LMFDB - Embedded newform 7865.2.a.bm.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7865,2,Mod(1,7865)] 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("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7865 = 5 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7865.a (trivial)

Newform invariants

comment:select newform

sage:traces = [26,-1,9,31,-26,-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:	$62.8023411897$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 715)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	7865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.352577 q^{2} +2.29811 q^{3} -1.87569 q^{4} -1.00000 q^{5} +0.810261 q^{6} -1.01025 q^{7} -1.36648 q^{8} +2.28133 q^{9} -0.352577 q^{10} -4.31055 q^{12} +1.00000 q^{13} -0.356190 q^{14} -2.29811 q^{15} +3.26959 q^{16} +5.19650 q^{17} +0.804342 q^{18} +1.01604 q^{19} +1.87569 q^{20} -2.32167 q^{21} +8.26698 q^{23} -3.14032 q^{24} +1.00000 q^{25} +0.352577 q^{26} -1.65159 q^{27} +1.89491 q^{28} -3.63516 q^{29} -0.810261 q^{30} -5.64000 q^{31} +3.88574 q^{32} +1.83216 q^{34} +1.01025 q^{35} -4.27906 q^{36} -7.92615 q^{37} +0.358232 q^{38} +2.29811 q^{39} +1.36648 q^{40} -7.03652 q^{41} -0.818565 q^{42} -4.10772 q^{43} -2.28133 q^{45} +2.91474 q^{46} +4.12792 q^{47} +7.51389 q^{48} -5.97940 q^{49} +0.352577 q^{50} +11.9421 q^{51} -1.87569 q^{52} +6.10240 q^{53} -0.582313 q^{54} +1.38048 q^{56} +2.33498 q^{57} -1.28167 q^{58} +9.82561 q^{59} +4.31055 q^{60} -1.21453 q^{61} -1.98853 q^{62} -2.30471 q^{63} -5.16916 q^{64} -1.00000 q^{65} +0.455029 q^{67} -9.74701 q^{68} +18.9985 q^{69} +0.356190 q^{70} +12.4706 q^{71} -3.11738 q^{72} -0.0209630 q^{73} -2.79458 q^{74} +2.29811 q^{75} -1.90578 q^{76} +0.810261 q^{78} +15.3937 q^{79} -3.26959 q^{80} -10.6395 q^{81} -2.48091 q^{82} -4.74746 q^{83} +4.35473 q^{84} -5.19650 q^{85} -1.44828 q^{86} -8.35402 q^{87} +12.7455 q^{89} -0.804342 q^{90} -1.01025 q^{91} -15.5063 q^{92} -12.9614 q^{93} +1.45541 q^{94} -1.01604 q^{95} +8.92986 q^{96} +4.47793 q^{97} -2.10820 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - q^{2} + 9 q^{3} + 31 q^{4} - 26 q^{5} - 9 q^{6} - 3 q^{7} - 3 q^{8} + 29 q^{9} + q^{10} + 16 q^{12} + 26 q^{13} - 7 q^{14} - 9 q^{15} + 45 q^{16} + 8 q^{17} + 11 q^{18} + 5 q^{19} - 31 q^{20} + 3 q^{21}+ \cdots - 6 q^{98}+O(q^{100}) $ 26 * q - q^2 + 9 * q^3 + 31 * q^4 - 26 * q^5 - 9 * q^6 - 3 * q^7 - 3 * q^8 + 29 * q^9 + q^10 + 16 * q^12 + 26 * q^13 - 7 * q^14 - 9 * q^15 + 45 * q^16 + 8 * q^17 + 11 * q^18 + 5 * q^19 - 31 * q^20 + 3 * q^21 + 19 * q^23 - 36 * q^24 + 26 * q^25 - q^26 + 57 * q^27 - 45 * q^28 + 20 * q^29 + 9 * q^30 + 43 * q^31 - 17 * q^32 - 4 * q^34 + 3 * q^35 + 20 * q^36 + 23 * q^37 + 2 * q^38 + 9 * q^39 + 3 * q^40 + 19 * q^41 + 4 * q^42 + 32 * q^43 - 29 * q^45 - 43 * q^46 + 39 * q^47 + 62 * q^48 + 47 * q^49 - q^50 + 23 * q^51 + 31 * q^52 + 11 * q^53 - 26 * q^54 - 40 * q^56 - 46 * q^57 + 40 * q^58 - 8 * q^59 - 16 * q^60 + 9 * q^61 + 34 * q^62 - 23 * q^63 + 39 * q^64 - 26 * q^65 + 56 * q^67 + 5 * q^68 + 32 * q^69 + 7 * q^70 - 7 * q^71 - 91 * q^72 + 16 * q^73 + 78 * q^74 + 9 * q^75 + 24 * q^76 - 9 * q^78 + 13 * q^79 - 45 * q^80 + 26 * q^81 + 62 * q^82 - 34 * q^83 - 15 * q^84 - 8 * q^85 - 23 * q^86 + 17 * q^87 - 5 * q^89 - 11 * q^90 - 3 * q^91 + 112 * q^92 + 74 * q^93 - 10 * q^94 - 5 * q^95 - 31 * q^96 + 52 * q^97 - 6 * 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 $

$2$	0.352577	0.249309	0.124655	−	0.992200i	$-0.460218\pi$
$2$	0.352577	0.249309	0.124655	+	0.992200i	$0.460218\pi$
$3$	2.29811	1.32682	0.663408	−	0.748258i	$-0.269109\pi$
$3$	2.29811	1.32682	0.663408	+	0.748258i	$0.269109\pi$
$4$	−1.87569	−0.937845
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.810261	0.330788
$7$	−1.01025	−0.381838	−0.190919	−	0.981606i	$-0.561147\pi$
$7$	−1.01025	−0.381838	−0.190919	+	0.981606i	$0.561147\pi$
$8$	−1.36648	−0.483123
$9$	2.28133	0.760442
$10$	−0.352577	−0.111494
$11$	0	0
$11$	0	0
$12$	−4.31055	−1.24435
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.356190	−0.0951958
$15$	−2.29811	−0.593370
$16$	3.26959	0.817398
$17$	5.19650	1.26034	0.630168	−	0.776459i	$-0.282986\pi$
$17$	5.19650	1.26034	0.630168	+	0.776459i	$0.282986\pi$
$18$	0.804342	0.189585
$19$	1.01604	0.233096	0.116548	−	0.993185i	$-0.462817\pi$
$19$	1.01604	0.233096	0.116548	+	0.993185i	$0.462817\pi$
$20$	1.87569	0.419417
$21$	−2.32167	−0.506629
$22$	0	0
$23$	8.26698	1.72378	0.861892	−	0.507091i	$-0.169279\pi$
$23$	8.26698	1.72378	0.861892	+	0.507091i	$0.169279\pi$
$24$	−3.14032	−0.641015
$25$	1.00000	0.200000
$26$	0.352577	0.0691459
$27$	−1.65159	−0.317849
$28$	1.89491	0.358105
$29$	−3.63516	−0.675033	−0.337516	−	0.941320i	$-0.609587\pi$
$29$	−3.63516	−0.675033	−0.337516	+	0.941320i	$0.609587\pi$
$30$	−0.810261	−0.147933
$31$	−5.64000	−1.01297	−0.506487	−	0.862248i	$-0.669056\pi$
$31$	−5.64000	−1.01297	−0.506487	+	0.862248i	$0.669056\pi$
$32$	3.88574	0.686908
$33$	0	0
$34$	1.83216	0.314213
$35$	1.01025	0.170763
$36$	−4.27906	−0.713177
$37$	−7.92615	−1.30305	−0.651526	−	0.758626i	$-0.725871\pi$
$37$	−7.92615	−1.30305	−0.651526	+	0.758626i	$0.725871\pi$
$38$	0.358232	0.0581129
$39$	2.29811	0.367993
$40$	1.36648	0.216059
$41$	−7.03652	−1.09892	−0.549460	−	0.835520i	$-0.685166\pi$
$41$	−7.03652	−1.09892	−0.549460	+	0.835520i	$0.685166\pi$
$42$	−0.818565	−0.126307
$43$	−4.10772	−0.626421	−0.313211	−	0.949684i	$-0.601405\pi$
$43$	−4.10772	−0.626421	−0.313211	+	0.949684i	$0.601405\pi$
$44$	0	0
$45$	−2.28133	−0.340080
$46$	2.91474	0.429755
$47$	4.12792	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$47$	4.12792	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$48$	7.51389	1.08454
$49$	−5.97940	−0.854200
$50$	0.352577	0.0498618
$51$	11.9421	1.67223
$52$	−1.87569	−0.260111
$53$	6.10240	0.838229	0.419114	−	0.907933i	$-0.362341\pi$
$53$	6.10240	0.838229	0.419114	+	0.907933i	$0.362341\pi$
$54$	−0.582313	−0.0792428
$55$	0	0
$56$	1.38048	0.184475
$57$	2.33498	0.309275
$58$	−1.28167	−0.168292
$59$	9.82561	1.27919	0.639593	−	0.768714i	$-0.279103\pi$
$59$	9.82561	1.27919	0.639593	+	0.768714i	$0.279103\pi$
$60$	4.31055	0.556489
$61$	−1.21453	−0.155504	−0.0777522	−	0.996973i	$-0.524774\pi$
$61$	−1.21453	−0.155504	−0.0777522	+	0.996973i	$0.524774\pi$
$62$	−1.98853	−0.252544
$63$	−2.30471	−0.290366
$64$	−5.16916	−0.646146
$65$	−1.00000	−0.124035
$66$	0	0
$67$	0.455029	0.0555907	0.0277953	−	0.999614i	$-0.491151\pi$
$67$	0.455029	0.0555907	0.0277953	+	0.999614i	$0.491151\pi$
$68$	−9.74701	−1.18200
$69$	18.9985	2.28715
$70$	0.356190	0.0425728
$71$	12.4706	1.47998	0.739991	−	0.672617i	$-0.234830\pi$
$71$	12.4706	1.47998	0.739991	+	0.672617i	$0.234830\pi$
$72$	−3.11738	−0.367387
$73$	−0.0209630	−0.00245354	−0.00122677	−	0.999999i	$-0.500390\pi$
$73$	−0.0209630	−0.00245354	−0.00122677	+	0.999999i	$0.500390\pi$
$74$	−2.79458	−0.324863
$75$	2.29811	0.265363
$76$	−1.90578	−0.218607
$77$	0	0
$78$	0.810261	0.0917440
$79$	15.3937	1.73193	0.865963	−	0.500108i	$-0.166706\pi$
$79$	15.3937	1.73193	0.865963	+	0.500108i	$0.166706\pi$
$80$	−3.26959	−0.365551
$81$	−10.6395	−1.18217
$82$	−2.48091	−0.273971
$83$	−4.74746	−0.521101	−0.260551	−	0.965460i	$-0.583904\pi$
$83$	−4.74746	−0.521101	−0.260551	+	0.965460i	$0.583904\pi$
$84$	4.35473	0.475139
$85$	−5.19650	−0.563639
$86$	−1.44828	−0.156173
$87$	−8.35402	−0.895644
$88$	0	0
$89$	12.7455	1.35102	0.675509	−	0.737351i	$-0.263924\pi$
$89$	12.7455	1.35102	0.675509	+	0.737351i	$0.263924\pi$
$90$	−0.804342	−0.0847851
$91$	−1.01025	−0.105903
$92$	−15.5063	−1.61664
$93$	−12.9614	−1.34403
$94$	1.45541	0.150114
$95$	−1.01604	−0.104243
$96$	8.92986	0.911400
$97$	4.47793	0.454664	0.227332	−	0.973817i	$-0.427000\pi$
$97$	4.47793	0.454664	0.227332	+	0.973817i	$0.427000\pi$
$98$	−2.10820	−0.212960
$99$	0	0
$100$	−1.87569	−0.187569
$101$	0.140101	0.0139406	0.00697029	−	0.999976i	$-0.497781\pi$
$101$	0.140101	0.0139406	0.00697029	+	0.999976i	$0.497781\pi$
$102$	4.21052	0.416903
$103$	−3.64965	−0.359611	−0.179805	−	0.983702i	$-0.557547\pi$
$103$	−3.64965	−0.359611	−0.179805	+	0.983702i	$0.557547\pi$
$104$	−1.36648	−0.133994
$105$	2.32167	0.226571
$106$	2.15156	0.208978
$107$	4.71376	0.455697	0.227848	−	0.973697i	$-0.426831\pi$
$107$	4.71376	0.455697	0.227848	+	0.973697i	$0.426831\pi$
$108$	3.09788	0.298093
$109$	10.5939	1.01471	0.507356	−	0.861736i	$-0.330623\pi$
$109$	10.5939	1.01471	0.507356	+	0.861736i	$0.330623\pi$
$110$	0	0
$111$	−18.2152	−1.72891
$112$	−3.30310	−0.312114
$113$	17.1164	1.61017	0.805087	−	0.593157i	$-0.202119\pi$
$113$	17.1164	1.61017	0.805087	+	0.593157i	$0.202119\pi$
$114$	0.823257	0.0771051
$115$	−8.26698	−0.770900
$116$	6.81844	0.633076
$117$	2.28133	0.210909
$118$	3.46428	0.318913
$119$	−5.24975	−0.481244
$120$	3.14032	0.286671
$121$	0	0
$122$	−0.428214	−0.0387687
$123$	−16.1707	−1.45807
$124$	10.5789	0.950012
$125$	−1.00000	−0.0894427
$126$	−0.812585	−0.0723909
$127$	−9.13869	−0.810928	−0.405464	−	0.914111i	$-0.632890\pi$
$127$	−9.13869	−0.810928	−0.405464	+	0.914111i	$0.632890\pi$
$128$	−9.59400	−0.847998
$129$	−9.44000	−0.831146
$130$	−0.352577	−0.0309230
$131$	13.7829	1.20421	0.602107	−	0.798415i	$-0.294328\pi$
$131$	13.7829	1.20421	0.602107	+	0.798415i	$0.294328\pi$
$132$	0	0
$133$	−1.02645	−0.0890047
$134$	0.160433	0.0138593
$135$	1.65159	0.142147
$136$	−7.10089	−0.608897
$137$	1.55444	0.132805	0.0664025	−	0.997793i	$-0.478848\pi$
$137$	1.55444	0.132805	0.0664025	+	0.997793i	$0.478848\pi$
$138$	6.69841	0.570207
$139$	−13.8505	−1.17479	−0.587393	−	0.809302i	$-0.699846\pi$
$139$	−13.8505	−1.17479	−0.587393	+	0.809302i	$0.699846\pi$
$140$	−1.89491	−0.160149
$141$	9.48644	0.798902
$142$	4.39682	0.368973
$143$	0	0
$144$	7.45901	0.621584
$145$	3.63516	0.301884
$146$	−0.00739107	−0.000611689 0
$147$	−13.7413	−1.13337
$148$	14.8670	1.22206
$149$	13.0800	1.07155	0.535777	−	0.844360i	$-0.320019\pi$
$149$	13.0800	1.07155	0.535777	+	0.844360i	$0.320019\pi$
$150$	0.810261	0.0661575
$151$	−22.7954	−1.85506	−0.927532	−	0.373743i	$-0.878074\pi$
$151$	−22.7954	−1.85506	−0.927532	+	0.373743i	$0.878074\pi$
$152$	−1.38840	−0.112614
$153$	11.8549	0.958412
$154$	0	0
$155$	5.64000	0.453015
$156$	−4.31055	−0.345120
$157$	19.7511	1.57631	0.788154	−	0.615478i	$-0.211037\pi$
$157$	19.7511	1.57631	0.788154	+	0.615478i	$0.211037\pi$
$158$	5.42746	0.431785
$159$	14.0240	1.11218
$160$	−3.88574	−0.307194
$161$	−8.35171	−0.658207
$162$	−3.75125	−0.294726
$163$	8.02659	0.628691	0.314346	−	0.949309i	$-0.398215\pi$
$163$	8.02659	0.628691	0.314346	+	0.949309i	$0.398215\pi$
$164$	13.1983	1.03062
$165$	0	0
$166$	−1.67384	−0.129915
$167$	−19.5795	−1.51511	−0.757554	−	0.652772i	$-0.773606\pi$
$167$	−19.5795	−1.51511	−0.757554	+	0.652772i	$0.773606\pi$
$168$	3.17250	0.244764
$169$	1.00000	0.0769231
$170$	−1.83216	−0.140520
$171$	2.31792	0.177256
$172$	7.70480	0.587486
$173$	20.7575	1.57816	0.789080	−	0.614290i	$-0.210558\pi$
$173$	20.7575	1.57816	0.789080	+	0.614290i	$0.210558\pi$
$174$	−2.94543	−0.223292
$175$	−1.01025	−0.0763676
$176$	0	0
$177$	22.5804	1.69724
$178$	4.49376	0.336821
$179$	9.68199	0.723666	0.361833	−	0.932243i	$-0.382151\pi$
$179$	9.68199	0.723666	0.361833	+	0.932243i	$0.382151\pi$
$180$	4.27906	0.318942
$181$	12.3283	0.916359	0.458180	−	0.888860i	$-0.348502\pi$
$181$	12.3283	0.916359	0.458180	+	0.888860i	$0.348502\pi$
$182$	−0.356190	−0.0264026
$183$	−2.79112	−0.206326
$184$	−11.2966	−0.832799
$185$	7.92615	0.582742
$186$	−4.56987	−0.335079
$187$	0	0
$188$	−7.74271	−0.564695
$189$	1.66852	0.121367
$190$	−0.358232	−0.0259889
$191$	14.8784	1.07656	0.538281	−	0.842765i	$-0.319074\pi$
$191$	14.8784	1.07656	0.538281	+	0.842765i	$0.319074\pi$
$192$	−11.8793	−0.857317
$193$	−1.27494	−0.0917722	−0.0458861	−	0.998947i	$-0.514611\pi$
$193$	−1.27494	−0.0917722	−0.0458861	+	0.998947i	$0.514611\pi$
$194$	1.57881	0.113352
$195$	−2.29811	−0.164571
$196$	11.2155	0.801107
$197$	17.7733	1.26629	0.633147	−	0.774031i	$-0.281763\pi$
$197$	17.7733	1.26629	0.633147	+	0.774031i	$0.281763\pi$
$198$	0	0
$199$	11.3730	0.806208	0.403104	−	0.915154i	$-0.367931\pi$
$199$	11.3730	0.806208	0.403104	+	0.915154i	$0.367931\pi$
$200$	−1.36648	−0.0966245
$201$	1.04571	0.0737586
$202$	0.0493964	0.00347552
$203$	3.67242	0.257753
$204$	−22.3997	−1.56830
$205$	7.03652	0.491452
$206$	−1.28678	−0.0896543
$207$	18.8597	1.31084
$208$	3.26959	0.226705
$209$	0	0
$210$	0.818565	0.0564863
$211$	17.8881	1.23147	0.615735	−	0.787953i	$-0.288859\pi$
$211$	17.8881	1.23147	0.615735	+	0.787953i	$0.288859\pi$
$212$	−11.4462	−0.786128
$213$	28.6587	1.96367
$214$	1.66196	0.113609
$215$	4.10772	0.280144
$216$	2.25687	0.153560
$217$	5.69780	0.386792
$218$	3.73516	0.252977
$219$	−0.0481754	−0.00325539
$220$	0	0
$221$	5.19650	0.349554
$222$	−6.42225	−0.431033
$223$	4.87979	0.326775	0.163388	−	0.986562i	$-0.447758\pi$
$223$	4.87979	0.326775	0.163388	+	0.986562i	$0.447758\pi$
$224$	−3.92556	−0.262287
$225$	2.28133	0.152088
$226$	6.03484	0.401431
$227$	24.0929	1.59910	0.799552	−	0.600596i	$-0.205070\pi$
$227$	24.0929	1.59910	0.799552	+	0.600596i	$0.205070\pi$
$228$	−4.37969	−0.290052
$229$	12.5817	0.831424	0.415712	−	0.909496i	$-0.363532\pi$
$229$	12.5817	0.831424	0.415712	+	0.909496i	$0.363532\pi$
$230$	−2.91474	−0.192192
$231$	0	0
$232$	4.96737	0.326124
$233$	3.60218	0.235987	0.117993	−	0.993014i	$-0.462354\pi$
$233$	3.60218	0.235987	0.117993	+	0.993014i	$0.462354\pi$
$234$	0.804342	0.0525815
$235$	−4.12792	−0.269276
$236$	−18.4298	−1.19968
$237$	35.3765	2.29795
$238$	−1.85094	−0.119979
$239$	0.920844	0.0595644	0.0297822	−	0.999556i	$-0.490519\pi$
$239$	0.920844	0.0595644	0.0297822	+	0.999556i	$0.490519\pi$
$240$	−7.51389	−0.485020
$241$	4.21776	0.271690	0.135845	−	0.990730i	$-0.456625\pi$
$241$	4.21776	0.271690	0.135845	+	0.990730i	$0.456625\pi$
$242$	0	0
$243$	−19.4961	−1.25067
$244$	2.27808	0.145839
$245$	5.97940	0.382010
$246$	−5.70142	−0.363509
$247$	1.01604	0.0646491
$248$	7.70693	0.489390
$249$	−10.9102	−0.691406
$250$	−0.352577	−0.0222989
$251$	−8.17462	−0.515977	−0.257989	−	0.966148i	$-0.583060\pi$
$251$	−8.17462	−0.515977	−0.257989	+	0.966148i	$0.583060\pi$
$252$	4.32291	0.272318
$253$	0	0
$254$	−3.22209	−0.202172
$255$	−11.9421	−0.747846
$256$	6.95571	0.434732
$257$	21.1350	1.31836	0.659182	−	0.751983i	$-0.270903\pi$
$257$	21.1350	1.31836	0.659182	+	0.751983i	$0.270903\pi$
$258$	−3.32832	−0.207212
$259$	8.00739	0.497555
$260$	1.87569	0.116325
$261$	−8.29299	−0.513323
$262$	4.85951	0.300222
$263$	17.9537	1.10707	0.553537	−	0.832824i	$-0.313278\pi$
$263$	17.9537	1.10707	0.553537	+	0.832824i	$0.313278\pi$
$264$	0	0
$265$	−6.10240	−0.374867
$266$	−0.361903	−0.0221897
$267$	29.2906	1.79255
$268$	−0.853494	−0.0521354
$269$	12.0971	0.737574	0.368787	−	0.929514i	$-0.379773\pi$
$269$	12.0971	0.737574	0.368787	+	0.929514i	$0.379773\pi$
$270$	0.582313	0.0354384
$271$	3.71785	0.225843	0.112922	−	0.993604i	$-0.463979\pi$
$271$	3.71785	0.225843	0.112922	+	0.993604i	$0.463979\pi$
$272$	16.9904	1.03020
$273$	−2.32167	−0.140514
$274$	0.548060	0.0331095
$275$	0	0
$276$	−35.6352	−2.14499
$277$	−14.2568	−0.856611	−0.428305	−	0.903634i	$-0.640889\pi$
$277$	−14.2568	−0.856611	−0.428305	+	0.903634i	$0.640889\pi$
$278$	−4.88337	−0.292885
$279$	−12.8667	−0.770308
$280$	−1.38048	−0.0824996
$281$	−23.6807	−1.41267	−0.706337	−	0.707876i	$-0.749654\pi$
$281$	−23.6807	−1.41267	−0.706337	+	0.707876i	$0.749654\pi$
$282$	3.34470	0.199174
$283$	1.90819	0.113430	0.0567150	−	0.998390i	$-0.481937\pi$
$283$	1.90819	0.113430	0.0567150	+	0.998390i	$0.481937\pi$
$284$	−23.3909	−1.38799
$285$	−2.33498	−0.138312
$286$	0	0
$287$	7.10863	0.419609
$288$	8.86463	0.522353
$289$	10.0036	0.588445
$290$	1.28167	0.0752624
$291$	10.2908	0.603256
$292$	0.0393201	0.00230104
$293$	−24.4473	−1.42823	−0.714113	−	0.700031i	$-0.753170\pi$
$293$	−24.4473	−1.42823	−0.714113	+	0.700031i	$0.753170\pi$
$294$	−4.84487	−0.282559
$295$	−9.82561	−0.572069
$296$	10.8309	0.629534
$297$	0	0
$298$	4.61169	0.267148
$299$	8.26698	0.478092
$300$	−4.31055	−0.248870
$301$	4.14982	0.239191
$302$	−8.03713	−0.462485
$303$	0.321968	0.0184966
$304$	3.32204	0.190532
$305$	1.21453	0.0695436
$306$	4.17976	0.238941
$307$	−32.0846	−1.83117	−0.915583	−	0.402130i	$-0.868270\pi$
$307$	−32.0846	−1.83117	−0.915583	+	0.402130i	$0.868270\pi$
$308$	0	0
$309$	−8.38732	−0.477138
$310$	1.98853	0.112941
$311$	−16.4859	−0.934828	−0.467414	−	0.884039i	$-0.654814\pi$
$311$	−16.4859	−0.934828	−0.467414	+	0.884039i	$0.654814\pi$
$312$	−3.14032	−0.177786
$313$	19.4868	1.10146	0.550730	−	0.834683i	$-0.314349\pi$
$313$	19.4868	1.10146	0.550730	+	0.834683i	$0.314349\pi$
$314$	6.96377	0.392988
$315$	2.30471	0.129856
$316$	−28.8738	−1.62428
$317$	−25.9810	−1.45924	−0.729620	−	0.683852i	$-0.760303\pi$
$317$	−25.9810	−1.45924	−0.729620	+	0.683852i	$0.760303\pi$
$318$	4.94453	0.277276
$319$	0	0
$320$	5.16916	0.288965
$321$	10.8328	0.604626
$322$	−2.94462	−0.164097
$323$	5.27985	0.293779
$324$	19.9565	1.10869
$325$	1.00000	0.0554700
$326$	2.82999	0.156738
$327$	24.3460	1.34634
$328$	9.61524	0.530913
$329$	−4.17023	−0.229912
$330$	0	0
$331$	7.52505	0.413614	0.206807	−	0.978382i	$-0.433693\pi$
$331$	7.52505	0.413614	0.206807	+	0.978382i	$0.433693\pi$
$332$	8.90476	0.488712
$333$	−18.0821	−0.990895
$334$	−6.90328	−0.377730
$335$	−0.455029	−0.0248609
$336$	−7.59090	−0.414118
$337$	−27.3812	−1.49155	−0.745775	−	0.666198i	$-0.767920\pi$
$337$	−27.3812	−1.49155	−0.745775	+	0.666198i	$0.767920\pi$
$338$	0.352577	0.0191776
$339$	39.3354	2.13641
$340$	9.74701	0.528606
$341$	0	0
$342$	0.817244	0.0441915
$343$	13.1124	0.708004
$344$	5.61310	0.302638
$345$	−18.9985	−1.02284
$346$	7.31859	0.393450
$347$	−7.83021	−0.420348	−0.210174	−	0.977664i	$-0.567403\pi$
$347$	−7.83021	−0.420348	−0.210174	+	0.977664i	$0.567403\pi$
$348$	15.6695	0.839976
$349$	−0.934782	−0.0500377	−0.0250189	−	0.999687i	$-0.507965\pi$
$349$	−0.934782	−0.0500377	−0.0250189	+	0.999687i	$0.507965\pi$
$350$	−0.356190	−0.0190392
$351$	−1.65159	−0.0881555
$352$	0	0
$353$	−5.96801	−0.317645	−0.158823	−	0.987307i	$-0.550770\pi$
$353$	−5.96801	−0.317645	−0.158823	+	0.987307i	$0.550770\pi$
$354$	7.96131	0.423139
$355$	−12.4706	−0.661868
$356$	−23.9066	−1.26705
$357$	−12.0645	−0.638522
$358$	3.41364	0.180417
$359$	−13.5952	−0.717528	−0.358764	−	0.933428i	$-0.616802\pi$
$359$	−13.5952	−0.717528	−0.358764	+	0.933428i	$0.616802\pi$
$360$	3.11738	0.164300
$361$	−17.9677	−0.945666
$362$	4.34669	0.228457
$363$	0	0
$364$	1.89491	0.0993204
$365$	0.0209630	0.00109725
$366$	−0.984084	−0.0514389
$367$	31.0999	1.62340	0.811700	−	0.584075i	$-0.198542\pi$
$367$	31.0999	1.62340	0.811700	+	0.584075i	$0.198542\pi$
$368$	27.0297	1.40902
$369$	−16.0526	−0.835665
$370$	2.79458	0.145283
$371$	−6.16494	−0.320068
$372$	24.3115	1.26049
$373$	−3.51303	−0.181898	−0.0909490	−	0.995856i	$-0.528990\pi$
$373$	−3.51303	−0.181898	−0.0909490	+	0.995856i	$0.528990\pi$
$374$	0	0
$375$	−2.29811	−0.118674
$376$	−5.64071	−0.290898
$377$	−3.63516	−0.187220
$378$	0.588281	0.0302579
$379$	19.1420	0.983257	0.491628	−	0.870805i	$-0.336402\pi$
$379$	19.1420	0.983257	0.491628	+	0.870805i	$0.336402\pi$
$380$	1.90578	0.0977642
$381$	−21.0017	−1.07595
$382$	5.24577	0.268397
$383$	−14.6669	−0.749446	−0.374723	−	0.927137i	$-0.622262\pi$
$383$	−14.6669	−0.749446	−0.374723	+	0.927137i	$0.622262\pi$
$384$	−22.0481	−1.12514
$385$	0	0
$386$	−0.449514	−0.0228797
$387$	−9.37105	−0.476357
$388$	−8.39920	−0.426405
$389$	17.0500	0.864467	0.432234	−	0.901762i	$-0.357726\pi$
$389$	17.0500	0.864467	0.432234	+	0.901762i	$0.357726\pi$
$390$	−0.810261	−0.0410292
$391$	42.9593	2.17255
$392$	8.17071	0.412683
$393$	31.6746	1.59777
$394$	6.26645	0.315699
$395$	−15.3937	−0.774541
$396$	0	0
$397$	−30.3251	−1.52197	−0.760986	−	0.648768i	$-0.775285\pi$
$397$	−30.3251	−1.52197	−0.760986	+	0.648768i	$0.775285\pi$
$398$	4.00984	0.200995
$399$	−2.35891	−0.118093
$400$	3.26959	0.163480
$401$	17.7520	0.886491	0.443246	−	0.896400i	$-0.353827\pi$
$401$	17.7520	0.886491	0.443246	+	0.896400i	$0.353827\pi$
$402$	0.368692	0.0183887
$403$	−5.64000	−0.280948
$404$	−0.262786	−0.0130741
$405$	10.6395	0.528682
$406$	1.29481	0.0642602
$407$	0	0
$408$	−16.3187	−0.807894
$409$	−15.7843	−0.780484	−0.390242	−	0.920712i	$-0.627609\pi$
$409$	−15.7843	−0.780484	−0.390242	+	0.920712i	$0.627609\pi$
$410$	2.48091	0.122524
$411$	3.57228	0.176208
$412$	6.84562	0.337259
$413$	−9.92631	−0.488442
$414$	6.64948	0.326804
$415$	4.74746	0.233044
$416$	3.88574	0.190514
$417$	−31.8301	−1.55873
$418$	0	0
$419$	−10.5491	−0.515359	−0.257679	−	0.966230i	$-0.582958\pi$
$419$	−10.5491	−0.515359	−0.257679	+	0.966230i	$0.582958\pi$
$420$	−4.35473	−0.212489
$421$	18.8585	0.919107	0.459554	−	0.888150i	$-0.348009\pi$
$421$	18.8585	0.919107	0.459554	+	0.888150i	$0.348009\pi$
$422$	6.30694	0.307017
$423$	9.41714	0.457877
$424$	−8.33879	−0.404967
$425$	5.19650	0.252067
$426$	10.1044	0.489560
$427$	1.22697	0.0593775
$428$	−8.84156	−0.427373
$429$	0	0
$430$	1.44828	0.0698425
$431$	−37.7626	−1.81896	−0.909480	−	0.415747i	$-0.863520\pi$
$431$	−37.7626	−1.81896	−0.909480	+	0.415747i	$0.863520\pi$
$432$	−5.40004	−0.259809
$433$	−16.3528	−0.785864	−0.392932	−	0.919567i	$-0.628539\pi$
$433$	−16.3528	−0.785864	−0.392932	+	0.919567i	$0.628539\pi$
$434$	2.00891	0.0964308
$435$	8.35402	0.400544
$436$	−19.8709	−0.951643
$437$	8.39958	0.401807
$438$	−0.0169855	−0.000811599 0
$439$	36.8040	1.75656	0.878279	−	0.478149i	$-0.158692\pi$
$439$	36.8040	1.75656	0.878279	+	0.478149i	$0.158692\pi$
$440$	0	0
$441$	−13.6410	−0.649569
$442$	1.83216	0.0871471
$443$	−5.65673	−0.268759	−0.134380	−	0.990930i	$-0.542904\pi$
$443$	−5.65673	−0.268759	−0.134380	+	0.990930i	$0.542904\pi$
$444$	34.1661	1.62145
$445$	−12.7455	−0.604194
$446$	1.72050	0.0814680
$447$	30.0593	1.42175
$448$	5.22214	0.246723
$449$	1.05268	0.0496792	0.0248396	−	0.999691i	$-0.492092\pi$
$449$	1.05268	0.0496792	0.0248396	+	0.999691i	$0.492092\pi$
$450$	0.804342	0.0379171
$451$	0	0
$452$	−32.1050	−1.51009
$453$	−52.3864	−2.46133
$454$	8.49461	0.398672
$455$	1.01025	0.0473612
$456$	−3.19069	−0.149418
$457$	11.4819	0.537101	0.268551	−	0.963266i	$-0.413455\pi$
$457$	11.4819	0.537101	0.268551	+	0.963266i	$0.413455\pi$
$458$	4.43602	0.207282
$459$	−8.58250	−0.400597
$460$	15.5063	0.722985
$461$	1.91111	0.0890094	0.0445047	−	0.999009i	$-0.485829\pi$
$461$	1.91111	0.0890094	0.0445047	+	0.999009i	$0.485829\pi$
$462$	0	0
$463$	11.5832	0.538316	0.269158	−	0.963096i	$-0.413255\pi$
$463$	11.5832	0.538316	0.269158	+	0.963096i	$0.413255\pi$
$464$	−11.8855	−0.551770
$465$	12.9614	0.601068
$466$	1.27004	0.0588337
$467$	−31.6490	−1.46454	−0.732270	−	0.681015i	$-0.761539\pi$
$467$	−31.6490	−1.46454	−0.732270	+	0.681015i	$0.761539\pi$
$468$	−4.27906	−0.197800
$469$	−0.459693	−0.0212266
$470$	−1.45541	−0.0671330
$471$	45.3902	2.09147
$472$	−13.4265	−0.618003
$473$	0	0
$474$	12.4729	0.572900
$475$	1.01604	0.0466191
$476$	9.84691	0.451332
$477$	13.9216	0.637424
$478$	0.324668	0.0148500
$479$	3.50261	0.160038	0.0800191	−	0.996793i	$-0.474502\pi$
$479$	3.50261	0.160038	0.0800191	+	0.996793i	$0.474502\pi$
$480$	−8.92986	−0.407591
$481$	−7.92615	−0.361402
$482$	1.48708	0.0677348
$483$	−19.1932	−0.873319
$484$	0	0
$485$	−4.47793	−0.203332
$486$	−6.87386	−0.311804
$487$	−1.22529	−0.0555234	−0.0277617	−	0.999615i	$-0.508838\pi$
$487$	−1.22529	−0.0555234	−0.0277617	+	0.999615i	$0.508838\pi$
$488$	1.65962	0.0751277
$489$	18.4460	0.834158
$490$	2.10820	0.0952386
$491$	12.9163	0.582903	0.291452	−	0.956586i	$-0.405862\pi$
$491$	12.9163	0.582903	0.291452	+	0.956586i	$0.405862\pi$
$492$	30.3313	1.36744
$493$	−18.8901	−0.850767
$494$	0.358232	0.0161176
$495$	0	0
$496$	−18.4405	−0.828002
$497$	−12.5984	−0.565114
$498$	−3.84668	−0.172374
$499$	5.95353	0.266517	0.133258	−	0.991081i	$-0.457456\pi$
$499$	5.95353	0.266517	0.133258	+	0.991081i	$0.457456\pi$
$500$	1.87569	0.0838834
$501$	−44.9959	−2.01027
$502$	−2.88218	−0.128638
$503$	−14.1880	−0.632610	−0.316305	−	0.948658i	$-0.602442\pi$
$503$	−14.1880	−0.632610	−0.316305	+	0.948658i	$0.602442\pi$
$504$	3.14933	0.140282
$505$	−0.140101	−0.00623442
$506$	0	0
$507$	2.29811	0.102063
$508$	17.1413	0.760524
$509$	32.1246	1.42390	0.711949	−	0.702231i	$-0.247813\pi$
$509$	32.1246	1.42390	0.711949	+	0.702231i	$0.247813\pi$
$510$	−4.21052	−0.186445
$511$	0.0211779	0.000936853 0
$512$	21.6404	0.956380
$513$	−1.67809	−0.0740893
$514$	7.45170	0.328680
$515$	3.64965	0.160823
$516$	17.7065	0.779486
$517$	0	0
$518$	2.82322	0.124045
$519$	47.7030	2.09393
$520$	1.36648	0.0599240
$521$	24.4208	1.06989	0.534946	−	0.844886i	$-0.320332\pi$
$521$	24.4208	1.06989	0.534946	+	0.844886i	$0.320332\pi$
$522$	−2.92391	−0.127976
$523$	−0.534093	−0.0233543	−0.0116771	−	0.999932i	$-0.503717\pi$
$523$	−0.534093	−0.0233543	−0.0116771	+	0.999932i	$0.503717\pi$
$524$	−25.8524	−1.12937
$525$	−2.32167	−0.101326
$526$	6.33006	0.276004
$527$	−29.3082	−1.27669
$528$	0	0
$529$	45.3430	1.97143
$530$	−2.15156	−0.0934579
$531$	22.4154	0.972746
$532$	1.92531	0.0834726
$533$	−7.03652	−0.304786
$534$	10.3272	0.446900
$535$	−4.71376	−0.203794
$536$	−0.621787	−0.0268571
$537$	22.2503	0.960171
$538$	4.26516	0.183884
$539$	0	0
$540$	−3.09788	−0.133311
$541$	−45.8729	−1.97223	−0.986114	−	0.166070i	$-0.946892\pi$
$541$	−45.8729	−1.97223	−0.986114	+	0.166070i	$0.946892\pi$
$542$	1.31083	0.0563048
$543$	28.3319	1.21584
$544$	20.1922	0.865734
$545$	−10.5939	−0.453793
$546$	−0.818565	−0.0350313
$547$	12.5569	0.536896	0.268448	−	0.963294i	$-0.413489\pi$
$547$	12.5569	0.536896	0.268448	+	0.963294i	$0.413489\pi$
$548$	−2.91565	−0.124550
$549$	−2.77073	−0.118252
$550$	0	0
$551$	−3.69347	−0.157347
$552$	−25.9610	−1.10497
$553$	−15.5515	−0.661315
$554$	−5.02663	−0.213561
$555$	18.2152	0.773192
$556$	25.9793	1.10177
$557$	37.2692	1.57915	0.789574	−	0.613656i	$-0.210302\pi$
$557$	37.2692	1.57915	0.789574	+	0.613656i	$0.210302\pi$
$558$	−4.53649	−0.192045
$559$	−4.10772	−0.173738
$560$	3.30310	0.139581
$561$	0	0
$562$	−8.34927	−0.352193
$563$	−39.5039	−1.66489	−0.832446	−	0.554107i	$-0.813060\pi$
$563$	−39.5039	−1.66489	−0.832446	+	0.554107i	$0.813060\pi$
$564$	−17.7936	−0.749246
$565$	−17.1164	−0.720092
$566$	0.672783	0.0282792
$567$	10.7486	0.451397
$568$	−17.0407	−0.715013
$569$	−32.3207	−1.35495	−0.677476	−	0.735544i	$-0.736926\pi$
$569$	−32.3207	−1.35495	−0.677476	+	0.735544i	$0.736926\pi$
$570$	−0.823257	−0.0344825
$571$	3.35611	0.140449	0.0702244	−	0.997531i	$-0.477628\pi$
$571$	3.35611	0.140449	0.0702244	+	0.997531i	$0.477628\pi$
$572$	0	0
$573$	34.1922	1.42840
$574$	2.50634	0.104613
$575$	8.26698	0.344757
$576$	−11.7926	−0.491356
$577$	−18.8033	−0.782790	−0.391395	−	0.920223i	$-0.628007\pi$
$577$	−18.8033	−0.782790	−0.391395	+	0.920223i	$0.628007\pi$
$578$	3.52702	0.146705
$579$	−2.92996	−0.121765
$580$	−6.81844	−0.283120
$581$	4.79611	0.198976
$582$	3.62829	0.150397
$583$	0	0
$584$	0.0286455	0.00118536
$585$	−2.28133	−0.0943212
$586$	−8.61954	−0.356070
$587$	−13.6710	−0.564262	−0.282131	−	0.959376i	$-0.591041\pi$
$587$	−13.6710	−0.564262	−0.282131	+	0.959376i	$0.591041\pi$
$588$	25.7745	1.06292
$589$	−5.73046	−0.236120
$590$	−3.46428	−0.142622
$591$	40.8450	1.68014
$592$	−25.9153	−1.06511
$593$	34.9825	1.43656	0.718280	−	0.695754i	$-0.244929\pi$
$593$	34.9825	1.43656	0.718280	+	0.695754i	$0.244929\pi$
$594$	0	0
$595$	5.24975	0.215219
$596$	−24.5340	−1.00495
$597$	26.1364	1.06969
$598$	2.91474	0.119193
$599$	−36.5209	−1.49220	−0.746101	−	0.665833i	$-0.768076\pi$
$599$	−36.5209	−1.49220	−0.746101	+	0.665833i	$0.768076\pi$
$600$	−3.14032	−0.128203
$601$	−27.3418	−1.11530	−0.557648	−	0.830077i	$-0.688296\pi$
$601$	−27.3418	−1.11530	−0.557648	+	0.830077i	$0.688296\pi$
$602$	1.46313	0.0596326
$603$	1.03807	0.0422735
$604$	42.7571	1.73976
$605$	0	0
$606$	0.113518	0.00461137
$607$	−32.8217	−1.33219	−0.666097	−	0.745865i	$-0.732036\pi$
$607$	−32.8217	−1.33219	−0.666097	+	0.745865i	$0.732036\pi$
$608$	3.94806	0.160115
$609$	8.43963	0.341991
$610$	0.428214	0.0173379
$611$	4.12792	0.166998
$612$	−22.2361	−0.898842
$613$	−23.3452	−0.942904	−0.471452	−	0.881892i	$-0.656270\pi$
$613$	−23.3452	−0.942904	−0.471452	+	0.881892i	$0.656270\pi$
$614$	−11.3123	−0.456526
$615$	16.1707	0.652067
$616$	0	0
$617$	−14.2346	−0.573064	−0.286532	−	0.958071i	$-0.592502\pi$
$617$	−14.2346	−0.573064	−0.286532	+	0.958071i	$0.592502\pi$
$618$	−2.95717	−0.118955
$619$	6.90352	0.277476	0.138738	−	0.990329i	$-0.455695\pi$
$619$	6.90352	0.277476	0.138738	+	0.990329i	$0.455695\pi$
$620$	−10.5789	−0.424858
$621$	−13.6537	−0.547904
$622$	−5.81253	−0.233061
$623$	−12.8761	−0.515870
$624$	7.51389	0.300796
$625$	1.00000	0.0400000
$626$	6.87060	0.274604
$627$	0	0
$628$	−37.0469	−1.47833
$629$	−41.1882	−1.64228
$630$	0.812585	0.0323742
$631$	19.0834	0.759700	0.379850	−	0.925048i	$-0.375976\pi$
$631$	19.0834	0.759700	0.379850	+	0.925048i	$0.375976\pi$
$632$	−21.0351	−0.836733
$633$	41.1090	1.63394
$634$	−9.16031	−0.363802
$635$	9.13869	0.362658
$636$	−26.3047	−1.04305
$637$	−5.97940	−0.236912
$638$	0	0
$639$	28.4494	1.12544
$640$	9.59400	0.379236
$641$	−27.8895	−1.10157	−0.550784	−	0.834648i	$-0.685671\pi$
$641$	−27.8895	−1.10157	−0.550784	+	0.834648i	$0.685671\pi$
$642$	3.81938	0.150739
$643$	2.16160	0.0852454	0.0426227	−	0.999091i	$-0.486429\pi$
$643$	2.16160	0.0852454	0.0426227	+	0.999091i	$0.486429\pi$
$644$	15.6652	0.617296
$645$	9.44000	0.371700
$646$	1.86155	0.0732417
$647$	35.3023	1.38788	0.693938	−	0.720035i	$-0.255874\pi$
$647$	35.3023	1.38788	0.693938	+	0.720035i	$0.255874\pi$
$648$	14.5387	0.571133
$649$	0	0
$650$	0.352577	0.0138292
$651$	13.0942	0.513202
$652$	−15.0554	−0.589615
$653$	−12.2376	−0.478894	−0.239447	−	0.970909i	$-0.576966\pi$
$653$	−12.2376	−0.478894	−0.239447	+	0.970909i	$0.576966\pi$
$654$	8.58383	0.335654
$655$	−13.7829	−0.538541
$656$	−23.0065	−0.898255
$657$	−0.0478235	−0.00186577
$658$	−1.47033	−0.0573192
$659$	−33.1387	−1.29090	−0.645450	−	0.763802i	$-0.723330\pi$
$659$	−33.1387	−1.29090	−0.645450	+	0.763802i	$0.723330\pi$
$660$	0	0
$661$	−16.3609	−0.636367	−0.318184	−	0.948029i	$-0.603073\pi$
$661$	−16.3609	−0.636367	−0.318184	+	0.948029i	$0.603073\pi$
$662$	2.65316	0.103118
$663$	11.9421	0.463794
$664$	6.48729	0.251756
$665$	1.02645	0.0398041
$666$	−6.37534	−0.247039
$667$	−30.0518	−1.16361
$668$	36.7251	1.42094
$669$	11.2143	0.433571
$670$	−0.160433	−0.00619805
$671$	0	0
$672$	−9.02138	−0.348007
$673$	7.70585	0.297039	0.148519	−	0.988910i	$-0.452549\pi$
$673$	7.70585	0.297039	0.148519	+	0.988910i	$0.452549\pi$
$674$	−9.65397	−0.371857
$675$	−1.65159	−0.0635699
$676$	−1.87569	−0.0721419
$677$	41.8077	1.60680	0.803399	−	0.595440i	$-0.203022\pi$
$677$	41.8077	1.60680	0.803399	+	0.595440i	$0.203022\pi$
$678$	13.8687	0.532626
$679$	−4.52382	−0.173608
$680$	7.10089	0.272307
$681$	55.3683	2.12172
$682$	0	0
$683$	−12.0932	−0.462735	−0.231368	−	0.972866i	$-0.574320\pi$
$683$	−12.0932	−0.462735	−0.231368	+	0.972866i	$0.574320\pi$
$684$	−4.34770	−0.166238
$685$	−1.55444	−0.0593922
$686$	4.62313	0.176512
$687$	28.9142	1.10315
$688$	−13.4306	−0.512035
$689$	6.10240	0.232483
$690$	−6.69841	−0.255004
$691$	−18.1717	−0.691284	−0.345642	−	0.938366i	$-0.612339\pi$
$691$	−18.1717	−0.691284	−0.345642	+	0.938366i	$0.612339\pi$
$692$	−38.9346	−1.48007
$693$	0	0
$694$	−2.76075	−0.104797
$695$	13.8505	0.525381
$696$	11.4156	0.432706
$697$	−36.5652	−1.38501
$698$	−0.329582	−0.0124749
$699$	8.27822	0.313111
$700$	1.89491	0.0716210
$701$	30.7956	1.16313	0.581566	−	0.813499i	$-0.302440\pi$
$701$	30.7956	1.16313	0.581566	+	0.813499i	$0.302440\pi$
$702$	−0.582313	−0.0219780
$703$	−8.05329	−0.303736
$704$	0	0
$705$	−9.48644	−0.357280
$706$	−2.10418	−0.0791919
$707$	−0.141537	−0.00532304
$708$	−42.3538	−1.59175
$709$	30.2679	1.13673	0.568367	−	0.822775i	$-0.307575\pi$
$709$	30.2679	1.13673	0.568367	+	0.822775i	$0.307575\pi$
$710$	−4.39682	−0.165010
$711$	35.1180	1.31703
$712$	−17.4164	−0.652708
$713$	−46.6257	−1.74615
$714$	−4.25367	−0.159190
$715$	0	0
$716$	−18.1604	−0.678686
$717$	2.11620	0.0790311
$718$	−4.79335	−0.178886
$719$	3.47981	0.129775	0.0648874	−	0.997893i	$-0.479331\pi$
$719$	3.47981	0.129775	0.0648874	+	0.997893i	$0.479331\pi$
$720$	−7.45901	−0.277981
$721$	3.68706	0.137313
$722$	−6.33498	−0.235763
$723$	9.69290	0.360483
$724$	−23.1242	−0.859403
$725$	−3.63516	−0.135007
$726$	0	0
$727$	−14.5774	−0.540644	−0.270322	−	0.962770i	$-0.587130\pi$
$727$	−14.5774	−0.540644	−0.270322	+	0.962770i	$0.587130\pi$
$728$	1.38048	0.0511641
$729$	−12.8856	−0.477244
$730$	0.00739107	0.000273556 0
$731$	−21.3457	−0.789501
$732$	5.23528	0.193502
$733$	−5.11376	−0.188881	−0.0944404	−	0.995531i	$-0.530106\pi$
$733$	−5.11376	−0.188881	−0.0944404	+	0.995531i	$0.530106\pi$
$734$	10.9651	0.404729
$735$	13.7413	0.506857
$736$	32.1233	1.18408
$737$	0	0
$738$	−5.65977	−0.208339
$739$	5.71529	0.210241	0.105120	−	0.994460i	$-0.466477\pi$
$739$	5.71529	0.210241	0.105120	+	0.994460i	$0.466477\pi$
$740$	−14.8670	−0.546522
$741$	2.33498	0.0857775
$742$	−2.17361	−0.0797958
$743$	43.5737	1.59856	0.799282	−	0.600956i	$-0.205213\pi$
$743$	43.5737	1.59856	0.799282	+	0.600956i	$0.205213\pi$
$744$	17.7114	0.649331
$745$	−13.0800	−0.479213
$746$	−1.23861	−0.0453489
$747$	−10.8305	−0.396267
$748$	0	0
$749$	−4.76207	−0.174002
$750$	−0.810261	−0.0295865
$751$	−6.51762	−0.237831	−0.118916	−	0.992904i	$-0.537942\pi$
$751$	−6.51762	−0.237831	−0.118916	+	0.992904i	$0.537942\pi$
$752$	13.4966	0.492171
$753$	−18.7862	−0.684607
$754$	−1.28167	−0.0466758
$755$	22.7954	0.829610
$756$	−3.12963	−0.113823
$757$	−27.2355	−0.989891	−0.494945	−	0.868924i	$-0.664812\pi$
$757$	−27.2355	−0.989891	−0.494945	+	0.868924i	$0.664812\pi$
$758$	6.74901	0.245135
$759$	0	0
$760$	1.38840	0.0503624
$761$	35.4612	1.28547	0.642734	−	0.766089i	$-0.277800\pi$
$761$	35.4612	1.28547	0.642734	+	0.766089i	$0.277800\pi$
$762$	−7.40472	−0.268245
$763$	−10.7025	−0.387456
$764$	−27.9072	−1.00965
$765$	−11.8549	−0.428615
$766$	−5.17122	−0.186844
$767$	9.82561	0.354782
$768$	15.9850	0.576810
$769$	38.8812	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$769$	38.8812	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$770$	0	0
$771$	48.5706	1.74923
$772$	2.39139	0.0860681
$773$	36.4065	1.30945	0.654726	−	0.755866i	$-0.272784\pi$
$773$	36.4065	1.30945	0.654726	+	0.755866i	$0.272784\pi$
$774$	−3.30401	−0.118760
$775$	−5.64000	−0.202595
$776$	−6.11898	−0.219659
$777$	18.4019	0.660164
$778$	6.01141	0.215520
$779$	−7.14939	−0.256153
$780$	4.31055	0.154342
$781$	0	0
$782$	15.1465	0.541636
$783$	6.00381	0.214559
$784$	−19.5502	−0.698221
$785$	−19.7511	−0.704947
$786$	11.1677	0.398339
$787$	−13.8936	−0.495252	−0.247626	−	0.968856i	$-0.579650\pi$
$787$	−13.8936	−0.495252	−0.247626	+	0.968856i	$0.579650\pi$
$788$	−33.3372	−1.18759
$789$	41.2597	1.46888
$790$	−5.42746	−0.193100
$791$	−17.2918	−0.614826
$792$	0	0
$793$	−1.21453	−0.0431291
$794$	−10.6919	−0.379442
$795$	−14.0240	−0.497380
$796$	−21.3322	−0.756099
$797$	12.1062	0.428823	0.214411	−	0.976743i	$-0.431217\pi$
$797$	12.1062	0.428823	0.214411	+	0.976743i	$0.431217\pi$
$798$	−0.831695	−0.0294417
$799$	21.4507	0.758873
$800$	3.88574	0.137382
$801$	29.0766	1.02737
$802$	6.25893	0.221011
$803$	0	0
$804$	−1.96143	−0.0691742
$805$	8.35171	0.294359
$806$	−1.98853	−0.0700430
$807$	27.8006	0.978626
$808$	−0.191445	−0.00673501
$809$	−3.88807	−0.136697	−0.0683487	−	0.997661i	$-0.521773\pi$
$809$	−3.88807	−0.136697	−0.0683487	+	0.997661i	$0.521773\pi$
$810$	3.75125	0.131805
$811$	−0.560893	−0.0196956	−0.00984781	−	0.999952i	$-0.503135\pi$
$811$	−0.560893	−0.0196956	−0.00984781	+	0.999952i	$0.503135\pi$
$812$	−6.88832	−0.241732
$813$	8.54405	0.299653
$814$	0	0
$815$	−8.02659	−0.281159
$816$	39.0459	1.36688
$817$	−4.17361	−0.146016
$818$	−5.56518	−0.194582
$819$	−2.30471	−0.0805330
$820$	−13.1983	−0.460906
$821$	11.9834	0.418224	0.209112	−	0.977892i	$-0.432943\pi$
$821$	11.9834	0.418224	0.209112	+	0.977892i	$0.432943\pi$
$822$	1.25950	0.0439302
$823$	0.466032	0.0162448	0.00812242	−	0.999967i	$-0.497415\pi$
$823$	0.466032	0.0162448	0.00812242	+	0.999967i	$0.497415\pi$
$824$	4.98717	0.173736
$825$	0	0
$826$	−3.49978	−0.121773
$827$	2.78295	0.0967725	0.0483863	−	0.998829i	$-0.484592\pi$
$827$	2.78295	0.0967725	0.0483863	+	0.998829i	$0.484592\pi$
$828$	−35.3749	−1.22936
$829$	42.7951	1.48634	0.743168	−	0.669105i	$-0.233322\pi$
$829$	42.7951	1.48634	0.743168	+	0.669105i	$0.233322\pi$
$830$	1.67384	0.0580999
$831$	−32.7638	−1.13657
$832$	−5.16916	−0.179209
$833$	−31.0719	−1.07658
$834$	−11.2225	−0.388605
$835$	19.5795	0.677577
$836$	0	0
$837$	9.31498	0.321973
$838$	−3.71938	−0.128484
$839$	−9.39261	−0.324269	−0.162134	−	0.986769i	$-0.551838\pi$
$839$	−9.39261	−0.324269	−0.162134	+	0.986769i	$0.551838\pi$
$840$	−3.17250	−0.109462
$841$	−15.7856	−0.544331
$842$	6.64907	0.229142
$843$	−54.4210	−1.87436
$844$	−33.5526	−1.15493
$845$	−1.00000	−0.0344010
$846$	3.32026	0.114153
$847$	0	0
$848$	19.9523	0.685166
$849$	4.38524	0.150501
$850$	1.83216	0.0628426
$851$	−65.5254	−2.24618
$852$	−53.7549	−1.84161
$853$	−27.7852	−0.951347	−0.475674	−	0.879622i	$-0.657796\pi$
$853$	−27.7852	−0.951347	−0.475674	+	0.879622i	$0.657796\pi$
$854$	0.432602	0.0148034
$855$	−2.31792	−0.0792711
$856$	−6.44125	−0.220157
$857$	−48.0083	−1.63993	−0.819967	−	0.572411i	$-0.806008\pi$
$857$	−48.0083	−1.63993	−0.819967	+	0.572411i	$0.806008\pi$
$858$	0	0
$859$	19.5360	0.666560	0.333280	−	0.942828i	$-0.391845\pi$
$859$	19.5360	0.666560	0.333280	+	0.942828i	$0.391845\pi$
$860$	−7.70480	−0.262732
$861$	16.3364	0.556745
$862$	−13.3142	−0.453484
$863$	50.6644	1.72464	0.862318	−	0.506368i	$-0.169012\pi$
$863$	50.6644	1.72464	0.862318	+	0.506368i	$0.169012\pi$
$864$	−6.41766	−0.218333
$865$	−20.7575	−0.705775
$866$	−5.76561	−0.195923
$867$	22.9893	0.780758
$868$	−10.6873	−0.362751
$869$	0	0
$870$	2.94543	0.0998594
$871$	0.455029	0.0154181
$872$	−14.4763	−0.490231
$873$	10.2156	0.345746
$874$	2.96150	0.100174
$875$	1.01025	0.0341526
$876$	0.0903621	0.00305305
$877$	−45.7787	−1.54584	−0.772918	−	0.634506i	$-0.781203\pi$
$877$	−45.7787	−1.54584	−0.772918	+	0.634506i	$0.781203\pi$
$878$	12.9762	0.437926
$879$	−56.1826	−1.89499
$880$	0	0
$881$	−0.502588	−0.0169326	−0.00846631	−	0.999964i	$-0.502695\pi$
$881$	−0.502588	−0.0169326	−0.00846631	+	0.999964i	$0.502695\pi$
$882$	−4.80948	−0.161944
$883$	−8.13923	−0.273907	−0.136953	−	0.990577i	$-0.543731\pi$
$883$	−8.13923	−0.273907	−0.136953	+	0.990577i	$0.543731\pi$
$884$	−9.74701	−0.327828
$885$	−22.5804	−0.759031
$886$	−1.99443	−0.0670042
$887$	38.3856	1.28886	0.644431	−	0.764662i	$-0.277094\pi$
$887$	38.3856	1.28886	0.644431	+	0.764662i	$0.277094\pi$
$888$	24.8907	0.835276
$889$	9.23235	0.309643
$890$	−4.49376	−0.150631
$891$	0	0
$892$	−9.15298	−0.306464
$893$	4.19414	0.140351
$894$	10.5982	0.354457
$895$	−9.68199	−0.323633
$896$	9.69232	0.323798
$897$	18.9985	0.634340
$898$	0.371152	0.0123855
$899$	20.5023	0.683790
$900$	−4.27906	−0.142635
$901$	31.7111	1.05645
$902$	0	0
$903$	9.53675	0.317363
$904$	−23.3892	−0.777912
$905$	−12.3283	−0.409808
$906$	−18.4702	−0.613632
$907$	−42.9084	−1.42475	−0.712375	−	0.701799i	$-0.752380\pi$
$907$	−42.9084	−1.42475	−0.712375	+	0.701799i	$0.752380\pi$
$908$	−45.1909	−1.49971
$909$	0.319616	0.0106010
$910$	0.356190	0.0118076
$911$	−50.5402	−1.67447	−0.837236	−	0.546841i	$-0.815830\pi$
$911$	−50.5402	−1.67447	−0.837236	+	0.546841i	$0.815830\pi$
$912$	7.63442	0.252801
$913$	0	0
$914$	4.04825	0.133904
$915$	2.79112	0.0922717
$916$	−23.5994	−0.779747
$917$	−13.9241	−0.459815
$918$	−3.02599	−0.0998725
$919$	50.0538	1.65112	0.825561	−	0.564312i	$-0.190859\pi$
$919$	50.0538	1.65112	0.825561	+	0.564312i	$0.190859\pi$
$920$	11.2966	0.372439
$921$	−73.7341	−2.42962
$922$	0.673814	0.0221909
$923$	12.4706	0.410473
$924$	0	0
$925$	−7.92615	−0.260610
$926$	4.08396	0.134207
$927$	−8.32605	−0.273463
$928$	−14.1253	−0.463685
$929$	22.6499	0.743120	0.371560	−	0.928409i	$-0.378823\pi$
$929$	22.6499	0.743120	0.371560	+	0.928409i	$0.378823\pi$
$930$	4.56987	0.149852
$931$	−6.07531	−0.199110
$932$	−6.75658	−0.221319
$933$	−37.8864	−1.24034
$934$	−11.1587	−0.365123
$935$	0	0
$936$	−3.11738	−0.101895
$937$	53.4744	1.74693	0.873466	−	0.486885i	$-0.161867\pi$
$937$	53.4744	1.74693	0.873466	+	0.486885i	$0.161867\pi$
$938$	−0.162077	−0.00529200
$939$	44.7830	1.46144
$940$	7.74271	0.252539
$941$	−27.9330	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$941$	−27.9330	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$942$	16.0035	0.521423
$943$	−58.1708	−1.89430
$944$	32.1257	1.04560
$945$	−1.66852	−0.0542770
$946$	0	0
$947$	−5.89506	−0.191564	−0.0957819	−	0.995402i	$-0.530535\pi$
$947$	−5.89506	−0.191564	−0.0957819	+	0.995402i	$0.530535\pi$
$948$	−66.3553	−2.15512
$949$	−0.0209630	−0.000680488 0
$950$	0.358232	0.0116226
$951$	−59.7074	−1.93614
$952$	7.17367	0.232500
$953$	−18.4199	−0.596679	−0.298339	−	0.954460i	$-0.596433\pi$
$953$	−18.4199	−0.596679	−0.298339	+	0.954460i	$0.596433\pi$
$954$	4.90841	0.158916
$955$	−14.8784	−0.481453
$956$	−1.72722	−0.0558622
$957$	0	0
$958$	1.23494	0.0398990
$959$	−1.57037	−0.0507100
$960$	11.8793	0.383404
$961$	0.809564	0.0261150
$962$	−2.79458	−0.0901007
$963$	10.7536	0.346531
$964$	−7.91121	−0.254803
$965$	1.27494	0.0410418
$966$	−6.76706	−0.217727
$967$	32.3610	1.04066	0.520330	−	0.853965i	$-0.325809\pi$
$967$	32.3610	1.04066	0.520330	+	0.853965i	$0.325809\pi$
$968$	0	0
$969$	12.1337	0.389790
$970$	−1.57881	−0.0506926
$971$	−12.1960	−0.391389	−0.195695	−	0.980665i	$-0.562696\pi$
$971$	−12.1960	−0.391389	−0.195695	+	0.980665i	$0.562696\pi$
$972$	36.5686	1.17294
$973$	13.9925	0.448578
$974$	−0.432010	−0.0138425
$975$	2.29811	0.0735985
$976$	−3.97101	−0.127109
$977$	−7.50263	−0.240030	−0.120015	−	0.992772i	$-0.538294\pi$
$977$	−7.50263	−0.240030	−0.120015	+	0.992772i	$0.538294\pi$
$978$	6.50363	0.207963
$979$	0	0
$980$	−11.2155	−0.358266
$981$	24.1682	0.771630
$982$	4.55397	0.145323
$983$	−18.8299	−0.600579	−0.300290	−	0.953848i	$-0.597083\pi$
$983$	−18.8299	−0.600579	−0.300290	+	0.953848i	$0.597083\pi$
$984$	22.0969	0.704424
$985$	−17.7733	−0.566304
$986$	−6.66021	−0.212104
$987$	−9.58366	−0.305051
$988$	−1.90578	−0.0606308
$989$	−33.9584	−1.07982
$990$	0	0
$991$	−54.1808	−1.72111	−0.860556	−	0.509357i	$-0.829883\pi$
$991$	−54.1808	−1.72111	−0.860556	+	0.509357i	$0.829883\pi$
$992$	−21.9155	−0.695819
$993$	17.2934	0.548790
$994$	−4.44189	−0.140888
$995$	−11.3730	−0.360547
$996$	20.4642	0.648431
$997$	17.4158	0.551565	0.275783	−	0.961220i	$-0.411063\pi$
$997$	17.4158	0.551565	0.275783	+	0.961220i	$0.411063\pi$
$998$	2.09908	0.0664451
$999$	13.0908	0.414174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7865.2.a.bm.1.15		26
11.5	even	5		715.2.v.d.586.6	yes	52
11.9	even	5		715.2.v.d.521.6	✓	52
11.10	odd	2		7865.2.a.bn.1.12		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
715.2.v.d.521.6	✓	52	11.9	even	5
715.2.v.d.586.6	yes	52	11.5	even	5
7865.2.a.bm.1.15		26	1.1	even	1	trivial
7865.2.a.bn.1.12		26	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:	\( N \)	\(=\)	\( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7865.a (trivial)

\(f(q)\)	\(=\)	\(q+0.352577 q^{2} +2.29811 q^{3} -1.87569 q^{4} -1.00000 q^{5} +0.810261 q^{6} -1.01025 q^{7} -1.36648 q^{8} +2.28133 q^{9} -0.352577 q^{10} -4.31055 q^{12} +1.00000 q^{13} -0.356190 q^{14} -2.29811 q^{15} +3.26959 q^{16} +5.19650 q^{17} +0.804342 q^{18} +1.01604 q^{19} +1.87569 q^{20} -2.32167 q^{21} +8.26698 q^{23} -3.14032 q^{24} +1.00000 q^{25} +0.352577 q^{26} -1.65159 q^{27} +1.89491 q^{28} -3.63516 q^{29} -0.810261 q^{30} -5.64000 q^{31} +3.88574 q^{32} +1.83216 q^{34} +1.01025 q^{35} -4.27906 q^{36} -7.92615 q^{37} +0.358232 q^{38} +2.29811 q^{39} +1.36648 q^{40} -7.03652 q^{41} -0.818565 q^{42} -4.10772 q^{43} -2.28133 q^{45} +2.91474 q^{46} +4.12792 q^{47} +7.51389 q^{48} -5.97940 q^{49} +0.352577 q^{50} +11.9421 q^{51} -1.87569 q^{52} +6.10240 q^{53} -0.582313 q^{54} +1.38048 q^{56} +2.33498 q^{57} -1.28167 q^{58} +9.82561 q^{59} +4.31055 q^{60} -1.21453 q^{61} -1.98853 q^{62} -2.30471 q^{63} -5.16916 q^{64} -1.00000 q^{65} +0.455029 q^{67} -9.74701 q^{68} +18.9985 q^{69} +0.356190 q^{70} +12.4706 q^{71} -3.11738 q^{72} -0.0209630 q^{73} -2.79458 q^{74} +2.29811 q^{75} -1.90578 q^{76} +0.810261 q^{78} +15.3937 q^{79} -3.26959 q^{80} -10.6395 q^{81} -2.48091 q^{82} -4.74746 q^{83} +4.35473 q^{84} -5.19650 q^{85} -1.44828 q^{86} -8.35402 q^{87} +12.7455 q^{89} -0.804342 q^{90} -1.01025 q^{91} -15.5063 q^{92} -12.9614 q^{93} +1.45541 q^{94} -1.01604 q^{95} +8.92986 q^{96} +4.47793 q^{97} -2.10820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - q^{2} + 9 q^{3} + 31 q^{4} - 26 q^{5} - 9 q^{6} - 3 q^{7} - 3 q^{8} + 29 q^{9} + q^{10} + 16 q^{12} + 26 q^{13} - 7 q^{14} - 9 q^{15} + 45 q^{16} + 8 q^{17} + 11 q^{18} + 5 q^{19} - 31 q^{20} + 3 q^{21}+ \cdots - 6 q^{98}+O(q^{100}) \) 26 * q - q^2 + 9 * q^3 + 31 * q^4 - 26 * q^5 - 9 * q^6 - 3 * q^7 - 3 * q^8 + 29 * q^9 + q^10 + 16 * q^12 + 26 * q^13 - 7 * q^14 - 9 * q^15 + 45 * q^16 + 8 * q^17 + 11 * q^18 + 5 * q^19 - 31 * q^20 + 3 * q^21 + 19 * q^23 - 36 * q^24 + 26 * q^25 - q^26 + 57 * q^27 - 45 * q^28 + 20 * q^29 + 9 * q^30 + 43 * q^31 - 17 * q^32 - 4 * q^34 + 3 * q^35 + 20 * q^36 + 23 * q^37 + 2 * q^38 + 9 * q^39 + 3 * q^40 + 19 * q^41 + 4 * q^42 + 32 * q^43 - 29 * q^45 - 43 * q^46 + 39 * q^47 + 62 * q^48 + 47 * q^49 - q^50 + 23 * q^51 + 31 * q^52 + 11 * q^53 - 26 * q^54 - 40 * q^56 - 46 * q^57 + 40 * q^58 - 8 * q^59 - 16 * q^60 + 9 * q^61 + 34 * q^62 - 23 * q^63 + 39 * q^64 - 26 * q^65 + 56 * q^67 + 5 * q^68 + 32 * q^69 + 7 * q^70 - 7 * q^71 - 91 * q^72 + 16 * q^73 + 78 * q^74 + 9 * q^75 + 24 * q^76 - 9 * q^78 + 13 * q^79 - 45 * q^80 + 26 * q^81 + 62 * q^82 - 34 * q^83 - 15 * q^84 - 8 * q^85 - 23 * q^86 + 17 * q^87 - 5 * q^89 - 11 * q^90 - 3 * q^91 + 112 * q^92 + 74 * q^93 - 10 * q^94 - 5 * q^95 - 31 * q^96 + 52 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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