LMFDB - Embedded newform 4032.2.v.d.1583.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$36$
Relative dimension:	$18$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.6
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.18126 - 1.18126i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-1.18126 - 1.18126i) q^{5} +1.00000 q^{7} +(1.81893 - 1.81893i) q^{11} +(-3.25298 - 3.25298i) q^{13} -1.32435i q^{17} +(-5.18163 + 5.18163i) q^{19} +4.26726i q^{23} -2.20926i q^{25} +(-4.51100 + 4.51100i) q^{29} +5.06204i q^{31} +(-1.18126 - 1.18126i) q^{35} +(-3.79530 + 3.79530i) q^{37} -0.186678 q^{41} +(-7.38788 - 7.38788i) q^{43} +4.80511 q^{47} +1.00000 q^{49} +(9.62181 + 9.62181i) q^{53} -4.29724 q^{55} +(3.57284 - 3.57284i) q^{59} +(2.12648 + 2.12648i) q^{61} +7.68522i q^{65} +(-4.70916 + 4.70916i) q^{67} +0.828539i q^{71} -7.95417i q^{73} +(1.81893 - 1.81893i) q^{77} +1.04387i q^{79} +(9.07047 + 9.07047i) q^{83} +(-1.56439 + 1.56439i) q^{85} +10.2447 q^{89} +(-3.25298 - 3.25298i) q^{91} +12.2417 q^{95} +17.6113 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{7}+O(q^{10}) $ 36 * q + 36 * q^7	$ 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) $ 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	−1.18126	−	1.18126i	−0.528274	−	0.528274i	0.391783	−	0.920058i	$-0.371858\pi$
$5$	−1.18126	−	1.18126i	−0.528274	−	0.528274i	−0.920058	+	0.391783i	$0.871858\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.81893	−	1.81893i	0.548427	−	0.548427i	−0.377558	−	0.925986i	$-0.623236\pi$
$11$	1.81893	−	1.81893i	0.548427	−	0.548427i	0.925986	+	0.377558i	$0.123236\pi$
$12$		0			0
$13$	−3.25298	−	3.25298i	−0.902215	−	0.902215i	0.0934126	−	0.995627i	$-0.470222\pi$
$13$	−3.25298	−	3.25298i	−0.902215	−	0.902215i	−0.995627	+	0.0934126i	$0.970222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.32435i		−	0.321201i	−0.987019	−	0.160601i	$-0.948657\pi$
$17$		−	1.32435i		−	0.321201i	0.987019	−	0.160601i	$-0.0513431\pi$
$18$		0			0
$19$	−5.18163	+	5.18163i	−1.18875	+	1.18875i	−0.211334	+	0.977414i	$0.567781\pi$
$19$	−5.18163	+	5.18163i	−1.18875	+	1.18875i	−0.977414	+	0.211334i	$0.932219\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.26726i			0.889786i	0.895584	+	0.444893i	$0.146758\pi$
$23$			4.26726i			0.889786i	−0.895584	+	0.444893i	$0.853242\pi$
$24$		0			0
$25$		−	2.20926i		−	0.441853i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.51100	+	4.51100i	−0.837672	+	0.837672i	−0.988552	−	0.150880i	$-0.951789\pi$
$29$	−4.51100	+	4.51100i	−0.837672	+	0.837672i	0.150880	+	0.988552i	$0.451789\pi$
$30$		0			0
$31$			5.06204i			0.909169i	0.890704	+	0.454584i	$0.150212\pi$
$31$			5.06204i			0.909169i	−0.890704	+	0.454584i	$0.849788\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.18126	−	1.18126i	−0.199669	−	0.199669i
$36$		0			0
$37$	−3.79530	+	3.79530i	−0.623943	+	0.623943i	−0.946537	−	0.322594i	$-0.895445\pi$
$37$	−3.79530	+	3.79530i	−0.623943	+	0.623943i	0.322594	+	0.946537i	$0.395445\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.186678			−0.0291542			−0.0145771	−	0.999894i	$-0.504640\pi$
$41$	−0.186678			−0.0291542			−0.0145771	+	0.999894i	$0.504640\pi$
$42$		0			0
$43$	−7.38788	−	7.38788i	−1.12664	−	1.12664i	−0.990720	−	0.135921i	$-0.956601\pi$
$43$	−7.38788	−	7.38788i	−1.12664	−	1.12664i	−0.135921	−	0.990720i	$-0.543399\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.80511			0.700898			0.350449	−	0.936582i	$-0.386029\pi$
$47$	4.80511			0.700898			0.350449	+	0.936582i	$0.386029\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.62181	+	9.62181i	1.32166	+	1.32166i	0.912434	+	0.409224i	$0.134200\pi$
$53$	9.62181	+	9.62181i	1.32166	+	1.32166i	0.409224	+	0.912434i	$0.365800\pi$
$54$		0			0
$55$	−4.29724			−0.579440
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.57284	−	3.57284i	0.465144	−	0.465144i	−0.435193	−	0.900337i	$-0.643320\pi$
$59$	3.57284	−	3.57284i	0.465144	−	0.465144i	0.900337	+	0.435193i	$0.143320\pi$
$60$		0			0
$61$	2.12648	+	2.12648i	0.272267	+	0.272267i	0.830012	−	0.557745i	$-0.188333\pi$
$61$	2.12648	+	2.12648i	0.272267	+	0.272267i	−0.557745	+	0.830012i	$0.688333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			7.68522i			0.953234i
$66$		0			0
$67$	−4.70916	+	4.70916i	−0.575316	+	0.575316i	−0.933609	−	0.358293i	$-0.883359\pi$
$67$	−4.70916	+	4.70916i	−0.575316	+	0.575316i	0.358293	+	0.933609i	$0.383359\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.828539i			0.0983295i	0.998791	+	0.0491648i	$0.0156559\pi$
$71$			0.828539i			0.0983295i	−0.998791	+	0.0491648i	$0.984344\pi$
$72$		0			0
$73$		−	7.95417i		−	0.930965i	−0.885057	−	0.465483i	$-0.845881\pi$
$73$		−	7.95417i		−	0.930965i	0.885057	−	0.465483i	$-0.154119\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.81893	−	1.81893i	0.207286	−	0.207286i
$78$		0			0
$79$			1.04387i			0.117444i	0.998274	+	0.0587222i	$0.0187026\pi$
$79$			1.04387i			0.117444i	−0.998274	+	0.0587222i	$0.981297\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.07047	+	9.07047i	0.995613	+	0.995613i	0.999990	−	0.00437697i	$-0.00139324\pi$
$83$	9.07047	+	9.07047i	0.995613	+	0.995613i	−0.00437697	+	0.999990i	$0.501393\pi$
$84$		0			0
$85$	−1.56439	+	1.56439i	−0.169682	+	0.169682i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.2447			1.08594			0.542969	−	0.839753i	$-0.317300\pi$
$89$	10.2447			1.08594			0.542969	+	0.839753i	$0.317300\pi$
$90$		0			0
$91$	−3.25298	−	3.25298i	−0.341005	−	0.341005i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.2417			1.25597
$96$		0			0
$97$	17.6113			1.78816			0.894079	−	0.447909i	$-0.147831\pi$
$97$	17.6113			1.78816			0.894079	+	0.447909i	$0.147831\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.86273	−	6.86273i	−0.682867	−	0.682867i	0.277778	−	0.960645i	$-0.410402\pi$
$101$	−6.86273	−	6.86273i	−0.682867	−	0.682867i	−0.960645	+	0.277778i	$0.910402\pi$
$102$		0			0
$103$	−14.1126			−1.39056			−0.695279	−	0.718740i	$-0.744719\pi$
$103$	−14.1126			−1.39056			−0.695279	+	0.718740i	$0.744719\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.14587	+	7.14587i	−0.690817	+	0.690817i	−0.962412	−	0.271595i	$-0.912449\pi$
$107$	−7.14587	+	7.14587i	−0.690817	+	0.690817i	0.271595	+	0.962412i	$0.412449\pi$
$108$		0			0
$109$	7.03518	+	7.03518i	0.673848	+	0.673848i	0.958601	−	0.284753i	$-0.0919116\pi$
$109$	7.03518	+	7.03518i	0.673848	+	0.673848i	−0.284753	+	0.958601i	$0.591912\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.1882i			1.24064i	0.784348	+	0.620321i	$0.212998\pi$
$113$			13.1882i			1.24064i	−0.784348	+	0.620321i	$0.787002\pi$
$114$		0			0
$115$	5.04073	−	5.04073i	0.470051	−	0.470051i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.32435i		−	0.121403i
$120$		0			0
$121$			4.38300i			0.398455i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.51599	+	8.51599i	−0.761694	+	0.761694i
$126$		0			0
$127$			16.7722i			1.48829i	0.668018	+	0.744145i	$0.267143\pi$
$127$			16.7722i			1.48829i	−0.668018	+	0.744145i	$0.732857\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.9315	−	11.9315i	−1.04246	−	1.04246i	−0.999058	−	0.0434055i	$-0.986179\pi$
$131$	−11.9315	−	11.9315i	−1.04246	−	1.04246i	−0.0434055	−	0.999058i	$-0.513821\pi$
$132$		0			0
$133$	−5.18163	+	5.18163i	−0.449304	+	0.449304i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.29568			−0.281569			−0.140785	−	0.990040i	$-0.544962\pi$
$137$	−3.29568			−0.281569			−0.140785	+	0.990040i	$0.544962\pi$
$138$		0			0
$139$	−8.41038	−	8.41038i	−0.713359	−	0.713359i	0.253877	−	0.967236i	$-0.418294\pi$
$139$	−8.41038	−	8.41038i	−0.713359	−	0.713359i	−0.967236	+	0.253877i	$0.918294\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.8339			−0.989599
$144$		0			0
$145$	10.6573			0.885041
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.49655	+	3.49655i	0.286449	+	0.286449i	0.835674	−	0.549225i	$-0.185077\pi$
$149$	3.49655	+	3.49655i	0.286449	+	0.286449i	−0.549225	+	0.835674i	$0.685077\pi$
$150$		0			0
$151$	−4.29802			−0.349768			−0.174884	−	0.984589i	$-0.555955\pi$
$151$	−4.29802			−0.349768			−0.174884	+	0.984589i	$0.555955\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.97957	−	5.97957i	0.480290	−	0.480290i
$156$		0			0
$157$	−7.99344	−	7.99344i	−0.637946	−	0.637946i	0.312102	−	0.950048i	$-0.398967\pi$
$157$	−7.99344	−	7.99344i	−0.637946	−	0.637946i	−0.950048	+	0.312102i	$0.898967\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.26726i			0.336307i
$162$		0			0
$163$	−1.98294	+	1.98294i	−0.155316	+	0.155316i	−0.780487	−	0.625172i	$-0.785029\pi$
$163$	−1.98294	+	1.98294i	−0.155316	+	0.155316i	0.625172	+	0.780487i	$0.285029\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.90536i			0.456970i	0.973547	+	0.228485i	$0.0733772\pi$
$167$			5.90536i			0.456970i	−0.973547	+	0.228485i	$0.926623\pi$
$168$		0			0
$169$			8.16378i			0.627983i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.64152	−	2.64152i	0.200831	−	0.200831i	−0.599525	−	0.800356i	$-0.704644\pi$
$173$	2.64152	−	2.64152i	0.200831	−	0.200831i	0.800356	+	0.599525i	$0.204644\pi$
$174$		0			0
$175$		−	2.20926i		−	0.167005i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.04105	+	9.04105i	0.675760	+	0.675760i	0.959038	−	0.283278i	$-0.0914219\pi$
$179$	9.04105	+	9.04105i	0.675760	+	0.675760i	−0.283278	+	0.959038i	$0.591422\pi$
$180$		0			0
$181$	−3.95280	+	3.95280i	−0.293809	+	0.293809i	−0.838583	−	0.544774i	$-0.816616\pi$
$181$	−3.95280	+	3.95280i	−0.293809	+	0.293809i	0.544774	+	0.838583i	$0.316616\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.96645			0.659226
$186$		0			0
$187$	−2.40889	−	2.40889i	−0.176156	−	0.176156i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.3508			−0.821315			−0.410658	−	0.911790i	$-0.634701\pi$
$191$	−11.3508			−0.821315			−0.410658	+	0.911790i	$0.634701\pi$
$192$		0			0
$193$	−23.2398			−1.67284			−0.836420	−	0.548089i	$-0.815356\pi$
$193$	−23.2398			−1.67284			−0.836420	+	0.548089i	$0.815356\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.8708	+	10.8708i	0.774515	+	0.774515i	0.978892	−	0.204377i	$-0.0655169\pi$
$197$	10.8708	+	10.8708i	0.774515	+	0.774515i	−0.204377	+	0.978892i	$0.565517\pi$
$198$		0			0
$199$	17.9111			1.26969			0.634843	−	0.772641i	$-0.281065\pi$
$199$	17.9111			1.26969			0.634843	+	0.772641i	$0.281065\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.51100	+	4.51100i	−0.316610	+	0.316610i
$204$		0			0
$205$	0.220515	+	0.220515i	0.0154014	+	0.0154014i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			18.8500i			1.30388i
$210$		0			0
$211$	−0.953583	+	0.953583i	−0.0656474	+	0.0656474i	−0.739168	−	0.673521i	$-0.764781\pi$
$211$	−0.953583	+	0.953583i	−0.0656474	+	0.0656474i	0.673521	+	0.739168i	$0.264781\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			17.4540i			1.19035i
$216$		0			0
$217$			5.06204i			0.343633i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.30807	+	4.30807i	−0.289792	+	0.289792i
$222$		0			0
$223$			16.2577i			1.08870i	0.838858	+	0.544350i	$0.183224\pi$
$223$			16.2577i			1.08870i	−0.838858	+	0.544350i	$0.816776\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.02940	+	8.02940i	0.532930	+	0.532930i	0.921443	−	0.388513i	$-0.127011\pi$
$227$	8.02940	+	8.02940i	0.532930	+	0.532930i	−0.388513	+	0.921443i	$0.627011\pi$
$228$		0			0
$229$	−7.10191	+	7.10191i	−0.469308	+	0.469308i	−0.901690	−	0.432383i	$-0.857673\pi$
$229$	−7.10191	+	7.10191i	−0.469308	+	0.469308i	0.432383	+	0.901690i	$0.357673\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.77335			0.574762			0.287381	−	0.957816i	$-0.407215\pi$
$233$	8.77335			0.574762			0.287381	+	0.957816i	$0.407215\pi$
$234$		0			0
$235$	−5.67607	−	5.67607i	−0.370266	−	0.370266i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.2430			−0.856618			−0.428309	−	0.903632i	$-0.640891\pi$
$239$	−13.2430			−0.856618			−0.428309	+	0.903632i	$0.640891\pi$
$240$		0			0
$241$	−14.6118			−0.941228			−0.470614	−	0.882339i	$-0.655968\pi$
$241$	−14.6118			−0.941228			−0.470614	+	0.882339i	$0.655968\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.18126	−	1.18126i	−0.0754677	−	0.0754677i
$246$		0			0
$247$	33.7115			2.14501
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.02166	−	6.02166i	0.380084	−	0.380084i	−0.491048	−	0.871132i	$-0.663386\pi$
$251$	6.02166	−	6.02166i	0.380084	−	0.380084i	0.871132	+	0.491048i	$0.163386\pi$
$252$		0			0
$253$	7.76184	+	7.76184i	0.487983	+	0.487983i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			25.5042i			1.59091i	0.606013	+	0.795455i	$0.292768\pi$
$257$			25.5042i			1.59091i	−0.606013	+	0.795455i	$0.707232\pi$
$258$		0			0
$259$	−3.79530	+	3.79530i	−0.235828	+	0.235828i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			27.9171i			1.72144i	0.509075	+	0.860722i	$0.329988\pi$
$263$			27.9171i			1.72144i	−0.509075	+	0.860722i	$0.670012\pi$
$264$		0			0
$265$		−	22.7317i		−	1.39640i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.41299	−	4.41299i	0.269065	−	0.269065i	−0.559659	−	0.828723i	$-0.689068\pi$
$269$	4.41299	−	4.41299i	0.269065	−	0.269065i	0.828723	+	0.559659i	$0.189068\pi$
$270$		0			0
$271$		−	13.5641i		−	0.823959i	−0.911193	−	0.411979i	$-0.864838\pi$
$271$		−	13.5641i		−	0.823959i	0.911193	−	0.411979i	$-0.135162\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.01849	−	4.01849i	−0.242324	−	0.242324i
$276$		0			0
$277$	−21.4157	+	21.4157i	−1.28674	+	1.28674i	−0.349988	+	0.936754i	$0.613814\pi$
$277$	−21.4157	+	21.4157i	−1.28674	+	1.28674i	−0.936754	+	0.349988i	$0.886186\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.6764			1.23345			0.616726	−	0.787178i	$-0.288458\pi$
$281$	20.6764			1.23345			0.616726	+	0.787178i	$0.288458\pi$
$282$		0			0
$283$	−13.4690	−	13.4690i	−0.800650	−	0.800650i	0.182547	−	0.983197i	$-0.441566\pi$
$283$	−13.4690	−	13.4690i	−0.800650	−	0.800650i	−0.983197	+	0.182547i	$0.941566\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.186678			−0.0110193
$288$		0			0
$289$	15.2461			0.896830
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.4847	−	16.4847i	−0.963047	−	0.963047i	0.0362939	−	0.999341i	$-0.488445\pi$
$293$	−16.4847	−	16.4847i	−0.963047	−	0.963047i	−0.999341	+	0.0362939i	$0.988445\pi$
$294$		0			0
$295$	−8.44088			−0.491447
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.8813	−	13.8813i	0.802778	−	0.802778i
$300$		0			0
$301$	−7.38788	−	7.38788i	−0.425830	−	0.425830i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.02383i		−	0.287664i
$306$		0			0
$307$	−12.0728	+	12.0728i	−0.689031	+	0.689031i	−0.962018	−	0.272987i	$-0.911988\pi$
$307$	−12.0728	+	12.0728i	−0.689031	+	0.689031i	0.272987	+	0.962018i	$0.411988\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.19521i		−	0.351298i	−0.984453	−	0.175649i	$-0.943798\pi$
$311$		−	6.19521i		−	0.351298i	0.984453	−	0.175649i	$-0.0562024\pi$
$312$		0			0
$313$		−	0.261631i		−	0.0147882i	−0.999973	−	0.00739412i	$-0.997646\pi$
$313$		−	0.261631i		−	0.0147882i	0.999973	−	0.00739412i	$-0.00235364\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.8287	−	18.8287i	1.05752	−	1.05752i	0.0592815	−	0.998241i	$-0.481119\pi$
$317$	18.8287	−	18.8287i	1.05752	−	1.05752i	0.998241	−	0.0592815i	$-0.0188809\pi$
$318$		0			0
$319$			16.4104i			0.918805i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.86227	+	6.86227i	0.381827	+	0.381827i
$324$		0			0
$325$	−7.18669	+	7.18669i	−0.398646	+	0.398646i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.80511			0.264914
$330$		0			0
$331$	11.8526	+	11.8526i	0.651479	+	0.651479i	0.953349	−	0.301870i	$-0.0976108\pi$
$331$	11.8526	+	11.8526i	0.651479	+	0.651479i	−0.301870	+	0.953349i	$0.597611\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.1255			0.607849
$336$		0			0
$337$	2.36736			0.128958			0.0644790	−	0.997919i	$-0.479461\pi$
$337$	2.36736			0.128958			0.0644790	+	0.997919i	$0.479461\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.20748	+	9.20748i	0.498613	+	0.498613i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.38908	−	4.38908i	0.235618	−	0.235618i	−0.579415	−	0.815033i	$-0.696719\pi$
$347$	4.38908	−	4.38908i	0.235618	−	0.235618i	0.815033	+	0.579415i	$0.196719\pi$
$348$		0			0
$349$	15.1457	+	15.1457i	0.810728	+	0.810728i	0.984743	−	0.174015i	$-0.0556740\pi$
$349$	15.1457	+	15.1457i	0.810728	+	0.810728i	−0.174015	+	0.984743i	$0.555674\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	15.6317i		−	0.831993i	−0.909366	−	0.415996i	$-0.863433\pi$
$353$		−	15.6317i		−	0.831993i	0.909366	−	0.415996i	$-0.136567\pi$
$354$		0			0
$355$	0.978718	−	0.978718i	0.0519449	−	0.0519449i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.0997i			0.585822i	0.956140	+	0.292911i	$0.0946240\pi$
$359$			11.0997i			0.585822i	−0.956140	+	0.292911i	$0.905376\pi$
$360$		0			0
$361$		−	34.6986i		−	1.82624i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.39592	+	9.39592i	−0.491805	+	0.491805i
$366$		0			0
$367$			18.4077i			0.960875i	0.877029	+	0.480437i	$0.159522\pi$
$367$			18.4077i			0.960875i	−0.877029	+	0.480437i	$0.840478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.62181	+	9.62181i	0.499540	+	0.499540i
$372$		0			0
$373$	−17.4756	+	17.4756i	−0.904852	+	0.904852i	−0.995851	−	0.0909985i	$-0.970994\pi$
$373$	−17.4756	+	17.4756i	−0.904852	+	0.904852i	0.0909985	+	0.995851i	$0.470994\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	29.3484			1.51152
$378$		0			0
$379$	−16.5126	−	16.5126i	−0.848194	−	0.848194i	0.141714	−	0.989908i	$-0.454739\pi$
$379$	−16.5126	−	16.5126i	−0.848194	−	0.848194i	−0.989908	+	0.141714i	$0.954739\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−21.6247			−1.10497			−0.552484	−	0.833523i	$-0.686320\pi$
$383$	−21.6247			−1.10497			−0.552484	+	0.833523i	$0.686320\pi$
$384$		0			0
$385$	−4.29724			−0.219008
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.1549	−	22.1549i	−1.12330	−	1.12330i	−0.991242	−	0.132057i	$-0.957842\pi$
$389$	−22.1549	−	22.1549i	−1.12330	−	1.12330i	−0.132057	−	0.991242i	$-0.542158\pi$
$390$		0			0
$391$	5.65133			0.285800
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.23308	−	1.23308i	0.0620428	−	0.0620428i
$396$		0			0
$397$	−10.9902	−	10.9902i	−0.551583	−	0.551583i	0.375315	−	0.926897i	$-0.377535\pi$
$397$	−10.9902	−	10.9902i	−0.551583	−	0.551583i	−0.926897	+	0.375315i	$0.877535\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	16.9904i		−	0.848458i	−0.905555	−	0.424229i	$-0.860545\pi$
$401$		−	16.9904i		−	0.848458i	0.905555	−	0.424229i	$-0.139455\pi$
$402$		0			0
$403$	16.4667	−	16.4667i	0.820266	−	0.820266i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.8067i			0.684375i
$408$		0			0
$409$		−	7.92579i		−	0.391905i	−0.980613	−	0.195953i	$-0.937220\pi$
$409$		−	7.92579i		−	0.391905i	0.980613	−	0.195953i	$-0.0627799\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.57284	−	3.57284i	0.175808	−	0.175808i
$414$		0			0
$415$		−	21.4291i		−	1.05191i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.9079	+	20.9079i	1.02142	+	1.02142i	0.999766	+	0.0216529i	$0.00689287\pi$
$419$	20.9079	+	20.9079i	1.02142	+	1.02142i	0.0216529	+	0.999766i	$0.493107\pi$
$420$		0			0
$421$	−7.04431	+	7.04431i	−0.343319	+	0.343319i	−0.857613	−	0.514295i	$-0.828054\pi$
$421$	−7.04431	+	7.04431i	−0.343319	+	0.343319i	0.514295	+	0.857613i	$0.328054\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.92583			−0.141924
$426$		0			0
$427$	2.12648	+	2.12648i	0.102907	+	0.102907i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.3255			−0.882709			−0.441354	−	0.897333i	$-0.645502\pi$
$431$	−18.3255			−0.882709			−0.441354	+	0.897333i	$0.645502\pi$
$432$		0			0
$433$	−29.3425			−1.41011			−0.705055	−	0.709153i	$-0.749078\pi$
$433$	−29.3425			−1.41011			−0.705055	+	0.709153i	$0.749078\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.1114	−	22.1114i	−1.05773	−	1.05773i
$438$		0			0
$439$	−0.611550			−0.0291877			−0.0145938	−	0.999894i	$-0.504646\pi$
$439$	−0.611550			−0.0291877			−0.0145938	+	0.999894i	$0.504646\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.85882	−	8.85882i	0.420895	−	0.420895i	−0.464617	−	0.885512i	$-0.653808\pi$
$443$	8.85882	−	8.85882i	0.420895	−	0.420895i	0.885512	+	0.464617i	$0.153808\pi$
$444$		0			0
$445$	−12.1016	−	12.1016i	−0.573673	−	0.573673i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			17.4632i			0.824140i	0.911152	+	0.412070i	$0.135194\pi$
$449$			17.4632i			0.824140i	−0.911152	+	0.412070i	$0.864806\pi$
$450$		0			0
$451$	−0.339554	+	0.339554i	−0.0159890	+	0.0159890i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			7.68522i			0.360288i
$456$		0			0
$457$		−	9.19404i		−	0.430079i	−0.976605	−	0.215040i	$-0.931012\pi$
$457$		−	9.19404i		−	0.430079i	0.976605	−	0.215040i	$-0.0689880\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.6896	+	19.6896i	−0.917036	+	0.917036i	−0.996813	−	0.0797764i	$-0.974579\pi$
$461$	−19.6896	+	19.6896i	−0.917036	+	0.917036i	0.0797764	+	0.996813i	$0.474579\pi$
$462$		0			0
$463$		−	25.4746i		−	1.18391i	−0.805973	−	0.591953i	$-0.798357\pi$
$463$		−	25.4746i		−	1.18391i	0.805973	−	0.591953i	$-0.201643\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.0379	−	18.0379i	−0.834695	−	0.834695i	0.153460	−	0.988155i	$-0.450959\pi$
$467$	−18.0379	−	18.0379i	−0.834695	−	0.834695i	−0.988155	+	0.153460i	$0.950959\pi$
$468$		0			0
$469$	−4.70916	+	4.70916i	−0.217449	+	0.217449i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−26.8760			−1.23576
$474$		0			0
$475$	11.4476	+	11.4476i	0.525251	+	0.525251i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.25775			0.103159			0.0515797	−	0.998669i	$-0.483574\pi$
$479$	2.25775			0.103159			0.0515797	+	0.998669i	$0.483574\pi$
$480$		0			0
$481$	24.6921			1.12586
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.8035	−	20.8035i	−0.944638	−	0.944638i
$486$		0			0
$487$	−3.44406			−0.156065			−0.0780326	−	0.996951i	$-0.524864\pi$
$487$	−3.44406			−0.156065			−0.0780326	+	0.996951i	$0.524864\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.94392	−	2.94392i	0.132857	−	0.132857i	−0.637551	−	0.770408i	$-0.720052\pi$
$491$	2.94392	−	2.94392i	0.132857	−	0.132857i	0.770408	+	0.637551i	$0.220052\pi$
$492$		0			0
$493$	5.97413	+	5.97413i	0.269061	+	0.269061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			0.828539i			0.0371651i
$498$		0			0
$499$	17.9097	−	17.9097i	0.801749	−	0.801749i	−0.181620	−	0.983369i	$-0.558134\pi$
$499$	17.9097	−	17.9097i	0.801749	−	0.801749i	0.983369	+	0.181620i	$0.0581341\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	2.95559i		−	0.131783i	−0.997827	−	0.0658915i	$-0.979011\pi$
$503$		−	2.95559i		−	0.131783i	0.997827	−	0.0658915i	$-0.0209891\pi$
$504$		0			0
$505$			16.2133i			0.721483i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.43654	+	4.43654i	−0.196646	+	0.196646i	−0.798561	−	0.601914i	$-0.794405\pi$
$509$	−4.43654	+	4.43654i	−0.196646	+	0.196646i	0.601914	+	0.798561i	$0.294405\pi$
$510$		0			0
$511$		−	7.95417i		−	0.351872i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.6706	+	16.6706i	0.734596	+	0.734596i
$516$		0			0
$517$	8.74015	−	8.74015i	0.384392	−	0.384392i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.5612			−1.20748			−0.603739	−	0.797182i	$-0.706323\pi$
$521$	−27.5612			−1.20748			−0.603739	+	0.797182i	$0.706323\pi$
$522$		0			0
$523$	−24.7297	−	24.7297i	−1.08135	−	1.08135i	−0.996383	−	0.0849713i	$-0.972920\pi$
$523$	−24.7297	−	24.7297i	−1.08135	−	1.08135i	−0.0849713	−	0.996383i	$-0.527080\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.70389			0.292026
$528$		0			0
$529$	4.79047			0.208281
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.607261	+	0.607261i	0.0263034	+	0.0263034i
$534$		0			0
$535$	16.8822			0.729881
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.81893	−	1.81893i	0.0783468	−	0.0783468i
$540$		0			0
$541$	−4.57184	−	4.57184i	−0.196559	−	0.196559i	0.601964	−	0.798523i	$-0.294385\pi$
$541$	−4.57184	−	4.57184i	−0.196559	−	0.196559i	−0.798523	+	0.601964i	$0.794385\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	16.6207i		−	0.711953i
$546$		0			0
$547$	−27.6872	+	27.6872i	−1.18382	+	1.18382i	−0.205071	+	0.978747i	$0.565743\pi$
$547$	−27.6872	+	27.6872i	−1.18382	+	1.18382i	−0.978747	+	0.205071i	$0.934257\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	46.7487i		−	1.99156i
$552$		0			0
$553$			1.04387i			0.0443898i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.8994	−	29.8994i	1.26688	−	1.26688i	0.319187	−	0.947692i	$-0.396590\pi$
$557$	29.8994	−	29.8994i	1.26688	−	1.26688i	0.947692	−	0.319187i	$-0.103410\pi$
$558$		0			0
$559$			48.0653i			2.03294i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.9564	−	14.9564i	−0.630335	−	0.630335i	0.317817	−	0.948152i	$-0.397050\pi$
$563$	−14.9564	−	14.9564i	−0.630335	−	0.630335i	−0.948152	+	0.317817i	$0.897050\pi$
$564$		0			0
$565$	15.5787	−	15.5787i	0.655399	−	0.655399i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−38.3512			−1.60776			−0.803882	−	0.594788i	$-0.797236\pi$
$569$	−38.3512			−1.60776			−0.803882	+	0.594788i	$0.797236\pi$
$570$		0			0
$571$	10.1756	+	10.1756i	0.425834	+	0.425834i	0.887206	−	0.461373i	$-0.152643\pi$
$571$	10.1756	+	10.1756i	0.425834	+	0.425834i	−0.461373	+	0.887206i	$0.652643\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.42751			0.393154
$576$		0			0
$577$	34.3232			1.42889			0.714447	−	0.699690i	$-0.246678\pi$
$577$	34.3232			1.42889			0.714447	+	0.699690i	$0.246678\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.07047	+	9.07047i	0.376307	+	0.376307i
$582$		0			0
$583$	35.0028			1.44967
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.00308	+	6.00308i	−0.247773	+	0.247773i	−0.820056	−	0.572283i	$-0.806058\pi$
$587$	−6.00308	+	6.00308i	−0.247773	+	0.247773i	0.572283	+	0.820056i	$0.306058\pi$
$588$		0			0
$589$	−26.2296	−	26.2296i	−1.08077	−	1.08077i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.77161i		−	0.154881i	−0.996997	−	0.0774407i	$-0.975325\pi$
$593$		−	3.77161i		−	0.154881i	0.996997	−	0.0774407i	$-0.0246749\pi$
$594$		0			0
$595$	−1.56439	+	1.56439i	−0.0641339	+	0.0641339i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			27.4473i			1.12146i	0.827997	+	0.560732i	$0.189480\pi$
$599$			27.4473i			1.12146i	−0.827997	+	0.560732i	$0.810520\pi$
$600$		0			0
$601$			22.0134i			0.897945i	0.893545	+	0.448973i	$0.148210\pi$
$601$			22.0134i			0.897945i	−0.893545	+	0.448973i	$0.851790\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.17745	−	5.17745i	0.210493	−	0.210493i
$606$		0			0
$607$			23.9911i			0.973769i	0.873466	+	0.486885i	$0.161867\pi$
$607$			23.9911i			0.973769i	−0.873466	+	0.486885i	$0.838133\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15.6309	−	15.6309i	−0.632360	−	0.632360i
$612$		0			0
$613$	−12.9018	+	12.9018i	−0.521099	+	0.521099i	−0.917903	−	0.396804i	$-0.870119\pi$
$613$	−12.9018	+	12.9018i	−0.521099	+	0.521099i	0.396804	+	0.917903i	$0.370119\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.1836			1.21515			0.607574	−	0.794263i	$-0.292143\pi$
$617$	30.1836			1.21515			0.607574	+	0.794263i	$0.292143\pi$
$618$		0			0
$619$	14.0047	+	14.0047i	0.562897	+	0.562897i	0.930129	−	0.367232i	$-0.119695\pi$
$619$	14.0047	+	14.0047i	0.562897	+	0.562897i	−0.367232	+	0.930129i	$0.619695\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.2447			0.410446
$624$		0			0
$625$	9.07284			0.362913
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.02629	+	5.02629i	0.200411	+	0.200411i
$630$		0			0
$631$	31.0229			1.23500			0.617500	−	0.786571i	$-0.288146\pi$
$631$	31.0229			1.23500			0.617500	+	0.786571i	$0.288146\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.8123	−	19.8123i	0.786225	−	0.786225i
$636$		0			0
$637$	−3.25298	−	3.25298i	−0.128888	−	0.128888i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			29.9301i			1.18217i	0.806610	+	0.591084i	$0.201300\pi$
$641$			29.9301i			1.18217i	−0.806610	+	0.591084i	$0.798700\pi$
$642$		0			0
$643$	15.7405	−	15.7405i	0.620746	−	0.620746i	−0.324976	−	0.945722i	$-0.605356\pi$
$643$	15.7405	−	15.7405i	0.620746	−	0.620746i	0.945722	+	0.324976i	$0.105356\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	41.9517i		−	1.64929i	−0.565649	−	0.824646i	$-0.691374\pi$
$647$		−	41.9517i		−	1.64929i	0.565649	−	0.824646i	$-0.308626\pi$
$648$		0			0
$649$		−	12.9975i		−	0.510195i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.97963	+	6.97963i	−0.273134	+	0.273134i	−0.830360	−	0.557227i	$-0.811865\pi$
$653$	−6.97963	+	6.97963i	−0.273134	+	0.273134i	0.557227	+	0.830360i	$0.311865\pi$
$654$		0			0
$655$			28.1884i			1.10141i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.54536	−	9.54536i	−0.371834	−	0.371834i	0.496311	−	0.868145i	$-0.334688\pi$
$659$	−9.54536	−	9.54536i	−0.371834	−	0.371834i	−0.868145	+	0.496311i	$0.834688\pi$
$660$		0			0
$661$	18.2084	−	18.2084i	0.708226	−	0.708226i	−0.257936	−	0.966162i	$-0.583042\pi$
$661$	18.2084	−	18.2084i	0.708226	−	0.708226i	0.966162	+	0.257936i	$0.0830424\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.2417			0.474712
$666$		0			0
$667$	−19.2496	−	19.2496i	−0.745349	−	0.745349i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.73581			0.298638
$672$		0			0
$673$	10.8430			0.417968			0.208984	−	0.977919i	$-0.432984\pi$
$673$	10.8430			0.417968			0.208984	+	0.977919i	$0.432984\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.5002	−	16.5002i	−0.634154	−	0.634154i	0.314953	−	0.949107i	$-0.398011\pi$
$677$	−16.5002	−	16.5002i	−0.634154	−	0.634154i	−0.949107	+	0.314953i	$0.898011\pi$
$678$		0			0
$679$	17.6113			0.675860
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.51214	−	5.51214i	0.210916	−	0.210916i	−0.593741	−	0.804657i	$-0.702349\pi$
$683$	5.51214	−	5.51214i	0.210916	−	0.210916i	0.804657	+	0.593741i	$0.202349\pi$
$684$		0			0
$685$	3.89305	+	3.89305i	0.148746	+	0.148746i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	62.5992i		−	2.38484i
$690$		0			0
$691$	−15.5329	+	15.5329i	−0.590900	+	0.590900i	−0.937875	−	0.346975i	$-0.887209\pi$
$691$	−15.5329	+	15.5329i	−0.590900	+	0.590900i	0.346975	+	0.937875i	$0.387209\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			19.8696i			0.753699i
$696$		0			0
$697$			0.247227i			0.00936438i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28.0234	−	28.0234i	1.05843	−	1.05843i	0.0602442	−	0.998184i	$-0.480812\pi$
$701$	28.0234	−	28.0234i	1.05843	−	1.05843i	0.998184	−	0.0602442i	$-0.0191879\pi$
$702$		0			0
$703$		−	39.3317i		−	1.48342i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.86273	−	6.86273i	−0.258100	−	0.258100i
$708$		0			0
$709$	4.36517	−	4.36517i	0.163937	−	0.163937i	−0.620371	−	0.784308i	$-0.713018\pi$
$709$	4.36517	−	4.36517i	0.163937	−	0.163937i	0.784308	+	0.620371i	$0.213018\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.6010			−0.808965
$714$		0			0
$715$	13.9789	+	13.9789i	0.522779	+	0.522779i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.71617			0.0640022			0.0320011	−	0.999488i	$-0.489812\pi$
$719$	1.71617			0.0640022			0.0320011	+	0.999488i	$0.489812\pi$
$720$		0			0
$721$	−14.1126			−0.525582
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.96599	+	9.96599i	0.370128	+	0.370128i
$726$		0			0
$727$	−43.4253			−1.61056			−0.805278	−	0.592897i	$-0.797984\pi$
$727$	−43.4253			−1.61056			−0.805278	+	0.592897i	$0.797984\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.78411	+	9.78411i	−0.361878	+	0.361878i
$732$		0			0
$733$	24.6931	+	24.6931i	0.912061	+	0.912061i	0.996434	−	0.0843735i	$-0.0268889\pi$
$733$	24.6931	+	24.6931i	0.912061	+	0.912061i	−0.0843735	+	0.996434i	$0.526889\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			17.1313i			0.631038i
$738$		0			0
$739$	28.3412	−	28.3412i	1.04255	−	1.04255i	0.0434958	−	0.999054i	$-0.486150\pi$
$739$	28.3412	−	28.3412i	1.04255	−	1.04255i	0.999054	−	0.0434958i	$-0.0138495\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			37.3822i			1.37142i	0.727876	+	0.685709i	$0.240508\pi$
$743$			37.3822i			1.37142i	−0.727876	+	0.685709i	$0.759492\pi$
$744$		0			0
$745$		−	8.26066i		−	0.302647i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.14587	+	7.14587i	−0.261104	+	0.261104i
$750$		0			0
$751$			1.99372i			0.0727519i	0.999338	+	0.0363760i	$0.0115814\pi$
$751$			1.99372i			0.0727519i	−0.999338	+	0.0363760i	$0.988419\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.07707	+	5.07707i	0.184773	+	0.184773i
$756$		0			0
$757$	−1.85886	+	1.85886i	−0.0675613	+	0.0675613i	−0.740080	−	0.672519i	$-0.765212\pi$
$757$	−1.85886	+	1.85886i	−0.0675613	+	0.0675613i	0.672519	+	0.740080i	$0.265212\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.8726			−0.937882			−0.468941	−	0.883230i	$-0.655364\pi$
$761$	−25.8726			−0.937882			−0.468941	+	0.883230i	$0.655364\pi$
$762$		0			0
$763$	7.03518	+	7.03518i	0.254691	+	0.254691i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.2448			−0.839320
$768$		0			0
$769$	−42.4405			−1.53044			−0.765222	−	0.643767i	$-0.777371\pi$
$769$	−42.4405			−1.53044			−0.765222	+	0.643767i	$0.777371\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.7217	−	26.7217i	−0.961113	−	0.961113i	0.0381585	−	0.999272i	$-0.487851\pi$
$773$	−26.7217	−	26.7217i	−0.961113	−	0.961113i	−0.999272	+	0.0381585i	$0.987851\pi$
$774$		0			0
$775$	11.1834			0.401719
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.967298	−	0.967298i	0.0346570	−	0.0346570i
$780$		0			0
$781$	1.50705	+	1.50705i	0.0539266	+	0.0539266i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			18.8846i			0.674021i
$786$		0			0
$787$	33.7616	−	33.7616i	1.20347	−	1.20347i	0.230367	−	0.973104i	$-0.426007\pi$
$787$	33.7616	−	33.7616i	1.20347	−	1.20347i	0.973104	−	0.230367i	$-0.0739926\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			13.1882i			0.468918i
$792$		0			0
$793$		−	13.8348i		−	0.491287i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.2038	+	18.2038i	−0.644811	+	0.644811i	−0.951734	−	0.306923i	$-0.900701\pi$
$797$	−18.2038	+	18.2038i	−0.644811	+	0.644811i	0.306923	+	0.951734i	$0.400701\pi$
$798$		0			0
$799$		−	6.36363i		−	0.225129i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.4681	−	14.4681i	−0.510567	−	0.510567i
$804$		0			0
$805$	5.04073	−	5.04073i	0.177663	−	0.177663i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	44.8570			1.57709			0.788543	−	0.614979i	$-0.210836\pi$
$809$	44.8570			1.57709			0.788543	+	0.614979i	$0.210836\pi$
$810$		0			0
$811$	−18.1777	−	18.1777i	−0.638307	−	0.638307i	0.311831	−	0.950138i	$-0.399058\pi$
$811$	−18.1777	−	18.1777i	−0.638307	−	0.638307i	−0.950138	+	0.311831i	$0.899058\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.68472			0.164099
$816$		0			0
$817$	76.5625			2.67858
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.11903	+	2.11903i	0.0739548	+	0.0739548i	0.743117	−	0.669162i	$-0.233347\pi$
$821$	2.11903	+	2.11903i	0.0739548	+	0.0739548i	−0.669162	+	0.743117i	$0.733347\pi$
$822$		0			0
$823$	3.25817			0.113573			0.0567863	−	0.998386i	$-0.481915\pi$
$823$	3.25817			0.113573			0.0567863	+	0.998386i	$0.481915\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.0131	+	39.0131i	−1.35662	+	1.35662i	−0.478571	+	0.878049i	$0.658845\pi$
$827$	−39.0131	+	39.0131i	−1.35662	+	1.35662i	−0.878049	+	0.478571i	$0.841155\pi$
$828$		0			0
$829$	33.1928	+	33.1928i	1.15283	+	1.15283i	0.985982	+	0.166852i	$0.0533602\pi$
$829$	33.1928	+	33.1928i	1.15283	+	1.15283i	0.166852	+	0.985982i	$0.446640\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.32435i		−	0.0458859i
$834$		0			0
$835$	6.97574	−	6.97574i	0.241406	−	0.241406i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	41.6480i		−	1.43785i	−0.695089	−	0.718924i	$-0.744635\pi$
$839$		−	41.6480i		−	1.43785i	0.695089	−	0.718924i	$-0.255365\pi$
$840$		0			0
$841$		−	11.6983i		−	0.403389i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.64353	−	9.64353i	0.331747	−	0.331747i
$846$		0			0
$847$			4.38300i			0.150602i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.1955	−	16.1955i	−0.555176	−	0.555176i
$852$		0			0
$853$	−0.280765	+	0.280765i	−0.00961321	+	0.00961321i	−0.711897	−	0.702284i	$-0.752164\pi$
$853$	−0.280765	+	0.280765i	−0.00961321	+	0.00961321i	0.702284	+	0.711897i	$0.252164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.0555			1.43659			0.718294	−	0.695740i	$-0.244923\pi$
$857$	42.0555			1.43659			0.718294	+	0.695740i	$0.244923\pi$
$858$		0			0
$859$	−8.23245	−	8.23245i	−0.280888	−	0.280888i	0.552575	−	0.833463i	$-0.313645\pi$
$859$	−8.23245	−	8.23245i	−0.280888	−	0.280888i	−0.833463	+	0.552575i	$0.813645\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.77432			−0.332722			−0.166361	−	0.986065i	$-0.553202\pi$
$863$	−9.77432			−0.332722			−0.166361	+	0.986065i	$0.553202\pi$
$864$		0			0
$865$	−6.24064			−0.212188
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.89872	+	1.89872i	0.0644097	+	0.0644097i
$870$		0			0
$871$	30.6377			1.03812
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.51599	+	8.51599i	−0.287893	+	0.287893i
$876$		0			0
$877$	−26.7105	−	26.7105i	−0.901951	−	0.901951i	0.0936539	−	0.995605i	$-0.470145\pi$
$877$	−26.7105	−	26.7105i	−0.901951	−	0.901951i	−0.995605	+	0.0936539i	$0.970145\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	22.2702i		−	0.750303i	−0.926964	−	0.375151i	$-0.877591\pi$
$881$		−	22.2702i		−	0.750303i	0.926964	−	0.375151i	$-0.122409\pi$
$882$		0			0
$883$	−23.0709	+	23.0709i	−0.776396	+	0.776396i	−0.979216	−	0.202820i	$-0.934989\pi$
$883$	−23.0709	+	23.0709i	−0.776396	+	0.776396i	0.202820	+	0.979216i	$0.434989\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.84617i		−	0.0619882i	−0.999520	−	0.0309941i	$-0.990133\pi$
$887$		−	1.84617i		−	0.0619882i	0.999520	−	0.0309941i	$-0.00986731\pi$
$888$		0			0
$889$			16.7722i			0.562521i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.8983	+	24.8983i	−0.833191	+	0.833191i
$894$		0			0
$895$		−	21.3596i		−	0.713973i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.8349	−	22.8349i	−0.761585	−	0.761585i
$900$		0			0
$901$	12.7426	−	12.7426i	0.424518	−	0.424518i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.33854			0.310424
$906$		0			0
$907$	3.28905	+	3.28905i	0.109211	+	0.109211i	0.759601	−	0.650390i	$-0.225394\pi$
$907$	3.28905	+	3.28905i	0.109211	+	0.109211i	−0.650390	+	0.759601i	$0.725394\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34.3242			1.13721			0.568606	−	0.822610i	$-0.307483\pi$
$911$	34.3242			1.13721			0.568606	+	0.822610i	$0.307483\pi$
$912$		0			0
$913$	32.9971			1.09204
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.9315	−	11.9315i	−0.394014	−	0.394014i
$918$		0			0
$919$	−21.1603			−0.698012			−0.349006	−	0.937120i	$-0.613481\pi$
$919$	−21.1603			−0.698012			−0.349006	+	0.937120i	$0.613481\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.69522	−	2.69522i	0.0887143	−	0.0887143i
$924$		0			0
$925$	8.38482	+	8.38482i	0.275691	+	0.275691i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.64400i			0.152365i	0.997094	+	0.0761823i	$0.0242731\pi$
$929$			4.64400i			0.152365i	−0.997094	+	0.0761823i	$0.975727\pi$
$930$		0			0
$931$	−5.18163	+	5.18163i	−0.169821	+	0.169821i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.69104i			0.186117i
$936$		0			0
$937$		−	45.1315i		−	1.47438i	−0.675685	−	0.737191i	$-0.736152\pi$
$937$		−	45.1315i		−	1.47438i	0.675685	−	0.737191i	$-0.263848\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.7501	+	35.7501i	−1.16542	+	1.16542i	−0.182149	+	0.983271i	$0.558305\pi$
$941$	−35.7501	+	35.7501i	−1.16542	+	1.16542i	−0.983271	+	0.182149i	$0.941695\pi$
$942$		0			0
$943$		−	0.796605i		−	0.0259410i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.9742	−	23.9742i	−0.779058	−	0.779058i	0.200612	−	0.979671i	$-0.435707\pi$
$947$	−23.9742	−	23.9742i	−0.779058	−	0.779058i	−0.979671	+	0.200612i	$0.935707\pi$
$948$		0			0
$949$	−25.8748	+	25.8748i	−0.839931	+	0.839931i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.9210			−1.35796			−0.678978	−	0.734159i	$-0.737577\pi$
$953$	−41.9210			−1.35796			−0.678978	+	0.734159i	$0.737577\pi$
$954$		0			0
$955$	13.4082	+	13.4082i	0.433880	+	0.433880i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.29568			−0.106423
$960$		0			0
$961$	5.37578			0.173412
$962$		0			0
$963$		0			0
$964$		0			0
$965$	27.4522	+	27.4522i	0.883718	+	0.883718i
$966$		0			0
$967$	−40.0296			−1.28727			−0.643633	−	0.765335i	$-0.722574\pi$
$967$	−40.0296			−1.28727			−0.643633	+	0.765335i	$0.722574\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.3073	−	10.3073i	0.330778	−	0.330778i	−0.522104	−	0.852882i	$-0.674853\pi$
$971$	10.3073	−	10.3073i	0.330778	−	0.330778i	0.852882	+	0.522104i	$0.174853\pi$
$972$		0			0
$973$	−8.41038	−	8.41038i	−0.269624	−	0.269624i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	57.0154i		−	1.82408i	−0.410097	−	0.912042i	$-0.634505\pi$
$977$		−	57.0154i		−	1.82408i	0.410097	−	0.912042i	$-0.365495\pi$
$978$		0			0
$979$	18.6344	−	18.6344i	0.595558	−	0.595558i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	59.3951i		−	1.89441i	−0.320628	−	0.947205i	$-0.603894\pi$
$983$		−	59.3951i		−	1.89441i	0.320628	−	0.947205i	$-0.396106\pi$
$984$		0			0
$985$		−	25.6825i		−	0.818313i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.5260	−	31.5260i	1.00247	−	1.00247i
$990$		0			0
$991$			13.0593i			0.414842i	0.978252	+	0.207421i	$0.0665069\pi$
$991$			13.0593i			0.414842i	−0.978252	+	0.207421i	$0.933493\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.1576	−	21.1576i	−0.670742	−	0.670742i
$996$		0			0
$997$	16.5393	−	16.5393i	0.523806	−	0.523806i	−0.394912	−	0.918719i	$-0.629225\pi$
$997$	16.5393	−	16.5393i	0.523806	−	0.523806i	0.918719	+	0.394912i	$0.129225\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.d.1583.6		36
3.2	odd	2	inner	4032.2.v.d.1583.13		36
4.3	odd	2		1008.2.v.d.323.17	yes	36
12.11	even	2		1008.2.v.d.323.2	✓	36
16.5	even	4		1008.2.v.d.827.2	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.13		36
48.5	odd	4		1008.2.v.d.827.17	yes	36
48.11	even	4	inner	4032.2.v.d.3599.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.2	✓	36	12.11	even	2
1008.2.v.d.323.17	yes	36	4.3	odd	2
1008.2.v.d.827.2	yes	36	16.5	even	4
1008.2.v.d.827.17	yes	36	48.5	odd	4
4032.2.v.d.1583.6		36	1.1	even	1	trivial
4032.2.v.d.1583.13		36	3.2	odd	2	inner
4032.2.v.d.3599.6		36	48.11	even	4	inner
4032.2.v.d.3599.13		36	16.11	odd	4	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.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.18126 - 1.18126i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-1.18126 - 1.18126i) q^{5} +1.00000 q^{7} +(1.81893 - 1.81893i) q^{11} +(-3.25298 - 3.25298i) q^{13} -1.32435i q^{17} +(-5.18163 + 5.18163i) q^{19} +4.26726i q^{23} -2.20926i q^{25} +(-4.51100 + 4.51100i) q^{29} +5.06204i q^{31} +(-1.18126 - 1.18126i) q^{35} +(-3.79530 + 3.79530i) q^{37} -0.186678 q^{41} +(-7.38788 - 7.38788i) q^{43} +4.80511 q^{47} +1.00000 q^{49} +(9.62181 + 9.62181i) q^{53} -4.29724 q^{55} +(3.57284 - 3.57284i) q^{59} +(2.12648 + 2.12648i) q^{61} +7.68522i q^{65} +(-4.70916 + 4.70916i) q^{67} +0.828539i q^{71} -7.95417i q^{73} +(1.81893 - 1.81893i) q^{77} +1.04387i q^{79} +(9.07047 + 9.07047i) q^{83} +(-1.56439 + 1.56439i) q^{85} +10.2447 q^{89} +(-3.25298 - 3.25298i) q^{91} +12.2417 q^{95} +17.6113 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 36 * q + 36 * q^7	\( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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