LMFDB - Embedded newform 576.8.d.i.289.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,8,Mod(289,576)]

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, 0]))

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

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	576.d (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:	$179.933774679$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5249 x^{10} + 20722017 x^{8} - 34316449184 x^{6} + 42622339324672 x^{4} + \cdots + 58\!\cdots\!56 $ x^12 - 5249x^10 + 20722017x^8 - 34316449184x^6 + 42622339324672x^4 - 5239065008185344*x^2 + 588393954824749056
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{64}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.11
Root			$-9.63634 + 5.56354i$ of defining polynomial
Character	$\chi$	$=$	576.289
Dual form			576.8.d.i.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+446.148i q^{5} -1632.95 q^{7} +O(q^{10})$	$q+446.148i q^{5} -1632.95 q^{7} +7538.21i q^{11} -5165.38i q^{13} -2933.39 q^{17} -46958.3i q^{19} +50612.9 q^{23} -120923. q^{25} +16476.9i q^{29} -41854.4 q^{31} -728538. i q^{35} -342702. i q^{37} -642067. q^{41} -572494. i q^{43} -804266. q^{47} +1.84299e6 q^{49} +529950. i q^{53} -3.36316e6 q^{55} -1.71444e6i q^{59} +2.91477e6i q^{61} +2.30452e6 q^{65} -2.41857e6i q^{67} +2.16822e6 q^{71} +2.11408e6 q^{73} -1.23095e7i q^{77} +220374. q^{79} +3.13522e6i q^{83} -1.30873e6i q^{85} -3.97529e6 q^{89} +8.43481e6i q^{91} +2.09503e7 q^{95} +1.67890e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4872 q^{17} - 441972 q^{25} - 356664 q^{41} + 6446076 q^{49} - 11543616 q^{65} + 26806872 q^{73} - 45367560 q^{89} + 82412136 q^{97}+O(q^{100}) $ 12 * q - 4872 * q^17 - 441972 * q^25 - 356664 * q^41 + 6446076 * q^49 - 11543616 * q^65 + 26806872 * q^73 - 45367560 * q^89 + 82412136 * 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
$5$	446.148i	1.59619i	0.602533	+	0.798094i	$0.294158\pi$
$5$	446.148i	1.59619i	−0.602533	+	0.798094i	$0.705842\pi$
$6$	0	0
$7$	−1632.95	−1.79941	−0.899705	−	0.436499i	$-0.856218\pi$
$7$	−1632.95	−1.79941	−0.899705	+	0.436499i	$0.856218\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7538.21i	1.70763i	0.520576	+	0.853815i	$0.325717\pi$
$11$	7538.21i	1.70763i	−0.520576	+	0.853815i	$0.674283\pi$
$12$	0	0
$13$	− 5165.38i	− 0.652080i	−0.945356	−	0.326040i	$-0.894286\pi$
$13$	− 5165.38i	− 0.652080i	0.945356	−	0.326040i	$-0.105714\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2933.39	−0.144810	−0.0724051	−	0.997375i	$-0.523067\pi$
$17$	−2933.39	−0.144810	−0.0724051	+	0.997375i	$0.523067\pi$
$18$	0	0
$19$	− 46958.3i	− 1.57063i	−0.619095	−	0.785316i	$-0.712500\pi$
$19$	− 46958.3i	− 1.57063i	0.619095	−	0.785316i	$-0.287500\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	50612.9	0.867389	0.433694	−	0.901060i	$-0.357210\pi$
$23$	50612.9	0.867389	0.433694	+	0.901060i	$0.357210\pi$
$24$	0	0
$25$	−120923.	−1.54782
$26$	0	0
$27$	0	0
$28$	0	0
$29$	16476.9i	0.125453i	0.998031	+	0.0627267i	$0.0199796\pi$
$29$	16476.9i	0.125453i	−0.998031	+	0.0627267i	$0.980020\pi$
$30$	0	0
$31$	−41854.4	−0.252334	−0.126167	−	0.992009i	$-0.540267\pi$
$31$	−41854.4	−0.252334	−0.126167	+	0.992009i	$0.540267\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 728538.i	− 2.87220i
$36$	0	0
$37$	− 342702.i	− 1.11227i	−0.831092	−	0.556135i	$-0.812284\pi$
$37$	− 342702.i	− 1.11227i	0.831092	−	0.556135i	$-0.187716\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−642067.	−1.45491	−0.727456	−	0.686154i	$-0.759298\pi$
$41$	−642067.	−1.45491	−0.727456	+	0.686154i	$0.759298\pi$
$42$	0	0
$43$	− 572494.i	− 1.09807i	−0.835798	−	0.549036i	$-0.814995\pi$
$43$	− 572494.i	− 1.09807i	0.835798	−	0.549036i	$-0.185005\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−804266.	−1.12994	−0.564972	−	0.825110i	$-0.691113\pi$
$47$	−804266.	−1.12994	−0.564972	+	0.825110i	$0.691113\pi$
$48$	0	0
$49$	1.84299e6	2.23787
$50$	0	0
$51$	0	0
$52$	0	0
$53$	529950.i	0.488956i	0.969655	+	0.244478i	$0.0786165\pi$
$53$	529950.i	0.488956i	−0.969655	+	0.244478i	$0.921383\pi$
$54$	0	0
$55$	−3.36316e6	−2.72570
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.71444e6i	− 1.08678i	−0.839481	−	0.543389i	$-0.817141\pi$
$59$	− 1.71444e6i	− 1.08678i	0.839481	−	0.543389i	$-0.182859\pi$
$60$	0	0
$61$	2.91477e6i	1.64418i	0.569356	+	0.822091i	$0.307192\pi$
$61$	2.91477e6i	1.64418i	−0.569356	+	0.822091i	$0.692808\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.30452e6	1.04084
$66$	0	0
$67$	− 2.41857e6i	− 0.982420i	−0.871041	−	0.491210i	$-0.836555\pi$
$67$	− 2.41857e6i	− 0.982420i	0.871041	−	0.491210i	$-0.163445\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.16822e6	0.718949	0.359475	−	0.933155i	$-0.382956\pi$
$71$	2.16822e6	0.718949	0.359475	+	0.933155i	$0.382956\pi$
$72$	0	0
$73$	2.11408e6	0.636049	0.318025	−	0.948082i	$-0.396981\pi$
$73$	2.11408e6	0.636049	0.318025	+	0.948082i	$0.396981\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.23095e7i	− 3.07273i
$78$	0	0
$79$	220374.	0.0502881	0.0251441	−	0.999684i	$-0.491996\pi$
$79$	220374.	0.0502881	0.0251441	+	0.999684i	$0.491996\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.13522e6i	0.601858i	0.953646	+	0.300929i	$0.0972968\pi$
$83$	3.13522e6i	0.601858i	−0.953646	+	0.300929i	$0.902703\pi$
$84$	0	0
$85$	− 1.30873e6i	− 0.231144i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.97529e6	−0.597728	−0.298864	−	0.954296i	$-0.596608\pi$
$89$	−3.97529e6	−0.597728	−0.298864	+	0.954296i	$0.596608\pi$
$90$	0	0
$91$	8.43481e6i	1.17336i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.09503e7	2.50702
$96$	0	0
$97$	1.67890e7	1.86777	0.933883	−	0.357578i	$-0.116397\pi$
$97$	1.67890e7	1.86777	0.933883	+	0.357578i	$0.116397\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.91624e6i	0.571374i	0.958323	+	0.285687i	$0.0922218\pi$
$101$	5.91624e6i	0.571374i	−0.958323	+	0.285687i	$0.907778\pi$
$102$	0	0
$103$	−3.46876e6	−0.312783	−0.156392	−	0.987695i	$-0.549986\pi$
$103$	−3.46876e6	−0.312783	−0.156392	+	0.987695i	$0.549986\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.10858e7i	− 0.874827i	−0.899261	−	0.437413i	$-0.855895\pi$
$107$	− 1.10858e7i	− 0.874827i	0.899261	−	0.437413i	$-0.144105\pi$
$108$	0	0
$109$	7.63708e6i	0.564852i	0.959289	+	0.282426i	$0.0911391\pi$
$109$	7.63708e6i	0.564852i	−0.959289	+	0.282426i	$0.908861\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	436803.	0.0284781	0.0142390	−	0.999899i	$-0.495467\pi$
$113$	436803.	0.0284781	0.0142390	+	0.999899i	$0.495467\pi$
$114$	0	0
$115$	2.25809e7i	1.38452i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.79009e6	0.260573
$120$	0	0
$121$	−3.73374e7	−1.91600
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.90943e7i	− 0.874417i
$126$	0	0
$127$	−1.40357e7	−0.608025	−0.304013	−	0.952668i	$-0.598327\pi$
$127$	−1.40357e7	−0.608025	−0.304013	+	0.952668i	$0.598327\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.82637e6i	− 0.0709807i	−0.999370	−	0.0354903i	$-0.988701\pi$
$131$	− 1.82637e6i	− 0.0709807i	0.999370	−	0.0354903i	$-0.0112993\pi$
$132$	0	0
$133$	7.66806e7i	2.82621i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.46792e7	−0.487733	−0.243866	−	0.969809i	$-0.578416\pi$
$137$	−1.46792e7	−0.487733	−0.243866	+	0.969809i	$0.578416\pi$
$138$	0	0
$139$	2.12414e7i	0.670858i	0.942065	+	0.335429i	$0.108881\pi$
$139$	2.12414e7i	0.670858i	−0.942065	+	0.335429i	$0.891119\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.89377e7	1.11351
$144$	0	0
$145$	−7.35114e6	−0.200247
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.39677e7i	1.83185i	0.401348	+	0.915926i	$0.368542\pi$
$149$	7.39677e7i	1.83185i	−0.401348	+	0.915926i	$0.631458\pi$
$150$	0	0
$151$	−2.25560e7	−0.533143	−0.266571	−	0.963815i	$-0.585891\pi$
$151$	−2.25560e7	−0.533143	−0.266571	+	0.963815i	$0.585891\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.86733e7i	− 0.402772i
$156$	0	0
$157$	− 2.10790e7i	− 0.434712i	−0.976092	−	0.217356i	$-0.930257\pi$
$157$	− 2.10790e7i	− 0.434712i	0.976092	−	0.217356i	$-0.0697433\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.26484e7	−1.56079
$162$	0	0
$163$	− 8.22257e7i	− 1.48714i	−0.668660	−	0.743568i	$-0.733132\pi$
$163$	− 8.22257e7i	− 1.48714i	0.668660	−	0.743568i	$-0.266868\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.41401e7	−0.733375	−0.366687	−	0.930344i	$-0.619508\pi$
$167$	−4.41401e7	−0.733375	−0.366687	+	0.930344i	$0.619508\pi$
$168$	0	0
$169$	3.60674e7	0.574792
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.28432e7i	0.775939i	0.921672	+	0.387969i	$0.126823\pi$
$173$	5.28432e7i	0.775939i	−0.921672	+	0.387969i	$0.873177\pi$
$174$	0	0
$175$	1.97462e8	2.78515
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.35457e7i	− 0.697813i	−0.937158	−	0.348906i	$-0.886553\pi$
$179$	− 5.35457e7i	− 0.697813i	0.937158	−	0.348906i	$-0.113447\pi$
$180$	0	0
$181$	− 2.29886e7i	− 0.288163i	−0.989566	−	0.144081i	$-0.953977\pi$
$181$	− 2.29886e7i	− 0.288163i	0.989566	−	0.144081i	$-0.0460227\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.52896e8	1.77539
$186$	0	0
$187$	− 2.21125e7i	− 0.247282i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.47098e7	−0.568130	−0.284065	−	0.958805i	$-0.591683\pi$
$191$	−5.47098e7	−0.568130	−0.284065	+	0.958805i	$0.591683\pi$
$192$	0	0
$193$	2.87636e7	0.288000	0.144000	−	0.989578i	$-0.454003\pi$
$193$	2.87636e7	0.288000	0.144000	+	0.989578i	$0.454003\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.57182e7i	0.332857i	0.986053	+	0.166429i	$0.0532236\pi$
$197$	3.57182e7i	0.332857i	−0.986053	+	0.166429i	$0.946776\pi$
$198$	0	0
$199$	−5.51627e7	−0.496203	−0.248102	−	0.968734i	$-0.579807\pi$
$199$	−5.51627e7	−0.496203	−0.248102	+	0.968734i	$0.579807\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.69060e7i	− 0.225742i
$204$	0	0
$205$	− 2.86457e8i	− 2.32231i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.53981e8	2.68206
$210$	0	0
$211$	− 1.70446e8i	− 1.24910i	−0.780984	−	0.624551i	$-0.785282\pi$
$211$	− 1.70446e8i	− 1.24910i	0.780984	−	0.624551i	$-0.214718\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.55417e8	1.75273
$216$	0	0
$217$	6.83462e7	0.454052
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.51521e7i	0.0944278i
$222$	0	0
$223$	2.14524e8	1.29541	0.647706	−	0.761890i	$-0.275728\pi$
$223$	2.14524e8	1.29541	0.647706	+	0.761890i	$0.275728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.04532e8i	1.16057i	0.814413	+	0.580286i	$0.197059\pi$
$227$	2.04532e8i	1.16057i	−0.814413	+	0.580286i	$0.802941\pi$
$228$	0	0
$229$	6.39571e7i	0.351936i	0.984396	+	0.175968i	$0.0563056\pi$
$229$	6.39571e7i	0.351936i	−0.984396	+	0.175968i	$0.943694\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.11374e8	1.61264	0.806319	−	0.591481i	$-0.201457\pi$
$233$	3.11374e8	1.61264	0.806319	+	0.591481i	$0.201457\pi$
$234$	0	0
$235$	− 3.58822e8i	− 1.80360i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.97462e8	0.935599	0.467800	−	0.883835i	$-0.345047\pi$
$239$	1.97462e8	0.935599	0.467800	+	0.883835i	$0.345047\pi$
$240$	0	0
$241$	2.42051e8	1.11390	0.556951	−	0.830546i	$-0.311971\pi$
$241$	2.42051e8	1.11390	0.556951	+	0.830546i	$0.311971\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.22244e8i	3.57207i
$246$	0	0
$247$	−2.42557e8	−1.02418
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.04661e7i	0.121607i	0.998150	+	0.0608036i	$0.0193663\pi$
$251$	3.04661e7i	0.121607i	−0.998150	+	0.0608036i	$0.980634\pi$
$252$	0	0
$253$	3.81531e8i	1.48118i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.65386e8	−0.975240	−0.487620	−	0.873056i	$-0.662135\pi$
$257$	−2.65386e8	−0.975240	−0.487620	+	0.873056i	$0.662135\pi$
$258$	0	0
$259$	5.59615e8i	2.00143i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.78634e8	−1.96137	−0.980683	−	0.195605i	$-0.937333\pi$
$263$	−5.78634e8	−1.96137	−0.980683	+	0.195605i	$0.937333\pi$
$264$	0	0
$265$	−2.36436e8	−0.780465
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.03486e8i	0.637384i	0.947858	+	0.318692i	$0.103244\pi$
$269$	2.03486e8i	0.637384i	−0.947858	+	0.318692i	$0.896756\pi$
$270$	0	0
$271$	5.43999e8	1.66037	0.830187	−	0.557485i	$-0.188234\pi$
$271$	5.43999e8	1.66037	0.830187	+	0.557485i	$0.188234\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9.11544e8i	− 2.64310i
$276$	0	0
$277$	− 1.49580e8i	− 0.422858i	−0.977393	−	0.211429i	$-0.932188\pi$
$277$	− 1.49580e8i	− 0.422858i	0.977393	−	0.211429i	$-0.0678117\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.86281e7	0.265173	0.132586	−	0.991171i	$-0.457672\pi$
$281$	9.86281e7	0.265173	0.132586	+	0.991171i	$0.457672\pi$
$282$	0	0
$283$	1.13961e8i	0.298886i	0.988770	+	0.149443i	$0.0477481\pi$
$283$	1.13961e8i	0.298886i	−0.988770	+	0.149443i	$0.952252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.04846e9	2.61798
$288$	0	0
$289$	−4.01734e8	−0.979030
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.94610e8i	− 0.451989i	−0.974129	−	0.225995i	$-0.927437\pi$
$293$	− 1.94610e8i	− 0.451989i	0.974129	−	0.225995i	$-0.0725632\pi$
$294$	0	0
$295$	7.64896e8	1.73470
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.61435e8i	− 0.565607i
$300$	0	0
$301$	9.34854e8i	1.97588i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.30042e9	−2.62442
$306$	0	0
$307$	− 8.85804e8i	− 1.74724i	−0.486605	−	0.873622i	$-0.661765\pi$
$307$	− 8.85804e8i	− 1.74724i	0.486605	−	0.873622i	$-0.338235\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.31268e8	1.37853	0.689263	−	0.724511i	$-0.257934\pi$
$311$	7.31268e8	1.37853	0.689263	+	0.724511i	$0.257934\pi$
$312$	0	0
$313$	−7.29695e7	−0.134504	−0.0672521	−	0.997736i	$-0.521423\pi$
$313$	−7.29695e7	−0.134504	−0.0672521	+	0.997736i	$0.521423\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.44123e8i	1.13569i	0.823134	+	0.567847i	$0.192224\pi$
$317$	6.44123e8i	1.13569i	−0.823134	+	0.567847i	$0.807776\pi$
$318$	0	0
$319$	−1.24206e8	−0.214228
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.37747e8i	0.227444i
$324$	0	0
$325$	6.24614e8i	1.00930i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.31333e9	2.03323
$330$	0	0
$331$	3.55203e8i	0.538367i	0.963089	+	0.269183i	$0.0867537\pi$
$331$	3.55203e8i	0.538367i	−0.963089	+	0.269183i	$0.913246\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.07904e9	1.56813
$336$	0	0
$337$	9.33174e8	1.32818	0.664091	−	0.747651i	$-0.268819\pi$
$337$	9.33174e8	1.32818	0.664091	+	0.747651i	$0.268819\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.15507e8i	− 0.430893i
$342$	0	0
$343$	−1.66470e9	−2.22744
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.94810e8i	− 0.250298i	−0.992138	−	0.125149i	$-0.960059\pi$
$347$	− 1.94810e8i	− 0.250298i	0.992138	−	0.125149i	$-0.0399410\pi$
$348$	0	0
$349$	5.63457e8i	0.709532i	0.934955	+	0.354766i	$0.115439\pi$
$349$	5.63457e8i	0.709532i	−0.934955	+	0.354766i	$0.884561\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.20360e8	0.871642	0.435821	−	0.900033i	$-0.356458\pi$
$353$	7.20360e8	0.871642	0.435821	+	0.900033i	$0.356458\pi$
$354$	0	0
$355$	9.67345e8i	1.14758i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.11053e9	1.26678	0.633388	−	0.773835i	$-0.281664\pi$
$359$	1.11053e9	1.26678	0.633388	+	0.773835i	$0.281664\pi$
$360$	0	0
$361$	−1.31121e9	−1.46689
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.43191e8i	1.01525i
$366$	0	0
$367$	7.67438e8	0.810424	0.405212	−	0.914223i	$-0.367198\pi$
$367$	7.67438e8	0.810424	0.405212	+	0.914223i	$0.367198\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 8.65383e8i	− 0.879831i
$372$	0	0
$373$	7.09842e8i	0.708240i	0.935200	+	0.354120i	$0.115220\pi$
$373$	7.09842e8i	0.708240i	−0.935200	+	0.354120i	$0.884780\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.51094e7	0.0818056
$378$	0	0
$379$	8.45020e8i	0.797314i	0.917100	+	0.398657i	$0.130524\pi$
$379$	8.45020e8i	0.797314i	−0.917100	+	0.398657i	$0.869476\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.57877e8	−0.507391	−0.253695	−	0.967284i	$-0.581646\pi$
$383$	−5.57877e8	−0.507391	−0.253695	+	0.967284i	$0.581646\pi$
$384$	0	0
$385$	5.49187e9	4.90465
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.71789e9i	− 1.47970i	−0.672774	−	0.739848i	$-0.734897\pi$
$389$	− 1.71789e9i	− 1.47970i	0.672774	−	0.739848i	$-0.265103\pi$
$390$	0	0
$391$	−1.48468e8	−0.125607
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.83195e7i	0.0802693i
$396$	0	0
$397$	− 8.89044e7i	− 0.0713110i	−0.999364	−	0.0356555i	$-0.988648\pi$
$397$	− 8.89044e7i	− 0.0713110i	0.999364	−	0.0356555i	$-0.0113519\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.49306e9	1.15630	0.578151	−	0.815930i	$-0.303774\pi$
$401$	1.49306e9	1.15630	0.578151	+	0.815930i	$0.303774\pi$
$402$	0	0
$403$	2.16194e8i	0.164542i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.58336e9	1.89935
$408$	0	0
$409$	−5.78573e8	−0.418145	−0.209072	−	0.977900i	$-0.567044\pi$
$409$	−5.78573e8	−0.418145	−0.209072	+	0.977900i	$0.567044\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.79960e9i	1.95556i
$414$	0	0
$415$	−1.39877e9	−0.960679
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.06482e9i	− 0.707177i	−0.935401	−	0.353588i	$-0.884961\pi$
$419$	− 1.06482e9i	− 0.707177i	0.935401	−	0.353588i	$-0.115039\pi$
$420$	0	0
$421$	− 2.32538e9i	− 1.51882i	−0.650610	−	0.759412i	$-0.725487\pi$
$421$	− 2.32538e9i	− 1.51882i	0.650610	−	0.759412i	$-0.274513\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.54715e8	0.224140
$426$	0	0
$427$	− 4.75968e9i	− 2.95856i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.88162e6	−0.00474182	−0.00237091	−	0.999997i	$-0.500755\pi$
$431$	−7.88162e6	−0.00474182	−0.00237091	+	0.999997i	$0.500755\pi$
$432$	0	0
$433$	−1.60056e9	−0.947465	−0.473733	−	0.880669i	$-0.657094\pi$
$433$	−1.60056e9	−0.947465	−0.473733	+	0.880669i	$0.657094\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.37670e9i	− 1.36235i
$438$	0	0
$439$	2.74109e9	1.54632	0.773158	−	0.634214i	$-0.218676\pi$
$439$	2.74109e9	1.54632	0.773158	+	0.634214i	$0.218676\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.57747e8i	− 0.468755i	−0.972146	−	0.234378i	$-0.924695\pi$
$443$	− 8.57747e8i	− 0.468755i	0.972146	−	0.234378i	$-0.0753052\pi$
$444$	0	0
$445$	− 1.77357e9i	− 0.954086i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.02392e9	−1.57655	−0.788275	−	0.615323i	$-0.789025\pi$
$449$	−3.02392e9	−1.57655	−0.788275	+	0.615323i	$0.789025\pi$
$450$	0	0
$451$	− 4.84004e9i	− 2.48445i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.76318e9	−1.87290
$456$	0	0
$457$	−1.57050e9	−0.769718	−0.384859	−	0.922975i	$-0.625750\pi$
$457$	−1.57050e9	−0.769718	−0.384859	+	0.922975i	$0.625750\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.46063e9i	− 0.694366i	−0.937797	−	0.347183i	$-0.887138\pi$
$461$	− 1.46063e9i	− 0.694366i	0.937797	−	0.347183i	$-0.112862\pi$
$462$	0	0
$463$	−5.97897e8	−0.279958	−0.139979	−	0.990154i	$-0.544703\pi$
$463$	−5.97897e8	−0.279958	−0.139979	+	0.990154i	$0.544703\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.63504e9i	1.65158i	0.563976	+	0.825791i	$0.309271\pi$
$467$	3.63504e9i	1.65158i	−0.563976	+	0.825791i	$0.690729\pi$
$468$	0	0
$469$	3.94941e9i	1.76778i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.31558e9	1.87510
$474$	0	0
$475$	5.67834e9i	2.43105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.48056e9	−1.86277	−0.931383	−	0.364040i	$-0.881397\pi$
$479$	−4.48056e9	−1.86277	−0.931383	+	0.364040i	$0.881397\pi$
$480$	0	0
$481$	−1.77018e9	−0.725289
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.49036e9i	2.98131i
$486$	0	0
$487$	4.35035e9	1.70676	0.853381	−	0.521287i	$-0.174548\pi$
$487$	4.35035e9	1.70676	0.853381	+	0.521287i	$0.174548\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.60376e9i	− 1.37395i	−0.726681	−	0.686975i	$-0.758938\pi$
$491$	− 3.60376e9i	− 1.37395i	0.726681	−	0.686975i	$-0.241062\pi$
$492$	0	0
$493$	− 4.83332e7i	− 0.0181669i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.54059e9	−1.29368
$498$	0	0
$499$	1.08597e9i	0.391262i	0.980678	+	0.195631i	$0.0626754\pi$
$499$	1.08597e9i	0.391262i	−0.980678	+	0.195631i	$0.937325\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.97906e8	−0.104374	−0.0521868	−	0.998637i	$-0.516619\pi$
$503$	−2.97906e8	−0.104374	−0.0521868	+	0.998637i	$0.516619\pi$
$504$	0	0
$505$	−2.63952e9	−0.912021
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.79758e8i	− 0.127642i	−0.997961	−	0.0638211i	$-0.979671\pi$
$509$	− 3.79758e8i	− 0.127642i	0.997961	−	0.0638211i	$-0.0203287\pi$
$510$	0	0
$511$	−3.45218e9	−1.14451
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.54758e9i	− 0.499261i
$516$	0	0
$517$	− 6.06273e9i	− 1.92953i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.16629e9	−0.980887	−0.490444	−	0.871473i	$-0.663165\pi$
$521$	−3.16629e9	−0.980887	−0.490444	+	0.871473i	$0.663165\pi$
$522$	0	0
$523$	1.68667e9i	0.515554i	0.966204	+	0.257777i	$0.0829900\pi$
$523$	1.68667e9i	0.515554i	−0.966204	+	0.257777i	$0.917010\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.22775e8	0.0365405
$528$	0	0
$529$	−8.43159e8	−0.247636
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.31652e9i	0.948719i
$534$	0	0
$535$	4.94589e9	1.39639
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.38928e10i	3.82146i
$540$	0	0
$541$	− 3.90472e8i	− 0.106023i	−0.998594	−	0.0530114i	$-0.983118\pi$
$541$	− 3.90472e8i	− 0.106023i	0.998594	−	0.0530114i	$-0.0168820\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.40727e9	−0.901610
$546$	0	0
$547$	2.81706e9i	0.735936i	0.929839	+	0.367968i	$0.119946\pi$
$547$	2.81706e9i	0.735936i	−0.929839	+	0.367968i	$0.880054\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.73727e8	0.197041
$552$	0	0
$553$	−3.59860e8	−0.0904890
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.15717e9i	− 1.01930i	−0.860380	−	0.509652i	$-0.829774\pi$
$557$	− 4.15717e9i	− 1.01930i	0.860380	−	0.509652i	$-0.170226\pi$
$558$	0	0
$559$	−2.95715e9	−0.716031
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9.58429e8i	− 0.226350i	−0.993575	−	0.113175i	$-0.963898\pi$
$563$	− 9.58429e8i	− 0.226350i	0.993575	−	0.113175i	$-0.0361021\pi$
$564$	0	0
$565$	1.94879e8i	0.0454564i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.33925e9	1.21503	0.607516	−	0.794307i	$-0.292166\pi$
$569$	5.33925e9	1.21503	0.607516	+	0.794307i	$0.292166\pi$
$570$	0	0
$571$	1.92364e9i	0.432412i	0.976348	+	0.216206i	$0.0693683\pi$
$571$	1.92364e9i	0.432412i	−0.976348	+	0.216206i	$0.930632\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.12027e9	−1.34256
$576$	0	0
$577$	−3.39535e9	−0.735816	−0.367908	−	0.929862i	$-0.619926\pi$
$577$	−3.39535e9	−0.735816	−0.367908	+	0.929862i	$0.619926\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.11966e9i	− 1.08299i
$582$	0	0
$583$	−3.99488e9	−0.834955
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.82382e9i	0.780305i	0.920750	+	0.390152i	$0.127578\pi$
$587$	3.82382e9i	0.780305i	−0.920750	+	0.390152i	$0.872422\pi$
$588$	0	0
$589$	1.96541e9i	0.396323i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.62862e9	0.911508	0.455754	−	0.890106i	$-0.349370\pi$
$593$	4.62862e9	0.911508	0.455754	+	0.890106i	$0.349370\pi$
$594$	0	0
$595$	2.13709e9i	0.415923i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.55935e9	−0.486560	−0.243280	−	0.969956i	$-0.578223\pi$
$599$	−2.55935e9	−0.486560	−0.243280	+	0.969956i	$0.578223\pi$
$600$	0	0
$601$	2.00392e8	0.0376548	0.0188274	−	0.999823i	$-0.494007\pi$
$601$	2.00392e8	0.0376548	0.0188274	+	0.999823i	$0.494007\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.66580e10i	− 3.05830i
$606$	0	0
$607$	7.49379e8	0.136001	0.0680004	−	0.997685i	$-0.478338\pi$
$607$	7.49379e8	0.136001	0.0680004	+	0.997685i	$0.478338\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.15434e9i	0.736814i
$612$	0	0
$613$	− 5.66498e9i	− 0.993315i	−0.867947	−	0.496657i	$-0.834561\pi$
$613$	− 5.66498e9i	− 0.993315i	0.867947	−	0.496657i	$-0.165439\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.90759e9	1.52673	0.763365	−	0.645967i	$-0.223546\pi$
$617$	8.90759e9	1.52673	0.763365	+	0.645967i	$0.223546\pi$
$618$	0	0
$619$	− 6.04104e9i	− 1.02375i	−0.859060	−	0.511875i	$-0.828951\pi$
$619$	− 6.04104e9i	− 1.02375i	0.859060	−	0.511875i	$-0.171049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.49145e9	1.07556
$624$	0	0
$625$	−9.28233e8	−0.152082
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.00528e9i	0.161068i
$630$	0	0
$631$	−4.73797e8	−0.0750740	−0.0375370	−	0.999295i	$-0.511951\pi$
$631$	−4.73797e8	−0.0750740	−0.0375370	+	0.999295i	$0.511951\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.26201e9i	− 0.970523i
$636$	0	0
$637$	− 9.51972e9i	− 1.45927i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.16058e10	−1.74049	−0.870247	−	0.492615i	$-0.836041\pi$
$641$	−1.16058e10	−1.74049	−0.870247	+	0.492615i	$0.836041\pi$
$642$	0	0
$643$	− 8.03379e8i	− 0.119174i	−0.998223	−	0.0595871i	$-0.981022\pi$
$643$	− 8.03379e8i	− 0.119174i	0.998223	−	0.0595871i	$-0.0189784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.64870e9	0.239318	0.119659	−	0.992815i	$-0.461820\pi$
$647$	1.64870e9	0.239318	0.119659	+	0.992815i	$0.461820\pi$
$648$	0	0
$649$	1.29238e10	1.85582
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.72978e9i	− 0.945812i	−0.881113	−	0.472906i	$-0.843205\pi$
$653$	− 6.72978e9i	− 0.945812i	0.881113	−	0.472906i	$-0.156795\pi$
$654$	0	0
$655$	8.14833e8	0.113298
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 9.49369e9i	− 1.29222i	−0.763245	−	0.646109i	$-0.776395\pi$
$659$	− 9.49369e9i	− 1.29222i	0.763245	−	0.646109i	$-0.223605\pi$
$660$	0	0
$661$	1.24247e10i	1.67332i	0.547720	+	0.836662i	$0.315496\pi$
$661$	1.24247e10i	1.67332i	−0.547720	+	0.836662i	$0.684504\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.42109e10	−4.51116
$666$	0	0
$667$	8.33944e8i	0.108817i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.19721e10	−2.80765
$672$	0	0
$673$	6.94039e9	0.877670	0.438835	−	0.898568i	$-0.355391\pi$
$673$	6.94039e9	0.877670	0.438835	+	0.898568i	$0.355391\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.48654e10i	− 1.84126i	−0.390436	−	0.920630i	$-0.627676\pi$
$677$	− 1.48654e10i	− 1.84126i	0.390436	−	0.920630i	$-0.372324\pi$
$678$	0	0
$679$	−2.74155e10	−3.36088
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.84896e9i	0.582340i	0.956671	+	0.291170i	$0.0940445\pi$
$683$	4.84896e9i	0.582340i	−0.956671	+	0.291170i	$0.905956\pi$
$684$	0	0
$685$	− 6.54912e9i	− 0.778513i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.73739e9	0.318838
$690$	0	0
$691$	− 8.03331e9i	− 0.926236i	−0.886297	−	0.463118i	$-0.846731\pi$
$691$	− 8.03331e9i	− 0.926236i	0.886297	−	0.463118i	$-0.153269\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.47680e9	−1.07082
$696$	0	0
$697$	1.88344e9	0.210686
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.10940e9i	0.340929i	0.985364	+	0.170465i	$0.0545268\pi$
$701$	3.10940e9i	0.340929i	−0.985364	+	0.170465i	$0.945473\pi$
$702$	0	0
$703$	−1.60927e10	−1.74697
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.66092e9i	− 1.02814i
$708$	0	0
$709$	1.31275e10i	1.38332i	0.722225	+	0.691658i	$0.243119\pi$
$709$	1.31275e10i	1.38332i	−0.722225	+	0.691658i	$0.756881\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.11837e9	−0.218871
$714$	0	0
$715$	1.73720e10i	1.77737i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.85693e10	1.86314	0.931570	−	0.363562i	$-0.118439\pi$
$719$	1.85693e10	1.86314	0.931570	+	0.363562i	$0.118439\pi$
$720$	0	0
$721$	5.66431e9	0.562825
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.99244e9i	− 0.194179i
$726$	0	0
$727$	1.99655e9	0.192713	0.0963563	−	0.995347i	$-0.469281\pi$
$727$	1.99655e9	0.192713	0.0963563	+	0.995347i	$0.469281\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.67935e9i	0.159012i
$732$	0	0
$733$	1.88169e8i	0.0176475i	0.999961	+	0.00882376i	$0.00280873\pi$
$733$	1.88169e8i	0.0176475i	−0.999961	+	0.00882376i	$0.997191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.82317e10	1.67761
$738$	0	0
$739$	6.74247e9i	0.614559i	0.951619	+	0.307279i	$0.0994186\pi$
$739$	6.74247e9i	0.614559i	−0.951619	+	0.307279i	$0.900581\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.08296e9	−0.454627	−0.227314	−	0.973822i	$-0.572994\pi$
$743$	−5.08296e9	−0.454627	−0.227314	+	0.973822i	$0.572994\pi$
$744$	0	0
$745$	−3.30005e10	−2.92398
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.81025e10i	1.57417i
$750$	0	0
$751$	3.09724e9	0.266830	0.133415	−	0.991060i	$-0.457406\pi$
$751$	3.09724e9	0.266830	0.133415	+	0.991060i	$0.457406\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.00633e10i	− 0.850996i
$756$	0	0
$757$	− 2.37742e9i	− 0.199191i	−0.995028	−	0.0995955i	$-0.968245\pi$
$757$	− 2.37742e9i	− 0.199191i	0.995028	−	0.0995955i	$-0.0317549\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.24219e9	−0.266681	−0.133340	−	0.991070i	$-0.542570\pi$
$761$	−3.24219e9	−0.266681	−0.133340	+	0.991070i	$0.542570\pi$
$762$	0	0
$763$	− 1.24710e10i	− 1.01640i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.85575e9	−0.708666
$768$	0	0
$769$	−2.85908e9	−0.226717	−0.113359	−	0.993554i	$-0.536161\pi$
$769$	−2.85908e9	−0.226717	−0.113359	+	0.993554i	$0.536161\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.45287e10i	− 1.91006i	−0.296508	−	0.955030i	$-0.595822\pi$
$773$	− 2.45287e10i	− 1.91006i	0.296508	−	0.955030i	$-0.404178\pi$
$774$	0	0
$775$	5.06116e9	0.390566
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.01504e10i	2.28513i
$780$	0	0
$781$	1.63445e10i	1.22770i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.40437e9	0.693883
$786$	0	0
$787$	6.91999e9i	0.506050i	0.967460	+	0.253025i	$0.0814255\pi$
$787$	6.91999e9i	0.506050i	−0.967460	+	0.253025i	$0.918574\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.13278e8	−0.0512438
$792$	0	0
$793$	1.50559e10	1.07214
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.92150e10i	− 1.34443i	−0.740358	−	0.672213i	$-0.765344\pi$
$797$	− 1.92150e10i	− 1.34443i	0.740358	−	0.672213i	$-0.234656\pi$
$798$	0	0
$799$	2.35923e9	0.163628
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.59363e10i	1.08614i
$804$	0	0
$805$	− 3.68734e10i	− 2.49131i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.56032e10	−1.70010	−0.850049	−	0.526704i	$-0.823428\pi$
$809$	−2.56032e10	−1.70010	−0.850049	+	0.526704i	$0.823428\pi$
$810$	0	0
$811$	1.60576e10i	1.05708i	0.848908	+	0.528541i	$0.177261\pi$
$811$	1.60576e10i	1.05708i	−0.848908	+	0.528541i	$0.822739\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.66848e10	2.37375
$816$	0	0
$817$	−2.68833e10	−1.72467
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.23543e10i	1.40981i	0.709302	+	0.704905i	$0.249010\pi$
$821$	2.23543e10i	1.40981i	−0.709302	+	0.704905i	$0.750990\pi$
$822$	0	0
$823$	1.71952e10	1.07524	0.537622	−	0.843186i	$-0.319323\pi$
$823$	1.71952e10	1.07524	0.537622	+	0.843186i	$0.319323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.78313e9i	0.355544i	0.984072	+	0.177772i	$0.0568890\pi$
$827$	5.78313e9i	0.355544i	−0.984072	+	0.177772i	$0.943111\pi$
$828$	0	0
$829$	− 1.66129e10i	− 1.01276i	−0.862312	−	0.506378i	$-0.830984\pi$
$829$	− 1.66129e10i	− 1.01276i	0.862312	−	0.506378i	$-0.169016\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.40620e9	−0.324067
$834$	0	0
$835$	− 1.96930e10i	− 1.17060i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.55158e10	−0.906998	−0.453499	−	0.891257i	$-0.649824\pi$
$839$	−1.55158e10	−0.906998	−0.453499	+	0.891257i	$0.649824\pi$
$840$	0	0
$841$	1.69784e10	0.984261
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.60914e10i	0.917476i
$846$	0	0
$847$	6.09702e10	3.44767
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.73451e10i	− 0.964771i
$852$	0	0
$853$	− 4.23518e9i	− 0.233642i	−0.993153	−	0.116821i	$-0.962730\pi$
$853$	− 4.23518e9i	− 0.233642i	0.993153	−	0.116821i	$-0.0372704\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.69406e10	1.46209	0.731046	−	0.682329i	$-0.239033\pi$
$857$	2.69406e10	1.46209	0.731046	+	0.682329i	$0.239033\pi$
$858$	0	0
$859$	2.11033e10i	1.13599i	0.823032	+	0.567995i	$0.192281\pi$
$859$	2.11033e10i	1.13599i	−0.823032	+	0.567995i	$0.807719\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.32732e10	−0.702973	−0.351486	−	0.936193i	$-0.614324\pi$
$863$	−1.32732e10	−0.702973	−0.351486	+	0.936193i	$0.614324\pi$
$864$	0	0
$865$	−2.35759e10	−1.23854
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.66123e9i	0.0858736i
$870$	0	0
$871$	−1.24928e10	−0.640616
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.11801e10i	1.57343i
$876$	0	0
$877$	2.74442e10i	1.37389i	0.726708	+	0.686946i	$0.241049\pi$
$877$	2.74442e10i	1.37389i	−0.726708	+	0.686946i	$0.758951\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.14250e10	0.562910	0.281455	−	0.959574i	$-0.409183\pi$
$881$	1.14250e10	0.562910	0.281455	+	0.959574i	$0.409183\pi$
$882$	0	0
$883$	2.19187e10i	1.07140i	0.844407	+	0.535702i	$0.179953\pi$
$883$	2.19187e10i	1.07140i	−0.844407	+	0.535702i	$0.820047\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.66919e7	−0.00320878	−0.00160439	−	0.999999i	$-0.500511\pi$
$887$	−6.66919e7	−0.00320878	−0.00160439	+	0.999999i	$0.500511\pi$
$888$	0	0
$889$	2.29196e10	1.09409
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.77670e10i	1.77473i
$894$	0	0
$895$	2.38893e10	1.11384
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 6.89631e8i	− 0.0316561i
$900$	0	0
$901$	− 1.55455e9i	− 0.0708058i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.02563e10	0.459962
$906$	0	0
$907$	− 3.51074e9i	− 0.156233i	−0.996944	−	0.0781167i	$-0.975109\pi$
$907$	− 3.51074e9i	− 0.156233i	0.996944	−	0.0781167i	$-0.0248907\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.42261e10	−1.93805	−0.969024	−	0.246966i	$-0.920566\pi$
$911$	−4.42261e10	−1.93805	−0.969024	+	0.246966i	$0.920566\pi$
$912$	0	0
$913$	−2.36339e10	−1.02775
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.98238e9i	0.127723i
$918$	0	0
$919$	−3.16533e10	−1.34529	−0.672644	−	0.739967i	$-0.734841\pi$
$919$	−3.16533e10	−1.34529	−0.672644	+	0.739967i	$0.734841\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.11997e10i	− 0.468812i
$924$	0	0
$925$	4.14406e10i	1.72159i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.43472e10	1.40552	0.702760	−	0.711427i	$-0.251951\pi$
$929$	3.43472e10	1.40552	0.702760	+	0.711427i	$0.251951\pi$
$930$	0	0
$931$	− 8.65434e10i	− 3.51488i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.86547e9	0.394709
$936$	0	0
$937$	4.78118e10	1.89866	0.949328	−	0.314286i	$-0.101765\pi$
$937$	4.78118e10	1.89866	0.949328	+	0.314286i	$0.101765\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.04638e10i	1.58308i	0.611117	+	0.791540i	$0.290720\pi$
$941$	4.04638e10i	1.58308i	−0.611117	+	0.791540i	$0.709280\pi$
$942$	0	0
$943$	−3.24969e10	−1.26198
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.80486e10i	− 0.690588i	−0.938494	−	0.345294i	$-0.887779\pi$
$947$	− 1.80486e10i	− 0.690588i	0.938494	−	0.345294i	$-0.112221\pi$
$948$	0	0
$949$	− 1.09200e10i	− 0.414755i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.21258e10	0.453821	0.226911	−	0.973916i	$-0.427137\pi$
$953$	1.21258e10	0.453821	0.226911	+	0.973916i	$0.427137\pi$
$954$	0	0
$955$	− 2.44087e10i	− 0.906843i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.39705e10	0.877630
$960$	0	0
$961$	−2.57608e10	−0.936328
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.28328e10i	0.459703i
$966$	0	0
$967$	−8.21287e9	−0.292080	−0.146040	−	0.989279i	$-0.546653\pi$
$967$	−8.21287e9	−0.292080	−0.146040	+	0.989279i	$0.546653\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 4.68214e10i	− 1.64126i	−0.571460	−	0.820630i	$-0.693623\pi$
$971$	− 4.68214e10i	− 1.64126i	0.571460	−	0.820630i	$-0.306377\pi$
$972$	0	0
$973$	− 3.46861e10i	− 1.20715i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.65189e10	−0.909754	−0.454877	−	0.890554i	$-0.650317\pi$
$977$	−2.65189e10	−0.909754	−0.454877	+	0.890554i	$0.650317\pi$
$978$	0	0
$979$	− 2.99665e10i	− 1.02070i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.89532e10	1.64378	0.821890	−	0.569647i	$-0.192920\pi$
$983$	4.89532e10	1.64378	0.821890	+	0.569647i	$0.192920\pi$
$984$	0	0
$985$	−1.59356e10	−0.531303
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.89756e10i	− 0.952456i
$990$	0	0
$991$	1.45153e10	0.473770	0.236885	−	0.971538i	$-0.423873\pi$
$991$	1.45153e10	0.473770	0.236885	+	0.971538i	$0.423873\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.46107e10i	− 0.792034i
$996$	0	0
$997$	− 4.50489e10i	− 1.43963i	−0.694166	−	0.719815i	$-0.744227\pi$
$997$	− 4.50489e10i	− 1.43963i	0.694166	−	0.719815i	$-0.255773\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.8.d.i.289.11		12
3.2	odd	2		192.8.d.d.97.1	✓	12
4.3	odd	2	inner	576.8.d.i.289.12		12
8.3	odd	2	inner	576.8.d.i.289.2		12
8.5	even	2	inner	576.8.d.i.289.1		12
12.11	even	2		192.8.d.d.97.7	yes	12
24.5	odd	2		192.8.d.d.97.12	yes	12
24.11	even	2		192.8.d.d.97.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.8.d.d.97.1	✓	12	3.2	odd	2
192.8.d.d.97.6	yes	12	24.11	even	2
192.8.d.d.97.7	yes	12	12.11	even	2
192.8.d.d.97.12	yes	12	24.5	odd	2
576.8.d.i.289.1		12	8.5	even	2	inner
576.8.d.i.289.2		12	8.3	odd	2	inner
576.8.d.i.289.11		12	1.1	even	1	trivial
576.8.d.i.289.12		12	4.3	odd	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+446.148i q^{5} -1632.95 q^{7} +O(q^{10})\)	\(q+446.148i q^{5} -1632.95 q^{7} +7538.21i q^{11} -5165.38i q^{13} -2933.39 q^{17} -46958.3i q^{19} +50612.9 q^{23} -120923. q^{25} +16476.9i q^{29} -41854.4 q^{31} -728538. i q^{35} -342702. i q^{37} -642067. q^{41} -572494. i q^{43} -804266. q^{47} +1.84299e6 q^{49} +529950. i q^{53} -3.36316e6 q^{55} -1.71444e6i q^{59} +2.91477e6i q^{61} +2.30452e6 q^{65} -2.41857e6i q^{67} +2.16822e6 q^{71} +2.11408e6 q^{73} -1.23095e7i q^{77} +220374. q^{79} +3.13522e6i q^{83} -1.30873e6i q^{85} -3.97529e6 q^{89} +8.43481e6i q^{91} +2.09503e7 q^{95} +1.67890e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4872 q^{17} - 441972 q^{25} - 356664 q^{41} + 6446076 q^{49} - 11543616 q^{65} + 26806872 q^{73} - 45367560 q^{89} + 82412136 q^{97}+O(q^{100}) \) 12 * q - 4872 * q^17 - 441972 * q^25 - 356664 * q^41 + 6446076 * q^49 - 11543616 * q^65 + 26806872 * q^73 - 45367560 * q^89 + 82412136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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