371$	0.417434	0.0216721
$372$	−10.8005	−0.559980
$373$	−0.0195746	−0.00101354	−0.000506768	−	1.00000i	$-0.500161\pi$
$373$	−0.0195746	−0.00101354	−0.000506768		1.00000i	$0.500161\pi$
$374$	−38.7525	−2.00384
$375$	3.78810	0.195616
$376$	76.4297	3.94156
$377$	0	0
$378$	−0.498827	−0.0256569
$379$	−5.23573	−0.268941	−0.134471	−	0.990918i	$-0.542933\pi$
$379$	−5.23573	−0.268941	−0.134471	+	0.990918i	$0.542933\pi$
$380$	8.78860	0.450845
$381$	−2.56469	−0.131393
$382$	−53.2935	−2.72673
$383$	−0.274844	−0.0140438	−0.00702192	−	0.999975i	$-0.502235\pi$
$383$	−0.274844	−0.0140438	−0.00702192	+	0.999975i	$0.502235\pi$
$384$	−0.989633	−0.0505020
$385$	0.732510	0.0373322
$386$	8.15711	0.415186
$387$	23.3362	1.18625
$388$	−54.4903	−2.76633
$389$	−7.53699	−0.382140	−0.191070	−	0.981576i	$-0.561196\pi$
$389$	−7.53699	−0.382140	−0.191070	+	0.981576i	$0.561196\pi$
$390$	2.59923	0.131617
$391$	4.64118	0.234715
$392$	50.4696	2.54910
$393$	2.91036	0.146808
$394$	−7.39197	−0.372402
$395$	34.0346	1.71246
$396$	−50.6181	−2.54366
$397$	16.7344	0.839877	0.419938	−	0.907553i	$-0.362052\pi$
$397$	16.7344	0.839877	0.419938	+	0.907553i	$0.362052\pi$
$398$	49.7558	2.49403
$399$	−0.0235551	−0.00117923
$400$	15.3141	0.765704
$401$	−8.05748	−0.402371	−0.201186	−	0.979553i	$-0.564479\pi$
$401$	−8.05748	−0.402371	−0.201186	+	0.979553i	$0.564479\pi$
$402$	5.64766	0.281680
$403$	4.53789	0.226049
$404$	45.0459	2.24112
$405$	18.8373	0.936034
$406$	0	0
$407$	−11.6657	−0.578247
$408$	12.4939	0.618540
$409$	−9.50361	−0.469923	−0.234962	−	0.972005i	$-0.575496\pi$
$409$	−9.50361	−0.469923	−0.234962	+	0.972005i	$0.575496\pi$
$410$	−44.7999	−2.21251
$411$	−0.684845	−0.0337809
$412$	−43.8285	−2.15927
$413$	0.745214	0.0366696
$414$	8.60267	0.422798
$415$	43.8321	2.15163
$416$	−8.45522	−0.414551
$417$	−4.67526	−0.228948
$418$	−7.01695	−0.343210
$419$	23.2480	1.13574	0.567870	−	0.823118i	$-0.307768\pi$
$419$	23.2480	1.13574	0.567870	+	0.823118i	$0.307768\pi$
$420$	−0.406512	−0.0198358
$421$	17.4532	0.850618	0.425309	−	0.905048i	$-0.360166\pi$
$421$	17.4532	0.850618	0.425309	+	0.905048i	$0.360166\pi$
$422$	−46.5669	−2.26684
$423$	29.7317	1.44560
$424$	−40.0914	−1.94701
$425$	6.53662	0.317073
$426$	5.61022	0.271816
$427$	0.268018	0.0129703
$428$	−23.2344	−1.12308
$429$	−1.46243	−0.0706069
$430$	55.8294	2.69233
$431$	−32.7850	−1.57920	−0.789598	−	0.613624i	$-0.789711\pi$
$431$	−32.7850	−1.57920	−0.789598	+	0.613624i	$0.789711\pi$
$432$	23.5561	1.13334
$433$	−5.02086	−0.241287	−0.120644	−	0.992696i	$-0.538496\pi$
$433$	−5.02086	−0.241287	−0.120644	+	0.992696i	$0.538496\pi$
$434$	−1.00712	−0.0483433
$435$	0	0
$436$	20.6859	0.990677
$437$	0.840384	0.0402010
$438$	−10.1783	−0.486338
$439$	−21.3691	−1.01989	−0.509946	−	0.860207i	$-0.670334\pi$
$439$	−21.3691	−1.01989	−0.509946	+	0.860207i	$0.670334\pi$
$440$	−70.3520	−3.35390
$441$	19.6330	0.934906
$442$	−9.03587	−0.429793
$443$	39.8292	1.89234	0.946171	−	0.323668i	$-0.104916\pi$
$443$	39.8292	1.89234	0.946171	+	0.323668i	$0.104916\pi$
$444$	6.47397	0.307241
$445$	17.5380	0.831380
$446$	−46.1435	−2.18496
$447$	−2.38672	−0.112888
$448$	0.489111	0.0231083
$449$	−9.92661	−0.468466	−0.234233	−	0.972181i	$-0.575258\pi$
$449$	−9.92661	−0.468466	−0.234233	+	0.972181i	$0.575258\pi$
$450$	12.1160	0.571152
$451$	25.2063	1.18692
$452$	21.5873	1.01538
$453$	−7.83858	−0.368289
$454$	−57.8044	−2.71290
$455$	0.170799	0.00800716
$456$	2.26228	0.105941
$457$	14.6244	0.684102	0.342051	−	0.939681i	$-0.388878\pi$
$457$	14.6244	0.684102	0.342051	+	0.939681i	$0.388878\pi$
$458$	25.9414	1.21216
$459$	10.0546	0.469310
$460$	14.5033	0.676221
$461$	−18.8523	−0.878037	−0.439019	−	0.898478i	$-0.644674\pi$
$461$	−18.8523	−0.878037	−0.439019	+	0.898478i	$0.644674\pi$
$462$	0.324566	0.0151002
$463$	−35.4987	−1.64977	−0.824883	−	0.565304i	$-0.808759\pi$
$463$	−35.4987	−1.64977	−0.824883	+	0.565304i	$0.808759\pi$
$464$	0	0
$465$	−5.83944	−0.270798
$466$	52.2478	2.42033
$467$	−6.06786	−0.280787	−0.140394	−	0.990096i	$-0.544837\pi$
$467$	−6.06786	−0.280787	−0.140394	+	0.990096i	$0.544837\pi$
$468$	−11.8026	−0.545574
$469$	0.371115	0.0171365
$470$	71.1298	3.28097
$471$	−4.15773	−0.191578
$472$	−71.5721	−3.29437
$473$	−31.4119	−1.44432
$474$	15.0803	0.692660
$475$	1.18359	0.0543070
$476$	1.41319	0.0647732
$477$	−15.5958	−0.714083
$478$	3.07310	0.140560
$479$	−3.51910	−0.160792	−0.0803958	−	0.996763i	$-0.525618\pi$
$479$	−3.51910	−0.160792	−0.0803958	+	0.996763i	$0.525618\pi$
$480$	10.8803	0.496617
$481$	−2.72008	−0.124025
$482$	−30.7898	−1.40244
$483$	−0.0388716	−0.00176872
$484$	15.6352	0.710692
$485$	−29.4609	−1.33775
$486$	28.2649	1.28212
$487$	7.87745	0.356961	0.178481	−	0.983943i	$-0.442882\pi$
$487$	7.87745	0.356961	0.178481	+	0.983943i	$0.442882\pi$
$488$	−25.7411	−1.16524
$489$	−4.59451	−0.207771
$490$	46.9699	2.12188
$491$	−21.8303	−0.985187	−0.492594	−	0.870259i	$-0.663951\pi$
$491$	−21.8303	−0.985187	−0.492594	+	0.870259i	$0.663951\pi$
$492$	−13.9884	−0.630646
$493$	0	0
$494$	−1.63614	−0.0736132
$495$	−27.3674	−1.23007
$496$	47.5591	2.13547
$497$	0.368654	0.0165364
$498$	19.4214	0.870295
$499$	−29.6115	−1.32559	−0.662796	−	0.748800i	$-0.730630\pi$
$499$	−29.6115	−1.32559	−0.662796	+	0.748800i	$0.730630\pi$
$500$	−41.1514	−1.84035
$501$	−0.0748793	−0.00334536
$502$	19.9476	0.890306
$503$	14.0757	0.627603	0.313802	−	0.949489i	$-0.398397\pi$
$503$	14.0757	0.627603	0.313802	+	0.949489i	$0.398397\pi$
$504$	1.52174	0.0677839
$505$	24.3547	1.08377
$506$	−11.5797	−0.514779
$507$	5.37041	0.238508
$508$	27.8612	1.23614
$509$	−28.1153	−1.24619	−0.623095	−	0.782146i	$-0.714125\pi$
$509$	−28.1153	−1.24619	−0.623095	+	0.782146i	$0.714125\pi$
$510$	11.6275	0.514875
$511$	−0.668828	−0.0295872
$512$	44.6351	1.97261
$513$	1.82060	0.0803815
$514$	−29.5970	−1.30547
$515$	−23.6965	−1.04419
$516$	17.4323	0.767413
$517$	−40.0205	−1.76010
$518$	0.603682	0.0265243
$519$	5.30254	0.232756
$520$	−16.4039	−0.719359
$521$	30.6374	1.34225	0.671125	−	0.741344i	$-0.265811\pi$
$521$	30.6374	1.34225	0.671125	+	0.741344i	$0.265811\pi$
$522$	0	0
$523$	31.1728	1.36309	0.681546	−	0.731775i	$-0.261308\pi$
$523$	31.1728	1.36309	0.681546	+	0.731775i	$0.261308\pi$
$524$	−31.6163	−1.38116
$525$	−0.0547466	−0.00238934
$526$	43.5816	1.90025
$527$	20.3000	0.884283
$528$	−15.3269	−0.667019
$529$	−21.6132	−0.939703
$530$	−37.3113	−1.62070
$531$	−27.8420	−1.20824
$532$	0.255887	0.0110941
$533$	5.87732	0.254575
$534$	7.77085	0.336278
$535$	−12.5620	−0.543104
$536$	−35.6427	−1.53953
$537$	0.735817	0.0317529
$538$	57.0811	2.46094
$539$	−26.4272	−1.13830
$540$	31.4199	1.35210
$541$	22.3769	0.962058	0.481029	−	0.876705i	$-0.340263\pi$
$541$	22.3769	0.962058	0.481029	+	0.876705i	$0.340263\pi$
$542$	17.0312	0.731552
$543$	6.74940	0.289645
$544$	−37.8240	−1.62169
$545$	11.1841	0.479076
$546$	0.0756787	0.00323875
$547$	−12.2963	−0.525752	−0.262876	−	0.964830i	$-0.584671\pi$
$547$	−12.2963	−0.525752	−0.262876	+	0.964830i	$0.584671\pi$
$548$	7.43971	0.317809
$549$	−10.0134	−0.427363
$550$	−16.3088	−0.695408
$551$	0	0
$552$	3.73332	0.158901
$553$	0.990943	0.0421392
$554$	−56.0244	−2.38025
$555$	3.50024	0.148577
$556$	50.7890	2.15393
$557$	6.00766	0.254553	0.127276	−	0.991867i	$-0.459377\pi$
$557$	6.00766	0.254553	0.127276	+	0.991867i	$0.459377\pi$
$558$	37.6271	1.59288
$559$	−7.32428	−0.309784
$560$	1.79004	0.0756432
$561$	−6.54212	−0.276208
$562$	22.6338	0.954749
$563$	−28.0097	−1.18047	−0.590233	−	0.807233i	$-0.700964\pi$
$563$	−28.0097	−1.18047	−0.590233	+	0.807233i	$0.700964\pi$
$564$	22.2097	0.935197
$565$	11.6715	0.491023
$566$	−57.2241	−2.40531
$567$	0.548464	0.0230333
$568$	−35.4064	−1.48562
$569$	34.6764	1.45371	0.726856	−	0.686790i	$-0.240981\pi$
$569$	34.6764	1.45371	0.726856	+	0.686790i	$0.240981\pi$
$570$	2.10541	0.0881858
$571$	4.37818	0.183221	0.0916106	−	0.995795i	$-0.470799\pi$
$571$	4.37818	0.183221	0.0916106	+	0.995795i	$0.470799\pi$
$572$	15.8869	0.664266
$573$	−8.99690	−0.375851
$574$	−1.30439	−0.0544440
$575$	1.95322	0.0814548
$576$	−18.2737	−0.761406
$577$	−44.8517	−1.86720	−0.933600	−	0.358317i	$-0.883351\pi$
$577$	−44.8517	−1.86720	−0.933600	+	0.358317i	$0.883351\pi$
$578$	3.81999	0.158891
$579$	1.37707	0.0572289
$580$	0	0
$581$	1.27621	0.0529459
$582$	−13.0538	−0.541096
$583$	20.9928	0.869434
$584$	64.2359	2.65810
$585$	−6.38123	−0.263831
$586$	5.79327	0.239318
$587$	6.40778	0.264477	0.132239	−	0.991218i	$-0.457783\pi$
$587$	6.40778	0.264477	0.132239	+	0.991218i	$0.457783\pi$
$588$	14.6660	0.604814
$589$	3.67575	0.151457
$590$	−66.6090	−2.74225
$591$	−1.24790	−0.0513317
$592$	−28.5076	−1.17166
$593$	−1.41754	−0.0582114	−0.0291057	−	0.999576i	$-0.509266\pi$
$593$	−1.41754	−0.0582114	−0.0291057	+	0.999576i	$0.509266\pi$
$594$	−25.0861	−1.02930
$595$	0.764058	0.0313233
$596$	25.9278	1.06204
$597$	8.39968	0.343776
$598$	−2.70002	−0.110412
$599$	−37.2538	−1.52215	−0.761075	−	0.648663i	$-0.775328\pi$
$599$	−37.2538	−1.52215	−0.761075	+	0.648663i	$0.775328\pi$
$600$	5.25799	0.214657
$601$	25.3174	1.03272	0.516360	−	0.856372i	$-0.327287\pi$
$601$	25.3174	1.03272	0.516360	+	0.856372i	$0.327287\pi$
$602$	1.62552	0.0662511
$603$	−13.8653	−0.564637
$604$	85.1533	3.46484
$605$	8.45340	0.343679
$606$	10.7913	0.438365
$607$	29.8274	1.21066	0.605328	−	0.795976i	$-0.293042\pi$
$607$	29.8274	1.21066	0.605328	+	0.795976i	$0.293042\pi$
$608$	−6.84883	−0.277757
$609$	0	0
$610$	−23.9561	−0.969953
$611$	−9.33154	−0.377514
$612$	−52.7982	−2.13424
$613$	31.5885	1.27585	0.637923	−	0.770100i	$-0.279794\pi$
$613$	31.5885	1.27585	0.637923	+	0.770100i	$0.279794\pi$
$614$	36.1114	1.45734
$615$	−7.56303	−0.304971
$616$	−2.04835	−0.0825306
$617$	−19.2746	−0.775968	−0.387984	−	0.921666i	$-0.626828\pi$
$617$	−19.2746	−0.775968	−0.387984	+	0.921666i	$0.626828\pi$
$618$	−10.4996	−0.422356
$619$	−44.9116	−1.80515	−0.902574	−	0.430534i	$-0.858325\pi$
$619$	−44.9116	−1.80515	−0.902574	+	0.430534i	$0.858325\pi$
$620$	63.4359	2.54765
$621$	3.00443	0.120564
$622$	3.67509	0.147358
$623$	0.510632	0.0204580
$624$	−3.57377	−0.143065
$625$	−30.5420	−1.22168
$626$	74.8168	2.99028
$627$	−1.18459	−0.0473079
$628$	45.1669	1.80235
$629$	−12.1681	−0.485175
$630$	1.41622	0.0564236
$631$	−38.1217	−1.51760	−0.758800	−	0.651324i	$-0.774214\pi$
$631$	−38.1217	−1.51760	−0.758800	+	0.651324i	$0.774214\pi$
$632$	−95.1725	−3.78576
$633$	−7.86134	−0.312460
$634$	58.0871	2.30693
$635$	15.0635	0.597778
$636$	−11.6501	−0.461958
$637$	−6.16199	−0.244147
$638$	0	0
$639$	−13.7733	−0.544864
$640$	5.81254	0.229761
$641$	36.9753	1.46044	0.730219	−	0.683214i	$-0.239418\pi$
$641$	36.9753	1.46044	0.730219	+	0.683214i	$0.239418\pi$
$642$	−5.56607	−0.219675
$643$	−17.0863	−0.673816	−0.336908	−	0.941537i	$-0.609381\pi$
$643$	−17.0863	−0.673816	−0.336908	+	0.941537i	$0.609381\pi$
$644$	0.422276	0.0166400
$645$	9.42501	0.371109
$646$	−7.31916	−0.287969
$647$	6.19335	0.243486	0.121743	−	0.992562i	$-0.461152\pi$
$647$	6.19335	0.243486	0.121743	+	0.992562i	$0.461152\pi$
$648$	−52.6758	−2.06930
$649$	37.4769	1.47110
$650$	−3.80270	−0.149154
$651$	−0.170020	−0.00666361
$652$	49.9117	1.95469
$653$	15.7585	0.616677	0.308339	−	0.951277i	$-0.400227\pi$
$653$	15.7585	0.616677	0.308339	+	0.951277i	$0.400227\pi$
$654$	4.95555	0.193777
$655$	−17.0938	−0.667909
$656$	61.5968	2.40495
$657$	24.9882	0.974881
$658$	2.07100	0.0807360
$659$	−16.6107	−0.647059	−0.323530	−	0.946218i	$-0.604870\pi$
$659$	−16.6107	−0.647059	−0.323530	+	0.946218i	$0.604870\pi$
$660$	−20.4436	−0.795765
$661$	−40.0401	−1.55738	−0.778691	−	0.627408i	$-0.784116\pi$
$661$	−40.0401	−1.55738	−0.778691	+	0.627408i	$0.784116\pi$
$662$	−24.3315	−0.945672
$663$	−1.52542	−0.0592423
$664$	−122.570	−4.75663
$665$	0.138349	0.00536494
$666$	−22.5542	−0.873959
$667$	0	0
$668$	0.813440	0.0314729
$669$	−7.78985	−0.301173
$670$	−33.1711	−1.28151
$671$	13.4787	0.520338
$672$	0.316789	0.0122204
$673$	−13.2244	−0.509764	−0.254882	−	0.966972i	$-0.582037\pi$
$673$	−13.2244	−0.509764	−0.254882	+	0.966972i	$0.582037\pi$
$674$	40.1584	1.54684
$675$	4.23144	0.162868
$676$	−58.3407	−2.24387
$677$	−23.8101	−0.915095	−0.457548	−	0.889185i	$-0.651272\pi$
$677$	−23.8101	−0.915095	−0.457548	+	0.889185i	$0.651272\pi$
$678$	5.17149	0.198610
$679$	−0.857779	−0.0329185
$680$	−73.3820	−2.81407
$681$	−9.75843	−0.373944
$682$	−50.6482	−1.93942
$683$	28.7668	1.10073	0.550365	−	0.834924i	$-0.314489\pi$
$683$	28.7668	1.10073	0.550365	+	0.834924i	$0.314489\pi$
$684$	−9.56022	−0.365544
$685$	4.02238	0.153687
$686$	2.73623	0.104470
$687$	4.37938	0.167084
$688$	−76.7616	−2.92651
$689$	4.89488	0.186480
$690$	3.47443	0.132269
$691$	27.1116	1.03137	0.515686	−	0.856777i	$-0.327537\pi$
$691$	27.1116	1.03137	0.515686	+	0.856777i	$0.327537\pi$
$692$	−57.6033	−2.18975
$693$	−0.796823	−0.0302688
$694$	24.4613	0.928540
$695$	27.4598	1.04161
$696$	0	0
$697$	26.2919	0.995875
$698$	63.7786	2.41406
$699$	8.82037	0.333617
$700$	0.594732	0.0224787
$701$	−24.5115	−0.925788	−0.462894	−	0.886414i	$-0.653189\pi$
$701$	−24.5115	−0.925788	−0.462894	+	0.886414i	$0.653189\pi$
$702$	−5.84930	−0.220768
$703$	−2.20330	−0.0830989
$704$	24.5975	0.927053
$705$	12.0080	0.452247
$706$	62.8425	2.36511
$707$	0.709107	0.0266687
$708$	−20.7981	−0.781641
$709$	16.5914	0.623104	0.311552	−	0.950229i	$-0.399151\pi$
$709$	16.5914	0.623104	0.311552	+	0.950229i	$0.399151\pi$
$710$	−32.9512	−1.23664
$711$	−37.0227	−1.38846
$712$	−49.0423	−1.83794
$713$	6.06588	0.227169
$714$	0.338544	0.0126697
$715$	8.58949	0.321229
$716$	−7.99344	−0.298729
$717$	0.518795	0.0193747
$718$	39.6746	1.48064
$719$	40.5018	1.51046	0.755231	−	0.655458i	$-0.227525\pi$
$719$	40.5018	1.51046	0.755231	+	0.655458i	$0.227525\pi$
$720$	−66.8780	−2.49240
$721$	−0.689942	−0.0256948
$722$	48.1211	1.79088
$723$	−5.19787	−0.193311
$724$	−73.3212	−2.72496
$725$	0	0
$726$	3.74559	0.139012
$727$	−31.7298	−1.17679	−0.588397	−	0.808572i	$-0.700241\pi$
$727$	−31.7298	−1.17679	−0.588397	+	0.808572i	$0.700241\pi$
$728$	−0.477613	−0.0177015
$729$	−17.1286	−0.634394
$730$	59.7815	2.21261
$731$	−32.7648	−1.21185
$732$	−7.48009	−0.276472
$733$	1.48939	0.0550118	0.0275059	−	0.999622i	$-0.491243\pi$
$733$	1.48939	0.0550118	0.0275059	+	0.999622i	$0.491243\pi$
$734$	33.3165	1.22974
$735$	7.92936	0.292479
$736$	−11.3022	−0.416606
$737$	18.6634	0.687476
$738$	48.7333	1.79390
$739$	−36.7246	−1.35094	−0.675468	−	0.737389i	$-0.736058\pi$
$739$	−36.7246	−1.35094	−0.675468	+	0.737389i	$0.736058\pi$
$740$	−38.0244	−1.39780
$741$	−0.276209	−0.0101468
$742$	−1.08635	−0.0398810
$743$	−31.1757	−1.14373	−0.571863	−	0.820349i	$-0.693779\pi$
$743$	−31.1757	−1.14373	−0.571863	+	0.820349i	$0.693779\pi$
$744$	16.3291	0.598655
$745$	14.0182	0.513588
$746$	0.0509418	0.00186511
$747$	−47.6805	−1.74454
$748$	71.0693	2.59855
$749$	−0.365753	−0.0133643
$750$	−9.85829	−0.359974
$751$	45.2999	1.65302	0.826509	−	0.562923i	$-0.190323\pi$
$751$	45.2999	1.65302	0.826509	+	0.562923i	$0.190323\pi$
$752$	−97.7986	−3.56635
$753$	3.36752	0.122719
$754$	0	0
$755$	46.0393	1.67554
$756$	0.914815	0.0332715
$757$	18.8863	0.686435	0.343218	−	0.939256i	$-0.388483\pi$
$757$	18.8863	0.686435	0.343218	+	0.939256i	$0.388483\pi$
$758$	13.6257	0.494906
$759$	−1.95486	−0.0709568
$760$	−13.2874	−0.481983
$761$	17.4270	0.631727	0.315863	−	0.948805i	$-0.397706\pi$
$761$	17.4270	0.631727	0.315863	+	0.948805i	$0.397706\pi$
$762$	6.67446	0.241790
$763$	0.325635	0.0117888
$764$	97.7365	3.53598
$765$	−28.5461	−1.03209
$766$	0.715264	0.0258435
$767$	8.73846	0.315527
$768$	8.29574	0.299347
$769$	27.2971	0.984357	0.492179	−	0.870494i	$-0.336201\pi$
$769$	27.2971	0.984357	0.492179	+	0.870494i	$0.336201\pi$
$770$	−1.90631	−0.0686988
$771$	−4.99650	−0.179945
$772$	−14.9596	−0.538406
$773$	−5.89778	−0.212128	−0.106064	−	0.994359i	$-0.533825\pi$
$773$	−5.89778	−0.212128	−0.106064	+	0.994359i	$0.533825\pi$
$774$	−60.7311	−2.18294
$775$	8.54316	0.306879
$776$	82.3831	2.95738
$777$	0.101912	0.00365609
$778$	19.6146	0.703216
$779$	4.76070	0.170570
$780$	−4.76681	−0.170679
$781$	18.5397	0.663402
$782$	−12.0784	−0.431923
$783$	0	0
$784$	−64.5804	−2.30644
$785$	24.4201	0.871591
$786$	−7.57403	−0.270157
$787$	−43.7976	−1.56122	−0.780609	−	0.625020i	$-0.785091\pi$
$787$	−43.7976	−1.56122	−0.780609	+	0.625020i	$0.785091\pi$
$788$	13.5564	0.482925
$789$	7.35736	0.261929
$790$	−88.5729	−3.15128
$791$	0.339825	0.0120828
$792$	76.5288	2.71933
$793$	3.14281	0.111604
$794$	−43.5503	−1.54554
$795$	−6.29881	−0.223396
$796$	−91.2487	−3.23422
$797$	−36.6686	−1.29887	−0.649435	−	0.760417i	$-0.724994\pi$
$797$	−36.6686	−1.29887	−0.649435	+	0.760417i	$0.724994\pi$
$798$	0.0613006	0.00217002
$799$	−41.7441	−1.47680
$800$	−15.9180	−0.562787
$801$	−19.0778	−0.674081
$802$	20.9691	0.740445
$803$	−33.6355	−1.18697
$804$	−10.3574	−0.365278
$805$	0.228309	0.00804684
$806$	−11.8096	−0.415975
$807$	9.63632	0.339214
$808$	−68.1043	−2.39590
$809$	18.7725	0.660007	0.330004	−	0.943980i	$-0.392950\pi$
$809$	18.7725	0.660007	0.330004	+	0.943980i	$0.392950\pi$
$810$	−49.0230	−1.72249
$811$	−33.9249	−1.19127	−0.595633	−	0.803257i	$-0.703099\pi$
$811$	−33.9249	−1.19127	−0.595633	+	0.803257i	$0.703099\pi$
$812$	0	0
$813$	2.87517	0.100837
$814$	30.3593	1.06409
$815$	26.9855	0.945260
$816$	−15.9870	−0.559659
$817$	−5.93275	−0.207561
$818$	24.7326	0.864754
$819$	−0.185794	−0.00649218
$820$	82.1599	2.86915
$821$	−12.5182	−0.436887	−0.218443	−	0.975850i	$-0.570098\pi$
$821$	−12.5182	−0.436887	−0.218443	+	0.975850i	$0.570098\pi$
$822$	1.78227	0.0621637
$823$	27.6813	0.964911	0.482456	−	0.875920i	$-0.339745\pi$
$823$	27.6813	0.964911	0.482456	+	0.875920i	$0.339745\pi$
$824$	66.2636	2.30840
$825$	−2.75321	−0.0958546
$826$	−1.93937	−0.0674794
$827$	29.5496	1.02754	0.513771	−	0.857928i	$-0.328248\pi$
$827$	29.5496	1.02754	0.513771	+	0.857928i	$0.328248\pi$
$828$	−15.7767	−0.548278
$829$	−1.28760	−0.0447203	−0.0223601	−	0.999750i	$-0.507118\pi$
$829$	−1.28760	−0.0447203	−0.0223601	+	0.999750i	$0.507118\pi$
$830$	−114.070	−3.95944
$831$	−9.45793	−0.328092
$832$	5.73537	0.198838
$833$	−27.5653	−0.955083
$834$	12.1671	0.421311
$835$	0.439798	0.0152198
$836$	12.8686	0.445070
$837$	13.1411	0.454222
$838$	−60.5016	−2.08999
$839$	−16.3512	−0.564505	−0.282253	−	0.959340i	$-0.591082\pi$
$839$	−16.3512	−0.564505	−0.282253	+	0.959340i	$0.591082\pi$
$840$	0.614600	0.0212057
$841$	0	0
$842$	−45.4209	−1.56531
$843$	3.82099	0.131602
$844$	85.4005	2.93961
$845$	−31.5427	−1.08510
$846$	−77.3749	−2.66020
$847$	0.246127	0.00845704
$848$	51.3004	1.76166
$849$	−9.66047	−0.331546
$850$	−17.0112	−0.583478
$851$	−3.63597	−0.124640
$852$	−10.2887	−0.352487
$853$	51.4321	1.76100	0.880501	−	0.474045i	$-0.157207\pi$
$853$	51.4321	1.76100	0.880501	+	0.474045i	$0.157207\pi$
$854$	−0.697500	−0.0238680
$855$	−5.16887	−0.176772
$856$	35.1278	1.20064
$857$	−19.4262	−0.663588	−0.331794	−	0.943352i	$-0.607654\pi$
$857$	−19.4262	−0.663588	−0.331794	+	0.943352i	$0.607654\pi$
$858$	3.80589	0.129931
$859$	23.9218	0.816201	0.408101	−	0.912937i	$-0.366191\pi$
$859$	23.9218	0.816201	0.408101	+	0.912937i	$0.366191\pi$
$860$	−102.387	−3.49137
$861$	−0.220204	−0.00750452
$862$	85.3209	2.90604
$863$	−28.4361	−0.967977	−0.483989	−	0.875074i	$-0.660812\pi$
$863$	−28.4361	−0.967977	−0.483989	+	0.875074i	$0.660812\pi$
$864$	−24.4851	−0.832999
$865$	−31.1441	−1.05893
$866$	13.0665	0.444018
$867$	0.644883	0.0219014
$868$	1.84699	0.0626908
$869$	49.8347	1.69053
$870$	0	0
$871$	4.35173	0.147453
$872$	−31.2748	−1.05910
$873$	32.0476	1.08465
$874$	−2.18705	−0.0739780
$875$	−0.647800	−0.0218996
$876$	18.6663	0.630675
$877$	−3.71106	−0.125314	−0.0626568	−	0.998035i	$-0.519957\pi$
$877$	−3.71106	−0.125314	−0.0626568	+	0.998035i	$0.519957\pi$
$878$	55.6118	1.87681
$879$	0.978009	0.0329874
$880$	90.0216	3.03463
$881$	11.9260	0.401796	0.200898	−	0.979612i	$-0.435614\pi$
$881$	11.9260	0.401796	0.200898	+	0.979612i	$0.435614\pi$
$882$	−51.0938	−1.72042
$883$	5.89920	0.198524	0.0992619	−	0.995061i	$-0.468352\pi$
$883$	5.89920	0.198524	0.0992619	+	0.995061i	$0.468352\pi$
$884$	16.5712	0.557348
$885$	−11.2448	−0.377990
$886$	−103.653	−3.48229
$887$	24.5541	0.824445	0.412223	−	0.911083i	$-0.364753\pi$
$887$	24.5541	0.824445	0.412223	+	0.911083i	$0.364753\pi$
$888$	−9.78791	−0.328461
$889$	0.438587	0.0147097
$890$	−45.6415	−1.52991
$891$	27.5823	0.924043
$892$	84.6239	2.83342
$893$	−7.55866	−0.252941
$894$	6.21129	0.207737
$895$	−4.32177	−0.144461
$896$	0.169237	0.00565380
$897$	−0.455812	−0.0152191
$898$	25.8334	0.862072
$899$	0	0
$900$	−22.2198	−0.740661
$901$	21.8970	0.729494
$902$	−65.5978	−2.18417
$903$	0.274417	0.00913201
$904$	−32.6376	−1.08551
$905$	−39.6421	−1.31775
$906$	20.3994	0.677726
$907$	40.2160	1.33535	0.667675	−	0.744453i	$-0.267289\pi$
$907$	40.2160	1.33535	0.667675	+	0.744453i	$0.267289\pi$
$908$	106.009	3.51804
$909$	−26.4930	−0.878718
$910$	−0.444493	−0.0147348
$911$	13.5344	0.448415	0.224208	−	0.974541i	$-0.428021\pi$
$911$	13.5344	0.448415	0.224208	+	0.974541i	$0.428021\pi$
$912$	−2.89479	−0.0958562
$913$	64.1806	2.12407
$914$	−38.0592	−1.25889
$915$	−4.04422	−0.133698
$916$	−47.5748	−1.57191
$917$	−0.497699	−0.0164355
$918$	−26.1666	−0.863625
$919$	55.1125	1.81799	0.908997	−	0.416803i	$-0.136849\pi$
$919$	55.1125	1.81799	0.908997	+	0.416803i	$0.136849\pi$
$920$	−21.9274	−0.722924
$921$	6.09626	0.200879
$922$	49.0619	1.61577
$923$	4.32288	0.142289
$924$	−0.595231	−0.0195817
$925$	−5.12089	−0.168374
$926$	92.3833	3.03590
$927$	25.7770	0.846627
$928$	0	0
$929$	4.05714	0.133111	0.0665553	−	0.997783i	$-0.478799\pi$
$929$	4.05714	0.133111	0.0665553	+	0.997783i	$0.478799\pi$
$930$	15.1968	0.498322
$931$	−4.99129	−0.163583
$932$	−95.8188	−3.13865
$933$	0.620421	0.0203117
$934$	15.7912	0.516705
$935$	38.4246	1.25662
$936$	17.8441	0.583254
$937$	−30.1884	−0.986213	−0.493106	−	0.869969i	$-0.664139\pi$
$937$	−30.1884	−0.986213	−0.493106	+	0.869969i	$0.664139\pi$
$938$	−0.965804	−0.0315346
$939$	12.6304	0.412178
$940$	−130.447	−4.25471
$941$	−8.38002	−0.273181	−0.136590	−	0.990628i	$-0.543614\pi$
$941$	−8.38002	−0.273181	−0.136590	+	0.990628i	$0.543614\pi$
$942$	10.8202	0.352542
$943$	7.85630	0.255836
$944$	91.5828	2.98077
$945$	0.494607	0.0160896
$946$	81.7476	2.65784
$947$	1.82587	0.0593328	0.0296664	−	0.999560i	$-0.490556\pi$
$947$	1.82587	0.0593328	0.0296664	+	0.999560i	$0.490556\pi$
$948$	−27.6562	−0.898231
$949$	−7.84275	−0.254587
$950$	−3.08023	−0.0999359
$951$	9.80616	0.317986
$952$	−2.13657	−0.0692468
$953$	46.8893	1.51889	0.759447	−	0.650569i	$-0.225470\pi$
$953$	46.8893	1.51889	0.759447	+	0.650569i	$0.225470\pi$
$954$	40.5871	1.31406
$955$	52.8426	1.70995
$956$	−5.63585	−0.182276
$957$	0	0
$958$	9.15823	0.295889
$959$	0.117115	0.00378184
$960$	−7.38038	−0.238201
$961$	−4.46852	−0.144146
$962$	7.07884	0.228231
$963$	13.6650	0.440347
$964$	56.4664	1.81866
$965$	−8.08810	−0.260365
$966$	0.101161	0.00325480
$967$	20.1580	0.648239	0.324119	−	0.946016i	$-0.394932\pi$
$967$	20.1580	0.648239	0.324119	+	0.946016i	$0.394932\pi$
$968$	−23.6387	−0.759775
$969$	−1.23561	−0.0396934
$970$	76.6703	2.46174
$971$	−16.9188	−0.542949	−0.271474	−	0.962446i	$-0.587511\pi$
$971$	−16.9188	−0.542949	−0.271474	+	0.962446i	$0.587511\pi$
$972$	−51.8358	−1.66263
$973$	0.799513	0.0256312
$974$	−20.5006	−0.656881
$975$	−0.641964	−0.0205593
$976$	32.9380	1.05432
$977$	−1.03630	−0.0331543	−0.0165772	−	0.999863i	$-0.505277\pi$
$977$	−1.03630	−0.0331543	−0.0165772	+	0.999863i	$0.505277\pi$
$978$	11.9569	0.382340
$979$	25.6798	0.820729
$980$	−86.1395	−2.75162
$981$	−12.1661	−0.388434
$982$	56.8120	1.81294
$983$	−16.2306	−0.517676	−0.258838	−	0.965921i	$-0.583340\pi$
$983$	−16.2306	−0.517676	−0.258838	+	0.965921i	$0.583340\pi$
$984$	21.1489	0.674202
$985$	7.32943	0.233535
$986$	0	0
$987$	0.349622	0.0111286
$988$	3.00056	0.0954605
$989$	−9.79048	−0.311319
$990$	71.2220	2.26358
$991$	37.1422	1.17986	0.589930	−	0.807454i	$-0.299155\pi$
$991$	37.1422	1.17986	0.589930	+	0.807454i	$0.299155\pi$
$992$	−49.4347	−1.56955
$993$	−4.10760	−0.130351
$994$	−0.959400	−0.0304303
$995$	−49.3349	−1.56402
$996$	−35.6175	−1.12859
$997$	36.1372	1.14448	0.572238	−	0.820088i	$-0.306075\pi$
$997$	36.1372	1.14448	0.572238	+	0.820088i	$0.306075\pi$
$998$	77.0621	2.43936
$999$	−7.87695	−0.249216

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.2.a.k.1.1		12
3.2	odd	2		7569.2.a.bp.1.12		12
29.2	odd	28		841.2.e.e.236.1		12
29.3	odd	28		29.2.e.a.9.1	✓	12
29.4	even	14		841.2.d.m.190.1		24
29.5	even	14		841.2.d.k.605.1		24
29.6	even	14		841.2.d.k.645.1		24
29.7	even	7		841.2.d.m.571.4		24
29.8	odd	28		841.2.e.a.267.1		12
29.9	even	14		841.2.d.l.574.4		24
29.10	odd	28		29.2.e.a.13.1	yes	12
29.11	odd	28		841.2.e.a.63.1		12
29.12	odd	4		841.2.b.e.840.1		12
29.13	even	14		841.2.d.l.778.4		24
29.14	odd	28		841.2.e.f.196.2		12
29.15	odd	28		841.2.e.e.196.1		12
29.16	even	7		841.2.d.l.778.1		24
29.17	odd	4		841.2.b.e.840.12		12
29.18	odd	28		841.2.e.h.63.2		12
29.19	odd	28		841.2.e.i.651.2		12
29.20	even	7		841.2.d.l.574.1		24
29.21	odd	28		841.2.e.h.267.2		12
29.22	even	14		841.2.d.m.571.1		24
29.23	even	7		841.2.d.k.645.4		24
29.24	even	7		841.2.d.k.605.4		24
29.25	even	7		841.2.d.m.190.4		24
29.26	odd	28		841.2.e.i.270.2		12
29.27	odd	28		841.2.e.f.236.2		12
29.28	even	2	inner	841.2.a.k.1.12		12
87.32	even	28		261.2.o.a.154.2		12
87.68	even	28		261.2.o.a.100.2		12
87.86	odd	2		7569.2.a.bp.1.1		12
116.3	even	28		464.2.y.d.241.1		12
116.39	even	28		464.2.y.d.129.1		12
145.3	even	28		725.2.p.a.299.1		24
145.32	even	28		725.2.p.a.299.4		24
145.39	odd	28		725.2.q.a.651.2		12
145.68	even	28		725.2.p.a.274.4		24
145.97	even	28		725.2.p.a.274.1		24
145.119	odd	28		725.2.q.a.676.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.e.a.9.1	✓	12	29.3	odd	28
29.2.e.a.13.1	yes	12	29.10	odd	28
261.2.o.a.100.2		12	87.68	even	28
261.2.o.a.154.2		12	87.32	even	28
464.2.y.d.129.1		12	116.39	even	28
464.2.y.d.241.1		12	116.3	even	28
725.2.p.a.274.1		24	145.97	even	28
725.2.p.a.274.4		24	145.68	even	28
725.2.p.a.299.1		24	145.3	even	28
725.2.p.a.299.4		24	145.32	even	28
725.2.q.a.651.2		12	145.39	odd	28
725.2.q.a.676.2		12	145.119	odd	28
841.2.a.k.1.1		12	1.1	even	1	trivial
841.2.a.k.1.12		12	29.28	even	2	inner
841.2.b.e.840.1		12	29.12	odd	4
841.2.b.e.840.12		12	29.17	odd	4
841.2.d.k.605.1		24	29.5	even	14
841.2.d.k.605.4		24	29.24	even	7
841.2.d.k.645.1		24	29.6	even	14
841.2.d.k.645.4		24	29.23	even	7
841.2.d.l.574.1		24	29.20	even	7
841.2.d.l.574.4		24	29.9	even	14
841.2.d.l.778.1		24	29.16	even	7
841.2.d.l.778.4		24	29.13	even	14
841.2.d.m.190.1		24	29.4	even	14
841.2.d.m.190.4		24	29.25	even	7
841.2.d.m.571.1		24	29.22	even	14
841.2.d.m.571.4		24	29.7	even	7
841.2.e.a.63.1		12	29.11	odd	28
841.2.e.a.267.1		12	29.8	odd	28
841.2.e.e.196.1		12	29.15	odd	28
841.2.e.e.236.1		12	29.2	odd	28
841.2.e.f.196.2		12	29.14	odd	28
841.2.e.f.236.2		12	29.27	odd	28
841.2.e.h.63.2		12	29.18	odd	28
841.2.e.h.267.2		12	29.21	odd	28
841.2.e.i.270.2		12	29.26	odd	28
841.2.e.i.651.2		12	29.19	odd	28
7569.2.a.bp.1.1		12	87.86	odd	2
7569.2.a.bp.1.12		12	3.2	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	841.2.a.k.1.1

Level	$841$
Weight	$2$
Character	841.1
Self dual	yes
Analytic conductor	$6.715$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$