LMFDB - Embedded newform 4032.2.k.g.3905.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3905,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4032.3905");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.k (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:	$32.1956820950$
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} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3905.11
Root			$-3.49930i$ of defining polynomial
Character	$\chi$	$=$	4032.3905
Dual form			4032.2.k.g.3905.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239 q^{5} +(2.10100 - 1.60804i) q^{7} +O(q^{10})$	$q+1.08239 q^{5} +(2.10100 - 1.60804i) q^{7} +1.23074i q^{11} -3.69552i q^{13} +6.30864 q^{17} -7.76429i q^{19} -7.17327i q^{23} -3.82843 q^{25} +1.41421i q^{29} +4.54822i q^{31} +(2.27411 - 1.74053i) q^{35} -2.82843 q^{37} -9.37011 q^{41} -7.68306 q^{43} +10.9804 q^{47} +(1.82843 - 6.75699i) q^{49} -5.41421i q^{53} +1.33214i q^{55} -6.43215 q^{59} +9.55582i q^{61} -4.00000i q^{65} -14.3465 q^{67} -10.6543i q^{71} +4.59220i q^{73} +(1.97908 + 2.58579i) q^{77} +9.42359 q^{79} -1.88393 q^{83} +6.82843 q^{85} +5.04054 q^{89} +(-5.94253 - 7.76429i) q^{91} -8.40401i q^{95} +5.86030i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{25} - 16 q^{49} + 64 q^{85}+O(q^{100}) $ 16 * q - 16 * q^25 - 16 * q^49 + 64 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.08239			0.484061			0.242030	−	0.970269i	$-0.422187\pi$
$5$	1.08239			0.484061			0.242030	+	0.970269i	$0.422187\pi$
$6$		0			0
$7$	2.10100	−	1.60804i	0.794104	−	0.607781i
$7$	2.10100	−	1.60804i	0.794104	−	0.607781i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.23074i			0.371082i	0.982637	+	0.185541i	$0.0594037\pi$
$11$			1.23074i			0.371082i	−0.982637	+	0.185541i	$0.940596\pi$
$12$		0			0
$13$		−	3.69552i		−	1.02495i	−0.858701	−	0.512476i	$-0.828728\pi$
$13$		−	3.69552i		−	1.02495i	0.858701	−	0.512476i	$-0.171272\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.30864			1.53007			0.765035	−	0.643988i	$-0.222721\pi$
$17$	6.30864			1.53007			0.765035	+	0.643988i	$0.222721\pi$
$18$		0			0
$19$		−	7.76429i		−	1.78125i	−0.454737	−	0.890626i	$-0.650267\pi$
$19$		−	7.76429i		−	1.78125i	0.454737	−	0.890626i	$-0.349733\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.17327i		−	1.49573i	−0.663850	−	0.747865i	$-0.731079\pi$
$23$		−	7.17327i		−	1.49573i	0.663850	−	0.747865i	$-0.268921\pi$
$24$		0			0
$25$	−3.82843			−0.765685
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$			4.54822i			0.816884i	0.912784	+	0.408442i	$0.133928\pi$
$31$			4.54822i			0.816884i	−0.912784	+	0.408442i	$0.866072\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.27411	−	1.74053i	0.384395	−	0.294203i
$36$		0			0
$37$	−2.82843			−0.464991			−0.232495	−	0.972598i	$-0.574689\pi$
$37$	−2.82843			−0.464991			−0.232495	+	0.972598i	$0.574689\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.37011			−1.46337			−0.731683	−	0.681645i	$-0.761265\pi$
$41$	−9.37011			−1.46337			−0.731683	+	0.681645i	$0.761265\pi$
$42$		0			0
$43$	−7.68306			−1.17166			−0.585828	−	0.810435i	$-0.699231\pi$
$43$	−7.68306			−1.17166			−0.585828	+	0.810435i	$0.699231\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.9804			1.60165			0.800826	−	0.598897i	$-0.204394\pi$
$47$	10.9804			1.60165			0.800826	+	0.598897i	$0.204394\pi$
$48$		0			0
$49$	1.82843	−	6.75699i	0.261204	−	0.965284i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.41421i		−	0.743699i	−0.928293	−	0.371850i	$-0.878724\pi$
$53$		−	5.41421i		−	0.743699i	0.928293	−	0.371850i	$-0.121276\pi$
$54$		0			0
$55$			1.33214i			0.179626i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.43215			−0.837395			−0.418697	−	0.908126i	$-0.637513\pi$
$59$	−6.43215			−0.837395			−0.418697	+	0.908126i	$0.637513\pi$
$60$		0			0
$61$			9.55582i			1.22350i	0.791052	+	0.611749i	$0.209534\pi$
$61$			9.55582i			1.22350i	−0.791052	+	0.611749i	$0.790466\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.00000i		−	0.496139i
$66$		0			0
$67$	−14.3465			−1.75271			−0.876355	−	0.481666i	$-0.840032\pi$
$67$	−14.3465			−1.75271			−0.876355	+	0.481666i	$0.840032\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.6543i		−	1.26444i	−0.774790	−	0.632218i	$-0.782145\pi$
$71$		−	10.6543i		−	1.26444i	0.774790	−	0.632218i	$-0.217855\pi$
$72$		0			0
$73$			4.59220i			0.537476i	0.963213	+	0.268738i	$0.0866067\pi$
$73$			4.59220i			0.537476i	−0.963213	+	0.268738i	$0.913393\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.97908	+	2.58579i	0.225537	+	0.294678i
$78$		0			0
$79$	9.42359			1.06024			0.530118	−	0.847924i	$-0.322147\pi$
$79$	9.42359			1.06024			0.530118	+	0.847924i	$0.322147\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.88393			−0.206789			−0.103394	−	0.994640i	$-0.532970\pi$
$83$	−1.88393			−0.206789			−0.103394	+	0.994640i	$0.532970\pi$
$84$		0			0
$85$	6.82843			0.740647
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.04054			0.534296			0.267148	−	0.963655i	$-0.413919\pi$
$89$	5.04054			0.534296			0.267148	+	0.963655i	$0.413919\pi$
$90$		0			0
$91$	−5.94253	−	7.76429i	−0.622947	−	0.813919i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	8.40401i		−	0.862233i
$96$		0			0
$97$			5.86030i			0.595024i	0.954718	+	0.297512i	$0.0961568\pi$
$97$			5.86030i			0.595024i	−0.954718	+	0.297512i	$0.903843\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.24718			0.323106			0.161553	−	0.986864i	$-0.448350\pi$
$101$	3.24718			0.323106			0.161553	+	0.986864i	$0.448350\pi$
$102$		0			0
$103$		−	1.88393i		−	0.185630i	−0.995683	−	0.0928148i	$-0.970414\pi$
$103$		−	1.88393i		−	0.185630i	0.995683	−	0.0928148i	$-0.0295864\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.17327i		−	0.693466i	−0.937964	−	0.346733i	$-0.887291\pi$
$107$		−	7.17327i		−	0.693466i	0.937964	−	0.346733i	$-0.112709\pi$
$108$		0			0
$109$	7.31371			0.700526			0.350263	−	0.936651i	$-0.386092\pi$
$109$	7.31371			0.700526			0.350263	+	0.936651i	$0.386092\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.0711i			1.04148i	0.853716	+	0.520739i	$0.174344\pi$
$113$			11.0711i			1.04148i	−0.853716	+	0.520739i	$0.825656\pi$
$114$		0			0
$115$		−	7.76429i		−	0.724024i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.2545	−	10.1445i	1.21504	−	0.929948i
$120$		0			0
$121$	9.48528			0.862298
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.55582			−0.854699
$126$		0			0
$127$	−2.46148			−0.218421			−0.109210	−	0.994019i	$-0.534832\pi$
$127$	−2.46148			−0.218421			−0.109210	+	0.994019i	$0.534832\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.6447			1.19214			0.596070	−	0.802933i	$-0.296728\pi$
$131$	13.6447			1.19214			0.596070	+	0.802933i	$0.296728\pi$
$132$		0			0
$133$	−12.4853	−	16.3128i	−1.08261	−	1.41450i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.92893i			0.421107i	0.977582	+	0.210554i	$0.0675266\pi$
$137$			4.92893i			0.421107i	−0.977582	+	0.210554i	$0.932473\pi$
$138$		0			0
$139$		−	15.5286i		−	1.31712i	−0.752529	−	0.658559i	$-0.771166\pi$
$139$		−	15.5286i		−	1.31712i	0.752529	−	0.658559i	$-0.228834\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.54822			0.380341
$144$		0			0
$145$			1.53073i			0.127121i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			9.41421i			0.771242i	0.922657	+	0.385621i	$0.126013\pi$
$149$			9.41421i			0.771242i	−0.922657	+	0.385621i	$0.873987\pi$
$150$		0			0
$151$	−0.720950			−0.0586701			−0.0293350	−	0.999570i	$-0.509339\pi$
$151$	−0.720950			−0.0586701			−0.0293350	+	0.999570i	$0.509339\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			4.92296i			0.395421i
$156$		0			0
$157$			3.95815i			0.315895i	0.987448	+	0.157947i	$0.0504876\pi$
$157$			3.95815i			0.315895i	−0.987448	+	0.157947i	$0.949512\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.5349	−	15.0711i	−0.909077	−	1.18777i
$162$		0			0
$163$	−19.2695			−1.50930			−0.754652	−	0.656125i	$-0.772194\pi$
$163$	−19.2695			−1.50930			−0.754652	+	0.656125i	$0.772194\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.0768			1.55359			0.776795	−	0.629754i	$-0.216844\pi$
$167$	20.0768			1.55359			0.776795	+	0.629754i	$0.216844\pi$
$168$		0			0
$169$	−0.656854			−0.0505272
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.5964			−1.10974			−0.554870	−	0.831937i	$-0.687232\pi$
$173$	−14.5964			−1.10974			−0.554870	+	0.831937i	$0.687232\pi$
$174$		0			0
$175$	−8.04354	+	6.15626i	−0.608034	+	0.465369i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	13.1158i		−	0.980321i	−0.871632	−	0.490161i	$-0.836938\pi$
$179$		−	13.1158i		−	0.980321i	0.871632	−	0.490161i	$-0.163062\pi$
$180$		0			0
$181$			17.2095i			1.27917i	0.768720	+	0.639586i	$0.220894\pi$
$181$			17.2095i			1.27917i	−0.768720	+	0.639586i	$0.779106\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.06147			−0.225084
$186$		0			0
$187$			7.76429i			0.567781i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.63475i		−	0.697146i	−0.937282	−	0.348573i	$-0.886666\pi$
$191$		−	9.63475i		−	0.697146i	0.937282	−	0.348573i	$-0.113334\pi$
$192$		0			0
$193$	−23.7990			−1.71309			−0.856544	−	0.516073i	$-0.827393\pi$
$193$	−23.7990			−1.71309			−0.856544	+	0.516073i	$0.827393\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.72792i			0.621839i	0.950436	+	0.310919i	$0.100637\pi$
$197$			8.72792i			0.621839i	−0.950436	+	0.310919i	$0.899363\pi$
$198$		0			0
$199$		−	9.64823i		−	0.683945i	−0.939710	−	0.341972i	$-0.888905\pi$
$199$		−	9.64823i		−	0.683945i	0.939710	−	0.341972i	$-0.111095\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.27411	+	2.97127i	0.159611	+	0.208542i
$204$		0			0
$205$	−10.1421			−0.708357
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.55582			0.660990
$210$		0			0
$211$	−9.12496			−0.628188			−0.314094	−	0.949392i	$-0.601701\pi$
$211$	−9.12496			−0.628188			−0.314094	+	0.949392i	$0.601701\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.31609			−0.567152
$216$		0			0
$217$	7.31371	+	9.55582i	0.496487	+	0.648691i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	23.3137i		−	1.56825i
$222$		0			0
$223$			3.21608i			0.215364i	0.994185	+	0.107682i	$0.0343429\pi$
$223$			3.21608i			0.215364i	−0.994185	+	0.107682i	$0.965657\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.6250			1.63442			0.817210	−	0.576340i	$-0.195519\pi$
$227$	24.6250			1.63442			0.817210	+	0.576340i	$0.195519\pi$
$228$		0			0
$229$		−	15.9414i		−	1.05344i	−0.850040	−	0.526718i	$-0.823422\pi$
$229$		−	15.9414i		−	1.05344i	0.850040	−	0.526718i	$-0.176578\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.928932i		−	0.0608564i	−0.999537	−	0.0304282i	$-0.990313\pi$
$233$		−	0.928932i		−	0.0608564i	0.999537	−	0.0304282i	$-0.00968709\pi$
$234$		0			0
$235$	11.8851			0.775296
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.25032i			0.145561i	0.997348	+	0.0727804i	$0.0231872\pi$
$239$			2.25032i			0.145561i	−0.997348	+	0.0727804i	$0.976813\pi$
$240$		0			0
$241$		−	10.1899i		−	0.656387i	−0.944610	−	0.328194i	$-0.893560\pi$
$241$		−	10.1899i		−	0.656387i	0.944610	−	0.328194i	$-0.106440\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.97908	−	7.31371i	0.126438	−	0.467256i
$246$		0			0
$247$	−28.6931			−1.82570
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.4125			1.09907			0.549534	−	0.835471i	$-0.314805\pi$
$251$	17.4125			1.09907			0.549534	+	0.835471i	$0.314805\pi$
$252$		0			0
$253$	8.82843			0.555038
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.08239			−0.0675178			−0.0337589	−	0.999430i	$-0.510748\pi$
$257$	−1.08239			−0.0675178			−0.0337589	+	0.999430i	$0.510748\pi$
$258$		0			0
$259$	−5.94253	+	4.54822i	−0.369251	+	0.282613i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.5394i		−	1.38984i	−0.719088	−	0.694919i	$-0.755440\pi$
$263$		−	22.5394i		−	1.38984i	0.719088	−	0.694919i	$-0.244560\pi$
$264$		0			0
$265$		−	5.86030i		−	0.359996i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.4203			1.54990			0.774951	−	0.632021i	$-0.217774\pi$
$269$	25.4203			1.54990			0.774951	+	0.632021i	$0.217774\pi$
$270$		0			0
$271$		−	4.54822i		−	0.276285i	−0.990412	−	0.138142i	$-0.955887\pi$
$271$		−	4.54822i		−	0.276285i	0.990412	−	0.138142i	$-0.0441131\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.71179i		−	0.284132i
$276$		0			0
$277$	23.4558			1.40933			0.704663	−	0.709543i	$-0.251098\pi$
$277$	23.4558			1.40933			0.704663	+	0.709543i	$0.251098\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8.72792i			0.520664i	0.965519	+	0.260332i	$0.0838320\pi$
$281$			8.72792i			0.520664i	−0.965519	+	0.260332i	$0.916168\pi$
$282$		0			0
$283$			1.33214i			0.0791876i	0.999216	+	0.0395938i	$0.0126064\pi$
$283$			1.33214i			0.0791876i	−0.999216	+	0.0395938i	$0.987394\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.6866	+	15.0675i	−1.16207	+	0.889406i
$288$		0			0
$289$	22.7990			1.34112
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.2555			1.35860			0.679300	−	0.733860i	$-0.262283\pi$
$293$	23.2555			1.35860			0.679300	+	0.733860i	$0.262283\pi$
$294$		0			0
$295$	−6.96211			−0.405350
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−26.5090			−1.53305
$300$		0			0
$301$	−16.1421	+	12.3547i	−0.930417	+	0.712111i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.3431i			0.592247i
$306$		0			0
$307$			10.4286i			0.595190i	0.954692	+	0.297595i	$0.0961846\pi$
$307$			10.4286i			0.595190i	−0.954692	+	0.297595i	$0.903815\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.6250			1.39636			0.698179	−	0.715924i	$-0.253994\pi$
$311$	24.6250			1.39636			0.698179	+	0.715924i	$0.253994\pi$
$312$		0			0
$313$		−	27.0279i		−	1.52771i	−0.645388	−	0.763855i	$-0.723304\pi$
$313$		−	27.0279i		−	1.52771i	0.645388	−	0.763855i	$-0.276696\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.8995i			1.45466i	0.686288	+	0.727330i	$0.259239\pi$
$317$			25.8995i			1.45466i	−0.686288	+	0.727330i	$0.740761\pi$
$318$		0			0
$319$	−1.74053			−0.0974509
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	48.9822i		−	2.72544i
$324$		0			0
$325$			14.1480i			0.784791i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	23.0698	−	17.6569i	1.27188	−	0.973454i
$330$		0			0
$331$	21.0100			1.15482			0.577408	−	0.816456i	$-0.304064\pi$
$331$	21.0100			1.15482			0.577408	+	0.816456i	$0.304064\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.5286			−0.848417
$336$		0			0
$337$	10.3431			0.563427			0.281714	−	0.959499i	$-0.409097\pi$
$337$	10.3431			0.563427			0.281714	+	0.959499i	$0.409097\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.59767			−0.303131
$342$		0			0
$343$	−7.02396	−	17.1366i	−0.379258	−	0.925291i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	6.15370i		−	0.330348i	−0.986264	−	0.165174i	$-0.947181\pi$
$347$		−	6.15370i		−	0.330348i	0.986264	−	0.165174i	$-0.0528185\pi$
$348$		0			0
$349$		−	23.0698i		−	1.23490i	−0.786611	−	0.617449i	$-0.788166\pi$
$349$		−	23.0698i		−	1.23490i	0.786611	−	0.617449i	$-0.211834\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.24718			0.172830			0.0864149	−	0.996259i	$-0.472459\pi$
$353$	3.24718			0.172830			0.0864149	+	0.996259i	$0.472459\pi$
$354$		0			0
$355$		−	11.5322i		−	0.612064i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8.61517i			0.454691i	0.973814	+	0.227346i	$0.0730047\pi$
$359$			8.61517i			0.454691i	−0.973814	+	0.227346i	$0.926995\pi$
$360$		0			0
$361$	−41.2843			−2.17286
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.97056i			0.260171i
$366$		0			0
$367$		−	14.9768i		−	0.781782i	−0.920437	−	0.390891i	$-0.872167\pi$
$367$		−	14.9768i		−	0.781782i	0.920437	−	0.390891i	$-0.127833\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.70626	−	11.3753i	−0.452007	−	0.590575i
$372$		0			0
$373$	−18.9706			−0.982259			−0.491129	−	0.871087i	$-0.663416\pi$
$373$	−18.9706			−0.982259			−0.491129	+	0.871087i	$0.663416\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.22625			0.269166
$378$		0			0
$379$	12.6060			0.647528			0.323764	−	0.946138i	$-0.395052\pi$
$379$	12.6060			0.647528			0.323764	+	0.946138i	$0.395052\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.54822			−0.232403			−0.116202	−	0.993226i	$-0.537072\pi$
$383$	−4.54822			−0.232403			−0.116202	+	0.993226i	$0.537072\pi$
$384$		0			0
$385$	2.14214	+	2.79884i	0.109173	+	0.142642i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	35.0711i		−	1.77817i	−0.457739	−	0.889087i	$-0.651340\pi$
$389$		−	35.0711i		−	1.77817i	0.457739	−	0.889087i	$-0.348660\pi$
$390$		0			0
$391$		−	45.2536i		−	2.28857i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.2000			0.513219
$396$		0			0
$397$		−	28.6675i		−	1.43878i	−0.694607	−	0.719389i	$-0.744422\pi$
$397$		−	28.6675i		−	1.43878i	0.694607	−	0.719389i	$-0.255578\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	13.4142i		−	0.669874i	−0.942241	−	0.334937i	$-0.891285\pi$
$401$		−	13.4142i		−	0.669874i	0.942241	−	0.334937i	$-0.108715\pi$
$402$		0			0
$403$	16.8080			0.837267
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.48106i		−	0.172550i
$408$		0			0
$409$			28.0334i			1.38616i	0.720859	+	0.693081i	$0.243747\pi$
$409$			28.0334i			1.38616i	−0.720859	+	0.693081i	$0.756253\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13.5140	+	10.3431i	−0.664979	+	0.508953i
$414$		0			0
$415$	−2.03916			−0.100098
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.09644			0.444390			0.222195	−	0.975002i	$-0.428678\pi$
$419$	9.09644			0.444390			0.222195	+	0.975002i	$0.428678\pi$
$420$		0			0
$421$	26.4853			1.29081			0.645407	−	0.763839i	$-0.276688\pi$
$421$	26.4853			1.29081			0.645407	+	0.763839i	$0.276688\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.1522			−1.17155
$426$		0			0
$427$	15.3661	+	20.0768i	0.743619	+	0.971585i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			40.7893i			1.96475i	0.186914	+	0.982376i	$0.440151\pi$
$431$			40.7893i			1.96475i	−0.186914	+	0.982376i	$0.559849\pi$
$432$		0			0
$433$			13.5140i			0.649440i	0.945810	+	0.324720i	$0.105270\pi$
$433$			13.5140i			0.649440i	−0.945810	+	0.324720i	$0.894730\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−55.6954			−2.66427
$438$		0			0
$439$			9.64823i			0.460485i	0.973133	+	0.230242i	$0.0739519\pi$
$439$			9.64823i			0.460485i	−0.973133	+	0.230242i	$0.926048\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			7.17327i			0.340812i	0.985374	+	0.170406i	$0.0545080\pi$
$443$			7.17327i			0.340812i	−0.985374	+	0.170406i	$0.945492\pi$
$444$		0			0
$445$	5.45584			0.258632
$446$		0			0
$447$		0			0
$448$		0			0
$449$			36.0416i			1.70091i	0.526048	+	0.850455i	$0.323673\pi$
$449$			36.0416i			1.70091i	−0.526048	+	0.850455i	$0.676327\pi$
$450$		0			0
$451$		−	11.5322i		−	0.543028i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.43215	−	8.40401i	−0.301544	−	0.393986i
$456$		0			0
$457$	24.9706			1.16807			0.584037	−	0.811727i	$-0.301472\pi$
$457$	24.9706			1.16807			0.584037	+	0.811727i	$0.301472\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.6997			0.638058			0.319029	−	0.947745i	$-0.396643\pi$
$461$	13.6997			0.638058			0.319029	+	0.947745i	$0.396643\pi$
$462$		0			0
$463$	−2.46148			−0.114395			−0.0571973	−	0.998363i	$-0.518216\pi$
$463$	−2.46148			−0.114395			−0.0571973	+	0.998363i	$0.518216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.2768			1.40104			0.700522	−	0.713631i	$-0.252950\pi$
$467$	30.2768			1.40104			0.700522	+	0.713631i	$0.252950\pi$
$468$		0			0
$469$	−30.1421	+	23.0698i	−1.39183	+	1.06526i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.45584i		−	0.434780i
$474$		0			0
$475$			29.7250i			1.36388i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.1929			−0.831254			−0.415627	−	0.909535i	$-0.636438\pi$
$479$	−18.1929			−0.831254			−0.415627	+	0.909535i	$0.636438\pi$
$480$		0			0
$481$			10.4525i			0.476593i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6.34315i			0.288027i
$486$		0			0
$487$	−27.9721			−1.26754			−0.633769	−	0.773522i	$-0.718493\pi$
$487$	−27.9721			−1.26754			−0.633769	+	0.773522i	$0.718493\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			28.4819i			1.28537i	0.766130	+	0.642686i	$0.222180\pi$
$491$			28.4819i			1.28537i	−0.766130	+	0.642686i	$0.777820\pi$
$492$		0			0
$493$			8.92177i			0.401816i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−17.1326	−	22.3848i	−0.768501	−	1.00409i
$498$		0			0
$499$	0.720950			0.0322742			0.0161371	−	0.999870i	$-0.494863\pi$
$499$	0.720950			0.0322742			0.0161371	+	0.999870i	$0.494863\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.7482			−0.657591			−0.328796	−	0.944401i	$-0.606643\pi$
$503$	−14.7482			−0.657591			−0.328796	+	0.944401i	$0.606643\pi$
$504$		0			0
$505$	3.51472			0.156403
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.9874			0.974574			0.487287	−	0.873242i	$-0.337986\pi$
$509$	21.9874			0.974574			0.487287	+	0.873242i	$0.337986\pi$
$510$		0			0
$511$	7.38443	+	9.64823i	0.326668	+	0.426812i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2.03916i		−	0.0898559i
$516$		0			0
$517$			13.5140i			0.594344i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.35049			0.102977			0.0514885	−	0.998674i	$-0.483603\pi$
$521$	2.35049			0.102977			0.0514885	+	0.998674i	$0.483603\pi$
$522$		0			0
$523$			21.9607i			0.960276i	0.877193	+	0.480138i	$0.159413\pi$
$523$			21.9607i			0.960276i	−0.877193	+	0.480138i	$0.840587\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28.6931i			1.24989i
$528$		0			0
$529$	−28.4558			−1.23721
$530$		0			0
$531$		0			0
$532$		0			0
$533$			34.6274i			1.49988i
$534$		0			0
$535$		−	7.76429i		−	0.335680i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.31609	+	2.25032i	0.358199	+	0.0969280i
$540$		0			0
$541$	−5.79899			−0.249318			−0.124659	−	0.992200i	$-0.539784\pi$
$541$	−5.79899			−0.249318			−0.124659	+	0.992200i	$0.539784\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.91630			0.339097
$546$		0			0
$547$	33.1937			1.41926			0.709631	−	0.704574i	$-0.248862\pi$
$547$	33.1937			1.41926			0.709631	+	0.704574i	$0.248862\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.9804			0.467780
$552$		0			0
$553$	19.7990	−	15.1535i	0.841939	−	0.644392i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	42.8701i		−	1.81646i	−0.418468	−	0.908231i	$-0.637433\pi$
$557$		−	42.8701i		−	1.81646i	0.418468	−	0.908231i	$-0.362567\pi$
$558$		0			0
$559$			28.3929i			1.20089i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.9804			−0.462767			−0.231384	−	0.972863i	$-0.574325\pi$
$563$	−10.9804			−0.462767			−0.231384	+	0.972863i	$0.574325\pi$
$564$		0			0
$565$			11.9832i			0.504139i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11.2721i		−	0.472550i	−0.971686	−	0.236275i	$-0.924073\pi$
$569$		−	11.2721i		−	0.472550i	0.971686	−	0.236275i	$-0.0759266\pi$
$570$		0			0
$571$	−2.46148			−0.103010			−0.0515048	−	0.998673i	$-0.516402\pi$
$571$	−2.46148			−0.103010			−0.0515048	+	0.998673i	$0.516402\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			27.4624i			1.14526i
$576$		0			0
$577$		−	5.59767i		−	0.233034i	−0.993189	−	0.116517i	$-0.962827\pi$
$577$		−	5.59767i		−	0.233034i	0.993189	−	0.116517i	$-0.0371730\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.95815	+	3.02944i	−0.164212	+	0.125682i
$582$		0			0
$583$	6.66348			0.275973
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.1732			−1.20411			−0.602054	−	0.798455i	$-0.705651\pi$
$587$	−29.1732			−1.20411			−0.602054	+	0.798455i	$0.705651\pi$
$588$		0			0
$589$	35.3137			1.45508
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.3897			−0.673046			−0.336523	−	0.941675i	$-0.609251\pi$
$593$	−16.3897			−0.673046			−0.336523	+	0.941675i	$0.609251\pi$
$594$		0			0
$595$	14.3465	−	10.9804i	0.588151	−	0.450151i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			0.808416i			0.0330310i	0.999864	+	0.0165155i	$0.00525728\pi$
$599$			0.808416i			0.0330310i	−0.999864	+	0.0165155i	$0.994743\pi$
$600$		0			0
$601$		−	5.59767i		−	0.228334i	−0.993462	−	0.114167i	$-0.963580\pi$
$601$		−	5.59767i		−	0.228334i	0.993462	−	0.114167i	$-0.0364199\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.2668			0.417405
$606$		0			0
$607$		−	21.4090i		−	0.868962i	−0.900681	−	0.434481i	$-0.856932\pi$
$607$		−	21.4090i		−	0.868962i	0.900681	−	0.434481i	$-0.143068\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	40.5782i		−	1.64162i
$612$		0			0
$613$	22.2843			0.900053			0.450027	−	0.893015i	$-0.351414\pi$
$613$	22.2843			0.900053			0.450027	+	0.893015i	$0.351414\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			44.0416i			1.77305i	0.462681	+	0.886525i	$0.346887\pi$
$617$			44.0416i			1.77305i	−0.462681	+	0.886525i	$0.653113\pi$
$618$		0			0
$619$			32.1608i			1.29265i	0.763062	+	0.646325i	$0.223695\pi$
$619$			32.1608i			1.29265i	−0.763062	+	0.646325i	$0.776305\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.5902	−	8.10538i	0.424287	−	0.324735i
$624$		0			0
$625$	8.79899			0.351960
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−17.8435			−0.711469
$630$		0			0
$631$	−24.1925			−0.963087			−0.481543	−	0.876422i	$-0.659924\pi$
$631$	−24.1925			−0.963087			−0.481543	+	0.876422i	$0.659924\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.66428			−0.105729
$636$		0			0
$637$	−24.9706	−	6.75699i	−0.989370	−	0.267722i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			27.7574i			1.09635i	0.836364	+	0.548175i	$0.184677\pi$
$641$			27.7574i			1.09635i	−0.836364	+	0.548175i	$0.815323\pi$
$642$		0			0
$643$		−	36.1572i		−	1.42590i	−0.701215	−	0.712950i	$-0.747359\pi$
$643$		−	36.1572i		−	1.42590i	0.701215	−	0.712950i	$-0.252641\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−41.2572			−1.62199			−0.810994	−	0.585055i	$-0.801073\pi$
$647$	−41.2572			−1.62199			−0.810994	+	0.585055i	$0.801073\pi$
$648$		0			0
$649$		−	7.91630i		−	0.310742i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16.7279i			0.654614i	0.944918	+	0.327307i	$0.106141\pi$
$653$			16.7279i			0.654614i	−0.944918	+	0.327307i	$0.893859\pi$
$654$		0			0
$655$	14.7689			0.577067
$656$		0			0
$657$		0			0
$658$		0			0
$659$			42.8285i			1.66836i	0.551492	+	0.834180i	$0.314059\pi$
$659$			42.8285i			1.66836i	−0.551492	+	0.834180i	$0.685941\pi$
$660$		0			0
$661$		−	42.1814i		−	1.64067i	−0.571885	−	0.820334i	$-0.693788\pi$
$661$		−	42.1814i		−	1.64067i	0.571885	−	0.820334i	$-0.306212\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13.5140	−	17.6569i	−0.524049	−	0.684703i
$666$		0			0
$667$	10.1445			0.392798
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11.7607			−0.454018
$672$		0			0
$673$	17.1716			0.661915			0.330958	−	0.943646i	$-0.392628\pi$
$673$	17.1716			0.661915			0.330958	+	0.943646i	$0.392628\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.3897			0.629908			0.314954	−	0.949107i	$-0.398011\pi$
$677$	16.3897			0.629908			0.314954	+	0.949107i	$0.398011\pi$
$678$		0			0
$679$	9.42359	+	12.3125i	0.361644	+	0.472511i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.5590i			0.901459i	0.892661	+	0.450730i	$0.148836\pi$
$683$			23.5590i			0.901459i	−0.892661	+	0.450730i	$0.851164\pi$
$684$		0			0
$685$			5.33504i			0.203841i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−20.0083			−0.762256
$690$		0			0
$691$			19.2965i			0.734072i	0.930207	+	0.367036i	$0.119627\pi$
$691$			19.2965i			0.734072i	−0.930207	+	0.367036i	$0.880373\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	16.8080i		−	0.637565i
$696$		0			0
$697$	−59.1127			−2.23905
$698$		0			0
$699$		0			0
$700$		0			0
$701$			16.0416i			0.605884i	0.953009	+	0.302942i	$0.0979689\pi$
$701$			16.0416i			0.605884i	−0.953009	+	0.302942i	$0.902031\pi$
$702$		0			0
$703$			21.9607i			0.828265i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.82233	−	5.22158i	0.256580	−	0.196378i
$708$		0			0
$709$	−32.2843			−1.21246			−0.606231	−	0.795289i	$-0.707319\pi$
$709$	−32.2843			−1.21246			−0.606231	+	0.795289i	$0.707319\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	32.6256			1.22184
$714$		0			0
$715$	4.92296			0.184108
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.1536			−1.49748			−0.748739	−	0.662865i	$-0.769340\pi$
$719$	−40.1536			−1.49748			−0.748739	+	0.662865i	$0.769340\pi$
$720$		0			0
$721$	−3.02944	−	3.95815i	−0.112822	−	0.147409i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	5.41421i		−	0.201079i
$726$		0			0
$727$			45.8054i			1.69883i	0.527726	+	0.849414i	$0.323045\pi$
$727$			45.8054i			1.69883i	−0.527726	+	0.849414i	$0.676955\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.4697			−1.79272
$732$		0			0
$733$			48.5670i			1.79386i	0.442169	+	0.896932i	$0.354209\pi$
$733$			48.5670i			1.79386i	−0.442169	+	0.896932i	$0.645791\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	17.6569i		−	0.650399i
$738$		0			0
$739$	−43.0396			−1.58324			−0.791619	−	0.611015i	$-0.790762\pi$
$739$	−43.0396			−1.58324			−0.791619	+	0.611015i	$0.790762\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	4.71179i		−	0.172859i	−0.996258	−	0.0864295i	$-0.972454\pi$
$743$		−	4.71179i		−	0.172859i	0.996258	−	0.0864295i	$-0.0275457\pi$
$744$		0			0
$745$			10.1899i			0.373328i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.5349	−	15.0711i	−0.421476	−	0.550685i
$750$		0			0
$751$	19.5681			0.714051			0.357026	−	0.934095i	$-0.383791\pi$
$751$	19.5681			0.714051			0.357026	+	0.934095i	$0.383791\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.780351			−0.0283999
$756$		0			0
$757$	17.6569			0.641749			0.320875	−	0.947122i	$-0.396023\pi$
$757$	17.6569			0.641749			0.320875	+	0.947122i	$0.396023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.7080			1.22191			0.610957	−	0.791664i	$-0.290785\pi$
$761$	33.7080			1.22191			0.610957	+	0.791664i	$0.290785\pi$
$762$		0			0
$763$	15.3661	−	11.7607i	0.556291	−	0.425767i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			23.7701i			0.858290i
$768$		0			0
$769$		−	40.5419i		−	1.46198i	−0.682389	−	0.730989i	$-0.739059\pi$
$769$		−	40.5419i		−	1.46198i	0.682389	−	0.730989i	$-0.260941\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.9343			1.40037			0.700184	−	0.713963i	$-0.253101\pi$
$773$	38.9343			1.40037			0.700184	+	0.713963i	$0.253101\pi$
$774$		0			0
$775$		−	17.4125i		−	0.625476i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			72.7523i			2.60662i
$780$		0			0
$781$	13.1127			0.469209
$782$		0			0
$783$		0			0
$784$		0			0
$785$			4.28427i			0.152912i
$786$		0			0
$787$			31.0572i			1.10707i	0.832826	+	0.553534i	$0.186721\pi$
$787$			31.0572i			1.10707i	−0.832826	+	0.553534i	$0.813279\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	17.8027	+	23.2603i	0.632991	+	0.827043i
$792$		0			0
$793$	35.3137			1.25403
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.2137			−0.963957			−0.481978	−	0.876183i	$-0.660082\pi$
$797$	−27.2137			−0.963957			−0.481978	+	0.876183i	$0.660082\pi$
$798$		0			0
$799$	69.2713			2.45064
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.65180			−0.199448
$804$		0			0
$805$	−12.4853	−	16.3128i	−0.440048	−	0.574951i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	27.3553i		−	0.961763i	−0.876786	−	0.480881i	$-0.840317\pi$
$809$		−	27.3553i		−	0.961763i	0.876786	−	0.480881i	$-0.159683\pi$
$810$		0			0
$811$			25.7286i			0.903454i	0.892156	+	0.451727i	$0.149192\pi$
$811$			25.7286i			0.903454i	−0.892156	+	0.451727i	$0.850808\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.8572			−0.730594
$816$		0			0
$817$			59.6536i			2.08701i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			13.4142i			0.468159i	0.972217	+	0.234080i	$0.0752076\pi$
$821$			13.4142i			0.468159i	−0.972217	+	0.234080i	$0.924792\pi$
$822$		0			0
$823$	−0.422323			−0.0147213			−0.00736063	−	0.999973i	$-0.502343\pi$
$823$	−0.422323			−0.0147213			−0.00736063	+	0.999973i	$0.502343\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11.0767i		−	0.385173i	−0.981280	−	0.192586i	$-0.938312\pi$
$827$		−	11.0767i		−	0.385173i	0.981280	−	0.192586i	$-0.0616876\pi$
$828$		0			0
$829$		−	8.55035i		−	0.296966i	−0.988915	−	0.148483i	$-0.952561\pi$
$829$		−	8.55035i		−	0.296966i	0.988915	−	0.148483i	$-0.0474390\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.5349	−	42.6274i	0.399660	−	1.47695i
$834$		0			0
$835$	21.7310			0.752032
$836$		0			0
$837$		0			0
$838$		0			0
$839$	43.1411			1.48940			0.744699	−	0.667401i	$-0.232593\pi$
$839$	43.1411			1.48940			0.744699	+	0.667401i	$0.232593\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.710974			−0.0244582
$846$		0			0
$847$	19.9286	−	15.2527i	0.684755	−	0.524089i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			20.2891i			0.695501i
$852$		0			0
$853$		−	9.55582i		−	0.327185i	−0.986528	−	0.163593i	$-0.947692\pi$
$853$		−	9.55582i		−	0.327185i	0.986528	−	0.163593i	$-0.0523082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.84485			−0.302134			−0.151067	−	0.988524i	$-0.548271\pi$
$857$	−8.84485			−0.302134			−0.151067	+	0.988524i	$0.548271\pi$
$858$		0			0
$859$		−	19.5250i		−	0.666185i	−0.942894	−	0.333092i	$-0.891908\pi$
$859$		−	19.5250i		−	0.666185i	0.942894	−	0.333092i	$-0.108092\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			33.4049i			1.13712i	0.822643	+	0.568558i	$0.192498\pi$
$863$			33.4049i			1.13712i	−0.822643	+	0.568558i	$0.807502\pi$
$864$		0			0
$865$	−15.7990			−0.537182
$866$		0			0
$867$		0			0
$868$		0			0
$869$			11.5980i			0.393435i
$870$		0			0
$871$			53.0179i			1.79644i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20.0768	+	15.3661i	−0.678720	+	0.519470i
$876$		0			0
$877$	−13.3137			−0.449572			−0.224786	−	0.974408i	$-0.572168\pi$
$877$	−13.3137			−0.449572			−0.224786	+	0.974408i	$0.572168\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.9678			−0.504277			−0.252139	−	0.967691i	$-0.581134\pi$
$881$	−14.9678			−0.504277			−0.252139	+	0.967691i	$0.581134\pi$
$882$		0			0
$883$	−43.3383			−1.45845			−0.729224	−	0.684275i	$-0.760119\pi$
$883$	−43.3383			−1.45845			−0.729224	+	0.684275i	$0.760119\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	32.9411			1.10605			0.553027	−	0.833163i	$-0.313473\pi$
$887$	32.9411			1.10605			0.553027	+	0.833163i	$0.313473\pi$
$888$		0			0
$889$	−5.17157	+	3.95815i	−0.173449	+	0.132752i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	85.2548i		−	2.85294i
$894$		0			0
$895$		−	14.1964i		−	0.474535i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.43215			−0.214524
$900$		0			0
$901$		−	34.1563i		−	1.13791i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			18.6274i			0.619196i
$906$		0			0
$907$	37.8181			1.25573			0.627864	−	0.778323i	$-0.283929\pi$
$907$	37.8181			1.25573			0.627864	+	0.778323i	$0.283929\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			3.26989i			0.108336i	0.998532	+	0.0541682i	$0.0172507\pi$
$911$			3.26989i			0.108336i	−0.998532	+	0.0541682i	$0.982749\pi$
$912$		0			0
$913$		−	2.31863i		−	0.0767355i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	28.6675	−	21.9411i	0.946683	−	0.724560i
$918$		0			0
$919$	19.5681			0.645493			0.322747	−	0.946485i	$-0.395394\pi$
$919$	19.5681			0.645493			0.322747	+	0.946485i	$0.395394\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−39.3733			−1.29599
$924$		0			0
$925$	10.8284			0.356036
$926$		0			0
$927$		0			0
$928$		0			0
$929$	37.1409			1.21855			0.609277	−	0.792958i	$-0.291460\pi$
$929$	37.1409			1.21855			0.609277	+	0.792958i	$0.291460\pi$
$930$		0			0
$931$	−52.4632	−	14.1964i	−1.71941	−	0.465270i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			8.40401i			0.274841i
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.35049			−0.0766239			−0.0383119	−	0.999266i	$-0.512198\pi$
$941$	−2.35049			−0.0766239			−0.0383119	+	0.999266i	$0.512198\pi$
$942$		0			0
$943$			67.2144i			2.18880i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.8859i			1.19863i	0.800513	+	0.599316i	$0.204561\pi$
$947$			36.8859i			1.19863i	−0.800513	+	0.599316i	$0.795439\pi$
$948$		0			0
$949$	16.9706			0.550888
$950$		0			0
$951$		0			0
$952$		0			0
$953$			38.8701i			1.25912i	0.776950	+	0.629562i	$0.216766\pi$
$953$			38.8701i			1.25912i	−0.776950	+	0.629562i	$0.783234\pi$
$954$		0			0
$955$		−	10.4286i		−	0.337461i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.92591	+	10.3557i	0.255941	+	0.334403i
$960$		0			0
$961$	10.3137			0.332700
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25.7598			−0.829239
$966$		0			0
$967$	−9.42359			−0.303042			−0.151521	−	0.988454i	$-0.548417\pi$
$967$	−9.42359			−0.303042			−0.151521	+	0.988454i	$0.548417\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.7607			−0.377420			−0.188710	−	0.982033i	$-0.560431\pi$
$971$	−11.7607			−0.377420			−0.188710	+	0.982033i	$0.560431\pi$
$972$		0			0
$973$	−24.9706	−	32.6256i	−0.800519	−	1.04593i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			5.41421i			0.173216i	0.996242	+	0.0866080i	$0.0276028\pi$
$977$			5.41421i			0.173216i	−0.996242	+	0.0866080i	$0.972397\pi$
$978$		0			0
$979$			6.20359i			0.198268i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	45.8054			1.46097			0.730483	−	0.682931i	$-0.239295\pi$
$983$	45.8054			1.46097			0.730483	+	0.682931i	$0.239295\pi$
$984$		0			0
$985$			9.44703i			0.301008i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			55.1127i			1.75248i
$990$		0			0
$991$	51.7423			1.64365			0.821824	−	0.569742i	$-0.192957\pi$
$991$	51.7423			1.64365			0.821824	+	0.569742i	$0.192957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	10.4432i		−	0.331071i
$996$		0			0
$997$			47.7791i			1.51318i	0.653890	+	0.756590i	$0.273136\pi$
$997$			47.7791i			1.51318i	−0.653890	+	0.756590i	$0.726864\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.k.g.3905.11		16
3.2	odd	2	inner	4032.2.k.g.3905.7		16
4.3	odd	2	inner	4032.2.k.g.3905.10		16
7.6	odd	2	inner	4032.2.k.g.3905.8		16
8.3	odd	2		2016.2.k.a.1889.6	yes	16
8.5	even	2		2016.2.k.a.1889.7	yes	16
12.11	even	2	inner	4032.2.k.g.3905.6		16
21.20	even	2	inner	4032.2.k.g.3905.12		16
24.5	odd	2		2016.2.k.a.1889.11	yes	16
24.11	even	2		2016.2.k.a.1889.10	yes	16
28.27	even	2	inner	4032.2.k.g.3905.5		16
56.13	odd	2		2016.2.k.a.1889.12	yes	16
56.27	even	2		2016.2.k.a.1889.9	yes	16
84.83	odd	2	inner	4032.2.k.g.3905.9		16
168.83	odd	2		2016.2.k.a.1889.5	✓	16
168.125	even	2		2016.2.k.a.1889.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.k.a.1889.5	✓	16	168.83	odd	2
2016.2.k.a.1889.6	yes	16	8.3	odd	2
2016.2.k.a.1889.7	yes	16	8.5	even	2
2016.2.k.a.1889.8	yes	16	168.125	even	2
2016.2.k.a.1889.9	yes	16	56.27	even	2
2016.2.k.a.1889.10	yes	16	24.11	even	2
2016.2.k.a.1889.11	yes	16	24.5	odd	2
2016.2.k.a.1889.12	yes	16	56.13	odd	2
4032.2.k.g.3905.5		16	28.27	even	2	inner
4032.2.k.g.3905.6		16	12.11	even	2	inner
4032.2.k.g.3905.7		16	3.2	odd	2	inner
4032.2.k.g.3905.8		16	7.6	odd	2	inner
4032.2.k.g.3905.9		16	84.83	odd	2	inner
4032.2.k.g.3905.10		16	4.3	odd	2	inner
4032.2.k.g.3905.11		16	1.1	even	1	trivial
4032.2.k.g.3905.12		16	21.20	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.08239 q^{5} +(2.10100 - 1.60804i) q^{7} +O(q^{10})\)	\(q+1.08239 q^{5} +(2.10100 - 1.60804i) q^{7} +1.23074i q^{11} -3.69552i q^{13} +6.30864 q^{17} -7.76429i q^{19} -7.17327i q^{23} -3.82843 q^{25} +1.41421i q^{29} +4.54822i q^{31} +(2.27411 - 1.74053i) q^{35} -2.82843 q^{37} -9.37011 q^{41} -7.68306 q^{43} +10.9804 q^{47} +(1.82843 - 6.75699i) q^{49} -5.41421i q^{53} +1.33214i q^{55} -6.43215 q^{59} +9.55582i q^{61} -4.00000i q^{65} -14.3465 q^{67} -10.6543i q^{71} +4.59220i q^{73} +(1.97908 + 2.58579i) q^{77} +9.42359 q^{79} -1.88393 q^{83} +6.82843 q^{85} +5.04054 q^{89} +(-5.94253 - 7.76429i) q^{91} -8.40401i q^{95} +5.86030i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{25} - 16 q^{49} + 64 q^{85}+O(q^{100}) \) 16 * q - 16 * q^25 - 16 * q^49 + 64 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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