LMFDB - Embedded newform 576.8.c.g.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,8,Mod(575,576)]

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([1, 0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("576.575");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	576.c (of order $2$, degree $1$, not 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:	$179.933774679$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 1360 x^{14} + 710372 x^{12} - 180776904 x^{10} + 23779111124 x^{8} - 1575523372272 x^{6} + \cdots + 18\!\cdots\!76 $ x^16 - 1360x^14 + 710372x^12 - 180776904x^10 + 23779111124x^8 - 1575523372272x^6 + 47596139390384x^4 - 481767002289088*x^2 + 1836428975987776
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{98}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.1
Root			$-20.5167 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	576.575
Dual form			576.8.c.g.575.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-445.722i q^{5} -201.952i q^{7} +O(q^{10})$	$q-445.722i q^{5} -201.952i q^{7} -5367.66 q^{11} -1173.08 q^{13} -7661.23i q^{17} +29821.5i q^{19} +78582.2 q^{23} -120543. q^{25} +250673. i q^{29} +12849.8i q^{31} -90014.2 q^{35} +98047.2 q^{37} -120977. i q^{41} +722010. i q^{43} +803354. q^{47} +782759. q^{49} -962157. i q^{53} +2.39248e6i q^{55} +3.09055e6 q^{59} -775974. q^{61} +522869. i q^{65} -252638. i q^{67} +163095. q^{71} +2.39257e6 q^{73} +1.08401e6i q^{77} +6.38854e6i q^{79} -3.03995e6 q^{83} -3.41478e6 q^{85} -2.42945e6i q^{89} +236906. i q^{91} +1.32921e7 q^{95} -2.12402e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 33280 q^{13} - 55248 q^{25} - 869664 q^{37} + 1042032 q^{49} - 5525344 q^{61} + 29037312 q^{73} - 56734176 q^{85} + 82141696 q^{97}+O(q^{100}) $ 16 * q - 33280 * q^13 - 55248 * q^25 - 869664 * q^37 + 1042032 * q^49 - 5525344 * q^61 + 29037312 * q^73 - 56734176 * q^85 + 82141696 * 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
$5$	− 445.722i	− 1.59466i	−0.603541	−	0.797332i	$-0.706244\pi$
$5$	− 445.722i	− 1.59466i	0.603541	−	0.797332i	$-0.293756\pi$
$6$	0	0
$7$	− 201.952i	− 0.222538i	−0.993790	−	0.111269i	$-0.964508\pi$
$7$	− 201.952i	− 0.222538i	0.993790	−	0.111269i	$-0.0354915\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5367.66	−1.21594	−0.607968	−	0.793961i	$-0.708015\pi$
$11$	−5367.66	−1.21594	−0.607968	+	0.793961i	$0.708015\pi$
$12$	0	0
$13$	−1173.08	−0.148090	−0.0740452	−	0.997255i	$-0.523591\pi$
$13$	−1173.08	−0.148090	−0.0740452	+	0.997255i	$0.523591\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7661.23i	− 0.378205i	−0.981957	−	0.189103i	$-0.939442\pi$
$17$	− 7661.23i	− 0.378205i	0.981957	−	0.189103i	$-0.0605579\pi$
$18$	0	0
$19$	29821.5i	0.997451i	0.866760	+	0.498725i	$0.166198\pi$
$19$	29821.5i	0.997451i	−0.866760	+	0.498725i	$0.833802\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	78582.2	1.34672	0.673359	−	0.739316i	$-0.264851\pi$
$23$	78582.2	1.34672	0.673359	+	0.739316i	$0.264851\pi$
$24$	0	0
$25$	−120543.	−1.54295
$26$	0	0
$27$	0	0
$28$	0	0
$29$	250673.i	1.90859i	0.298859	+	0.954297i	$0.403394\pi$
$29$	250673.i	1.90859i	−0.298859	+	0.954297i	$0.596606\pi$
$30$	0	0
$31$	12849.8i	0.0774696i	0.999250	+	0.0387348i	$0.0123328\pi$
$31$	12849.8i	0.0774696i	−0.999250	+	0.0387348i	$0.987667\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−90014.2	−0.354873
$36$	0	0
$37$	98047.2	0.318221	0.159111	−	0.987261i	$-0.449137\pi$
$37$	98047.2	0.318221	0.159111	+	0.987261i	$0.449137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 120977.i	− 0.274132i	−0.990562	−	0.137066i	$-0.956233\pi$
$41$	− 120977.i	− 0.274132i	0.990562	−	0.137066i	$-0.0437673\pi$
$42$	0	0
$43$	722010.i	1.38485i	0.721488	+	0.692426i	$0.243458\pi$
$43$	722010.i	1.38485i	−0.721488	+	0.692426i	$0.756542\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	803354.	1.12866	0.564332	−	0.825548i	$-0.309134\pi$
$47$	803354.	1.12866	0.564332	+	0.825548i	$0.309134\pi$
$48$	0	0
$49$	782759.	0.950477
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 962157.i	− 0.887729i	−0.896094	−	0.443864i	$-0.853607\pi$
$53$	− 962157.i	− 0.887729i	0.896094	−	0.443864i	$-0.146393\pi$
$54$	0	0
$55$	2.39248e6i	1.93901i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.09055e6	1.95909	0.979543	−	0.201236i	$-0.0644958\pi$
$59$	3.09055e6	1.95909	0.979543	+	0.201236i	$0.0644958\pi$
$60$	0	0
$61$	−775974.	−0.437717	−0.218858	−	0.975757i	$-0.570233\pi$
$61$	−775974.	−0.437717	−0.218858	+	0.975757i	$0.570233\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	522869.i	0.236154i
$66$	0	0
$67$	− 252638.i	− 0.102621i	−0.998683	−	0.0513106i	$-0.983660\pi$
$67$	− 252638.i	− 0.102621i	0.998683	−	0.0513106i	$-0.0163398\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	163095.	0.0540799	0.0270400	−	0.999634i	$-0.491392\pi$
$71$	163095.	0.0540799	0.0270400	+	0.999634i	$0.491392\pi$
$72$	0	0
$73$	2.39257e6	0.719839	0.359919	−	0.932983i	$-0.382804\pi$
$73$	2.39257e6	0.719839	0.359919	+	0.932983i	$0.382804\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.08401e6i	0.270592i
$78$	0	0
$79$	6.38854e6i	1.45783i	0.684605	+	0.728915i	$0.259975\pi$
$79$	6.38854e6i	1.45783i	−0.684605	+	0.728915i	$0.740025\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.03995e6	−0.583571	−0.291786	−	0.956484i	$-0.594249\pi$
$83$	−3.03995e6	−0.583571	−0.291786	+	0.956484i	$0.594249\pi$
$84$	0	0
$85$	−3.41478e6	−0.603110
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.42945e6i	− 0.365294i	−0.983179	−	0.182647i	$-0.941533\pi$
$89$	− 2.42945e6i	− 0.365294i	0.983179	−	0.182647i	$-0.0584665\pi$
$90$	0	0
$91$	236906.i	0.0329557i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.32921e7	1.59060
$96$	0	0
$97$	−2.12402e6	−0.236297	−0.118148	−	0.992996i	$-0.537696\pi$
$97$	−2.12402e6	−0.236297	−0.118148	+	0.992996i	$0.537696\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.19855e7i	1.15753i	0.815495	+	0.578764i	$0.196465\pi$
$101$	1.19855e7i	1.15753i	−0.815495	+	0.578764i	$0.803535\pi$
$102$	0	0
$103$	− 1.34950e7i	− 1.21686i	−0.793606	−	0.608432i	$-0.791799\pi$
$103$	− 1.34950e7i	− 1.21686i	0.793606	−	0.608432i	$-0.208201\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.51239e7	−1.19350	−0.596749	−	0.802428i	$-0.703541\pi$
$107$	−1.51239e7	−1.19350	−0.596749	+	0.802428i	$0.703541\pi$
$108$	0	0
$109$	−1.49832e7	−1.10818	−0.554091	−	0.832456i	$-0.686934\pi$
$109$	−1.49832e7	−1.10818	−0.554091	+	0.832456i	$0.686934\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.58082e7i	− 1.68261i	−0.540560	−	0.841306i	$-0.681787\pi$
$113$	− 2.58082e7i	− 1.68261i	0.540560	−	0.841306i	$-0.318213\pi$
$114$	0	0
$115$	− 3.50258e7i	− 2.14756i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.54720e6	−0.0841650
$120$	0	0
$121$	9.32463e6	0.478501
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.89067e7i	0.865825i
$126$	0	0
$127$	1.30174e7i	0.563913i	0.959427	+	0.281956i	$0.0909833\pi$
$127$	1.30174e7i	0.563913i	−0.959427	+	0.281956i	$0.909017\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.59125e7	−0.618426	−0.309213	−	0.950993i	$-0.600066\pi$
$131$	−1.59125e7	−0.618426	−0.309213	+	0.950993i	$0.600066\pi$
$132$	0	0
$133$	6.02249e6	0.221971
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.22241e7i	0.406156i	0.979163	+	0.203078i	$0.0650946\pi$
$137$	1.22241e7i	0.406156i	−0.979163	+	0.203078i	$0.934905\pi$
$138$	0	0
$139$	− 5.36512e7i	− 1.69445i	−0.531238	−	0.847223i	$-0.678273\pi$
$139$	− 5.36512e7i	− 1.69445i	0.531238	−	0.847223i	$-0.321727\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.29671e6	0.180068
$144$	0	0
$145$	1.11730e8	3.04357
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.26997e7i	− 1.05748i	−0.848783	−	0.528741i	$-0.822664\pi$
$149$	− 4.26997e7i	− 1.05748i	0.848783	−	0.528741i	$-0.177336\pi$
$150$	0	0
$151$	9.72945e6i	0.229969i	0.993367	+	0.114984i	$0.0366818\pi$
$151$	9.72945e6i	0.229969i	−0.993367	+	0.114984i	$0.963318\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.72745e6	0.123538
$156$	0	0
$157$	−8.74323e7	−1.80311	−0.901557	−	0.432661i	$-0.857575\pi$
$157$	−8.74323e7	−1.80311	−0.901557	+	0.432661i	$0.857575\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.58698e7i	− 0.299696i
$162$	0	0
$163$	− 1.07309e8i	− 1.94080i	−0.241498	−	0.970401i	$-0.577639\pi$
$163$	− 1.07309e8i	− 1.94080i	0.241498	−	0.970401i	$-0.422361\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.78896e7	0.961817	0.480909	−	0.876771i	$-0.340307\pi$
$167$	5.78896e7	0.961817	0.480909	+	0.876771i	$0.340307\pi$
$168$	0	0
$169$	−6.13724e7	−0.978069
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.07783e8i	1.58267i	0.611384	+	0.791334i	$0.290613\pi$
$173$	1.07783e8i	1.58267i	−0.611384	+	0.791334i	$0.709387\pi$
$174$	0	0
$175$	2.43439e7i	0.343365i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.65555e7	0.606716	0.303358	−	0.952877i	$-0.401892\pi$
$179$	4.65555e7	0.606716	0.303358	+	0.952877i	$0.401892\pi$
$180$	0	0
$181$	−4.00341e7	−0.501828	−0.250914	−	0.968009i	$-0.580731\pi$
$181$	−4.00341e7	−0.501828	−0.250914	+	0.968009i	$0.580731\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.37018e7i	− 0.507456i
$186$	0	0
$187$	4.11229e7i	0.459873i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.76797e8	1.83593	0.917967	−	0.396657i	$-0.129830\pi$
$191$	1.76797e8	1.83593	0.917967	+	0.396657i	$0.129830\pi$
$192$	0	0
$193$	3.32665e7	0.333087	0.166543	−	0.986034i	$-0.446739\pi$
$193$	3.32665e7	0.333087	0.166543	+	0.986034i	$0.446739\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.11714e7i	0.756434i	0.925717	+	0.378217i	$0.123463\pi$
$197$	8.11714e7i	0.756434i	−0.925717	+	0.378217i	$0.876537\pi$
$198$	0	0
$199$	1.43040e8i	1.28668i	0.765580	+	0.643341i	$0.222452\pi$
$199$	1.43040e8i	1.28668i	−0.765580	+	0.643341i	$0.777548\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.06237e7	0.424735
$204$	0	0
$205$	−5.39223e7	−0.437149
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.60072e8i	− 1.21284i
$210$	0	0
$211$	1.51433e8i	1.10977i	0.831928	+	0.554884i	$0.187237\pi$
$211$	1.51433e8i	1.10977i	−0.831928	+	0.554884i	$0.812763\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.21816e8	2.20837
$216$	0	0
$217$	2.59504e6	0.0172399
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.98726e6i	0.0560086i
$222$	0	0
$223$	− 2.36793e8i	− 1.42989i	−0.699181	−	0.714945i	$-0.746452\pi$
$223$	− 2.36793e8i	− 1.42989i	0.699181	−	0.714945i	$-0.253548\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.75301e8	0.994705	0.497353	−	0.867548i	$-0.334306\pi$
$227$	1.75301e8	0.994705	0.497353	+	0.867548i	$0.334306\pi$
$228$	0	0
$229$	3.08750e8	1.69896	0.849480	−	0.527621i	$-0.176916\pi$
$229$	3.08750e8	1.69896	0.849480	+	0.527621i	$0.176916\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.91707e8i	0.992869i	0.868074	+	0.496435i	$0.165358\pi$
$233$	1.91707e8i	0.992869i	−0.868074	+	0.496435i	$0.834642\pi$
$234$	0	0
$235$	− 3.58072e8i	− 1.79984i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.13941e8	0.539868	0.269934	−	0.962879i	$-0.412998\pi$
$239$	1.13941e8	0.539868	0.269934	+	0.962879i	$0.412998\pi$
$240$	0	0
$241$	1.68043e8	0.773323	0.386662	−	0.922222i	$-0.373628\pi$
$241$	1.68043e8	0.773323	0.386662	+	0.922222i	$0.373628\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.48893e8i	− 1.51569i
$246$	0	0
$247$	− 3.49831e7i	− 0.147713i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.42282e8	1.36624	0.683118	−	0.730308i	$-0.260623\pi$
$251$	3.42282e8	1.36624	0.683118	+	0.730308i	$0.260623\pi$
$252$	0	0
$253$	−4.21803e8	−1.63752
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.19015e8i	− 0.437355i	−0.975797	−	0.218678i	$-0.929826\pi$
$257$	− 1.19015e8i	− 0.437355i	0.975797	−	0.218678i	$-0.0701743\pi$
$258$	0	0
$259$	− 1.98008e7i	− 0.0708163i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.17137e8	−1.07498	−0.537492	−	0.843269i	$-0.680628\pi$
$263$	−3.17137e8	−1.07498	−0.537492	+	0.843269i	$0.680628\pi$
$264$	0	0
$265$	−4.28854e8	−1.41563
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.03521e7i	0.283012i	0.989937	+	0.141506i	$0.0451945\pi$
$269$	9.03521e7i	0.283012i	−0.989937	+	0.141506i	$0.954805\pi$
$270$	0	0
$271$	− 1.29807e8i	− 0.396191i	−0.980183	−	0.198095i	$-0.936524\pi$
$271$	− 1.29807e8i	− 0.396191i	0.980183	−	0.198095i	$-0.0634756\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.47035e8	1.87613
$276$	0	0
$277$	5.01224e8	1.41694	0.708472	−	0.705739i	$-0.249385\pi$
$277$	5.01224e8	1.41694	0.708472	+	0.705739i	$0.249385\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.39332e8i	1.45005i	0.688720	+	0.725027i	$0.258173\pi$
$281$	5.39332e8i	1.45005i	−0.688720	+	0.725027i	$0.741827\pi$
$282$	0	0
$283$	− 2.61973e8i	− 0.687076i	−0.939139	−	0.343538i	$-0.888375\pi$
$283$	− 2.61973e8i	− 0.687076i	0.939139	−	0.343538i	$-0.111625\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.44316e7	−0.0610049
$288$	0	0
$289$	3.51644e8	0.856961
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4.42675e8i	− 1.02813i	−0.857751	−	0.514065i	$-0.828139\pi$
$293$	− 4.42675e8i	− 1.02813i	0.857751	−	0.514065i	$-0.171861\pi$
$294$	0	0
$295$	− 1.37753e9i	− 3.12408i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.21834e7	−0.199436
$300$	0	0
$301$	1.45811e8	0.308182
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.45869e8i	0.698011i
$306$	0	0
$307$	− 6.00549e8i	− 1.18458i	−0.805725	−	0.592290i	$-0.798224\pi$
$307$	− 6.00549e8i	− 1.18458i	0.805725	−	0.592290i	$-0.201776\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.60256e8	1.43317	0.716586	−	0.697499i	$-0.245704\pi$
$311$	7.60256e8	1.43317	0.716586	+	0.697499i	$0.245704\pi$
$312$	0	0
$313$	−9.79664e8	−1.80581	−0.902905	−	0.429840i	$-0.858570\pi$
$313$	−9.79664e8	−1.80581	−0.902905	+	0.429840i	$0.858570\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.73197e8i	1.01064i	0.862932	+	0.505320i	$0.168626\pi$
$317$	5.73197e8i	1.01064i	−0.862932	+	0.505320i	$0.831374\pi$
$318$	0	0
$319$	− 1.34553e9i	− 2.32073i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.28469e8	0.377241
$324$	0	0
$325$	1.41407e8	0.228496
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.62239e8i	− 0.251170i
$330$	0	0
$331$	6.30547e8i	0.955695i	0.878443	+	0.477847i	$0.158583\pi$
$331$	6.30547e8i	0.955695i	−0.878443	+	0.477847i	$0.841417\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.12606e8	−0.163646
$336$	0	0
$337$	−8.22055e8	−1.17003	−0.585014	−	0.811023i	$-0.698911\pi$
$337$	−8.22055e8	−1.17003	−0.585014	+	0.811023i	$0.698911\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.89735e7i	− 0.0941981i
$342$	0	0
$343$	− 3.24395e8i	− 0.434055i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.29721e9	1.66670	0.833351	−	0.552744i	$-0.186419\pi$
$347$	1.29721e9	1.66670	0.833351	+	0.552744i	$0.186419\pi$
$348$	0	0
$349$	1.22063e9	1.53707	0.768536	−	0.639807i	$-0.220986\pi$
$349$	1.22063e9	1.53707	0.768536	+	0.639807i	$0.220986\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.98374e8i	− 0.482036i	−0.970521	−	0.241018i	$-0.922519\pi$
$353$	− 3.98374e8i	− 0.482036i	0.970521	−	0.241018i	$-0.0774813\pi$
$354$	0	0
$355$	− 7.26950e7i	− 0.0862393i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.74326e8	0.426992	0.213496	−	0.976944i	$-0.431515\pi$
$359$	3.74326e8	0.426992	0.213496	+	0.976944i	$0.431515\pi$
$360$	0	0
$361$	4.55145e6	0.00509184
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.06642e9i	− 1.14790i
$366$	0	0
$367$	3.89260e8i	0.411063i	0.978650	+	0.205532i	$0.0658924\pi$
$367$	3.89260e8i	0.411063i	−0.978650	+	0.205532i	$0.934108\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.94309e8	−0.197553
$372$	0	0
$373$	2.20587e8	0.220089	0.110044	−	0.993927i	$-0.464901\pi$
$373$	2.20587e8	0.220089	0.110044	+	0.993927i	$0.464901\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.94060e8i	− 0.282645i
$378$	0	0
$379$	3.58994e8i	0.338727i	0.985554	+	0.169364i	$0.0541712\pi$
$379$	3.58994e8i	0.338727i	−0.985554	+	0.169364i	$0.945829\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.11450e9	1.01364	0.506822	−	0.862051i	$-0.330820\pi$
$383$	1.11450e9	1.01364	0.506822	+	0.862051i	$0.330820\pi$
$384$	0	0
$385$	4.83166e8	0.431503
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7.82474e8i	− 0.673979i	−0.941508	−	0.336989i	$-0.890591\pi$
$389$	− 7.82474e8i	− 0.673979i	0.941508	−	0.336989i	$-0.109409\pi$
$390$	0	0
$391$	− 6.02036e8i	− 0.509336i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.84751e9	2.32475
$396$	0	0
$397$	−1.16495e8	−0.0934415	−0.0467207	−	0.998908i	$-0.514877\pi$
$397$	−1.16495e8	−0.0934415	−0.0467207	+	0.998908i	$0.514877\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.72842e9i	1.33858i	0.743000	+	0.669291i	$0.233402\pi$
$401$	1.72842e9i	1.33858i	−0.743000	+	0.669291i	$0.766598\pi$
$402$	0	0
$403$	− 1.50739e7i	− 0.0114725i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.26284e8	−0.386937
$408$	0	0
$409$	1.63857e9	1.18422	0.592111	−	0.805857i	$-0.298295\pi$
$409$	1.63857e9	1.18422	0.592111	+	0.805857i	$0.298295\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.24141e8i	− 0.435971i
$414$	0	0
$415$	1.35497e9i	0.930600i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.04990e9	1.36139	0.680697	−	0.732565i	$-0.261677\pi$
$419$	2.04990e9	1.36139	0.680697	+	0.732565i	$0.261677\pi$
$420$	0	0
$421$	−1.85898e9	−1.21420	−0.607098	−	0.794627i	$-0.707666\pi$
$421$	−1.85898e9	−1.21420	−0.607098	+	0.794627i	$0.707666\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.23509e8i	0.583552i
$426$	0	0
$427$	1.56709e8i	0.0974085i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.25651e9	1.35758	0.678792	−	0.734330i	$-0.262504\pi$
$431$	2.25651e9	1.35758	0.678792	+	0.734330i	$0.262504\pi$
$432$	0	0
$433$	−1.19933e9	−0.709954	−0.354977	−	0.934875i	$-0.615511\pi$
$433$	−1.19933e9	−0.709954	−0.354977	+	0.934875i	$0.615511\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.34344e9i	1.34329i
$438$	0	0
$439$	2.26073e9i	1.27533i	0.770312	+	0.637667i	$0.220100\pi$
$439$	2.26073e9i	1.27533i	−0.770312	+	0.637667i	$0.779900\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.14557e9	−1.17255	−0.586274	−	0.810113i	$-0.699406\pi$
$443$	−2.14557e9	−1.17255	−0.586274	+	0.810113i	$0.699406\pi$
$444$	0	0
$445$	−1.08286e9	−0.582521
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8.44899e8i	− 0.440496i	−0.975444	−	0.220248i	$-0.929313\pi$
$449$	− 8.44899e8i	− 0.440496i	0.975444	−	0.220248i	$-0.0706867\pi$
$450$	0	0
$451$	6.49365e8i	0.333328i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.05594e8	0.0525533
$456$	0	0
$457$	−8.46931e8	−0.415089	−0.207544	−	0.978226i	$-0.566547\pi$
$457$	−8.46931e8	−0.415089	−0.207544	+	0.978226i	$0.566547\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.29935e9i	0.617694i	0.951112	+	0.308847i	$0.0999432\pi$
$461$	1.29935e9i	0.617694i	−0.951112	+	0.308847i	$0.900057\pi$
$462$	0	0
$463$	5.18884e8i	0.242961i	0.992594	+	0.121481i	$0.0387642\pi$
$463$	5.18884e8i	0.242961i	−0.992594	+	0.121481i	$0.961236\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.69058e9	−0.768118	−0.384059	−	0.923309i	$-0.625474\pi$
$467$	−1.69058e9	−0.768118	−0.384059	+	0.923309i	$0.625474\pi$
$468$	0	0
$469$	−5.10206e7	−0.0228371
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.87551e9i	− 1.68389i
$474$	0	0
$475$	− 3.59477e9i	− 1.53902i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.84920e9	−0.768792	−0.384396	−	0.923168i	$-0.625590\pi$
$479$	−1.84920e9	−0.768792	−0.384396	+	0.923168i	$0.625590\pi$
$480$	0	0
$481$	−1.15018e8	−0.0471255
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.46724e8i	0.376814i
$486$	0	0
$487$	− 3.02324e9i	− 1.18610i	−0.805166	−	0.593050i	$-0.797924\pi$
$487$	− 3.02324e9i	− 1.18610i	0.805166	−	0.593050i	$-0.202076\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.78758e9	−1.06278	−0.531388	−	0.847129i	$-0.678329\pi$
$491$	−2.78758e9	−1.06278	−0.531388	+	0.847129i	$0.678329\pi$
$492$	0	0
$493$	1.92046e9	0.721840
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.29373e7i	− 0.0120348i
$498$	0	0
$499$	− 1.09189e9i	− 0.393392i	−0.980465	−	0.196696i	$-0.936979\pi$
$499$	− 1.09189e9i	− 0.393392i	0.980465	−	0.196696i	$-0.0630212\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.91716e9	1.37241	0.686204	−	0.727410i	$-0.259276\pi$
$503$	3.91716e9	1.37241	0.686204	+	0.727410i	$0.259276\pi$
$504$	0	0
$505$	5.34220e9	1.84587
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.25898e9i	0.423163i	0.977360	+	0.211581i	$0.0678613\pi$
$509$	1.25898e9i	0.423163i	−0.977360	+	0.211581i	$0.932139\pi$
$510$	0	0
$511$	− 4.83184e8i	− 0.160191i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.01501e9	−1.94049
$516$	0	0
$517$	−4.31213e9	−1.37238
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 4.11117e9i	− 1.27360i	−0.771029	−	0.636800i	$-0.780258\pi$
$521$	− 4.11117e9i	− 1.27360i	0.771029	−	0.636800i	$-0.219742\pi$
$522$	0	0
$523$	2.91561e6i	0 0.000891197i	1.00000		0.000445599i	$0.000141838\pi$
$523$	2.91561e6i	0 0.000891197i	−1.00000		0.000445599i	$0.999858\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.84455e7	0.0292994
$528$	0	0
$529$	2.77034e9	0.813650
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.41916e8i	0.0405964i
$534$	0	0
$535$	6.74107e9i	1.90323i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.20158e9	−1.15572
$540$	0	0
$541$	4.88925e9	1.32755	0.663777	−	0.747931i	$-0.268952\pi$
$541$	4.88925e9	1.32755	0.663777	+	0.747931i	$0.268952\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.67833e9i	1.76718i
$546$	0	0
$547$	− 3.00310e9i	− 0.784539i	−0.919850	−	0.392270i	$-0.871690\pi$
$547$	− 3.00310e9i	− 0.784539i	0.919850	−	0.392270i	$-0.128310\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.47542e9	−1.90373
$552$	0	0
$553$	1.29018e9	0.324422
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.50407e8i	0.0368786i	0.999830	+	0.0184393i	$0.00586974\pi$
$557$	1.50407e8i	0.0368786i	−0.999830	+	0.0184393i	$0.994130\pi$
$558$	0	0
$559$	− 8.46978e8i	− 0.205083i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.06608e9	0.960276	0.480138	−	0.877193i	$-0.340587\pi$
$563$	4.06608e9	0.960276	0.480138	+	0.877193i	$0.340587\pi$
$564$	0	0
$565$	−1.15033e10	−2.68320
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.52046e9i	− 1.71140i	−0.517472	−	0.855700i	$-0.673127\pi$
$569$	− 7.52046e9i	− 1.71140i	0.517472	−	0.855700i	$-0.326873\pi$
$570$	0	0
$571$	− 3.43788e9i	− 0.772795i	−0.922332	−	0.386398i	$-0.873719\pi$
$571$	− 3.43788e9i	− 0.772795i	0.922332	−	0.386398i	$-0.126281\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.47254e9	−2.07792
$576$	0	0
$577$	−3.81016e9	−0.825711	−0.412855	−	0.910797i	$-0.635469\pi$
$577$	−3.81016e9	−0.825711	−0.412855	+	0.910797i	$0.635469\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.13923e8i	0.129867i
$582$	0	0
$583$	5.16453e9i	1.07942i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.98289e9	−1.62902	−0.814511	−	0.580149i	$-0.802994\pi$
$587$	−7.98289e9	−1.62902	−0.814511	+	0.580149i	$0.802994\pi$
$588$	0	0
$589$	−3.83201e8	−0.0772721
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.09226e9i	0.215096i	0.994200	+	0.107548i	$0.0343000\pi$
$593$	1.09226e9i	0.215096i	−0.994200	+	0.107548i	$0.965700\pi$
$594$	0	0
$595$	6.89620e8i	0.134215i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.96369e9	0.373318	0.186659	−	0.982425i	$-0.440234\pi$
$599$	1.96369e9	0.373318	0.186659	+	0.982425i	$0.440234\pi$
$600$	0	0
$601$	7.03584e9	1.32207	0.661036	−	0.750354i	$-0.270117\pi$
$601$	7.03584e9	1.32207	0.661036	+	0.750354i	$0.270117\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.15619e9i	− 0.763048i
$606$	0	0
$607$	3.97206e9i	0.720868i	0.932785	+	0.360434i	$0.117371\pi$
$607$	3.97206e9i	0.720868i	−0.932785	+	0.360434i	$0.882629\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.42401e8	−0.167144
$612$	0	0
$613$	1.11811e9	0.196054	0.0980268	−	0.995184i	$-0.468747\pi$
$613$	1.11811e9	0.196054	0.0980268	+	0.995184i	$0.468747\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.93863e9i	1.36065i	0.732909	+	0.680327i	$0.238162\pi$
$617$	7.93863e9i	1.36065i	−0.732909	+	0.680327i	$0.761838\pi$
$618$	0	0
$619$	5.53963e9i	0.938779i	0.882991	+	0.469389i	$0.155526\pi$
$619$	5.53963e9i	0.938779i	−0.882991	+	0.469389i	$0.844474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.90631e8	−0.0812917
$624$	0	0
$625$	−9.90310e8	−0.162252
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 7.51163e8i	− 0.120353i
$630$	0	0
$631$	2.90603e9i	0.460466i	0.973136	+	0.230233i	$0.0739489\pi$
$631$	2.90603e9i	0.460466i	−0.973136	+	0.230233i	$0.926051\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.80215e9	0.899251
$636$	0	0
$637$	−9.18241e8	−0.140757
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2.87322e9i	− 0.430889i	−0.976516	−	0.215444i	$-0.930880\pi$
$641$	− 2.87322e9i	− 0.430889i	0.976516	−	0.215444i	$-0.0691200\pi$
$642$	0	0
$643$	1.04113e10i	1.54443i	0.635364	+	0.772213i	$0.280850\pi$
$643$	1.04113e10i	1.54443i	−0.635364	+	0.772213i	$0.719150\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.77707e9	0.693420	0.346710	−	0.937972i	$-0.387299\pi$
$647$	4.77707e9	0.693420	0.346710	+	0.937972i	$0.387299\pi$
$648$	0	0
$649$	−1.65890e10	−2.38212
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.13626e9i	0.721856i	0.932594	+	0.360928i	$0.117540\pi$
$653$	5.13626e9i	0.721856i	−0.932594	+	0.360928i	$0.882460\pi$
$654$	0	0
$655$	7.09253e9i	0.986181i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.18123e10	−1.60781	−0.803906	−	0.594757i	$-0.797248\pi$
$659$	−1.18123e10	−1.60781	−0.803906	+	0.594757i	$0.797248\pi$
$660$	0	0
$661$	6.45411e9	0.869223	0.434612	−	0.900618i	$-0.356886\pi$
$661$	6.45411e9	0.869223	0.434612	+	0.900618i	$0.356886\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.68436e9i	− 0.353968i
$666$	0	0
$667$	1.96984e10i	2.57034i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.16517e9	0.532235
$672$	0	0
$673$	7.48253e9	0.946228	0.473114	−	0.881001i	$-0.343130\pi$
$673$	7.48253e9	0.946228	0.473114	+	0.881001i	$0.343130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.62586e9i	0.449108i	0.974462	+	0.224554i	$0.0720925\pi$
$677$	3.62586e9i	0.449108i	−0.974462	+	0.224554i	$0.927908\pi$
$678$	0	0
$679$	4.28950e8i	0.0525850i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.06326e9	−0.487980	−0.243990	−	0.969778i	$-0.578456\pi$
$683$	−4.06326e9	−0.487980	−0.243990	+	0.969778i	$0.578456\pi$
$684$	0	0
$685$	5.44853e9	0.647683
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.12869e9i	0.131464i
$690$	0	0
$691$	1.03231e10i	1.19025i	0.803635	+	0.595123i	$0.202897\pi$
$691$	1.03231e10i	1.19025i	−0.803635	+	0.595123i	$0.797103\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.39135e10	−2.70207
$696$	0	0
$697$	−9.26835e8	−0.103678
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.70619e9i	0.954586i	0.878744	+	0.477293i	$0.158382\pi$
$701$	8.70619e9i	0.954586i	−0.878744	+	0.477293i	$0.841618\pi$
$702$	0	0
$703$	2.92391e9i	0.317410i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.42049e9	0.257594
$708$	0	0
$709$	1.36353e10	1.43682	0.718410	−	0.695620i	$-0.244870\pi$
$709$	1.36353e10	1.43682	0.718410	+	0.695620i	$0.244870\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.00977e9i	0.104330i
$714$	0	0
$715$	− 2.80658e9i	− 0.287149i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.82007e9	0.182615	0.0913075	−	0.995823i	$-0.470895\pi$
$719$	1.82007e9	0.182615	0.0913075	+	0.995823i	$0.470895\pi$
$720$	0	0
$721$	−2.72533e9	−0.270799
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.02168e10i	− 2.94487i
$726$	0	0
$727$	− 1.04415e10i	− 1.00784i	−0.863750	−	0.503920i	$-0.831891\pi$
$727$	− 1.04415e10i	− 1.00784i	0.863750	−	0.503920i	$-0.168109\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.53149e9	0.523758
$732$	0	0
$733$	7.80483e9	0.731981	0.365990	−	0.930619i	$-0.380730\pi$
$733$	7.80483e9	0.731981	0.365990	+	0.930619i	$0.380730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.35608e9i	0.124781i
$738$	0	0
$739$	3.97888e9i	0.362664i	0.983422	+	0.181332i	$0.0580409\pi$
$739$	3.97888e9i	0.362664i	−0.983422	+	0.181332i	$0.941959\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.20404e10	−1.07691	−0.538455	−	0.842654i	$-0.680992\pi$
$743$	−1.20404e10	−1.07691	−0.538455	+	0.842654i	$0.680992\pi$
$744$	0	0
$745$	−1.90322e10	−1.68633
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.05430e9i	0.265598i
$750$	0	0
$751$	− 4.23550e9i	− 0.364893i	−0.983216	−	0.182446i	$-0.941598\pi$
$751$	− 4.23550e9i	− 0.364893i	0.983216	−	0.182446i	$-0.0584016\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.33663e9	0.366723
$756$	0	0
$757$	−8.69080e9	−0.728155	−0.364078	−	0.931369i	$-0.618616\pi$
$757$	−8.69080e9	−0.728155	−0.364078	+	0.931369i	$0.618616\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 7.08162e9i	− 0.582487i	−0.956649	−	0.291243i	$-0.905931\pi$
$761$	− 7.08162e9i	− 0.582487i	0.956649	−	0.291243i	$-0.0940690\pi$
$762$	0	0
$763$	3.02587e9i	0.246612i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.62547e9	−0.290122
$768$	0	0
$769$	1.91172e10	1.51594	0.757969	−	0.652290i	$-0.226192\pi$
$769$	1.91172e10	1.51594	0.757969	+	0.652290i	$0.226192\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.56030e10i	− 1.21501i	−0.794316	−	0.607505i	$-0.792170\pi$
$773$	− 1.56030e10i	− 1.21501i	0.794316	−	0.607505i	$-0.207830\pi$
$774$	0	0
$775$	− 1.54896e9i	− 0.119532i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.60772e9	0.273434
$780$	0	0
$781$	−8.75438e8	−0.0657578
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.89705e10i	2.87536i
$786$	0	0
$787$	− 2.59880e9i	− 0.190047i	−0.995475	−	0.0950234i	$-0.969707\pi$
$787$	− 2.59880e9i	− 0.190047i	0.995475	−	0.0950234i	$-0.0302926\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.21201e9	−0.374445
$792$	0	0
$793$	9.10282e8	0.0648216
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.40901e8i	0.0168552i	0.999964	+	0.00842760i	$0.00268262\pi$
$797$	2.40901e8i	0.0168552i	−0.999964	+	0.00842760i	$0.997317\pi$
$798$	0	0
$799$	− 6.15468e9i	− 0.426866i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.28425e10	−0.875278
$804$	0	0
$805$	−7.07352e9	−0.477914
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.61647e10i	1.07337i	0.843783	+	0.536684i	$0.180323\pi$
$809$	1.61647e10i	1.07337i	−0.843783	+	0.536684i	$0.819677\pi$
$810$	0	0
$811$	− 3.42220e9i	− 0.225285i	−0.993636	−	0.112642i	$-0.964069\pi$
$811$	− 3.42220e9i	− 0.225285i	0.993636	−	0.112642i	$-0.0359315\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.78302e10	−3.09493
$816$	0	0
$817$	−2.15314e10	−1.38132
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.94367e9i	0.248714i	0.992238	+	0.124357i	$0.0396868\pi$
$821$	3.94367e9i	0.248714i	−0.992238	+	0.124357i	$0.960313\pi$
$822$	0	0
$823$	− 2.27646e10i	− 1.42351i	−0.702427	−	0.711755i	$-0.747900\pi$
$823$	− 2.27646e10i	− 1.42351i	0.702427	−	0.711755i	$-0.252100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.20469e9	−0.319982	−0.159991	−	0.987118i	$-0.551147\pi$
$827$	−5.20469e9	−0.319982	−0.159991	+	0.987118i	$0.551147\pi$
$828$	0	0
$829$	−6.42188e9	−0.391491	−0.195745	−	0.980655i	$-0.562713\pi$
$829$	−6.42188e9	−0.391491	−0.195745	+	0.980655i	$0.562713\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.99690e9i	− 0.359475i
$834$	0	0
$835$	− 2.58026e10i	− 1.53377i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.63779e10	0.957396	0.478698	−	0.877980i	$-0.341109\pi$
$839$	1.63779e10	0.957396	0.478698	+	0.877980i	$0.341109\pi$
$840$	0	0
$841$	−4.55868e10	−2.64273
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.73550e10i	1.55969i
$846$	0	0
$847$	− 1.88312e9i	− 0.106485i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.70477e9	0.428555
$852$	0	0
$853$	2.95535e10	1.63037	0.815187	−	0.579198i	$-0.196634\pi$
$853$	2.95535e10	1.63037	0.815187	+	0.579198i	$0.196634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.36480e10i	− 1.82611i	−0.407840	−	0.913054i	$-0.633718\pi$
$857$	− 3.36480e10i	− 1.82611i	0.407840	−	0.913054i	$-0.366282\pi$
$858$	0	0
$859$	1.03722e10i	0.558336i	0.960242	+	0.279168i	$0.0900586\pi$
$859$	1.03722e10i	0.558336i	−0.960242	+	0.279168i	$0.909941\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.13251e10	1.12942	0.564708	−	0.825291i	$-0.308989\pi$
$863$	2.13251e10	1.12942	0.564708	+	0.825291i	$0.308989\pi$
$864$	0	0
$865$	4.80414e10	2.52382
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.42915e10i	− 1.77263i
$870$	0	0
$871$	2.96365e8i	0.0151972i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.81823e9	0.192679
$876$	0	0
$877$	7.22632e9	0.361759	0.180879	−	0.983505i	$-0.442106\pi$
$877$	7.22632e9	0.361759	0.180879	+	0.983505i	$0.442106\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 2.44432e10i	− 1.20432i	−0.798376	−	0.602159i	$-0.794307\pi$
$881$	− 2.44432e10i	− 1.20432i	0.798376	−	0.602159i	$-0.205693\pi$
$882$	0	0
$883$	− 4.58740e9i	− 0.224235i	−0.993695	−	0.112118i	$-0.964237\pi$
$883$	− 4.58740e9i	− 0.224235i	0.993695	−	0.112118i	$-0.0357633\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.97698e10	0.951196	0.475598	−	0.879663i	$-0.342232\pi$
$887$	1.97698e10	0.951196	0.475598	+	0.879663i	$0.342232\pi$
$888$	0	0
$889$	2.62889e9	0.125492
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.39572e10i	1.12579i
$894$	0	0
$895$	− 2.07508e10i	− 0.967508i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.22110e9	−0.147858
$900$	0	0
$901$	−7.37131e9	−0.335744
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.78441e10i	0.800247i
$906$	0	0
$907$	1.22458e10i	0.544957i	0.962162	+	0.272479i	$0.0878434\pi$
$907$	1.22458e10i	0.544957i	−0.962162	+	0.272479i	$0.912157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.56276e10	1.56125	0.780625	−	0.625000i	$-0.214901\pi$
$911$	3.56276e10	1.56125	0.780625	+	0.625000i	$0.214901\pi$
$912$	0	0
$913$	1.63175e10	0.709585
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.21354e9i	0.137623i
$918$	0	0
$919$	1.02923e10i	0.437430i	0.975789	+	0.218715i	$0.0701865\pi$
$919$	1.02923e10i	0.437430i	−0.975789	+	0.218715i	$0.929814\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.91324e8	−0.00800872
$924$	0	0
$925$	−1.18189e10	−0.491000
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.80492e10i	− 0.738590i	−0.929312	−	0.369295i	$-0.879599\pi$
$929$	− 1.80492e10i	− 0.738590i	0.929312	−	0.369295i	$-0.120401\pi$
$930$	0	0
$931$	2.33430e10i	0.948054i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.83294e10	0.733343
$936$	0	0
$937$	−2.03829e10	−0.809425	−0.404712	−	0.914444i	$-0.632628\pi$
$937$	−2.03829e10	−0.809425	−0.404712	+	0.914444i	$0.632628\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 4.71378e8i	− 0.0184419i	−0.999957	−	0.00922094i	$-0.997065\pi$
$941$	− 4.71378e8i	− 0.0184419i	0.999957	−	0.00922094i	$-0.00293516\pi$
$942$	0	0
$943$	− 9.50666e9i	− 0.369179i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.91127e10	0.731302	0.365651	−	0.930752i	$-0.380846\pi$
$947$	1.91127e10	0.731302	0.365651	+	0.930752i	$0.380846\pi$
$948$	0	0
$949$	−2.80669e9	−0.106601
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.26355e10i	− 0.847159i	−0.905859	−	0.423580i	$-0.860773\pi$
$953$	− 2.26355e10i	− 0.847159i	0.905859	−	0.423580i	$-0.139227\pi$
$954$	0	0
$955$	− 7.88021e10i	− 2.92770i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.46867e9	0.0903852
$960$	0	0
$961$	2.73475e10	0.993998
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.48276e10i	− 0.531161i
$966$	0	0
$967$	5.25242e10i	1.86796i	0.357328	+	0.933979i	$0.383688\pi$
$967$	5.25242e10i	1.86796i	−0.357328	+	0.933979i	$0.616312\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.09805e9	−0.143651	−0.0718257	−	0.997417i	$-0.522883\pi$
$971$	−4.09805e9	−0.143651	−0.0718257	+	0.997417i	$0.522883\pi$
$972$	0	0
$973$	−1.08349e10	−0.377078
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.11726e10i	1.75552i	0.479097	+	0.877762i	$0.340964\pi$
$977$	5.11726e10i	1.75552i	−0.479097	+	0.877762i	$0.659036\pi$
$978$	0	0
$979$	1.30405e10i	0.444174i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.41382e10	1.14631	0.573157	−	0.819446i	$-0.305719\pi$
$983$	3.41382e10	1.14631	0.573157	+	0.819446i	$0.305719\pi$
$984$	0	0
$985$	3.61799e10	1.20626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.67371e10i	1.86501i
$990$	0	0
$991$	5.50572e10i	1.79703i	0.438938	+	0.898517i	$0.355355\pi$
$991$	5.50572e10i	1.79703i	−0.438938	+	0.898517i	$0.644645\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.37560e10	2.05182
$996$	0	0
$997$	−4.48885e10	−1.43450	−0.717252	−	0.696814i	$-0.754600\pi$
$997$	−4.48885e10	−1.43450	−0.717252	+	0.696814i	$0.754600\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.8.c.g.575.1		16
3.2	odd	2	inner	576.8.c.g.575.15		16
4.3	odd	2	inner	576.8.c.g.575.2		16
8.3	odd	2		288.8.c.b.287.16	yes	16
8.5	even	2		288.8.c.b.287.15	yes	16
12.11	even	2	inner	576.8.c.g.575.16		16
24.5	odd	2		288.8.c.b.287.1	✓	16
24.11	even	2		288.8.c.b.287.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.8.c.b.287.1	✓	16	24.5	odd	2
288.8.c.b.287.2	yes	16	24.11	even	2
288.8.c.b.287.15	yes	16	8.5	even	2
288.8.c.b.287.16	yes	16	8.3	odd	2
576.8.c.g.575.1		16	1.1	even	1	trivial
576.8.c.g.575.2		16	4.3	odd	2	inner
576.8.c.g.575.15		16	3.2	odd	2	inner
576.8.c.g.575.16		16	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	576.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-445.722i q^{5} -201.952i q^{7} +O(q^{10})\)	\(q-445.722i q^{5} -201.952i q^{7} -5367.66 q^{11} -1173.08 q^{13} -7661.23i q^{17} +29821.5i q^{19} +78582.2 q^{23} -120543. q^{25} +250673. i q^{29} +12849.8i q^{31} -90014.2 q^{35} +98047.2 q^{37} -120977. i q^{41} +722010. i q^{43} +803354. q^{47} +782759. q^{49} -962157. i q^{53} +2.39248e6i q^{55} +3.09055e6 q^{59} -775974. q^{61} +522869. i q^{65} -252638. i q^{67} +163095. q^{71} +2.39257e6 q^{73} +1.08401e6i q^{77} +6.38854e6i q^{79} -3.03995e6 q^{83} -3.41478e6 q^{85} -2.42945e6i q^{89} +236906. i q^{91} +1.32921e7 q^{95} -2.12402e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 33280 q^{13} - 55248 q^{25} - 869664 q^{37} + 1042032 q^{49} - 5525344 q^{61} + 29037312 q^{73} - 56734176 q^{85} + 82141696 q^{97}+O(q^{100}) \) 16 * q - 33280 * q^13 - 55248 * q^25 - 869664 * q^37 + 1042032 * q^49 - 5525344 * q^61 + 29037312 * q^73 - 56734176 * q^85 + 82141696 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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