LMFDB - Embedded newform 576.7.h.b.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.h (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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Character	$\chi$	$=$	576.161
Dual form			576.7.h.b.161.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-108.406 q^{5} +167.464 q^{7} +O(q^{10})$	$q-108.406 q^{5} +167.464 q^{7} +1016.29 q^{11} -2272.52i q^{13} +865.606i q^{17} +8674.42i q^{19} +7261.66i q^{23} -3873.24 q^{25} +21010.9 q^{29} -12998.3 q^{31} -18154.1 q^{35} -3255.75i q^{37} -112661. i q^{41} +59336.4i q^{43} -117122. i q^{47} -89604.7 q^{49} +11719.7 q^{53} -110171. q^{55} -47019.2 q^{59} -35243.8i q^{61} +246354. i q^{65} +123187. i q^{67} +184149. i q^{71} -294641. q^{73} +170192. q^{77} +404395. q^{79} +43028.4 q^{83} -93836.5i q^{85} +308883. i q^{89} -380567. i q^{91} -940355. i q^{95} +1.21046e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) $ 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−108.406	−0.867244	−0.433622	−	0.901095i	$-0.642765\pi$
$5$	−108.406	−0.867244	−0.433622	+	0.901095i	$0.642765\pi$
$6$	0	0
$7$	167.464	0.488234	0.244117	−	0.969746i	$-0.421502\pi$
$7$	167.464	0.488234	0.244117	+	0.969746i	$0.421502\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1016.29	0.763551	0.381776	−	0.924255i	$-0.375313\pi$
$11$	1016.29	0.763551	0.381776	+	0.924255i	$0.375313\pi$
$12$	0	0
$13$	− 2272.52i	− 1.03438i	−0.855872	−	0.517188i	$-0.826979\pi$
$13$	− 2272.52i	− 1.03438i	0.855872	−	0.517188i	$-0.173021\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	865.606i	0.176187i	0.996112	+	0.0880934i	$0.0280774\pi$
$17$	865.606i	0.176187i	−0.996112	+	0.0880934i	$0.971923\pi$
$18$	0	0
$19$	8674.42i	1.26468i	0.774692	+	0.632339i	$0.217905\pi$
$19$	8674.42i	1.26468i	−0.774692	+	0.632339i	$0.782095\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7261.66i	0.596832i	0.954436	+	0.298416i	$0.0964583\pi$
$23$	7261.66i	0.596832i	−0.954436	+	0.298416i	$0.903542\pi$
$24$	0	0
$25$	−3873.24	−0.247888
$26$	0	0
$27$	0	0
$28$	0	0
$29$	21010.9	0.861491	0.430746	−	0.902473i	$-0.358251\pi$
$29$	21010.9	0.861491	0.430746	+	0.902473i	$0.358251\pi$
$30$	0	0
$31$	−12998.3	−0.436316	−0.218158	−	0.975913i	$-0.570005\pi$
$31$	−12998.3	−0.436316	−0.218158	+	0.975913i	$0.570005\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−18154.1	−0.423418
$36$	0	0
$37$	− 3255.75i	− 0.0642755i	−0.999483	−	0.0321378i	$-0.989768\pi$
$37$	− 3255.75i	− 0.0642755i	0.999483	−	0.0321378i	$-0.0102315\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 112661.i	− 1.63464i	−0.576186	−	0.817318i	$-0.695460\pi$
$41$	− 112661.i	− 1.63464i	0.576186	−	0.817318i	$-0.304540\pi$
$42$	0	0
$43$	59336.4i	0.746304i	0.927770	+	0.373152i	$0.121723\pi$
$43$	59336.4i	0.746304i	−0.927770	+	0.373152i	$0.878277\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 117122.i	− 1.12809i	−0.825744	−	0.564045i	$-0.809245\pi$
$47$	− 117122.i	− 1.12809i	0.825744	−	0.564045i	$-0.190755\pi$
$48$	0	0
$49$	−89604.7	−0.761628
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11719.7	0.0787208	0.0393604	−	0.999225i	$-0.487468\pi$
$53$	11719.7	0.0787208	0.0393604	+	0.999225i	$0.487468\pi$
$54$	0	0
$55$	−110171.	−0.662185
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−47019.2	−0.228939	−0.114469	−	0.993427i	$-0.536517\pi$
$59$	−47019.2	−0.228939	−0.114469	+	0.993427i	$0.536517\pi$
$60$	0	0
$61$	− 35243.8i	− 0.155272i	−0.996982	−	0.0776359i	$-0.975263\pi$
$61$	− 35243.8i	− 0.155272i	0.996982	−	0.0776359i	$-0.0247372\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	246354.i	0.897056i
$66$	0	0
$67$	123187.i	0.409581i	0.978806	+	0.204791i	$0.0656513\pi$
$67$	123187.i	0.409581i	−0.978806	+	0.204791i	$0.934349\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	184149.i	0.514511i	0.966343	+	0.257255i	$0.0828182\pi$
$71$	184149.i	0.514511i	−0.966343	+	0.257255i	$0.917182\pi$
$72$	0	0
$73$	−294641.	−0.757398	−0.378699	−	0.925520i	$-0.623628\pi$
$73$	−294641.	−0.757398	−0.378699	+	0.925520i	$0.623628\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	170192.	0.372792
$78$	0	0
$79$	404395.	0.820209	0.410105	−	0.912039i	$-0.365492\pi$
$79$	404395.	0.820209	0.410105	+	0.912039i	$0.365492\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	43028.4	0.0752525	0.0376263	−	0.999292i	$-0.488020\pi$
$83$	43028.4	0.0752525	0.0376263	+	0.999292i	$0.488020\pi$
$84$	0	0
$85$	− 93836.5i	− 0.152797i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	308883.i	0.438151i	0.975708	+	0.219076i	$0.0703041\pi$
$89$	308883.i	0.438151i	−0.975708	+	0.219076i	$0.929696\pi$
$90$	0	0
$91$	− 380567.i	− 0.505017i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 940355.i	− 1.09678i
$96$	0	0
$97$	1.21046e6	1.32628	0.663139	−	0.748496i	$-0.269224\pi$
$97$	1.21046e6	1.32628	0.663139	+	0.748496i	$0.269224\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12652.2	0.0122801	0.00614006	−	0.999981i	$-0.498046\pi$
$101$	12652.2	0.0122801	0.00614006	+	0.999981i	$0.498046\pi$
$102$	0	0
$103$	1.15864e6	1.06032	0.530158	−	0.847899i	$-0.322133\pi$
$103$	1.15864e6	1.06032	0.530158	+	0.847899i	$0.322133\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	899526.	0.734282	0.367141	−	0.930165i	$-0.380337\pi$
$107$	899526.	0.734282	0.367141	+	0.930165i	$0.380337\pi$
$108$	0	0
$109$	1.99024e6i	1.53683i	0.639950	+	0.768416i	$0.278955\pi$
$109$	1.99024e6i	1.53683i	−0.639950	+	0.768416i	$0.721045\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	300140.i	0.208012i	0.994577	+	0.104006i	$0.0331662\pi$
$113$	300140.i	0.208012i	−0.994577	+	0.104006i	$0.966834\pi$
$114$	0	0
$115$	− 787204.i	− 0.517599i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	144958.i	0.0860204i
$120$	0	0
$121$	−738723.	−0.416990
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.11372e6	1.08222
$126$	0	0
$127$	−1.52556e6	−0.744765	−0.372383	−	0.928079i	$-0.621459\pi$
$127$	−1.52556e6	−0.744765	−0.372383	+	0.928079i	$0.621459\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.64651e6	1.17722	0.588612	−	0.808415i	$-0.299674\pi$
$131$	2.64651e6	1.17722	0.588612	+	0.808415i	$0.299674\pi$
$132$	0	0
$133$	1.45266e6i	0.617459i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.92479e6i	0.748551i	0.927318	+	0.374276i	$0.122109\pi$
$137$	1.92479e6i	0.748551i	−0.927318	+	0.374276i	$0.877891\pi$
$138$	0	0
$139$	2.59236e6i	0.965276i	0.875820	+	0.482638i	$0.160321\pi$
$139$	2.59236e6i	0.965276i	−0.875820	+	0.482638i	$0.839679\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 2.30953e6i	− 0.789799i
$144$	0	0
$145$	−2.27770e6	−0.747123
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.38355e6	0.418252	0.209126	−	0.977889i	$-0.432938\pi$
$149$	1.38355e6	0.418252	0.209126	+	0.977889i	$0.432938\pi$
$150$	0	0
$151$	6.22138e6	1.80699	0.903495	−	0.428599i	$-0.140993\pi$
$151$	6.22138e6	1.80699	0.903495	+	0.428599i	$0.140993\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.40909e6	0.378393
$156$	0	0
$157$	− 369736.i	− 0.0955417i	−0.998858	−	0.0477709i	$-0.984788\pi$
$157$	− 369736.i	− 0.0955417i	0.998858	−	0.0477709i	$-0.0152117\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.21607e6i	0.291394i
$162$	0	0
$163$	5.02401e6i	1.16008i	0.814588	+	0.580040i	$0.196963\pi$
$163$	5.02401e6i	1.16008i	−0.814588	+	0.580040i	$0.803037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.12372e6i	− 1.52953i	−0.644311	−	0.764764i	$-0.722856\pi$
$167$	− 7.12372e6i	− 1.52953i	0.644311	−	0.764764i	$-0.277144\pi$
$168$	0	0
$169$	−337555.	−0.0699333
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.14826e6	1.76685	0.883426	−	0.468571i	$-0.155231\pi$
$173$	9.14826e6	1.76685	0.883426	+	0.468571i	$0.155231\pi$
$174$	0	0
$175$	−648630.	−0.121027
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.71580e6	−0.822235	−0.411117	−	0.911582i	$-0.634861\pi$
$179$	−4.71580e6	−0.822235	−0.411117	+	0.911582i	$0.634861\pi$
$180$	0	0
$181$	1.61763e6i	0.272800i	0.990654	+	0.136400i	$0.0435532\pi$
$181$	1.61763e6i	0.272800i	−0.990654	+	0.136400i	$0.956447\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	352941.i	0.0557426i
$186$	0	0
$187$	879704.i	0.134528i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.01300e7i	− 1.45381i	−0.686737	−	0.726906i	$-0.740958\pi$
$191$	− 1.01300e7i	− 1.45381i	0.686737	−	0.726906i	$-0.259042\pi$
$192$	0	0
$193$	−1.31577e7	−1.83024	−0.915120	−	0.403181i	$-0.867904\pi$
$193$	−1.31577e7	−1.83024	−0.915120	+	0.403181i	$0.867904\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.11089e7	1.45302	0.726512	−	0.687154i	$-0.241140\pi$
$197$	1.11089e7	1.45302	0.726512	+	0.687154i	$0.241140\pi$
$198$	0	0
$199$	9.20132e6	1.16759	0.583795	−	0.811901i	$-0.301567\pi$
$199$	9.20132e6	1.16759	0.583795	+	0.811901i	$0.301567\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.51858e6	0.420609
$204$	0	0
$205$	1.22131e7i	1.41763i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.81570e6i	0.965646i
$210$	0	0
$211$	1.58526e6i	0.168753i	0.996434	+	0.0843766i	$0.0268899\pi$
$211$	1.58526e6i	0.168753i	−0.996434	+	0.0843766i	$0.973110\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 6.43239e6i	− 0.647228i
$216$	0	0
$217$	−2.17675e6	−0.213024
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.96711e6	0.182243
$222$	0	0
$223$	−60439.3	−0.00545010	−0.00272505	−	0.999996i	$-0.500867\pi$
$223$	−60439.3	−0.00545010	−0.00272505	+	0.999996i	$0.500867\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.09366e7	1.78990	0.894952	−	0.446163i	$-0.147210\pi$
$227$	2.09366e7	1.78990	0.894952	+	0.446163i	$0.147210\pi$
$228$	0	0
$229$	9.68612e6i	0.806573i	0.915074	+	0.403286i	$0.132132\pi$
$229$	9.68612e6i	0.806573i	−0.915074	+	0.403286i	$0.867868\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.32063e7i	1.04403i	0.852937	+	0.522014i	$0.174819\pi$
$233$	1.32063e7i	1.04403i	−0.852937	+	0.522014i	$0.825181\pi$
$234$	0	0
$235$	1.26966e7i	0.978329i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.45983e7i	− 1.80182i	−0.434006	−	0.900910i	$-0.642900\pi$
$239$	− 2.45983e7i	− 1.80182i	0.434006	−	0.900910i	$-0.357100\pi$
$240$	0	0
$241$	3.47566e6	0.248305	0.124153	−	0.992263i	$-0.460379\pi$
$241$	3.47566e6	0.248305	0.124153	+	0.992263i	$0.460379\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.71365e6	0.660517
$246$	0	0
$247$	1.97128e7	1.30815
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.98804e7	1.25720	0.628599	−	0.777729i	$-0.283629\pi$
$251$	1.98804e7	1.25720	0.628599	+	0.777729i	$0.283629\pi$
$252$	0	0
$253$	7.37993e6i	0.455712i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.06716e6i	0.121780i	0.998144	+	0.0608898i	$0.0193938\pi$
$257$	2.06716e6i	0.121780i	−0.998144	+	0.0608898i	$0.980606\pi$
$258$	0	0
$259$	− 545221.i	− 0.0313815i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.11532e7i	0.613100i	0.951855	+	0.306550i	$0.0991746\pi$
$263$	1.11532e7i	0.613100i	−0.951855	+	0.306550i	$0.900825\pi$
$264$	0	0
$265$	−1.27048e6	−0.0682702
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.93896e7	1.50986	0.754930	−	0.655805i	$-0.227671\pi$
$269$	2.93896e7	1.50986	0.754930	+	0.655805i	$0.227671\pi$
$270$	0	0
$271$	2.42141e7	1.21663	0.608317	−	0.793694i	$-0.291845\pi$
$271$	2.42141e7	1.21663	0.608317	+	0.793694i	$0.291845\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.93632e6	−0.189275
$276$	0	0
$277$	2.59678e7i	1.22179i	0.791712	+	0.610895i	$0.209190\pi$
$277$	2.59678e7i	1.22179i	−0.791712	+	0.610895i	$0.790810\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.40037e7i	1.08183i	0.841077	+	0.540916i	$0.181922\pi$
$281$	2.40037e7i	1.08183i	−0.841077	+	0.540916i	$0.818078\pi$
$282$	0	0
$283$	− 1.31588e7i	− 0.580574i	−0.956940	−	0.290287i	$-0.906249\pi$
$283$	− 1.31588e7i	− 0.580574i	0.956940	−	0.290287i	$-0.0937508\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.88667e7i	− 0.798085i
$288$	0	0
$289$	2.33883e7	0.968958
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.01348e7	1.59558	0.797790	−	0.602935i	$-0.206002\pi$
$293$	4.01348e7	1.59558	0.797790	+	0.602935i	$0.206002\pi$
$294$	0	0
$295$	5.09714e6	0.198546
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.65023e7	0.617349
$300$	0	0
$301$	9.93672e6i	0.364371i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.82062e6i	0.134659i
$306$	0	0
$307$	6.27888e6i	0.217004i	0.994096	+	0.108502i	$0.0346054\pi$
$307$	6.27888e6i	0.217004i	−0.994096	+	0.108502i	$0.965395\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.40106e7i	− 1.13066i	−0.824864	−	0.565332i	$-0.808748\pi$
$311$	− 3.40106e7i	− 1.13066i	0.824864	−	0.565332i	$-0.191252\pi$
$312$	0	0
$313$	−3.31466e7	−1.08095	−0.540476	−	0.841359i	$-0.681756\pi$
$313$	−3.31466e7	−1.08095	−0.540476	+	0.841359i	$0.681756\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.78191e7	−0.873305	−0.436652	−	0.899630i	$-0.643836\pi$
$317$	−2.78191e7	−0.873305	−0.436652	+	0.899630i	$0.643836\pi$
$318$	0	0
$319$	2.13531e7	0.657792
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.50863e6	−0.222820
$324$	0	0
$325$	8.80204e6i	0.256409i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.96137e7i	− 0.550772i
$330$	0	0
$331$	4.84270e7i	1.33538i	0.744442	+	0.667688i	$0.232716\pi$
$331$	4.84270e7i	1.33538i	−0.744442	+	0.667688i	$0.767284\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.33541e7i	− 0.355207i
$336$	0	0
$337$	−3.71914e7	−0.971745	−0.485873	−	0.874030i	$-0.661498\pi$
$337$	−3.71914e7	−0.971745	−0.485873	+	0.874030i	$0.661498\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.32100e7	−0.333150
$342$	0	0
$343$	−3.47076e7	−0.860086
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.33386e6	0.0797918	0.0398959	−	0.999204i	$-0.487297\pi$
$347$	3.33386e6	0.0797918	0.0398959	+	0.999204i	$0.487297\pi$
$348$	0	0
$349$	− 3.31165e7i	− 0.779056i	−0.921015	−	0.389528i	$-0.872638\pi$
$349$	− 3.31165e7i	− 0.779056i	0.921015	−	0.389528i	$-0.127362\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.48754e7i	1.24754i	0.781609	+	0.623769i	$0.214399\pi$
$353$	5.48754e7i	1.24754i	−0.781609	+	0.623769i	$0.785601\pi$
$354$	0	0
$355$	− 1.99628e7i	− 0.446207i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 4.05268e7i	− 0.875910i	−0.898997	−	0.437955i	$-0.855703\pi$
$359$	− 4.05268e7i	− 0.875910i	0.898997	−	0.437955i	$-0.144297\pi$
$360$	0	0
$361$	−2.81997e7	−0.599409
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.19407e7	0.656849
$366$	0	0
$367$	−1.17570e7	−0.237848	−0.118924	−	0.992903i	$-0.537944\pi$
$367$	−1.17570e7	−0.237848	−0.118924	+	0.992903i	$0.537944\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.96263e6	0.0384342
$372$	0	0
$373$	− 5.53125e6i	− 0.106585i	−0.998579	−	0.0532926i	$-0.983028\pi$
$373$	− 5.53125e6i	− 0.106585i	0.998579	−	0.0532926i	$-0.0169716\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.77478e7i	− 0.891105i
$378$	0	0
$379$	5.25982e6i	0.0966169i	0.998832	+	0.0483085i	$0.0153830\pi$
$379$	5.25982e6i	0.0966169i	−0.998832	+	0.0483085i	$0.984617\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.04442e7i	1.43185i	0.698175	+	0.715927i	$0.253996\pi$
$383$	8.04442e7i	1.43185i	−0.698175	+	0.715927i	$0.746004\pi$
$384$	0	0
$385$	−1.84497e7	−0.323301
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.65628e7	0.281375	0.140688	−	0.990054i	$-0.455069\pi$
$389$	1.65628e7	0.281375	0.140688	+	0.990054i	$0.455069\pi$
$390$	0	0
$391$	−6.28574e6	−0.105154
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.38387e7	−0.711322
$396$	0	0
$397$	2.39116e7i	0.382153i	0.981575	+	0.191077i	$0.0611979\pi$
$397$	2.39116e7i	0.382153i	−0.981575	+	0.191077i	$0.938802\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.47399e7i	− 0.693845i	−0.937894	−	0.346922i	$-0.887227\pi$
$401$	− 4.47399e7i	− 0.693845i	0.937894	−	0.346922i	$-0.112773\pi$
$402$	0	0
$403$	2.95389e7i	0.451315i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.30877e6i	− 0.0490776i
$408$	0	0
$409$	8.44529e7	1.23437	0.617184	−	0.786819i	$-0.288273\pi$
$409$	8.44529e7	1.23437	0.617184	+	0.786819i	$0.288273\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.87403e6	−0.111776
$414$	0	0
$415$	−4.66452e6	−0.0652623
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.18395e8	−1.60950	−0.804750	−	0.593614i	$-0.797701\pi$
$419$	−1.18395e8	−1.60950	−0.804750	+	0.593614i	$0.797701\pi$
$420$	0	0
$421$	− 7.97937e7i	− 1.06936i	−0.845056	−	0.534678i	$-0.820433\pi$
$421$	− 7.97937e7i	− 1.06936i	0.845056	−	0.534678i	$-0.179567\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.35270e6i	− 0.0436745i
$426$	0	0
$427$	− 5.90207e6i	− 0.0758090i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.11850e7i	0.639309i	0.947534	+	0.319654i	$0.103567\pi$
$431$	5.11850e7i	0.639309i	−0.947534	+	0.319654i	$0.896433\pi$
$432$	0	0
$433$	6.36319e7	0.783811	0.391906	−	0.920005i	$-0.371816\pi$
$433$	6.36319e7	0.783811	0.391906	+	0.920005i	$0.371816\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.29907e7	−0.754801
$438$	0	0
$439$	−9.70188e6	−0.114673	−0.0573367	−	0.998355i	$-0.518261\pi$
$439$	−9.70188e6	−0.114673	−0.0573367	+	0.998355i	$0.518261\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.59558e8	1.83530	0.917651	−	0.397387i	$-0.130083\pi$
$443$	1.59558e8	1.83530	0.917651	+	0.397387i	$0.130083\pi$
$444$	0	0
$445$	− 3.34846e7i	− 0.379984i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9.87088e7i	− 1.09048i	−0.838281	−	0.545239i	$-0.816439\pi$
$449$	− 9.87088e7i	− 1.09048i	0.838281	−	0.545239i	$-0.183561\pi$
$450$	0	0
$451$	− 1.14496e8i	− 1.24813i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.12555e7i	0.437973i
$456$	0	0
$457$	6.39206e7	0.669718	0.334859	−	0.942268i	$-0.391311\pi$
$457$	6.39206e7	0.669718	0.334859	+	0.942268i	$0.391311\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.56965e8	−1.60214	−0.801068	−	0.598574i	$-0.795734\pi$
$461$	−1.56965e8	−1.60214	−0.801068	+	0.598574i	$0.795734\pi$
$462$	0	0
$463$	1.41552e7	0.142617	0.0713086	−	0.997454i	$-0.477282\pi$
$463$	1.41552e7	0.142617	0.0713086	+	0.997454i	$0.477282\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.88086e6	−0.0577418	−0.0288709	−	0.999583i	$-0.509191\pi$
$467$	−5.88086e6	−0.0577418	−0.0288709	+	0.999583i	$0.509191\pi$
$468$	0	0
$469$	2.06294e7i	0.199971i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.03028e7i	0.569841i
$474$	0	0
$475$	− 3.35982e7i	− 0.313498i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.02069e8i	1.83862i	0.393531	+	0.919311i	$0.371253\pi$
$479$	2.02069e8i	1.83862i	−0.393531	+	0.919311i	$0.628747\pi$
$480$	0	0
$481$	−7.39876e6	−0.0664850
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.31220e8	−1.15021
$486$	0	0
$487$	−2.07112e8	−1.79316	−0.896581	−	0.442881i	$-0.853956\pi$
$487$	−2.07112e8	−1.79316	−0.896581	+	0.442881i	$0.853956\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.77240e7	−0.234214	−0.117107	−	0.993119i	$-0.537362\pi$
$491$	−2.77240e7	−0.234214	−0.117107	+	0.993119i	$0.537362\pi$
$492$	0	0
$493$	1.81872e7i	0.151783i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.08384e7i	0.251202i
$498$	0	0
$499$	− 1.20361e8i	− 0.968690i	−0.874877	−	0.484345i	$-0.839058\pi$
$499$	− 1.20361e8i	− 0.968690i	0.874877	−	0.484345i	$-0.160942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.35139e7i	0.734805i	0.930062	+	0.367402i	$0.119753\pi$
$503$	9.35139e7i	0.734805i	−0.930062	+	0.367402i	$0.880247\pi$
$504$	0	0
$505$	−1.37157e6	−0.0106499
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.78114e7	0.210896	0.105448	−	0.994425i	$-0.466372\pi$
$509$	2.78114e7	0.210896	0.105448	+	0.994425i	$0.466372\pi$
$510$	0	0
$511$	−4.93418e7	−0.369787
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.25602e8	−0.919552
$516$	0	0
$517$	− 1.19029e8i	− 0.861354i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.21387e7i	0.0858336i	0.999079	+	0.0429168i	$0.0136650\pi$
$521$	1.21387e7i	0.0858336i	−0.999079	+	0.0429168i	$0.986335\pi$
$522$	0	0
$523$	6.09328e6i	0.0425938i	0.999773	+	0.0212969i	$0.00677952\pi$
$523$	6.09328e6i	0.0425938i	−0.999773	+	0.0212969i	$0.993220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.12514e7i	− 0.0768732i
$528$	0	0
$529$	9.53042e7	0.643791
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.56024e8	−1.69083
$534$	0	0
$535$	−9.75136e7	−0.636801
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.10641e7	−0.581542
$540$	0	0
$541$	− 1.95052e8i	− 1.23185i	−0.787804	−	0.615926i	$-0.788782\pi$
$541$	− 1.95052e8i	− 1.23185i	0.787804	−	0.615926i	$-0.211218\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.15753e8i	− 1.33281i
$546$	0	0
$547$	− 1.88786e8i	− 1.15347i	−0.816930	−	0.576737i	$-0.804326\pi$
$547$	− 1.88786e8i	− 1.15347i	0.816930	−	0.576737i	$-0.195674\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.82257e8i	1.08951i
$552$	0	0
$553$	6.77217e7	0.400454
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.27981e8	−0.740594	−0.370297	−	0.928913i	$-0.620744\pi$
$557$	−1.27981e8	−0.740594	−0.370297	+	0.928913i	$0.620744\pi$
$558$	0	0
$559$	1.34843e8	0.771959
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.03932e8	−1.70314	−0.851571	−	0.524239i	$-0.824350\pi$
$563$	−3.03932e8	−1.70314	−0.851571	+	0.524239i	$0.824350\pi$
$564$	0	0
$565$	− 3.25369e7i	− 0.180397i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.54208e8i	− 0.837087i	−0.908197	−	0.418543i	$-0.862541\pi$
$569$	− 1.54208e8i	− 0.837087i	0.908197	−	0.418543i	$-0.137459\pi$
$570$	0	0
$571$	1.81493e8i	0.974880i	0.873156	+	0.487440i	$0.162069\pi$
$571$	1.81493e8i	0.974880i	−0.873156	+	0.487440i	$0.837931\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2.81262e7i	− 0.147947i
$576$	0	0
$577$	3.73075e7	0.194209	0.0971044	−	0.995274i	$-0.469042\pi$
$577$	3.73075e7	0.194209	0.0971044	+	0.995274i	$0.469042\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.20572e6	0.0367408
$582$	0	0
$583$	1.19106e7	0.0601074
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.23294e8	0.609576	0.304788	−	0.952420i	$-0.401414\pi$
$587$	1.23294e8	0.609576	0.304788	+	0.952420i	$0.401414\pi$
$588$	0	0
$589$	− 1.12753e8i	− 0.551799i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5.26663e7i	− 0.252562i	−0.991994	−	0.126281i	$-0.959696\pi$
$593$	− 5.26663e7i	− 0.252562i	0.991994	−	0.126281i	$-0.0403042\pi$
$594$	0	0
$595$	− 1.57143e7i	− 0.0746007i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.67475e8i	0.779237i	0.920976	+	0.389619i	$0.127393\pi$
$599$	1.67475e8i	0.779237i	−0.920976	+	0.389619i	$0.872607\pi$
$600$	0	0
$601$	−5.04027e7	−0.232183	−0.116091	−	0.993239i	$-0.537037\pi$
$601$	−5.04027e7	−0.232183	−0.116091	+	0.993239i	$0.537037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.00816e7	0.361632
$606$	0	0
$607$	−4.13800e8	−1.85023	−0.925113	−	0.379693i	$-0.876030\pi$
$607$	−4.13800e8	−1.85023	−0.925113	+	0.379693i	$0.876030\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.66162e8	−1.16687
$612$	0	0
$613$	− 4.18881e8i	− 1.81848i	−0.416268	−	0.909242i	$-0.636662\pi$
$613$	− 4.18881e8i	− 1.81848i	0.416268	−	0.909242i	$-0.363338\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.43886e8i	− 1.03832i	−0.854677	−	0.519161i	$-0.826245\pi$
$617$	− 2.43886e8i	− 1.03832i	0.854677	−	0.519161i	$-0.173755\pi$
$618$	0	0
$619$	6.06565e7i	0.255744i	0.991791	+	0.127872i	$0.0408146\pi$
$619$	6.06565e7i	0.255744i	−0.991791	+	0.127872i	$0.959185\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.17269e7i	0.213920i
$624$	0	0
$625$	−1.68619e8	−0.690664
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.81819e6	0.0113245
$630$	0	0
$631$	−1.14374e8	−0.455239	−0.227619	−	0.973750i	$-0.573094\pi$
$631$	−1.14374e8	−0.455239	−0.227619	+	0.973750i	$0.573094\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.65380e8	0.645893
$636$	0	0
$637$	2.03629e8i	0.787809i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.66320e8i	1.01118i	0.862773	+	0.505592i	$0.168726\pi$
$641$	2.66320e8i	1.01118i	−0.862773	+	0.505592i	$0.831274\pi$
$642$	0	0
$643$	1.18317e8i	0.445056i	0.974926	+	0.222528i	$0.0714310\pi$
$643$	1.18317e8i	0.445056i	−0.974926	+	0.222528i	$0.928569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.63390e8i	0.972491i	0.873822	+	0.486246i	$0.161634\pi$
$647$	2.63390e8i	0.972491i	−0.873822	+	0.486246i	$0.838366\pi$
$648$	0	0
$649$	−4.77850e7	−0.174806
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.96726e8	−1.42479	−0.712396	−	0.701778i	$-0.752390\pi$
$653$	−3.96726e8	−1.42479	−0.712396	+	0.701778i	$0.752390\pi$
$654$	0	0
$655$	−2.86896e8	−1.02094
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.73867e8	−0.607520	−0.303760	−	0.952749i	$-0.598242\pi$
$659$	−1.73867e8	−0.607520	−0.303760	+	0.952749i	$0.598242\pi$
$660$	0	0
$661$	3.86993e8i	1.33998i	0.742369	+	0.669991i	$0.233702\pi$
$661$	3.86993e8i	1.33998i	−0.742369	+	0.669991i	$0.766298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.57476e8i	− 0.535487i
$666$	0	0
$667$	1.52574e8i	0.514166i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 3.58178e7i	− 0.118558i
$672$	0	0
$673$	1.44017e8	0.472463	0.236231	−	0.971697i	$-0.424088\pi$
$673$	1.44017e8	0.472463	0.236231	+	0.971697i	$0.424088\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.59250e7	0.309148	0.154574	−	0.987981i	$-0.450600\pi$
$677$	9.59250e7	0.309148	0.154574	+	0.987981i	$0.450600\pi$
$678$	0	0
$679$	2.02708e8	0.647534
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.27860e8	1.65675	0.828374	−	0.560176i	$-0.189266\pi$
$683$	5.27860e8	1.65675	0.828374	+	0.560176i	$0.189266\pi$
$684$	0	0
$685$	− 2.08658e8i	− 0.649177i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2.66333e7i	− 0.0814269i
$690$	0	0
$691$	− 5.54597e8i	− 1.68091i	−0.541884	−	0.840453i	$-0.682289\pi$
$691$	− 5.54597e8i	− 1.68091i	0.541884	−	0.840453i	$-0.317711\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 2.81027e8i	− 0.837130i
$696$	0	0
$697$	9.75199e7	0.288002
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.17969e8	0.342463	0.171232	−	0.985231i	$-0.445225\pi$
$701$	1.17969e8	0.342463	0.171232	+	0.985231i	$0.445225\pi$
$702$	0	0
$703$	2.82417e7	0.0812878
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.11879e6	0.00599557
$708$	0	0
$709$	− 8.62629e7i	− 0.242039i	−0.992650	−	0.121020i	$-0.961384\pi$
$709$	− 8.62629e7i	− 0.242039i	0.992650	−	0.121020i	$-0.0386164\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 9.43892e7i	− 0.260408i
$714$	0	0
$715$	2.50366e8i	0.684948i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 8.41415e7i	− 0.226372i	−0.993574	−	0.113186i	$-0.963894\pi$
$719$	− 8.41415e7i	− 0.226372i	0.993574	−	0.113186i	$-0.0361057\pi$
$720$	0	0
$721$	1.94030e8	0.517682
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.13803e7	−0.213553
$726$	0	0
$727$	7.23120e8	1.88195	0.940973	−	0.338483i	$-0.109914\pi$
$727$	7.23120e8	1.88195	0.940973	+	0.338483i	$0.109914\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.13619e7	−0.131489
$732$	0	0
$733$	4.34884e7i	0.110423i	0.998475	+	0.0552117i	$0.0175834\pi$
$733$	4.34884e7i	0.110423i	−0.998475	+	0.0552117i	$0.982417\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.25193e8i	0.312736i
$738$	0	0
$739$	− 4.17057e8i	− 1.03338i	−0.856171	−	0.516692i	$-0.827163\pi$
$739$	− 4.17057e8i	− 1.03338i	0.856171	−	0.516692i	$-0.172837\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.30520e8i	− 0.562009i	−0.959706	−	0.281004i	$-0.909332\pi$
$743$	− 2.30520e8i	− 0.562009i	0.959706	−	0.281004i	$-0.0906676\pi$
$744$	0	0
$745$	−1.49985e8	−0.362726
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.50639e8	0.358501
$750$	0	0
$751$	1.46588e8	0.346081	0.173041	−	0.984915i	$-0.444641\pi$
$751$	1.46588e8	0.346081	0.173041	+	0.984915i	$0.444641\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.74432e8	−1.56710
$756$	0	0
$757$	6.22315e8i	1.43457i	0.696778	+	0.717286i	$0.254616\pi$
$757$	6.22315e8i	1.43457i	−0.696778	+	0.717286i	$0.745384\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.99870e8i	1.81495i	0.420102	+	0.907477i	$0.361994\pi$
$761$	7.99870e8i	1.81495i	−0.420102	+	0.907477i	$0.638006\pi$
$762$	0	0
$763$	3.33295e8i	0.750334i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.06852e8i	0.236809i
$768$	0	0
$769$	2.96019e8	0.650940	0.325470	−	0.945552i	$-0.394478\pi$
$769$	2.96019e8	0.650940	0.325470	+	0.945552i	$0.394478\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.14258e8	1.32988	0.664939	−	0.746897i	$-0.268457\pi$
$773$	6.14258e8	1.32988	0.664939	+	0.746897i	$0.268457\pi$
$774$	0	0
$775$	5.03456e7	0.108157
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.77267e8	2.06729
$780$	0	0
$781$	1.87148e8i	0.392855i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00814e7i	0.0828580i
$786$	0	0
$787$	− 3.58778e8i	− 0.736040i	−0.929818	−	0.368020i	$-0.880036\pi$
$787$	− 3.58778e8i	− 0.736040i	0.929818	−	0.368020i	$-0.119964\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.02628e7i	0.101559i
$792$	0	0
$793$	−8.00923e7	−0.160609
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.16661e8	0.427963	0.213981	−	0.976838i	$-0.431357\pi$
$797$	2.16661e8	0.427963	0.213981	+	0.976838i	$0.431357\pi$
$798$	0	0
$799$	1.01381e8	0.198755
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.99439e8	−0.578312
$804$	0	0
$805$	− 1.31829e8i	− 0.252710i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.25793e8i	1.55964i	0.626001	+	0.779822i	$0.284691\pi$
$809$	8.25793e8i	1.55964i	−0.626001	+	0.779822i	$0.715309\pi$
$810$	0	0
$811$	− 8.72696e6i	− 0.0163606i	−0.999967	−	0.00818032i	$-0.997396\pi$
$811$	− 8.72696e6i	− 0.0163606i	0.999967	−	0.00818032i	$-0.00260391\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 5.44631e8i	− 1.00607i
$816$	0	0
$817$	−5.14709e8	−0.943834
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.19753e8	0.216400	0.108200	−	0.994129i	$-0.465491\pi$
$821$	1.19753e8	0.216400	0.108200	+	0.994129i	$0.465491\pi$
$822$	0	0
$823$	4.19710e8	0.752921	0.376460	−	0.926433i	$-0.377141\pi$
$823$	4.19710e8	0.752921	0.376460	+	0.926433i	$0.377141\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.01240e9	1.78993	0.894964	−	0.446138i	$-0.147201\pi$
$827$	1.01240e9	1.78993	0.894964	+	0.446138i	$0.147201\pi$
$828$	0	0
$829$	2.27028e7i	0.0398488i	0.999801	+	0.0199244i	$0.00634256\pi$
$829$	2.27028e7i	0.0398488i	−0.999801	+	0.0199244i	$0.993657\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 7.75624e7i	− 0.134189i
$834$	0	0
$835$	7.72250e8i	1.32647i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 5.45149e8i	− 0.923058i	−0.887125	−	0.461529i	$-0.847301\pi$
$839$	− 5.45149e8i	− 0.923058i	0.887125	−	0.461529i	$-0.152699\pi$
$840$	0	0
$841$	−1.53365e8	−0.257833
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.65928e7	0.0606492
$846$	0	0
$847$	−1.23710e8	−0.203589
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.36421e7	0.0383617
$852$	0	0
$853$	4.55247e8i	0.733499i	0.930320	+	0.366750i	$0.119529\pi$
$853$	4.55247e8i	0.733499i	−0.930320	+	0.366750i	$0.880471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.98993e8i	− 0.792778i	−0.918083	−	0.396389i	$-0.870263\pi$
$857$	− 4.98993e8i	− 0.792778i	0.918083	−	0.396389i	$-0.129737\pi$
$858$	0	0
$859$	9.96593e8i	1.57231i	0.618029	+	0.786155i	$0.287931\pi$
$859$	9.96593e8i	1.57231i	−0.618029	+	0.786155i	$0.712069\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.23182e8i	− 1.43633i	−0.695871	−	0.718167i	$-0.744982\pi$
$863$	− 9.23182e8i	− 1.43633i	0.695871	−	0.718167i	$-0.255018\pi$
$864$	0	0
$865$	−9.91722e8	−1.53229
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.10981e8	0.626272
$870$	0	0
$871$	2.79945e8	0.423661
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.53972e8	0.528378
$876$	0	0
$877$	7.38117e8i	1.09427i	0.837043	+	0.547137i	$0.184282\pi$
$877$	7.38117e8i	1.09427i	−0.837043	+	0.547137i	$0.815718\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.10844e6i	0.00454584i	0.999997	+	0.00227292i	$0.000723494\pi$
$881$	3.10844e6i	0.00454584i	−0.999997	+	0.00227292i	$0.999277\pi$
$882$	0	0
$883$	− 6.69574e8i	− 0.972560i	−0.873803	−	0.486280i	$-0.838353\pi$
$883$	− 6.69574e8i	− 0.972560i	0.873803	−	0.486280i	$-0.161647\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.58571e8i	− 0.370517i	−0.982690	−	0.185259i	$-0.940688\pi$
$887$	− 2.58571e8i	− 0.370517i	0.982690	−	0.185259i	$-0.0593123\pi$
$888$	0	0
$889$	−2.55478e8	−0.363620
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.01596e9	1.42667
$894$	0	0
$895$	5.11218e8	0.713078
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.73106e8	−0.375883
$900$	0	0
$901$	1.01447e7i	0.0138696i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.75360e8i	− 0.236584i
$906$	0	0
$907$	1.18881e9i	1.59328i	0.604456	+	0.796639i	$0.293391\pi$
$907$	1.18881e9i	1.59328i	−0.604456	+	0.796639i	$0.706609\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 8.70158e8i	− 1.15091i	−0.817832	−	0.575457i	$-0.804824\pi$
$911$	− 8.70158e8i	− 1.15091i	0.817832	−	0.575457i	$-0.195176\pi$
$912$	0	0
$913$	4.37292e7	0.0574591
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.43195e8	0.574761
$918$	0	0
$919$	−9.37383e8	−1.20773	−0.603866	−	0.797086i	$-0.706374\pi$
$919$	−9.37383e8	−1.20773	−0.603866	+	0.797086i	$0.706374\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.18483e8	0.532198
$924$	0	0
$925$	1.26103e7i	0.0159331i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.13347e7i	0.0266096i	0.999911	+	0.0133048i	$0.00423518\pi$
$929$	2.13347e7i	0.0266096i	−0.999911	+	0.0133048i	$0.995765\pi$
$930$	0	0
$931$	− 7.77269e8i	− 0.963213i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 9.53648e7i	− 0.116668i
$936$	0	0
$937$	9.68741e8	1.17758	0.588788	−	0.808288i	$-0.299605\pi$
$937$	9.68741e8	1.17758	0.588788	+	0.808288i	$0.299605\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.60277e8	1.15246	0.576232	−	0.817286i	$-0.304522\pi$
$941$	9.60277e8	1.15246	0.576232	+	0.817286i	$0.304522\pi$
$942$	0	0
$943$	8.18104e8	0.975604
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.72326e8	−1.14489	−0.572443	−	0.819945i	$-0.694004\pi$
$947$	−9.72326e8	−1.14489	−0.572443	+	0.819945i	$0.694004\pi$
$948$	0	0
$949$	6.69578e8i	0.783434i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.62051e8i	0.187228i	0.995609	+	0.0936142i	$0.0298420\pi$
$953$	1.62051e8i	0.187228i	−0.995609	+	0.0936142i	$0.970158\pi$
$954$	0	0
$955$	1.09814e9i	1.26081i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.22333e8i	0.365468i
$960$	0	0
$961$	−7.18548e8	−0.809628
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.42637e9	1.58727
$966$	0	0
$967$	4.00115e8	0.442492	0.221246	−	0.975218i	$-0.428988\pi$
$967$	4.00115e8	0.442492	0.221246	+	0.975218i	$0.428988\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.04286e9	1.13912	0.569560	−	0.821950i	$-0.307114\pi$
$971$	1.04286e9	1.13912	0.569560	+	0.821950i	$0.307114\pi$
$972$	0	0
$973$	4.34128e8i	0.471281i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.20778e9i	1.29511i	0.762021	+	0.647553i	$0.224207\pi$
$977$	1.20778e9i	1.29511i	−0.762021	+	0.647553i	$0.775793\pi$
$978$	0	0
$979$	3.13914e8i	0.334551i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 7.86498e8i	− 0.828013i	−0.910274	−	0.414007i	$-0.864129\pi$
$983$	− 7.86498e8i	− 0.828013i	0.910274	−	0.414007i	$-0.135871\pi$
$984$	0	0
$985$	−1.20427e9	−1.26013
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.30881e8	−0.445418
$990$	0	0
$991$	1.16459e9	1.19661	0.598304	−	0.801269i	$-0.295841\pi$
$991$	1.16459e9	1.19661	0.598304	+	0.801269i	$0.295841\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.97473e8	−1.01259
$996$	0	0
$997$	− 5.86089e8i	− 0.591396i	−0.955281	−	0.295698i	$-0.904448\pi$
$997$	− 5.86089e8i	− 0.591396i	0.955281	−	0.295698i	$-0.0955522\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.h.b.161.5	✓	32
3.2	odd	2	inner	576.7.h.b.161.27	yes	32
4.3	odd	2	inner	576.7.h.b.161.6	yes	32
8.3	odd	2	inner	576.7.h.b.161.25	yes	32
8.5	even	2	inner	576.7.h.b.161.26	yes	32
12.11	even	2	inner	576.7.h.b.161.28	yes	32
24.5	odd	2	inner	576.7.h.b.161.8	yes	32
24.11	even	2	inner	576.7.h.b.161.7	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.7.h.b.161.5	✓	32	1.1	even	1	trivial
576.7.h.b.161.6	yes	32	4.3	odd	2	inner
576.7.h.b.161.7	yes	32	24.11	even	2	inner
576.7.h.b.161.8	yes	32	24.5	odd	2	inner
576.7.h.b.161.25	yes	32	8.3	odd	2	inner
576.7.h.b.161.26	yes	32	8.5	even	2	inner
576.7.h.b.161.27	yes	32	3.2	odd	2	inner
576.7.h.b.161.28	yes	32	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-108.406 q^{5} +167.464 q^{7} +O(q^{10})\)	\(q-108.406 q^{5} +167.464 q^{7} +1016.29 q^{11} -2272.52i q^{13} +865.606i q^{17} +8674.42i q^{19} +7261.66i q^{23} -3873.24 q^{25} +21010.9 q^{29} -12998.3 q^{31} -18154.1 q^{35} -3255.75i q^{37} -112661. i q^{41} +59336.4i q^{43} -117122. i q^{47} -89604.7 q^{49} +11719.7 q^{53} -110171. q^{55} -47019.2 q^{59} -35243.8i q^{61} +246354. i q^{65} +123187. i q^{67} +184149. i q^{71} -294641. q^{73} +170192. q^{77} +404395. q^{79} +43028.4 q^{83} -93836.5i q^{85} +308883. i q^{89} -380567. i q^{91} -940355. i q^{95} +1.21046e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) \) 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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