LMFDB - Embedded newform 784.4.f.j.783.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.16
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30410 q^{3} +13.4612i q^{5} -16.0830 q^{9} +O(q^{10})$	$q+3.30410 q^{3} +13.4612i q^{5} -16.0830 q^{9} +28.0363i q^{11} -27.3479i q^{13} +44.4772i q^{15} +1.11444i q^{17} -150.534 q^{19} -67.4421i q^{23} -56.2048 q^{25} -142.350 q^{27} +217.296 q^{29} -42.9820 q^{31} +92.6348i q^{33} -238.364 q^{37} -90.3602i q^{39} -21.4009i q^{41} -219.906i q^{43} -216.496i q^{45} -156.752 q^{47} +3.68222i q^{51} -666.721 q^{53} -377.404 q^{55} -497.378 q^{57} +782.018 q^{59} -112.475i q^{61} +368.137 q^{65} -1026.00i q^{67} -222.835i q^{69} -621.964i q^{71} -272.144i q^{73} -185.706 q^{75} +837.469i q^{79} -36.0990 q^{81} -105.709 q^{83} -15.0017 q^{85} +717.967 q^{87} +1004.67i q^{89} -142.017 q^{93} -2026.37i q^{95} +1846.54i q^{97} -450.907i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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	0
$2$	0	0
$3$	3.30410	0.635874	0.317937	−	0.948112i	$-0.397010\pi$
$3$	3.30410	0.635874	0.317937	+	0.948112i	$0.397010\pi$
$4$	0	0
$5$	13.4612i	1.20401i	0.798493	+	0.602005i	$0.205631\pi$
$5$	13.4612i	1.20401i	−0.798493	+	0.602005i	$0.794369\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−16.0830	−0.595665
$10$	0	0
$11$	28.0363i	0.768480i	0.923233	+	0.384240i	$0.125536\pi$
$11$	28.0363i	0.768480i	−0.923233	+	0.384240i	$0.874464\pi$
$12$	0	0
$13$	− 27.3479i	− 0.583458i	−0.956501	−	0.291729i	$-0.905769\pi$
$13$	− 27.3479i	− 0.583458i	0.956501	−	0.291729i	$-0.0942306\pi$
$14$	0	0
$15$	44.4772i	0.765598i
$16$	0	0
$17$	1.11444i	0.0158995i	0.999968	+	0.00794975i	$0.00253051\pi$
$17$	1.11444i	0.0158995i	−0.999968	+	0.00794975i	$0.997469\pi$
$18$	0	0
$19$	−150.534	−1.81762	−0.908810	−	0.417209i	$-0.863008\pi$
$19$	−150.534	−1.81762	−0.908810	+	0.417209i	$0.863008\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 67.4421i	− 0.611420i	−0.952125	−	0.305710i	$-0.901106\pi$
$23$	− 67.4421i	− 0.611420i	0.952125	−	0.305710i	$-0.0988937\pi$
$24$	0	0
$25$	−56.2048	−0.449639
$26$	0	0
$27$	−142.350	−1.01464
$28$	0	0
$29$	217.296	1.39141	0.695704	−	0.718328i	$-0.255092\pi$
$29$	217.296	1.39141	0.695704	+	0.718328i	$0.255092\pi$
$30$	0	0
$31$	−42.9820	−0.249026	−0.124513	−	0.992218i	$-0.539737\pi$
$31$	−42.9820	−0.249026	−0.124513	+	0.992218i	$0.539737\pi$
$32$	0	0
$33$	92.6348i	0.488656i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−238.364	−1.05910	−0.529552	−	0.848277i	$-0.677640\pi$
$37$	−238.364	−1.05910	−0.529552	+	0.848277i	$0.677640\pi$
$38$	0	0
$39$	− 90.3602i	− 0.371006i
$40$	0	0
$41$	− 21.4009i	− 0.0815184i	−0.999169	−	0.0407592i	$-0.987022\pi$
$41$	− 21.4009i	− 0.0815184i	0.999169	−	0.0407592i	$-0.0129777\pi$
$42$	0	0
$43$	− 219.906i	− 0.779893i	−0.920837	−	0.389947i	$-0.872493\pi$
$43$	− 219.906i	− 0.779893i	0.920837	−	0.389947i	$-0.127507\pi$
$44$	0	0
$45$	− 216.496i	− 0.717186i
$46$	0	0
$47$	−156.752	−0.486482	−0.243241	−	0.969966i	$-0.578211\pi$
$47$	−156.752	−0.486482	−0.243241	+	0.969966i	$0.578211\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.68222i	0.0101101i
$52$	0	0
$53$	−666.721	−1.72795	−0.863973	−	0.503538i	$-0.832031\pi$
$53$	−666.721	−1.72795	−0.863973	+	0.503538i	$0.832031\pi$
$54$	0	0
$55$	−377.404	−0.925257
$56$	0	0
$57$	−497.378	−1.15578
$58$	0	0
$59$	782.018	1.72559	0.862797	−	0.505551i	$-0.168711\pi$
$59$	782.018	1.72559	0.862797	+	0.505551i	$0.168711\pi$
$60$	0	0
$61$	− 112.475i	− 0.236082i	−0.993009	−	0.118041i	$-0.962339\pi$
$61$	− 112.475i	− 0.236082i	0.993009	−	0.118041i	$-0.0376614\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	368.137	0.702489
$66$	0	0
$67$	− 1026.00i	− 1.87082i	−0.353559	−	0.935412i	$-0.615029\pi$
$67$	− 1026.00i	− 1.87082i	0.353559	−	0.935412i	$-0.384971\pi$
$68$	0	0
$69$	− 222.835i	− 0.388786i
$70$	0	0
$71$	− 621.964i	− 1.03963i	−0.854280	−	0.519813i	$-0.826001\pi$
$71$	− 621.964i	− 1.03963i	0.854280	−	0.519813i	$-0.173999\pi$
$72$	0	0
$73$	− 272.144i	− 0.436329i	−0.975912	−	0.218164i	$-0.929993\pi$
$73$	− 272.144i	− 0.436329i	0.975912	−	0.218164i	$-0.0700069\pi$
$74$	0	0
$75$	−185.706	−0.285913
$76$	0	0
$77$	0	0
$78$	0	0
$79$	837.469i	1.19269i	0.802728	+	0.596346i	$0.203381\pi$
$79$	837.469i	1.19269i	−0.802728	+	0.596346i	$0.796619\pi$
$80$	0	0
$81$	−36.0990	−0.0495186
$82$	0	0
$83$	−105.709	−0.139796	−0.0698980	−	0.997554i	$-0.522267\pi$
$83$	−105.709	−0.139796	−0.0698980	+	0.997554i	$0.522267\pi$
$84$	0	0
$85$	−15.0017	−0.0191432
$86$	0	0
$87$	717.967	0.884760
$88$	0	0
$89$	1004.67i	1.19658i	0.801281	+	0.598288i	$0.204152\pi$
$89$	1004.67i	1.19658i	−0.801281	+	0.598288i	$0.795848\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−142.017	−0.158349
$94$	0	0
$95$	− 2026.37i	− 2.18843i
$96$	0	0
$97$	1846.54i	1.93287i	0.256919	+	0.966433i	$0.417293\pi$
$97$	1846.54i	1.93287i	−0.256919	+	0.966433i	$0.582707\pi$
$98$	0	0
$99$	− 450.907i	− 0.457756i
$100$	0	0
$101$	− 802.281i	− 0.790395i	−0.918596	−	0.395198i	$-0.870676\pi$
$101$	− 802.281i	− 0.790395i	0.918596	−	0.395198i	$-0.129324\pi$
$102$	0	0
$103$	−1629.54	−1.55887	−0.779434	−	0.626485i	$-0.784493\pi$
$103$	−1629.54	−1.55887	−0.779434	+	0.626485i	$0.784493\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1789.05i	− 1.61639i	−0.588914	−	0.808196i	$-0.700444\pi$
$107$	− 1789.05i	− 1.61639i	0.588914	−	0.808196i	$-0.299556\pi$
$108$	0	0
$109$	−1243.49	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$109$	−1243.49	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$110$	0	0
$111$	−787.578	−0.673456
$112$	0	0
$113$	−849.364	−0.707093	−0.353546	−	0.935417i	$-0.615024\pi$
$113$	−849.364	−0.707093	−0.353546	+	0.935417i	$0.615024\pi$
$114$	0	0
$115$	907.854	0.736155
$116$	0	0
$117$	439.836i	0.347545i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	544.963	0.409439
$122$	0	0
$123$	− 70.7105i	− 0.0518354i
$124$	0	0
$125$	926.068i	0.662640i
$126$	0	0
$127$	1539.77i	1.07585i	0.842993	+	0.537925i	$0.180792\pi$
$127$	1539.77i	1.07585i	−0.842993	+	0.537925i	$0.819208\pi$
$128$	0	0
$129$	− 726.592i	− 0.495913i
$130$	0	0
$131$	−2087.36	−1.39216	−0.696081	−	0.717963i	$-0.745074\pi$
$131$	−2087.36	−1.39216	−0.696081	+	0.717963i	$0.745074\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1916.21i	− 1.22164i
$136$	0	0
$137$	1330.74	0.829872	0.414936	−	0.909851i	$-0.363804\pi$
$137$	1330.74	0.829872	0.414936	+	0.909851i	$0.363804\pi$
$138$	0	0
$139$	−1310.39	−0.799613	−0.399806	−	0.916600i	$-0.630923\pi$
$139$	−1310.39	−0.799613	−0.399806	+	0.916600i	$0.630923\pi$
$140$	0	0
$141$	−517.924	−0.309341
$142$	0	0
$143$	766.736	0.448376
$144$	0	0
$145$	2925.07i	1.67527i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2722.73	−1.49701	−0.748507	−	0.663127i	$-0.769229\pi$
$149$	−2722.73	−1.49701	−0.748507	+	0.663127i	$0.769229\pi$
$150$	0	0
$151$	− 3114.44i	− 1.67847i	−0.543767	−	0.839236i	$-0.683002\pi$
$151$	− 3114.44i	− 1.67847i	0.543767	−	0.839236i	$-0.316998\pi$
$152$	0	0
$153$	− 17.9235i	− 0.00947078i
$154$	0	0
$155$	− 578.591i	− 0.299829i
$156$	0	0
$157$	1998.99i	1.01616i	0.861310	+	0.508079i	$0.169644\pi$
$157$	1998.99i	1.01616i	−0.861310	+	0.508079i	$0.830356\pi$
$158$	0	0
$159$	−2202.91	−1.09876
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1049.95i	0.504529i	0.967658	+	0.252265i	$0.0811754\pi$
$163$	1049.95i	0.504529i	−0.967658	+	0.252265i	$0.918825\pi$
$164$	0	0
$165$	−1246.98	−0.588346
$166$	0	0
$167$	−833.219	−0.386086	−0.193043	−	0.981190i	$-0.561836\pi$
$167$	−833.219	−0.386086	−0.193043	+	0.981190i	$0.561836\pi$
$168$	0	0
$169$	1449.09	0.659577
$170$	0	0
$171$	2421.03	1.08269
$172$	0	0
$173$	486.217i	0.213678i	0.994276	+	0.106839i	$0.0340730\pi$
$173$	486.217i	0.213678i	−0.994276	+	0.106839i	$0.965927\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2583.86	1.09726
$178$	0	0
$179$	300.249i	0.125372i	0.998033	+	0.0626862i	$0.0199667\pi$
$179$	300.249i	0.125372i	−0.998033	+	0.0626862i	$0.980033\pi$
$180$	0	0
$181$	879.807i	0.361301i	0.983547	+	0.180651i	$0.0578203\pi$
$181$	879.807i	0.361301i	−0.983547	+	0.180651i	$0.942180\pi$
$182$	0	0
$183$	− 371.629i	− 0.150118i
$184$	0	0
$185$	− 3208.68i	− 1.27517i
$186$	0	0
$187$	−31.2448	−0.0122184
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2322.81i	0.879963i	0.898007	+	0.439981i	$0.145015\pi$
$191$	2322.81i	0.879963i	−0.898007	+	0.439981i	$0.854985\pi$
$192$	0	0
$193$	−3232.46	−1.20558	−0.602791	−	0.797899i	$-0.705945\pi$
$193$	−3232.46	−1.20558	−0.602791	+	0.797899i	$0.705945\pi$
$194$	0	0
$195$	1216.36	0.446694
$196$	0	0
$197$	−1415.25	−0.511841	−0.255920	−	0.966698i	$-0.582379\pi$
$197$	−1415.25	−0.511841	−0.255920	+	0.966698i	$0.582379\pi$
$198$	0	0
$199$	−2794.93	−0.995616	−0.497808	−	0.867287i	$-0.665862\pi$
$199$	−2794.93	−0.995616	−0.497808	+	0.867287i	$0.665862\pi$
$200$	0	0
$201$	− 3389.99i	− 1.18961i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	288.082	0.0981489
$206$	0	0
$207$	1084.67i	0.364201i
$208$	0	0
$209$	− 4220.41i	− 1.39680i
$210$	0	0
$211$	− 2369.52i	− 0.773103i	−0.922268	−	0.386551i	$-0.873666\pi$
$211$	− 2369.52i	− 0.773103i	0.922268	−	0.386551i	$-0.126334\pi$
$212$	0	0
$213$	− 2055.03i	− 0.661071i
$214$	0	0
$215$	2960.21	0.938999
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 899.188i	− 0.277450i
$220$	0	0
$221$	30.4777	0.00927669
$222$	0	0
$223$	−1363.08	−0.409320	−0.204660	−	0.978833i	$-0.565609\pi$
$223$	−1363.08	−0.409320	−0.204660	+	0.978833i	$0.565609\pi$
$224$	0	0
$225$	903.940	0.267834
$226$	0	0
$227$	2164.86	0.632982	0.316491	−	0.948596i	$-0.397495\pi$
$227$	2164.86	0.632982	0.316491	+	0.948596i	$0.397495\pi$
$228$	0	0
$229$	− 2633.48i	− 0.759936i	−0.925000	−	0.379968i	$-0.875935\pi$
$229$	− 2633.48i	− 0.759936i	0.925000	−	0.379968i	$-0.124065\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1340.74	−0.376973	−0.188486	−	0.982076i	$-0.560358\pi$
$233$	−1340.74	−0.376973	−0.188486	+	0.982076i	$0.560358\pi$
$234$	0	0
$235$	− 2110.08i	− 0.585729i
$236$	0	0
$237$	2767.08i	0.758401i
$238$	0	0
$239$	− 5371.34i	− 1.45374i	−0.686777	−	0.726868i	$-0.740975\pi$
$239$	− 5371.34i	− 1.45374i	0.686777	−	0.726868i	$-0.259025\pi$
$240$	0	0
$241$	1752.16i	0.468326i	0.972197	+	0.234163i	$0.0752349\pi$
$241$	1752.16i	0.468326i	−0.972197	+	0.234163i	$0.924765\pi$
$242$	0	0
$243$	3724.18	0.983154
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4116.79i	1.06051i
$248$	0	0
$249$	−349.273	−0.0888926
$250$	0	0
$251$	634.002	0.159434	0.0797168	−	0.996818i	$-0.474598\pi$
$251$	634.002	0.159434	0.0797168	+	0.996818i	$0.474598\pi$
$252$	0	0
$253$	1890.83	0.469864
$254$	0	0
$255$	−49.5672	−0.0121726
$256$	0	0
$257$	1665.41i	0.404223i	0.979363	+	0.202111i	$0.0647803\pi$
$257$	1665.41i	0.404223i	−0.979363	+	0.202111i	$0.935220\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3494.76	−0.828813
$262$	0	0
$263$	5307.93i	1.24449i	0.782823	+	0.622245i	$0.213779\pi$
$263$	5307.93i	1.24449i	−0.782823	+	0.622245i	$0.786221\pi$
$264$	0	0
$265$	− 8974.89i	− 2.08046i
$266$	0	0
$267$	3319.54i	0.760871i
$268$	0	0
$269$	7778.53i	1.76307i	0.472120	+	0.881534i	$0.343489\pi$
$269$	7778.53i	1.76307i	−0.472120	+	0.881534i	$0.656511\pi$
$270$	0	0
$271$	4760.96	1.06719	0.533594	−	0.845741i	$-0.320841\pi$
$271$	4760.96	1.06719	0.533594	+	0.845741i	$0.320841\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1575.78i	− 0.345538i
$276$	0	0
$277$	4243.97	0.920560	0.460280	−	0.887774i	$-0.347749\pi$
$277$	4243.97	0.920560	0.460280	+	0.887774i	$0.347749\pi$
$278$	0	0
$279$	691.278	0.148336
$280$	0	0
$281$	1037.16	0.220183	0.110092	−	0.993921i	$-0.464886\pi$
$281$	1037.16	0.220183	0.110092	+	0.993921i	$0.464886\pi$
$282$	0	0
$283$	−7507.89	−1.57702	−0.788512	−	0.615020i	$-0.789148\pi$
$283$	−7507.89	−1.57702	−0.788512	+	0.615020i	$0.789148\pi$
$284$	0	0
$285$	− 6695.32i	− 1.39157i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4911.76	0.999747
$290$	0	0
$291$	6101.15i	1.22906i
$292$	0	0
$293$	5514.65i	1.09955i	0.835311	+	0.549777i	$0.185287\pi$
$293$	5514.65i	1.09955i	−0.835311	+	0.549777i	$0.814713\pi$
$294$	0	0
$295$	10526.9i	2.07763i
$296$	0	0
$297$	− 3990.98i	− 0.779731i
$298$	0	0
$299$	−1844.40	−0.356738
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 2650.81i	− 0.502591i
$304$	0	0
$305$	1514.06	0.284244
$306$	0	0
$307$	737.420	0.137090	0.0685452	−	0.997648i	$-0.478164\pi$
$307$	737.420	0.137090	0.0685452	+	0.997648i	$0.478164\pi$
$308$	0	0
$309$	−5384.16	−0.991243
$310$	0	0
$311$	−5638.29	−1.02803	−0.514016	−	0.857780i	$-0.671843\pi$
$311$	−5638.29	−1.02803	−0.514016	+	0.857780i	$0.671843\pi$
$312$	0	0
$313$	5677.30i	1.02524i	0.858616	+	0.512620i	$0.171325\pi$
$313$	5677.30i	1.02524i	−0.858616	+	0.512620i	$0.828675\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−911.918	−0.161572	−0.0807861	−	0.996731i	$-0.525743\pi$
$317$	−911.918	−0.161572	−0.0807861	+	0.996731i	$0.525743\pi$
$318$	0	0
$319$	6092.19i	1.06927i
$320$	0	0
$321$	− 5911.19i	− 1.02782i
$322$	0	0
$323$	− 167.761i	− 0.0288993i
$324$	0	0
$325$	1537.09i	0.262345i
$326$	0	0
$327$	−4108.62	−0.694823
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2444.24i	0.405884i	0.979191	+	0.202942i	$0.0650503\pi$
$331$	2444.24i	0.405884i	−0.979191	+	0.202942i	$0.934950\pi$
$332$	0	0
$333$	3833.60	0.630871
$334$	0	0
$335$	13811.2	2.25249
$336$	0	0
$337$	10364.7	1.67537	0.837686	−	0.546152i	$-0.183908\pi$
$337$	10364.7	1.67537	0.837686	+	0.546152i	$0.183908\pi$
$338$	0	0
$339$	−2806.38	−0.449621
$340$	0	0
$341$	− 1205.06i	− 0.191371i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2999.64	0.468102
$346$	0	0
$347$	9192.48i	1.42213i	0.703128	+	0.711063i	$0.251786\pi$
$347$	9192.48i	1.42213i	−0.703128	+	0.711063i	$0.748214\pi$
$348$	0	0
$349$	11131.7i	1.70735i	0.520809	+	0.853673i	$0.325630\pi$
$349$	11131.7i	1.70735i	−0.520809	+	0.853673i	$0.674370\pi$
$350$	0	0
$351$	3892.99i	0.592001i
$352$	0	0
$353$	5971.26i	0.900334i	0.892944	+	0.450167i	$0.148636\pi$
$353$	5971.26i	0.900334i	−0.892944	+	0.450167i	$0.851364\pi$
$354$	0	0
$355$	8372.40	1.25172
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 5799.58i	− 0.852619i	−0.904577	−	0.426310i	$-0.859813\pi$
$359$	− 5799.58i	− 0.852619i	0.904577	−	0.426310i	$-0.140187\pi$
$360$	0	0
$361$	15801.4	2.30375
$362$	0	0
$363$	1800.61	0.260351
$364$	0	0
$365$	3663.39	0.525344
$366$	0	0
$367$	1273.10	0.181077	0.0905386	−	0.995893i	$-0.471141\pi$
$367$	1273.10	0.181077	0.0905386	+	0.995893i	$0.471141\pi$
$368$	0	0
$369$	344.189i	0.0485576i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6201.55	−0.860869	−0.430434	−	0.902622i	$-0.641640\pi$
$373$	−6201.55	−0.860869	−0.430434	+	0.902622i	$0.641640\pi$
$374$	0	0
$375$	3059.82i	0.421355i
$376$	0	0
$377$	− 5942.60i	− 0.811829i
$378$	0	0
$379$	5064.08i	0.686345i	0.939272	+	0.343172i	$0.111501\pi$
$379$	5064.08i	0.686345i	−0.939272	+	0.343172i	$0.888499\pi$
$380$	0	0
$381$	5087.56i	0.684104i
$382$	0	0
$383$	7814.34	1.04254	0.521272	−	0.853391i	$-0.325458\pi$
$383$	7814.34	1.04254	0.521272	+	0.853391i	$0.325458\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3536.74i	0.464555i
$388$	0	0
$389$	6713.69	0.875058	0.437529	−	0.899204i	$-0.355854\pi$
$389$	6713.69	0.875058	0.437529	+	0.899204i	$0.355854\pi$
$390$	0	0
$391$	75.1602	0.00972127
$392$	0	0
$393$	−6896.83	−0.885239
$394$	0	0
$395$	−11273.4	−1.43601
$396$	0	0
$397$	− 9882.90i	− 1.24939i	−0.780868	−	0.624696i	$-0.785223\pi$
$397$	− 9882.90i	− 1.24939i	0.780868	−	0.624696i	$-0.214777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9230.17	1.14946	0.574729	−	0.818344i	$-0.305107\pi$
$401$	9230.17	1.14946	0.574729	+	0.818344i	$0.305107\pi$
$402$	0	0
$403$	1175.47i	0.145296i
$404$	0	0
$405$	− 485.938i	− 0.0596208i
$406$	0	0
$407$	− 6682.86i	− 0.813900i
$408$	0	0
$409$	− 5502.17i	− 0.665195i	−0.943069	−	0.332597i	$-0.892075\pi$
$409$	− 5502.17i	− 0.665195i	0.943069	−	0.332597i	$-0.107925\pi$
$410$	0	0
$411$	4396.88	0.527694
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 1422.97i	− 0.168316i
$416$	0	0
$417$	−4329.67	−0.508453
$418$	0	0
$419$	6463.00	0.753552	0.376776	−	0.926304i	$-0.377033\pi$
$419$	6463.00	0.753552	0.376776	+	0.926304i	$0.377033\pi$
$420$	0	0
$421$	−10065.7	−1.16526	−0.582629	−	0.812738i	$-0.697976\pi$
$421$	−10065.7	−1.16526	−0.582629	+	0.812738i	$0.697976\pi$
$422$	0	0
$423$	2521.04	0.289780
$424$	0	0
$425$	− 62.6370i	− 0.00714903i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2533.37	0.285110
$430$	0	0
$431$	− 4385.75i	− 0.490149i	−0.969504	−	0.245074i	$-0.921188\pi$
$431$	− 4385.75i	− 0.490149i	0.969504	−	0.245074i	$-0.0788124\pi$
$432$	0	0
$433$	− 699.884i	− 0.0776774i	−0.999245	−	0.0388387i	$-0.987634\pi$
$433$	− 699.884i	− 0.0776774i	0.999245	−	0.0388387i	$-0.0123658\pi$
$434$	0	0
$435$	9664.72i	1.06526i
$436$	0	0
$437$	10152.3i	1.11133i
$438$	0	0
$439$	10403.7	1.13107	0.565537	−	0.824723i	$-0.308669\pi$
$439$	10403.7	1.13107	0.565537	+	0.824723i	$0.308669\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5903.68i	0.633165i	0.948565	+	0.316582i	$0.102535\pi$
$443$	5903.68i	0.633165i	−0.948565	+	0.316582i	$0.897465\pi$
$444$	0	0
$445$	−13524.2	−1.44069
$446$	0	0
$447$	−8996.18	−0.951912
$448$	0	0
$449$	−6329.16	−0.665237	−0.332619	−	0.943061i	$-0.607932\pi$
$449$	−6329.16	−0.665237	−0.332619	+	0.943061i	$0.607932\pi$
$450$	0	0
$451$	600.002	0.0626452
$452$	0	0
$453$	− 10290.4i	− 1.06730i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8748.66	0.895504	0.447752	−	0.894158i	$-0.352225\pi$
$457$	8748.66	0.895504	0.447752	+	0.894158i	$0.352225\pi$
$458$	0	0
$459$	− 158.641i	− 0.0161323i
$460$	0	0
$461$	− 2975.68i	− 0.300632i	−0.988638	−	0.150316i	$-0.951971\pi$
$461$	− 2975.68i	− 0.300632i	0.988638	−	0.150316i	$-0.0480291\pi$
$462$	0	0
$463$	− 12437.7i	− 1.24844i	−0.781248	−	0.624220i	$-0.785417\pi$
$463$	− 12437.7i	− 1.24844i	0.781248	−	0.624220i	$-0.214583\pi$
$464$	0	0
$465$	− 1911.72i	− 0.190654i
$466$	0	0
$467$	8001.86	0.792895	0.396447	−	0.918057i	$-0.370243\pi$
$467$	8001.86	0.792895	0.396447	+	0.918057i	$0.370243\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6604.86i	0.646148i
$472$	0	0
$473$	6165.37	0.599332
$474$	0	0
$475$	8460.72	0.817273
$476$	0	0
$477$	10722.8	1.02928
$478$	0	0
$479$	−5999.60	−0.572294	−0.286147	−	0.958186i	$-0.592375\pi$
$479$	−5999.60	−0.572294	−0.286147	+	0.958186i	$0.592375\pi$
$480$	0	0
$481$	6518.77i	0.617943i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24856.7	−2.32719
$486$	0	0
$487$	13599.5i	1.26540i	0.774396	+	0.632701i	$0.218054\pi$
$487$	13599.5i	1.26540i	−0.774396	+	0.632701i	$0.781946\pi$
$488$	0	0
$489$	3469.13i	0.320817i
$490$	0	0
$491$	− 7896.69i	− 0.725810i	−0.931826	−	0.362905i	$-0.881785\pi$
$491$	− 7896.69i	− 0.725810i	0.931826	−	0.362905i	$-0.118215\pi$
$492$	0	0
$493$	242.164i	0.0221227i
$494$	0	0
$495$	6069.77	0.551143
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6551.02i	0.587703i	0.955851	+	0.293851i	$0.0949371\pi$
$499$	6551.02i	0.587703i	−0.955851	+	0.293851i	$0.905063\pi$
$500$	0	0
$501$	−2753.03	−0.245502
$502$	0	0
$503$	3768.11	0.334020	0.167010	−	0.985955i	$-0.446589\pi$
$503$	3768.11	0.334020	0.167010	+	0.985955i	$0.446589\pi$
$504$	0	0
$505$	10799.7	0.951643
$506$	0	0
$507$	4787.93	0.419407
$508$	0	0
$509$	2039.48i	0.177600i	0.996049	+	0.0887998i	$0.0283031\pi$
$509$	2039.48i	0.177600i	−0.996049	+	0.0887998i	$0.971697\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	21428.5	1.84423
$514$	0	0
$515$	− 21935.6i	− 1.87689i
$516$	0	0
$517$	− 4394.76i	− 0.373851i
$518$	0	0
$519$	1606.51i	0.135872i
$520$	0	0
$521$	− 20137.7i	− 1.69338i	−0.532089	−	0.846688i	$-0.678593\pi$
$521$	− 20137.7i	− 1.69338i	0.532089	−	0.846688i	$-0.321407\pi$
$522$	0	0
$523$	−14310.2	−1.19645	−0.598225	−	0.801328i	$-0.704127\pi$
$523$	−14310.2	−1.19645	−0.598225	+	0.801328i	$0.704127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 47.9009i	− 0.00395939i
$528$	0	0
$529$	7618.56	0.626166
$530$	0	0
$531$	−12577.2	−1.02788
$532$	0	0
$533$	−585.270	−0.0475626
$534$	0	0
$535$	24082.8	1.94615
$536$	0	0
$537$	992.051i	0.0797210i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17950.3	−1.42651	−0.713257	−	0.700903i	$-0.752781\pi$
$541$	−17950.3	−1.42651	−0.713257	+	0.700903i	$0.752781\pi$
$542$	0	0
$543$	2906.97i	0.229742i
$544$	0	0
$545$	− 16739.0i	− 1.31563i
$546$	0	0
$547$	− 7231.41i	− 0.565252i	−0.959230	−	0.282626i	$-0.908795\pi$
$547$	− 7231.41i	− 0.565252i	0.959230	−	0.282626i	$-0.0912055\pi$
$548$	0	0
$549$	1808.93i	0.140625i
$550$	0	0
$551$	−32710.4	−2.52905
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 10601.8i	− 0.810848i
$556$	0	0
$557$	−21447.3	−1.63151	−0.815755	−	0.578397i	$-0.803678\pi$
$557$	−21447.3	−1.63151	−0.815755	+	0.578397i	$0.803678\pi$
$558$	0	0
$559$	−6013.99	−0.455035
$560$	0	0
$561$	−103.236	−0.00776939
$562$	0	0
$563$	−15864.0	−1.18754	−0.593772	−	0.804633i	$-0.702362\pi$
$563$	−15864.0	−1.18754	−0.593772	+	0.804633i	$0.702362\pi$
$564$	0	0
$565$	− 11433.5i	− 0.851346i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9703.47	0.714922	0.357461	−	0.933928i	$-0.383642\pi$
$569$	9703.47	0.714922	0.357461	+	0.933928i	$0.383642\pi$
$570$	0	0
$571$	9807.28i	0.718777i	0.933188	+	0.359389i	$0.117015\pi$
$571$	9807.28i	0.718777i	−0.933188	+	0.359389i	$0.882985\pi$
$572$	0	0
$573$	7674.80i	0.559545i
$574$	0	0
$575$	3790.57i	0.274918i
$576$	0	0
$577$	2204.49i	0.159054i	0.996833	+	0.0795271i	$0.0253410\pi$
$577$	2204.49i	0.159054i	−0.996833	+	0.0795271i	$0.974659\pi$
$578$	0	0
$579$	−10680.4	−0.766598
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 18692.4i	− 1.32789i
$584$	0	0
$585$	−5920.73	−0.418448
$586$	0	0
$587$	−13099.5	−0.921082	−0.460541	−	0.887639i	$-0.652344\pi$
$587$	−13099.5	−0.921082	−0.460541	+	0.887639i	$0.652344\pi$
$588$	0	0
$589$	6470.24	0.452634
$590$	0	0
$591$	−4676.13	−0.325466
$592$	0	0
$593$	23577.0i	1.63270i	0.577558	+	0.816350i	$0.304006\pi$
$593$	23577.0i	1.63270i	−0.577558	+	0.816350i	$0.695994\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9234.73	−0.633086
$598$	0	0
$599$	− 5404.03i	− 0.368619i	−0.982868	−	0.184309i	$-0.940995\pi$
$599$	− 5404.03i	− 0.368619i	0.982868	−	0.184309i	$-0.0590048\pi$
$600$	0	0
$601$	10606.7i	0.719893i	0.932973	+	0.359946i	$0.117205\pi$
$601$	10606.7i	0.719893i	−0.932973	+	0.359946i	$0.882795\pi$
$602$	0	0
$603$	16501.0i	1.11438i
$604$	0	0
$605$	7335.88i	0.492968i
$606$	0	0
$607$	454.341	0.0303808	0.0151904	−	0.999885i	$-0.495165\pi$
$607$	454.341	0.0303808	0.0151904	+	0.999885i	$0.495165\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4286.85i	0.283842i
$612$	0	0
$613$	−9527.79	−0.627771	−0.313886	−	0.949461i	$-0.601631\pi$
$613$	−9527.79	−0.627771	−0.313886	+	0.949461i	$0.601631\pi$
$614$	0	0
$615$	951.851	0.0624103
$616$	0	0
$617$	−8072.43	−0.526716	−0.263358	−	0.964698i	$-0.584830\pi$
$617$	−8072.43	−0.526716	−0.263358	+	0.964698i	$0.584830\pi$
$618$	0	0
$619$	27389.0	1.77845	0.889224	−	0.457473i	$-0.151245\pi$
$619$	27389.0	1.77845	0.889224	+	0.457473i	$0.151245\pi$
$620$	0	0
$621$	9600.40i	0.620371i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19491.6	−1.24746
$626$	0	0
$627$	− 13944.7i	− 0.888191i
$628$	0	0
$629$	− 265.643i	− 0.0168392i
$630$	0	0
$631$	12763.2i	0.805219i	0.915372	+	0.402610i	$0.131897\pi$
$631$	12763.2i	0.805219i	−0.915372	+	0.402610i	$0.868103\pi$
$632$	0	0
$633$	− 7829.13i	− 0.491595i
$634$	0	0
$635$	−20727.3	−1.29533
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10003.0i	0.619269i
$640$	0	0
$641$	−1044.98	−0.0643904	−0.0321952	−	0.999482i	$-0.510250\pi$
$641$	−1044.98	−0.0643904	−0.0321952	+	0.999482i	$0.510250\pi$
$642$	0	0
$643$	−29419.0	−1.80431	−0.902155	−	0.431413i	$-0.858015\pi$
$643$	−29419.0	−1.80431	−0.902155	+	0.431413i	$0.858015\pi$
$644$	0	0
$645$	9780.82	0.597085
$646$	0	0
$647$	−9910.27	−0.602184	−0.301092	−	0.953595i	$-0.597351\pi$
$647$	−9910.27	−0.602184	−0.301092	+	0.953595i	$0.597351\pi$
$648$	0	0
$649$	21924.9i	1.32608i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7151.21	−0.428558	−0.214279	−	0.976772i	$-0.568740\pi$
$653$	−7151.21	−0.428558	−0.214279	+	0.976772i	$0.568740\pi$
$654$	0	0
$655$	− 28098.4i	− 1.67618i
$656$	0	0
$657$	4376.87i	0.259906i
$658$	0	0
$659$	− 30748.1i	− 1.81757i	−0.417267	−	0.908784i	$-0.637012\pi$
$659$	− 30748.1i	− 1.81757i	0.417267	−	0.908784i	$-0.362988\pi$
$660$	0	0
$661$	− 17239.0i	− 1.01440i	−0.861829	−	0.507200i	$-0.830681\pi$
$661$	− 17239.0i	− 1.01440i	0.861829	−	0.507200i	$-0.169319\pi$
$662$	0	0
$663$	100.701	0.00589880
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 14654.9i	− 0.850735i
$668$	0	0
$669$	−4503.73	−0.260275
$670$	0	0
$671$	3153.39	0.181424
$672$	0	0
$673$	−11342.2	−0.649640	−0.324820	−	0.945776i	$-0.605304\pi$
$673$	−11342.2	−0.649640	−0.324820	+	0.945776i	$0.605304\pi$
$674$	0	0
$675$	8000.77	0.456222
$676$	0	0
$677$	12121.5i	0.688135i	0.938945	+	0.344068i	$0.111805\pi$
$677$	12121.5i	0.688135i	−0.938945	+	0.344068i	$0.888195\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7152.91	0.402496
$682$	0	0
$683$	− 17871.3i	− 1.00121i	−0.865676	−	0.500605i	$-0.833111\pi$
$683$	− 17871.3i	− 1.00121i	0.865676	−	0.500605i	$-0.166889\pi$
$684$	0	0
$685$	17913.4i	0.999174i
$686$	0	0
$687$	− 8701.28i	− 0.483223i
$688$	0	0
$689$	18233.4i	1.00818i
$690$	0	0
$691$	−4169.88	−0.229565	−0.114783	−	0.993391i	$-0.536617\pi$
$691$	−4169.88	−0.229565	−0.114783	+	0.993391i	$0.536617\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 17639.5i	− 0.962742i
$696$	0	0
$697$	23.8500	0.00129610
$698$	0	0
$699$	−4429.93	−0.239707
$700$	0	0
$701$	−19476.2	−1.04937	−0.524684	−	0.851297i	$-0.675817\pi$
$701$	−19476.2	−1.04937	−0.524684	+	0.851297i	$0.675817\pi$
$702$	0	0
$703$	35881.9	1.92505
$704$	0	0
$705$	− 6971.90i	− 0.372449i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14716.6	0.779538	0.389769	−	0.920913i	$-0.372555\pi$
$709$	14716.6	0.779538	0.389769	+	0.920913i	$0.372555\pi$
$710$	0	0
$711$	− 13469.0i	− 0.710444i
$712$	0	0
$713$	2898.80i	0.152259i
$714$	0	0
$715$	10321.2i	0.539849i
$716$	0	0
$717$	− 17747.4i	− 0.924392i
$718$	0	0
$719$	−17273.9	−0.895977	−0.447989	−	0.894039i	$-0.647860\pi$
$719$	−17273.9	−0.895977	−0.447989	+	0.894039i	$0.647860\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5789.30i	0.297796i
$724$	0	0
$725$	−12213.1	−0.625631
$726$	0	0
$727$	−515.093	−0.0262775	−0.0131388	−	0.999914i	$-0.504182\pi$
$727$	−515.093	−0.0262775	−0.0131388	+	0.999914i	$0.504182\pi$
$728$	0	0
$729$	13279.7	0.674680
$730$	0	0
$731$	245.073	0.0123999
$732$	0	0
$733$	− 38253.5i	− 1.92759i	−0.266644	−	0.963795i	$-0.585915\pi$
$733$	− 38253.5i	− 1.92759i	0.266644	−	0.963795i	$-0.414085\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28765.2	1.43769
$738$	0	0
$739$	− 3737.90i	− 0.186063i	−0.995663	−	0.0930317i	$-0.970344\pi$
$739$	− 3737.90i	− 0.186063i	0.995663	−	0.0930317i	$-0.0296558\pi$
$740$	0	0
$741$	13602.3i	0.674347i
$742$	0	0
$743$	− 32057.7i	− 1.58288i	−0.611245	−	0.791441i	$-0.709331\pi$
$743$	− 32057.7i	− 1.58288i	0.611245	−	0.791441i	$-0.290669\pi$
$744$	0	0
$745$	− 36651.4i	− 1.80242i
$746$	0	0
$747$	1700.11	0.0832716
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18678.3i	0.907565i	0.891113	+	0.453782i	$0.149926\pi$
$751$	18678.3i	0.907565i	−0.891113	+	0.453782i	$0.850074\pi$
$752$	0	0
$753$	2094.80	0.101380
$754$	0	0
$755$	41924.2	2.02090
$756$	0	0
$757$	−27149.4	−1.30352	−0.651759	−	0.758426i	$-0.725969\pi$
$757$	−27149.4	−1.30352	−0.651759	+	0.758426i	$0.725969\pi$
$758$	0	0
$759$	6247.48	0.298774
$760$	0	0
$761$	4340.88i	0.206777i	0.994641	+	0.103388i	$0.0329684\pi$
$761$	4340.88i	0.206777i	−0.994641	+	0.103388i	$0.967032\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	241.272	0.0114029
$766$	0	0
$767$	− 21386.6i	− 1.00681i
$768$	0	0
$769$	− 10291.8i	− 0.482618i	−0.970448	−	0.241309i	$-0.922423\pi$
$769$	− 10291.8i	− 0.482618i	0.970448	−	0.241309i	$-0.0775767\pi$
$770$	0	0
$771$	5502.67i	0.257035i
$772$	0	0
$773$	29177.0i	1.35760i	0.734324	+	0.678799i	$0.237499\pi$
$773$	29177.0i	1.35760i	−0.734324	+	0.678799i	$0.762501\pi$
$774$	0	0
$775$	2415.80	0.111972
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3221.55i	0.148170i
$780$	0	0
$781$	17437.6	0.798932
$782$	0	0
$783$	−30932.1	−1.41178
$784$	0	0
$785$	−26908.9	−1.22346
$786$	0	0
$787$	31488.4	1.42623	0.713113	−	0.701049i	$-0.247285\pi$
$787$	31488.4	1.42623	0.713113	+	0.701049i	$0.247285\pi$
$788$	0	0
$789$	17537.9i	0.791338i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3075.97	−0.137744
$794$	0	0
$795$	− 29653.9i	− 1.32291i
$796$	0	0
$797$	36101.3i	1.60448i	0.596999	+	0.802242i	$0.296360\pi$
$797$	36101.3i	1.60448i	−0.596999	+	0.802242i	$0.703640\pi$
$798$	0	0
$799$	− 174.691i	− 0.00773482i
$800$	0	0
$801$	− 16158.1i	− 0.712758i
$802$	0	0
$803$	7629.91	0.335310
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25701.0i	1.12109i
$808$	0	0
$809$	−32668.9	−1.41975	−0.709874	−	0.704328i	$-0.751248\pi$
$809$	−32668.9	−1.41975	−0.709874	+	0.704328i	$0.751248\pi$
$810$	0	0
$811$	4155.57	0.179928	0.0899642	−	0.995945i	$-0.471325\pi$
$811$	4155.57	0.179928	0.0899642	+	0.995945i	$0.471325\pi$
$812$	0	0
$813$	15730.7	0.678596
$814$	0	0
$815$	−14133.6	−0.607458
$816$	0	0
$817$	33103.3i	1.41755i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3276.19	−0.139269	−0.0696346	−	0.997573i	$-0.522183\pi$
$821$	−3276.19	−0.139269	−0.0696346	+	0.997573i	$0.522183\pi$
$822$	0	0
$823$	− 14977.8i	− 0.634376i	−0.948363	−	0.317188i	$-0.897261\pi$
$823$	− 14977.8i	− 0.634376i	0.948363	−	0.317188i	$-0.102739\pi$
$824$	0	0
$825$	− 5206.52i	− 0.219719i
$826$	0	0
$827$	− 26008.3i	− 1.09359i	−0.837267	−	0.546795i	$-0.815848\pi$
$827$	− 26008.3i	− 1.09359i	0.837267	−	0.546795i	$-0.184152\pi$
$828$	0	0
$829$	− 22291.2i	− 0.933904i	−0.884283	−	0.466952i	$-0.845352\pi$
$829$	− 22291.2i	− 0.933904i	0.884283	−	0.466952i	$-0.154648\pi$
$830$	0	0
$831$	14022.5	0.585360
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 11216.2i	− 0.464851i
$836$	0	0
$837$	6118.50	0.252672
$838$	0	0
$839$	−16141.7	−0.664210	−0.332105	−	0.943242i	$-0.607759\pi$
$839$	−16141.7	−0.664210	−0.332105	+	0.943242i	$0.607759\pi$
$840$	0	0
$841$	22828.6	0.936019
$842$	0	0
$843$	3426.86	0.140009
$844$	0	0
$845$	19506.5i	0.794137i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24806.8	−1.00279
$850$	0	0
$851$	16075.8i	0.647557i
$852$	0	0
$853$	− 3816.56i	− 0.153196i	−0.997062	−	0.0765981i	$-0.975594\pi$
$853$	− 3816.56i	− 0.153196i	0.997062	−	0.0765981i	$-0.0244058\pi$
$854$	0	0
$855$	32590.0i	1.30357i
$856$	0	0
$857$	38911.5i	1.55098i	0.631359	+	0.775491i	$0.282498\pi$
$857$	38911.5i	1.55098i	−0.631359	+	0.775491i	$0.717502\pi$
$858$	0	0
$859$	13966.1	0.554735	0.277368	−	0.960764i	$-0.410538\pi$
$859$	13966.1	0.554735	0.277368	+	0.960764i	$0.410538\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 20893.4i	− 0.824123i	−0.911156	−	0.412062i	$-0.864809\pi$
$863$	− 20893.4i	− 0.824123i	0.911156	−	0.412062i	$-0.135191\pi$
$864$	0	0
$865$	−6545.08	−0.257271
$866$	0	0
$867$	16228.9	0.635713
$868$	0	0
$869$	−23479.6	−0.916559
$870$	0	0
$871$	−28058.9	−1.09155
$872$	0	0
$873$	− 29697.9i	− 1.15134i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32606.0	1.25544	0.627722	−	0.778437i	$-0.283987\pi$
$877$	32606.0	1.25544	0.627722	+	0.778437i	$0.283987\pi$
$878$	0	0
$879$	18220.9i	0.699177i
$880$	0	0
$881$	− 33651.2i	− 1.28687i	−0.765499	−	0.643437i	$-0.777508\pi$
$881$	− 33651.2i	− 1.28687i	0.765499	−	0.643437i	$-0.222492\pi$
$882$	0	0
$883$	13005.1i	0.495646i	0.968805	+	0.247823i	$0.0797151\pi$
$883$	13005.1i	0.495646i	−0.968805	+	0.247823i	$0.920285\pi$
$884$	0	0
$885$	34782.0i	1.32111i
$886$	0	0
$887$	−39980.0	−1.51341	−0.756706	−	0.653755i	$-0.773193\pi$
$887$	−39980.0	−1.51341	−0.756706	+	0.653755i	$0.773193\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1012.08i	− 0.0380540i
$892$	0	0
$893$	23596.5	0.884240
$894$	0	0
$895$	−4041.72	−0.150950
$896$	0	0
$897$	−6094.08	−0.226840
$898$	0	0
$899$	−9339.82	−0.346497
$900$	0	0
$901$	− 743.021i	− 0.0274735i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11843.3	−0.435010
$906$	0	0
$907$	40461.3i	1.48125i	0.671917	+	0.740627i	$0.265471\pi$
$907$	40461.3i	1.48125i	−0.671917	+	0.740627i	$0.734529\pi$
$908$	0	0
$909$	12903.0i	0.470811i
$910$	0	0
$911$	− 531.853i	− 0.0193425i	−0.999953	−	0.00967127i	$-0.996921\pi$
$911$	− 531.853i	− 0.0193425i	0.999953	−	0.00967127i	$-0.00307851\pi$
$912$	0	0
$913$	− 2963.69i	− 0.107430i
$914$	0	0
$915$	5002.58	0.180744
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 14537.1i	− 0.521800i	−0.965366	−	0.260900i	$-0.915981\pi$
$919$	− 14537.1i	− 0.521800i	0.965366	−	0.260900i	$-0.0840193\pi$
$920$	0	0
$921$	2436.51	0.0871722
$922$	0	0
$923$	−17009.4	−0.606579
$924$	0	0
$925$	13397.2	0.476214
$926$	0	0
$927$	26207.8	0.928563
$928$	0	0
$929$	− 50518.4i	− 1.78413i	−0.451908	−	0.892065i	$-0.649256\pi$
$929$	− 50518.4i	− 1.78413i	0.451908	−	0.892065i	$-0.350744\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18629.5	−0.653699
$934$	0	0
$935$	− 420.594i	− 0.0147111i
$936$	0	0
$937$	9465.31i	0.330009i	0.986293	+	0.165004i	$0.0527638\pi$
$937$	9465.31i	0.330009i	−0.986293	+	0.165004i	$0.947236\pi$
$938$	0	0
$939$	18758.3i	0.651923i
$940$	0	0
$941$	− 46034.6i	− 1.59478i	−0.603466	−	0.797388i	$-0.706214\pi$
$941$	− 46034.6i	− 1.59478i	0.603466	−	0.797388i	$-0.293786\pi$
$942$	0	0
$943$	−1443.32	−0.0498419
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26311.4i	− 0.902856i	−0.892308	−	0.451428i	$-0.850915\pi$
$947$	− 26311.4i	− 0.902856i	0.892308	−	0.451428i	$-0.149085\pi$
$948$	0	0
$949$	−7442.57	−0.254580
$950$	0	0
$951$	−3013.06	−0.102740
$952$	0	0
$953$	35312.1	1.20028	0.600141	−	0.799894i	$-0.295111\pi$
$953$	35312.1	1.20028	0.600141	+	0.799894i	$0.295111\pi$
$954$	0	0
$955$	−31267.9	−1.05948
$956$	0	0
$957$	20129.2i	0.679920i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27943.5	−0.937986
$962$	0	0
$963$	28773.2i	0.962828i
$964$	0	0
$965$	− 43512.9i	− 1.45153i
$966$	0	0
$967$	26359.4i	0.876590i	0.898831	+	0.438295i	$0.144417\pi$
$967$	26359.4i	0.876590i	−0.898831	+	0.438295i	$0.855583\pi$
$968$	0	0
$969$	− 554.298i	− 0.0183763i
$970$	0	0
$971$	−37879.3	−1.25191	−0.625955	−	0.779859i	$-0.715291\pi$
$971$	−37879.3	−1.25191	−0.625955	+	0.779859i	$0.715291\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5078.68i	0.166818i
$976$	0	0
$977$	7239.38	0.237061	0.118530	−	0.992950i	$-0.462182\pi$
$977$	7239.38	0.237061	0.118530	+	0.992950i	$0.462182\pi$
$978$	0	0
$979$	−28167.4	−0.919544
$980$	0	0
$981$	19999.0	0.650887
$982$	0	0
$983$	−58748.7	−1.90620	−0.953099	−	0.302659i	$-0.902126\pi$
$983$	−58748.7	−1.90620	−0.953099	+	0.302659i	$0.902126\pi$
$984$	0	0
$985$	− 19051.1i	− 0.616261i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14831.0	−0.476842
$990$	0	0
$991$	21247.8i	0.681088i	0.940229	+	0.340544i	$0.110611\pi$
$991$	21247.8i	0.681088i	−0.940229	+	0.340544i	$0.889389\pi$
$992$	0	0
$993$	8076.01i	0.258091i
$994$	0	0
$995$	− 37623.3i	− 1.19873i
$996$	0	0
$997$	43206.3i	1.37248i	0.727377	+	0.686238i	$0.240739\pi$
$997$	43206.3i	1.37248i	−0.727377	+	0.686238i	$0.759261\pi$
$998$	0	0
$999$	33931.2	1.07461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.16	yes	24
4.3	odd	2	inner	784.4.f.j.783.10	yes	24
7.6	odd	2	inner	784.4.f.j.783.9	✓	24
28.27	even	2	inner	784.4.f.j.783.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.9	✓	24	7.6	odd	2	inner
784.4.f.j.783.10	yes	24	4.3	odd	2	inner
784.4.f.j.783.15	yes	24	28.27	even	2	inner
784.4.f.j.783.16	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.30410 q^{3} +13.4612i q^{5} -16.0830 q^{9} +O(q^{10})\)	\(q+3.30410 q^{3} +13.4612i q^{5} -16.0830 q^{9} +28.0363i q^{11} -27.3479i q^{13} +44.4772i q^{15} +1.11444i q^{17} -150.534 q^{19} -67.4421i q^{23} -56.2048 q^{25} -142.350 q^{27} +217.296 q^{29} -42.9820 q^{31} +92.6348i q^{33} -238.364 q^{37} -90.3602i q^{39} -21.4009i q^{41} -219.906i q^{43} -216.496i q^{45} -156.752 q^{47} +3.68222i q^{51} -666.721 q^{53} -377.404 q^{55} -497.378 q^{57} +782.018 q^{59} -112.475i q^{61} +368.137 q^{65} -1026.00i q^{67} -222.835i q^{69} -621.964i q^{71} -272.144i q^{73} -185.706 q^{75} +837.469i q^{79} -36.0990 q^{81} -105.709 q^{83} -15.0017 q^{85} +717.967 q^{87} +1004.67i q^{89} -142.017 q^{93} -2026.37i q^{95} +1846.54i q^{97} -450.907i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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