LMFDB - Embedded newform 4608.2.d.e.2305.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2305,4608)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("4608.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.d (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:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2305.3
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	4608.2305
Dual form			4608.2.d.e.2305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{5} -1.41421 q^{7} +O(q^{10})$	$q+2.44949i q^{5} -1.41421 q^{7} -3.46410i q^{11} +4.89898i q^{13} -4.00000 q^{17} -6.92820i q^{19} +5.65685 q^{23} -1.00000 q^{25} -2.44949i q^{29} -1.41421 q^{31} -3.46410i q^{35} -4.89898i q^{37} +4.00000 q^{41} -6.92820i q^{43} +5.65685 q^{47} -5.00000 q^{49} -7.34847i q^{53} +8.48528 q^{55} +13.8564i q^{59} +4.89898i q^{61} -12.0000 q^{65} +11.3137 q^{71} +4.00000 q^{73} +4.89898i q^{77} +7.07107 q^{79} +10.3923i q^{83} -9.79796i q^{85} +16.0000 q^{89} -6.92820i q^{91} +16.9706 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{17} - 4 q^{25} + 16 q^{41} - 20 q^{49} - 48 q^{65} + 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100}) $ 4 * q - 16 * q^17 - 4 * q^25 + 16 * q^41 - 20 * q^49 - 48 * q^65 + 16 * q^73 + 64 * q^89 + 24 * q^97

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
$5$	2.44949i	1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$	2.44949i	1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	4.89898i	1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$	4.89898i	1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.44949i	− 0.454859i	−0.973795	−	0.227429i	$-0.926968\pi$
$29$	− 2.44949i	− 0.454859i	0.973795	−	0.227429i	$-0.0730321\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.46410i	− 0.585540i
$36$	0	0
$37$	− 4.89898i	− 0.805387i	−0.915335	−	0.402694i	$-0.868074\pi$
$37$	− 4.89898i	− 0.805387i	0.915335	−	0.402694i	$-0.131926\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	− 6.92820i	− 1.05654i	−0.849076	−	0.528271i	$-0.822841\pi$
$43$	− 6.92820i	− 1.05654i	0.849076	−	0.528271i	$-0.177159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.34847i	− 1.00939i	−0.863298	−	0.504695i	$-0.831605\pi$
$53$	− 7.34847i	− 1.00939i	0.863298	−	0.504695i	$-0.168395\pi$
$54$	0	0
$55$	8.48528	1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.8564i	1.80395i	0.431788	+	0.901975i	$0.357883\pi$
$59$	13.8564i	1.80395i	−0.431788	+	0.901975i	$0.642117\pi$
$60$	0	0
$61$	4.89898i	0.627250i	0.949547	+	0.313625i	$0.101543\pi$
$61$	4.89898i	0.627250i	−0.949547	+	0.313625i	$0.898457\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.89898i	0.558291i
$78$	0	0
$79$	7.07107	0.795557	0.397779	−	0.917481i	$-0.369781\pi$
$79$	7.07107	0.795557	0.397779	+	0.917481i	$0.369781\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3923i	1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$	10.3923i	1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$	0	0
$85$	− 9.79796i	− 1.06274i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	− 6.92820i	− 0.726273i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.9706	1.74114
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.44949i	− 0.243733i	−0.992546	−	0.121867i	$-0.961112\pi$
$101$	− 2.44949i	− 0.243733i	0.992546	−	0.121867i	$-0.0388880\pi$
$102$	0	0
$103$	−18.3848	−1.81151	−0.905753	−	0.423806i	$-0.860694\pi$
$103$	−18.3848	−1.81151	−0.905753	+	0.423806i	$0.860694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.8564i	− 1.33955i	−0.742564	−	0.669775i	$-0.766391\pi$
$107$	− 13.8564i	− 1.33955i	0.742564	−	0.669775i	$-0.233609\pi$
$108$	0	0
$109$	− 14.6969i	− 1.40771i	−0.710343	−	0.703856i	$-0.751460\pi$
$109$	− 14.6969i	− 1.40771i	0.710343	−	0.703856i	$-0.248540\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	13.8564i	1.29212i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.65685	0.518563
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.79796i	0.876356i
$126$	0	0
$127$	−9.89949	−0.878438	−0.439219	−	0.898380i	$-0.644745\pi$
$127$	−9.89949	−0.878438	−0.439219	+	0.898380i	$0.644745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	9.79796i	0.849591i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	0	0
$139$	13.8564i	1.17529i	0.809121	+	0.587643i	$0.199944\pi$
$139$	13.8564i	1.17529i	−0.809121	+	0.587643i	$0.800056\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.9706	1.41915
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.0454i	1.80603i	0.429609	+	0.903015i	$0.358651\pi$
$149$	22.0454i	1.80603i	−0.429609	+	0.903015i	$0.641349\pi$
$150$	0	0
$151$	18.3848	1.49613	0.748066	−	0.663624i	$-0.230983\pi$
$151$	18.3848	1.49613	0.748066	+	0.663624i	$0.230983\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.46410i	− 0.278243i
$156$	0	0
$157$	− 4.89898i	− 0.390981i	−0.980706	−	0.195491i	$-0.937370\pi$
$157$	− 4.89898i	− 0.390981i	0.980706	−	0.195491i	$-0.0626299\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	− 6.92820i	− 0.542659i	−0.962487	−	0.271329i	$-0.912537\pi$
$163$	− 6.92820i	− 0.542659i	0.962487	−	0.271329i	$-0.0874633\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.9706	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$167$	16.9706	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 17.1464i	− 1.30362i	−0.758383	−	0.651809i	$-0.774010\pi$
$173$	− 17.1464i	− 1.30362i	0.758383	−	0.651809i	$-0.225990\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 13.8564i	− 1.03568i	−0.855479	−	0.517838i	$-0.826737\pi$
$179$	− 13.8564i	− 1.03568i	0.855479	−	0.517838i	$-0.173263\pi$
$180$	0	0
$181$	24.4949i	1.82069i	0.413849	+	0.910346i	$0.364184\pi$
$181$	24.4949i	1.82069i	−0.413849	+	0.910346i	$0.635816\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	13.8564i	1.01328i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.34847i	− 0.523557i	−0.965128	−	0.261778i	$-0.915691\pi$
$197$	− 7.34847i	− 0.523557i	0.965128	−	0.261778i	$-0.0843089\pi$
$198$	0	0
$199$	−18.3848	−1.30326	−0.651631	−	0.758536i	$-0.725915\pi$
$199$	−18.3848	−1.30326	−0.651631	+	0.758536i	$0.725915\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.46410i	0.243132i
$204$	0	0
$205$	9.79796i	0.684319i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	13.8564i	0.953914i	0.878927	+	0.476957i	$0.158260\pi$
$211$	13.8564i	0.953914i	−0.878927	+	0.476957i	$0.841740\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.9706	1.15738
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 19.5959i	− 1.31816i
$222$	0	0
$223$	−7.07107	−0.473514	−0.236757	−	0.971569i	$-0.576084\pi$
$223$	−7.07107	−0.473514	−0.236757	+	0.971569i	$0.576084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.3923i	− 0.689761i	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	− 10.3923i	− 0.689761i	0.938647	−	0.344881i	$-0.112081\pi$
$228$	0	0
$229$	4.89898i	0.323734i	0.986813	+	0.161867i	$0.0517515\pi$
$229$	4.89898i	0.323734i	−0.986813	+	0.161867i	$0.948248\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	13.8564i	0.903892i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.2474i	− 0.782461i
$246$	0	0
$247$	33.9411	2.15962
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.46410i	− 0.218652i	−0.994006	−	0.109326i	$-0.965131\pi$
$251$	− 3.46410i	− 0.218652i	0.994006	−	0.109326i	$-0.0348693\pi$
$252$	0	0
$253$	− 19.5959i	− 1.23198i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	6.92820i	0.430498i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.2474i	0.746740i	0.927682	+	0.373370i	$0.121798\pi$
$269$	12.2474i	0.746740i	−0.927682	+	0.373370i	$0.878202\pi$
$270$	0	0
$271$	26.8701	1.63224	0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701	1.63224	0.816120	+	0.577883i	$0.196121\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46410i	0.208893i
$276$	0	0
$277$	4.89898i	0.294351i	0.989110	+	0.147176i	$0.0470182\pi$
$277$	4.89898i	0.294351i	−0.989110	+	0.147176i	$0.952982\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	13.8564i	0.823678i	0.911257	+	0.411839i	$0.135113\pi$
$283$	13.8564i	0.823678i	−0.911257	+	0.411839i	$0.864887\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.34847i	− 0.429302i	−0.976691	−	0.214651i	$-0.931139\pi$
$293$	− 7.34847i	− 0.429302i	0.976691	−	0.214651i	$-0.0688614\pi$
$294$	0	0
$295$	−33.9411	−1.97613
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.7128i	1.60267i
$300$	0	0
$301$	9.79796i	0.564745i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	− 27.7128i	− 1.58165i	−0.612040	−	0.790827i	$-0.709651\pi$
$307$	− 27.7128i	− 1.58165i	0.612040	−	0.790827i	$-0.290349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.9411	1.92462	0.962312	−	0.271947i	$-0.0876674\pi$
$311$	33.9411	1.92462	0.962312	+	0.271947i	$0.0876674\pi$
$312$	0	0
$313$	12.0000	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.8434i	1.78850i	0.447566	+	0.894251i	$0.352291\pi$
$317$	31.8434i	1.78850i	−0.447566	+	0.894251i	$0.647709\pi$
$318$	0	0
$319$	−8.48528	−0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.7128i	1.54198i
$324$	0	0
$325$	− 4.89898i	− 0.271746i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	− 27.7128i	− 1.52323i	−0.648027	−	0.761617i	$-0.724406\pi$
$331$	− 27.7128i	− 1.52323i	0.648027	−	0.761617i	$-0.275594\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.89898i	0.265295i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.2487i	− 1.30174i	−0.759190	−	0.650870i	$-0.774404\pi$
$347$	− 24.2487i	− 1.30174i	0.759190	−	0.650870i	$-0.225596\pi$
$348$	0	0
$349$	4.89898i	0.262236i	0.991367	+	0.131118i	$0.0418567\pi$
$349$	4.89898i	0.262236i	−0.991367	+	0.131118i	$0.958143\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.0000	−1.70319	−0.851594	−	0.524202i	$-0.824364\pi$
$353$	−32.0000	−1.70319	−0.851594	+	0.524202i	$0.824364\pi$
$354$	0	0
$355$	27.7128i	1.47084i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.79796i	0.512849i
$366$	0	0
$367$	−1.41421	−0.0738213	−0.0369107	−	0.999319i	$-0.511752\pi$
$367$	−1.41421	−0.0738213	−0.0369107	+	0.999319i	$0.511752\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3923i	0.539542i
$372$	0	0
$373$	− 34.2929i	− 1.77562i	−0.460213	−	0.887808i	$-0.652227\pi$
$373$	− 34.2929i	− 1.77562i	0.460213	−	0.887808i	$-0.347773\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 34.6410i	− 1.77939i	−0.456556	−	0.889695i	$-0.650917\pi$
$379$	− 34.6410i	− 1.77939i	0.456556	−	0.889695i	$-0.349083\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.9706	0.867155	0.433578	−	0.901116i	$-0.357251\pi$
$383$	16.9706	0.867155	0.433578	+	0.901116i	$0.357251\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.9444i	1.36613i	0.730355	+	0.683067i	$0.239354\pi$
$389$	26.9444i	1.36613i	−0.730355	+	0.683067i	$0.760646\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.3205i	0.871489i
$396$	0	0
$397$	− 24.4949i	− 1.22936i	−0.788775	−	0.614682i	$-0.789284\pi$
$397$	− 24.4949i	− 1.22936i	0.788775	−	0.614682i	$-0.210716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	0	0
$403$	− 6.92820i	− 0.345118i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 19.5959i	− 0.964252i
$414$	0	0
$415$	−25.4558	−1.24958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.3205i	0.846162i	0.906092	+	0.423081i	$0.139051\pi$
$419$	17.3205i	0.846162i	−0.906092	+	0.423081i	$0.860949\pi$
$420$	0	0
$421$	− 14.6969i	− 0.716285i	−0.933667	−	0.358142i	$-0.883410\pi$
$421$	− 14.6969i	− 0.716285i	0.933667	−	0.358142i	$-0.116590\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	− 6.92820i	− 0.335279i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.65685	−0.272481	−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685	−0.272481	−0.136241	+	0.990676i	$0.543502\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 39.1918i	− 1.87480i
$438$	0	0
$439$	26.8701	1.28244	0.641219	−	0.767358i	$-0.278429\pi$
$439$	26.8701	1.28244	0.641219	+	0.767358i	$0.278429\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.2487i	− 1.15209i	−0.817418	−	0.576046i	$-0.804595\pi$
$443$	− 24.2487i	− 1.15209i	0.817418	−	0.576046i	$-0.195405\pi$
$444$	0	0
$445$	39.1918i	1.85787i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	− 13.8564i	− 0.652473i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.9706	0.795592
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.44949i	0.114084i	0.998372	+	0.0570421i	$0.0181669\pi$
$461$	2.44949i	0.114084i	−0.998372	+	0.0570421i	$0.981833\pi$
$462$	0	0
$463$	1.41421	0.0657241	0.0328620	−	0.999460i	$-0.489538\pi$
$463$	1.41421	0.0657241	0.0328620	+	0.999460i	$0.489538\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 38.1051i	− 1.76329i	−0.471909	−	0.881647i	$-0.656435\pi$
$467$	− 38.1051i	− 1.76329i	0.471909	−	0.881647i	$-0.343565\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.6969i	0.667354i
$486$	0	0
$487$	−24.0416	−1.08943	−0.544715	−	0.838621i	$-0.683362\pi$
$487$	−24.0416	−1.08943	−0.544715	+	0.838621i	$0.683362\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	9.79796i	0.441278i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	− 13.8564i	− 0.620298i	−0.950688	−	0.310149i	$-0.899621\pi$
$499$	− 13.8564i	− 0.620298i	0.950688	−	0.310149i	$-0.100379\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.6274	−1.00891	−0.504453	−	0.863439i	$-0.668306\pi$
$503$	−22.6274	−1.00891	−0.504453	+	0.863439i	$0.668306\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.0454i	− 0.977146i	−0.872523	−	0.488573i	$-0.837518\pi$
$509$	− 22.0454i	− 0.977146i	0.872523	−	0.488573i	$-0.162482\pi$
$510$	0	0
$511$	−5.65685	−0.250244
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 45.0333i	− 1.98441i
$516$	0	0
$517$	− 19.5959i	− 0.861827i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	− 20.7846i	− 0.908848i	−0.890786	−	0.454424i	$-0.849845\pi$
$523$	− 20.7846i	− 0.908848i	0.890786	−	0.454424i	$-0.150155\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.65685	0.246416
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	19.5959i	0.848793i
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.3205i	0.746047i
$540$	0	0
$541$	14.6969i	0.631871i	0.948781	+	0.315935i	$0.102318\pi$
$541$	14.6969i	0.631871i	−0.948781	+	0.315935i	$0.897682\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	36.0000	1.54207
$546$	0	0
$547$	− 34.6410i	− 1.48114i	−0.671978	−	0.740571i	$-0.734555\pi$
$547$	− 34.6410i	− 1.48114i	0.671978	−	0.740571i	$-0.265445\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.9706	−0.722970
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.8434i	− 1.34925i	−0.738162	−	0.674623i	$-0.764306\pi$
$557$	− 31.8434i	− 1.34925i	0.738162	−	0.674623i	$-0.235694\pi$
$558$	0	0
$559$	33.9411	1.43556
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 10.3923i	− 0.437983i	−0.975727	−	0.218992i	$-0.929723\pi$
$563$	− 10.3923i	− 0.437983i	0.975727	−	0.218992i	$-0.0702768\pi$
$564$	0	0
$565$	− 19.5959i	− 0.824406i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.6969i	− 0.609732i
$582$	0	0
$583$	−25.4558	−1.05427
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	9.79796i	0.403718i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	0	0
$595$	13.8564i	0.568057i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.44949i	− 0.0995859i
$606$	0	0
$607$	24.0416	0.975820	0.487910	−	0.872894i	$-0.337759\pi$
$607$	24.0416	0.975820	0.487910	+	0.872894i	$0.337759\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.7128i	1.12114i
$612$	0	0
$613$	44.0908i	1.78081i	0.455168	+	0.890406i	$0.349579\pi$
$613$	44.0908i	1.78081i	−0.455168	+	0.890406i	$0.650421\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.0000	−0.644136	−0.322068	−	0.946717i	$-0.604378\pi$
$617$	−16.0000	−0.644136	−0.322068	+	0.946717i	$0.604378\pi$
$618$	0	0
$619$	27.7128i	1.11387i	0.830555	+	0.556936i	$0.188023\pi$
$619$	27.7128i	1.11387i	−0.830555	+	0.556936i	$0.811977\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22.6274	−0.906548
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.5959i	0.781340i
$630$	0	0
$631$	9.89949	0.394093	0.197046	−	0.980394i	$-0.436865\pi$
$631$	9.89949	0.394093	0.197046	+	0.980394i	$0.436865\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 24.2487i	− 0.962281i
$636$	0	0
$637$	− 24.4949i	− 0.970523i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	0	0
$643$	− 20.7846i	− 0.819665i	−0.912161	−	0.409832i	$-0.865587\pi$
$643$	− 20.7846i	− 0.819665i	0.912161	−	0.409832i	$-0.134413\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.2843	−1.11197	−0.555985	−	0.831193i	$-0.687659\pi$
$647$	−28.2843	−1.11197	−0.555985	+	0.831193i	$0.687659\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.0454i	0.862703i	0.902184	+	0.431352i	$0.141963\pi$
$653$	22.0454i	0.862703i	−0.902184	+	0.431352i	$0.858037\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.7128i	1.07954i	0.841813	+	0.539769i	$0.181488\pi$
$659$	27.7128i	1.07954i	−0.841813	+	0.539769i	$0.818512\pi$
$660$	0	0
$661$	44.0908i	1.71493i	0.514539	+	0.857467i	$0.327963\pi$
$661$	44.0908i	1.71493i	−0.514539	+	0.857467i	$0.672037\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	− 13.8564i	− 0.536522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.1464i	− 0.658991i	−0.944157	−	0.329495i	$-0.893121\pi$
$677$	− 17.1464i	− 0.658991i	0.944157	−	0.329495i	$-0.106879\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.46410i	− 0.132550i	−0.997801	−	0.0662751i	$-0.978889\pi$
$683$	− 3.46410i	− 0.132550i	0.997801	−	0.0662751i	$-0.0211115\pi$
$684$	0	0
$685$	48.9898i	1.87180i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	20.7846i	0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$691$	20.7846i	0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.9411	−1.28746
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.8434i	1.20271i	0.798983	+	0.601354i	$0.205372\pi$
$701$	31.8434i	1.20271i	−0.798983	+	0.601354i	$0.794628\pi$
$702$	0	0
$703$	−33.9411	−1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.46410i	0.130281i
$708$	0	0
$709$	− 14.6969i	− 0.551955i	−0.961164	−	0.275978i	$-0.910998\pi$
$709$	− 14.6969i	− 0.551955i	0.961164	−	0.275978i	$-0.0890015\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	41.5692i	1.55460i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6274	−0.843860	−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274	−0.843860	−0.421930	+	0.906628i	$0.638647\pi$
$720$	0	0
$721$	26.0000	0.968291
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.44949i	0.0909718i
$726$	0	0
$727$	7.07107	0.262251	0.131126	−	0.991366i	$-0.458141\pi$
$727$	7.07107	0.262251	0.131126	+	0.991366i	$0.458141\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	27.7128i	1.02500i
$732$	0	0
$733$	− 24.4949i	− 0.904740i	−0.891830	−	0.452370i	$-0.850579\pi$
$733$	− 24.4949i	− 0.904740i	0.891830	−	0.452370i	$-0.149421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.9706	−0.622590	−0.311295	−	0.950313i	$-0.600763\pi$
$743$	−16.9706	−0.622590	−0.311295	+	0.950313i	$0.600763\pi$
$744$	0	0
$745$	−54.0000	−1.97841
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.5959i	0.716019i
$750$	0	0
$751$	−15.5563	−0.567659	−0.283830	−	0.958875i	$-0.591605\pi$
$751$	−15.5563	−0.567659	−0.283830	+	0.958875i	$0.591605\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.0333i	1.63893i
$756$	0	0
$757$	4.89898i	0.178056i	0.996029	+	0.0890282i	$0.0283761\pi$
$757$	4.89898i	0.178056i	−0.996029	+	0.0890282i	$0.971624\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.0000	−1.88500	−0.942499	−	0.334208i	$-0.891531\pi$
$761$	−52.0000	−1.88500	−0.942499	+	0.334208i	$0.891531\pi$
$762$	0	0
$763$	20.7846i	0.752453i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−67.8823	−2.45109
$768$	0	0
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.44949i	− 0.0881020i	−0.999029	−	0.0440510i	$-0.985974\pi$
$773$	− 2.44949i	− 0.0881020i	0.999029	−	0.0440510i	$-0.0140264\pi$
$774$	0	0
$775$	1.41421	0.0508001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 27.7128i	− 0.992915i
$780$	0	0
$781$	− 39.1918i	− 1.40239i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	20.7846i	0.740891i	0.928854	+	0.370446i	$0.120795\pi$
$787$	20.7846i	0.740891i	−0.928854	+	0.370446i	$0.879205\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.3137	0.402269
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.5403i	1.64854i	0.566195	+	0.824271i	$0.308415\pi$
$797$	46.5403i	1.64854i	−0.566195	+	0.824271i	$0.691585\pi$
$798$	0	0
$799$	−22.6274	−0.800500
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 13.8564i	− 0.488982i
$804$	0	0
$805$	− 19.5959i	− 0.690665i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	− 20.7846i	− 0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$	− 20.7846i	− 0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.9706	0.594453
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.44949i	− 0.0854878i	−0.999086	−	0.0427439i	$-0.986390\pi$
$821$	− 2.44949i	− 0.0854878i	0.999086	−	0.0427439i	$-0.0136099\pi$
$822$	0	0
$823$	−1.41421	−0.0492964	−0.0246482	−	0.999696i	$-0.507847\pi$
$823$	−1.41421	−0.0492964	−0.0246482	+	0.999696i	$0.507847\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.5692i	1.44550i	0.691108	+	0.722752i	$0.257123\pi$
$827$	41.5692i	1.44550i	−0.691108	+	0.722752i	$0.742877\pi$
$828$	0	0
$829$	− 34.2929i	− 1.19104i	−0.803340	−	0.595520i	$-0.796946\pi$
$829$	− 34.2929i	− 1.19104i	0.803340	−	0.595520i	$-0.203054\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	41.5692i	1.43856i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.2548	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$839$	−45.2548	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 26.9444i	− 0.926915i
$846$	0	0
$847$	1.41421	0.0485930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27.7128i	− 0.949983i
$852$	0	0
$853$	4.89898i	0.167738i	0.996477	+	0.0838689i	$0.0267277\pi$
$853$	4.89898i	0.167738i	−0.996477	+	0.0838689i	$0.973272\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	− 6.92820i	− 0.236387i	−0.992991	−	0.118194i	$-0.962290\pi$
$859$	− 6.92820i	− 0.236387i	0.992991	−	0.118194i	$-0.0377103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.9706	−0.577685	−0.288842	−	0.957377i	$-0.593270\pi$
$863$	−16.9706	−0.577685	−0.288842	+	0.957377i	$0.593270\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 24.4949i	− 0.830932i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 13.8564i	− 0.468432i
$876$	0	0
$877$	− 34.2929i	− 1.15799i	−0.815332	−	0.578994i	$-0.803446\pi$
$877$	− 34.2929i	− 1.15799i	0.815332	−	0.578994i	$-0.196554\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	0	0
$883$	34.6410i	1.16576i	0.812557	+	0.582882i	$0.198075\pi$
$883$	34.6410i	1.16576i	−0.812557	+	0.582882i	$0.801925\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 39.1918i	− 1.31150i
$894$	0	0
$895$	33.9411	1.13453
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.46410i	0.115534i
$900$	0	0
$901$	29.3939i	0.979252i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.0000	−1.99447
$906$	0	0
$907$	20.7846i	0.690142i	0.938577	+	0.345071i	$0.112145\pi$
$907$	20.7846i	0.690142i	−0.938577	+	0.345071i	$0.887855\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.9117	−1.68678	−0.843390	−	0.537302i	$-0.819443\pi$
$911$	−50.9117	−1.68678	−0.843390	+	0.537302i	$0.819443\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.41421	−0.0466506	−0.0233253	−	0.999728i	$-0.507425\pi$
$919$	−1.41421	−0.0466506	−0.0233253	+	0.999728i	$0.507425\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	55.4256i	1.82436i
$924$	0	0
$925$	4.89898i	0.161077i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	0	0
$931$	34.6410i	1.13531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−33.9411	−1.10999
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 17.1464i	− 0.558958i	−0.960152	−	0.279479i	$-0.909838\pi$
$941$	− 17.1464i	− 0.558958i	0.960152	−	0.279479i	$-0.0901617\pi$
$942$	0	0
$943$	22.6274	0.736850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.8564i	0.450273i	0.974327	+	0.225136i	$0.0722828\pi$
$947$	13.8564i	0.450273i	−0.974327	+	0.225136i	$0.927717\pi$
$948$	0	0
$949$	19.5959i	0.636110i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	41.5692i	1.34515i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.2843	−0.913347
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 29.3939i	− 0.946222i
$966$	0	0
$967$	−32.5269	−1.04599	−0.522997	−	0.852334i	$-0.675186\pi$
$967$	−32.5269	−1.04599	−0.522997	+	0.852334i	$0.675186\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 31.1769i	− 1.00051i	−0.865877	−	0.500257i	$-0.833239\pi$
$971$	− 31.1769i	− 1.00051i	0.865877	−	0.500257i	$-0.166761\pi$
$972$	0	0
$973$	− 19.5959i	− 0.628216i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	− 55.4256i	− 1.77141i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.9411	−1.08255	−0.541277	−	0.840844i	$-0.682059\pi$
$983$	−33.9411	−1.08255	−0.541277	+	0.840844i	$0.682059\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 39.1918i	− 1.24623i
$990$	0	0
$991$	−15.5563	−0.494164	−0.247082	−	0.968995i	$-0.579472\pi$
$991$	−15.5563	−0.494164	−0.247082	+	0.968995i	$0.579472\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 45.0333i	− 1.42765i
$996$	0	0
$997$	14.6969i	0.465457i	0.972542	+	0.232728i	$0.0747653\pi$
$997$	14.6969i	0.465457i	−0.972542	+	0.232728i	$0.925235\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.d.e.2305.3		4
3.2	odd	2		4608.2.d.m.2305.1		4
4.3	odd	2	inner	4608.2.d.e.2305.4		4
8.3	odd	2	inner	4608.2.d.e.2305.2		4
8.5	even	2	inner	4608.2.d.e.2305.1		4
12.11	even	2		4608.2.d.m.2305.2		4
16.3	odd	4		4608.2.a.v.1.3	yes	4
16.5	even	4		4608.2.a.v.1.2	yes	4
16.11	odd	4		4608.2.a.v.1.1	✓	4
16.13	even	4		4608.2.a.v.1.4	yes	4
24.5	odd	2		4608.2.d.m.2305.3		4
24.11	even	2		4608.2.d.m.2305.4		4
48.5	odd	4		4608.2.a.z.1.4	yes	4
48.11	even	4		4608.2.a.z.1.3	yes	4
48.29	odd	4		4608.2.a.z.1.2	yes	4
48.35	even	4		4608.2.a.z.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.a.v.1.1	✓	4	16.11	odd	4
4608.2.a.v.1.2	yes	4	16.5	even	4
4608.2.a.v.1.3	yes	4	16.3	odd	4
4608.2.a.v.1.4	yes	4	16.13	even	4
4608.2.a.z.1.1	yes	4	48.35	even	4
4608.2.a.z.1.2	yes	4	48.29	odd	4
4608.2.a.z.1.3	yes	4	48.11	even	4
4608.2.a.z.1.4	yes	4	48.5	odd	4
4608.2.d.e.2305.1		4	8.5	even	2	inner
4608.2.d.e.2305.2		4	8.3	odd	2	inner
4608.2.d.e.2305.3		4	1.1	even	1	trivial
4608.2.d.e.2305.4		4	4.3	odd	2	inner
4608.2.d.m.2305.1		4	3.2	odd	2
4608.2.d.m.2305.2		4	12.11	even	2
4608.2.d.m.2305.3		4	24.5	odd	2
4608.2.d.m.2305.4		4	24.11	even	2

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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{5} -1.41421 q^{7} +O(q^{10})\)	\(q+2.44949i q^{5} -1.41421 q^{7} -3.46410i q^{11} +4.89898i q^{13} -4.00000 q^{17} -6.92820i q^{19} +5.65685 q^{23} -1.00000 q^{25} -2.44949i q^{29} -1.41421 q^{31} -3.46410i q^{35} -4.89898i q^{37} +4.00000 q^{41} -6.92820i q^{43} +5.65685 q^{47} -5.00000 q^{49} -7.34847i q^{53} +8.48528 q^{55} +13.8564i q^{59} +4.89898i q^{61} -12.0000 q^{65} +11.3137 q^{71} +4.00000 q^{73} +4.89898i q^{77} +7.07107 q^{79} +10.3923i q^{83} -9.79796i q^{85} +16.0000 q^{89} -6.92820i q^{91} +16.9706 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{17} - 4 q^{25} + 16 q^{41} - 20 q^{49} - 48 q^{65} + 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100}) \) 4 * q - 16 * q^17 - 4 * q^25 + 16 * q^41 - 20 * q^49 - 48 * q^65 + 16 * q^73 + 64 * q^89 + 24 * q^97

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