LMFDB - Embedded newform 1656.2.m.a.1241.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(1241,1656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1656, 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("1656.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.m (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:	$13.2232265747$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.20
Character	$\chi$	$=$	1656.1241
Dual form			1656.2.m.a.1241.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83341 q^{5} +1.64300i q^{7} +O(q^{10})$	$q+1.83341 q^{5} +1.64300i q^{7} -3.26043 q^{11} -3.78789 q^{13} -6.03797 q^{17} -2.35482i q^{19} +(-3.44514 + 3.33632i) q^{23} -1.63861 q^{25} -7.92037i q^{29} -7.60302 q^{31} +3.01229i q^{35} -0.936880i q^{37} -8.18881i q^{41} +11.8379i q^{43} -5.94267i q^{47} +4.30056 q^{49} +1.87763 q^{53} -5.97770 q^{55} -7.09194i q^{59} -2.92383i q^{61} -6.94475 q^{65} +13.4241i q^{67} +1.06962i q^{71} +13.8630 q^{73} -5.35688i q^{77} -0.0827410i q^{79} +13.4890 q^{83} -11.0701 q^{85} -13.8396 q^{89} -6.22349i q^{91} -4.31735i q^{95} -4.44021i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 40 q^{25} - 16 q^{31} - 40 q^{49} - 64 q^{55} + 48 q^{85}+O(q^{100}) $ 24 * q + 40 * q^25 - 16 * q^31 - 40 * q^49 - 64 * q^55 + 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1656\mathbb{Z}\right)^\times$.

$n$	$415$	$649$	$829$	$1289$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	1.83341			0.819925			0.409963	−	0.912102i	$-0.365542\pi$
$5$	1.83341			0.819925			0.409963	+	0.912102i	$0.365542\pi$
$6$		0			0
$7$			1.64300i			0.620995i	0.950574	+	0.310497i	$0.100496\pi$
$7$			1.64300i			0.620995i	−0.950574	+	0.310497i	$0.899504\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.26043			−0.983057			−0.491528	−	0.870862i	$-0.663562\pi$
$11$	−3.26043			−0.983057			−0.491528	+	0.870862i	$0.663562\pi$
$12$		0			0
$13$	−3.78789			−1.05057			−0.525286	−	0.850926i	$-0.676041\pi$
$13$	−3.78789			−1.05057			−0.525286	+	0.850926i	$0.676041\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.03797			−1.46442			−0.732211	−	0.681078i	$-0.761512\pi$
$17$	−6.03797			−1.46442			−0.732211	+	0.681078i	$0.761512\pi$
$18$		0			0
$19$		−	2.35482i		−	0.540233i	−0.962828	−	0.270116i	$-0.912938\pi$
$19$		−	2.35482i		−	0.540233i	0.962828	−	0.270116i	$-0.0870622\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.44514	+	3.33632i	−0.718361	+	0.695671i
$23$	−3.44514	+	3.33632i	−0.718361	+	0.695671i
$24$		0			0
$25$	−1.63861			−0.327723
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.92037i		−	1.47078i	−0.677646	−	0.735388i	$-0.737000\pi$
$29$		−	7.92037i		−	1.47078i	0.677646	−	0.735388i	$-0.263000\pi$
$30$		0			0
$31$	−7.60302			−1.36554			−0.682772	−	0.730632i	$-0.739226\pi$
$31$	−7.60302			−1.36554			−0.682772	+	0.730632i	$0.739226\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.01229i			0.509169i
$36$		0			0
$37$		−	0.936880i		−	0.154022i	−0.997030	−	0.0770111i	$-0.975462\pi$
$37$		−	0.936880i		−	0.154022i	0.997030	−	0.0770111i	$-0.0245377\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.18881i		−	1.27888i	−0.768842	−	0.639439i	$-0.779167\pi$
$41$		−	8.18881i		−	1.27888i	0.768842	−	0.639439i	$-0.220833\pi$
$42$		0			0
$43$			11.8379i			1.80527i	0.430410	+	0.902633i	$0.358369\pi$
$43$			11.8379i			1.80527i	−0.430410	+	0.902633i	$0.641631\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.94267i		−	0.866827i	−0.901195	−	0.433414i	$-0.857309\pi$
$47$		−	5.94267i		−	0.866827i	0.901195	−	0.433414i	$-0.142691\pi$
$48$		0			0
$49$	4.30056			0.614365
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.87763			0.257913			0.128956	−	0.991650i	$-0.458837\pi$
$53$	1.87763			0.257913			0.128956	+	0.991650i	$0.458837\pi$
$54$		0			0
$55$	−5.97770			−0.806033
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.09194i		−	0.923292i	−0.887064	−	0.461646i	$-0.847259\pi$
$59$		−	7.09194i		−	0.923292i	0.887064	−	0.461646i	$-0.152741\pi$
$60$		0			0
$61$		−	2.92383i		−	0.374358i	−0.982326	−	0.187179i	$-0.940065\pi$
$61$		−	2.92383i		−	0.374358i	0.982326	−	0.187179i	$-0.0599345\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.94475			−0.861390
$66$		0			0
$67$			13.4241i			1.64001i	0.572354	+	0.820007i	$0.306030\pi$
$67$			13.4241i			1.64001i	−0.572354	+	0.820007i	$0.693970\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.06962i			0.126940i	0.997984	+	0.0634702i	$0.0202168\pi$
$71$			1.06962i			0.126940i	−0.997984	+	0.0634702i	$0.979783\pi$
$72$		0			0
$73$	13.8630			1.62255			0.811273	−	0.584667i	$-0.198775\pi$
$73$	13.8630			1.62255			0.811273	+	0.584667i	$0.198775\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	5.35688i		−	0.610473i
$78$		0			0
$79$		−	0.0827410i		−	0.00930909i	−0.999989	−	0.00465455i	$-0.998518\pi$
$79$		−	0.0827410i		−	0.00930909i	0.999989	−	0.00465455i	$-0.00148159\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.4890			1.48061			0.740305	−	0.672272i	$-0.234681\pi$
$83$	13.4890			1.48061			0.740305	+	0.672272i	$0.234681\pi$
$84$		0			0
$85$	−11.0701			−1.20072
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.8396			−1.46699			−0.733495	−	0.679695i	$-0.762112\pi$
$89$	−13.8396			−1.46699			−0.733495	+	0.679695i	$0.762112\pi$
$90$		0			0
$91$		−	6.22349i		−	0.652399i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.31735i		−	0.442951i
$96$		0			0
$97$		−	4.44021i		−	0.450835i	−0.974262	−	0.225417i	$-0.927625\pi$
$97$		−	4.44021i		−	0.450835i	0.974262	−	0.225417i	$-0.0723746\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.264939i			0.0263624i	0.999913	+	0.0131812i	$0.00419582\pi$
$101$			0.264939i			0.0263624i	−0.999913	+	0.0131812i	$0.995804\pi$
$102$		0			0
$103$			7.26539i			0.715880i	0.933745	+	0.357940i	$0.116521\pi$
$103$			7.26539i			0.715880i	−0.933745	+	0.357940i	$0.883479\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.4330			−1.00860			−0.504300	−	0.863528i	$-0.668250\pi$
$107$	−10.4330			−1.00860			−0.504300	+	0.863528i	$0.668250\pi$
$108$		0			0
$109$			18.7964i			1.80037i	0.435505	+	0.900186i	$0.356570\pi$
$109$			18.7964i			1.80037i	−0.435505	+	0.900186i	$0.643430\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.6326			−1.47059			−0.735296	−	0.677746i	$-0.762957\pi$
$113$	−15.6326			−1.47059			−0.735296	+	0.677746i	$0.762957\pi$
$114$		0			0
$115$	−6.31634	+	6.11684i	−0.589002	+	0.570398i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.92037i		−	0.909399i
$120$		0			0
$121$	−0.369590			−0.0335991
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.1713			−1.08863
$126$		0			0
$127$	−9.88534			−0.877182			−0.438591	−	0.898687i	$-0.644522\pi$
$127$	−9.88534			−0.877182			−0.438591	+	0.898687i	$0.644522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.21889i			0.543347i	0.962389	+	0.271674i	$0.0875771\pi$
$131$			6.21889i			0.543347i	−0.962389	+	0.271674i	$0.912423\pi$
$132$		0			0
$133$	3.86897			0.335482
$134$		0			0
$135$		0			0
$136$		0			0
$137$	21.0930			1.80210			0.901050	−	0.433716i	$-0.142798\pi$
$137$	21.0930			1.80210			0.901050	+	0.433716i	$0.142798\pi$
$138$		0			0
$139$	−8.37267			−0.710161			−0.355080	−	0.934836i	$-0.615546\pi$
$139$	−8.37267			−0.710161			−0.355080	+	0.934836i	$0.615546\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.3501			1.03277
$144$		0			0
$145$		−	14.5213i		−	1.20593i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.58572			−0.621446			−0.310723	−	0.950500i	$-0.600571\pi$
$149$	−7.58572			−0.621446			−0.310723	+	0.950500i	$0.600571\pi$
$150$		0			0
$151$	6.42650			0.522981			0.261491	−	0.965206i	$-0.415786\pi$
$151$	6.42650			0.522981			0.261491	+	0.965206i	$0.415786\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.9394			−1.11964
$156$		0			0
$157$		−	11.6186i		−	0.927263i	−0.886028	−	0.463631i	$-0.846546\pi$
$157$		−	11.6186i		−	0.927263i	0.886028	−	0.463631i	$-0.153454\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.48157	−	5.66035i	−0.432008	−	0.446098i
$162$		0			0
$163$	−11.0934			−0.868902			−0.434451	−	0.900696i	$-0.643058\pi$
$163$	−11.0934			−0.868902			−0.434451	+	0.900696i	$0.643058\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			21.5199i			1.66526i	0.553830	+	0.832630i	$0.313166\pi$
$167$			21.5199i			1.66526i	−0.553830	+	0.832630i	$0.686834\pi$
$168$		0			0
$169$	1.34809			0.103700
$170$		0			0
$171$		0			0
$172$		0			0
$173$			0.0350333i			0.00266353i	0.999999	+	0.00133177i	$0.000423914\pi$
$173$			0.0350333i			0.00266353i	−0.999999	+	0.00133177i	$0.999576\pi$
$174$		0			0
$175$		−	2.69224i		−	0.203514i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.54275i		−	0.115311i	−0.998337	−	0.0576554i	$-0.981638\pi$
$179$		−	1.54275i		−	0.115311i	0.998337	−	0.0576554i	$-0.0183625\pi$
$180$		0			0
$181$		−	6.24099i		−	0.463889i	−0.972729	−	0.231945i	$-0.925491\pi$
$181$		−	6.24099i		−	0.463889i	0.972729	−	0.231945i	$-0.0745088\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.71768i		−	0.126287i
$186$		0			0
$187$	19.6864			1.43961
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.486425			−0.0351965			−0.0175982	−	0.999845i	$-0.505602\pi$
$191$	−0.486425			−0.0351965			−0.0175982	+	0.999845i	$0.505602\pi$
$192$		0			0
$193$	9.56757			0.688689			0.344344	−	0.938843i	$-0.388101\pi$
$193$	9.56757			0.688689			0.344344	+	0.938843i	$0.388101\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.26352i			0.161269i	0.996744	+	0.0806344i	$0.0256946\pi$
$197$			2.26352i			0.161269i	−0.996744	+	0.0806344i	$0.974305\pi$
$198$		0			0
$199$			8.22640i			0.583154i	0.956547	+	0.291577i	$0.0941800\pi$
$199$			8.22640i			0.583154i	−0.956547	+	0.291577i	$0.905820\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.0132			0.913344
$204$		0			0
$205$		−	15.0134i		−	1.04858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7.67773i			0.531080i
$210$		0			0
$211$	−20.3281			−1.39944			−0.699721	−	0.714416i	$-0.746692\pi$
$211$	−20.3281			−1.39944			−0.699721	+	0.714416i	$0.746692\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			21.7038i			1.48018i
$216$		0			0
$217$		−	12.4918i		−	0.847995i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.8711			1.53848
$222$		0			0
$223$	−13.1472			−0.880403			−0.440202	−	0.897899i	$-0.645093\pi$
$223$	−13.1472			−0.880403			−0.440202	+	0.897899i	$0.645093\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.848435			0.0563126			0.0281563	−	0.999604i	$-0.491036\pi$
$227$	0.848435			0.0563126			0.0281563	+	0.999604i	$0.491036\pi$
$228$		0			0
$229$			25.8440i			1.70782i	0.520422	+	0.853909i	$0.325775\pi$
$229$			25.8440i			1.70782i	−0.520422	+	0.853909i	$0.674225\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	25.2880i		−	1.65667i	−0.560234	−	0.828334i	$-0.689289\pi$
$233$		−	25.2880i		−	1.65667i	0.560234	−	0.828334i	$-0.310711\pi$
$234$		0			0
$235$		−	10.8953i		−	0.710734i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.37062i		−	0.541450i	−0.962657	−	0.270725i	$-0.912737\pi$
$239$		−	8.37062i		−	0.541450i	0.962657	−	0.270725i	$-0.0872635\pi$
$240$		0			0
$241$			5.40397i			0.348100i	0.984737	+	0.174050i	$0.0556854\pi$
$241$			5.40397i			0.348100i	−0.984737	+	0.174050i	$0.944315\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.88468			0.503734
$246$		0			0
$247$			8.91980i			0.567553i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.6489			1.55582			0.777912	−	0.628373i	$-0.216279\pi$
$251$	24.6489			1.55582			0.777912	+	0.628373i	$0.216279\pi$
$252$		0			0
$253$	11.2326	−	10.8778i	0.706189	−	0.683884i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.95413i			0.309030i	0.987990	+	0.154515i	$0.0493815\pi$
$257$			4.95413i			0.309030i	−0.987990	+	0.154515i	$0.950618\pi$
$258$		0			0
$259$	1.53929			0.0956470
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.65718			−0.595488			−0.297744	−	0.954646i	$-0.596234\pi$
$263$	−9.65718			−0.595488			−0.297744	+	0.954646i	$0.596234\pi$
$264$		0			0
$265$	3.44247			0.211469
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.6218i			0.830537i	0.909699	+	0.415269i	$0.136312\pi$
$269$			13.6218i			0.830537i	−0.909699	+	0.415269i	$0.863688\pi$
$270$		0			0
$271$	15.7179			0.954795			0.477398	−	0.878687i	$-0.341580\pi$
$271$	15.7179			0.954795			0.477398	+	0.878687i	$0.341580\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.34258			0.322170
$276$		0			0
$277$	24.0906			1.44746			0.723731	−	0.690082i	$-0.242425\pi$
$277$	24.0906			1.44746			0.723731	+	0.690082i	$0.242425\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.0123			−1.25349			−0.626745	−	0.779224i	$-0.715613\pi$
$281$	−21.0123			−1.25349			−0.626745	+	0.779224i	$0.715613\pi$
$282$		0			0
$283$		−	10.8524i		−	0.645107i	−0.946551	−	0.322554i	$-0.895459\pi$
$283$		−	10.8524i		−	0.645107i	0.946551	−	0.322554i	$-0.104541\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.4542			0.794176
$288$		0			0
$289$	19.4571			1.14453
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.09447			−0.531305			−0.265652	−	0.964069i	$-0.585587\pi$
$293$	−9.09447			−0.531305			−0.265652	+	0.964069i	$0.585587\pi$
$294$		0			0
$295$		−	13.0024i		−	0.757031i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.0498	−	12.6376i	0.754689	−	0.730852i
$300$		0			0
$301$	−19.4497			−1.12106
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.36058i		−	0.306946i
$306$		0			0
$307$	16.4407			0.938322			0.469161	−	0.883113i	$-0.344556\pi$
$307$	16.4407			0.938322			0.469161	+	0.883113i	$0.344556\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.13574i		−	0.461335i	−0.973033	−	0.230668i	$-0.925909\pi$
$311$		−	8.13574i		−	0.461335i	0.973033	−	0.230668i	$-0.0740910\pi$
$312$		0			0
$313$		−	19.4789i		−	1.10101i	−0.834832	−	0.550505i	$-0.814435\pi$
$313$		−	19.4789i		−	1.10101i	0.834832	−	0.550505i	$-0.185565\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.5980i			0.932234i	0.884723	+	0.466117i	$0.154347\pi$
$317$			16.5980i			0.932234i	−0.884723	+	0.466117i	$0.845653\pi$
$318$		0			0
$319$			25.8238i			1.44586i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			14.2183i			0.791129i
$324$		0			0
$325$	6.20688			0.344296
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.76379			0.538295
$330$		0			0
$331$	−7.35830			−0.404449			−0.202224	−	0.979339i	$-0.564817\pi$
$331$	−7.35830			−0.404449			−0.202224	+	0.979339i	$0.564817\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			24.6118i			1.34469i
$336$		0			0
$337$			10.1645i			0.553697i	0.960914	+	0.276848i	$0.0892900\pi$
$337$			10.1645i			0.553697i	−0.960914	+	0.276848i	$0.910710\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.7891			1.34241
$342$		0			0
$343$			18.5668i			1.00251i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			24.2664i			1.30269i	0.758782	+	0.651345i	$0.225795\pi$
$347$			24.2664i			1.30269i	−0.758782	+	0.651345i	$0.774205\pi$
$348$		0			0
$349$	−8.73710			−0.467686			−0.233843	−	0.972274i	$-0.575130\pi$
$349$	−8.73710			−0.467686			−0.233843	+	0.972274i	$0.575130\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.58579i			0.350526i	0.984522	+	0.175263i	$0.0560776\pi$
$353$			6.58579i			0.350526i	−0.984522	+	0.175263i	$0.943922\pi$
$354$		0			0
$355$			1.96105i			0.104082i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.12019			0.0591212			0.0295606	−	0.999563i	$-0.490589\pi$
$359$	1.12019			0.0591212			0.0295606	+	0.999563i	$0.490589\pi$
$360$		0			0
$361$	13.4548			0.708148
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.4166			1.33037
$366$		0			0
$367$		−	34.1663i		−	1.78347i	−0.452560	−	0.891734i	$-0.649489\pi$
$367$		−	34.1663i		−	1.78347i	0.452560	−	0.891734i	$-0.350511\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			3.08495i			0.160162i
$372$		0			0
$373$			9.49561i			0.491664i	0.969312	+	0.245832i	$0.0790612\pi$
$373$			9.49561i			0.491664i	−0.969312	+	0.245832i	$0.920939\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			30.0015i			1.54515i
$378$		0			0
$379$		−	18.0220i		−	0.925730i	−0.886429	−	0.462865i	$-0.846822\pi$
$379$		−	18.0220i		−	0.925730i	0.886429	−	0.462865i	$-0.153178\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.3420			−0.886134			−0.443067	−	0.896489i	$-0.646110\pi$
$383$	−17.3420			−0.886134			−0.443067	+	0.896489i	$0.646110\pi$
$384$		0			0
$385$		−	9.82135i		−	0.500543i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.68070			−0.237321			−0.118660	−	0.992935i	$-0.537860\pi$
$389$	−4.68070			−0.237321			−0.118660	+	0.992935i	$0.537860\pi$
$390$		0			0
$391$	20.8016	−	20.1446i	1.05198	−	1.01876i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	0.151698i		−	0.00763276i
$396$		0			0
$397$	−37.4640			−1.88026			−0.940132	−	0.340809i	$-0.889299\pi$
$397$	−37.4640			−1.88026			−0.940132	+	0.340809i	$0.889299\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.1044			1.00396			0.501982	−	0.864878i	$-0.332604\pi$
$401$	20.1044			1.00396			0.501982	+	0.864878i	$0.332604\pi$
$402$		0			0
$403$	28.7994			1.43460
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.05463i			0.151413i
$408$		0			0
$409$	20.6143			1.01931			0.509655	−	0.860379i	$-0.329773\pi$
$409$	20.6143			1.01931			0.509655	+	0.860379i	$0.329773\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.6521			0.573360
$414$		0			0
$415$	24.7308			1.21399
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−21.8836			−1.06908			−0.534542	−	0.845142i	$-0.679516\pi$
$419$	−21.8836			−1.06908			−0.534542	+	0.845142i	$0.679516\pi$
$420$		0			0
$421$			6.42783i			0.313273i	0.987656	+	0.156637i	$0.0500651\pi$
$421$			6.42783i			0.313273i	−0.987656	+	0.156637i	$0.949935\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.89389			0.479924
$426$		0			0
$427$	4.80385			0.232475
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.64393			−0.0791852			−0.0395926	−	0.999216i	$-0.512606\pi$
$431$	−1.64393			−0.0791852			−0.0395926	+	0.999216i	$0.512606\pi$
$432$		0			0
$433$		−	28.8872i		−	1.38823i	−0.719864	−	0.694115i	$-0.755796\pi$
$433$		−	28.8872i		−	1.38823i	0.719864	−	0.694115i	$-0.244204\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.85644	+	8.11268i	0.375824	+	0.388082i
$438$		0			0
$439$	4.95621			0.236547			0.118273	−	0.992981i	$-0.462264\pi$
$439$	4.95621			0.236547			0.118273	+	0.992981i	$0.462264\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			13.7015i			0.650976i	0.945546	+	0.325488i	$0.105528\pi$
$443$			13.7015i			0.650976i	−0.945546	+	0.325488i	$0.894472\pi$
$444$		0			0
$445$	−25.3736			−1.20282
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	11.9274i		−	0.562887i	−0.959578	−	0.281444i	$-0.909187\pi$
$449$		−	11.9274i		−	0.562887i	0.959578	−	0.281444i	$-0.0908133\pi$
$450$		0			0
$451$			26.6990i			1.25721i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	11.4102i		−	0.534919i
$456$		0			0
$457$		−	38.6779i		−	1.80927i	−0.426182	−	0.904637i	$-0.640142\pi$
$457$		−	38.6779i		−	1.80927i	0.426182	−	0.904637i	$-0.359858\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.94111i			0.136981i	0.997652	+	0.0684905i	$0.0218183\pi$
$461$			2.94111i			0.136981i	−0.997652	+	0.0684905i	$0.978182\pi$
$462$		0			0
$463$	−11.0885			−0.515326			−0.257663	−	0.966235i	$-0.582952\pi$
$463$	−11.0885			−0.515326			−0.257663	+	0.966235i	$0.582952\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.01236			0.0931211			0.0465606	−	0.998915i	$-0.485174\pi$
$467$	2.01236			0.0931211			0.0465606	+	0.998915i	$0.485174\pi$
$468$		0			0
$469$	−22.0558			−1.01844
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	38.5967i		−	1.77468i
$474$		0			0
$475$			3.85864i			0.177047i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.8890			0.725988			0.362994	−	0.931792i	$-0.381755\pi$
$479$	15.8890			0.725988			0.362994	+	0.931792i	$0.381755\pi$
$480$		0			0
$481$			3.54880i			0.161811i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.14071i		−	0.369651i
$486$		0			0
$487$	34.2815			1.55344			0.776721	−	0.629845i	$-0.216882\pi$
$487$	34.2815			1.55344			0.776721	+	0.629845i	$0.216882\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	26.7288i		−	1.20625i	−0.797646	−	0.603126i	$-0.793921\pi$
$491$		−	26.7288i		−	1.20625i	0.797646	−	0.603126i	$-0.206079\pi$
$492$		0			0
$493$			47.8229i			2.15384i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.75738			−0.0788293
$498$		0			0
$499$	−41.8916			−1.87533			−0.937663	−	0.347545i	$-0.887015\pi$
$499$	−41.8916			−1.87533			−0.937663	+	0.347545i	$0.887015\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.76403			0.390769			0.195384	−	0.980727i	$-0.437405\pi$
$503$	8.76403			0.390769			0.195384	+	0.980727i	$0.437405\pi$
$504$		0			0
$505$			0.485741i			0.0216152i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			23.4825i			1.04085i	0.853909	+	0.520423i	$0.174226\pi$
$509$			23.4825i			1.04085i	−0.853909	+	0.520423i	$0.825774\pi$
$510$		0			0
$511$			22.7770i			1.00759i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			13.3204i			0.586968i
$516$		0			0
$517$			19.3757i			0.852141i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.3887			−1.46279			−0.731393	−	0.681956i	$-0.761129\pi$
$521$	−33.3887			−1.46279			−0.731393	+	0.681956i	$0.761129\pi$
$522$		0			0
$523$		−	35.5008i		−	1.55234i	−0.630523	−	0.776170i	$-0.717160\pi$
$523$		−	35.5008i		−	1.55234i	0.630523	−	0.776170i	$-0.282840\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	45.9068			1.99973
$528$		0			0
$529$	0.737933	−	22.9882i	0.0320840	−	0.999485i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			31.0183i			1.34355i
$534$		0			0
$535$	−19.1280			−0.826977
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.0217			−0.603956
$540$		0			0
$541$	20.3788			0.876154			0.438077	−	0.898937i	$-0.355660\pi$
$541$	20.3788			0.876154			0.438077	+	0.898937i	$0.355660\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			34.4616i			1.47617i
$546$		0			0
$547$	8.71055			0.372436			0.186218	−	0.982508i	$-0.440377\pi$
$547$	8.71055			0.372436			0.186218	+	0.982508i	$0.440377\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.6511			−0.794562
$552$		0			0
$553$	0.135943			0.00578090
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.56710			−0.363000			−0.181500	−	0.983391i	$-0.558095\pi$
$557$	−8.56710			−0.363000			−0.181500	+	0.983391i	$0.558095\pi$
$558$		0			0
$559$		−	44.8407i		−	1.89656i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.3873			−0.859224			−0.429612	−	0.903014i	$-0.641350\pi$
$563$	−20.3873			−0.859224			−0.429612	+	0.903014i	$0.641350\pi$
$564$		0			0
$565$	−28.6610			−1.20578
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.41420			0.310820			0.155410	−	0.987850i	$-0.450330\pi$
$569$	7.41420			0.310820			0.155410	+	0.987850i	$0.450330\pi$
$570$		0			0
$571$			4.21519i			0.176400i	0.996103	+	0.0882002i	$0.0281115\pi$
$571$			4.21519i			0.176400i	−0.996103	+	0.0882002i	$0.971888\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.64525	−	5.46694i	0.235423	−	0.227987i
$576$		0			0
$577$	13.7320			0.571673			0.285836	−	0.958278i	$-0.407729\pi$
$577$	13.7320			0.571673			0.285836	+	0.958278i	$0.407729\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			22.1624i			0.919451i
$582$		0			0
$583$	−6.12189			−0.253543
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	14.5548i		−	0.600739i	−0.953823	−	0.300370i	$-0.902890\pi$
$587$		−	14.5548i		−	0.600739i	0.953823	−	0.300370i	$-0.0971100\pi$
$588$		0			0
$589$			17.9038i			0.737711i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	36.2120i		−	1.48705i	−0.668710	−	0.743524i	$-0.733153\pi$
$593$		−	36.2120i		−	1.48705i	0.668710	−	0.743524i	$-0.266847\pi$
$594$		0			0
$595$		−	18.1881i		−	0.745639i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	43.8367i		−	1.79112i	−0.444941	−	0.895560i	$-0.646775\pi$
$599$		−	43.8367i		−	1.79112i	0.444941	−	0.895560i	$-0.353225\pi$
$600$		0			0
$601$	18.2824			0.745755			0.372877	−	0.927881i	$-0.378371\pi$
$601$	18.2824			0.745755			0.372877	+	0.927881i	$0.378371\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.677610			−0.0275488
$606$		0			0
$607$	−39.3525			−1.59727			−0.798634	−	0.601817i	$-0.794443\pi$
$607$	−39.3525			−1.59727			−0.798634	+	0.601817i	$0.794443\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			22.5102i			0.910664i
$612$		0			0
$613$		−	8.93724i		−	0.360972i	−0.983578	−	0.180486i	$-0.942233\pi$
$613$		−	8.93724i		−	0.360972i	0.983578	−	0.180486i	$-0.0577670\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.9402			−1.08457			−0.542285	−	0.840195i	$-0.682441\pi$
$617$	−26.9402			−1.08457			−0.542285	+	0.840195i	$0.682441\pi$
$618$		0			0
$619$		−	37.9100i		−	1.52373i	−0.647736	−	0.761865i	$-0.724284\pi$
$619$		−	37.9100i		−	1.52373i	0.647736	−	0.761865i	$-0.275716\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	22.7384i		−	0.910993i
$624$		0			0
$625$	−14.1219			−0.564875
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.65685i			0.225554i
$630$		0			0
$631$		−	26.8047i		−	1.06708i	−0.845775	−	0.533540i	$-0.820862\pi$
$631$		−	26.8047i		−	1.06708i	0.845775	−	0.533540i	$-0.179138\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.1239			−0.719224
$636$		0			0
$637$	−16.2900			−0.645434
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.09827			0.0433793			0.0216896	−	0.999765i	$-0.493095\pi$
$641$	1.09827			0.0433793			0.0216896	+	0.999765i	$0.493095\pi$
$642$		0			0
$643$			19.7449i			0.778663i	0.921098	+	0.389331i	$0.127294\pi$
$643$			19.7449i			0.778663i	−0.921098	+	0.389331i	$0.872706\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	11.6980i		−	0.459894i	−0.973203	−	0.229947i	$-0.926145\pi$
$647$		−	11.6980i		−	0.459894i	0.973203	−	0.229947i	$-0.0738553\pi$
$648$		0			0
$649$			23.1228i			0.907649i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			45.0547i			1.76313i	0.472064	+	0.881564i	$0.343509\pi$
$653$			45.0547i			1.76313i	−0.472064	+	0.881564i	$0.656491\pi$
$654$		0			0
$655$			11.4018i			0.445504i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.444334			0.0173088			0.00865440	−	0.999963i	$-0.497245\pi$
$659$	0.444334			0.0173088			0.00865440	+	0.999963i	$0.497245\pi$
$660$		0			0
$661$			26.4896i			1.03033i	0.857092	+	0.515163i	$0.172268\pi$
$661$			26.4896i			1.03033i	−0.857092	+	0.515163i	$0.827732\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.09340			0.275070
$666$		0			0
$667$	26.4249	+	27.2868i	1.02318	+	1.05655i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.53296i			0.368016i
$672$		0			0
$673$	−29.2611			−1.12793			−0.563966	−	0.825798i	$-0.690725\pi$
$673$	−29.2611			−1.12793			−0.563966	+	0.825798i	$0.690725\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.0525			0.809113			0.404557	−	0.914513i	$-0.367426\pi$
$677$	21.0525			0.809113			0.404557	+	0.914513i	$0.367426\pi$
$678$		0			0
$679$	7.29525			0.279966
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13.9450i		−	0.533590i	−0.963753	−	0.266795i	$-0.914035\pi$
$683$		−	13.9450i		−	0.533590i	0.963753	−	0.266795i	$-0.0859646\pi$
$684$		0			0
$685$	38.6721			1.47759
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.11226			−0.270955
$690$		0			0
$691$	27.5291			1.04726			0.523629	−	0.851947i	$-0.324578\pi$
$691$	27.5291			1.04726			0.523629	+	0.851947i	$0.324578\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15.3505			−0.582279
$696$		0			0
$697$			49.4438i			1.87282i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.51124			−0.170387			−0.0851936	−	0.996364i	$-0.527151\pi$
$701$	−4.51124			−0.170387			−0.0851936	+	0.996364i	$0.527151\pi$
$702$		0			0
$703$	−2.20619			−0.0832079
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.435294			−0.0163709
$708$		0			0
$709$			50.7563i			1.90619i	0.302665	+	0.953097i	$0.402124\pi$
$709$			50.7563i			1.90619i	−0.302665	+	0.953097i	$0.597876\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	26.1935	−	25.3661i	0.980952	−	0.949969i
$714$		0			0
$715$	22.6429			0.846795
$716$		0			0
$717$		0			0
$718$		0			0
$719$			9.99328i			0.372687i	0.982485	+	0.186343i	$0.0596636\pi$
$719$			9.99328i			0.372687i	−0.982485	+	0.186343i	$0.940336\pi$
$720$		0			0
$721$	−11.9370			−0.444558
$722$		0			0
$723$		0			0
$724$		0			0
$725$			12.9784i			0.482006i
$726$		0			0
$727$		−	31.9656i		−	1.18554i	−0.805372	−	0.592769i	$-0.798035\pi$
$727$		−	31.9656i		−	1.18554i	0.805372	−	0.592769i	$-0.201965\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	71.4770i		−	2.64367i
$732$		0			0
$733$			52.2061i			1.92828i	0.265398	+	0.964139i	$0.414497\pi$
$733$			52.2061i			1.92828i	−0.265398	+	0.964139i	$0.585503\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	43.7683i		−	1.61223i
$738$		0			0
$739$	30.5729			1.12464			0.562322	−	0.826919i	$-0.309908\pi$
$739$	30.5729			1.12464			0.562322	+	0.826919i	$0.309908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.0389			−1.24877			−0.624383	−	0.781118i	$-0.714649\pi$
$743$	−34.0389			−1.24877			−0.624383	+	0.781118i	$0.714649\pi$
$744$		0			0
$745$	−13.9077			−0.509540
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	17.1415i		−	0.626336i
$750$		0			0
$751$			45.7451i			1.66926i	0.550809	+	0.834632i	$0.314319\pi$
$751$			45.7451i			1.66926i	−0.550809	+	0.834632i	$0.685681\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.7824			0.428806
$756$		0			0
$757$			0.175770i			0.00638846i	0.999995	+	0.00319423i	$0.00101676\pi$
$757$			0.175770i			0.00638846i	−0.999995	+	0.00319423i	$0.998983\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17.7402i		−	0.643083i	−0.946895	−	0.321542i	$-0.895799\pi$
$761$		−	17.7402i		−	0.643083i	0.946895	−	0.321542i	$-0.104201\pi$
$762$		0			0
$763$	−30.8825			−1.11802
$764$		0			0
$765$		0			0
$766$		0			0
$767$			26.8635i			0.969984i
$768$		0			0
$769$		−	27.2120i		−	0.981290i	−0.871360	−	0.490645i	$-0.836761\pi$
$769$		−	27.2120i		−	0.981290i	0.871360	−	0.490645i	$-0.163239\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.1598			−0.437358			−0.218679	−	0.975797i	$-0.570175\pi$
$773$	−12.1598			−0.437358			−0.218679	+	0.975797i	$0.570175\pi$
$774$		0			0
$775$	12.4584			0.447519
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.2832			−0.690892
$780$		0			0
$781$		−	3.48742i		−	0.124790i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	21.3016i		−	0.760286i
$786$		0			0
$787$		−	16.9104i		−	0.602791i	−0.953499	−	0.301395i	$-0.902548\pi$
$787$		−	16.9104i		−	0.602791i	0.953499	−	0.301395i	$-0.0974524\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	25.6844i		−	0.913231i
$792$		0			0
$793$			11.0752i			0.393290i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.8466			0.455049			0.227524	−	0.973772i	$-0.426937\pi$
$797$	12.8466			0.455049			0.227524	+	0.973772i	$0.426937\pi$
$798$		0			0
$799$			35.8816i			1.26940i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−45.1995			−1.59506
$804$		0			0
$805$	−10.0500	−	10.3777i	−0.354214	−	0.365767i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	5.05407i		−	0.177692i	−0.996045	−	0.0888459i	$-0.971682\pi$
$809$		−	5.05407i		−	0.177692i	0.996045	−	0.0888459i	$-0.0283179\pi$
$810$		0			0
$811$	−3.90170			−0.137007			−0.0685037	−	0.997651i	$-0.521822\pi$
$811$	−3.90170			−0.137007			−0.0685037	+	0.997651i	$0.521822\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.3387			−0.712434
$816$		0			0
$817$	27.8762			0.975264
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	7.50574i		−	0.261952i	−0.991386	−	0.130976i	$-0.958189\pi$
$821$		−	7.50574i		−	0.261952i	0.991386	−	0.130976i	$-0.0418111\pi$
$822$		0			0
$823$	0.132861			0.00463123			0.00231561	−	0.999997i	$-0.499263\pi$
$823$	0.132861			0.00463123			0.00231561	+	0.999997i	$0.499263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.9679			0.624807			0.312403	−	0.949950i	$-0.398866\pi$
$827$	17.9679			0.624807			0.312403	+	0.949950i	$0.398866\pi$
$828$		0			0
$829$	−41.3252			−1.43528			−0.717642	−	0.696412i	$-0.754779\pi$
$829$	−41.3252			−1.43528			−0.717642	+	0.696412i	$0.754779\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−25.9666			−0.899690
$834$		0			0
$835$			39.4548i			1.36539i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−48.5625			−1.67656			−0.838281	−	0.545238i	$-0.816439\pi$
$839$	−48.5625			−1.67656			−0.838281	+	0.545238i	$0.816439\pi$
$840$		0			0
$841$	−33.7323			−1.16318
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.47161			0.0850259
$846$		0			0
$847$		−	0.607236i		−	0.0208649i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.12573	+	3.22768i	0.107149	+	0.110643i
$852$		0			0
$853$	−25.3187			−0.866894			−0.433447	−	0.901179i	$-0.642703\pi$
$853$	−25.3187			−0.866894			−0.433447	+	0.901179i	$0.642703\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			45.3227i			1.54819i	0.633067	+	0.774097i	$0.281796\pi$
$857$			45.3227i			1.54819i	−0.633067	+	0.774097i	$0.718204\pi$
$858$		0			0
$859$	−18.5382			−0.632516			−0.316258	−	0.948673i	$-0.602426\pi$
$859$	−18.5382			−0.632516			−0.316258	+	0.948673i	$0.602426\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	31.3303i		−	1.06650i	−0.845959	−	0.533248i	$-0.820971\pi$
$863$		−	31.3303i		−	1.06650i	0.845959	−	0.533248i	$-0.179029\pi$
$864$		0			0
$865$			0.0642303i			0.00218390i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			0.269771i			0.00915137i
$870$		0			0
$871$		−	50.8489i		−	1.72295i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	19.9974i		−	0.676036i
$876$		0			0
$877$	1.79605			0.0606484			0.0303242	−	0.999540i	$-0.490346\pi$
$877$	1.79605			0.0606484			0.0303242	+	0.999540i	$0.490346\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.3937			0.451247			0.225623	−	0.974215i	$-0.427558\pi$
$881$	13.3937			0.451247			0.225623	+	0.974215i	$0.427558\pi$
$882$		0			0
$883$	−52.9170			−1.78080			−0.890399	−	0.455181i	$-0.849574\pi$
$883$	−52.9170			−1.78080			−0.890399	+	0.455181i	$0.849574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	25.5935i		−	0.859346i	−0.902985	−	0.429673i	$-0.858629\pi$
$887$		−	25.5935i		−	0.859346i	0.902985	−	0.429673i	$-0.141371\pi$
$888$		0			0
$889$		−	16.2416i		−	0.544726i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.9939			−0.468289
$894$		0			0
$895$		−	2.82850i		−	0.0945462i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			60.2188i			2.00841i
$900$		0			0
$901$	−11.3371			−0.377693
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.4423i		−	0.380354i
$906$		0			0
$907$		−	23.8309i		−	0.791292i	−0.918403	−	0.395646i	$-0.870521\pi$
$907$		−	23.8309i		−	0.791292i	0.918403	−	0.395646i	$-0.129479\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−52.7203			−1.74670			−0.873351	−	0.487091i	$-0.838058\pi$
$911$	−52.7203			−1.74670			−0.873351	+	0.487091i	$0.838058\pi$
$912$		0			0
$913$	−43.9799			−1.45552
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.2176			−0.337416
$918$		0			0
$919$		−	27.7988i		−	0.916996i	−0.888695	−	0.458498i	$-0.848388\pi$
$919$		−	27.7988i		−	0.916996i	0.888695	−	0.458498i	$-0.151612\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.05159i		−	0.133360i
$924$		0			0
$925$			1.53518i			0.0504766i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	0.351342i		−	0.0115271i	−0.999983	−	0.00576357i	$-0.998165\pi$
$929$		−	0.351342i		−	0.0115271i	0.999983	−	0.00576357i	$-0.00183461\pi$
$930$		0			0
$931$		−	10.1270i		−	0.331900i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	36.0932			1.18037
$936$		0			0
$937$		−	28.9702i		−	0.946417i	−0.880951	−	0.473208i	$-0.843096\pi$
$937$		−	28.9702i		−	0.946417i	0.880951	−	0.473208i	$-0.156904\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.5124			−0.505689			−0.252845	−	0.967507i	$-0.581366\pi$
$941$	−15.5124			−0.505689			−0.252845	+	0.967507i	$0.581366\pi$
$942$		0			0
$943$	27.3205	+	28.2116i	0.889678	+	0.918695i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	43.2686i		−	1.40604i	−0.711171	−	0.703019i	$-0.751835\pi$
$947$		−	43.2686i		−	1.40604i	0.711171	−	0.703019i	$-0.248165\pi$
$948$		0			0
$949$	−52.5116			−1.70460
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.9491			0.516641			0.258321	−	0.966059i	$-0.416831\pi$
$953$	15.9491			0.516641			0.258321	+	0.966059i	$0.416831\pi$
$954$		0			0
$955$	−0.891816			−0.0288585
$956$		0			0
$957$		0			0
$958$		0			0
$959$			34.6558i			1.11909i
$960$		0			0
$961$	26.8059			0.864708
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.5413			0.564673
$966$		0			0
$967$	38.6824			1.24394			0.621971	−	0.783040i	$-0.286332\pi$
$967$	38.6824			1.24394			0.621971	+	0.783040i	$0.286332\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.2803			0.618733			0.309367	−	0.950943i	$-0.399883\pi$
$971$	19.2803			0.618733			0.309367	+	0.950943i	$0.399883\pi$
$972$		0			0
$973$		−	13.7563i		−	0.441006i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.50902			0.240235			0.120117	−	0.992760i	$-0.461673\pi$
$977$	7.50902			0.240235			0.120117	+	0.992760i	$0.461673\pi$
$978$		0			0
$979$	45.1229			1.44213
$980$		0			0
$981$		0			0
$982$		0			0
$983$	56.4177			1.79945			0.899723	−	0.436462i	$-0.143769\pi$
$983$	56.4177			1.79945			0.899723	+	0.436462i	$0.143769\pi$
$984$		0			0
$985$			4.14995i			0.132228i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−39.4951	−	40.7833i	−1.25587	−	1.29683i
$990$		0			0
$991$	−55.9951			−1.77874			−0.889371	−	0.457187i	$-0.848857\pi$
$991$	−55.9951			−1.77874			−0.889371	+	0.457187i	$0.848857\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			15.0824i			0.478143i
$996$		0			0
$997$	16.9850			0.537919			0.268960	−	0.963151i	$-0.413320\pi$
$997$	16.9850			0.537919			0.268960	+	0.963151i	$0.413320\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.m.a.1241.20	yes	24
3.2	odd	2	inner	1656.2.m.a.1241.6	yes	24
4.3	odd	2		3312.2.m.d.2897.20		24
12.11	even	2		3312.2.m.d.2897.6		24
23.22	odd	2	inner	1656.2.m.a.1241.5	✓	24
69.68	even	2	inner	1656.2.m.a.1241.19	yes	24
92.91	even	2		3312.2.m.d.2897.5		24
276.275	odd	2		3312.2.m.d.2897.19		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.m.a.1241.5	✓	24	23.22	odd	2	inner
1656.2.m.a.1241.6	yes	24	3.2	odd	2	inner
1656.2.m.a.1241.19	yes	24	69.68	even	2	inner
1656.2.m.a.1241.20	yes	24	1.1	even	1	trivial
3312.2.m.d.2897.5		24	92.91	even	2
3312.2.m.d.2897.6		24	12.11	even	2
3312.2.m.d.2897.19		24	276.275	odd	2
3312.2.m.d.2897.20		24	4.3	odd	2

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:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.83341 q^{5} +1.64300i q^{7} +O(q^{10})\)	\(q+1.83341 q^{5} +1.64300i q^{7} -3.26043 q^{11} -3.78789 q^{13} -6.03797 q^{17} -2.35482i q^{19} +(-3.44514 + 3.33632i) q^{23} -1.63861 q^{25} -7.92037i q^{29} -7.60302 q^{31} +3.01229i q^{35} -0.936880i q^{37} -8.18881i q^{41} +11.8379i q^{43} -5.94267i q^{47} +4.30056 q^{49} +1.87763 q^{53} -5.97770 q^{55} -7.09194i q^{59} -2.92383i q^{61} -6.94475 q^{65} +13.4241i q^{67} +1.06962i q^{71} +13.8630 q^{73} -5.35688i q^{77} -0.0827410i q^{79} +13.4890 q^{83} -11.0701 q^{85} -13.8396 q^{89} -6.22349i q^{91} -4.31735i q^{95} -4.44021i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 40 q^{25} - 16 q^{31} - 40 q^{49} - 64 q^{55} + 48 q^{85}+O(q^{100}) \) 24 * q + 40 * q^25 - 16 * q^31 - 40 * q^49 - 64 * q^55 + 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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