LMFDB - Embedded newform 4032.2.i.c.1889.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1889,4032)]

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, 1, 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("4032.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.i (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.10
Character	$\chi$	$=$	4032.1889
Dual form			4032.2.i.c.1889.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21807i q^{5} +(0.355387 - 2.62177i) q^{7} +O(q^{10})$	$q+1.21807i q^{5} +(0.355387 - 2.62177i) q^{7} -4.06403 q^{11} +1.42934 q^{13} -4.74513 q^{17} +4.84714 q^{19} -6.03391i q^{23} +3.51630 q^{25} +3.45468 q^{29} +3.15195i q^{31} +(3.19351 + 0.432887i) q^{35} +9.24401i q^{37} -1.24399 q^{41} -3.23110i q^{43} -9.43689 q^{47} +(-6.74740 - 1.86349i) q^{49} -2.44949 q^{53} -4.95027i q^{55} -10.2086i q^{59} -10.1938 q^{61} +1.74103i q^{65} +6.51630i q^{67} +2.14012i q^{71} -12.6779i q^{73} +(-1.44430 + 10.6550i) q^{77} -10.6656 q^{79} -16.3452i q^{83} -5.77991i q^{85} -14.9538 q^{89} +(0.507968 - 3.74740i) q^{91} +5.90417i q^{95} -11.4536i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 80 q^{25} - 16 q^{49}+O(q^{100}) $ 48 * q - 80 * q^25 - 16 * q^49

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
$5$			1.21807i			0.544738i	0.962193	+	0.272369i	$0.0878071\pi$
$5$			1.21807i			0.544738i	−0.962193	+	0.272369i	$0.912193\pi$
$6$		0			0
$7$	0.355387	−	2.62177i	0.134324	−	0.990937i
$7$	0.355387	−	2.62177i	0.134324	−	0.990937i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.06403			−1.22535			−0.612675	−	0.790335i	$-0.709906\pi$
$11$	−4.06403			−1.22535			−0.612675	+	0.790335i	$0.709906\pi$
$12$		0			0
$13$	1.42934			0.396427			0.198213	−	0.980159i	$-0.436486\pi$
$13$	1.42934			0.396427			0.198213	+	0.980159i	$0.436486\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.74513			−1.15086			−0.575432	−	0.817850i	$-0.695166\pi$
$17$	−4.74513			−1.15086			−0.575432	+	0.817850i	$0.695166\pi$
$18$		0			0
$19$	4.84714			1.11201			0.556005	−	0.831179i	$-0.312333\pi$
$19$	4.84714			1.11201			0.556005	+	0.831179i	$0.312333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.03391i		−	1.25816i	−0.777342	−	0.629079i	$-0.783432\pi$
$23$		−	6.03391i		−	1.25816i	0.777342	−	0.629079i	$-0.216568\pi$
$24$		0			0
$25$	3.51630			0.703260
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.45468			0.641517			0.320759	−	0.947161i	$-0.396062\pi$
$29$	3.45468			0.641517			0.320759	+	0.947161i	$0.396062\pi$
$30$		0			0
$31$			3.15195i			0.566107i	0.959104	+	0.283053i	$0.0913474\pi$
$31$			3.15195i			0.566107i	−0.959104	+	0.283053i	$0.908653\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.19351	+	0.432887i	0.539801	+	0.0731713i
$36$		0			0
$37$			9.24401i			1.51971i	0.650095	+	0.759853i	$0.274729\pi$
$37$			9.24401i			1.51971i	−0.650095	+	0.759853i	$0.725271\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.24399			−0.194278			−0.0971391	−	0.995271i	$-0.530969\pi$
$41$	−1.24399			−0.194278			−0.0971391	+	0.995271i	$0.530969\pi$
$42$		0			0
$43$		−	3.23110i		−	0.492738i	−0.969176	−	0.246369i	$-0.920763\pi$
$43$		−	3.23110i		−	0.492738i	0.969176	−	0.246369i	$-0.0792375\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.43689			−1.37651			−0.688256	−	0.725468i	$-0.741623\pi$
$47$	−9.43689			−1.37651			−0.688256	+	0.725468i	$0.741623\pi$
$48$		0			0
$49$	−6.74740	−	1.86349i	−0.963914	−	0.266213i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.44949			−0.336463			−0.168232	−	0.985747i	$-0.553806\pi$
$53$	−2.44949			−0.336463			−0.168232	+	0.985747i	$0.553806\pi$
$54$		0			0
$55$		−	4.95027i		−	0.667495i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	10.2086i		−	1.32905i	−0.747265	−	0.664526i	$-0.768633\pi$
$59$		−	10.2086i		−	1.32905i	0.747265	−	0.664526i	$-0.231367\pi$
$60$		0			0
$61$	−10.1938			−1.30519			−0.652593	−	0.757709i	$-0.726319\pi$
$61$	−10.1938			−1.30519			−0.652593	+	0.757709i	$0.726319\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.74103i			0.215949i
$66$		0			0
$67$			6.51630i			0.796093i	0.917365	+	0.398047i	$0.130312\pi$
$67$			6.51630i			0.796093i	−0.917365	+	0.398047i	$0.869688\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.14012i			0.253985i	0.991904	+	0.126993i	$0.0405325\pi$
$71$			2.14012i			0.253985i	−0.991904	+	0.126993i	$0.959468\pi$
$72$		0			0
$73$		−	12.6779i		−	1.48384i	−0.670488	−	0.741920i	$-0.733915\pi$
$73$		−	12.6779i		−	1.48384i	0.670488	−	0.741920i	$-0.266085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.44430	+	10.6550i	−0.164594	+	1.21425i
$78$		0			0
$79$	−10.6656			−1.19997			−0.599985	−	0.800011i	$-0.704827\pi$
$79$	−10.6656			−1.19997			−0.599985	+	0.800011i	$0.704827\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.3452i		−	1.79411i	−0.441914	−	0.897057i	$-0.645700\pi$
$83$		−	16.3452i		−	1.79411i	0.441914	−	0.897057i	$-0.354300\pi$
$84$		0			0
$85$		−	5.77991i		−	0.626920i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.9538			−1.58510			−0.792549	−	0.609809i	$-0.791246\pi$
$89$	−14.9538			−1.58510			−0.792549	+	0.609809i	$0.791246\pi$
$90$		0			0
$91$	0.507968	−	3.74740i	0.0532495	−	0.392834i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.90417i			0.605755i
$96$		0			0
$97$		−	11.4536i		−	1.16293i	−0.813571	−	0.581466i	$-0.802479\pi$
$97$		−	11.4536i		−	1.16293i	0.813571	−	0.581466i	$-0.197521\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	3.65421i		−	0.363608i	−0.983335	−	0.181804i	$-0.941806\pi$
$101$		−	3.65421i		−	0.363608i	0.983335	−	0.181804i	$-0.0581936\pi$
$102$		0			0
$103$		−	3.73850i		−	0.368366i	−0.982892	−	0.184183i	$-0.941036\pi$
$103$		−	3.73850i		−	0.368366i	0.982892	−	0.184183i	$-0.0589638\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.80506			−0.561196			−0.280598	−	0.959825i	$-0.590533\pi$
$107$	−5.80506			−0.561196			−0.280598	+	0.959825i	$0.590533\pi$
$108$		0			0
$109$		−	2.04255i		−	0.195641i	−0.995204	−	0.0978205i	$-0.968813\pi$
$109$		−	2.04255i		−	0.195641i	0.995204	−	0.0978205i	$-0.0311871\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.50161i		−	0.235331i	−0.993053	−	0.117666i	$-0.962459\pi$
$113$		−	2.50161i		−	0.235331i	0.993053	−	0.117666i	$-0.0375411\pi$
$114$		0			0
$115$	7.34973			0.685366
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.68636	+	12.4407i	−0.154588	+	1.14043i
$120$		0			0
$121$	5.51630			0.501482
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.3735i			0.927831i
$126$		0			0
$127$	16.1722			1.43505			0.717526	−	0.696532i	$-0.245274\pi$
$127$	16.1722			1.43505			0.717526	+	0.696532i	$0.245274\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	8.62450i		−	0.753526i	−0.926310	−	0.376763i	$-0.877037\pi$
$131$		−	8.62450i		−	0.753526i	0.926310	−	0.376763i	$-0.122963\pi$
$132$		0			0
$133$	1.72261	−	12.7081i	0.149369	−	1.10193i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.1117i			1.20565i	0.797875	+	0.602823i	$0.205958\pi$
$137$			14.1117i			1.20565i	−0.797875	+	0.602823i	$0.794042\pi$
$138$		0			0
$139$	13.4213			1.13838			0.569189	−	0.822207i	$-0.307257\pi$
$139$	13.4213			1.13838			0.569189	+	0.822207i	$0.307257\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.80886			−0.485762
$144$		0			0
$145$			4.20804i			0.349459i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.6235			−0.870313			−0.435156	−	0.900355i	$-0.643307\pi$
$149$	−10.6235			−0.870313			−0.435156	+	0.900355i	$0.643307\pi$
$150$		0			0
$151$	−9.06053			−0.737335			−0.368668	−	0.929561i	$-0.620186\pi$
$151$	−9.06053			−0.737335			−0.368668	+	0.929561i	$0.620186\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.83930			−0.308380
$156$		0			0
$157$	−17.0843			−1.36347			−0.681737	−	0.731598i	$-0.738775\pi$
$157$	−17.0843			−1.36347			−0.681737	+	0.731598i	$0.738775\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.8196	−	2.14438i	−1.24676	−	0.169000i
$162$		0			0
$163$			2.51630i			0.197092i	0.995132	+	0.0985460i	$0.0314192\pi$
$163$			2.51630i			0.197092i	−0.995132	+	0.0985460i	$0.968581\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.8730			−0.918763			−0.459381	−	0.888239i	$-0.651929\pi$
$167$	−11.8730			−0.918763			−0.459381	+	0.888239i	$0.651929\pi$
$168$		0			0
$169$	−10.9570			−0.842846
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0277i			1.06650i	0.845956	+	0.533252i	$0.179030\pi$
$173$			14.0277i			1.06650i	−0.845956	+	0.533252i	$0.820970\pi$
$174$		0			0
$175$	1.24965	−	9.21895i	0.0944646	−	0.696887i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	17.1953			1.28524			0.642618	−	0.766187i	$-0.277848\pi$
$179$	17.1953			1.28524			0.642618	+	0.766187i	$0.277848\pi$
$180$		0			0
$181$	−4.87456			−0.362323			−0.181162	−	0.983453i	$-0.557986\pi$
$181$	−4.87456			−0.362323			−0.181162	+	0.983453i	$0.557986\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.2599			−0.827842
$186$		0			0
$187$	19.2843			1.41021
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.13493i			0.0821208i	0.999157	+	0.0410604i	$0.0130736\pi$
$191$			1.13493i			0.0821208i	−0.999157	+	0.0410604i	$0.986926\pi$
$192$		0			0
$193$	−18.0111			−1.29647			−0.648234	−	0.761441i	$-0.724492\pi$
$193$	−18.0111			−1.29647			−0.648234	+	0.761441i	$0.724492\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	24.3153			1.73239			0.866196	−	0.499705i	$-0.166558\pi$
$197$	24.3153			1.73239			0.866196	+	0.499705i	$0.166558\pi$
$198$		0			0
$199$		−	4.65700i		−	0.330126i	−0.986283	−	0.165063i	$-0.947217\pi$
$199$		−	4.65700i		−	0.330126i	0.986283	−	0.165063i	$-0.0527827\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.22775	−	9.05738i	0.0861710	−	0.635704i
$204$		0			0
$205$		−	1.51527i		−	0.105831i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−19.6989			−1.36260
$210$		0			0
$211$		−	12.2637i		−	0.844268i	−0.906533	−	0.422134i	$-0.861281\pi$
$211$		−	12.2637i		−	0.844268i	0.906533	−	0.422134i	$-0.138719\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.93571			0.268413
$216$		0			0
$217$	8.26370	+	1.12016i	0.560977	+	0.0760416i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.78240			−0.456233
$222$		0			0
$223$		−	8.68877i		−	0.581843i	−0.956747	−	0.290922i	$-0.906038\pi$
$223$		−	8.68877i		−	0.581843i	0.956747	−	0.290922i	$-0.0939619\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	14.4282i		−	0.957631i	−0.877916	−	0.478816i	$-0.841066\pi$
$227$		−	14.4282i		−	0.957631i	0.877916	−	0.478816i	$-0.158934\pi$
$228$		0			0
$229$	15.3617			1.01513			0.507564	−	0.861614i	$-0.330546\pi$
$229$	15.3617			1.01513			0.507564	+	0.861614i	$0.330546\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3.88541i			0.254542i	0.991868	+	0.127271i	$0.0406217\pi$
$233$			3.88541i			0.254542i	−0.991868	+	0.127271i	$0.959378\pi$
$234$		0			0
$235$		−	11.4948i		−	0.749838i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	18.7205i		−	1.21093i	−0.795873	−	0.605463i	$-0.792988\pi$
$239$		−	18.7205i		−	1.21093i	0.795873	−	0.605463i	$-0.207012\pi$
$240$		0			0
$241$		−	17.4209i		−	1.12218i	−0.827756	−	0.561088i	$-0.810383\pi$
$241$		−	17.4209i		−	1.12218i	0.827756	−	0.561088i	$-0.189617\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.26986	−	8.21881i	0.145016	−	0.525081i
$246$		0			0
$247$	6.92820			0.440831
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.57327i		−	0.478021i	−0.971017	−	0.239010i	$-0.923177\pi$
$251$		−	7.57327i		−	0.478021i	0.971017	−	0.239010i	$-0.0768230\pi$
$252$		0			0
$253$			24.5220i			1.54168i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.46351			−0.340804			−0.170402	−	0.985375i	$-0.554507\pi$
$257$	−5.46351			−0.340804			−0.170402	+	0.985375i	$0.554507\pi$
$258$		0			0
$259$	24.2357	+	3.28520i	1.50593	+	0.204133i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.75886i			0.170119i	0.996376	+	0.0850593i	$0.0271080\pi$
$263$			2.75886i			0.170119i	−0.996376	+	0.0850593i	$0.972892\pi$
$264$		0			0
$265$		−	2.98365i		−	0.183284i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			30.4653i			1.85750i	0.370704	+	0.928751i	$0.379117\pi$
$269$			30.4653i			1.85750i	−0.370704	+	0.928751i	$0.620883\pi$
$270$		0			0
$271$		−	9.60727i		−	0.583600i	−0.956479	−	0.291800i	$-0.905746\pi$
$271$		−	9.60727i		−	0.583600i	0.956479	−	0.291800i	$-0.0942542\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.2903			−0.861740
$276$		0			0
$277$		−	2.56984i		−	0.154407i	−0.997015	−	0.0772034i	$-0.975401\pi$
$277$		−	2.56984i		−	0.154407i	0.997015	−	0.0772034i	$-0.0245991\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.2430i		−	1.62518i	−0.582835	−	0.812590i	$-0.698057\pi$
$281$		−	27.2430i		−	1.62518i	0.582835	−	0.812590i	$-0.301943\pi$
$282$		0			0
$283$	−20.5087			−1.21912			−0.609559	−	0.792741i	$-0.708653\pi$
$283$	−20.5087			−1.21912			−0.609559	+	0.792741i	$0.708653\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.442097	+	3.26145i	−0.0260962	+	0.192518i
$288$		0			0
$289$	5.51630			0.324488
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.15539i		−	0.534864i	−0.963577	−	0.267432i	$-0.913825\pi$
$293$		−	9.15539i		−	0.534864i	0.963577	−	0.267432i	$-0.0861750\pi$
$294$		0			0
$295$	12.4349			0.723985
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	8.62450i		−	0.498767i
$300$		0			0
$301$	−8.47121	−	1.14829i	−0.488272	−	0.0661864i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.4168i		−	0.710984i
$306$		0			0
$307$	−9.79851			−0.559231			−0.279615	−	0.960112i	$-0.590207\pi$
$307$	−9.79851			−0.559231			−0.279615	+	0.960112i	$0.590207\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−29.8102			−1.69038			−0.845191	−	0.534464i	$-0.820514\pi$
$311$	−29.8102			−1.69038			−0.845191	+	0.534464i	$0.820514\pi$
$312$		0			0
$313$		−	26.8425i		−	1.51723i	−0.651539	−	0.758615i	$-0.725876\pi$
$313$		−	26.8425i		−	1.51723i	0.651539	−	0.758615i	$-0.274124\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.4422			1.37281			0.686406	−	0.727218i	$-0.259187\pi$
$317$	24.4422			1.37281			0.686406	+	0.727218i	$0.259187\pi$
$318$		0			0
$319$	−14.0399			−0.786083
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.0003			−1.27977
$324$		0			0
$325$	5.02598			0.278791
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.35375	+	24.7414i	−0.184898	+	1.36404i
$330$		0			0
$331$			9.80151i			0.538740i	0.963037	+	0.269370i	$0.0868154\pi$
$331$			9.80151i			0.538740i	−0.963037	+	0.269370i	$0.913185\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.93732			−0.433662
$336$		0			0
$337$	30.9896			1.68811			0.844056	−	0.536256i	$-0.180162\pi$
$337$	30.9896			1.68811			0.844056	+	0.536256i	$0.180162\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	12.8096i		−	0.693679i
$342$		0			0
$343$	−7.28359	+	17.0279i	−0.393277	+	0.919420i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.06403			−0.218168			−0.109084	−	0.994033i	$-0.534792\pi$
$347$	−4.06403			−0.218168			−0.109084	+	0.994033i	$0.534792\pi$
$348$		0			0
$349$	−31.1680			−1.66839			−0.834193	−	0.551473i	$-0.814066\pi$
$349$	−31.1680			−1.66839			−0.834193	+	0.551473i	$0.814066\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.6853			0.888070			0.444035	−	0.896009i	$-0.353547\pi$
$353$	16.6853			0.888070			0.444035	+	0.896009i	$0.353547\pi$
$354$		0			0
$355$	−2.60682			−0.138356
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.14012i			0.112951i	0.998404	+	0.0564756i	$0.0179863\pi$
$359$			2.14012i			0.112951i	−0.998404	+	0.0564756i	$0.982014\pi$
$360$		0			0
$361$	4.49480			0.236568
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.4426			0.808304
$366$		0			0
$367$		−	5.98152i		−	0.312233i	−0.987739	−	0.156116i	$-0.950103\pi$
$367$		−	5.98152i		−	0.312233i	0.987739	−	0.156116i	$-0.0498975\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.870517	+	6.42201i	−0.0451950	+	0.333414i
$372$		0			0
$373$			2.66356i			0.137914i	0.997620	+	0.0689569i	$0.0219671\pi$
$373$			2.66356i			0.137914i	−0.997620	+	0.0689569i	$0.978033\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.93790			0.254315
$378$		0			0
$379$		−	3.73630i		−	0.191921i	−0.995385	−	0.0959604i	$-0.969408\pi$
$379$		−	3.73630i		−	0.191921i	0.995385	−	0.0959604i	$-0.0305922\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.9381			0.763299			0.381650	−	0.924307i	$-0.375356\pi$
$383$	14.9381			0.763299			0.381650	+	0.924307i	$0.375356\pi$
$384$		0			0
$385$	−12.9785	−	1.75926i	−0.661445	−	0.0896604i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−33.2350			−1.68508			−0.842541	−	0.538633i	$-0.818941\pi$
$389$	−33.2350			−1.68508			−0.842541	+	0.538633i	$0.818941\pi$
$390$		0			0
$391$			28.6317i			1.44797i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12.9914i		−	0.653669i
$396$		0			0
$397$	8.86930			0.445137			0.222569	−	0.974917i	$-0.428556\pi$
$397$	8.86930			0.445137			0.222569	+	0.974917i	$0.428556\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.14438i			0.107085i	0.998566	+	0.0535425i	$0.0170513\pi$
$401$			2.14438i			0.107085i	−0.998566	+	0.0535425i	$0.982949\pi$
$402$		0			0
$403$			4.50520i			0.224420i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	37.5679i		−	1.86217i
$408$		0			0
$409$			10.4376i			0.516107i	0.966131	+	0.258053i	$0.0830811\pi$
$409$			10.4376i			0.516107i	−0.966131	+	0.258053i	$0.916919\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−26.7648	−	3.62802i	−1.31701	−	0.178523i
$414$		0			0
$415$	19.9096			0.977323
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.2218i			0.548222i	0.961698	+	0.274111i	$0.0883835\pi$
$419$			11.2218i			0.548222i	−0.961698	+	0.274111i	$0.911616\pi$
$420$		0			0
$421$			25.1430i			1.22539i	0.790318	+	0.612697i	$0.209915\pi$
$421$			25.1430i			1.22539i	−0.790318	+	0.612697i	$0.790085\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.6853			−0.809357
$426$		0			0
$427$	−3.62275	+	26.7259i	−0.175317	+	1.29336i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			27.6402i			1.33138i	0.746228	+	0.665691i	$0.231863\pi$
$431$			27.6402i			1.33138i	−0.746228	+	0.665691i	$0.768137\pi$
$432$		0			0
$433$			3.51853i			0.169090i	0.996420	+	0.0845448i	$0.0269436\pi$
$433$			3.51853i			0.169090i	−0.996420	+	0.0845448i	$0.973056\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	29.2472i		−	1.39908i
$438$		0			0
$439$		−	23.2076i		−	1.10764i	−0.832636	−	0.553820i	$-0.813169\pi$
$439$		−	23.2076i		−	1.10764i	0.832636	−	0.553820i	$-0.186831\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.5464			1.45131			0.725653	−	0.688061i	$-0.241538\pi$
$443$	30.5464			1.45131			0.725653	+	0.688061i	$0.241538\pi$
$444$		0			0
$445$		−	18.2148i		−	0.863463i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.0432i			0.851513i	0.904838	+	0.425757i	$0.139992\pi$
$449$			18.0432i			0.851513i	−0.904838	+	0.425757i	$0.860008\pi$
$450$		0			0
$451$	5.05560			0.238059
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.56460	+	0.618742i	0.213992	+	0.0290071i
$456$		0			0
$457$	−18.9896			−0.888296			−0.444148	−	0.895953i	$-0.646494\pi$
$457$	−18.9896			−0.888296			−0.444148	+	0.895953i	$0.646494\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			20.0918i			0.935770i	0.883789	+	0.467885i	$0.154984\pi$
$461$			20.0918i			0.935770i	−0.883789	+	0.467885i	$0.845016\pi$
$462$		0			0
$463$	−13.8756			−0.644855			−0.322428	−	0.946594i	$-0.604499\pi$
$463$	−13.8756			−0.644855			−0.322428	+	0.946594i	$0.604499\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	0.865774i		−	0.0400632i	−0.999799	−	0.0200316i	$-0.993623\pi$
$467$		−	0.865774i		−	0.0400632i	0.999799	−	0.0200316i	$-0.00637669\pi$
$468$		0			0
$469$	17.0843	+	2.31581i	0.788878	+	0.106934i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			13.1313i			0.603776i
$474$		0			0
$475$	17.0440			0.782033
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.5653			0.528434			0.264217	−	0.964463i	$-0.414886\pi$
$479$	11.5653			0.528434			0.264217	+	0.964463i	$0.414886\pi$
$480$		0			0
$481$			13.2128i			0.602452i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.9512			0.633493
$486$		0			0
$487$	−11.7241			−0.531269			−0.265634	−	0.964074i	$-0.585581\pi$
$487$	−11.7241			−0.531269			−0.265634	+	0.964074i	$0.585581\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.1655			−0.503892			−0.251946	−	0.967741i	$-0.581071\pi$
$491$	−11.1655			−0.503892			−0.251946	+	0.967741i	$0.581071\pi$
$492$		0			0
$493$	−16.3929			−0.738299
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.61091	+	0.760571i	0.251684	+	0.0341163i
$498$		0			0
$499$		−	33.2963i		−	1.49055i	−0.666759	−	0.745274i	$-0.732319\pi$
$499$		−	33.2963i		−	1.49055i	0.666759	−	0.745274i	$-0.267681\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.307685			−0.0137190			−0.00685950	−	0.999976i	$-0.502183\pi$
$503$	−0.307685			−0.0137190			−0.00685950	+	0.999976i	$0.502183\pi$
$504$		0			0
$505$	4.45109			0.198071
$506$		0			0
$507$		0			0
$508$		0			0
$509$			38.9656i			1.72712i	0.504246	+	0.863560i	$0.331771\pi$
$509$			38.9656i			1.72712i	−0.504246	+	0.863560i	$0.668229\pi$
$510$		0			0
$511$	−33.2387	−	4.50558i	−1.47039	−	0.199315i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.55376			0.200663
$516$		0			0
$517$	38.3517			1.68671
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.5062			0.854580			0.427290	−	0.904115i	$-0.359468\pi$
$521$	19.5062			0.854580			0.427290	+	0.904115i	$0.359468\pi$
$522$		0			0
$523$	−13.2128			−0.577756			−0.288878	−	0.957366i	$-0.593282\pi$
$523$	−13.2128			−0.577756			−0.288878	+	0.957366i	$0.593282\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	14.9564i		−	0.651512i
$528$		0			0
$529$	−13.4081			−0.582960
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.77808			−0.0770171
$534$		0			0
$535$		−	7.07098i		−	0.305705i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.4216	+	7.57327i	1.18113	+	0.326204i
$540$		0			0
$541$		−	15.5512i		−	0.668599i	−0.942467	−	0.334299i	$-0.891500\pi$
$541$		−	15.5512i		−	0.668599i	0.942467	−	0.334299i	$-0.108500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.48797			0.106573
$546$		0			0
$547$		−	0.451093i		−	0.0192874i	−0.999953	−	0.00964368i	$-0.996930\pi$
$547$		−	0.451093i		−	0.0192874i	0.999953	−	0.00964368i	$-0.00306973\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16.7453			0.713374
$552$		0			0
$553$	−3.79041	+	27.9627i	−0.161184	+	1.18909i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.96202			−0.294990			−0.147495	−	0.989063i	$-0.547121\pi$
$557$	−6.96202			−0.294990			−0.147495	+	0.989063i	$0.547121\pi$
$558$		0			0
$559$		−	4.61833i		−	0.195334i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			28.3234i			1.19369i	0.802357	+	0.596845i	$0.203579\pi$
$563$			28.3234i			1.19369i	−0.802357	+	0.596845i	$0.796421\pi$
$564$		0			0
$565$	3.04713			0.128194
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.9155i		−	1.37989i	−0.723861	−	0.689946i	$-0.757634\pi$
$569$		−	32.9155i		−	1.37989i	0.723861	−	0.689946i	$-0.242366\pi$
$570$		0			0
$571$			27.9681i			1.17043i	0.810879	+	0.585214i	$0.198990\pi$
$571$			27.9681i			1.17043i	−0.810879	+	0.585214i	$0.801010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	21.2171i		−	0.884812i
$576$		0			0
$577$			8.46990i			0.352606i	0.984336	+	0.176303i	$0.0564139\pi$
$577$			8.46990i			0.352606i	−0.984336	+	0.176303i	$0.943586\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.8533	−	5.80886i	−1.77786	−	0.240992i
$582$		0			0
$583$	9.95479			0.412285
$584$		0			0
$585$		0			0
$586$		0			0
$587$			34.2745i			1.41466i	0.706884	+	0.707330i	$0.250100\pi$
$587$			34.2745i			1.41466i	−0.706884	+	0.707330i	$0.749900\pi$
$588$		0			0
$589$			15.2780i			0.629517i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25.1624			1.03330			0.516648	−	0.856198i	$-0.327179\pi$
$593$	25.1624			1.03330			0.516648	+	0.856198i	$0.327179\pi$
$594$		0			0
$595$	−15.1536	−	2.05411i	−0.621238	−	0.0842102i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			45.6122i			1.86366i	0.362890	+	0.931832i	$0.381790\pi$
$599$			45.6122i			1.86366i	−0.362890	+	0.931832i	$0.618210\pi$
$600$		0			0
$601$			1.22439i			0.0499439i	0.999688	+	0.0249719i	$0.00794964\pi$
$601$			1.22439i			0.0499439i	−0.999688	+	0.0249719i	$0.992050\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			6.71925i			0.273176i
$606$		0			0
$607$			39.9986i			1.62349i	0.584009	+	0.811747i	$0.301483\pi$
$607$			39.9986i			1.62349i	−0.584009	+	0.811747i	$0.698517\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.4885			−0.545686
$612$		0			0
$613$		−	35.1875i		−	1.42121i	−0.703591	−	0.710606i	$-0.748421\pi$
$613$		−	35.1875i		−	1.42121i	0.703591	−	0.710606i	$-0.251579\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.5970i		−	0.909721i	−0.890563	−	0.454861i	$-0.849689\pi$
$617$		−	22.5970i		−	0.909721i	0.890563	−	0.454861i	$-0.150311\pi$
$618$		0			0
$619$	14.4372			0.580280			0.290140	−	0.956984i	$-0.406298\pi$
$619$	14.4372			0.580280			0.290140	+	0.956984i	$0.406298\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.31438	+	39.2054i	−0.212916	+	1.57073i
$624$		0			0
$625$	4.94589			0.197836
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	43.8641i		−	1.74898i
$630$		0			0
$631$	19.8903			0.791822			0.395911	−	0.918289i	$-0.370429\pi$
$631$	19.8903			0.791822			0.395911	+	0.918289i	$0.370429\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			19.6989i			0.781728i
$636$		0			0
$637$	−9.64431	−	2.66356i	−0.382122	−	0.105534i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			38.2760i			1.51181i	0.654681	+	0.755905i	$0.272803\pi$
$641$			38.2760i			1.51181i	−0.654681	+	0.755905i	$0.727197\pi$
$642$		0			0
$643$	−35.2082			−1.38848			−0.694238	−	0.719745i	$-0.744259\pi$
$643$	−35.2082			−1.38848			−0.694238	+	0.719745i	$0.744259\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	43.8641			1.72448			0.862238	−	0.506504i	$-0.169062\pi$
$647$	43.8641			1.72448			0.862238	+	0.506504i	$0.169062\pi$
$648$		0			0
$649$			41.4882i			1.62855i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.44949			0.0958559			0.0479280	−	0.998851i	$-0.484738\pi$
$653$	2.44949			0.0958559			0.0479280	+	0.998851i	$0.484738\pi$
$654$		0			0
$655$	10.5053			0.410474
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.51952			−0.253964			−0.126982	−	0.991905i	$-0.540529\pi$
$659$	−6.51952			−0.253964			−0.126982	+	0.991905i	$0.540529\pi$
$660$		0			0
$661$	35.1998			1.36911			0.684557	−	0.728960i	$-0.259996\pi$
$661$	35.1998			1.36911			0.684557	+	0.728960i	$0.259996\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.4794	+	2.09827i	0.600265	+	0.0813672i
$666$		0			0
$667$		−	20.8452i		−	0.807130i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	41.4279			1.59931
$672$		0			0
$673$	41.5059			1.59993			0.799967	−	0.600043i	$-0.204850\pi$
$673$	41.5059			1.59993			0.799967	+	0.600043i	$0.204850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	49.3391i		−	1.89625i	−0.317891	−	0.948127i	$-0.602974\pi$
$677$		−	49.3391i		−	1.89625i	0.317891	−	0.948127i	$-0.397026\pi$
$678$		0			0
$679$	−30.0286	−	4.07045i	−1.15239	−	0.156209i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.7039			0.830478			0.415239	−	0.909712i	$-0.363698\pi$
$683$	21.7039			0.830478			0.415239	+	0.909712i	$0.363698\pi$
$684$		0			0
$685$	−17.1891			−0.656761
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.50115			−0.133383
$690$		0			0
$691$	−2.44878			−0.0931559			−0.0465780	−	0.998915i	$-0.514832\pi$
$691$	−2.44878			−0.0931559			−0.0465780	+	0.998915i	$0.514832\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.3481i			0.620117i
$696$		0			0
$697$	5.90289			0.223588
$698$		0			0
$699$		0			0
$700$		0			0
$701$	44.9164			1.69647			0.848234	−	0.529622i	$-0.177666\pi$
$701$	44.9164			1.69647			0.848234	+	0.529622i	$0.177666\pi$
$702$		0			0
$703$			44.8071i			1.68993i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.58052	−	1.29866i	−0.360313	−	0.0488412i
$708$		0			0
$709$		−	16.4455i		−	0.617623i	−0.951123	−	0.308811i	$-0.900069\pi$
$709$		−	16.4455i		−	0.617623i	0.951123	−	0.308811i	$-0.0999312\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	19.0186			0.712252
$714$		0			0
$715$		−	7.07561i		−	0.264613i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−49.9283			−1.86201			−0.931005	−	0.365007i	$-0.881067\pi$
$719$	−49.9283			−1.86201			−0.931005	+	0.365007i	$0.881067\pi$
$720$		0			0
$721$	−9.80151	−	1.32862i	−0.365027	−	0.0494802i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.1477			0.451154
$726$		0			0
$727$		−	19.5078i		−	0.723505i	−0.932274	−	0.361752i	$-0.882179\pi$
$727$		−	19.5078i		−	0.723505i	0.932274	−	0.361752i	$-0.117821\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			15.3320i			0.567074i
$732$		0			0
$733$	−23.9377			−0.884159			−0.442079	−	0.896976i	$-0.645759\pi$
$733$	−23.9377			−0.884159			−0.442079	+	0.896976i	$0.645759\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	26.4824i		−	0.975492i
$738$		0			0
$739$			44.4733i			1.63598i	0.575235	+	0.817988i	$0.304911\pi$
$739$			44.4733i			1.63598i	−0.575235	+	0.817988i	$0.695089\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	44.3475i		−	1.62695i	−0.581598	−	0.813476i	$-0.697572\pi$
$743$		−	44.3475i		−	1.62695i	0.581598	−	0.813476i	$-0.302428\pi$
$744$		0			0
$745$		−	12.9402i		−	0.474092i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.06304	+	15.2196i	−0.0753820	+	0.556111i
$750$		0			0
$751$	41.4050			1.51089			0.755444	−	0.655213i	$-0.227421\pi$
$751$	41.4050			1.51089			0.755444	+	0.655213i	$0.227421\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	11.0364i		−	0.401655i
$756$		0			0
$757$			28.3338i			1.02981i	0.857247	+	0.514905i	$0.172173\pi$
$757$			28.3338i			1.02981i	−0.857247	+	0.514905i	$0.827827\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	48.3625			1.75314			0.876569	−	0.481276i	$-0.159826\pi$
$761$	48.3625			1.75314			0.876569	+	0.481276i	$0.159826\pi$
$762$		0			0
$763$	−5.35511	−	0.725897i	−0.193868	−	0.0262792i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	14.5916i		−	0.526872i
$768$		0			0
$769$			18.1642i			0.655017i	0.944848	+	0.327509i	$0.106209\pi$
$769$			18.1642i			0.655017i	−0.944848	+	0.327509i	$0.893791\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	43.2749i		−	1.55649i	−0.627961	−	0.778245i	$-0.716110\pi$
$773$		−	43.2749i		−	1.55649i	0.627961	−	0.778245i	$-0.283890\pi$
$774$		0			0
$775$			11.0832i			0.398121i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.02979			−0.216039
$780$		0			0
$781$		−	8.69750i		−	0.311221i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20.8099i		−	0.742736i
$786$		0			0
$787$	−35.3124			−1.25875			−0.629376	−	0.777101i	$-0.716689\pi$
$787$	−35.3124			−1.25875			−0.629376	+	0.777101i	$0.716689\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.55865	−	0.889039i	−0.233199	−	0.0316106i
$792$		0			0
$793$	−14.5704			−0.517410
$794$		0			0
$795$		0			0
$796$		0			0
$797$			21.3361i			0.755764i	0.925854	+	0.377882i	$0.123347\pi$
$797$			21.3361i			0.755764i	−0.925854	+	0.377882i	$0.876653\pi$
$798$		0			0
$799$	44.7793			1.58418
$800$		0			0
$801$		0			0
$802$		0			0
$803$			51.5235i			1.81822i
$804$		0			0
$805$	2.61200	−	19.2693i	0.0920610	−	0.679155i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.59987i			0.161723i	0.996725	+	0.0808614i	$0.0257671\pi$
$809$			4.59987i			0.161723i	−0.996725	+	0.0808614i	$0.974233\pi$
$810$		0			0
$811$	−45.0067			−1.58040			−0.790200	−	0.612849i	$-0.790023\pi$
$811$	−45.0067			−1.58040			−0.790200	+	0.612849i	$0.790023\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.06504			−0.107364
$816$		0			0
$817$		−	15.6616i		−	0.547930i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34.3727			1.19962			0.599808	−	0.800144i	$-0.295244\pi$
$821$	34.3727			1.19962			0.599808	+	0.800144i	$0.295244\pi$
$822$		0			0
$823$	6.58046			0.229380			0.114690	−	0.993401i	$-0.463412\pi$
$823$	6.58046			0.229380			0.114690	+	0.993401i	$0.463412\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−46.9399			−1.63226			−0.816130	−	0.577869i	$-0.803885\pi$
$827$	−46.9399			−1.63226			−0.816130	+	0.577869i	$0.803885\pi$
$828$		0			0
$829$	18.8069			0.653190			0.326595	−	0.945164i	$-0.394099\pi$
$829$	18.8069			0.653190			0.326595	+	0.945164i	$0.394099\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32.0173	+	8.84251i	1.10933	+	0.306375i
$834$		0			0
$835$		−	14.4622i		−	0.500485i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.2465			0.768034			0.384017	−	0.923326i	$-0.374540\pi$
$839$	22.2465			0.768034			0.384017	+	0.923326i	$0.374540\pi$
$840$		0			0
$841$	−17.0652			−0.588455
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	13.3464i		−	0.459130i
$846$		0			0
$847$	1.96042	−	14.4625i	0.0673609	−	0.496937i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	55.7776			1.91203
$852$		0			0
$853$	32.8536			1.12489			0.562443	−	0.826836i	$-0.309862\pi$
$853$	32.8536			1.12489			0.562443	+	0.826836i	$0.309862\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.230815			0.00788449			0.00394224	−	0.999992i	$-0.498745\pi$
$857$	0.230815			0.00788449			0.00394224	+	0.999992i	$0.498745\pi$
$858$		0			0
$859$	−34.9459			−1.19234			−0.596170	−	0.802858i	$-0.703312\pi$
$859$	−34.9459			−1.19234			−0.596170	+	0.802858i	$0.703312\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	16.9640i		−	0.577462i	−0.957410	−	0.288731i	$-0.906767\pi$
$863$		−	16.9640i		−	0.577462i	0.957410	−	0.288731i	$-0.0932333\pi$
$864$		0			0
$865$	−17.0867			−0.580966
$866$		0			0
$867$		0			0
$868$		0			0
$869$	43.3451			1.47038
$870$		0			0
$871$			9.31399i			0.315593i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	27.1969	+	3.68660i	0.919422	+	0.124630i
$876$		0			0
$877$			47.6224i			1.60809i	0.594566	+	0.804047i	$0.297324\pi$
$877$			47.6224i			1.60809i	−0.594566	+	0.804047i	$0.702676\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	50.0940			1.68771			0.843855	−	0.536571i	$-0.180280\pi$
$881$	50.0940			1.68771			0.843855	+	0.536571i	$0.180280\pi$
$882$		0			0
$883$		−	3.84451i		−	0.129378i	−0.997905	−	0.0646891i	$-0.979394\pi$
$883$		−	3.84451i		−	0.129378i	0.997905	−	0.0646891i	$-0.0206056\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.6176			0.725848			0.362924	−	0.931819i	$-0.381778\pi$
$887$	21.6176			0.725848			0.362924	+	0.931819i	$0.381778\pi$
$888$		0			0
$889$	5.74740	−	42.3999i	0.192762	−	1.42205i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−45.7419			−1.53070
$894$		0			0
$895$			20.9451i			0.700117i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			10.8890i			0.363167i
$900$		0			0
$901$	11.6232			0.387224
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	5.93757i		−	0.197371i
$906$		0			0
$907$		−	40.6829i		−	1.35085i	−0.737427	−	0.675427i	$-0.763960\pi$
$907$		−	40.6829i		−	1.35085i	0.737427	−	0.675427i	$-0.236040\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	44.4801i		−	1.47369i	−0.676062	−	0.736845i	$-0.736315\pi$
$911$		−	44.4801i		−	1.47369i	0.676062	−	0.736845i	$-0.263685\pi$
$912$		0			0
$913$			66.4272i			2.19842i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.6115	−	3.06504i	−0.746697	−	0.101216i
$918$		0			0
$919$	−26.1270			−0.861850			−0.430925	−	0.902388i	$-0.641813\pi$
$919$	−26.1270			−0.861850			−0.430925	+	0.902388i	$0.641813\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			3.05895i			0.100687i
$924$		0			0
$925$			32.5047i			1.06875i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.19506			0.236062			0.118031	−	0.993010i	$-0.462342\pi$
$929$	7.19506			0.236062			0.118031	+	0.993010i	$0.462342\pi$
$930$		0			0
$931$	−32.7056	−	9.03260i	−1.07188	−	0.296032i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			23.4897i			0.768196i
$936$		0			0
$937$			19.1801i			0.626587i	0.949656	+	0.313294i	$0.101432\pi$
$937$			19.1801i			0.626587i	−0.949656	+	0.313294i	$0.898568\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	18.8476i		−	0.614414i	−0.951643	−	0.307207i	$-0.900606\pi$
$941$		−	18.8476i		−	0.614414i	0.951643	−	0.307207i	$-0.0993943\pi$
$942$		0			0
$943$			7.50611i			0.244433i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.13572			−0.166888			−0.0834442	−	0.996512i	$-0.526592\pi$
$947$	−5.13572			−0.166888			−0.0834442	+	0.996512i	$0.526592\pi$
$948$		0			0
$949$		−	18.1211i		−	0.588234i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	43.0496i		−	1.39451i	−0.716821	−	0.697257i	$-0.754403\pi$
$953$		−	43.0496i		−	1.39451i	0.716821	−	0.697257i	$-0.245597\pi$
$954$		0			0
$955$	−1.38243			−0.0447343
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.9978	+	5.01513i	1.19472	+	0.161947i
$960$		0			0
$961$	21.0652			0.679523
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	21.9388i		−	0.706235i
$966$		0			0
$967$	45.8531			1.47454			0.737268	−	0.675600i	$-0.236115\pi$
$967$	45.8531			1.47454			0.737268	+	0.675600i	$0.236115\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			30.2404i			0.970461i	0.874386	+	0.485230i	$0.161264\pi$
$971$			30.2404i			0.970461i	−0.874386	+	0.485230i	$0.838736\pi$
$972$		0			0
$973$	4.76975	−	35.1875i	0.152911	−	1.12806i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.55796i		−	0.305786i	−0.988243	−	0.152893i	$-0.951141\pi$
$977$		−	9.55796i		−	0.305786i	0.988243	−	0.152893i	$-0.0488590\pi$
$978$		0			0
$979$	60.7725			1.94230
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.8730			0.378691			0.189345	−	0.981911i	$-0.439363\pi$
$983$	11.8730			0.378691			0.189345	+	0.981911i	$0.439363\pi$
$984$		0			0
$985$			29.6177i			0.943700i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.4962			−0.619942
$990$		0			0
$991$	−61.1350			−1.94202			−0.971009	−	0.239045i	$-0.923166\pi$
$991$	−61.1350			−1.94202			−0.971009	+	0.239045i	$0.923166\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.67255			0.179832
$996$		0			0
$997$	7.92170			0.250883			0.125441	−	0.992101i	$-0.459965\pi$
$997$	7.92170			0.250883			0.125441	+	0.992101i	$0.459965\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.i.c.1889.10	yes	48
3.2	odd	2	inner	4032.2.i.c.1889.3	✓	48
4.3	odd	2	inner	4032.2.i.c.1889.26	yes	48
7.6	odd	2	inner	4032.2.i.c.1889.23	yes	48
8.3	odd	2	inner	4032.2.i.c.1889.45	yes	48
8.5	even	2	inner	4032.2.i.c.1889.27	yes	48
12.11	even	2	inner	4032.2.i.c.1889.21	yes	48
21.20	even	2	inner	4032.2.i.c.1889.28	yes	48
24.5	odd	2	inner	4032.2.i.c.1889.24	yes	48
24.11	even	2	inner	4032.2.i.c.1889.40	yes	48
28.27	even	2	inner	4032.2.i.c.1889.39	yes	48
56.13	odd	2	inner	4032.2.i.c.1889.4	yes	48
56.27	even	2	inner	4032.2.i.c.1889.22	yes	48
84.83	odd	2	inner	4032.2.i.c.1889.46	yes	48
168.83	odd	2	inner	4032.2.i.c.1889.25	yes	48
168.125	even	2	inner	4032.2.i.c.1889.9	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.i.c.1889.3	✓	48	3.2	odd	2	inner
4032.2.i.c.1889.4	yes	48	56.13	odd	2	inner
4032.2.i.c.1889.9	yes	48	168.125	even	2	inner
4032.2.i.c.1889.10	yes	48	1.1	even	1	trivial
4032.2.i.c.1889.21	yes	48	12.11	even	2	inner
4032.2.i.c.1889.22	yes	48	56.27	even	2	inner
4032.2.i.c.1889.23	yes	48	7.6	odd	2	inner
4032.2.i.c.1889.24	yes	48	24.5	odd	2	inner
4032.2.i.c.1889.25	yes	48	168.83	odd	2	inner
4032.2.i.c.1889.26	yes	48	4.3	odd	2	inner
4032.2.i.c.1889.27	yes	48	8.5	even	2	inner
4032.2.i.c.1889.28	yes	48	21.20	even	2	inner
4032.2.i.c.1889.39	yes	48	28.27	even	2	inner
4032.2.i.c.1889.40	yes	48	24.11	even	2	inner
4032.2.i.c.1889.45	yes	48	8.3	odd	2	inner
4032.2.i.c.1889.46	yes	48	84.83	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.21807i q^{5} +(0.355387 - 2.62177i) q^{7} +O(q^{10})\)	\(q+1.21807i q^{5} +(0.355387 - 2.62177i) q^{7} -4.06403 q^{11} +1.42934 q^{13} -4.74513 q^{17} +4.84714 q^{19} -6.03391i q^{23} +3.51630 q^{25} +3.45468 q^{29} +3.15195i q^{31} +(3.19351 + 0.432887i) q^{35} +9.24401i q^{37} -1.24399 q^{41} -3.23110i q^{43} -9.43689 q^{47} +(-6.74740 - 1.86349i) q^{49} -2.44949 q^{53} -4.95027i q^{55} -10.2086i q^{59} -10.1938 q^{61} +1.74103i q^{65} +6.51630i q^{67} +2.14012i q^{71} -12.6779i q^{73} +(-1.44430 + 10.6550i) q^{77} -10.6656 q^{79} -16.3452i q^{83} -5.77991i q^{85} -14.9538 q^{89} +(0.507968 - 3.74740i) q^{91} +5.90417i q^{95} -11.4536i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 80 q^{25} - 16 q^{49}+O(q^{100}) \) 48 * q - 80 * q^25 - 16 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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