LMFDB - Embedded newform 4320.2.h.c.2591.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(2591,4320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4320.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 523 x^{12} - 1120 x^{11} + 2214 x^{10} - 3524 x^{9} + \cdots + 2116 $ x^16 - 8x^15 + 44x^14 - 168x^13 + 523x^12 - 1120x^11 + 2214x^10 - 3524x^9 + 6063x^8 - 18332x^7 + 43638x^6 - 65248x^5 + 6502x^4 - 2712x^3 + 160676x^2 + 36616*x + 2116
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Root			$-1.60613 + 1.23243i$ of defining polynomial
Character	$\chi$	$=$	4320.2591
Dual form			4320.2.h.c.2591.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +1.44838i q^{7} +0.989142 q^{11} -4.22656 q^{13} +6.64295i q^{17} +6.97614i q^{19} -5.18043 q^{23} -1.00000 q^{25} -4.27342i q^{29} -6.73826i q^{31} +1.44838 q^{35} -5.91809 q^{37} -9.85940i q^{41} -10.1760i q^{43} +8.13408 q^{47} +4.90221 q^{49} -9.06019i q^{53} -0.989142i q^{55} +0.602718 q^{59} -4.48694 q^{61} +4.22656i q^{65} -11.1758i q^{67} +11.7911 q^{71} -10.2031 q^{73} +1.43265i q^{77} -8.57174i q^{79} -8.49507 q^{83} +6.64295 q^{85} +9.79200i q^{89} -6.12165i q^{91} +6.97614 q^{95} -13.7017 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{11} - 8 q^{13} - 32 q^{23} - 16 q^{25} + 8 q^{37} - 16 q^{47} - 8 q^{49} + 32 q^{59} + 8 q^{61} + 32 q^{71} - 8 q^{73} + 32 q^{83} - 8 q^{97}+O(q^{100}) $ 16 * q + 8 * q^11 - 8 * q^13 - 32 * q^23 - 16 * q^25 + 8 * q^37 - 16 * q^47 - 8 * q^49 + 32 * q^59 + 8 * q^61 + 32 * q^71 - 8 * q^73 + 32 * q^83 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.44838i	0.547434i	0.961810	+	0.273717i	$0.0882532\pi$
$7$	1.44838i	0.547434i	−0.961810	+	0.273717i	$0.911747\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.989142	0.298238	0.149119	−	0.988819i	$-0.452356\pi$
$11$	0.989142	0.298238	0.149119	+	0.988819i	$0.452356\pi$
$12$	0	0
$13$	−4.22656	−1.17224	−0.586118	−	0.810225i	$-0.699345\pi$
$13$	−4.22656	−1.17224	−0.586118	+	0.810225i	$0.699345\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.64295i	1.61115i	0.592492	+	0.805576i	$0.298144\pi$
$17$	6.64295i	1.61115i	−0.592492	+	0.805576i	$0.701856\pi$
$18$	0	0
$19$	6.97614i	1.60044i	0.599708	+	0.800219i	$0.295283\pi$
$19$	6.97614i	1.60044i	−0.599708	+	0.800219i	$0.704717\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.18043	−1.08019	−0.540097	−	0.841603i	$-0.681612\pi$
$23$	−5.18043	−1.08019	−0.540097	+	0.841603i	$0.681612\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.27342i	− 0.793554i	−0.917915	−	0.396777i	$-0.870129\pi$
$29$	− 4.27342i	− 0.793554i	0.917915	−	0.396777i	$-0.129871\pi$
$30$	0	0
$31$	− 6.73826i	− 1.21023i	−0.796139	−	0.605114i	$-0.793128\pi$
$31$	− 6.73826i	− 1.21023i	0.796139	−	0.605114i	$-0.206872\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.44838	0.244820
$36$	0	0
$37$	−5.91809	−0.972928	−0.486464	−	0.873701i	$-0.661713\pi$
$37$	−5.91809	−0.972928	−0.486464	+	0.873701i	$0.661713\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.85940i	− 1.53978i	−0.638177	−	0.769890i	$-0.720311\pi$
$41$	− 9.85940i	− 1.53978i	0.638177	−	0.769890i	$-0.279689\pi$
$42$	0	0
$43$	− 10.1760i	− 1.55183i	−0.630837	−	0.775915i	$-0.717289\pi$
$43$	− 10.1760i	− 1.55183i	0.630837	−	0.775915i	$-0.282711\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.13408	1.18648	0.593239	−	0.805027i	$-0.297849\pi$
$47$	8.13408	1.18648	0.593239	+	0.805027i	$0.297849\pi$
$48$	0	0
$49$	4.90221	0.700316
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.06019i	− 1.24451i	−0.782813	−	0.622257i	$-0.786216\pi$
$53$	− 9.06019i	− 1.24451i	0.782813	−	0.622257i	$-0.213784\pi$
$54$	0	0
$55$	− 0.989142i	− 0.133376i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.602718	0.0784672	0.0392336	−	0.999230i	$-0.487508\pi$
$59$	0.602718	0.0784672	0.0392336	+	0.999230i	$0.487508\pi$
$60$	0	0
$61$	−4.48694	−0.574494	−0.287247	−	0.957857i	$-0.592740\pi$
$61$	−4.48694	−0.574494	−0.287247	+	0.957857i	$0.592740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.22656i	0.524240i
$66$	0	0
$67$	− 11.1758i	− 1.36534i	−0.730727	−	0.682670i	$-0.760819\pi$
$67$	− 11.1758i	− 1.36534i	0.730727	−	0.682670i	$-0.239181\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.7911	1.39934	0.699672	−	0.714465i	$-0.253330\pi$
$71$	11.7911	1.39934	0.699672	+	0.714465i	$0.253330\pi$
$72$	0	0
$73$	−10.2031	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$73$	−10.2031	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.43265i	0.163266i
$78$	0	0
$79$	− 8.57174i	− 0.964396i	−0.876062	−	0.482198i	$-0.839839\pi$
$79$	− 8.57174i	− 0.964396i	0.876062	−	0.482198i	$-0.160161\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.49507	−0.932455	−0.466228	−	0.884665i	$-0.654387\pi$
$83$	−8.49507	−0.932455	−0.466228	+	0.884665i	$0.654387\pi$
$84$	0	0
$85$	6.64295	0.720529
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.79200i	1.03795i	0.854790	+	0.518975i	$0.173686\pi$
$89$	9.79200i	1.03795i	−0.854790	+	0.518975i	$0.826314\pi$
$90$	0	0
$91$	− 6.12165i	− 0.641723i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.97614	0.715737
$96$	0	0
$97$	−13.7017	−1.39119	−0.695597	−	0.718432i	$-0.744860\pi$
$97$	−13.7017	−1.39119	−0.695597	+	0.718432i	$0.744860\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.93373i	− 0.192413i	−0.995361	−	0.0962065i	$-0.969329\pi$
$101$	− 1.93373i	− 0.192413i	0.995361	−	0.0962065i	$-0.0306709\pi$
$102$	0	0
$103$	4.73829i	0.466877i	0.972372	+	0.233439i	$0.0749978\pi$
$103$	4.73829i	0.466877i	−0.972372	+	0.233439i	$0.925002\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.16495	−0.789335	−0.394668	−	0.918824i	$-0.629140\pi$
$107$	−8.16495	−0.789335	−0.394668	+	0.918824i	$0.629140\pi$
$108$	0	0
$109$	−12.5251	−1.19969	−0.599845	−	0.800117i	$-0.704771\pi$
$109$	−12.5251	−1.19969	−0.599845	+	0.800117i	$0.704771\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.7076i	1.85394i	0.375141	+	0.926968i	$0.377594\pi$
$113$	19.7076i	1.85394i	−0.375141	+	0.926968i	$0.622406\pi$
$114$	0	0
$115$	5.18043i	0.483077i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.62149	−0.882000
$120$	0	0
$121$	−10.0216	−0.911054
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	− 5.05957i	− 0.448964i	−0.974478	−	0.224482i	$-0.927931\pi$
$127$	− 5.05957i	− 0.448964i	0.974478	−	0.224482i	$-0.0720691\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.84767	0.336173	0.168086	−	0.985772i	$-0.446241\pi$
$131$	3.84767	0.336173	0.168086	+	0.985772i	$0.446241\pi$
$132$	0	0
$133$	−10.1041	−0.876134
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.44997i	− 0.721930i	−0.932579	−	0.360965i	$-0.882447\pi$
$137$	− 8.44997i	− 0.721930i	0.932579	−	0.360965i	$-0.117553\pi$
$138$	0	0
$139$	− 10.5386i	− 0.893873i	−0.894566	−	0.446936i	$-0.852515\pi$
$139$	− 10.5386i	− 0.893873i	0.894566	−	0.446936i	$-0.147485\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.18067	−0.349605
$144$	0	0
$145$	−4.27342	−0.354888
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.83806i	0.478272i	0.970986	+	0.239136i	$0.0768643\pi$
$149$	5.83806i	0.478272i	−0.970986	+	0.239136i	$0.923136\pi$
$150$	0	0
$151$	− 9.48344i	− 0.771751i	−0.922551	−	0.385876i	$-0.873899\pi$
$151$	− 9.48344i	− 0.771751i	0.922551	−	0.385876i	$-0.126101\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.73826	−0.541230
$156$	0	0
$157$	16.3612	1.30577	0.652885	−	0.757457i	$-0.273559\pi$
$157$	16.3612	1.30577	0.652885	+	0.757457i	$0.273559\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.50320i	− 0.591335i
$162$	0	0
$163$	− 6.00049i	− 0.469995i	−0.971996	−	0.234997i	$-0.924492\pi$
$163$	− 6.00049i	− 0.469995i	0.971996	−	0.234997i	$-0.0755081\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.6377	−1.51961	−0.759807	−	0.650149i	$-0.774707\pi$
$167$	−19.6377	−1.51961	−0.759807	+	0.650149i	$0.774707\pi$
$168$	0	0
$169$	4.86381	0.374139
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.785648i	− 0.0597317i	−0.999554	−	0.0298659i	$-0.990492\pi$
$173$	− 0.785648i	− 0.0597317i	0.999554	−	0.0298659i	$-0.00950801\pi$
$174$	0	0
$175$	− 1.44838i	− 0.109487i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.34957	0.698820	0.349410	−	0.936970i	$-0.386382\pi$
$179$	9.34957	0.698820	0.349410	+	0.936970i	$0.386382\pi$
$180$	0	0
$181$	−2.03967	−0.151607	−0.0758036	−	0.997123i	$-0.524152\pi$
$181$	−2.03967	−0.151607	−0.0758036	+	0.997123i	$0.524152\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.91809i	0.435107i
$186$	0	0
$187$	6.57082i	0.480506i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.55565	−0.401992	−0.200996	−	0.979592i	$-0.564418\pi$
$191$	−5.55565	−0.401992	−0.200996	+	0.979592i	$0.564418\pi$
$192$	0	0
$193$	17.3479	1.24873	0.624365	−	0.781133i	$-0.285358\pi$
$193$	17.3479	1.24873	0.624365	+	0.781133i	$0.285358\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.66466i	− 0.403590i	−0.979428	−	0.201795i	$-0.935322\pi$
$197$	− 5.66466i	− 0.403590i	0.979428	−	0.201795i	$-0.0646776\pi$
$198$	0	0
$199$	− 14.3899i	− 1.02007i	−0.860152	−	0.510037i	$-0.829632\pi$
$199$	− 14.3899i	− 1.02007i	0.860152	−	0.510037i	$-0.170368\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.18952	0.434419
$204$	0	0
$205$	−9.85940	−0.688610
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.90040i	0.477311i
$210$	0	0
$211$	− 6.31585i	− 0.434801i	−0.976082	−	0.217400i	$-0.930242\pi$
$211$	− 6.31585i	− 0.434801i	0.976082	−	0.217400i	$-0.0697578\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.1760	−0.693999
$216$	0	0
$217$	9.75954	0.662520
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 28.0768i	− 1.88865i
$222$	0	0
$223$	− 2.38685i	− 0.159835i	−0.996801	−	0.0799176i	$-0.974534\pi$
$223$	− 2.38685i	− 0.159835i	0.996801	−	0.0799176i	$-0.0254657\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.2616	0.946573	0.473286	−	0.880909i	$-0.343068\pi$
$227$	14.2616	0.946573	0.473286	+	0.880909i	$0.343068\pi$
$228$	0	0
$229$	−22.8474	−1.50980	−0.754900	−	0.655840i	$-0.772314\pi$
$229$	−22.8474	−1.50980	−0.754900	+	0.655840i	$0.772314\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0315i	1.05026i	0.851023	+	0.525128i	$0.175983\pi$
$233$	16.0315i	1.05026i	−0.851023	+	0.525128i	$0.824017\pi$
$234$	0	0
$235$	− 8.13408i	− 0.530609i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.2399	1.37390	0.686949	−	0.726706i	$-0.258950\pi$
$239$	21.2399	1.37390	0.686949	+	0.726706i	$0.258950\pi$
$240$	0	0
$241$	−14.3874	−0.926775	−0.463387	−	0.886156i	$-0.653366\pi$
$241$	−14.3874	−0.926775	−0.463387	+	0.886156i	$0.653366\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.90221i	− 0.313191i
$246$	0	0
$247$	− 29.4851i	− 1.87609i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.47745	0.0932557	0.0466278	−	0.998912i	$-0.485153\pi$
$251$	1.47745	0.0932557	0.0466278	+	0.998912i	$0.485153\pi$
$252$	0	0
$253$	−5.12418	−0.322154
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.55243i	− 0.595864i	−0.954587	−	0.297932i	$-0.903703\pi$
$257$	− 9.55243i	− 0.595864i	0.954587	−	0.297932i	$-0.0962969\pi$
$258$	0	0
$259$	− 8.57161i	− 0.532614i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.70846	−0.536986	−0.268493	−	0.963282i	$-0.586526\pi$
$263$	−8.70846	−0.536986	−0.268493	+	0.963282i	$0.586526\pi$
$264$	0	0
$265$	−9.06019	−0.556563
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.72070i	− 0.531710i	−0.964013	−	0.265855i	$-0.914346\pi$
$269$	− 8.72070i	− 0.531710i	0.964013	−	0.265855i	$-0.0856543\pi$
$270$	0	0
$271$	25.7850i	1.56633i	0.621816	+	0.783163i	$0.286395\pi$
$271$	25.7850i	1.56633i	−0.621816	+	0.783163i	$0.713605\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.989142	−0.0596475
$276$	0	0
$277$	−22.1970	−1.33369	−0.666845	−	0.745196i	$-0.732356\pi$
$277$	−22.1970	−1.33369	−0.666845	+	0.745196i	$0.732356\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.4827i	1.81844i	0.416311	+	0.909222i	$0.363323\pi$
$281$	30.4827i	1.81844i	−0.416311	+	0.909222i	$0.636677\pi$
$282$	0	0
$283$	− 25.7569i	− 1.53109i	−0.643383	−	0.765545i	$-0.722470\pi$
$283$	− 25.7569i	− 1.53109i	0.643383	−	0.765545i	$-0.277530\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.2801	0.842928
$288$	0	0
$289$	−27.1288	−1.59581
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.08936i	0.0636410i	0.999494	+	0.0318205i	$0.0101305\pi$
$293$	1.08936i	0.0636410i	−0.999494	+	0.0318205i	$0.989870\pi$
$294$	0	0
$295$	− 0.602718i	− 0.0350916i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.8954	1.26624
$300$	0	0
$301$	14.7387	0.849525
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.48694i	0.256922i
$306$	0	0
$307$	11.3121i	0.645617i	0.946464	+	0.322808i	$0.104627\pi$
$307$	11.3121i	0.645617i	−0.946464	+	0.322808i	$0.895373\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.99400	−0.510003	−0.255001	−	0.966941i	$-0.582076\pi$
$311$	−8.99400	−0.510003	−0.255001	+	0.966941i	$0.582076\pi$
$312$	0	0
$313$	−9.04520	−0.511265	−0.255633	−	0.966774i	$-0.582284\pi$
$313$	−9.04520	−0.511265	−0.255633	+	0.966774i	$0.582284\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 31.2803i	− 1.75687i	−0.477858	−	0.878437i	$-0.658587\pi$
$317$	− 31.2803i	− 1.75687i	0.477858	−	0.878437i	$-0.341413\pi$
$318$	0	0
$319$	− 4.22702i	− 0.236668i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−46.3422	−2.57855
$324$	0	0
$325$	4.22656	0.234447
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.7812i	0.649518i
$330$	0	0
$331$	20.3977i	1.12116i	0.828100	+	0.560581i	$0.189422\pi$
$331$	20.3977i	1.12116i	−0.828100	+	0.560581i	$0.810578\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.1758	−0.610598
$336$	0	0
$337$	−10.6458	−0.579913	−0.289956	−	0.957040i	$-0.593641\pi$
$337$	−10.6458	−0.579913	−0.289956	+	0.957040i	$0.593641\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.66510i	− 0.360935i
$342$	0	0
$343$	17.2389i	0.930811i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.1561	1.02836	0.514178	−	0.857684i	$-0.328097\pi$
$347$	19.1561	1.02836	0.514178	+	0.857684i	$0.328097\pi$
$348$	0	0
$349$	−5.76577	−0.308634	−0.154317	−	0.988021i	$-0.549318\pi$
$349$	−5.76577	−0.308634	−0.154317	+	0.988021i	$0.549318\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.6865i	1.63328i	0.577148	+	0.816640i	$0.304166\pi$
$353$	30.6865i	1.63328i	−0.577148	+	0.816640i	$0.695834\pi$
$354$	0	0
$355$	− 11.7911i	− 0.625805i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.5315	1.13639	0.568195	−	0.822894i	$-0.307642\pi$
$359$	21.5315	1.13639	0.568195	+	0.822894i	$0.307642\pi$
$360$	0	0
$361$	−29.6666	−1.56140
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.2031i	0.534055i
$366$	0	0
$367$	9.83094i	0.513171i	0.966522	+	0.256586i	$0.0825975\pi$
$367$	9.83094i	0.513171i	−0.966522	+	0.256586i	$0.917402\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.1226	0.681289
$372$	0	0
$373$	0.772880	0.0400182	0.0200091	−	0.999800i	$-0.493630\pi$
$373$	0.772880	0.0400182	0.0200091	+	0.999800i	$0.493630\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0619i	0.930233i
$378$	0	0
$379$	− 10.8276i	− 0.556176i	−0.960556	−	0.278088i	$-0.910299\pi$
$379$	− 10.8276i	− 0.556176i	0.960556	−	0.278088i	$-0.0897007\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.9657	−1.22459	−0.612295	−	0.790629i	$-0.709754\pi$
$383$	−23.9657	−1.22459	−0.612295	+	0.790629i	$0.709754\pi$
$384$	0	0
$385$	1.43265	0.0730146
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.8091i	− 0.852254i	−0.904663	−	0.426127i	$-0.859878\pi$
$389$	− 16.8091i	− 0.852254i	0.904663	−	0.426127i	$-0.140122\pi$
$390$	0	0
$391$	− 34.4133i	− 1.74036i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.57174	−0.431291
$396$	0	0
$397$	−31.3550	−1.57366	−0.786832	−	0.617167i	$-0.788280\pi$
$397$	−31.3550	−1.57366	−0.786832	+	0.617167i	$0.788280\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.6448i	0.581514i	0.956797	+	0.290757i	$0.0939071\pi$
$401$	11.6448i	0.581514i	−0.956797	+	0.290757i	$0.906093\pi$
$402$	0	0
$403$	28.4797i	1.41867i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.85383	−0.290164
$408$	0	0
$409$	16.1936	0.800720	0.400360	−	0.916358i	$-0.368885\pi$
$409$	16.1936	0.800720	0.400360	+	0.916358i	$0.368885\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.872962i	0.0429556i
$414$	0	0
$415$	8.49507i	0.417007i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−38.6422	−1.88779	−0.943896	−	0.330242i	$-0.892870\pi$
$419$	−38.6422	−1.88779	−0.943896	+	0.330242i	$0.892870\pi$
$420$	0	0
$421$	29.5138	1.43842	0.719208	−	0.694795i	$-0.244505\pi$
$421$	29.5138	1.43842	0.719208	+	0.694795i	$0.244505\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.64295i	− 0.322230i
$426$	0	0
$427$	− 6.49878i	− 0.314498i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.9116	1.92247	0.961237	−	0.275725i	$-0.0889179\pi$
$431$	39.9116	1.92247	0.961237	+	0.275725i	$0.0889179\pi$
$432$	0	0
$433$	−4.25927	−0.204688	−0.102344	−	0.994749i	$-0.532634\pi$
$433$	−4.25927	−0.204688	−0.102344	+	0.994749i	$0.532634\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 36.1394i	− 1.72878i
$438$	0	0
$439$	13.8706i	0.662009i	0.943629	+	0.331005i	$0.107388\pi$
$439$	13.8706i	0.662009i	−0.943629	+	0.331005i	$0.892612\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.3341	0.823569	0.411784	−	0.911281i	$-0.364906\pi$
$443$	17.3341	0.823569	0.411784	+	0.911281i	$0.364906\pi$
$444$	0	0
$445$	9.79200	0.464185
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14.3299i	− 0.676269i	−0.941098	−	0.338134i	$-0.890204\pi$
$449$	− 14.3299i	− 0.676269i	0.941098	−	0.338134i	$-0.109796\pi$
$450$	0	0
$451$	− 9.75235i	− 0.459220i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.12165	−0.286987
$456$	0	0
$457$	20.8428	0.974984	0.487492	−	0.873128i	$-0.337912\pi$
$457$	20.8428	0.974984	0.487492	+	0.873128i	$0.337912\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 25.4393i	− 1.18483i	−0.805635	−	0.592413i	$-0.798176\pi$
$461$	− 25.4393i	− 1.18483i	0.805635	−	0.592413i	$-0.201824\pi$
$462$	0	0
$463$	− 29.1907i	− 1.35660i	−0.734783	−	0.678302i	$-0.762716\pi$
$463$	− 29.1907i	− 1.35660i	0.734783	−	0.678302i	$-0.237284\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.26473	0.104799	0.0523996	−	0.998626i	$-0.483313\pi$
$467$	2.26473	0.104799	0.0523996	+	0.998626i	$0.483313\pi$
$468$	0	0
$469$	16.1867	0.747434
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.0655i	− 0.462814i
$474$	0	0
$475$	− 6.97614i	− 0.320087i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.2543	−1.56512	−0.782559	−	0.622576i	$-0.786086\pi$
$479$	−34.2543	−1.56512	−0.782559	+	0.622576i	$0.786086\pi$
$480$	0	0
$481$	25.0132	1.14050
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.7017i	0.622161i
$486$	0	0
$487$	0.183116i	0.00829776i	0.999991	+	0.00414888i	$0.00132063\pi$
$487$	0.183116i	0.00829776i	−0.999991	+	0.00414888i	$0.998679\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.2108	0.731582	0.365791	−	0.930697i	$-0.380798\pi$
$491$	16.2108	0.731582	0.365791	+	0.930697i	$0.380798\pi$
$492$	0	0
$493$	28.3881	1.27854
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.0779i	0.766049i
$498$	0	0
$499$	− 2.80133i	− 0.125405i	−0.998032	−	0.0627024i	$-0.980028\pi$
$499$	− 2.80133i	− 0.125405i	0.998032	−	0.0627024i	$-0.0199719\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.55379	0.292219	0.146110	−	0.989268i	$-0.453325\pi$
$503$	6.55379	0.292219	0.146110	+	0.989268i	$0.453325\pi$
$504$	0	0
$505$	−1.93373	−0.0860497
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 10.3371i	− 0.458183i	−0.973405	−	0.229091i	$-0.926425\pi$
$509$	− 10.3371i	− 0.458183i	0.973405	−	0.229091i	$-0.0735755\pi$
$510$	0	0
$511$	− 14.7779i	− 0.653737i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.73829	0.208794
$516$	0	0
$517$	8.04576	0.353852
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 6.59861i	− 0.289090i	−0.989498	−	0.144545i	$-0.953828\pi$
$521$	− 6.59861i	− 0.289090i	0.989498	−	0.144545i	$-0.0461719\pi$
$522$	0	0
$523$	39.0957i	1.70954i	0.519010	+	0.854768i	$0.326301\pi$
$523$	39.0957i	1.70954i	−0.519010	+	0.854768i	$0.673699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	44.7620	1.94986
$528$	0	0
$529$	3.83681	0.166818
$530$	0	0
$531$	0	0
$532$	0	0
$533$	41.6713i	1.80499i
$534$	0	0
$535$	8.16495i	0.353001i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.84898	0.208860
$540$	0	0
$541$	−12.7050	−0.546232	−0.273116	−	0.961981i	$-0.588054\pi$
$541$	−12.7050	−0.546232	−0.273116	+	0.961981i	$0.588054\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.5251i	0.536517i
$546$	0	0
$547$	− 15.1980i	− 0.649821i	−0.945745	−	0.324910i	$-0.894666\pi$
$547$	− 15.1980i	− 0.649821i	0.945745	−	0.324910i	$-0.105334\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.8120	1.27003
$552$	0	0
$553$	12.4151	0.527943
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20.0914i	− 0.851300i	−0.904888	−	0.425650i	$-0.860046\pi$
$557$	− 20.0914i	− 0.851300i	0.904888	−	0.425650i	$-0.139954\pi$
$558$	0	0
$559$	43.0096i	1.81911i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.91785	−0.0808278	−0.0404139	−	0.999183i	$-0.512868\pi$
$563$	−1.91785	−0.0808278	−0.0404139	+	0.999183i	$0.512868\pi$
$564$	0	0
$565$	19.7076	0.829105
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 26.4900i	− 1.11052i	−0.831677	−	0.555260i	$-0.812619\pi$
$569$	− 26.4900i	− 1.11052i	0.831677	−	0.555260i	$-0.187381\pi$
$570$	0	0
$571$	12.4925i	0.522794i	0.965231	+	0.261397i	$0.0841832\pi$
$571$	12.4925i	0.522794i	−0.965231	+	0.261397i	$0.915817\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.18043	0.216039
$576$	0	0
$577$	4.72825	0.196839	0.0984197	−	0.995145i	$-0.468621\pi$
$577$	4.72825	0.196839	0.0984197	+	0.995145i	$0.468621\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 12.3041i	− 0.510458i
$582$	0	0
$583$	− 8.96182i	− 0.371161i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.3987	1.04832	0.524158	−	0.851621i	$-0.324380\pi$
$587$	25.3987	1.04832	0.524158	+	0.851621i	$0.324380\pi$
$588$	0	0
$589$	47.0071	1.93689
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.17007i	− 0.0891143i	−0.999007	−	0.0445571i	$-0.985812\pi$
$593$	− 2.17007i	− 0.0891143i	0.999007	−	0.0445571i	$-0.0141877\pi$
$594$	0	0
$595$	9.62149i	0.394443i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.83305	0.197473	0.0987365	−	0.995114i	$-0.468520\pi$
$599$	4.83305	0.197473	0.0987365	+	0.995114i	$0.468520\pi$
$600$	0	0
$601$	−39.4289	−1.60834	−0.804168	−	0.594402i	$-0.797389\pi$
$601$	−39.4289	−1.60834	−0.804168	+	0.594402i	$0.797389\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.0216i	0.407436i
$606$	0	0
$607$	33.0559i	1.34170i	0.741594	+	0.670849i	$0.234070\pi$
$607$	33.0559i	1.34170i	−0.741594	+	0.670849i	$0.765930\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.3792	−1.39083
$612$	0	0
$613$	−9.94455	−0.401657	−0.200828	−	0.979626i	$-0.564363\pi$
$613$	−9.94455	−0.401657	−0.200828	+	0.979626i	$0.564363\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.1535i	0.771091i	0.922689	+	0.385545i	$0.125987\pi$
$617$	19.1535i	0.771091i	−0.922689	+	0.385545i	$0.874013\pi$
$618$	0	0
$619$	20.7016i	0.832068i	0.909349	+	0.416034i	$0.136580\pi$
$619$	20.7016i	0.832068i	−0.909349	+	0.416034i	$0.863420\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.1825	−0.568209
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 39.3136i	− 1.56754i
$630$	0	0
$631$	8.61632i	0.343010i	0.985183	+	0.171505i	$0.0548630\pi$
$631$	8.61632i	0.343010i	−0.985183	+	0.171505i	$0.945137\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.05957	−0.200783
$636$	0	0
$637$	−20.7195	−0.820936
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 33.8782i	− 1.33811i	−0.743214	−	0.669054i	$-0.766700\pi$
$641$	− 33.8782i	− 1.33811i	0.743214	−	0.669054i	$-0.233300\pi$
$642$	0	0
$643$	− 33.9371i	− 1.33835i	−0.743105	−	0.669175i	$-0.766648\pi$
$643$	− 33.9371i	− 1.33835i	0.743105	−	0.669175i	$-0.233352\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.0484	−0.395042	−0.197521	−	0.980299i	$-0.563289\pi$
$647$	−10.0484	−0.395042	−0.197521	+	0.980299i	$0.563289\pi$
$648$	0	0
$649$	0.596174	0.0234019
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.2773i	1.02831i	0.857697	+	0.514156i	$0.171895\pi$
$653$	26.2773i	1.02831i	−0.857697	+	0.514156i	$0.828105\pi$
$654$	0	0
$655$	− 3.84767i	− 0.150341i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−47.4000	−1.84644	−0.923221	−	0.384269i	$-0.874454\pi$
$659$	−47.4000	−1.84644	−0.923221	+	0.384269i	$0.874454\pi$
$660$	0	0
$661$	28.8243	1.12113	0.560567	−	0.828109i	$-0.310583\pi$
$661$	28.8243	1.12113	0.560567	+	0.828109i	$0.310583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.1041i	0.391819i
$666$	0	0
$667$	22.1381i	0.857192i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.43823	−0.171336
$672$	0	0
$673$	−25.5567	−0.985140	−0.492570	−	0.870273i	$-0.663942\pi$
$673$	−25.5567	−0.985140	−0.492570	+	0.870273i	$0.663942\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.6608i	1.33212i	0.745897	+	0.666061i	$0.232021\pi$
$677$	34.6608i	1.33212i	−0.745897	+	0.666061i	$0.767979\pi$
$678$	0	0
$679$	− 19.8452i	− 0.761587i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.7355	−1.48217	−0.741087	−	0.671409i	$-0.765689\pi$
$683$	−38.7355	−1.48217	−0.741087	+	0.671409i	$0.765689\pi$
$684$	0	0
$685$	−8.44997	−0.322857
$686$	0	0
$687$	0	0
$688$	0	0
$689$	38.2935i	1.45886i
$690$	0	0
$691$	− 5.55847i	− 0.211454i	−0.994395	−	0.105727i	$-0.966283\pi$
$691$	− 5.55847i	− 0.211454i	0.994395	−	0.105727i	$-0.0337170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.5386	−0.399752
$696$	0	0
$697$	65.4955	2.48082
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 32.7779i	− 1.23800i	−0.785389	−	0.619002i	$-0.787537\pi$
$701$	− 32.7779i	− 1.23800i	0.785389	−	0.619002i	$-0.212463\pi$
$702$	0	0
$703$	− 41.2854i	− 1.55711i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.80076	0.105333
$708$	0	0
$709$	−40.7368	−1.52990	−0.764952	−	0.644088i	$-0.777237\pi$
$709$	−40.7368	−1.52990	−0.764952	+	0.644088i	$0.777237\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34.9071i	1.30728i
$714$	0	0
$715$	4.18067i	0.156348i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.95735	0.0729967	0.0364984	−	0.999334i	$-0.488380\pi$
$719$	1.95735	0.0729967	0.0364984	+	0.999334i	$0.488380\pi$
$720$	0	0
$721$	−6.86282	−0.255585
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.27342i	0.158711i
$726$	0	0
$727$	22.9128i	0.849787i	0.905243	+	0.424894i	$0.139688\pi$
$727$	22.9128i	0.849787i	−0.905243	+	0.424894i	$0.860312\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	67.5989	2.50023
$732$	0	0
$733$	−4.19364	−0.154896	−0.0774479	−	0.996996i	$-0.524677\pi$
$733$	−4.19364	−0.154896	−0.0774479	+	0.996996i	$0.524677\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.0544i	− 0.407195i
$738$	0	0
$739$	24.0714i	0.885481i	0.896650	+	0.442740i	$0.145994\pi$
$739$	24.0714i	0.885481i	−0.896650	+	0.442740i	$0.854006\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.3550	−1.81066	−0.905329	−	0.424711i	$-0.860376\pi$
$743$	−49.3550	−1.81066	−0.905329	+	0.424711i	$0.860376\pi$
$744$	0	0
$745$	5.83806	0.213890
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 11.8259i	− 0.432109i
$750$	0	0
$751$	− 4.86042i	− 0.177359i	−0.996060	−	0.0886796i	$-0.971735\pi$
$751$	− 4.86042i	− 0.177359i	0.996060	−	0.0886796i	$-0.0282647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.48344	−0.345138
$756$	0	0
$757$	−20.6917	−0.752052	−0.376026	−	0.926609i	$-0.622710\pi$
$757$	−20.6917	−0.752052	−0.376026	+	0.926609i	$0.622710\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 26.9583i	− 0.977236i	−0.872498	−	0.488618i	$-0.837501\pi$
$761$	− 26.9583i	− 0.977236i	0.872498	−	0.488618i	$-0.162499\pi$
$762$	0	0
$763$	− 18.1411i	− 0.656751i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.54742	−0.0919821
$768$	0	0
$769$	39.5131	1.42488	0.712439	−	0.701734i	$-0.247591\pi$
$769$	39.5131	1.42488	0.712439	+	0.701734i	$0.247591\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30.8685i	1.11026i	0.831763	+	0.555131i	$0.187332\pi$
$773$	30.8685i	1.11026i	−0.831763	+	0.555131i	$0.812668\pi$
$774$	0	0
$775$	6.73826i	0.242046i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	68.7806	2.46432
$780$	0	0
$781$	11.6630	0.417337
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 16.3612i	− 0.583958i
$786$	0	0
$787$	2.08529i	0.0743325i	0.999309	+	0.0371662i	$0.0118331\pi$
$787$	2.08529i	0.0743325i	−0.999309	+	0.0371662i	$0.988167\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.5440	−1.01491
$792$	0	0
$793$	18.9643	0.673443
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6863i	0.591060i	0.955333	+	0.295530i	$0.0954962\pi$
$797$	16.6863i	0.591060i	−0.955333	+	0.295530i	$0.904504\pi$
$798$	0	0
$799$	54.0343i	1.91160i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.0923	−0.356150
$804$	0	0
$805$	−7.50320	−0.264453
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.3954i	1.13896i	0.822004	+	0.569481i	$0.192856\pi$
$809$	32.3954i	1.13896i	−0.822004	+	0.569481i	$0.807144\pi$
$810$	0	0
$811$	20.8288i	0.731398i	0.930733	+	0.365699i	$0.119170\pi$
$811$	20.8288i	0.731398i	−0.930733	+	0.365699i	$0.880830\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.00049	−0.210188
$816$	0	0
$817$	70.9895	2.48361
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.4099i	− 1.68952i	−0.535149	−	0.844758i	$-0.679744\pi$
$821$	− 48.4099i	− 1.68952i	0.535149	−	0.844758i	$-0.320256\pi$
$822$	0	0
$823$	53.6158i	1.86893i	0.356056	+	0.934465i	$0.384121\pi$
$823$	53.6158i	1.86893i	−0.356056	+	0.934465i	$0.615879\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33.2303	−1.15553	−0.577766	−	0.816203i	$-0.696075\pi$
$827$	−33.2303	−1.15553	−0.577766	+	0.816203i	$0.696075\pi$
$828$	0	0
$829$	3.74209	0.129968	0.0649842	−	0.997886i	$-0.479300\pi$
$829$	3.74209	0.129968	0.0649842	+	0.997886i	$0.479300\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32.5651i	1.12832i
$834$	0	0
$835$	19.6377i	0.679592i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.4649	−1.05177	−0.525883	−	0.850557i	$-0.676265\pi$
$839$	−30.4649	−1.05177	−0.525883	+	0.850557i	$0.676265\pi$
$840$	0	0
$841$	10.7379	0.370272
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.86381i	− 0.167320i
$846$	0	0
$847$	− 14.5150i	− 0.498743i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.6582	1.05095
$852$	0	0
$853$	−12.5220	−0.428744	−0.214372	−	0.976752i	$-0.568770\pi$
$853$	−12.5220	−0.428744	−0.214372	+	0.976752i	$0.568770\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.14410i	− 0.0390816i	−0.999809	−	0.0195408i	$-0.993780\pi$
$857$	− 1.14410i	− 0.0390816i	0.999809	−	0.0195408i	$-0.00622043\pi$
$858$	0	0
$859$	− 29.2298i	− 0.997306i	−0.866802	−	0.498653i	$-0.833828\pi$
$859$	− 29.2298i	− 0.997306i	0.866802	−	0.498653i	$-0.166172\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.9436	1.56394	0.781970	−	0.623316i	$-0.214215\pi$
$863$	45.9436	1.56394	0.781970	+	0.623316i	$0.214215\pi$
$864$	0	0
$865$	−0.785648	−0.0267128
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.47867i	− 0.287619i
$870$	0	0
$871$	47.2351i	1.60050i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.44838	−0.0489640
$876$	0	0
$877$	−41.5295	−1.40235	−0.701176	−	0.712988i	$-0.747341\pi$
$877$	−41.5295	−1.40235	−0.701176	+	0.712988i	$0.747341\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.6899i	0.360153i	0.983653	+	0.180077i	$0.0576346\pi$
$881$	10.6899i	0.360153i	−0.983653	+	0.180077i	$0.942365\pi$
$882$	0	0
$883$	21.9209i	0.737696i	0.929490	+	0.368848i	$0.120248\pi$
$883$	21.9209i	0.737696i	−0.929490	+	0.368848i	$0.879752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.12408	−0.138473	−0.0692365	−	0.997600i	$-0.522056\pi$
$887$	−4.12408	−0.138473	−0.0692365	+	0.997600i	$0.522056\pi$
$888$	0	0
$889$	7.32816	0.245779
$890$	0	0
$891$	0	0
$892$	0	0
$893$	56.7445i	1.89888i
$894$	0	0
$895$	− 9.34957i	− 0.312522i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28.7954	−0.960381
$900$	0	0
$901$	60.1864	2.00510
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.03967i	0.0678008i
$906$	0	0
$907$	− 21.7948i	− 0.723685i	−0.932239	−	0.361842i	$-0.882148\pi$
$907$	− 21.7948i	− 0.723685i	0.932239	−	0.361842i	$-0.117852\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.0566	1.72471	0.862357	−	0.506301i	$-0.168988\pi$
$911$	52.0566	1.72471	0.862357	+	0.506301i	$0.168988\pi$
$912$	0	0
$913$	−8.40283	−0.278093
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.57288i	0.184033i
$918$	0	0
$919$	− 11.3346i	− 0.373894i	−0.982370	−	0.186947i	$-0.940141\pi$
$919$	− 11.3346i	− 0.373894i	0.982370	−	0.186947i	$-0.0598592\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−49.8357	−1.64036
$924$	0	0
$925$	5.91809	0.194586
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 19.7621i	− 0.648374i	−0.945993	−	0.324187i	$-0.894909\pi$
$929$	− 19.7621i	− 0.648374i	0.945993	−	0.324187i	$-0.105091\pi$
$930$	0	0
$931$	34.1985i	1.12081i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.57082	0.214889
$936$	0	0
$937$	13.0900	0.427631	0.213815	−	0.976874i	$-0.431411\pi$
$937$	13.0900	0.427631	0.213815	+	0.976874i	$0.431411\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.1482i	0.330823i	0.986225	+	0.165412i	$0.0528953\pi$
$941$	10.1482i	0.330823i	−0.986225	+	0.165412i	$0.947105\pi$
$942$	0	0
$943$	51.0759i	1.66326i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.7982	−0.643354	−0.321677	−	0.946850i	$-0.604246\pi$
$947$	−19.7982	−0.643354	−0.321677	+	0.946850i	$0.604246\pi$
$948$	0	0
$949$	43.1241	1.39987
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.5611i	0.763219i	0.924324	+	0.381610i	$0.124630\pi$
$953$	23.5611i	0.763219i	−0.924324	+	0.381610i	$0.875370\pi$
$954$	0	0
$955$	5.55565i	0.179776i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.2387	0.395209
$960$	0	0
$961$	−14.4042	−0.464652
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 17.3479i	− 0.558449i
$966$	0	0
$967$	− 12.2567i	− 0.394149i	−0.980389	−	0.197074i	$-0.936856\pi$
$967$	− 12.2567i	− 0.394149i	0.980389	−	0.197074i	$-0.0631440\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.7410	1.56417	0.782086	−	0.623171i	$-0.214156\pi$
$971$	48.7410	1.56417	0.782086	+	0.623171i	$0.214156\pi$
$972$	0	0
$973$	15.2639	0.489337
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.0401i	− 1.18502i	−0.805564	−	0.592509i	$-0.798138\pi$
$977$	− 37.0401i	− 1.18502i	0.805564	−	0.592509i	$-0.201862\pi$
$978$	0	0
$979$	9.68568i	0.309556i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.8219	−0.791694	−0.395847	−	0.918316i	$-0.629549\pi$
$983$	−24.8219	−0.791694	−0.395847	+	0.918316i	$0.629549\pi$
$984$	0	0
$985$	−5.66466	−0.180491
$986$	0	0
$987$	0	0
$988$	0	0
$989$	52.7162i	1.67628i
$990$	0	0
$991$	28.7539i	0.913399i	0.889621	+	0.456699i	$0.150968\pi$
$991$	28.7539i	0.913399i	−0.889621	+	0.456699i	$0.849032\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.3899	−0.456191
$996$	0	0
$997$	14.4011	0.456087	0.228043	−	0.973651i	$-0.426767\pi$
$997$	14.4011	0.456087	0.228043	+	0.973651i	$0.426767\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.h.c.2591.6	yes	16
3.2	odd	2		4320.2.h.b.2591.14	yes	16
4.3	odd	2		4320.2.h.b.2591.3	✓	16
12.11	even	2	inner	4320.2.h.c.2591.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.h.b.2591.3	✓	16	4.3	odd	2
4320.2.h.b.2591.14	yes	16	3.2	odd	2
4320.2.h.c.2591.6	yes	16	1.1	even	1	trivial
4320.2.h.c.2591.11	yes	16	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +1.44838i q^{7} +0.989142 q^{11} -4.22656 q^{13} +6.64295i q^{17} +6.97614i q^{19} -5.18043 q^{23} -1.00000 q^{25} -4.27342i q^{29} -6.73826i q^{31} +1.44838 q^{35} -5.91809 q^{37} -9.85940i q^{41} -10.1760i q^{43} +8.13408 q^{47} +4.90221 q^{49} -9.06019i q^{53} -0.989142i q^{55} +0.602718 q^{59} -4.48694 q^{61} +4.22656i q^{65} -11.1758i q^{67} +11.7911 q^{71} -10.2031 q^{73} +1.43265i q^{77} -8.57174i q^{79} -8.49507 q^{83} +6.64295 q^{85} +9.79200i q^{89} -6.12165i q^{91} +6.97614 q^{95} -13.7017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{11} - 8 q^{13} - 32 q^{23} - 16 q^{25} + 8 q^{37} - 16 q^{47} - 8 q^{49} + 32 q^{59} + 8 q^{61} + 32 q^{71} - 8 q^{73} + 32 q^{83} - 8 q^{97}+O(q^{100}) \) 16 * q + 8 * q^11 - 8 * q^13 - 32 * q^23 - 16 * q^25 + 8 * q^37 - 16 * q^47 - 8 * q^49 + 32 * q^59 + 8 * q^61 + 32 * q^71 - 8 * q^73 + 32 * q^83 - 8 * q^97

Introduction

L-functions

Modular forms

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Database

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