LMFDB - Embedded newform 924.2.l.b.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [924,2,Mod(881,924)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(924, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("924.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	924.l (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:	$7.37817714677$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	924.881
Dual form			924.2.l.b.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} +3.00000 q^{9} -1.00000i q^{11} +5.19615i q^{13} -3.00000 q^{15} -1.73205i q^{19} +(-3.46410 - 3.00000i) q^{21} -6.00000i q^{23} -2.00000 q^{25} -5.19615 q^{27} +9.00000i q^{29} +3.46410i q^{31} +1.73205i q^{33} +(3.46410 + 3.00000i) q^{35} +7.00000 q^{37} -9.00000i q^{39} +6.92820 q^{41} +10.0000 q^{43} +5.19615 q^{45} -8.66025 q^{47} +(1.00000 + 6.92820i) q^{49} -1.73205i q^{55} +3.00000i q^{57} -1.73205 q^{59} +13.8564i q^{61} +(6.00000 + 5.19615i) q^{63} +9.00000i q^{65} +5.00000 q^{67} +10.3923i q^{69} +12.0000i q^{71} -8.66025i q^{73} +3.46410 q^{75} +(1.73205 - 2.00000i) q^{77} -10.0000 q^{79} +9.00000 q^{81} +13.8564 q^{83} -15.5885i q^{87} +6.92820 q^{89} +(-9.00000 + 10.3923i) q^{91} -6.00000i q^{93} -3.00000i q^{95} -17.3205i q^{97} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} + 12 q^{9} - 12 q^{15} - 8 q^{25} + 28 q^{37} + 40 q^{43} + 4 q^{49} + 24 q^{63} + 20 q^{67} - 40 q^{79} + 36 q^{81} - 36 q^{91}+O(q^{100}) $ 4 * q + 8 * q^7 + 12 * q^9 - 12 * q^15 - 8 * q^25 + 28 * q^37 + 40 * q^43 + 4 * q^49 + 24 * q^63 + 20 * q^67 - 40 * q^79 + 36 * q^81 - 36 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/924\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$661$	$673$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$		0			0
$5$	1.73205			0.774597			0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205			0.774597			0.387298	+	0.921954i	$0.373408\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			5.19615i			1.44115i	0.693375	+	0.720577i	$0.256123\pi$
$13$			5.19615i			1.44115i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	−3.00000			−0.774597
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	1.73205i		−	0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$19$		−	1.73205i		−	0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$20$		0			0
$21$	−3.46410	−	3.00000i	−0.755929	−	0.654654i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$		0			0
$33$			1.73205i			0.301511i
$34$		0			0
$35$	3.46410	+	3.00000i	0.585540	+	0.507093i
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		−	9.00000i		−	1.44115i
$40$		0			0
$41$	6.92820			1.08200			0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820			1.08200			0.541002	+	0.841021i	$0.318045\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$	5.19615			0.774597
$46$		0			0
$47$	−8.66025			−1.26323			−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025			−1.26323			−0.631614	+	0.775283i	$0.717607\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	1.73205i		−	0.233550i
$56$		0			0
$57$			3.00000i			0.397360i
$58$		0			0
$59$	−1.73205			−0.225494			−0.112747	−	0.993624i	$-0.535965\pi$
$59$	−1.73205			−0.225494			−0.112747	+	0.993624i	$0.535965\pi$
$60$		0			0
$61$			13.8564i			1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$			13.8564i			1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$		0			0
$63$	6.00000	+	5.19615i	0.755929	+	0.654654i
$64$		0			0
$65$			9.00000i			1.11631i
$66$		0			0
$67$	5.00000			0.610847			0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000			0.610847			0.305424	+	0.952217i	$0.401202\pi$
$68$		0			0
$69$			10.3923i			1.25109i
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$		−	8.66025i		−	1.01361i	−0.862062	−	0.506803i	$-0.830827\pi$
$73$		−	8.66025i		−	1.01361i	0.862062	−	0.506803i	$-0.169173\pi$
$74$		0			0
$75$	3.46410			0.400000
$76$		0			0
$77$	1.73205	−	2.00000i	0.197386	−	0.227921i
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	13.8564			1.52094			0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564			1.52094			0.760469	+	0.649374i	$0.224969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	15.5885i		−	1.67126i
$88$		0			0
$89$	6.92820			0.734388			0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820			0.734388			0.367194	+	0.930144i	$0.380318\pi$
$90$		0			0
$91$	−9.00000	+	10.3923i	−0.943456	+	1.08941i
$92$		0			0
$93$		−	6.00000i		−	0.622171i
$94$		0			0
$95$		−	3.00000i		−	0.307794i
$96$		0			0
$97$		−	17.3205i		−	1.75863i	−0.476240	−	0.879316i	$-0.658000\pi$
$97$		−	17.3205i		−	1.75863i	0.476240	−	0.879316i	$-0.342000\pi$
$98$		0			0
$99$		−	3.00000i		−	0.301511i
$100$		0			0
$101$	3.46410			0.344691			0.172345	−	0.985037i	$-0.444865\pi$
$101$	3.46410			0.344691			0.172345	+	0.985037i	$0.444865\pi$
$102$		0			0
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$	−6.00000	−	5.19615i	−0.585540	−	0.507093i
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	−12.1244			−1.15079
$112$		0			0
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$		−	10.3923i		−	0.969087i
$116$		0			0
$117$			15.5885i			1.44115i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	−12.0000			−1.08200
$124$		0			0
$125$	−12.1244			−1.08444
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	−17.3205			−1.52499
$130$		0			0
$131$	6.92820			0.605320			0.302660	−	0.953099i	$-0.402125\pi$
$131$	6.92820			0.605320			0.302660	+	0.953099i	$0.402125\pi$
$132$		0			0
$133$	3.00000	−	3.46410i	0.260133	−	0.300376i
$134$		0			0
$135$	−9.00000			−0.774597
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$		−	10.3923i		−	0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$139$		−	10.3923i		−	0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$		0			0
$141$	15.0000			1.26323
$142$		0			0
$143$	5.19615			0.434524
$144$		0			0
$145$			15.5885i			1.29455i
$146$		0			0
$147$	−1.73205	−	12.0000i	−0.142857	−	0.989743i
$148$		0			0
$149$		−	21.0000i		−	1.72039i	−0.509968	−	0.860194i	$-0.670343\pi$
$149$		−	21.0000i		−	1.72039i	0.509968	−	0.860194i	$-0.329657\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			6.00000i			0.481932i
$156$		0			0
$157$		−	3.46410i		−	0.276465i	−0.990400	−	0.138233i	$-0.955858\pi$
$157$		−	3.46410i		−	0.276465i	0.990400	−	0.138233i	$-0.0441422\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.3923	−	12.0000i	0.819028	−	0.945732i
$162$		0			0
$163$	−7.00000			−0.548282			−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000			−0.548282			−0.274141	+	0.961689i	$0.588394\pi$
$164$		0			0
$165$			3.00000i			0.233550i
$166$		0			0
$167$	−10.3923			−0.804181			−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923			−0.804181			−0.402090	+	0.915600i	$0.631716\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		0			0
$171$		−	5.19615i		−	0.397360i
$172$		0			0
$173$	−17.3205			−1.31685			−0.658427	−	0.752645i	$-0.728778\pi$
$173$	−17.3205			−1.31685			−0.658427	+	0.752645i	$0.728778\pi$
$174$		0			0
$175$	−4.00000	−	3.46410i	−0.302372	−	0.261861i
$176$		0			0
$177$	3.00000			0.225494
$178$		0			0
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$		−	24.0000i		−	1.77413i
$184$		0			0
$185$	12.1244			0.891400
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−10.3923	−	9.00000i	−0.755929	−	0.654654i
$190$		0			0
$191$			24.0000i			1.73658i	0.496058	+	0.868290i	$0.334780\pi$
$191$			24.0000i			1.73658i	−0.496058	+	0.868290i	$0.665220\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$		−	15.5885i		−	1.11631i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$		−	24.2487i		−	1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$		−	24.2487i		−	1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$		0			0
$201$	−8.66025			−0.610847
$202$		0			0
$203$	−15.5885	+	18.0000i	−1.09410	+	1.26335i
$204$		0			0
$205$	12.0000			0.838116
$206$		0			0
$207$		−	18.0000i		−	1.25109i
$208$		0			0
$209$	−1.73205			−0.119808
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		−	20.7846i		−	1.42414i
$214$		0			0
$215$	17.3205			1.18125
$216$		0			0
$217$	−6.00000	+	6.92820i	−0.407307	+	0.470317i
$218$		0			0
$219$			15.0000i			1.01361i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	13.8564i		−	0.927894i	−0.885863	−	0.463947i	$-0.846433\pi$
$223$		−	13.8564i		−	0.927894i	0.885863	−	0.463947i	$-0.153567\pi$
$224$		0			0
$225$	−6.00000			−0.400000
$226$		0			0
$227$	6.92820			0.459841			0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820			0.459841			0.229920	+	0.973209i	$0.426153\pi$
$228$		0			0
$229$			10.3923i			0.686743i	0.939200	+	0.343371i	$0.111569\pi$
$229$			10.3923i			0.686743i	−0.939200	+	0.343371i	$0.888431\pi$
$230$		0			0
$231$	−3.00000	+	3.46410i	−0.197386	+	0.227921i
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	−15.0000			−0.978492
$236$		0			0
$237$	17.3205			1.12509
$238$		0			0
$239$		−	3.00000i		−	0.194054i	−0.995282	−	0.0970269i	$-0.969067\pi$
$239$		−	3.00000i		−	0.194054i	0.995282	−	0.0970269i	$-0.0309333\pi$
$240$		0			0
$241$		−	5.19615i		−	0.334714i	−0.985896	−	0.167357i	$-0.946477\pi$
$241$		−	5.19615i		−	0.334714i	0.985896	−	0.167357i	$-0.0535232\pi$
$242$		0			0
$243$	−15.5885			−1.00000
$244$		0			0
$245$	1.73205	+	12.0000i	0.110657	+	0.766652i
$246$		0			0
$247$	9.00000			0.572656
$248$		0			0
$249$	−24.0000			−1.52094
$250$		0			0
$251$	29.4449			1.85854			0.929272	−	0.369397i	$-0.120436\pi$
$251$	29.4449			1.85854			0.929272	+	0.369397i	$0.120436\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.73205			−0.108042			−0.0540212	−	0.998540i	$-0.517204\pi$
$257$	−1.73205			−0.108042			−0.0540212	+	0.998540i	$0.517204\pi$
$258$		0			0
$259$	14.0000	+	12.1244i	0.869918	+	0.753371i
$260$		0			0
$261$			27.0000i			1.67126i
$262$		0			0
$263$			3.00000i			0.184988i	0.995713	+	0.0924940i	$0.0294839\pi$
$263$			3.00000i			0.184988i	−0.995713	+	0.0924940i	$0.970516\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−12.0000			−0.734388
$268$		0			0
$269$	−13.8564			−0.844840			−0.422420	−	0.906400i	$-0.638819\pi$
$269$	−13.8564			−0.844840			−0.422420	+	0.906400i	$0.638819\pi$
$270$		0			0
$271$		−	15.5885i		−	0.946931i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$		−	15.5885i		−	0.946931i	0.880812	−	0.473466i	$-0.156997\pi$
$272$		0			0
$273$	15.5885	−	18.0000i	0.943456	−	1.08941i
$274$		0			0
$275$			2.00000i			0.120605i
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$			10.3923i			0.622171i
$280$		0			0
$281$			3.00000i			0.178965i	0.995988	+	0.0894825i	$0.0285213\pi$
$281$			3.00000i			0.178965i	−0.995988	+	0.0894825i	$0.971479\pi$
$282$		0			0
$283$		−	12.1244i		−	0.720718i	−0.932814	−	0.360359i	$-0.882654\pi$
$283$		−	12.1244i		−	0.720718i	0.932814	−	0.360359i	$-0.117346\pi$
$284$		0			0
$285$			5.19615i			0.307794i
$286$		0			0
$287$	13.8564	+	12.0000i	0.817918	+	0.708338i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$			30.0000i			1.75863i
$292$		0			0
$293$	−6.92820			−0.404750			−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820			−0.404750			−0.202375	+	0.979308i	$0.564866\pi$
$294$		0			0
$295$	−3.00000			−0.174667
$296$		0			0
$297$			5.19615i			0.301511i
$298$		0			0
$299$	31.1769			1.80301
$300$		0			0
$301$	20.0000	+	17.3205i	1.15278	+	0.998337i
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$			24.0000i			1.37424i
$306$		0			0
$307$			3.46410i			0.197707i	0.995102	+	0.0988534i	$0.0315175\pi$
$307$			3.46410i			0.197707i	−0.995102	+	0.0988534i	$0.968483\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$	−10.3923			−0.589294			−0.294647	−	0.955606i	$-0.595202\pi$
$311$	−10.3923			−0.589294			−0.294647	+	0.955606i	$0.595202\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	10.3923	+	9.00000i	0.585540	+	0.507093i
$316$		0			0
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$	9.00000			0.503903
$320$		0			0
$321$		−	5.19615i		−	0.290021i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	10.3923i		−	0.576461i
$326$		0			0
$327$	−17.3205			−0.957826
$328$		0			0
$329$	−17.3205	−	15.0000i	−0.954911	−	0.826977i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	21.0000			1.15079
$334$		0			0
$335$	8.66025			0.473160
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$		0			0
$339$		−	31.1769i		−	1.69330i
$340$		0			0
$341$	3.46410			0.187592
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$			18.0000i			0.969087i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$			19.0526i			1.01986i	0.860216	+	0.509930i	$0.170329\pi$
$349$			19.0526i			1.01986i	−0.860216	+	0.509930i	$0.829671\pi$
$350$		0			0
$351$		−	27.0000i		−	1.44115i
$352$		0			0
$353$	8.66025			0.460939			0.230469	−	0.973080i	$-0.425974\pi$
$353$	8.66025			0.460939			0.230469	+	0.973080i	$0.425974\pi$
$354$		0			0
$355$			20.7846i			1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	12.0000i		−	0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$359$		−	12.0000i		−	0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$360$		0			0
$361$	16.0000			0.842105
$362$		0			0
$363$	1.73205			0.0909091
$364$		0			0
$365$		−	15.0000i		−	0.785136i
$366$		0			0
$367$			10.3923i			0.542474i	0.962513	+	0.271237i	$0.0874327\pi$
$367$			10.3923i			0.542474i	−0.962513	+	0.271237i	$0.912567\pi$
$368$		0			0
$369$	20.7846			1.08200
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−16.0000			−0.828449			−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000			−0.828449			−0.414224	+	0.910175i	$0.635947\pi$
$374$		0			0
$375$	21.0000			1.08444
$376$		0			0
$377$	−46.7654			−2.40854
$378$		0			0
$379$	13.0000			0.667765			0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000			0.667765			0.333883	+	0.942615i	$0.391641\pi$
$380$		0			0
$381$	27.7128			1.41977
$382$		0			0
$383$	−10.3923			−0.531022			−0.265511	−	0.964108i	$-0.585541\pi$
$383$	−10.3923			−0.531022			−0.265511	+	0.964108i	$0.585541\pi$
$384$		0			0
$385$	3.00000	−	3.46410i	0.152894	−	0.176547i
$386$		0			0
$387$	30.0000			1.52499
$388$		0			0
$389$			24.0000i			1.21685i	0.793612	+	0.608424i	$0.208198\pi$
$389$			24.0000i			1.21685i	−0.793612	+	0.608424i	$0.791802\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−12.0000			−0.605320
$394$		0			0
$395$	−17.3205			−0.871489
$396$		0			0
$397$			13.8564i			0.695433i	0.937600	+	0.347717i	$0.113043\pi$
$397$			13.8564i			0.695433i	−0.937600	+	0.347717i	$0.886957\pi$
$398$		0			0
$399$	−5.19615	+	6.00000i	−0.260133	+	0.300376i
$400$		0			0
$401$		−	6.00000i		−	0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$		−	6.00000i		−	0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$		0			0
$403$	−18.0000			−0.896644
$404$		0			0
$405$	15.5885			0.774597
$406$		0			0
$407$		−	7.00000i		−	0.346977i
$408$		0			0
$409$		−	27.7128i		−	1.37031i	−0.728397	−	0.685155i	$-0.759734\pi$
$409$		−	27.7128i		−	1.37031i	0.728397	−	0.685155i	$-0.240266\pi$
$410$		0			0
$411$			20.7846i			1.02523i
$412$		0			0
$413$	−3.46410	−	3.00000i	−0.170457	−	0.147620i
$414$		0			0
$415$	24.0000			1.17811
$416$		0			0
$417$			18.0000i			0.881464i
$418$		0			0
$419$	−29.4449			−1.43848			−0.719238	−	0.694764i	$-0.755509\pi$
$419$	−29.4449			−1.43848			−0.719238	+	0.694764i	$0.755509\pi$
$420$		0			0
$421$	29.0000			1.41337			0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000			1.41337			0.706687	+	0.707527i	$0.250189\pi$
$422$		0			0
$423$	−25.9808			−1.26323
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.0000	+	27.7128i	−1.16144	+	1.34112i
$428$		0			0
$429$	−9.00000			−0.434524
$430$		0			0
$431$		−	21.0000i		−	1.01153i	−0.862670	−	0.505767i	$-0.831209\pi$
$431$		−	21.0000i		−	1.01153i	0.862670	−	0.505767i	$-0.168791\pi$
$432$		0			0
$433$		−	24.2487i		−	1.16532i	−0.812716	−	0.582659i	$-0.802012\pi$
$433$		−	24.2487i		−	1.16532i	0.812716	−	0.582659i	$-0.197988\pi$
$434$		0			0
$435$		−	27.0000i		−	1.29455i
$436$		0			0
$437$	−10.3923			−0.497131
$438$		0			0
$439$			8.66025i			0.413331i	0.978412	+	0.206666i	$0.0662612\pi$
$439$			8.66025i			0.413331i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$	3.00000	+	20.7846i	0.142857	+	0.989743i
$442$		0			0
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$			36.3731i			1.72039i
$448$		0			0
$449$		−	42.0000i		−	1.98210i	−0.133482	−	0.991051i	$-0.542616\pi$
$449$		−	42.0000i		−	1.98210i	0.133482	−	0.991051i	$-0.457384\pi$
$450$		0			0
$451$		−	6.92820i		−	0.326236i
$452$		0			0
$453$	13.8564			0.651031
$454$		0			0
$455$	−15.5885	+	18.0000i	−0.730798	+	0.843853i
$456$		0			0
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.3923			0.484018			0.242009	−	0.970274i	$-0.422194\pi$
$461$	10.3923			0.484018			0.242009	+	0.970274i	$0.422194\pi$
$462$		0			0
$463$	−19.0000			−0.883005			−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000			−0.883005			−0.441502	+	0.897260i	$0.645554\pi$
$464$		0			0
$465$		−	10.3923i		−	0.481932i
$466$		0			0
$467$	−25.9808			−1.20225			−0.601123	−	0.799156i	$-0.705280\pi$
$467$	−25.9808			−1.20225			−0.601123	+	0.799156i	$0.705280\pi$
$468$		0			0
$469$	10.0000	+	8.66025i	0.461757	+	0.399893i
$470$		0			0
$471$			6.00000i			0.276465i
$472$		0			0
$473$		−	10.0000i		−	0.459800i
$474$		0			0
$475$			3.46410i			0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	34.6410			1.58279			0.791394	−	0.611306i	$-0.209356\pi$
$479$	34.6410			1.58279			0.791394	+	0.611306i	$0.209356\pi$
$480$		0			0
$481$			36.3731i			1.65847i
$482$		0			0
$483$	−18.0000	+	20.7846i	−0.819028	+	0.945732i
$484$		0			0
$485$		−	30.0000i		−	1.36223i
$486$		0			0
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$		0			0
$489$	12.1244			0.548282
$490$		0			0
$491$			3.00000i			0.135388i	0.997706	+	0.0676941i	$0.0215642\pi$
$491$			3.00000i			0.135388i	−0.997706	+	0.0676941i	$0.978436\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	5.19615i		−	0.233550i
$496$		0			0
$497$	−20.7846	+	24.0000i	−0.932317	+	1.07655i
$498$		0			0
$499$	−23.0000			−1.02962			−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000			−1.02962			−0.514811	+	0.857304i	$0.672138\pi$
$500$		0			0
$501$	18.0000			0.804181
$502$		0			0
$503$	13.8564			0.617827			0.308913	−	0.951090i	$-0.400035\pi$
$503$	13.8564			0.617827			0.308913	+	0.951090i	$0.400035\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	24.2487			1.07692
$508$		0			0
$509$	−27.7128			−1.22835			−0.614174	−	0.789170i	$-0.710511\pi$
$509$	−27.7128			−1.22835			−0.614174	+	0.789170i	$0.710511\pi$
$510$		0			0
$511$	15.0000	−	17.3205i	0.663561	−	0.766214i
$512$		0			0
$513$			9.00000i			0.397360i
$514$		0			0
$515$		−	6.00000i		−	0.264392i
$516$		0			0
$517$			8.66025i			0.380878i
$518$		0			0
$519$	30.0000			1.31685
$520$		0			0
$521$	−25.9808			−1.13824			−0.569119	−	0.822255i	$-0.692716\pi$
$521$	−25.9808			−1.13824			−0.569119	+	0.822255i	$0.692716\pi$
$522$		0			0
$523$		−	29.4449i		−	1.28753i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$		−	29.4449i		−	1.28753i	0.765222	−	0.643767i	$-0.222629\pi$
$524$		0			0
$525$	6.92820	+	6.00000i	0.302372	+	0.261861i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	−5.19615			−0.225494
$532$		0			0
$533$			36.0000i			1.55933i
$534$		0			0
$535$			5.19615i			0.224649i
$536$		0			0
$537$		−	10.3923i		−	0.448461i
$538$		0			0
$539$	6.92820	−	1.00000i	0.298419	−	0.0430730i
$540$		0			0
$541$	32.0000			1.37579			0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000			1.37579			0.687894	+	0.725811i	$0.258536\pi$
$542$		0			0
$543$			12.0000i			0.514969i
$544$		0			0
$545$	17.3205			0.741929
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$		0			0
$549$			41.5692i			1.77413i
$550$		0			0
$551$	15.5885			0.664091
$552$		0			0
$553$	−20.0000	−	17.3205i	−0.850487	−	0.736543i
$554$		0			0
$555$	−21.0000			−0.891400
$556$		0			0
$557$		−	21.0000i		−	0.889799i	−0.895581	−	0.444899i	$-0.853239\pi$
$557$		−	21.0000i		−	0.889799i	0.895581	−	0.444899i	$-0.146761\pi$
$558$		0			0
$559$			51.9615i			2.19774i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.8564			0.583978			0.291989	−	0.956422i	$-0.405683\pi$
$563$	13.8564			0.583978			0.291989	+	0.956422i	$0.405683\pi$
$564$		0			0
$565$			31.1769i			1.31162i
$566$		0			0
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$		0			0
$569$		−	30.0000i		−	1.25767i	−0.777541	−	0.628833i	$-0.783533\pi$
$569$		−	30.0000i		−	1.25767i	0.777541	−	0.628833i	$-0.216467\pi$
$570$		0			0
$571$	14.0000			0.585882			0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000			0.585882			0.292941	+	0.956131i	$0.405366\pi$
$572$		0			0
$573$		−	41.5692i		−	1.73658i
$574$		0			0
$575$			12.0000i			0.500435i
$576$		0			0
$577$			3.46410i			0.144212i	0.997397	+	0.0721062i	$0.0229721\pi$
$577$			3.46410i			0.144212i	−0.997397	+	0.0721062i	$0.977028\pi$
$578$		0			0
$579$	−6.92820			−0.287926
$580$		0			0
$581$	27.7128	+	24.0000i	1.14972	+	0.995688i
$582$		0			0
$583$		0			0
$584$		0			0
$585$			27.0000i			1.11631i
$586$		0			0
$587$	−12.1244			−0.500426			−0.250213	−	0.968191i	$-0.580501\pi$
$587$	−12.1244			−0.500426			−0.250213	+	0.968191i	$0.580501\pi$
$588$		0			0
$589$	6.00000			0.247226
$590$		0			0
$591$			10.3923i			0.427482i
$592$		0			0
$593$	24.2487			0.995775			0.497888	−	0.867242i	$-0.334109\pi$
$593$	24.2487			0.995775			0.497888	+	0.867242i	$0.334109\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			42.0000i			1.71895i
$598$		0			0
$599$		−	6.00000i		−	0.245153i	−0.992459	−	0.122577i	$-0.960884\pi$
$599$		−	6.00000i		−	0.245153i	0.992459	−	0.122577i	$-0.0391157\pi$
$600$		0			0
$601$		−	29.4449i		−	1.20108i	−0.799594	−	0.600541i	$-0.794952\pi$
$601$		−	29.4449i		−	1.20108i	0.799594	−	0.600541i	$-0.205048\pi$
$602$		0			0
$603$	15.0000			0.610847
$604$		0			0
$605$	−1.73205			−0.0704179
$606$		0			0
$607$		−	15.5885i		−	0.632716i	−0.948640	−	0.316358i	$-0.897540\pi$
$607$		−	15.5885i		−	0.632716i	0.948640	−	0.316358i	$-0.102460\pi$
$608$		0			0
$609$	27.0000	−	31.1769i	1.09410	−	1.26335i
$610$		0			0
$611$		−	45.0000i		−	1.82051i
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$	−20.7846			−0.838116
$616$		0			0
$617$		−	30.0000i		−	1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$		−	30.0000i		−	1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$			31.1769i			1.25109i
$622$		0			0
$623$	13.8564	+	12.0000i	0.555145	+	0.480770i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	3.00000			0.119808
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$	−6.92820			−0.275371
$634$		0			0
$635$	−27.7128			−1.09975
$636$		0			0
$637$	−36.0000	+	5.19615i	−1.42637	+	0.205879i
$638$		0			0
$639$			36.0000i			1.42414i
$640$		0			0
$641$			42.0000i			1.65890i	0.558581	+	0.829450i	$0.311346\pi$
$641$			42.0000i			1.65890i	−0.558581	+	0.829450i	$0.688654\pi$
$642$		0			0
$643$		−	3.46410i		−	0.136611i	−0.997664	−	0.0683054i	$-0.978241\pi$
$643$		−	3.46410i		−	0.136611i	0.997664	−	0.0683054i	$-0.0217592\pi$
$644$		0			0
$645$	−30.0000			−1.18125
$646$		0			0
$647$	−19.0526			−0.749033			−0.374517	−	0.927220i	$-0.622191\pi$
$647$	−19.0526			−0.749033			−0.374517	+	0.927220i	$0.622191\pi$
$648$		0			0
$649$			1.73205i			0.0679889i
$650$		0			0
$651$	10.3923	−	12.0000i	0.407307	−	0.470317i
$652$		0			0
$653$		−	36.0000i		−	1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$		−	36.0000i		−	1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$		0			0
$655$	12.0000			0.468879
$656$		0			0
$657$		−	25.9808i		−	1.01361i
$658$		0			0
$659$			33.0000i			1.28550i	0.766077	+	0.642749i	$0.222206\pi$
$659$			33.0000i			1.28550i	−0.766077	+	0.642749i	$0.777794\pi$
$660$		0			0
$661$			34.6410i			1.34738i	0.739014	+	0.673690i	$0.235292\pi$
$661$			34.6410i			1.34738i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.19615	−	6.00000i	0.201498	−	0.232670i
$666$		0			0
$667$	54.0000			2.09089
$668$		0			0
$669$			24.0000i			0.927894i
$670$		0			0
$671$	13.8564			0.534921
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$	10.3923			0.400000
$676$		0			0
$677$	24.2487			0.931954			0.465977	−	0.884797i	$-0.345703\pi$
$677$	24.2487			0.931954			0.465977	+	0.884797i	$0.345703\pi$
$678$		0			0
$679$	30.0000	−	34.6410i	1.15129	−	1.32940i
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$		−	6.00000i		−	0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$		−	6.00000i		−	0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$		0			0
$685$		−	20.7846i		−	0.794139i
$686$		0			0
$687$		−	18.0000i		−	0.686743i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			3.46410i			0.131781i	0.997827	+	0.0658903i	$0.0209887\pi$
$691$			3.46410i			0.131781i	−0.997827	+	0.0658903i	$0.979011\pi$
$692$		0			0
$693$	5.19615	−	6.00000i	0.197386	−	0.227921i
$694$		0			0
$695$		−	18.0000i		−	0.682779i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			10.3923i			0.393073i
$700$		0			0
$701$			30.0000i			1.13308i	0.824033	+	0.566542i	$0.191719\pi$
$701$			30.0000i			1.13308i	−0.824033	+	0.566542i	$0.808281\pi$
$702$		0			0
$703$		−	12.1244i		−	0.457279i
$704$		0			0
$705$	25.9808			0.978492
$706$		0			0
$707$	6.92820	+	6.00000i	0.260562	+	0.225653i
$708$		0			0
$709$	19.0000			0.713560			0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000			0.713560			0.356780	+	0.934188i	$0.383875\pi$
$710$		0			0
$711$	−30.0000			−1.12509
$712$		0			0
$713$	20.7846			0.778390
$714$		0			0
$715$	9.00000			0.336581
$716$		0			0
$717$			5.19615i			0.194054i
$718$		0			0
$719$	−29.4449			−1.09811			−0.549054	−	0.835787i	$-0.685012\pi$
$719$	−29.4449			−1.09811			−0.549054	+	0.835787i	$0.685012\pi$
$720$		0			0
$721$	6.00000	−	6.92820i	0.223452	−	0.258020i
$722$		0			0
$723$			9.00000i			0.334714i
$724$		0			0
$725$		−	18.0000i		−	0.668503i
$726$		0			0
$727$			51.9615i			1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$727$			51.9615i			1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−3.00000	−	20.7846i	−0.110657	−	0.766652i
$736$		0			0
$737$		−	5.00000i		−	0.184177i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$		0			0
$741$	−15.5885			−0.572656
$742$		0			0
$743$		−	15.0000i		−	0.550297i	−0.961402	−	0.275148i	$-0.911273\pi$
$743$		−	15.0000i		−	0.550297i	0.961402	−	0.275148i	$-0.0887270\pi$
$744$		0			0
$745$		−	36.3731i		−	1.33261i
$746$		0			0
$747$	41.5692			1.52094
$748$		0			0
$749$	−5.19615	+	6.00000i	−0.189863	+	0.219235i
$750$		0			0
$751$	−5.00000			−0.182453			−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000			−0.182453			−0.0912263	+	0.995830i	$0.529079\pi$
$752$		0			0
$753$	−51.0000			−1.85854
$754$		0			0
$755$	−13.8564			−0.504286
$756$		0			0
$757$	1.00000			0.0363456			0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000			0.0363456			0.0181728	+	0.999835i	$0.494215\pi$
$758$		0			0
$759$	10.3923			0.377217
$760$		0			0
$761$	−38.1051			−1.38131			−0.690655	−	0.723185i	$-0.742678\pi$
$761$	−38.1051			−1.38131			−0.690655	+	0.723185i	$0.742678\pi$
$762$		0			0
$763$	20.0000	+	17.3205i	0.724049	+	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	9.00000i		−	0.324971i
$768$		0			0
$769$		−	39.8372i		−	1.43657i	−0.695752	−	0.718283i	$-0.744929\pi$
$769$		−	39.8372i		−	1.43657i	0.695752	−	0.718283i	$-0.255071\pi$
$770$		0			0
$771$	3.00000			0.108042
$772$		0			0
$773$	−36.3731			−1.30825			−0.654124	−	0.756387i	$-0.726963\pi$
$773$	−36.3731			−1.30825			−0.654124	+	0.756387i	$0.726963\pi$
$774$		0			0
$775$		−	6.92820i		−	0.248868i
$776$		0			0
$777$	−24.2487	−	21.0000i	−0.869918	−	0.753371i
$778$		0			0
$779$		−	12.0000i		−	0.429945i
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		−	46.7654i		−	1.67126i
$784$		0			0
$785$		−	6.00000i		−	0.214149i
$786$		0			0
$787$		−	5.19615i		−	0.185223i	−0.995702	−	0.0926114i	$-0.970479\pi$
$787$		−	5.19615i		−	0.185223i	0.995702	−	0.0926114i	$-0.0295214\pi$
$788$		0			0
$789$		−	5.19615i		−	0.184988i
$790$		0			0
$791$	−31.1769	+	36.0000i	−1.10852	+	1.28001i
$792$		0			0
$793$	−72.0000			−2.55679
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.73205			−0.0613524			−0.0306762	−	0.999529i	$-0.509766\pi$
$797$	−1.73205			−0.0613524			−0.0306762	+	0.999529i	$0.509766\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	20.7846			0.734388
$802$		0			0
$803$	−8.66025			−0.305614
$804$		0			0
$805$	18.0000	−	20.7846i	0.634417	−	0.732561i
$806$		0			0
$807$	24.0000			0.844840
$808$		0			0
$809$		−	27.0000i		−	0.949269i	−0.880183	−	0.474635i	$-0.842580\pi$
$809$		−	27.0000i		−	0.949269i	0.880183	−	0.474635i	$-0.157420\pi$
$810$		0			0
$811$			39.8372i			1.39887i	0.714695	+	0.699436i	$0.246565\pi$
$811$			39.8372i			1.39887i	−0.714695	+	0.699436i	$0.753435\pi$
$812$		0			0
$813$			27.0000i			0.946931i
$814$		0			0
$815$	−12.1244			−0.424698
$816$		0			0
$817$		−	17.3205i		−	0.605968i
$818$		0			0
$819$	−27.0000	+	31.1769i	−0.943456	+	1.08941i
$820$		0			0
$821$		−	21.0000i		−	0.732905i	−0.930437	−	0.366453i	$-0.880572\pi$
$821$		−	21.0000i		−	0.732905i	0.930437	−	0.366453i	$-0.119428\pi$
$822$		0			0
$823$	−49.0000			−1.70803			−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000			−1.70803			−0.854016	+	0.520246i	$0.825840\pi$
$824$		0			0
$825$		−	3.46410i		−	0.120605i
$826$		0			0
$827$			21.0000i			0.730242i	0.930960	+	0.365121i	$0.118972\pi$
$827$			21.0000i			0.730242i	−0.930960	+	0.365121i	$0.881028\pi$
$828$		0			0
$829$			6.92820i			0.240626i	0.992736	+	0.120313i	$0.0383899\pi$
$829$			6.92820i			0.240626i	−0.992736	+	0.120313i	$0.961610\pi$
$830$		0			0
$831$	38.1051			1.32185
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−18.0000			−0.622916
$836$		0			0
$837$		−	18.0000i		−	0.622171i
$838$		0			0
$839$	36.3731			1.25574			0.627869	−	0.778319i	$-0.283927\pi$
$839$	36.3731			1.25574			0.627869	+	0.778319i	$0.283927\pi$
$840$		0			0
$841$	−52.0000			−1.79310
$842$		0			0
$843$		−	5.19615i		−	0.178965i
$844$		0			0
$845$	−24.2487			−0.834181
$846$		0			0
$847$	−2.00000	−	1.73205i	−0.0687208	−	0.0595140i
$848$		0			0
$849$			21.0000i			0.720718i
$850$		0			0
$851$		−	42.0000i		−	1.43974i
$852$		0			0
$853$		−	27.7128i		−	0.948869i	−0.880291	−	0.474434i	$-0.842653\pi$
$853$		−	27.7128i		−	0.948869i	0.880291	−	0.474434i	$-0.157347\pi$
$854$		0			0
$855$		−	9.00000i		−	0.307794i
$856$		0			0
$857$	48.4974			1.65664			0.828320	−	0.560255i	$-0.189297\pi$
$857$	48.4974			1.65664			0.828320	+	0.560255i	$0.189297\pi$
$858$		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
$861$	−24.0000	−	20.7846i	−0.817918	−	0.708338i
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$	−30.0000			−1.02003
$866$		0			0
$867$	29.4449			1.00000
$868$		0			0
$869$			10.0000i			0.339227i
$870$		0			0
$871$			25.9808i			0.880325i
$872$		0			0
$873$		−	51.9615i		−	1.75863i
$874$		0			0
$875$	−24.2487	−	21.0000i	−0.819756	−	0.709930i
$876$		0			0
$877$	−4.00000			−0.135070			−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000			−0.135070			−0.0675352	+	0.997717i	$0.521513\pi$
$878$		0			0
$879$	12.0000			0.404750
$880$		0			0
$881$	−25.9808			−0.875314			−0.437657	−	0.899142i	$-0.644192\pi$
$881$	−25.9808			−0.875314			−0.437657	+	0.899142i	$0.644192\pi$
$882$		0			0
$883$	7.00000			0.235569			0.117784	−	0.993039i	$-0.462421\pi$
$883$	7.00000			0.235569			0.117784	+	0.993039i	$0.462421\pi$
$884$		0			0
$885$	5.19615			0.174667
$886$		0			0
$887$	31.1769			1.04682			0.523409	−	0.852081i	$-0.324660\pi$
$887$	31.1769			1.04682			0.523409	+	0.852081i	$0.324660\pi$
$888$		0			0
$889$	−32.0000	−	27.7128i	−1.07325	−	0.929458i
$890$		0			0
$891$		−	9.00000i		−	0.301511i
$892$		0			0
$893$			15.0000i			0.501956i
$894$		0			0
$895$			10.3923i			0.347376i
$896$		0			0
$897$	−54.0000			−1.80301
$898$		0			0
$899$	−31.1769			−1.03981
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−34.6410	−	30.0000i	−1.15278	−	0.998337i
$904$		0			0
$905$		−	12.0000i		−	0.398893i
$906$		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
$909$	10.3923			0.344691
$910$		0			0
$911$			54.0000i			1.78910i	0.446968	+	0.894550i	$0.352504\pi$
$911$			54.0000i			1.78910i	−0.446968	+	0.894550i	$0.647496\pi$
$912$		0			0
$913$		−	13.8564i		−	0.458580i
$914$		0			0
$915$		−	41.5692i		−	1.37424i
$916$		0			0
$917$	13.8564	+	12.0000i	0.457579	+	0.396275i
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$		0			0
$921$		−	6.00000i		−	0.197707i
$922$		0			0
$923$	−62.3538			−2.05240
$924$		0			0
$925$	−14.0000			−0.460317
$926$		0			0
$927$		−	10.3923i		−	0.341328i
$928$		0			0
$929$	−46.7654			−1.53432			−0.767161	−	0.641455i	$-0.778331\pi$
$929$	−46.7654			−1.53432			−0.767161	+	0.641455i	$0.778331\pi$
$930$		0			0
$931$	12.0000	−	1.73205i	0.393284	−	0.0567657i
$932$		0			0
$933$	18.0000			0.589294
$934$		0			0
$935$		0			0
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.46410			−0.112926			−0.0564632	−	0.998405i	$-0.517982\pi$
$941$	−3.46410			−0.112926			−0.0564632	+	0.998405i	$0.517982\pi$
$942$		0			0
$943$		−	41.5692i		−	1.35368i
$944$		0			0
$945$	−18.0000	−	15.5885i	−0.585540	−	0.507093i
$946$		0			0
$947$			18.0000i			0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$			18.0000i			0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$		0			0
$949$	45.0000			1.46076
$950$		0			0
$951$			41.5692i			1.34797i
$952$		0			0
$953$		−	3.00000i		−	0.0971795i	−0.998819	−	0.0485898i	$-0.984527\pi$
$953$		−	3.00000i		−	0.0971795i	0.998819	−	0.0485898i	$-0.0154727\pi$
$954$		0			0
$955$			41.5692i			1.34515i
$956$		0			0
$957$	−15.5885			−0.503903
$958$		0			0
$959$	20.7846	−	24.0000i	0.671170	−	0.775000i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$			9.00000i			0.290021i
$964$		0			0
$965$	6.92820			0.223027
$966$		0			0
$967$	50.0000			1.60789			0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000			1.60789			0.803946	+	0.594703i	$0.202730\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.9808			0.833762			0.416881	−	0.908961i	$-0.363123\pi$
$971$	25.9808			0.833762			0.416881	+	0.908961i	$0.363123\pi$
$972$		0			0
$973$	18.0000	−	20.7846i	0.577054	−	0.666324i
$974$		0			0
$975$			18.0000i			0.576461i
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$		−	6.92820i		−	0.221426i
$980$		0			0
$981$	30.0000			0.957826
$982$		0			0
$983$	38.1051			1.21536			0.607682	−	0.794180i	$-0.292099\pi$
$983$	38.1051			1.21536			0.607682	+	0.794180i	$0.292099\pi$
$984$		0			0
$985$		−	10.3923i		−	0.331126i
$986$		0			0
$987$	30.0000	+	25.9808i	0.954911	+	0.826977i
$988$		0			0
$989$		−	60.0000i		−	1.90789i
$990$		0			0
$991$	−5.00000			−0.158830			−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000			−0.158830			−0.0794151	+	0.996842i	$0.525305\pi$
$992$		0			0
$993$	−6.92820			−0.219860
$994$		0			0
$995$		−	42.0000i		−	1.33149i
$996$		0			0
$997$		−	27.7128i		−	0.877674i	−0.898567	−	0.438837i	$-0.855391\pi$
$997$		−	27.7128i		−	0.877674i	0.898567	−	0.438837i	$-0.144609\pi$
$998$		0			0
$999$	−36.3731			−1.15079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	924.2.l.b.881.2	yes	4
3.2	odd	2	inner	924.2.l.b.881.4	yes	4
7.6	odd	2	inner	924.2.l.b.881.3	yes	4
21.20	even	2	inner	924.2.l.b.881.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
924.2.l.b.881.1	✓	4	21.20	even	2	inner
924.2.l.b.881.2	yes	4	1.1	even	1	trivial
924.2.l.b.881.3	yes	4	7.6	odd	2	inner
924.2.l.b.881.4	yes	4	3.2	odd	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:	\( N \)	\(=\)	\( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	924.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} +3.00000 q^{9} -1.00000i q^{11} +5.19615i q^{13} -3.00000 q^{15} -1.73205i q^{19} +(-3.46410 - 3.00000i) q^{21} -6.00000i q^{23} -2.00000 q^{25} -5.19615 q^{27} +9.00000i q^{29} +3.46410i q^{31} +1.73205i q^{33} +(3.46410 + 3.00000i) q^{35} +7.00000 q^{37} -9.00000i q^{39} +6.92820 q^{41} +10.0000 q^{43} +5.19615 q^{45} -8.66025 q^{47} +(1.00000 + 6.92820i) q^{49} -1.73205i q^{55} +3.00000i q^{57} -1.73205 q^{59} +13.8564i q^{61} +(6.00000 + 5.19615i) q^{63} +9.00000i q^{65} +5.00000 q^{67} +10.3923i q^{69} +12.0000i q^{71} -8.66025i q^{73} +3.46410 q^{75} +(1.73205 - 2.00000i) q^{77} -10.0000 q^{79} +9.00000 q^{81} +13.8564 q^{83} -15.5885i q^{87} +6.92820 q^{89} +(-9.00000 + 10.3923i) q^{91} -6.00000i q^{93} -3.00000i q^{95} -17.3205i q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} + 12 q^{9} - 12 q^{15} - 8 q^{25} + 28 q^{37} + 40 q^{43} + 4 q^{49} + 24 q^{63} + 20 q^{67} - 40 q^{79} + 36 q^{81} - 36 q^{91}+O(q^{100}) \) 4 * q + 8 * q^7 + 12 * q^9 - 12 * q^15 - 8 * q^25 + 28 * q^37 + 40 * q^43 + 4 * q^49 + 24 * q^63 + 20 * q^67 - 40 * q^79 + 36 * q^81 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more