LMFDB - Embedded newform 4560.2.d.d.2431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.d.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +3.46410i q^{11} +1.00000 q^{15} +(-4.00000 + 1.73205i) q^{19} -3.46410i q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.92820i q^{29} -2.00000 q^{31} +3.46410i q^{33} +6.92820i q^{37} -3.46410i q^{41} +10.3923i q^{43} +1.00000 q^{45} +3.46410i q^{47} +7.00000 q^{49} +3.46410i q^{55} +(-4.00000 + 1.73205i) q^{57} -6.00000 q^{59} +10.0000 q^{61} +4.00000 q^{67} -3.46410i q^{69} -14.0000 q^{73} +1.00000 q^{75} +14.0000 q^{79} +1.00000 q^{81} +13.8564i q^{83} +6.92820i q^{87} +10.3923i q^{89} -2.00000 q^{93} +(-4.00000 + 1.73205i) q^{95} -6.92820i q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 2 q^{15} - 8 q^{19} + 2 q^{25} + 2 q^{27} - 4 q^{31} + 2 q^{45} + 14 q^{49} - 8 q^{57} - 12 q^{59} + 20 q^{61} + 8 q^{67} - 28 q^{73} + 2 q^{75} + 28 q^{79} + 2 q^{81} - 4 q^{93} - 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 2 * q^15 - 8 * q^19 + 2 * q^25 + 2 * q^27 - 4 * q^31 + 2 * q^45 + 14 * q^49 - 8 * q^57 - 12 * q^59 + 20 * q^61 + 8 * q^67 - 28 * q^73 + 2 * q^75 + 28 * q^79 + 2 * q^81 - 4 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$			3.46410i			0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$		0			0
$43$			10.3923i			1.58481i	0.609994	+	0.792406i	$0.291172\pi$
$43$			10.3923i			1.58481i	−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			3.46410i			0.467099i
$56$		0			0
$57$	−4.00000	+	1.73205i	−0.529813	+	0.229416i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		−	3.46410i		−	0.417029i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−14.0000			−1.63858			−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000			−1.63858			−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$		0			0
$78$		0			0
$79$	14.0000			1.57512			0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000			1.57512			0.787562	+	0.616236i	$0.211343\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			13.8564i			1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$			13.8564i			1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			6.92820i			0.742781i
$88$		0			0
$89$			10.3923i			1.10158i	0.834643	+	0.550791i	$0.185674\pi$
$89$			10.3923i			1.10158i	−0.834643	+	0.550791i	$0.814326\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.00000			−0.207390
$94$		0			0
$95$	−4.00000	+	1.73205i	−0.410391	+	0.177705i
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$		−	10.3923i		−	0.995402i	−0.867349	−	0.497701i	$-0.834178\pi$
$109$		−	10.3923i		−	0.995402i	0.867349	−	0.497701i	$-0.165822\pi$
$110$		0			0
$111$			6.92820i			0.657596i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	3.46410i		−	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	3.46410i		−	0.312348i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$			10.3923i			0.914991i
$130$		0			0
$131$		−	3.46410i		−	0.302660i	−0.988483	−	0.151330i	$-0.951644\pi$
$131$		−	3.46410i		−	0.302660i	0.988483	−	0.151330i	$-0.0483556\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$		0			0
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$			3.46410i			0.291730i
$142$		0			0
$143$		0			0
$144$		0			0
$145$			6.92820i			0.575356i
$146$		0			0
$147$	7.00000			0.577350
$148$		0			0
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	17.3205i		−	1.35665i	−0.734763	−	0.678323i	$-0.762707\pi$
$163$		−	17.3205i		−	1.35665i	0.734763	−	0.678323i	$-0.237293\pi$
$164$		0			0
$165$			3.46410i			0.269680i
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−4.00000	+	1.73205i	−0.305888	+	0.132453i
$172$		0			0
$173$			13.8564i			1.05348i	0.850026	+	0.526742i	$0.176586\pi$
$173$			13.8564i			1.05348i	−0.850026	+	0.526742i	$0.823414\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.00000			−0.450988
$178$		0			0
$179$	−6.00000			−0.448461			−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000			−0.448461			−0.224231	+	0.974536i	$0.571987\pi$
$180$		0			0
$181$		−	3.46410i		−	0.257485i	−0.991678	−	0.128742i	$-0.958906\pi$
$181$		−	3.46410i		−	0.257485i	0.991678	−	0.128742i	$-0.0410940\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$			6.92820i			0.509372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$		0			0
$193$		−	6.92820i		−	0.498703i	−0.968413	−	0.249351i	$-0.919783\pi$
$193$		−	6.92820i		−	0.498703i	0.968413	−	0.249351i	$-0.0802174\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	3.46410i		−	0.241943i
$206$		0			0
$207$		−	3.46410i		−	0.240772i
$208$		0			0
$209$	−6.00000	−	13.8564i	−0.415029	−	0.958468i
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			10.3923i			0.708749i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−14.0000			−0.946032
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			3.46410i			0.225973i
$236$		0			0
$237$	14.0000			0.909398
$238$		0			0
$239$			17.3205i			1.12037i	0.828367	+	0.560185i	$0.189270\pi$
$239$			17.3205i			1.12037i	−0.828367	+	0.560185i	$0.810730\pi$
$240$		0			0
$241$		−	6.92820i		−	0.446285i	−0.974786	−	0.223142i	$-0.928369\pi$
$241$		−	6.92820i		−	0.446285i	0.974786	−	0.223142i	$-0.0716315\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	7.00000			0.447214
$246$		0			0
$247$		0			0
$248$		0			0
$249$			13.8564i			0.878114i
$250$		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
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.7846i		−	1.29651i	−0.761424	−	0.648254i	$-0.775499\pi$
$257$		−	20.7846i		−	1.29651i	0.761424	−	0.648254i	$-0.224501\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$			6.92820i			0.428845i
$262$		0			0
$263$			3.46410i			0.213606i	0.994280	+	0.106803i	$0.0340614\pi$
$263$			3.46410i			0.213606i	−0.994280	+	0.106803i	$0.965939\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			10.3923i			0.635999i
$268$		0			0
$269$			6.92820i			0.422420i	0.977441	+	0.211210i	$0.0677404\pi$
$269$			6.92820i			0.422420i	−0.977441	+	0.211210i	$0.932260\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			3.46410i			0.208893i
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$		−	10.3923i		−	0.619953i	−0.950744	−	0.309976i	$-0.899679\pi$
$281$		−	10.3923i		−	0.619953i	0.950744	−	0.309976i	$-0.100321\pi$
$282$		0			0
$283$		−	3.46410i		−	0.205919i	−0.994686	−	0.102960i	$-0.967169\pi$
$283$		−	3.46410i		−	0.205919i	0.994686	−	0.102960i	$-0.0328313\pi$
$284$		0			0
$285$	−4.00000	+	1.73205i	−0.236940	+	0.102598i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	6.92820i		−	0.406138i
$292$		0			0
$293$			27.7128i			1.61900i	0.587120	+	0.809500i	$0.300262\pi$
$293$			27.7128i			1.61900i	−0.587120	+	0.809500i	$0.699738\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$		0			0
$297$			3.46410i			0.201008i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$		−	3.46410i		−	0.196431i	−0.995165	−	0.0982156i	$-0.968687\pi$
$311$		−	3.46410i		−	0.196431i	0.995165	−	0.0982156i	$-0.0313135\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	13.8564i		−	0.778253i	−0.921184	−	0.389127i	$-0.872777\pi$
$317$		−	13.8564i		−	0.778253i	0.921184	−	0.389127i	$-0.127223\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	10.3923i		−	0.574696i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$			6.92820i			0.379663i
$334$		0			0
$335$	4.00000			0.218543
$336$		0			0
$337$		−	6.92820i		−	0.377403i	−0.982034	−	0.188702i	$-0.939572\pi$
$337$		−	6.92820i		−	0.377403i	0.982034	−	0.188702i	$-0.0604279\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		−	3.46410i		−	0.186501i
$346$		0			0
$347$			6.92820i			0.371925i	0.982557	+	0.185963i	$0.0595404\pi$
$347$			6.92820i			0.371925i	−0.982557	+	0.185963i	$0.940460\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			31.1769i			1.64545i	0.568436	+	0.822727i	$0.307549\pi$
$359$			31.1769i			1.64545i	−0.568436	+	0.822727i	$0.692451\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−14.0000			−0.732793
$366$		0			0
$367$		−	6.92820i		−	0.361649i	−0.983515	−	0.180825i	$-0.942123\pi$
$367$		−	6.92820i		−	0.361649i	0.983515	−	0.180825i	$-0.0578766\pi$
$368$		0			0
$369$		−	3.46410i		−	0.180334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	34.6410i		−	1.79364i	−0.442392	−	0.896822i	$-0.645870\pi$
$373$		−	34.6410i		−	1.79364i	0.442392	−	0.896822i	$-0.354130\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	−20.0000			−1.02463
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			10.3923i			0.528271i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	3.46410i		−	0.174741i
$394$		0			0
$395$	14.0000			0.704416
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			24.2487i			1.21092i	0.795875	+	0.605461i	$0.207011\pi$
$401$			24.2487i			1.21092i	−0.795875	+	0.605461i	$0.792989\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$		−	34.6410i		−	1.71289i	−0.516240	−	0.856444i	$-0.672669\pi$
$409$		−	34.6410i		−	1.71289i	0.516240	−	0.856444i	$-0.327331\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		0			0
$414$		0			0
$415$			13.8564i			0.680184i
$416$		0			0
$417$			17.3205i			0.848189i
$418$		0			0
$419$			3.46410i			0.169232i	0.996414	+	0.0846162i	$0.0269664\pi$
$419$			3.46410i			0.169232i	−0.996414	+	0.0846162i	$0.973034\pi$
$420$		0			0
$421$		−	24.2487i		−	1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$		−	24.2487i		−	1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$		0			0
$423$			3.46410i			0.168430i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$			20.7846i			0.998845i	0.866359	+	0.499422i	$0.166454\pi$
$433$			20.7846i			0.998845i	−0.866359	+	0.499422i	$0.833546\pi$
$434$		0			0
$435$			6.92820i			0.332182i
$436$		0			0
$437$	6.00000	+	13.8564i	0.287019	+	0.662842i
$438$		0			0
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$		−	27.7128i		−	1.31668i	−0.752723	−	0.658338i	$-0.771260\pi$
$443$		−	27.7128i		−	1.31668i	0.752723	−	0.658338i	$-0.228740\pi$
$444$		0			0
$445$			10.3923i			0.492642i
$446$		0			0
$447$	18.0000			0.851371
$448$		0			0
$449$		−	3.46410i		−	0.163481i	−0.996654	−	0.0817405i	$-0.973952\pi$
$449$		−	3.46410i		−	0.163481i	0.996654	−	0.0817405i	$-0.0260479\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$	−10.0000			−0.469841
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.0000			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$		−	13.8564i		−	0.643962i	−0.946746	−	0.321981i	$-0.895651\pi$
$463$		−	13.8564i		−	0.643962i	0.946746	−	0.321981i	$-0.104349\pi$
$464$		0			0
$465$	−2.00000			−0.0927478
$466$		0			0
$467$			6.92820i			0.320599i	0.987068	+	0.160300i	$0.0512460\pi$
$467$			6.92820i			0.320599i	−0.987068	+	0.160300i	$0.948754\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4.00000			0.184310
$472$		0			0
$473$	−36.0000			−1.65528
$474$		0			0
$475$	−4.00000	+	1.73205i	−0.183533	+	0.0794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			24.2487i			1.10795i	0.832533	+	0.553976i	$0.186890\pi$
$479$			24.2487i			1.10795i	−0.832533	+	0.553976i	$0.813110\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	6.92820i		−	0.314594i
$486$		0			0
$487$	−32.0000			−1.45006			−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000			−1.45006			−0.725029	+	0.688718i	$0.758174\pi$
$488$		0			0
$489$		−	17.3205i		−	0.783260i
$490$		0			0
$491$		−	31.1769i		−	1.40699i	−0.710698	−	0.703497i	$-0.751621\pi$
$491$		−	31.1769i		−	1.40699i	0.710698	−	0.703497i	$-0.248379\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$			3.46410i			0.155700i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			17.3205i			0.775372i	0.921791	+	0.387686i	$0.126726\pi$
$499$			17.3205i			0.775372i	−0.921791	+	0.387686i	$0.873274\pi$
$500$		0			0
$501$	24.0000			1.07224
$502$		0			0
$503$		−	17.3205i		−	0.772283i	−0.922440	−	0.386142i	$-0.873808\pi$
$503$		−	17.3205i		−	0.772283i	0.922440	−	0.386142i	$-0.126192\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$	13.0000			0.577350
$508$		0			0
$509$			41.5692i			1.84252i	0.388943	+	0.921262i	$0.372840\pi$
$509$			41.5692i			1.84252i	−0.388943	+	0.921262i	$0.627160\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4.00000	+	1.73205i	−0.176604	+	0.0764719i
$514$		0			0
$515$	−4.00000			−0.176261
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$			13.8564i			0.608229i
$520$		0			0
$521$			10.3923i			0.455295i	0.973744	+	0.227648i	$0.0731034\pi$
$521$			10.3923i			0.455295i	−0.973744	+	0.227648i	$0.926897\pi$
$522$		0			0
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$	−6.00000			−0.260378
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−12.0000			−0.518805
$536$		0			0
$537$	−6.00000			−0.258919
$538$		0			0
$539$			24.2487i			1.04447i
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		−	3.46410i		−	0.148659i
$544$		0			0
$545$		−	10.3923i		−	0.445157i
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	−12.0000	−	27.7128i	−0.511217	−	1.18061i
$552$		0			0
$553$		0			0
$554$		0			0
$555$			6.92820i			0.294086i
$556$		0			0
$557$	42.0000			1.77960			0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000			1.77960			0.889799	+	0.456354i	$0.150845\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			45.0333i			1.88790i	0.330096	+	0.943948i	$0.392919\pi$
$569$			45.0333i			1.88790i	−0.330096	+	0.943948i	$0.607081\pi$
$570$		0			0
$571$		−	31.1769i		−	1.30471i	−0.757912	−	0.652357i	$-0.773780\pi$
$571$		−	31.1769i		−	1.30471i	0.757912	−	0.652357i	$-0.226220\pi$
$572$		0			0
$573$			10.3923i			0.434145i
$574$		0			0
$575$		−	3.46410i		−	0.144463i
$576$		0			0
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$		0			0
$579$		−	6.92820i		−	0.287926i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			6.92820i			0.285958i	0.989726	+	0.142979i	$0.0456681\pi$
$587$			6.92820i			0.285958i	−0.989726	+	0.142979i	$0.954332\pi$
$588$		0			0
$589$	8.00000	−	3.46410i	0.329634	−	0.142736i
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$	−36.0000			−1.47834			−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000			−1.47834			−0.739171	+	0.673517i	$0.764783\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			10.3923i			0.425329i
$598$		0			0
$599$	36.0000			1.47092			0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000			1.47092			0.735460	+	0.677568i	$0.236966\pi$
$600$		0			0
$601$		−	41.5692i		−	1.69564i	−0.530281	−	0.847822i	$-0.677914\pi$
$601$		−	41.5692i		−	1.69564i	0.530281	−	0.847822i	$-0.322086\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$	−1.00000			−0.0406558
$606$		0			0
$607$	16.0000			0.649420			0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000			0.649420			0.324710	+	0.945814i	$0.394733\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.00000			0.161558			0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000			0.161558			0.0807792	+	0.996732i	$0.474259\pi$
$614$		0			0
$615$		−	3.46410i		−	0.139686i
$616$		0			0
$617$	−48.0000			−1.93241			−0.966204	−	0.257780i	$-0.917009\pi$
$617$	−48.0000			−1.93241			−0.966204	+	0.257780i	$0.917009\pi$
$618$		0			0
$619$			38.1051i			1.53157i	0.643094	+	0.765787i	$0.277650\pi$
$619$			38.1051i			1.53157i	−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		−	3.46410i		−	0.139010i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−6.00000	−	13.8564i	−0.239617	−	0.553372i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	45.0333i		−	1.79275i	−0.443298	−	0.896374i	$-0.646192\pi$
$631$		−	45.0333i		−	1.79275i	0.443298	−	0.896374i	$-0.353808\pi$
$632$		0			0
$633$	−8.00000			−0.317971
$634$		0			0
$635$	−20.0000			−0.793676
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			10.3923i			0.410471i	0.978713	+	0.205236i	$0.0657961\pi$
$641$			10.3923i			0.410471i	−0.978713	+	0.205236i	$0.934204\pi$
$642$		0			0
$643$		−	10.3923i		−	0.409832i	−0.978780	−	0.204916i	$-0.934308\pi$
$643$		−	10.3923i		−	0.409832i	0.978780	−	0.204916i	$-0.0656922\pi$
$644$		0			0
$645$			10.3923i			0.409197i
$646$		0			0
$647$		−	45.0333i		−	1.77044i	−0.465170	−	0.885221i	$-0.654007\pi$
$647$		−	45.0333i		−	1.77044i	0.465170	−	0.885221i	$-0.345993\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		−	3.46410i		−	0.135354i
$656$		0			0
$657$	−14.0000			−0.546192
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$		0			0
$661$		−	10.3923i		−	0.404214i	−0.979363	−	0.202107i	$-0.935221\pi$
$661$		−	10.3923i		−	0.404214i	0.979363	−	0.202107i	$-0.0647788\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$	8.00000			0.309298
$670$		0			0
$671$			34.6410i			1.33730i
$672$		0			0
$673$		−	34.6410i		−	1.33531i	−0.744469	−	0.667657i	$-0.767297\pi$
$673$		−	34.6410i		−	1.33531i	0.744469	−	0.667657i	$-0.232703\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$	14.0000			0.534133
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	3.46410i		−	0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$		−	3.46410i		−	0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.3205i			0.657004i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	−12.0000	−	27.7128i	−0.452589	−	1.04521i
$704$		0			0
$705$			3.46410i			0.130466i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−14.0000			−0.525781			−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000			−0.525781			−0.262891	+	0.964826i	$0.584676\pi$
$710$		0			0
$711$	14.0000			0.525041
$712$		0			0
$713$			6.92820i			0.259463i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			17.3205i			0.646846i
$718$		0			0
$719$		−	10.3923i		−	0.387568i	−0.981044	−	0.193784i	$-0.937924\pi$
$719$		−	10.3923i		−	0.387568i	0.981044	−	0.193784i	$-0.0620760\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	6.92820i		−	0.257663i
$724$		0			0
$725$			6.92820i			0.257307i
$726$		0			0
$727$		−	13.8564i		−	0.513906i	−0.966424	−	0.256953i	$-0.917281\pi$
$727$		−	13.8564i		−	0.513906i	0.966424	−	0.256953i	$-0.0827185\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	40.0000			1.47743			0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000			1.47743			0.738717	+	0.674016i	$0.235432\pi$
$734$		0			0
$735$	7.00000			0.258199
$736$		0			0
$737$			13.8564i			0.510407i
$738$		0			0
$739$		−	17.3205i		−	0.637145i	−0.947898	−	0.318573i	$-0.896797\pi$
$739$		−	17.3205i		−	0.637145i	0.947898	−	0.318573i	$-0.103203\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	18.0000			0.659469
$746$		0			0
$747$			13.8564i			0.506979i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−22.0000			−0.802791			−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000			−0.802791			−0.401396	+	0.915905i	$0.631475\pi$
$752$		0			0
$753$		−	3.46410i		−	0.126239i
$754$		0			0
$755$	−10.0000			−0.363937
$756$		0			0
$757$	8.00000			0.290765			0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000			0.290765			0.145382	+	0.989376i	$0.453559\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$		−	20.7846i		−	0.748539i
$772$		0			0
$773$			27.7128i			0.996761i	0.866959	+	0.498380i	$0.166072\pi$
$773$			27.7128i			0.996761i	−0.866959	+	0.498380i	$0.833928\pi$
$774$		0			0
$775$	−2.00000			−0.0718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.00000	+	13.8564i	0.214972	+	0.496457i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			6.92820i			0.247594i
$784$		0			0
$785$	4.00000			0.142766
$786$		0			0
$787$	4.00000			0.142585			0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000			0.142585			0.0712923	+	0.997455i	$0.477288\pi$
$788$		0			0
$789$			3.46410i			0.123325i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	41.5692i		−	1.47246i	−0.676733	−	0.736229i	$-0.736605\pi$
$797$		−	41.5692i		−	1.47246i	0.676733	−	0.736229i	$-0.263395\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			10.3923i			0.367194i
$802$		0			0
$803$		−	48.4974i		−	1.71144i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			6.92820i			0.243884i
$808$		0			0
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$			10.3923i			0.364474i
$814$		0			0
$815$		−	17.3205i		−	0.606711i
$816$		0			0
$817$	−18.0000	−	41.5692i	−0.629740	−	1.45432i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−54.0000			−1.88461			−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000			−1.88461			−0.942306	+	0.334751i	$0.891348\pi$
$822$		0			0
$823$		−	41.5692i		−	1.44901i	−0.689269	−	0.724506i	$-0.742068\pi$
$823$		−	41.5692i		−	1.44901i	0.689269	−	0.724506i	$-0.257932\pi$
$824$		0			0
$825$			3.46410i			0.120605i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		−	3.46410i		−	0.120313i	−0.998189	−	0.0601566i	$-0.980840\pi$
$829$		−	3.46410i		−	0.120313i	0.998189	−	0.0601566i	$-0.0191600\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$		0			0
$833$		0			0
$834$		0			0
$835$	24.0000			0.830554
$836$		0			0
$837$	−2.00000			−0.0691301
$838$		0			0
$839$	−36.0000			−1.24286			−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000			−1.24286			−0.621429	+	0.783470i	$0.713448\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$		−	10.3923i		−	0.357930i
$844$		0			0
$845$	13.0000			0.447214
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	3.46410i		−	0.118888i
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$	−40.0000			−1.36957			−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000			−1.36957			−0.684787	+	0.728743i	$0.740105\pi$
$854$		0			0
$855$	−4.00000	+	1.73205i	−0.136797	+	0.0592349i
$856$		0			0
$857$			6.92820i			0.236663i	0.992974	+	0.118331i	$0.0377545\pi$
$857$			6.92820i			0.236663i	−0.992974	+	0.118331i	$0.962245\pi$
$858$		0			0
$859$		−	3.46410i		−	0.118194i	−0.998252	−	0.0590968i	$-0.981178\pi$
$859$		−	3.46410i		−	0.118194i	0.998252	−	0.0590968i	$-0.0188221\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$			13.8564i			0.471132i
$866$		0			0
$867$	−17.0000			−0.577350
$868$		0			0
$869$			48.4974i			1.64516i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	6.92820i		−	0.234484i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$			27.7128i			0.934730i
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$		−	24.2487i		−	0.816034i	−0.912974	−	0.408017i	$-0.866220\pi$
$883$		−	24.2487i		−	0.816034i	0.912974	−	0.408017i	$-0.133780\pi$
$884$		0			0
$885$	−6.00000			−0.201688
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			3.46410i			0.116052i
$892$		0			0
$893$	−6.00000	−	13.8564i	−0.200782	−	0.463687i
$894$		0			0
$895$	−6.00000			−0.200558
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	13.8564i		−	0.462137i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.46410i		−	0.115151i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	−48.0000			−1.58857
$914$		0			0
$915$	10.0000			0.330590
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	3.46410i		−	0.114270i	−0.998366	−	0.0571351i	$-0.981803\pi$
$919$		−	3.46410i		−	0.114270i	0.998366	−	0.0571351i	$-0.0181966\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$		0			0
$924$		0			0
$925$			6.92820i			0.227798i
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	−28.0000	+	12.1244i	−0.917663	+	0.397360i
$932$		0			0
$933$		−	3.46410i		−	0.113410i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	10.0000			0.326338
$940$		0			0
$941$			20.7846i			0.677559i	0.940866	+	0.338779i	$0.110014\pi$
$941$			20.7846i			0.677559i	−0.940866	+	0.338779i	$0.889986\pi$
$942$		0			0
$943$	−12.0000			−0.390774
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6.92820i		−	0.225136i	−0.993644	−	0.112568i	$-0.964092\pi$
$947$		−	6.92820i		−	0.225136i	0.993644	−	0.112568i	$-0.0359077\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		−	13.8564i		−	0.449325i
$952$		0			0
$953$		−	34.6410i		−	1.12213i	−0.827771	−	0.561066i	$-0.810391\pi$
$953$		−	34.6410i		−	1.12213i	0.827771	−	0.561066i	$-0.189609\pi$
$954$		0			0
$955$			10.3923i			0.336287i
$956$		0			0
$957$	−24.0000			−0.775810
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$		−	6.92820i		−	0.223027i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−30.0000			−0.962746			−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000			−0.962746			−0.481373	+	0.876516i	$0.659862\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.7846i		−	0.664959i	−0.943111	−	0.332479i	$-0.892115\pi$
$977$		−	20.7846i		−	0.664959i	0.943111	−	0.332479i	$-0.107885\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$		0			0
$981$		−	10.3923i		−	0.331801i
$982$		0			0
$983$	48.0000			1.53096			0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000			1.53096			0.765481	+	0.643458i	$0.222501\pi$
$984$		0			0
$985$	−6.00000			−0.191176
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.0000			1.14473
$990$		0			0
$991$	−2.00000			−0.0635321			−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000			−0.0635321			−0.0317660	+	0.999495i	$0.510113\pi$
$992$		0			0
$993$	−4.00000			−0.126936
$994$		0			0
$995$			10.3923i			0.329458i
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$			6.92820i			0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.d.2431.2	yes	2
4.3	odd	2		4560.2.d.b.2431.1	✓	2
19.18	odd	2		4560.2.d.b.2431.2	yes	2
76.75	even	2	inner	4560.2.d.d.2431.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.b.2431.1	✓	2	4.3	odd	2
4560.2.d.b.2431.2	yes	2	19.18	odd	2
4560.2.d.d.2431.1	yes	2	76.75	even	2	inner
4560.2.d.d.2431.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +3.46410i q^{11} +1.00000 q^{15} +(-4.00000 + 1.73205i) q^{19} -3.46410i q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.92820i q^{29} -2.00000 q^{31} +3.46410i q^{33} +6.92820i q^{37} -3.46410i q^{41} +10.3923i q^{43} +1.00000 q^{45} +3.46410i q^{47} +7.00000 q^{49} +3.46410i q^{55} +(-4.00000 + 1.73205i) q^{57} -6.00000 q^{59} +10.0000 q^{61} +4.00000 q^{67} -3.46410i q^{69} -14.0000 q^{73} +1.00000 q^{75} +14.0000 q^{79} +1.00000 q^{81} +13.8564i q^{83} +6.92820i q^{87} +10.3923i q^{89} -2.00000 q^{93} +(-4.00000 + 1.73205i) q^{95} -6.92820i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 2 q^{15} - 8 q^{19} + 2 q^{25} + 2 q^{27} - 4 q^{31} + 2 q^{45} + 14 q^{49} - 8 q^{57} - 12 q^{59} + 20 q^{61} + 8 q^{67} - 28 q^{73} + 2 q^{75} + 28 q^{79} + 2 q^{81} - 4 q^{93} - 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 2 * q^15 - 8 * q^19 + 2 * q^25 + 2 * q^27 - 4 * q^31 + 2 * q^45 + 14 * q^49 - 8 * q^57 - 12 * q^59 + 20 * q^61 + 8 * q^67 - 28 * q^73 + 2 * q^75 + 28 * q^79 + 2 * q^81 - 4 * q^93 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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