LMFDB - Embedded newform 3744.2.m.h.1585.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1585,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3744.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$29.8959905168$
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} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.5
Root			$0.556839 - 1.81878i$ of defining polynomial
Character	$\chi$	$=$	3744.1585
Dual form			3744.2.m.h.1585.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.66251 q^{5} -3.57266i q^{7} +O(q^{10})$	$q-1.66251 q^{5} -3.57266i q^{7} +1.02749 q^{11} +(-1.87826 - 3.07768i) q^{13} -5.05251 q^{17} +1.16083 q^{19} -8.17513 q^{23} -2.23607 q^{25} -4.29792i q^{29} +7.98872i q^{31} +5.93958i q^{35} +9.83470 q^{37} -1.62054i q^{41} +2.35114i q^{43} +7.73877i q^{47} -5.76393 q^{49} +11.2521i q^{53} -1.70820 q^{55} +9.73249 q^{59} -11.4127i q^{61} +(3.12262 + 5.11667i) q^{65} -7.23901 q^{67} +10.5672i q^{71} -4.41606i q^{73} -3.67086i q^{77} +6.94427 q^{79} +2.29753 q^{83} +8.39984 q^{85} +9.02546i q^{89} +(-10.9955 + 6.71040i) q^{91} -1.92989 q^{95} -11.5614i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100}) $ 16 * q - 128 * q^49 + 80 * q^55 - 32 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	−1.66251			−0.743496			−0.371748	−	0.928334i	$-0.621241\pi$
$5$	−1.66251			−0.743496			−0.371748	+	0.928334i	$0.621241\pi$
$6$		0			0
$7$		−	3.57266i		−	1.35034i	−0.737662	−	0.675170i	$-0.764070\pi$
$7$		−	3.57266i		−	1.35034i	0.737662	−	0.675170i	$-0.235930\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.02749			0.309799			0.154899	−	0.987930i	$-0.450495\pi$
$11$	1.02749			0.309799			0.154899	+	0.987930i	$0.450495\pi$
$12$		0			0
$13$	−1.87826	−	3.07768i	−0.520936	−	0.853596i
$13$	−1.87826	−	3.07768i	−0.520936	−	0.853596i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.05251			−1.22541			−0.612707	−	0.790310i	$-0.709919\pi$
$17$	−5.05251			−1.22541			−0.612707	+	0.790310i	$0.709919\pi$
$18$		0			0
$19$	1.16083			0.266312			0.133156	−	0.991095i	$-0.457489\pi$
$19$	1.16083			0.266312			0.133156	+	0.991095i	$0.457489\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.17513			−1.70463			−0.852317	−	0.523026i	$-0.824803\pi$
$23$	−8.17513			−1.70463			−0.852317	+	0.523026i	$0.824803\pi$
$24$		0			0
$25$	−2.23607			−0.447214
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.29792i		−	0.798104i	−0.916928	−	0.399052i	$-0.869339\pi$
$29$		−	4.29792i		−	0.798104i	0.916928	−	0.399052i	$-0.130661\pi$
$30$		0			0
$31$			7.98872i			1.43482i	0.696653	+	0.717408i	$0.254672\pi$
$31$			7.98872i			1.43482i	−0.696653	+	0.717408i	$0.745328\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.93958i			1.00397i
$36$		0			0
$37$	9.83470			1.61682			0.808408	−	0.588623i	$-0.200330\pi$
$37$	9.83470			1.61682			0.808408	+	0.588623i	$0.200330\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.62054i		−	0.253087i	−0.991961	−	0.126543i	$-0.959612\pi$
$41$		−	1.62054i		−	0.253087i	0.991961	−	0.126543i	$-0.0403883\pi$
$42$		0			0
$43$			2.35114i			0.358546i	0.983799	+	0.179273i	$0.0573745\pi$
$43$			2.35114i			0.358546i	−0.983799	+	0.179273i	$0.942626\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.73877i			1.12882i	0.825496	+	0.564408i	$0.190895\pi$
$47$			7.73877i			1.12882i	−0.825496	+	0.564408i	$0.809105\pi$
$48$		0			0
$49$	−5.76393			−0.823419
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.2521i			1.54560i	0.634652	+	0.772798i	$0.281143\pi$
$53$			11.2521i			1.54560i	−0.634652	+	0.772798i	$0.718857\pi$
$54$		0			0
$55$	−1.70820			−0.230334
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.73249			1.26706			0.633531	−	0.773717i	$-0.281605\pi$
$59$	9.73249			1.26706			0.633531	+	0.773717i	$0.281605\pi$
$60$		0			0
$61$		−	11.4127i		−	1.46124i	−0.682782	−	0.730622i	$-0.739230\pi$
$61$		−	11.4127i		−	1.46124i	0.682782	−	0.730622i	$-0.260770\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.12262	+	5.11667i	0.387314	+	0.634645i
$66$		0			0
$67$	−7.23901			−0.884386			−0.442193	−	0.896920i	$-0.645799\pi$
$67$	−7.23901			−0.884386			−0.442193	+	0.896920i	$0.645799\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.5672i			1.25410i	0.778981	+	0.627048i	$0.215737\pi$
$71$			10.5672i			1.25410i	−0.778981	+	0.627048i	$0.784263\pi$
$72$		0			0
$73$		−	4.41606i		−	0.516860i	−0.966030	−	0.258430i	$-0.916795\pi$
$73$		−	4.41606i		−	0.516860i	0.966030	−	0.258430i	$-0.0832052\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.67086i		−	0.418334i
$78$		0			0
$79$	6.94427			0.781292			0.390646	−	0.920541i	$-0.372252\pi$
$79$	6.94427			0.781292			0.390646	+	0.920541i	$0.372252\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.29753			0.252187			0.126093	−	0.992018i	$-0.459756\pi$
$83$	2.29753			0.252187			0.126093	+	0.992018i	$0.459756\pi$
$84$		0			0
$85$	8.39984			0.911090
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.02546i			0.956697i	0.878170	+	0.478349i	$0.158764\pi$
$89$			9.02546i			0.956697i	−0.878170	+	0.478349i	$0.841236\pi$
$90$		0			0
$91$	−10.9955	+	6.71040i	−1.15264	+	0.703441i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.92989			−0.198002
$96$		0			0
$97$		−	11.5614i		−	1.17388i	−0.809630	−	0.586940i	$-0.800332\pi$
$97$		−	11.5614i		−	1.17388i	0.809630	−	0.586940i	$-0.199668\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.96879i		−	0.792924i	−0.918051	−	0.396462i	$-0.870238\pi$
$101$		−	7.96879i		−	0.792924i	0.918051	−	0.396462i	$-0.129762\pi$
$102$		0			0
$103$	11.7082			1.15364			0.576822	−	0.816870i	$-0.304293\pi$
$103$	11.7082			1.15364			0.576822	+	0.816870i	$0.304293\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.65626i		−	0.256791i	−0.991723	−	0.128395i	$-0.959017\pi$
$107$		−	2.65626i		−	0.256791i	0.991723	−	0.128395i	$-0.0409826\pi$
$108$		0			0
$109$	−12.1564			−1.16437			−0.582184	−	0.813057i	$-0.697802\pi$
$109$	−12.1564			−1.16437			−0.582184	+	0.813057i	$0.697802\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.05251			−0.475300			−0.237650	−	0.971351i	$-0.576377\pi$
$113$	−5.05251			−0.475300			−0.237650	+	0.971351i	$0.576377\pi$
$114$		0			0
$115$	13.5912			1.26739
$116$		0			0
$117$		0			0
$118$		0			0
$119$			18.0509i			1.65473i
$120$		0			0
$121$	−9.94427			−0.904025
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0300			1.07600
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			16.5646i			1.44726i	0.690189	+	0.723629i	$0.257527\pi$
$131$			16.5646i			1.44726i	−0.690189	+	0.723629i	$0.742473\pi$
$132$		0			0
$133$		−	4.14725i		−	0.359612i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			13.6020i			1.16209i	0.813870	+	0.581047i	$0.197357\pi$
$137$			13.6020i			1.16209i	−0.813870	+	0.581047i	$0.802643\pi$
$138$		0			0
$139$			19.3642i			1.64245i	0.570607	+	0.821223i	$0.306708\pi$
$139$			19.3642i			1.64245i	−0.570607	+	0.821223i	$0.693292\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.92989	−	3.16228i	−0.161385	−	0.264443i
$144$		0			0
$145$			7.14533i			0.593387i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.04250			−0.576944			−0.288472	−	0.957488i	$-0.593147\pi$
$149$	−7.04250			−0.576944			−0.288472	+	0.957488i	$0.593147\pi$
$150$		0			0
$151$			3.57266i			0.290739i	0.989377	+	0.145370i	$0.0464371\pi$
$151$			3.57266i			0.290739i	−0.989377	+	0.145370i	$0.953563\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	13.2813i		−	1.06678i
$156$		0			0
$157$		−	0.555029i		−	0.0442961i	−0.999755	−	0.0221481i	$-0.992949\pi$
$157$		−	0.555029i		−	0.0442961i	0.999755	−	0.0221481i	$-0.00705053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			29.2070i			2.30183i
$162$		0			0
$163$	−7.23901			−0.567003			−0.283501	−	0.958972i	$-0.591496\pi$
$163$	−7.23901			−0.567003			−0.283501	+	0.958972i	$0.591496\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.32611i			0.566911i	0.958985	+	0.283456i	$0.0914809\pi$
$167$			7.32611i			0.566911i	−0.958985	+	0.283456i	$0.908519\pi$
$168$		0			0
$169$	−5.94427	+	11.5614i	−0.457252	+	0.889337i
$170$		0			0
$171$		0			0
$172$		0			0
$173$			11.2521i			0.855482i	0.903901	+	0.427741i	$0.140690\pi$
$173$			11.2521i			0.855482i	−0.903901	+	0.427741i	$0.859310\pi$
$174$		0			0
$175$			7.98872i			0.603891i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	21.8772i		−	1.63518i	−0.575804	−	0.817588i	$-0.695311\pi$
$179$		−	21.8772i		−	1.63518i	0.575804	−	0.817588i	$-0.304689\pi$
$180$		0			0
$181$			12.8658i			0.956305i	0.878277	+	0.478152i	$0.158693\pi$
$181$			12.8658i			0.956305i	−0.878277	+	0.478152i	$0.841307\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.3503			−1.20210
$186$		0			0
$187$	−5.19139			−0.379632
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.92989			−0.139642			−0.0698209	−	0.997560i	$-0.522243\pi$
$191$	−1.92989			−0.139642			−0.0698209	+	0.997560i	$0.522243\pi$
$192$		0			0
$193$		−	4.41606i		−	0.317875i	−0.987289	−	0.158937i	$-0.949193\pi$
$193$		−	4.41606i		−	0.317875i	0.987289	−	0.158937i	$-0.0508068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	23.1825			1.65168			0.825841	−	0.563903i	$-0.190701\pi$
$197$	23.1825			1.65168			0.825841	+	0.563903i	$0.190701\pi$
$198$		0			0
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−15.3550			−1.07771
$204$		0			0
$205$			2.69417i			0.188169i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.19274			0.0825033
$210$		0			0
$211$			10.5146i			0.723856i	0.932206	+	0.361928i	$0.117881\pi$
$211$			10.5146i			0.723856i	−0.932206	+	0.361928i	$0.882119\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.90879i		−	0.266577i
$216$		0			0
$217$	28.5410			1.93749
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.48993	+	15.5500i	0.638362	+	1.04601i
$222$		0			0
$223$		−	7.98872i		−	0.534964i	−0.963563	−	0.267482i	$-0.913808\pi$
$223$		−	7.98872i		−	0.534964i	0.963563	−	0.267482i	$-0.0861916\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.35250			−0.288886			−0.144443	−	0.989513i	$-0.546139\pi$
$227$	−4.35250			−0.288886			−0.144443	+	0.989513i	$0.546139\pi$
$228$		0			0
$229$	1.43486			0.0948185			0.0474092	−	0.998876i	$-0.484904\pi$
$229$	1.43486			0.0948185			0.0474092	+	0.998876i	$0.484904\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.3503			−1.07114			−0.535571	−	0.844490i	$-0.679904\pi$
$233$	−16.3503			−1.07114			−0.535571	+	0.844490i	$0.679904\pi$
$234$		0			0
$235$		−	12.8658i		−	0.839270i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.07107i		−	0.457389i	−0.973498	−	0.228695i	$-0.926554\pi$
$239$		−	7.07107i		−	0.457389i	0.973498	−	0.228695i	$-0.0734457\pi$
$240$		0			0
$241$			15.9774i			1.02920i	0.857431	+	0.514599i	$0.172059\pi$
$241$			15.9774i			1.02920i	−0.857431	+	0.514599i	$0.827941\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.58258			0.612209
$246$		0			0
$247$	−2.18034	−	3.57266i	−0.138732	−	0.227323i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			22.5042i			1.42045i	0.703973	+	0.710227i	$0.251408\pi$
$251$			22.5042i			1.42045i	−0.703973	+	0.710227i	$0.748592\pi$
$252$		0			0
$253$	−8.39984			−0.528093
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.5955			−1.40947			−0.704735	−	0.709471i	$-0.748934\pi$
$257$	−22.5955			−1.40947			−0.704735	+	0.709471i	$0.748934\pi$
$258$		0			0
$259$		−	35.1361i		−	2.18325i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.3503			−1.00820			−0.504100	−	0.863645i	$-0.668176\pi$
$263$	−16.3503			−1.00820			−0.504100	+	0.863645i	$0.668176\pi$
$264$		0			0
$265$		−	18.7067i		−	1.14914i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			7.58124i			0.462237i	0.972926	+	0.231118i	$0.0742384\pi$
$269$			7.58124i			0.462237i	−0.972926	+	0.231118i	$0.925762\pi$
$270$		0			0
$271$		−	7.98872i		−	0.485280i	−0.970116	−	0.242640i	$-0.921987\pi$
$271$		−	7.98872i		−	0.485280i	0.970116	−	0.242640i	$-0.0780134\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.29753			−0.138546
$276$		0			0
$277$			18.1231i			1.08891i	0.838790	+	0.544455i	$0.183263\pi$
$277$			18.1231i			1.08891i	−0.838790	+	0.544455i	$0.816737\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6.19704i		−	0.369684i	−0.982768	−	0.184842i	$-0.940823\pi$
$281$		−	6.19704i		−	0.369684i	0.982768	−	0.184842i	$-0.0591774\pi$
$282$		0			0
$283$			2.90617i			0.172754i	0.996263	+	0.0863769i	$0.0275289\pi$
$283$			2.90617i			0.172754i	−0.996263	+	0.0863769i	$0.972471\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.78966			−0.341753
$288$		0			0
$289$	8.52786			0.501639
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.877578			−0.0512687			−0.0256343	−	0.999671i	$-0.508161\pi$
$293$	−0.877578			−0.0512687			−0.0256343	+	0.999671i	$0.508161\pi$
$294$		0			0
$295$	−16.1803			−0.942056
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.3550	+	25.1605i	0.888005	+	1.45507i
$300$		0			0
$301$	8.39984			0.484159
$302$		0			0
$303$		0			0
$304$		0			0
$305$			18.9737i			1.08643i
$306$		0			0
$307$	14.7521			0.841944			0.420972	−	0.907074i	$-0.361689\pi$
$307$	14.7521			0.841944			0.420972	+	0.907074i	$0.361689\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.5254			−1.39071			−0.695354	−	0.718667i	$-0.744752\pi$
$311$	−24.5254			−1.39071			−0.695354	+	0.718667i	$0.744752\pi$
$312$		0			0
$313$	6.47214			0.365827			0.182913	−	0.983129i	$-0.441447\pi$
$313$	6.47214			0.365827			0.182913	+	0.983129i	$0.441447\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.1025			−1.74689			−0.873446	−	0.486921i	$-0.838120\pi$
$317$	−31.1025			−1.74689			−0.873446	+	0.486921i	$0.838120\pi$
$318$		0			0
$319$		−	4.41606i		−	0.247252i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.86510			−0.326343
$324$		0			0
$325$	4.19992	+	6.88191i	0.232970	+	0.381740i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	27.6480			1.52428
$330$		0			0
$331$	10.9955			0.604369			0.302185	−	0.953249i	$-0.402284\pi$
$331$	10.9955			0.604369			0.302185	+	0.953249i	$0.402284\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0349			0.657537
$336$		0			0
$337$	5.52786			0.301122			0.150561	−	0.988601i	$-0.451892\pi$
$337$	5.52786			0.301122			0.150561	+	0.988601i	$0.451892\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.20830i			0.444504i
$342$		0			0
$343$		−	4.41606i		−	0.238445i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.4750i		−	1.09916i	−0.835442	−	0.549578i	$-0.814788\pi$
$347$		−	20.4750i		−	1.09916i	0.835442	−	0.549578i	$-0.185212\pi$
$348$		0			0
$349$	−13.5912			−0.727522			−0.363761	−	0.931492i	$-0.618507\pi$
$349$	−13.5912			−0.727522			−0.363761	+	0.931492i	$0.618507\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.5942i		−	1.09612i	−0.836439	−	0.548060i	$-0.815367\pi$
$353$		−	20.5942i		−	1.09612i	0.836439	−	0.548060i	$-0.184633\pi$
$354$		0			0
$355$		−	17.5680i		−	0.932415i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			32.1142i			1.69493i	0.530855	+	0.847463i	$0.321871\pi$
$359$			32.1142i			1.69493i	−0.530855	+	0.847463i	$0.678129\pi$
$360$		0			0
$361$	−17.6525			−0.929078
$362$		0			0
$363$		0			0
$364$		0			0
$365$			7.34173i			0.384284i
$366$		0			0
$367$	−31.1246			−1.62469			−0.812346	−	0.583176i	$-0.801810\pi$
$367$	−31.1246			−1.62469			−0.812346	+	0.583176i	$0.801810\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	40.2000			2.08708
$372$		0			0
$373$		−	26.0746i		−	1.35009i	−0.737777	−	0.675045i	$-0.764124\pi$
$373$		−	26.0746i		−	1.35009i	0.737777	−	0.675045i	$-0.235876\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.2276	+	8.07262i	−0.681258	+	0.415761i
$378$		0			0
$379$	19.3954			0.996273			0.498137	−	0.867099i	$-0.334018\pi$
$379$	19.3954			0.996273			0.498137	+	0.867099i	$0.334018\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.57494i		−	0.182671i	−0.995820	−	0.0913354i	$-0.970886\pi$
$383$		−	3.57494i		−	0.182671i	0.995820	−	0.0913354i	$-0.0291135\pi$
$384$		0			0
$385$			6.10284i			0.311030i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	35.3980i		−	1.79475i	−0.441270	−	0.897374i	$-0.645472\pi$
$389$		−	35.3980i		−	1.79475i	0.441270	−	0.897374i	$-0.354528\pi$
$390$		0			0
$391$	41.3050			2.08888
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.5449			−0.580887
$396$		0			0
$397$	−20.5562			−1.03169			−0.515843	−	0.856683i	$-0.672521\pi$
$397$	−20.5562			−1.03169			−0.515843	+	0.856683i	$0.672521\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	30.4149i		−	1.51885i	−0.650596	−	0.759424i	$-0.725481\pi$
$401$		−	30.4149i		−	1.51885i	0.650596	−	0.759424i	$-0.274519\pi$
$402$		0			0
$403$	24.5868	−	15.0049i	1.22475	−	0.747447i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.1050			0.500887
$408$		0			0
$409$			34.6842i			1.71502i	0.514466	+	0.857511i	$0.327990\pi$
$409$			34.6842i			1.71502i	−0.514466	+	0.857511i	$0.672010\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	34.7709i		−	1.71097i
$414$		0			0
$415$	−3.81966			−0.187500
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	7.96879i		−	0.389301i	−0.980873	−	0.194650i	$-0.937643\pi$
$419$		−	7.96879i		−	0.389301i	0.980873	−	0.194650i	$-0.0623572\pi$
$420$		0			0
$421$	8.39984			0.409383			0.204692	−	0.978827i	$-0.434381\pi$
$421$	8.39984			0.409383			0.204692	+	0.978827i	$0.434381\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.2978			0.548022
$426$		0			0
$427$	−40.7737			−1.97318
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.82688i		−	0.0879975i	−0.999032	−	0.0439987i	$-0.985990\pi$
$431$		−	1.82688i		−	0.0879975i	0.999032	−	0.0439987i	$-0.0140098\pi$
$432$		0			0
$433$	−8.18034			−0.393122			−0.196561	−	0.980492i	$-0.562977\pi$
$433$	−8.18034			−0.393122			−0.196561	+	0.980492i	$0.562977\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.48993			−0.453965
$438$		0			0
$439$	−1.81966			−0.0868476			−0.0434238	−	0.999057i	$-0.513827\pi$
$439$	−1.81966			−0.0868476			−0.0434238	+	0.999057i	$0.513827\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			11.2521i			0.534604i	0.963613	+	0.267302i	$0.0861321\pi$
$443$			11.2521i			0.534604i	−0.963613	+	0.267302i	$0.913868\pi$
$444$		0			0
$445$		−	15.0049i		−	0.711301i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.86163i			0.229435i	0.993398	+	0.114717i	$0.0365962\pi$
$449$			4.86163i			0.229435i	−0.993398	+	0.114717i	$0.963404\pi$
$450$		0			0
$451$		−	1.66509i		−	0.0784059i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18.2802	−	11.1561i	0.856987	−	0.523005i
$456$		0			0
$457$		−	17.0199i		−	0.796159i	−0.917351	−	0.398079i	$-0.869677\pi$
$457$		−	17.0199i		−	0.796159i	0.917351	−	0.398079i	$-0.130323\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.9124			−1.02056			−0.510282	−	0.860007i	$-0.670459\pi$
$461$	−21.9124			−1.02056			−0.510282	+	0.860007i	$0.670459\pi$
$462$		0			0
$463$		−	3.57266i		−	0.166036i	−0.996548	−	0.0830179i	$-0.973544\pi$
$463$		−	3.57266i		−	0.166036i	0.996548	−	0.0830179i	$-0.0264559\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	8.59584i		−	0.397768i	−0.980023	−	0.198884i	$-0.936268\pi$
$467$		−	8.59584i		−	0.397768i	0.980023	−	0.198884i	$-0.0637317\pi$
$468$		0			0
$469$			25.8626i			1.19422i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.41577i			0.111077i
$474$		0			0
$475$	−2.59569			−0.119099
$476$		0			0
$477$		0			0
$478$		0			0
$479$			29.2858i			1.33810i	0.743216	+	0.669052i	$0.233300\pi$
$479$			29.2858i			1.33810i	−0.743216	+	0.669052i	$0.766700\pi$
$480$		0			0
$481$	−18.4721	−	30.2681i	−0.842257	−	1.38011i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			19.2209i			0.872776i
$486$		0			0
$487$		−	0.843392i		−	0.0382177i	−0.999817	−	0.0191089i	$-0.993917\pi$
$487$		−	0.843392i		−	0.0382177i	0.999817	−	0.0191089i	$-0.00608291\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2.02920i		−	0.0915767i	−0.998951	−	0.0457883i	$-0.985420\pi$
$491$		−	2.02920i		−	0.0915767i	0.998951	−	0.0457883i	$-0.0145800\pi$
$492$		0			0
$493$			21.7153i			0.978008i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	37.7530			1.69346
$498$		0			0
$499$	33.5347			1.50122			0.750609	−	0.660747i	$-0.229760\pi$
$499$	33.5347			1.50122			0.750609	+	0.660747i	$0.229760\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.24525			−0.278462			−0.139231	−	0.990260i	$-0.544463\pi$
$503$	−6.24525			−0.278462			−0.139231	+	0.990260i	$0.544463\pi$
$504$		0			0
$505$			13.2482i			0.589536i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.58258			0.424740			0.212370	−	0.977189i	$-0.431882\pi$
$509$	9.58258			0.424740			0.212370	+	0.977189i	$0.431882\pi$
$510$		0			0
$511$	−15.7771			−0.697937
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19.4650			−0.857729
$516$		0			0
$517$			7.95148i			0.349706i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.19274			0.0522547			0.0261274	−	0.999659i	$-0.491682\pi$
$521$	1.19274			0.0522547			0.0261274	+	0.999659i	$0.491682\pi$
$522$		0			0
$523$			24.0664i			1.05235i	0.850376	+	0.526176i	$0.176375\pi$
$523$			24.0664i			1.05235i	−0.850376	+	0.526176i	$0.823625\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	40.3631i		−	1.75824i
$528$		0			0
$529$	43.8328			1.90577
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.98752	+	3.04381i	−0.216034	+	0.131842i
$534$		0			0
$535$			4.41606i			0.190923i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.92236			−0.255094
$540$		0			0
$541$	3.20845			0.137942			0.0689711	−	0.997619i	$-0.478028\pi$
$541$	3.20845			0.137942			0.0689711	+	0.997619i	$0.478028\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.2100			0.865703
$546$		0			0
$547$		−	42.1895i		−	1.80389i	−0.431847	−	0.901947i	$-0.642138\pi$
$547$		−	42.1895i		−	1.80389i	0.431847	−	0.901947i	$-0.357862\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.98915i		−	0.212545i
$552$		0			0
$553$		−	24.8096i		−	1.05501i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.4525			1.03609			0.518043	−	0.855355i	$-0.326661\pi$
$557$	24.4525			1.03609			0.518043	+	0.855355i	$0.326661\pi$
$558$		0			0
$559$	7.23607	−	4.41606i	0.306053	−	0.186779i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	0.627058i		−	0.0264274i	−0.999913	−	0.0132137i	$-0.995794\pi$
$563$		−	0.627058i		−	0.0264274i	0.999913	−	0.0132137i	$-0.00420617\pi$
$564$		0			0
$565$	8.39984			0.353384
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.4553			1.10906			0.554532	−	0.832163i	$-0.312897\pi$
$569$	26.4553			1.10906			0.554532	+	0.832163i	$0.312897\pi$
$570$		0			0
$571$			24.6215i			1.03038i	0.857077	+	0.515188i	$0.172278\pi$
$571$			24.6215i			1.03038i	−0.857077	+	0.515188i	$0.827722\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.2802			0.762335
$576$		0			0
$577$			44.5588i			1.85501i	0.373816	+	0.927503i	$0.378049\pi$
$577$			44.5588i			1.85501i	−0.373816	+	0.927503i	$0.621951\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	8.20830i		−	0.340538i
$582$		0			0
$583$			11.5614i			0.478824i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.5174			0.434100			0.217050	−	0.976160i	$-0.430356\pi$
$587$	10.5174			0.434100			0.217050	+	0.976160i	$0.430356\pi$
$588$		0			0
$589$			9.27354i			0.382110i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	26.6637i		−	1.09495i	−0.836823	−	0.547474i	$-0.815589\pi$
$593$		−	26.6637i		−	1.09495i	0.836823	−	0.547474i	$-0.184411\pi$
$594$		0			0
$595$		−	30.0098i		−	1.23028i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	46.6653			1.90669			0.953347	−	0.301877i	$-0.0976132\pi$
$599$	46.6653			1.90669			0.953347	+	0.301877i	$0.0976132\pi$
$600$		0			0
$601$	19.4164			0.792012			0.396006	−	0.918248i	$-0.370396\pi$
$601$	19.4164			0.792012			0.396006	+	0.918248i	$0.370396\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	16.5324			0.672139
$606$		0			0
$607$	−15.7082			−0.637576			−0.318788	−	0.947826i	$-0.603276\pi$
$607$	−15.7082			−0.637576			−0.318788	+	0.947826i	$0.603276\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23.8175	−	14.5354i	0.963552	−	0.588040i
$612$		0			0
$613$	−6.96497			−0.281313			−0.140656	−	0.990058i	$-0.544921\pi$
$613$	−6.96497			−0.281313			−0.140656	+	0.990058i	$0.544921\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	3.36861i		−	0.135615i	−0.997698	−	0.0678075i	$-0.978400\pi$
$617$		−	3.36861i		−	0.135615i	0.997698	−	0.0678075i	$-0.0216004\pi$
$618$		0			0
$619$	−29.7781			−1.19688			−0.598442	−	0.801166i	$-0.704213\pi$
$619$	−29.7781			−1.19688			−0.598442	+	0.801166i	$0.704213\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	32.2450			1.29187
$624$		0			0
$625$	−8.81966			−0.352786
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−49.6899			−1.98127
$630$		0			0
$631$			22.2794i			0.886928i	0.896292	+	0.443464i	$0.146251\pi$
$631$			22.2794i			0.886928i	−0.896292	+	0.443464i	$0.853749\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.97505			−0.395848
$636$		0			0
$637$	10.8262	+	17.7396i	0.428948	+	0.702867i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−43.9983			−1.73783			−0.868914	−	0.494963i	$-0.835182\pi$
$641$	−43.9983			−1.73783			−0.868914	+	0.494963i	$0.835182\pi$
$642$		0			0
$643$	−33.8734			−1.33584			−0.667918	−	0.744235i	$-0.732814\pi$
$643$	−33.8734			−1.33584			−0.667918	+	0.744235i	$0.732814\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−36.5603			−1.43733			−0.718667	−	0.695354i	$-0.755247\pi$
$647$	−36.5603			−1.43733			−0.718667	+	0.695354i	$0.755247\pi$
$648$		0			0
$649$	10.0000			0.392534
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.3688i		−	1.30582i	−0.757435	−	0.652911i	$-0.773548\pi$
$653$		−	33.3688i		−	1.30582i	0.757435	−	0.652911i	$-0.226452\pi$
$654$		0			0
$655$		−	27.5388i		−	1.07603i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			36.4126i			1.41843i	0.704991	+	0.709216i	$0.250951\pi$
$659$			36.4126i			1.41843i	−0.704991	+	0.709216i	$0.749049\pi$
$660$		0			0
$661$	−46.3038			−1.80101			−0.900504	−	0.434847i	$-0.856802\pi$
$661$	−46.3038			−1.80101			−0.900504	+	0.434847i	$0.856802\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			6.89484i			0.267370i
$666$		0			0
$667$			35.1361i			1.36047i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	11.7264i		−	0.452692i
$672$		0			0
$673$	−12.6525			−0.487717			−0.243859	−	0.969811i	$-0.578413\pi$
$673$	−12.6525			−0.487717			−0.243859	+	0.969811i	$0.578413\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			17.8187i			0.684830i	0.939549	+	0.342415i	$0.111245\pi$
$677$			17.8187i			0.684830i	−0.939549	+	0.342415i	$0.888755\pi$
$678$		0			0
$679$	−41.3050			−1.58514
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5174			0.402438			0.201219	−	0.979546i	$-0.435510\pi$
$683$	10.5174			0.402438			0.201219	+	0.979546i	$0.435510\pi$
$684$		0			0
$685$		−	22.6134i		−	0.864012i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	34.6304	−	21.1344i	1.31931	−	0.805156i
$690$		0			0
$691$	4.91735			0.187065			0.0935324	−	0.995616i	$-0.470184\pi$
$691$	4.91735			0.187065			0.0935324	+	0.995616i	$0.470184\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	32.1931i		−	1.22115i
$696$		0			0
$697$			8.18782i			0.310136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			26.8021i			1.01230i	0.862445	+	0.506151i	$0.168932\pi$
$701$			26.8021i			1.01230i	−0.862445	+	0.506151i	$0.831068\pi$
$702$		0			0
$703$	11.4164			0.430578
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.4698			−1.07072
$708$		0			0
$709$	−8.39984			−0.315463			−0.157731	−	0.987482i	$-0.550418\pi$
$709$	−8.39984			−0.315463			−0.157731	+	0.987482i	$0.550418\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	65.3089i		−	2.44584i
$714$		0			0
$715$	3.20845	+	5.25731i	0.119989	+	0.196612i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.0349			−0.448826			−0.224413	−	0.974494i	$-0.572047\pi$
$719$	−12.0349			−0.448826			−0.224413	+	0.974494i	$0.572047\pi$
$720$		0			0
$721$		−	41.8295i		−	1.55781i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			9.61045i			0.356923i
$726$		0			0
$727$	22.3607			0.829312			0.414656	−	0.909978i	$-0.363902\pi$
$727$	22.3607			0.829312			0.414656	+	0.909978i	$0.363902\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	11.8792i		−	0.439367i
$732$		0			0
$733$	46.8519			1.73051			0.865256	−	0.501330i	$-0.167156\pi$
$733$	46.8519			1.73051			0.865256	+	0.501330i	$0.167156\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.43798			−0.273982
$738$		0			0
$739$	19.9434			0.733631			0.366816	−	0.930294i	$-0.380448\pi$
$739$	19.9434			0.733631			0.366816	+	0.930294i	$0.380448\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	47.5918i		−	1.74597i	−0.487744	−	0.872987i	$-0.662180\pi$
$743$		−	47.5918i		−	1.74597i	0.487744	−	0.872987i	$-0.337820\pi$
$744$		0			0
$745$	11.7082			0.428955
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.48993			−0.346755
$750$		0			0
$751$	54.5410			1.99023			0.995115	−	0.0987222i	$-0.0314755\pi$
$751$	54.5410			1.99023			0.995115	+	0.0987222i	$0.0314755\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	5.93958i		−	0.216164i
$756$		0			0
$757$		−	8.50651i		−	0.309174i	−0.987979	−	0.154587i	$-0.950595\pi$
$757$		−	8.50651i		−	0.309174i	0.987979	−	0.154587i	$-0.0494047\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	16.4304i		−	0.595601i	−0.954628	−	0.297800i	$-0.903747\pi$
$761$		−	16.4304i		−	0.595601i	0.954628	−	0.297800i	$-0.0962530\pi$
$762$		0			0
$763$			43.4306i			1.57229i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.2802	−	29.9535i	−0.660058	−	1.08156i
$768$		0			0
$769$			24.1653i			0.871422i	0.900087	+	0.435711i	$0.143503\pi$
$769$			24.1653i			0.871422i	−0.900087	+	0.435711i	$0.856497\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−30.3176			−1.09045			−0.545224	−	0.838290i	$-0.683555\pi$
$773$	−30.3176			−1.09045			−0.545224	+	0.838290i	$0.683555\pi$
$774$		0			0
$775$		−	17.8633i		−	0.641669i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.88118i		−	0.0674001i
$780$		0			0
$781$			10.8576i			0.388517i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			0.922740i			0.0329340i
$786$		0			0
$787$	40.4996			1.44366			0.721828	−	0.692072i	$-0.243302\pi$
$787$	40.4996			1.44366			0.721828	+	0.692072i	$0.243302\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			18.0509i			0.641817i
$792$		0			0
$793$	−35.1246	+	21.4360i	−1.24731	+	0.761214i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	21.4896i		−	0.761201i	−0.924740	−	0.380601i	$-0.875717\pi$
$797$		−	21.4896i		−	0.761201i	0.924740	−	0.380601i	$-0.124283\pi$
$798$		0			0
$799$		−	39.1002i		−	1.38327i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.53744i		−	0.160123i
$804$		0			0
$805$		−	48.5569i		−	1.71141i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27.6480			−0.972053			−0.486026	−	0.873944i	$-0.661554\pi$
$809$	−27.6480			−0.972053			−0.486026	+	0.873944i	$0.661554\pi$
$810$		0			0
$811$	−49.7863			−1.74823			−0.874116	−	0.485717i	$-0.838559\pi$
$811$	−49.7863			−1.74823			−0.874116	+	0.485717i	$0.838559\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0349			0.421564
$816$		0			0
$817$			2.72927i			0.0954852i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.5874			0.648705			0.324352	−	0.945936i	$-0.394854\pi$
$821$	18.5874			0.648705			0.324352	+	0.945936i	$0.394854\pi$
$822$		0			0
$823$	−14.1803			−0.494296			−0.247148	−	0.968978i	$-0.579493\pi$
$823$	−14.1803			−0.494296			−0.247148	+	0.968978i	$0.579493\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.3326			−1.26341			−0.631704	−	0.775209i	$-0.717644\pi$
$827$	−36.3326			−1.26341			−0.631704	+	0.775209i	$0.717644\pi$
$828$		0			0
$829$			31.3319i			1.08820i	0.839020	+	0.544100i	$0.183129\pi$
$829$			31.3319i			1.08820i	−0.839020	+	0.544100i	$0.816871\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	29.1223			1.00903
$834$		0			0
$835$		−	12.1797i		−	0.421496i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	45.0184i		−	1.55421i	−0.629372	−	0.777104i	$-0.716688\pi$
$839$		−	45.0184i		−	1.55421i	0.629372	−	0.777104i	$-0.283312\pi$
$840$		0			0
$841$	10.5279			0.363030
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.88240	−	19.2209i	0.339965	−	0.661219i
$846$		0			0
$847$			35.5275i			1.22074i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−80.4000			−2.75608
$852$		0			0
$853$	−29.5041			−1.01020			−0.505101	−	0.863060i	$-0.668545\pi$
$853$	−29.5041			−1.01020			−0.505101	+	0.863060i	$0.668545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.4905			−0.426667			−0.213334	−	0.976979i	$-0.568432\pi$
$857$	−12.4905			−0.426667			−0.213334	+	0.976979i	$0.568432\pi$
$858$		0			0
$859$			27.5276i			0.939231i	0.882871	+	0.469615i	$0.155607\pi$
$859$			27.5276i			0.939231i	−0.882871	+	0.469615i	$0.844393\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.0602i		−	0.410535i	−0.978706	−	0.205267i	$-0.934194\pi$
$863$		−	12.0602i		−	0.410535i	0.978706	−	0.205267i	$-0.0658064\pi$
$864$		0			0
$865$		−	18.7067i		−	0.636047i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.13514			0.242043
$870$		0			0
$871$	13.5967	+	22.2794i	0.460708	+	0.754908i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	42.9792i		−	1.45296i
$876$		0			0
$877$	21.4430			0.724078			0.362039	−	0.932163i	$-0.382081\pi$
$877$	21.4430			0.724078			0.362039	+	0.932163i	$0.382081\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.1050			−0.340447			−0.170223	−	0.985405i	$-0.554449\pi$
$881$	−10.1050			−0.340447			−0.170223	+	0.985405i	$0.554449\pi$
$882$		0			0
$883$			10.5146i			0.353845i	0.984225	+	0.176923i	$0.0566142\pi$
$883$			10.5146i			0.353845i	−0.984225	+	0.176923i	$0.943386\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	30.7706			1.03318			0.516589	−	0.856234i	$-0.327202\pi$
$887$	30.7706			1.03318			0.516589	+	0.856234i	$0.327202\pi$
$888$		0			0
$889$		−	21.4360i		−	0.718940i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			8.98339i			0.300618i
$894$		0			0
$895$			36.3709i			1.21575i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	34.3349			1.14513
$900$		0			0
$901$		−	56.8514i		−	1.89399i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	21.3894i		−	0.711009i
$906$		0			0
$907$			5.81234i			0.192996i	0.995333	+	0.0964978i	$0.0307641\pi$
$907$			5.81234i			0.192996i	−0.995333	+	0.0964978i	$0.969236\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.8757			−1.35427			−0.677136	−	0.735858i	$-0.736779\pi$
$911$	−40.8757			−1.35427			−0.677136	+	0.735858i	$0.736779\pi$
$912$		0			0
$913$	2.36068			0.0781271
$914$		0			0
$915$		0			0
$916$		0			0
$917$	59.1799			1.95429
$918$		0			0
$919$	−5.41641			−0.178671			−0.0893354	−	0.996002i	$-0.528474\pi$
$919$	−5.41641			−0.178671			−0.0893354	+	0.996002i	$0.528474\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	32.5225	−	19.8480i	1.07049	−	0.653303i
$924$		0			0
$925$	−21.9911			−0.723062
$926$		0			0
$927$		0			0
$928$		0			0
$929$			6.86474i			0.225225i	0.993639	+	0.112612i	$0.0359218\pi$
$929$			6.86474i			0.225225i	−0.993639	+	0.112612i	$0.964078\pi$
$930$		0			0
$931$	−6.69094			−0.219287
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.63072			0.282255
$936$		0			0
$937$	−30.6525			−1.00137			−0.500686	−	0.865629i	$-0.666919\pi$
$937$	−30.6525			−1.00137			−0.500686	+	0.865629i	$0.666919\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−36.4825			−1.18930			−0.594648	−	0.803986i	$-0.702709\pi$
$941$	−36.4825			−1.18930			−0.594648	+	0.803986i	$0.702709\pi$
$942$		0			0
$943$			13.2482i			0.431420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−49.9325			−1.62259			−0.811294	−	0.584638i	$-0.801236\pi$
$947$	−49.9325			−1.62259			−0.811294	+	0.584638i	$0.801236\pi$
$948$		0			0
$949$	−13.5912	+	8.29451i	−0.441190	+	0.269251i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.7005			1.05927			0.529637	−	0.848224i	$-0.322328\pi$
$953$	32.7005			1.05927			0.529637	+	0.848224i	$0.322328\pi$
$954$		0			0
$955$	3.20845			0.103823
$956$		0			0
$957$		0			0
$958$		0			0
$959$	48.5952			1.56922
$960$		0			0
$961$	−32.8197			−1.05870
$962$		0			0
$963$		0			0
$964$		0			0
$965$			7.34173i			0.236339i
$966$		0			0
$967$			5.25945i			0.169132i	0.996418	+	0.0845662i	$0.0269505\pi$
$967$			5.25945i			0.169132i	−0.996418	+	0.0845662i	$0.973050\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.05841i		−	0.130240i	−0.997877	−	0.0651202i	$-0.979257\pi$
$971$		−	4.05841i		−	0.130240i	0.997877	−	0.0651202i	$-0.0207431\pi$
$972$		0			0
$973$	69.1816			2.21786
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15.1924i		−	0.486048i	−0.970020	−	0.243024i	$-0.921861\pi$
$977$		−	15.1924i		−	0.486048i	0.970020	−	0.243024i	$-0.0781393\pi$
$978$		0			0
$979$			9.27354i			0.296384i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	24.4543i		−	0.779971i	−0.920821	−	0.389985i	$-0.872480\pi$
$983$		−	24.4543i		−	0.779971i	0.920821	−	0.389985i	$-0.127520\pi$
$984$		0			0
$985$	−38.5410			−1.22802
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	19.2209i		−	0.611189i
$990$		0			0
$991$	10.1803			0.323389			0.161695	−	0.986841i	$-0.448304\pi$
$991$	10.1803			0.323389			0.161695	+	0.986841i	$0.448304\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	29.9251			0.948691
$996$		0			0
$997$		−	12.9968i		−	0.411612i	−0.978593	−	0.205806i	$-0.934018\pi$
$997$		−	12.9968i		−	0.411612i	0.978593	−	0.205806i	$-0.0659817\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.m.h.1585.5		16
3.2	odd	2	inner	3744.2.m.h.1585.9		16
4.3	odd	2		936.2.m.h.181.2	yes	16
8.3	odd	2		936.2.m.h.181.14	yes	16
8.5	even	2	inner	3744.2.m.h.1585.10		16
12.11	even	2		936.2.m.h.181.16	yes	16
13.12	even	2	inner	3744.2.m.h.1585.12		16
24.5	odd	2	inner	3744.2.m.h.1585.6		16
24.11	even	2		936.2.m.h.181.4	yes	16
39.38	odd	2	inner	3744.2.m.h.1585.8		16
52.51	odd	2		936.2.m.h.181.15	yes	16
104.51	odd	2		936.2.m.h.181.3	yes	16
104.77	even	2	inner	3744.2.m.h.1585.7		16
156.155	even	2		936.2.m.h.181.1	✓	16
312.77	odd	2	inner	3744.2.m.h.1585.11		16
312.155	even	2		936.2.m.h.181.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.m.h.181.1	✓	16	156.155	even	2
936.2.m.h.181.2	yes	16	4.3	odd	2
936.2.m.h.181.3	yes	16	104.51	odd	2
936.2.m.h.181.4	yes	16	24.11	even	2
936.2.m.h.181.13	yes	16	312.155	even	2
936.2.m.h.181.14	yes	16	8.3	odd	2
936.2.m.h.181.15	yes	16	52.51	odd	2
936.2.m.h.181.16	yes	16	12.11	even	2
3744.2.m.h.1585.5		16	1.1	even	1	trivial
3744.2.m.h.1585.6		16	24.5	odd	2	inner
3744.2.m.h.1585.7		16	104.77	even	2	inner
3744.2.m.h.1585.8		16	39.38	odd	2	inner
3744.2.m.h.1585.9		16	3.2	odd	2	inner
3744.2.m.h.1585.10		16	8.5	even	2	inner
3744.2.m.h.1585.11		16	312.77	odd	2	inner
3744.2.m.h.1585.12		16	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.66251 q^{5} -3.57266i q^{7} +O(q^{10})\)	\(q-1.66251 q^{5} -3.57266i q^{7} +1.02749 q^{11} +(-1.87826 - 3.07768i) q^{13} -5.05251 q^{17} +1.16083 q^{19} -8.17513 q^{23} -2.23607 q^{25} -4.29792i q^{29} +7.98872i q^{31} +5.93958i q^{35} +9.83470 q^{37} -1.62054i q^{41} +2.35114i q^{43} +7.73877i q^{47} -5.76393 q^{49} +11.2521i q^{53} -1.70820 q^{55} +9.73249 q^{59} -11.4127i q^{61} +(3.12262 + 5.11667i) q^{65} -7.23901 q^{67} +10.5672i q^{71} -4.41606i q^{73} -3.67086i q^{77} +6.94427 q^{79} +2.29753 q^{83} +8.39984 q^{85} +9.02546i q^{89} +(-10.9955 + 6.71040i) q^{91} -1.92989 q^{95} -11.5614i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100}) \) 16 * q - 128 * q^49 + 80 * q^55 - 32 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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