LMFDB - Embedded newform 4608.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.a (trivial)

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:	yes
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	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
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	4.89898	1.35873	0.679366	−	0.733799i	$-0.262255\pi$
$13$	4.89898	1.35873	0.679366	+	0.733799i	$0.262255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	4.89898	0.805387	0.402694	−	0.915335i	$-0.368074\pi$
$37$	4.89898	0.805387	0.402694	+	0.915335i	$0.368074\pi$
$38$	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
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	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
$53$	−7.34847	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$53$	−7.34847	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$54$	0	0
$55$	8.48528	1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	4.89898	0.627250	0.313625	−	0.949547i	$-0.398457\pi$
$61$	4.89898	0.627250	0.313625	+	0.949547i	$0.398457\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.89898	−0.558291
$78$	0	0
$79$	−7.07107	−0.795557	−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107	−0.795557	−0.397779	+	0.917481i	$0.630219\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	9.79796	1.06274
$86$	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
$91$	−6.92820	−0.726273
$92$	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
$101$	−2.44949	−0.243733	−0.121867	−	0.992546i	$-0.538888\pi$
$101$	−2.44949	−0.243733	−0.121867	+	0.992546i	$0.538888\pi$
$102$	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
$107$	13.8564	1.33955	0.669775	−	0.742564i	$-0.266391\pi$
$107$	13.8564	1.33955	0.669775	+	0.742564i	$0.266391\pi$
$108$	0	0
$109$	−14.6969	−1.40771	−0.703856	−	0.710343i	$-0.748540\pi$
$109$	−14.6969	−1.40771	−0.703856	+	0.710343i	$0.748540\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−13.8564	−1.29212
$116$	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
$125$	−9.79796	−0.876356
$126$	0	0
$127$	9.89949	0.878438	0.439219	−	0.898380i	$-0.355255\pi$
$127$	9.89949	0.878438	0.439219	+	0.898380i	$0.355255\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−9.79796	−0.849591
$134$	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
$139$	13.8564	1.17529	0.587643	−	0.809121i	$-0.300056\pi$
$139$	13.8564	1.17529	0.587643	+	0.809121i	$0.300056\pi$
$140$	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
$149$	22.0454	1.80603	0.903015	−	0.429609i	$-0.141349\pi$
$149$	22.0454	1.80603	0.903015	+	0.429609i	$0.141349\pi$
$150$	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
$155$	3.46410	0.278243
$156$	0	0
$157$	−4.89898	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$157$	−4.89898	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	6.92820	0.542659	0.271329	−	0.962487i	$-0.412537\pi$
$163$	6.92820	0.542659	0.271329	+	0.962487i	$0.412537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.9706	−1.31322	−0.656611	−	0.754230i	$-0.728011\pi$
$167$	−16.9706	−1.31322	−0.656611	+	0.754230i	$0.728011\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.1464	1.30362	0.651809	−	0.758383i	$-0.274010\pi$
$173$	17.1464	1.30362	0.651809	+	0.758383i	$0.274010\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.8564	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$179$	−13.8564	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$180$	0	0
$181$	−24.4949	−1.82069	−0.910346	−	0.413849i	$-0.864184\pi$
$181$	−24.4949	−1.82069	−0.910346	+	0.413849i	$0.864184\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	13.8564	1.01328
$188$	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
$197$	−7.34847	−0.523557	−0.261778	−	0.965128i	$-0.584309\pi$
$197$	−7.34847	−0.523557	−0.261778	+	0.965128i	$0.584309\pi$
$198$	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
$203$	−3.46410	−0.243132
$204$	0	0
$205$	9.79796	0.684319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	−13.8564	−0.953914	−0.476957	−	0.878927i	$-0.658260\pi$
$211$	−13.8564	−0.953914	−0.476957	+	0.878927i	$0.658260\pi$
$212$	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
$221$	19.5959	1.31816
$222$	0	0
$223$	7.07107	0.473514	0.236757	−	0.971569i	$-0.423916\pi$
$223$	7.07107	0.473514	0.236757	+	0.971569i	$0.423916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−4.89898	−0.323734	−0.161867	−	0.986813i	$-0.551752\pi$
$229$	−4.89898	−0.323734	−0.161867	+	0.986813i	$0.551752\pi$
$230$	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
$235$	13.8564	0.903892
$236$	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
$245$	−12.2474	−0.782461
$246$	0	0
$247$	33.9411	2.15962
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	0	0
$253$	−19.5959	−1.23198
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	−6.92820	−0.430498
$260$	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
$269$	−12.2474	−0.746740	−0.373370	−	0.927682i	$-0.621798\pi$
$269$	−12.2474	−0.746740	−0.373370	+	0.927682i	$0.621798\pi$
$270$	0	0
$271$	−26.8701	−1.63224	−0.816120	−	0.577883i	$-0.803879\pi$
$271$	−26.8701	−1.63224	−0.816120	+	0.577883i	$0.803879\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46410	0.208893
$276$	0	0
$277$	−4.89898	−0.294351	−0.147176	−	0.989110i	$-0.547018\pi$
$277$	−4.89898	−0.294351	−0.147176	+	0.989110i	$0.547018\pi$
$278$	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
$283$	13.8564	0.823678	0.411839	−	0.911257i	$-0.364887\pi$
$283$	13.8564	0.823678	0.411839	+	0.911257i	$0.364887\pi$
$284$	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
$293$	−7.34847	−0.429302	−0.214651	−	0.976691i	$-0.568861\pi$
$293$	−7.34847	−0.429302	−0.214651	+	0.976691i	$0.568861\pi$
$294$	0	0
$295$	−33.9411	−1.97613
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.7128	−1.60267
$300$	0	0
$301$	9.79796	0.564745
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	27.7128	1.58165	0.790827	−	0.612040i	$-0.209651\pi$
$307$	27.7128	1.58165	0.790827	+	0.612040i	$0.209651\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.9411	−1.92462	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411	−1.92462	−0.962312	+	0.271947i	$0.912333\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.8434	−1.78850	−0.894251	−	0.447566i	$-0.852291\pi$
$317$	−31.8434	−1.78850	−0.894251	+	0.447566i	$0.852291\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.7128	1.54198
$324$	0	0
$325$	4.89898	0.271746
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−27.7128	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$331$	−27.7128	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$332$	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
$341$	4.89898	0.265295
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	4.89898	0.262236	0.131118	−	0.991367i	$-0.458143\pi$
$349$	4.89898	0.262236	0.131118	+	0.991367i	$0.458143\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.0000	1.70319	0.851594	−	0.524202i	$-0.175636\pi$
$353$	32.0000	1.70319	0.851594	+	0.524202i	$0.175636\pi$
$354$	0	0
$355$	−27.7128	−1.47084
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.79796	−0.512849
$366$	0	0
$367$	1.41421	0.0738213	0.0369107	−	0.999319i	$-0.488248\pi$
$367$	1.41421	0.0738213	0.0369107	+	0.999319i	$0.488248\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	34.2929	1.77562	0.887808	−	0.460213i	$-0.152227\pi$
$373$	34.2929	1.77562	0.887808	+	0.460213i	$0.152227\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−34.6410	−1.77939	−0.889695	−	0.456556i	$-0.849083\pi$
$379$	−34.6410	−1.77939	−0.889695	+	0.456556i	$0.849083\pi$
$380$	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
$389$	26.9444	1.36613	0.683067	−	0.730355i	$-0.260646\pi$
$389$	26.9444	1.36613	0.683067	+	0.730355i	$0.260646\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−17.3205	−0.871489
$396$	0	0
$397$	−24.4949	−1.22936	−0.614682	−	0.788775i	$-0.710716\pi$
$397$	−24.4949	−1.22936	−0.614682	+	0.788775i	$0.710716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	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	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.5959	0.964252
$414$	0	0
$415$	25.4558	1.24958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.3205	0.846162	0.423081	−	0.906092i	$-0.360949\pi$
$419$	17.3205	0.846162	0.423081	+	0.906092i	$0.360949\pi$
$420$	0	0
$421$	14.6969	0.716285	0.358142	−	0.933667i	$-0.383410\pi$
$421$	14.6969	0.716285	0.358142	+	0.933667i	$0.383410\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−6.92820	−0.335279
$428$	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
$437$	−39.1918	−1.87480
$438$	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
$443$	24.2487	1.15209	0.576046	−	0.817418i	$-0.304595\pi$
$443$	24.2487	1.15209	0.576046	+	0.817418i	$0.304595\pi$
$444$	0	0
$445$	39.1918	1.85787
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	13.8564	0.652473
$452$	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	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.44949	−0.114084	−0.0570421	−	0.998372i	$-0.518167\pi$
$461$	−2.44949	−0.114084	−0.0570421	+	0.998372i	$0.518167\pi$
$462$	0	0
$463$	−1.41421	−0.0657241	−0.0328620	−	0.999460i	$-0.510462\pi$
$463$	−1.41421	−0.0657241	−0.0328620	+	0.999460i	$0.510462\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.1051	−1.76329	−0.881647	−	0.471909i	$-0.843565\pi$
$467$	−38.1051	−1.76329	−0.881647	+	0.471909i	$0.843565\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	6.92820	0.317888
$476$	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
$485$	14.6969	0.667354
$486$	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		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	9.79796	0.441278
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	13.8564	0.620298	0.310149	−	0.950688i	$-0.399621\pi$
$499$	13.8564	0.620298	0.310149	+	0.950688i	$0.399621\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.6274	1.00891	0.504453	−	0.863439i	$-0.331694\pi$
$503$	22.6274	1.00891	0.504453	+	0.863439i	$0.331694\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.0454	0.977146	0.488573	−	0.872523i	$-0.337518\pi$
$509$	22.0454	0.977146	0.488573	+	0.872523i	$0.337518\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−45.0333	−1.98441
$516$	0	0
$517$	19.5959	0.861827
$518$	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
$523$	−20.7846	−0.908848	−0.454424	−	0.890786i	$-0.650155\pi$
$523$	−20.7846	−0.908848	−0.454424	+	0.890786i	$0.650155\pi$
$524$	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
$533$	19.5959	0.848793
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−17.3205	−0.746047
$540$	0	0
$541$	14.6969	0.631871	0.315935	−	0.948781i	$-0.397682\pi$
$541$	14.6969	0.631871	0.315935	+	0.948781i	$0.397682\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	34.6410	1.48114	0.740571	−	0.671978i	$-0.234555\pi$
$547$	34.6410	1.48114	0.740571	+	0.671978i	$0.234555\pi$
$548$	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
$557$	31.8434	1.34925	0.674623	−	0.738162i	$-0.264306\pi$
$557$	31.8434	1.34925	0.674623	+	0.738162i	$0.264306\pi$
$558$	0	0
$559$	−33.9411	−1.43556
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.3923	−0.437983	−0.218992	−	0.975727i	$-0.570277\pi$
$563$	−10.3923	−0.437983	−0.218992	+	0.975727i	$0.570277\pi$
$564$	0	0
$565$	19.5959	0.824406
$566$	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		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	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
$581$	−14.6969	−0.609732
$582$	0	0
$583$	−25.4558	−1.05427
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	9.79796	0.403718
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.00000	−0.328521	−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000	−0.328521	−0.164260	+	0.986417i	$0.552524\pi$
$594$	0	0
$595$	−13.8564	−0.568057
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.44949	0.0995859
$606$	0	0
$607$	−24.0416	−0.975820	−0.487910	−	0.872894i	$-0.662241\pi$
$607$	−24.0416	−0.975820	−0.487910	+	0.872894i	$0.662241\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.7128	1.12114
$612$	0	0
$613$	−44.0908	−1.78081	−0.890406	−	0.455168i	$-0.849579\pi$
$613$	−44.0908	−1.78081	−0.890406	+	0.455168i	$0.849579\pi$
$614$	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
$619$	27.7128	1.11387	0.556936	−	0.830555i	$-0.311977\pi$
$619$	27.7128	1.11387	0.556936	+	0.830555i	$0.311977\pi$
$620$	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
$629$	19.5959	0.781340
$630$	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
$635$	24.2487	0.962281
$636$	0	0
$637$	−24.4949	−0.970523
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28.0000	−1.10593	−0.552967	−	0.833203i	$-0.686504\pi$
$641$	−28.0000	−1.10593	−0.552967	+	0.833203i	$0.686504\pi$
$642$	0	0
$643$	20.7846	0.819665	0.409832	−	0.912161i	$-0.365587\pi$
$643$	20.7846	0.819665	0.409832	+	0.912161i	$0.365587\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.2843	1.11197	0.555985	−	0.831193i	$-0.312341\pi$
$647$	28.2843	1.11197	0.555985	+	0.831193i	$0.312341\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.0454	−0.862703	−0.431352	−	0.902184i	$-0.641963\pi$
$653$	−22.0454	−0.862703	−0.431352	+	0.902184i	$0.641963\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.7128	1.07954	0.539769	−	0.841813i	$-0.318512\pi$
$659$	27.7128	1.07954	0.539769	+	0.841813i	$0.318512\pi$
$660$	0	0
$661$	−44.0908	−1.71493	−0.857467	−	0.514539i	$-0.827963\pi$
$661$	−44.0908	−1.71493	−0.857467	+	0.514539i	$0.827963\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	−13.8564	−0.536522
$668$	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
$677$	−17.1464	−0.658991	−0.329495	−	0.944157i	$-0.606879\pi$
$677$	−17.1464	−0.658991	−0.329495	+	0.944157i	$0.606879\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.46410	0.132550	0.0662751	−	0.997801i	$-0.478889\pi$
$683$	3.46410	0.132550	0.0662751	+	0.997801i	$0.478889\pi$
$684$	0	0
$685$	48.9898	1.87180
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−20.7846	−0.790684	−0.395342	−	0.918534i	$-0.629374\pi$
$691$	−20.7846	−0.790684	−0.395342	+	0.918534i	$0.629374\pi$
$692$	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
$701$	−31.8434	−1.20271	−0.601354	−	0.798983i	$-0.705372\pi$
$701$	−31.8434	−1.20271	−0.601354	+	0.798983i	$0.705372\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.46410	0.130281
$708$	0	0
$709$	14.6969	0.551955	0.275978	−	0.961164i	$-0.410998\pi$
$709$	14.6969	0.551955	0.275978	+	0.961164i	$0.410998\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	41.5692	1.55460
$716$	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
$725$	2.44949	0.0909718
$726$	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
$731$	−27.7128	−1.02500
$732$	0	0
$733$	−24.4949	−0.904740	−0.452370	−	0.891830i	$-0.649421\pi$
$733$	−24.4949	−0.904740	−0.452370	+	0.891830i	$0.649421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.9706	0.622590	0.311295	−	0.950313i	$-0.399237\pi$
$743$	16.9706	0.622590	0.311295	+	0.950313i	$0.399237\pi$
$744$	0	0
$745$	54.0000	1.97841
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.5959	−0.716019
$750$	0	0
$751$	15.5563	0.567659	0.283830	−	0.958875i	$-0.408395\pi$
$751$	15.5563	0.567659	0.283830	+	0.958875i	$0.408395\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.0333	1.63893
$756$	0	0
$757$	−4.89898	−0.178056	−0.0890282	−	0.996029i	$-0.528376\pi$
$757$	−4.89898	−0.178056	−0.0890282	+	0.996029i	$0.528376\pi$
$758$	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
$763$	20.7846	0.752453
$764$	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
$773$	−2.44949	−0.0881020	−0.0440510	−	0.999029i	$-0.514026\pi$
$773$	−2.44949	−0.0881020	−0.0440510	+	0.999029i	$0.514026\pi$
$774$	0	0
$775$	1.41421	0.0508001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.7128	0.992915
$780$	0	0
$781$	−39.1918	−1.40239
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−20.7846	−0.740891	−0.370446	−	0.928854i	$-0.620795\pi$
$787$	−20.7846	−0.740891	−0.370446	+	0.928854i	$0.620795\pi$
$788$	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
$797$	−46.5403	−1.64854	−0.824271	−	0.566195i	$-0.808415\pi$
$797$	−46.5403	−1.64854	−0.824271	+	0.566195i	$0.808415\pi$
$798$	0	0
$799$	22.6274	0.800500
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.8564	−0.488982
$804$	0	0
$805$	19.5959	0.690665
$806$	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
$811$	−20.7846	−0.729846	−0.364923	−	0.931038i	$-0.618905\pi$
$811$	−20.7846	−0.729846	−0.364923	+	0.931038i	$0.618905\pi$
$812$	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
$821$	−2.44949	−0.0854878	−0.0427439	−	0.999086i	$-0.513610\pi$
$821$	−2.44949	−0.0854878	−0.0427439	+	0.999086i	$0.513610\pi$
$822$	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
$827$	−41.5692	−1.44550	−0.722752	−	0.691108i	$-0.757123\pi$
$827$	−41.5692	−1.44550	−0.722752	+	0.691108i	$0.757123\pi$
$828$	0	0
$829$	−34.2929	−1.19104	−0.595520	−	0.803340i	$-0.703054\pi$
$829$	−34.2929	−1.19104	−0.595520	+	0.803340i	$0.703054\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.0000	−0.692959
$834$	0	0
$835$	−41.5692	−1.43856
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.2548	1.56237	0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548	1.56237	0.781185	+	0.624299i	$0.214615\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26.9444	0.926915
$846$	0	0
$847$	−1.41421	−0.0485930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27.7128	−0.949983
$852$	0	0
$853$	−4.89898	−0.167738	−0.0838689	−	0.996477i	$-0.526728\pi$
$853$	−4.89898	−0.167738	−0.0838689	+	0.996477i	$0.526728\pi$
$854$	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
$859$	−6.92820	−0.236387	−0.118194	−	0.992991i	$-0.537710\pi$
$859$	−6.92820	−0.236387	−0.118194	+	0.992991i	$0.537710\pi$
$860$	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
$869$	−24.4949	−0.830932
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.8564	0.468432
$876$	0	0
$877$	−34.2929	−1.15799	−0.578994	−	0.815332i	$-0.696554\pi$
$877$	−34.2929	−1.15799	−0.578994	+	0.815332i	$0.696554\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	−34.6410	−1.16576	−0.582882	−	0.812557i	$-0.698075\pi$
$883$	−34.6410	−1.16576	−0.582882	+	0.812557i	$0.698075\pi$
$884$	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
$893$	39.1918	1.31150
$894$	0	0
$895$	−33.9411	−1.13453
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.46410	0.115534
$900$	0	0
$901$	−29.3939	−0.979252
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.0000	−1.99447
$906$	0	0
$907$	20.7846	0.690142	0.345071	−	0.938577i	$-0.387855\pi$
$907$	20.7846	0.690142	0.345071	+	0.938577i	$0.387855\pi$
$908$	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
$923$	−55.4256	−1.82436
$924$	0	0
$925$	4.89898	0.161077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	−34.6410	−1.13531
$932$	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	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.1464	0.558958	0.279479	−	0.960152i	$-0.409838\pi$
$941$	17.1464	0.558958	0.279479	+	0.960152i	$0.409838\pi$
$942$	0	0
$943$	−22.6274	−0.736850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.8564	0.450273	0.225136	−	0.974327i	$-0.427717\pi$
$947$	13.8564	0.450273	0.225136	+	0.974327i	$0.427717\pi$
$948$	0	0
$949$	−19.5959	−0.636110
$950$	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
$955$	41.5692	1.34515
$956$	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
$965$	−29.3939	−0.946222
$966$	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
$971$	31.1769	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$972$	0	0
$973$	−19.5959	−0.628216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	55.4256	1.77141
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.9411	1.08255	0.541277	−	0.840844i	$-0.317941\pi$
$983$	33.9411	1.08255	0.541277	+	0.840844i	$0.317941\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.1918	1.24623
$990$	0	0
$991$	15.5563	0.494164	0.247082	−	0.968995i	$-0.420528\pi$
$991$	15.5563	0.494164	0.247082	+	0.968995i	$0.420528\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−45.0333	−1.42765
$996$	0	0
$997$	−14.6969	−0.465457	−0.232728	−	0.972542i	$-0.574765\pi$
$997$	−14.6969	−0.465457	−0.232728	+	0.972542i	$0.574765\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more