LMFDB - Embedded newform 4608.2.f.p.2303.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2303,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4608.2303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.f (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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2303.4
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	4608.2303
Dual form			4608.2.f.p.2303.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03528 q^{5} +2.44949i q^{7} +O(q^{10})$	$q-1.03528 q^{5} +2.44949i q^{7} -1.46410i q^{11} +2.00000i q^{13} -3.48477i q^{17} +4.89898 q^{19} +0.535898 q^{23} -3.92820 q^{25} +3.86370 q^{29} -3.20736i q^{31} -2.53590i q^{35} +7.46410i q^{37} -0.656339i q^{41} +2.82843 q^{43} -7.46410 q^{47} +1.00000 q^{49} +11.5911 q^{53} +1.51575i q^{55} -6.92820i q^{59} +4.53590i q^{61} -2.07055i q^{65} -3.58630 q^{67} +10.3923 q^{71} -2.00000 q^{73} +3.58630 q^{77} -13.0053i q^{79} +13.4641i q^{83} +3.60770i q^{85} +1.41421i q^{89} -4.89898 q^{91} -5.07180 q^{95} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{23} + 24 q^{25} - 32 q^{47} + 8 q^{49} - 16 q^{73} - 96 q^{95}+O(q^{100}) $ 8 * q + 32 * q^23 + 24 * q^25 - 32 * q^47 + 8 * q^49 - 16 * q^73 - 96 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	−1.03528	−0.462990	−0.231495	−	0.972836i	$-0.574362\pi$
$5$	−1.03528	−0.462990	−0.231495	+	0.972836i	$0.574362\pi$
$6$	0	0
$7$	2.44949i	0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$	2.44949i	0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.46410i	− 0.441443i	−0.975337	−	0.220722i	$-0.929159\pi$
$11$	− 1.46410i	− 0.441443i	0.975337	−	0.220722i	$-0.0708412\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.48477i	− 0.845180i	−0.906321	−	0.422590i	$-0.861121\pi$
$17$	− 3.48477i	− 0.845180i	0.906321	−	0.422590i	$-0.138879\pi$
$18$	0	0
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	−3.92820	−0.785641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.86370	0.717472	0.358736	−	0.933439i	$-0.383208\pi$
$29$	3.86370	0.717472	0.358736	+	0.933439i	$0.383208\pi$
$30$	0	0
$31$	− 3.20736i	− 0.576060i	−0.957621	−	0.288030i	$-0.907000\pi$
$31$	− 3.20736i	− 0.576060i	0.957621	−	0.288030i	$-0.0930002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.53590i	− 0.428645i
$36$	0	0
$37$	7.46410i	1.22709i	0.789659	+	0.613545i	$0.210257\pi$
$37$	7.46410i	1.22709i	−0.789659	+	0.613545i	$0.789743\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.656339i	− 0.102503i	−0.998686	−	0.0512514i	$-0.983679\pi$
$41$	− 0.656339i	− 0.102503i	0.998686	−	0.0512514i	$-0.0163210\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.46410	−1.08875	−0.544376	−	0.838842i	$-0.683233\pi$
$47$	−7.46410	−1.08875	−0.544376	+	0.838842i	$0.683233\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.5911	1.59216	0.796081	−	0.605190i	$-0.206903\pi$
$53$	11.5911	1.59216	0.796081	+	0.605190i	$0.206903\pi$
$54$	0	0
$55$	1.51575i	0.204384i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.92820i	− 0.901975i	−0.892530	−	0.450988i	$-0.851072\pi$
$59$	− 6.92820i	− 0.901975i	0.892530	−	0.450988i	$-0.148928\pi$
$60$	0	0
$61$	4.53590i	0.580762i	0.956911	+	0.290381i	$0.0937821\pi$
$61$	4.53590i	0.580762i	−0.956911	+	0.290381i	$0.906218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.07055i	− 0.256820i
$66$	0	0
$67$	−3.58630	−0.438137	−0.219068	−	0.975710i	$-0.570302\pi$
$67$	−3.58630	−0.438137	−0.219068	+	0.975710i	$0.570302\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.58630	0.408697
$78$	0	0
$79$	− 13.0053i	− 1.46321i	−0.681727	−	0.731607i	$-0.738771\pi$
$79$	− 13.0053i	− 1.46321i	0.681727	−	0.731607i	$-0.261229\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.4641i	1.47788i	0.673773	+	0.738939i	$0.264673\pi$
$83$	13.4641i	1.47788i	−0.673773	+	0.738939i	$0.735327\pi$
$84$	0	0
$85$	3.60770i	0.391309i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.41421i	0.149906i	0.997187	+	0.0749532i	$0.0238807\pi$
$89$	1.41421i	0.149906i	−0.997187	+	0.0749532i	$0.976119\pi$
$90$	0	0
$91$	−4.89898	−0.513553
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.07180	−0.520355
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.79315	0.178425	0.0892126	−	0.996013i	$-0.471565\pi$
$101$	1.79315	0.178425	0.0892126	+	0.996013i	$0.471565\pi$
$102$	0	0
$103$	17.9043i	1.76416i	0.471096	+	0.882082i	$0.343858\pi$
$103$	17.9043i	1.76416i	−0.471096	+	0.882082i	$0.656142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9282i	1.05647i	0.849098	+	0.528235i	$0.177146\pi$
$107$	10.9282i	1.05647i	−0.849098	+	0.528235i	$0.822854\pi$
$108$	0	0
$109$	16.9282i	1.62143i	0.585443	+	0.810714i	$0.300921\pi$
$109$	16.9282i	1.62143i	−0.585443	+	0.810714i	$0.699079\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.2122i	1.05475i	0.849632	+	0.527376i	$0.176824\pi$
$113$	11.2122i	1.05475i	−0.849632	+	0.527376i	$0.823176\pi$
$114$	0	0
$115$	−0.554803	−0.0517356
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.53590	0.782485
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.24316	0.826733
$126$	0	0
$127$	− 8.86422i	− 0.786572i	−0.919416	−	0.393286i	$-0.871338\pi$
$127$	− 8.86422i	− 0.786572i	0.919416	−	0.393286i	$-0.128662\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 18.9282i	− 1.65376i	−0.562375	−	0.826882i	$-0.690112\pi$
$131$	− 18.9282i	− 1.65376i	0.562375	−	0.826882i	$-0.309888\pi$
$132$	0	0
$133$	12.0000i	1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.00052i	− 0.427223i	−0.976919	−	0.213611i	$-0.931477\pi$
$137$	− 5.00052i	− 0.427223i	0.976919	−	0.213611i	$-0.0685226\pi$
$138$	0	0
$139$	−2.07055	−0.175622	−0.0878110	−	0.996137i	$-0.527987\pi$
$139$	−2.07055	−0.175622	−0.0878110	+	0.996137i	$0.527987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.92820	0.244869
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.55103	0.208988	0.104494	−	0.994526i	$-0.466678\pi$
$149$	2.55103	0.208988	0.104494	+	0.994526i	$0.466678\pi$
$150$	0	0
$151$	14.5211i	1.18171i	0.806778	+	0.590854i	$0.201209\pi$
$151$	14.5211i	1.18171i	−0.806778	+	0.590854i	$0.798791\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.32051i	0.266710i
$156$	0	0
$157$	18.3923i	1.46787i	0.679222	+	0.733933i	$0.262317\pi$
$157$	18.3923i	1.46787i	−0.679222	+	0.733933i	$0.737683\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.31268i	0.103453i
$162$	0	0
$163$	−18.2832	−1.43205	−0.716027	−	0.698073i	$-0.754041\pi$
$163$	−18.2832	−1.43205	−0.716027	+	0.698073i	$0.754041\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.76268	0.666214	0.333107	−	0.942889i	$-0.391903\pi$
$173$	8.76268	0.666214	0.333107	+	0.942889i	$0.391903\pi$
$174$	0	0
$175$	− 9.62209i	− 0.727362i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.0000i	1.49487i	0.664335	+	0.747435i	$0.268715\pi$
$179$	20.0000i	1.49487i	−0.664335	+	0.747435i	$0.731285\pi$
$180$	0	0
$181$	22.7846i	1.69357i	0.531938	+	0.846783i	$0.321464\pi$
$181$	22.7846i	1.69357i	−0.531938	+	0.846783i	$0.678536\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7.72741i	− 0.568130i
$186$	0	0
$187$	−5.10205	−0.373099
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0	0
$193$	9.85641	0.709480	0.354740	−	0.934965i	$-0.384569\pi$
$193$	9.85641	0.709480	0.354740	+	0.934965i	$0.384569\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8343	1.48438	0.742190	−	0.670190i	$-0.233787\pi$
$197$	20.8343	1.48438	0.742190	+	0.670190i	$0.233787\pi$
$198$	0	0
$199$	3.96524i	0.281088i	0.990074	+	0.140544i	$0.0448852\pi$
$199$	3.96524i	0.281088i	−0.990074	+	0.140544i	$0.955115\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.46410i	0.664250i
$204$	0	0
$205$	0.679492i	0.0474578i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.17260i	− 0.496139i
$210$	0	0
$211$	19.0411	1.31084	0.655422	−	0.755263i	$-0.272491\pi$
$211$	19.0411	1.31084	0.655422	+	0.755263i	$0.272491\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.92820	−0.199702
$216$	0	0
$217$	7.85641	0.533328
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.96953	0.468821
$222$	0	0
$223$	− 13.7632i	− 0.921652i	−0.887491	−	0.460826i	$-0.847553\pi$
$223$	− 13.7632i	− 0.921652i	0.887491	−	0.460826i	$-0.152447\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.4641i	− 1.42462i	−0.701864	−	0.712311i	$-0.747648\pi$
$227$	− 21.4641i	− 1.42462i	0.701864	−	0.712311i	$-0.252352\pi$
$228$	0	0
$229$	7.07180i	0.467317i	0.972319	+	0.233659i	$0.0750699\pi$
$229$	7.07180i	0.467317i	−0.972319	+	0.233659i	$0.924930\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0101i	1.37642i	0.725512	+	0.688210i	$0.241603\pi$
$233$	21.0101i	1.37642i	−0.725512	+	0.688210i	$0.758397\pi$
$234$	0	0
$235$	7.72741	0.504080
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.7846	−1.60318	−0.801592	−	0.597872i	$-0.796013\pi$
$239$	−24.7846	−1.60318	−0.801592	+	0.597872i	$0.796013\pi$
$240$	0	0
$241$	3.07180	0.197872	0.0989359	−	0.995094i	$-0.468456\pi$
$241$	3.07180	0.197872	0.0989359	+	0.995094i	$0.468456\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.03528	−0.0661414
$246$	0	0
$247$	9.79796i	0.623429i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.53590i	− 0.412542i	−0.978495	−	0.206271i	$-0.933867\pi$
$251$	− 6.53590i	− 0.412542i	0.978495	−	0.206271i	$-0.0661329\pi$
$252$	0	0
$253$	− 0.784610i	− 0.0493280i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.8690i	1.05226i	0.850404	+	0.526130i	$0.176358\pi$
$257$	16.8690i	1.05226i	−0.850404	+	0.526130i	$0.823642\pi$
$258$	0	0
$259$	−18.2832	−1.13607
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.8343	1.27029	0.635144	−	0.772394i	$-0.280941\pi$
$269$	20.8343	1.27029	0.635144	+	0.772394i	$0.280941\pi$
$270$	0	0
$271$	− 13.7632i	− 0.836055i	−0.908434	−	0.418027i	$-0.862722\pi$
$271$	− 13.7632i	− 0.836055i	0.908434	−	0.418027i	$-0.137278\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.75129i	0.346816i
$276$	0	0
$277$	11.0718i	0.665240i	0.943061	+	0.332620i	$0.107933\pi$
$277$	11.0718i	0.665240i	−0.943061	+	0.332620i	$0.892067\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.38375i	0.500132i	0.968229	+	0.250066i	$0.0804524\pi$
$281$	8.38375i	0.500132i	−0.968229	+	0.250066i	$0.919548\pi$
$282$	0	0
$283$	−4.14110	−0.246163	−0.123082	−	0.992397i	$-0.539278\pi$
$283$	−4.14110	−0.246163	−0.123082	+	0.992397i	$0.539278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.60770	0.0948992
$288$	0	0
$289$	4.85641	0.285671
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.7322	0.919086	0.459543	−	0.888156i	$-0.348013\pi$
$293$	15.7322	0.919086	0.459543	+	0.888156i	$0.348013\pi$
$294$	0	0
$295$	7.17260i	0.417605i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.07180i	0.0619836i
$300$	0	0
$301$	6.92820i	0.399335i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.69591i	− 0.268887i
$306$	0	0
$307$	28.8391	1.64593	0.822966	−	0.568090i	$-0.192317\pi$
$307$	28.8391	1.64593	0.822966	+	0.568090i	$0.192317\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.7128	1.57145	0.785725	−	0.618576i	$-0.212290\pi$
$311$	27.7128	1.57145	0.785725	+	0.618576i	$0.212290\pi$
$312$	0	0
$313$	−23.7128	−1.34033	−0.670164	−	0.742213i	$-0.733776\pi$
$313$	−23.7128	−1.34033	−0.670164	+	0.742213i	$0.733776\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.0754	−0.565889	−0.282944	−	0.959136i	$-0.591311\pi$
$317$	−10.0754	−0.565889	−0.282944	+	0.959136i	$0.591311\pi$
$318$	0	0
$319$	− 5.65685i	− 0.316723i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 17.0718i	− 0.949900i
$324$	0	0
$325$	− 7.85641i	− 0.435795i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 18.2832i	− 1.00799i
$330$	0	0
$331$	32.9802	1.81275	0.906377	−	0.422469i	$-0.138837\pi$
$331$	32.9802	1.81275	0.906377	+	0.422469i	$0.138837\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.71281	0.202853
$336$	0	0
$337$	−4.14359	−0.225716	−0.112858	−	0.993611i	$-0.536001\pi$
$337$	−4.14359	−0.225716	−0.112858	+	0.993611i	$0.536001\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.69591	−0.254298
$342$	0	0
$343$	19.5959i	1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.2487i	− 1.62384i	−0.583772	−	0.811918i	$-0.698424\pi$
$347$	− 30.2487i	− 1.62384i	0.583772	−	0.811918i	$-0.301576\pi$
$348$	0	0
$349$	27.1769i	1.45475i	0.686242	+	0.727373i	$0.259259\pi$
$349$	27.1769i	1.45475i	−0.686242	+	0.727373i	$0.740741\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 23.8386i	− 1.26880i	−0.773006	−	0.634399i	$-0.781248\pi$
$353$	− 23.8386i	− 1.26880i	0.773006	−	0.634399i	$-0.218752\pi$
$354$	0	0
$355$	−10.7589	−0.571023
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.3205	−1.33637	−0.668183	−	0.743997i	$-0.732928\pi$
$359$	−25.3205	−1.33637	−0.668183	+	0.743997i	$0.732928\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.07055	0.108378
$366$	0	0
$367$	12.2474i	0.639312i	0.947534	+	0.319656i	$0.103567\pi$
$367$	12.2474i	0.639312i	−0.947534	+	0.319656i	$0.896433\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.3923i	1.47406i
$372$	0	0
$373$	10.3923i	0.538093i	0.963127	+	0.269047i	$0.0867086\pi$
$373$	10.3923i	0.538093i	−0.963127	+	0.269047i	$0.913291\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.72741i	0.397982i
$378$	0	0
$379$	0.757875	0.0389294	0.0194647	−	0.999811i	$-0.493804\pi$
$379$	0.757875	0.0389294	0.0194647	+	0.999811i	$0.493804\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.78461	0.448873	0.224436	−	0.974489i	$-0.427946\pi$
$383$	8.78461	0.448873	0.224436	+	0.974489i	$0.427946\pi$
$384$	0	0
$385$	−3.71281	−0.189222
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.55103	−0.129342	−0.0646711	−	0.997907i	$-0.520600\pi$
$389$	−2.55103	−0.129342	−0.0646711	+	0.997907i	$0.520600\pi$
$390$	0	0
$391$	− 1.86748i	− 0.0944425i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.4641i	0.677452i
$396$	0	0
$397$	14.3923i	0.722329i	0.932502	+	0.361165i	$0.117621\pi$
$397$	14.3923i	0.722329i	−0.932502	+	0.361165i	$0.882379\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.00052i	− 0.249714i	−0.992175	−	0.124857i	$-0.960153\pi$
$401$	− 5.00052i	− 0.249714i	0.992175	−	0.124857i	$-0.0398472\pi$
$402$	0	0
$403$	6.41473	0.319540
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9282	0.541691
$408$	0	0
$409$	1.07180	0.0529969	0.0264985	−	0.999649i	$-0.491564\pi$
$409$	1.07180	0.0529969	0.0264985	+	0.999649i	$0.491564\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.9706	0.835067
$414$	0	0
$415$	− 13.9391i	− 0.684242i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.5359i	− 1.29636i	−0.761486	−	0.648182i	$-0.775530\pi$
$419$	− 26.5359i	− 1.29636i	0.761486	−	0.648182i	$-0.224470\pi$
$420$	0	0
$421$	3.85641i	0.187950i	0.995575	+	0.0939749i	$0.0299573\pi$
$421$	3.85641i	0.187950i	−0.995575	+	0.0939749i	$0.970043\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.6889i	0.664008i
$426$	0	0
$427$	−11.1106	−0.537681
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.3205	0.641626	0.320813	−	0.947143i	$-0.396044\pi$
$431$	13.3205	0.641626	0.320813	+	0.947143i	$0.396044\pi$
$432$	0	0
$433$	23.8564	1.14647	0.573233	−	0.819393i	$-0.305689\pi$
$433$	23.8564	1.14647	0.573233	+	0.819393i	$0.305689\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.62536	0.125588
$438$	0	0
$439$	− 3.96524i	− 0.189251i	−0.995513	−	0.0946253i	$-0.969835\pi$
$439$	− 3.96524i	− 0.189251i	0.995513	−	0.0946253i	$-0.0301653\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.46410i	− 0.259607i	−0.991540	−	0.129804i	$-0.958565\pi$
$443$	− 5.46410i	− 0.259607i	0.991540	−	0.129804i	$-0.0414347\pi$
$444$	0	0
$445$	− 1.46410i	− 0.0694051i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.656339i	− 0.0309745i	−0.999880	−	0.0154873i	$-0.995070\pi$
$449$	− 0.656339i	− 0.0309745i	0.999880	−	0.0154873i	$-0.00492995\pi$
$450$	0	0
$451$	−0.960947	−0.0452492
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.07180	0.237769
$456$	0	0
$457$	−14.9282	−0.698312	−0.349156	−	0.937065i	$-0.613532\pi$
$457$	−14.9282	−0.698312	−0.349156	+	0.937065i	$0.613532\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.8744	−1.39139	−0.695694	−	0.718339i	$-0.744903\pi$
$461$	−29.8744	−1.39139	−0.695694	+	0.718339i	$0.744903\pi$
$462$	0	0
$463$	− 2.44949i	− 0.113837i	−0.998379	−	0.0569187i	$-0.981872\pi$
$463$	− 2.44949i	− 0.113837i	0.998379	−	0.0569187i	$-0.0181276\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.2487i	1.39974i	0.714269	+	0.699872i	$0.246760\pi$
$467$	30.2487i	1.39974i	−0.714269	+	0.699872i	$0.753240\pi$
$468$	0	0
$469$	− 8.78461i	− 0.405636i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.14110i	− 0.190408i
$474$	0	0
$475$	−19.2442	−0.882984
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	−14.9282	−0.680667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.17260	0.325691
$486$	0	0
$487$	− 8.10634i	− 0.367334i	−0.982989	−	0.183667i	$-0.941203\pi$
$487$	− 8.10634i	− 0.367334i	0.982989	−	0.183667i	$-0.0587967\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.78461i	− 0.215926i	−0.994155	−	0.107963i	$-0.965567\pi$
$491$	− 4.78461i	− 0.215926i	0.994155	−	0.107963i	$-0.0344329\pi$
$492$	0	0
$493$	− 13.4641i	− 0.606393i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.4558i	1.14185i
$498$	0	0
$499$	21.1117	0.945088	0.472544	−	0.881307i	$-0.343336\pi$
$499$	21.1117	0.945088	0.472544	+	0.881307i	$0.343336\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.4641	−1.22456	−0.612282	−	0.790640i	$-0.709748\pi$
$503$	−27.4641	−1.22456	−0.612282	+	0.790640i	$0.709748\pi$
$504$	0	0
$505$	−1.85641	−0.0826090
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.5911	−0.513767	−0.256883	−	0.966442i	$-0.582696\pi$
$509$	−11.5911	−0.513767	−0.256883	+	0.966442i	$0.582696\pi$
$510$	0	0
$511$	− 4.89898i	− 0.216718i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 18.5359i	− 0.816789i
$516$	0	0
$517$	10.9282i	0.480622i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 38.7386i	− 1.69717i	−0.529061	−	0.848584i	$-0.677456\pi$
$521$	− 38.7386i	− 1.69717i	0.529061	−	0.848584i	$-0.322544\pi$
$522$	0	0
$523$	−22.9791	−1.00481	−0.502404	−	0.864633i	$-0.667551\pi$
$523$	−22.9791	−1.00481	−0.502404	+	0.864633i	$0.667551\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.1769	−0.486874
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.31268	0.0568584
$534$	0	0
$535$	− 11.3137i	− 0.489134i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.46410i	− 0.0630633i
$540$	0	0
$541$	− 41.7128i	− 1.79337i	−0.442665	−	0.896687i	$-0.645967\pi$
$541$	− 41.7128i	− 1.79337i	0.442665	−	0.896687i	$-0.354033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 17.5254i	− 0.750704i
$546$	0	0
$547$	−6.96953	−0.297996	−0.148998	−	0.988838i	$-0.547605\pi$
$547$	−6.96953	−0.297996	−0.148998	+	0.988838i	$0.547605\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	31.8564	1.35467
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.7332	1.09035	0.545176	−	0.838321i	$-0.316463\pi$
$557$	25.7332	1.09035	0.545176	+	0.838321i	$0.316463\pi$
$558$	0	0
$559$	5.65685i	0.239259i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.3205i	1.32000i	0.751265	+	0.660001i	$0.229444\pi$
$563$	31.3205i	1.32000i	−0.751265	+	0.660001i	$0.770556\pi$
$564$	0	0
$565$	− 11.6077i	− 0.488339i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 10.6574i	− 0.446780i	−0.974729	−	0.223390i	$-0.928288\pi$
$569$	− 10.6574i	− 0.446780i	0.974729	−	0.223390i	$-0.0717124\pi$
$570$	0	0
$571$	−2.62536	−0.109868	−0.0549338	−	0.998490i	$-0.517495\pi$
$571$	−2.62536	−0.109868	−0.0549338	+	0.998490i	$0.517495\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.10512	−0.0877895
$576$	0	0
$577$	−5.85641	−0.243805	−0.121903	−	0.992542i	$-0.538900\pi$
$577$	−5.85641	−0.243805	−0.121903	+	0.992542i	$0.538900\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−32.9802	−1.36825
$582$	0	0
$583$	− 16.9706i	− 0.702849i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.0000i	0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$	20.0000i	0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$	0	0
$589$	− 15.7128i	− 0.647435i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 33.8396i	− 1.38963i	−0.719191	−	0.694813i	$-0.755487\pi$
$593$	− 33.8396i	− 1.38963i	0.719191	−	0.694813i	$-0.244513\pi$
$594$	0	0
$595$	−8.83701	−0.362282
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.535898	−0.0218962	−0.0109481	−	0.999940i	$-0.503485\pi$
$599$	−0.535898	−0.0218962	−0.0109481	+	0.999940i	$0.503485\pi$
$600$	0	0
$601$	21.8564	0.891541	0.445771	−	0.895147i	$-0.352930\pi$
$601$	21.8564	0.891541	0.445771	+	0.895147i	$0.352930\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.16883	−0.372766
$606$	0	0
$607$	29.2180i	1.18592i	0.805231	+	0.592961i	$0.202041\pi$
$607$	29.2180i	1.18592i	−0.805231	+	0.592961i	$0.797959\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 14.9282i	− 0.603930i
$612$	0	0
$613$	− 14.3923i	− 0.581300i	−0.956829	−	0.290650i	$-0.906129\pi$
$613$	− 14.3923i	− 0.581300i	0.956829	−	0.290650i	$-0.0938715\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.5563i	0.626275i	0.949708	+	0.313138i	$0.101380\pi$
$617$	15.5563i	0.626275i	−0.949708	+	0.313138i	$0.898620\pi$
$618$	0	0
$619$	16.9706	0.682105	0.341052	−	0.940044i	$-0.389217\pi$
$619$	16.9706	0.682105	0.341052	+	0.940044i	$0.389217\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.46410	−0.138786
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.0106	1.03711
$630$	0	0
$631$	7.34847i	0.292538i	0.989245	+	0.146269i	$0.0467265\pi$
$631$	7.34847i	0.292538i	−0.989245	+	0.146269i	$0.953274\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.17691i	0.364175i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.5670i	1.64180i	0.571074	+	0.820899i	$0.306527\pi$
$641$	41.5670i	1.64180i	−0.571074	+	0.820899i	$0.693473\pi$
$642$	0	0
$643$	−33.7381	−1.33050	−0.665249	−	0.746621i	$-0.731675\pi$
$643$	−33.7381	−1.33050	−0.665249	+	0.746621i	$0.731675\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.4641	−1.55149	−0.775747	−	0.631044i	$-0.782627\pi$
$647$	−39.4641	−1.55149	−0.775747	+	0.631044i	$0.782627\pi$
$648$	0	0
$649$	−10.1436	−0.398171
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.5216	−0.763939	−0.381969	−	0.924175i	$-0.624754\pi$
$653$	−19.5216	−0.763939	−0.381969	+	0.924175i	$0.624754\pi$
$654$	0	0
$655$	19.5959i	0.765676i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	0	0
$661$	− 36.5359i	− 1.42108i	−0.703656	−	0.710541i	$-0.748450\pi$
$661$	− 36.5359i	− 1.42108i	0.703656	−	0.710541i	$-0.251550\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 12.4233i	− 0.481755i
$666$	0	0
$667$	2.07055	0.0801721
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.64102	0.256374
$672$	0	0
$673$	−0.143594	−0.00553512	−0.00276756	−	0.999996i	$-0.500881\pi$
$673$	−0.143594	−0.00553512	−0.00276756	+	0.999996i	$0.500881\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.4607	−1.28600	−0.642999	−	0.765867i	$-0.722310\pi$
$677$	−33.4607	−1.28600	−0.642999	+	0.765867i	$0.722310\pi$
$678$	0	0
$679$	− 16.9706i	− 0.651270i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.39230i	0.321123i	0.987026	+	0.160561i	$0.0513304\pi$
$683$	8.39230i	0.321123i	−0.987026	+	0.160561i	$0.948670\pi$
$684$	0	0
$685$	5.17691i	0.197800i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.1822i	0.883172i
$690$	0	0
$691$	1.31268	0.0499366	0.0249683	−	0.999688i	$-0.492052\pi$
$691$	1.31268	0.0499366	0.0249683	+	0.999688i	$0.492052\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.14359	0.0813111
$696$	0	0
$697$	−2.28719	−0.0866334
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.5302	0.964261	0.482131	−	0.876099i	$-0.339863\pi$
$701$	25.5302	0.964261	0.482131	+	0.876099i	$0.339863\pi$
$702$	0	0
$703$	36.5665i	1.37913i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.39230i	0.165190i
$708$	0	0
$709$	− 0.143594i	− 0.00539277i	−0.999996	−	0.00269638i	$-0.999142\pi$
$709$	− 0.143594i	− 0.00539277i	0.999996	−	0.00269638i	$-0.000858287\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.71882i	− 0.0643704i
$714$	0	0
$715$	−3.03150	−0.113372
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.3923	1.87932	0.939658	−	0.342115i	$-0.111143\pi$
$719$	50.3923	1.87932	0.939658	+	0.342115i	$0.111143\pi$
$720$	0	0
$721$	−43.8564	−1.63330
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.1774	−0.563675
$726$	0	0
$727$	− 24.3190i	− 0.901943i	−0.892538	−	0.450971i	$-0.851078\pi$
$727$	− 24.3190i	− 0.901943i	0.892538	−	0.450971i	$-0.148922\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 9.85641i	− 0.364552i
$732$	0	0
$733$	− 4.14359i	− 0.153047i	−0.997068	−	0.0765236i	$-0.975618\pi$
$733$	− 4.14359i	− 0.153047i	0.997068	−	0.0765236i	$-0.0243820\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.25071i	0.193412i
$738$	0	0
$739$	−12.8295	−0.471939	−0.235970	−	0.971760i	$-0.575827\pi$
$739$	−12.8295	−0.471939	−0.235970	+	0.971760i	$0.575827\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.92820	−0.107425	−0.0537127	−	0.998556i	$-0.517106\pi$
$743$	−2.92820	−0.107425	−0.0537127	+	0.998556i	$0.517106\pi$
$744$	0	0
$745$	−2.64102	−0.0967593
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−26.7685	−0.978100
$750$	0	0
$751$	− 5.07484i	− 0.185184i	−0.995704	−	0.0925919i	$-0.970485\pi$
$751$	− 5.07484i	− 0.185184i	0.995704	−	0.0925919i	$-0.0295152\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 15.0333i	− 0.547119i
$756$	0	0
$757$	− 14.7846i	− 0.537356i	−0.963230	−	0.268678i	$-0.913413\pi$
$757$	− 14.7846i	− 0.537356i	0.963230	−	0.268678i	$-0.0865867\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.31319i	0.228853i	0.993432	+	0.114427i	$0.0365031\pi$
$761$	6.31319i	0.228853i	−0.993432	+	0.114427i	$0.963497\pi$
$762$	0	0
$763$	−41.4655	−1.50115
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8564	0.500326
$768$	0	0
$769$	31.7128	1.14359	0.571797	−	0.820395i	$-0.306247\pi$
$769$	31.7128	1.14359	0.571797	+	0.820395i	$0.306247\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.93426	−0.213440	−0.106720	−	0.994289i	$-0.534035\pi$
$773$	−5.93426	−0.213440	−0.106720	+	0.994289i	$0.534035\pi$
$774$	0	0
$775$	12.5992i	0.452576i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.21539i	− 0.115203i
$780$	0	0
$781$	− 15.2154i	− 0.544449i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 19.0411i	− 0.679607i
$786$	0	0
$787$	20.3538	0.725534	0.362767	−	0.931880i	$-0.381832\pi$
$787$	20.3538	0.725534	0.362767	+	0.931880i	$0.381832\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27.4641	−0.976511
$792$	0	0
$793$	−9.07180	−0.322149
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.8429	−0.950823	−0.475411	−	0.879764i	$-0.657701\pi$
$797$	−26.8429	−0.950823	−0.475411	+	0.879764i	$0.657701\pi$
$798$	0	0
$799$	26.0106i	0.920191i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.92820i	0.103334i
$804$	0	0
$805$	− 1.35898i	− 0.0478979i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.3650i	1.80590i	0.429750	+	0.902948i	$0.358602\pi$
$809$	51.3650i	1.80590i	−0.429750	+	0.902948i	$0.641398\pi$
$810$	0	0
$811$	17.3223	0.608268	0.304134	−	0.952629i	$-0.401633\pi$
$811$	17.3223	0.608268	0.304134	+	0.952629i	$0.401633\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.9282	0.663026
$816$	0	0
$817$	13.8564	0.484774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.4617	−1.51682	−0.758412	−	0.651776i	$-0.774024\pi$
$821$	−43.4617	−1.51682	−0.758412	+	0.651776i	$0.774024\pi$
$822$	0	0
$823$	− 31.0855i	− 1.08357i	−0.840516	−	0.541786i	$-0.817748\pi$
$823$	− 31.0855i	− 1.08357i	0.840516	−	0.541786i	$-0.182252\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16.0000i	− 0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$	− 16.0000i	− 0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$	0	0
$829$	− 23.0718i	− 0.801317i	−0.916228	−	0.400658i	$-0.868781\pi$
$829$	− 23.0718i	− 0.801317i	0.916228	−	0.400658i	$-0.131219\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.48477i	− 0.120740i
$834$	0	0
$835$	−5.25071	−0.181708
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.32051	0.0455890	0.0227945	−	0.999740i	$-0.492744\pi$
$839$	1.32051	0.0455890	0.0227945	+	0.999740i	$0.492744\pi$
$840$	0	0
$841$	−14.0718	−0.485234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.31749	−0.320531
$846$	0	0
$847$	21.6937i	0.745404i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.00000i	0.137118i
$852$	0	0
$853$	− 24.2487i	− 0.830260i	−0.909762	−	0.415130i	$-0.863736\pi$
$853$	− 24.2487i	− 0.830260i	0.909762	−	0.415130i	$-0.136264\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.7985i	0.505506i	0.967531	+	0.252753i	$0.0813360\pi$
$857$	14.7985i	0.505506i	−0.967531	+	0.252753i	$0.918664\pi$
$858$	0	0
$859$	43.5360	1.48543	0.742715	−	0.669608i	$-0.233538\pi$
$859$	43.5360	1.48543	0.742715	+	0.669608i	$0.233538\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.0718	1.26194	0.630969	−	0.775808i	$-0.282657\pi$
$863$	37.0718	1.26194	0.630969	+	0.775808i	$0.282657\pi$
$864$	0	0
$865$	−9.07180	−0.308450
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.0411	−0.645926
$870$	0	0
$871$	− 7.17260i	− 0.243034i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	22.6410i	0.765406i
$876$	0	0
$877$	− 28.2487i	− 0.953891i	−0.878933	−	0.476946i	$-0.841744\pi$
$877$	− 28.2487i	− 0.953891i	0.878933	−	0.476946i	$-0.158256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.7787i	1.60970i	0.593476	+	0.804852i	$0.297755\pi$
$881$	47.7787i	1.60970i	−0.593476	+	0.804852i	$0.702245\pi$
$882$	0	0
$883$	2.82843	0.0951842	0.0475921	−	0.998867i	$-0.484845\pi$
$883$	2.82843	0.0951842	0.0475921	+	0.998867i	$0.484845\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.7128	−1.19912	−0.599559	−	0.800330i	$-0.704658\pi$
$887$	−35.7128	−1.19912	−0.599559	+	0.800330i	$0.704658\pi$
$888$	0	0
$889$	21.7128	0.728224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−36.5665	−1.22365
$894$	0	0
$895$	− 20.7055i	− 0.692109i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 12.3923i	− 0.413307i
$900$	0	0
$901$	− 40.3923i	− 1.34566i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 23.5884i	− 0.784104i
$906$	0	0
$907$	42.4264	1.40875	0.704373	−	0.709830i	$-0.251228\pi$
$907$	42.4264	1.40875	0.704373	+	0.709830i	$0.251228\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.7846	−1.61631	−0.808153	−	0.588972i	$-0.799533\pi$
$911$	−48.7846	−1.61631	−0.808153	+	0.588972i	$0.799533\pi$
$912$	0	0
$913$	19.7128	0.652399
$914$	0	0
$915$	0	0
$916$	0	0
$917$	46.3644	1.53109
$918$	0	0
$919$	− 49.5718i	− 1.63522i	−0.575770	−	0.817611i	$-0.695298\pi$
$919$	− 49.5718i	− 1.63522i	0.575770	−	0.817611i	$-0.304702\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.7846i	0.684134i
$924$	0	0
$925$	− 29.3205i	− 0.964052i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 20.4553i	− 0.671118i	−0.942019	−	0.335559i	$-0.891075\pi$
$929$	− 20.4553i	− 0.671118i	0.942019	−	0.335559i	$-0.108925\pi$
$930$	0	0
$931$	4.89898	0.160558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.28203	0.172741
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.7038	−1.39210	−0.696052	−	0.717991i	$-0.745062\pi$
$941$	−42.7038	−1.39210	−0.696052	+	0.717991i	$0.745062\pi$
$942$	0	0
$943$	− 0.351731i	− 0.0114539i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 48.4974i	− 1.57595i	−0.615704	−	0.787977i	$-0.711128\pi$
$947$	− 48.4974i	− 1.57595i	0.615704	−	0.787977i	$-0.288872\pi$
$948$	0	0
$949$	− 4.00000i	− 0.129845i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 10.8604i	− 0.351804i	−0.984408	−	0.175902i	$-0.943716\pi$
$953$	− 10.8604i	− 0.351804i	0.984408	−	0.175902i	$-0.0562842\pi$
$954$	0	0
$955$	19.5959	0.634109
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.2487	0.395532
$960$	0	0
$961$	20.7128	0.668155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.2041	−0.328482
$966$	0	0
$967$	− 26.5927i	− 0.855162i	−0.903977	−	0.427581i	$-0.859366\pi$
$967$	− 26.5927i	− 0.855162i	0.903977	−	0.427581i	$-0.140634\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 39.0333i	− 1.25264i	−0.779567	−	0.626319i	$-0.784561\pi$
$971$	− 39.0333i	− 1.25264i	0.779567	−	0.626319i	$-0.215439\pi$
$972$	0	0
$973$	− 5.07180i	− 0.162594i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55.7091i	− 1.78229i	−0.453716	−	0.891147i	$-0.649902\pi$
$977$	− 55.7091i	− 1.78229i	0.453716	−	0.891147i	$-0.350098\pi$
$978$	0	0
$979$	2.07055	0.0661751
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−21.5692	−0.687252
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.51575	0.0481980
$990$	0	0
$991$	0.175865i	0.00558655i	0.999996	+	0.00279328i	$0.000889128\pi$
$991$	0.175865i	0.00558655i	−0.999996	+	0.00279328i	$0.999111\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.10512i	− 0.130141i
$996$	0	0
$997$	− 30.1051i	− 0.953439i	−0.879056	−	0.476719i	$-0.841826\pi$
$997$	− 30.1051i	− 0.953439i	0.879056	−	0.476719i	$-0.158174\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.f.p.2303.4		8
3.2	odd	2		4608.2.f.m.2303.6		8
4.3	odd	2		4608.2.f.m.2303.3		8
8.3	odd	2		4608.2.f.m.2303.5		8
8.5	even	2	inner	4608.2.f.p.2303.6		8
12.11	even	2	inner	4608.2.f.p.2303.5		8
16.3	odd	4		4608.2.c.i.4607.3	yes	4
16.5	even	4		4608.2.c.j.4607.2	yes	4
16.11	odd	4		4608.2.c.p.4607.2	yes	4
16.13	even	4		4608.2.c.o.4607.3	yes	4
24.5	odd	2		4608.2.f.m.2303.4		8
24.11	even	2	inner	4608.2.f.p.2303.3		8
48.5	odd	4		4608.2.c.p.4607.3	yes	4
48.11	even	4		4608.2.c.j.4607.3	yes	4
48.29	odd	4		4608.2.c.i.4607.2	✓	4
48.35	even	4		4608.2.c.o.4607.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.c.i.4607.2	✓	4	48.29	odd	4
4608.2.c.i.4607.3	yes	4	16.3	odd	4
4608.2.c.j.4607.2	yes	4	16.5	even	4
4608.2.c.j.4607.3	yes	4	48.11	even	4
4608.2.c.o.4607.2	yes	4	48.35	even	4
4608.2.c.o.4607.3	yes	4	16.13	even	4
4608.2.c.p.4607.2	yes	4	16.11	odd	4
4608.2.c.p.4607.3	yes	4	48.5	odd	4
4608.2.f.m.2303.3		8	4.3	odd	2
4608.2.f.m.2303.4		8	24.5	odd	2
4608.2.f.m.2303.5		8	8.3	odd	2
4608.2.f.m.2303.6		8	3.2	odd	2
4608.2.f.p.2303.3		8	24.11	even	2	inner
4608.2.f.p.2303.4		8	1.1	even	1	trivial
4608.2.f.p.2303.5		8	12.11	even	2	inner
4608.2.f.p.2303.6		8	8.5	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.03528 q^{5} +2.44949i q^{7} +O(q^{10})\)	\(q-1.03528 q^{5} +2.44949i q^{7} -1.46410i q^{11} +2.00000i q^{13} -3.48477i q^{17} +4.89898 q^{19} +0.535898 q^{23} -3.92820 q^{25} +3.86370 q^{29} -3.20736i q^{31} -2.53590i q^{35} +7.46410i q^{37} -0.656339i q^{41} +2.82843 q^{43} -7.46410 q^{47} +1.00000 q^{49} +11.5911 q^{53} +1.51575i q^{55} -6.92820i q^{59} +4.53590i q^{61} -2.07055i q^{65} -3.58630 q^{67} +10.3923 q^{71} -2.00000 q^{73} +3.58630 q^{77} -13.0053i q^{79} +13.4641i q^{83} +3.60770i q^{85} +1.41421i q^{89} -4.89898 q^{91} -5.07180 q^{95} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{23} + 24 q^{25} - 32 q^{47} + 8 q^{49} - 16 q^{73} - 96 q^{95}+O(q^{100}) \) 8 * q + 32 * q^23 + 24 * q^25 - 32 * q^47 + 8 * q^49 - 16 * q^73 - 96 * q^95

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