LMFDB - Embedded newform 4032.2.v.d.1583.15

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			1583.15
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.01396 + 2.01396i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(2.01396 + 2.01396i) q^{5} +1.00000 q^{7} +(-3.69809 + 3.69809i) q^{11} +(2.62173 + 2.62173i) q^{13} +6.01559i q^{17} +(-1.07444 + 1.07444i) q^{19} +3.99603i q^{23} +3.11208i q^{25} +(2.70229 - 2.70229i) q^{29} +1.17640i q^{31} +(2.01396 + 2.01396i) q^{35} +(3.35894 - 3.35894i) q^{37} -8.80066 q^{41} +(-7.99594 - 7.99594i) q^{43} -3.66298 q^{47} +1.00000 q^{49} +(-9.44391 - 9.44391i) q^{53} -14.8956 q^{55} +(3.29886 - 3.29886i) q^{59} +(5.05790 + 5.05790i) q^{61} +10.5601i q^{65} +(-2.32750 + 2.32750i) q^{67} +2.79177i q^{71} +4.79747i q^{73} +(-3.69809 + 3.69809i) q^{77} -1.49360i q^{79} +(5.86731 + 5.86731i) q^{83} +(-12.1152 + 12.1152i) q^{85} -7.45344 q^{89} +(2.62173 + 2.62173i) q^{91} -4.32777 q^{95} -5.33798 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.01396	+	2.01396i	0.900671	+	0.900671i	0.995494	−	0.0948231i	$-0.0302285\pi$
$5$	2.01396	+	2.01396i	0.900671	+	0.900671i	−0.0948231	+	0.995494i	$0.530229\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.69809	+	3.69809i	−1.11502	+	1.11502i	−0.122553	+	0.992462i	$0.539108\pi$
$11$	−3.69809	+	3.69809i	−1.11502	+	1.11502i	−0.992462	+	0.122553i	$0.960892\pi$
$12$		0			0
$13$	2.62173	+	2.62173i	0.727137	+	0.727137i	0.970048	−	0.242911i	$-0.0781025\pi$
$13$	2.62173	+	2.62173i	0.727137	+	0.727137i	−0.242911	+	0.970048i	$0.578102\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.01559i			1.45899i	0.683984	+	0.729497i	$0.260246\pi$
$17$			6.01559i			1.45899i	−0.683984	+	0.729497i	$0.739754\pi$
$18$		0			0
$19$	−1.07444	+	1.07444i	−0.246494	+	0.246494i	−0.819530	−	0.573036i	$-0.805765\pi$
$19$	−1.07444	+	1.07444i	−0.246494	+	0.246494i	0.573036	+	0.819530i	$0.305765\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.99603i			0.833230i	0.909083	+	0.416615i	$0.136784\pi$
$23$			3.99603i			0.833230i	−0.909083	+	0.416615i	$0.863216\pi$
$24$		0			0
$25$			3.11208i			0.622417i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.70229	−	2.70229i	0.501802	−	0.501802i	−0.410196	−	0.911998i	$-0.634540\pi$
$29$	2.70229	−	2.70229i	0.501802	−	0.501802i	0.911998	+	0.410196i	$0.134540\pi$
$30$		0			0
$31$			1.17640i			0.211287i	0.994404	+	0.105644i	$0.0336903\pi$
$31$			1.17640i			0.211287i	−0.994404	+	0.105644i	$0.966310\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.01396	+	2.01396i	0.340422	+	0.340422i
$36$		0			0
$37$	3.35894	−	3.35894i	0.552206	−	0.552206i	−0.374871	−	0.927077i	$-0.622313\pi$
$37$	3.35894	−	3.35894i	0.552206	−	0.552206i	0.927077	+	0.374871i	$0.122313\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.80066			−1.37443			−0.687216	−	0.726453i	$-0.741167\pi$
$41$	−8.80066			−1.37443			−0.687216	+	0.726453i	$0.741167\pi$
$42$		0			0
$43$	−7.99594	−	7.99594i	−1.21937	−	1.21937i	−0.967853	−	0.251516i	$-0.919071\pi$
$43$	−7.99594	−	7.99594i	−1.21937	−	1.21937i	−0.251516	−	0.967853i	$-0.580929\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.66298			−0.534301			−0.267150	−	0.963655i	$-0.586082\pi$
$47$	−3.66298			−0.534301			−0.267150	+	0.963655i	$0.586082\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.44391	−	9.44391i	−1.29722	−	1.29722i	−0.930221	−	0.366999i	$-0.880385\pi$
$53$	−9.44391	−	9.44391i	−1.29722	−	1.29722i	−0.366999	−	0.930221i	$-0.619615\pi$
$54$		0			0
$55$	−14.8956			−2.00852
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.29886	−	3.29886i	0.429475	−	0.429475i	−0.458974	−	0.888450i	$-0.651783\pi$
$59$	3.29886	−	3.29886i	0.429475	−	0.429475i	0.888450	+	0.458974i	$0.151783\pi$
$60$		0			0
$61$	5.05790	+	5.05790i	0.647598	+	0.647598i	0.952412	−	0.304814i	$-0.0985943\pi$
$61$	5.05790	+	5.05790i	0.647598	+	0.647598i	−0.304814	+	0.952412i	$0.598594\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			10.5601i			1.30982i
$66$		0			0
$67$	−2.32750	+	2.32750i	−0.284349	+	0.284349i	−0.834841	−	0.550491i	$-0.814440\pi$
$67$	−2.32750	+	2.32750i	−0.284349	+	0.284349i	0.550491	+	0.834841i	$0.314440\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.79177i			0.331322i	0.986183	+	0.165661i	$0.0529758\pi$
$71$			2.79177i			0.331322i	−0.986183	+	0.165661i	$0.947024\pi$
$72$		0			0
$73$			4.79747i			0.561501i	0.959781	+	0.280751i	$0.0905834\pi$
$73$			4.79747i			0.561501i	−0.959781	+	0.280751i	$0.909417\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.69809	+	3.69809i	−0.421436	+	0.421436i
$78$		0			0
$79$		−	1.49360i		−	0.168043i	−0.996464	−	0.0840215i	$-0.973224\pi$
$79$		−	1.49360i		−	0.168043i	0.996464	−	0.0840215i	$-0.0267764\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.86731	+	5.86731i	0.644020	+	0.644020i	0.951541	−	0.307521i	$-0.0994994\pi$
$83$	5.86731	+	5.86731i	0.644020	+	0.644020i	−0.307521	+	0.951541i	$0.599499\pi$
$84$		0			0
$85$	−12.1152	+	12.1152i	−1.31407	+	1.31407i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.45344			−0.790063			−0.395032	−	0.918667i	$-0.629266\pi$
$89$	−7.45344			−0.790063			−0.395032	+	0.918667i	$0.629266\pi$
$90$		0			0
$91$	2.62173	+	2.62173i	0.274832	+	0.274832i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.32777			−0.444020
$96$		0			0
$97$	−5.33798			−0.541990			−0.270995	−	0.962581i	$-0.587353\pi$
$97$	−5.33798			−0.541990			−0.270995	+	0.962581i	$0.587353\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.39628	−	3.39628i	−0.337943	−	0.337943i	0.517650	−	0.855593i	$-0.326807\pi$
$101$	−3.39628	−	3.39628i	−0.337943	−	0.337943i	−0.855593	+	0.517650i	$0.826807\pi$
$102$		0			0
$103$	14.1894			1.39812			0.699061	−	0.715062i	$-0.253602\pi$
$103$	14.1894			1.39812			0.699061	+	0.715062i	$0.253602\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.5230	−	12.5230i	1.21064	−	1.21064i	0.239826	−	0.970816i	$-0.422909\pi$
$107$	12.5230	−	12.5230i	1.21064	−	1.21064i	0.970816	−	0.239826i	$-0.0770905\pi$
$108$		0			0
$109$	−5.18580	−	5.18580i	−0.496710	−	0.496710i	0.413702	−	0.910412i	$-0.364235\pi$
$109$	−5.18580	−	5.18580i	−0.496710	−	0.496710i	−0.910412	+	0.413702i	$0.864235\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.4478i			1.54728i	0.633626	+	0.773640i	$0.281566\pi$
$113$			16.4478i			1.54728i	−0.633626	+	0.773640i	$0.718434\pi$
$114$		0			0
$115$	−8.04785	+	8.04785i	−0.750466	+	0.750466i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.01559i			0.551448i
$120$		0			0
$121$		−	16.3517i		−	1.48652i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.80219	−	3.80219i	0.340078	−	0.340078i
$126$		0			0
$127$		−	17.3122i		−	1.53621i	−0.640325	−	0.768104i	$-0.721200\pi$
$127$		−	17.3122i		−	1.53621i	0.640325	−	0.768104i	$-0.278800\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.1625	+	12.1625i	1.06264	+	1.06264i	0.997902	+	0.0647387i	$0.0206214\pi$
$131$	12.1625	+	12.1625i	1.06264	+	1.06264i	0.0647387	+	0.997902i	$0.479379\pi$
$132$		0			0
$133$	−1.07444	+	1.07444i	−0.0931660	+	0.0931660i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.2426			1.21683			0.608415	−	0.793619i	$-0.291806\pi$
$137$	14.2426			1.21683			0.608415	+	0.793619i	$0.291806\pi$
$138$		0			0
$139$	−14.4658	−	14.4658i	−1.22697	−	1.22697i	−0.965104	−	0.261865i	$-0.915662\pi$
$139$	−14.4658	−	14.4658i	−1.22697	−	1.22697i	−0.261865	−	0.965104i	$-0.584338\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.3908			−1.62154
$144$		0			0
$145$	10.8846			0.903917
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.629784	−	0.629784i	−0.0515939	−	0.0515939i	0.680839	−	0.732433i	$-0.261615\pi$
$149$	−0.629784	−	0.629784i	−0.0515939	−	0.0515939i	−0.732433	+	0.680839i	$0.761615\pi$
$150$		0			0
$151$	2.41962			0.196906			0.0984530	−	0.995142i	$-0.468611\pi$
$151$	2.41962			0.196906			0.0984530	+	0.995142i	$0.468611\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.36922	+	2.36922i	−0.190300	+	0.190300i
$156$		0			0
$157$	−0.182630	−	0.182630i	−0.0145754	−	0.0145754i	0.699782	−	0.714357i	$-0.253281\pi$
$157$	−0.182630	−	0.182630i	−0.0145754	−	0.0145754i	−0.714357	+	0.699782i	$0.753281\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.99603i			0.314931i
$162$		0			0
$163$	8.48064	−	8.48064i	0.664255	−	0.664255i	−0.292125	−	0.956380i	$-0.594362\pi$
$163$	8.48064	−	8.48064i	0.664255	−	0.664255i	0.956380	+	0.292125i	$0.0943623\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			18.0645i			1.39787i	0.715184	+	0.698937i	$0.246343\pi$
$167$			18.0645i			1.39787i	−0.715184	+	0.698937i	$0.753657\pi$
$168$		0			0
$169$			0.746933i			0.0574564i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−17.1706	+	17.1706i	−1.30546	+	1.30546i	−0.380800	+	0.924657i	$0.624351\pi$
$173$	−17.1706	+	17.1706i	−1.30546	+	1.30546i	−0.924657	+	0.380800i	$0.875649\pi$
$174$		0			0
$175$			3.11208i			0.235251i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.71369	+	5.71369i	0.427061	+	0.427061i	0.887626	−	0.460565i	$-0.152353\pi$
$179$	5.71369	+	5.71369i	0.427061	+	0.427061i	−0.460565	+	0.887626i	$0.652353\pi$
$180$		0			0
$181$	−17.2867	+	17.2867i	−1.28491	+	1.28491i	−0.347071	+	0.937839i	$0.612824\pi$
$181$	−17.2867	+	17.2867i	−1.28491	+	1.28491i	−0.937839	+	0.347071i	$0.887176\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.5295			0.994712
$186$		0			0
$187$	−22.2462	−	22.2462i	−1.62680	−	1.62680i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.70167			0.267844			0.133922	−	0.990992i	$-0.457243\pi$
$191$	3.70167			0.267844			0.133922	+	0.990992i	$0.457243\pi$
$192$		0			0
$193$	−7.58396			−0.545905			−0.272953	−	0.962027i	$-0.588000\pi$
$193$	−7.58396			−0.545905			−0.272953	+	0.962027i	$0.588000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.7959	+	18.7959i	1.33916	+	1.33916i	0.896878	+	0.442277i	$0.145829\pi$
$197$	18.7959	+	18.7959i	1.33916	+	1.33916i	0.442277	+	0.896878i	$0.354171\pi$
$198$		0			0
$199$	−8.50192			−0.602685			−0.301343	−	0.953516i	$-0.597435\pi$
$199$	−8.50192			−0.602685			−0.301343	+	0.953516i	$0.597435\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.70229	−	2.70229i	0.189663	−	0.189663i
$204$		0			0
$205$	−17.7242	−	17.7242i	−1.23791	−	1.23791i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	7.94677i		−	0.549689i
$210$		0			0
$211$	3.93290	−	3.93290i	0.270752	−	0.270752i	−0.558651	−	0.829403i	$-0.688681\pi$
$211$	3.93290	−	3.93290i	0.270752	−	0.270752i	0.829403	+	0.558651i	$0.188681\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	32.2070i		−	2.19650i
$216$		0			0
$217$			1.17640i			0.0798591i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.7712	+	15.7712i	−1.06089	+	1.06089i
$222$		0			0
$223$			18.3065i			1.22590i	0.790123	+	0.612948i	$0.210017\pi$
$223$			18.3065i			1.22590i	−0.790123	+	0.612948i	$0.789983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.97214	−	6.97214i	−0.462757	−	0.462757i	0.436801	−	0.899558i	$-0.356111\pi$
$227$	−6.97214	−	6.97214i	−0.462757	−	0.462757i	−0.899558	+	0.436801i	$0.856111\pi$
$228$		0			0
$229$	−2.92714	+	2.92714i	−0.193431	+	0.193431i	−0.797177	−	0.603746i	$-0.793674\pi$
$229$	−2.92714	+	2.92714i	−0.193431	+	0.193431i	0.603746	+	0.797177i	$0.293674\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.2669			1.06568			0.532842	−	0.846215i	$-0.321124\pi$
$233$	16.2669			1.06568			0.532842	+	0.846215i	$0.321124\pi$
$234$		0			0
$235$	−7.37711	−	7.37711i	−0.481229	−	0.481229i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.77692			0.503047			0.251523	−	0.967851i	$-0.419068\pi$
$239$	7.77692			0.503047			0.251523	+	0.967851i	$0.419068\pi$
$240$		0			0
$241$	29.0632			1.87213			0.936063	−	0.351833i	$-0.114442\pi$
$241$	29.0632			1.87213			0.936063	+	0.351833i	$0.114442\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.01396	+	2.01396i	0.128667	+	0.128667i
$246$		0			0
$247$	−5.63380			−0.358470
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.92556	−	3.92556i	0.247779	−	0.247779i	−0.572279	−	0.820059i	$-0.693941\pi$
$251$	3.92556	−	3.92556i	0.247779	−	0.247779i	0.820059	+	0.572279i	$0.193941\pi$
$252$		0			0
$253$	−14.7777	−	14.7777i	−0.929064	−	0.929064i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.3860i			1.33402i	0.745049	+	0.667010i	$0.232426\pi$
$257$			21.3860i			1.33402i	−0.745049	+	0.667010i	$0.767574\pi$
$258$		0			0
$259$	3.35894	−	3.35894i	0.208714	−	0.208714i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.1247i			0.809307i	0.914470	+	0.404653i	$0.132608\pi$
$263$			13.1247i			0.809307i	−0.914470	+	0.404653i	$0.867392\pi$
$264$		0			0
$265$		−	38.0393i		−	2.33674i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.81310	−	4.81310i	0.293460	−	0.293460i	−0.544986	−	0.838445i	$-0.683465\pi$
$269$	4.81310	−	4.81310i	0.293460	−	0.293460i	0.838445	+	0.544986i	$0.183465\pi$
$270$		0			0
$271$			21.0397i			1.27807i	0.769178	+	0.639034i	$0.220666\pi$
$271$			21.0397i			1.27807i	−0.769178	+	0.639034i	$0.779334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.5088	−	11.5088i	−0.694004	−	0.694004i
$276$		0			0
$277$	−3.65488	+	3.65488i	−0.219600	+	0.219600i	−0.808330	−	0.588730i	$-0.799628\pi$
$277$	−3.65488	+	3.65488i	−0.219600	+	0.219600i	0.588730	+	0.808330i	$0.299628\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−28.8508			−1.72109			−0.860546	−	0.509373i	$-0.829878\pi$
$281$	−28.8508			−1.72109			−0.860546	+	0.509373i	$0.829878\pi$
$282$		0			0
$283$	18.5087	+	18.5087i	1.10023	+	1.10023i	0.994383	+	0.105846i	$0.0337551\pi$
$283$	18.5087	+	18.5087i	1.10023	+	1.10023i	0.105846	+	0.994383i	$0.466245\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.80066			−0.519486
$288$		0			0
$289$	−19.1873			−1.12867
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.18838	−	3.18838i	−0.186267	−	0.186267i	0.607813	−	0.794080i	$-0.292047\pi$
$293$	−3.18838	−	3.18838i	−0.186267	−	0.186267i	−0.794080	+	0.607813i	$0.792047\pi$
$294$		0			0
$295$	13.2876			0.773632
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.4765	+	10.4765i	−0.605872	+	0.605872i
$300$		0			0
$301$	−7.99594	−	7.99594i	−0.460878	−	0.460878i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			20.3728i			1.16655i
$306$		0			0
$307$	12.0614	−	12.0614i	0.688382	−	0.688382i	−0.273492	−	0.961874i	$-0.588179\pi$
$307$	12.0614	−	12.0614i	0.688382	−	0.688382i	0.961874	+	0.273492i	$0.0881787\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.0636i			1.36452i	0.731108	+	0.682262i	$0.239004\pi$
$311$			24.0636i			1.36452i	−0.731108	+	0.682262i	$0.760996\pi$
$312$		0			0
$313$			14.5278i			0.821162i	0.911824	+	0.410581i	$0.134674\pi$
$313$			14.5278i			0.821162i	−0.911824	+	0.410581i	$0.865326\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.9051	−	18.9051i	1.06181	−	1.06181i	0.0638557	−	0.997959i	$-0.479660\pi$
$317$	18.9051	−	18.9051i	1.06181	−	1.06181i	0.997959	−	0.0638557i	$-0.0203397\pi$
$318$		0			0
$319$			19.9866i			1.11903i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.46341	−	6.46341i	−0.359634	−	0.359634i
$324$		0			0
$325$	−8.15904	+	8.15904i	−0.452582	+	0.452582i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.66298			−0.201947
$330$		0			0
$331$	−11.0020	−	11.0020i	−0.604725	−	0.604725i	0.336838	−	0.941563i	$-0.390643\pi$
$331$	−11.0020	−	11.0020i	−0.604725	−	0.604725i	−0.941563	+	0.336838i	$0.890643\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.37499			−0.512210
$336$		0			0
$337$	15.4189			0.839922			0.419961	−	0.907542i	$-0.362044\pi$
$337$	15.4189			0.839922			0.419961	+	0.907542i	$0.362044\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.35042	−	4.35042i	−0.235589	−	0.235589i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−23.7662	+	23.7662i	−1.27584	+	1.27584i	−0.332862	+	0.942975i	$0.608014\pi$
$347$	−23.7662	+	23.7662i	−1.27584	+	1.27584i	−0.942975	+	0.332862i	$0.891986\pi$
$348$		0			0
$349$	6.66814	+	6.66814i	0.356937	+	0.356937i	0.862683	−	0.505746i	$-0.168783\pi$
$349$	6.66814	+	6.66814i	0.356937	+	0.356937i	−0.505746	+	0.862683i	$0.668783\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	23.4209i		−	1.24657i	−0.781995	−	0.623284i	$-0.785798\pi$
$353$		−	23.4209i		−	1.24657i	0.781995	−	0.623284i	$-0.214202\pi$
$354$		0			0
$355$	−5.62252	+	5.62252i	−0.298413	+	0.298413i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.58905i			0.0838669i	0.999120	+	0.0419334i	$0.0133517\pi$
$359$			1.58905i			0.0838669i	−0.999120	+	0.0419334i	$0.986648\pi$
$360$		0			0
$361$			16.6911i			0.878481i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.66192	+	9.66192i	−0.505728	+	0.505728i
$366$		0			0
$367$			30.6018i			1.59740i	0.601728	+	0.798701i	$0.294479\pi$
$367$			30.6018i			1.59740i	−0.601728	+	0.798701i	$0.705521\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.44391	−	9.44391i	−0.490303	−	0.490303i
$372$		0			0
$373$	−0.0746952	+	0.0746952i	−0.00386757	+	0.00386757i	−0.709038	−	0.705170i	$-0.750871\pi$
$373$	−0.0746952	+	0.0746952i	−0.00386757	+	0.00386757i	0.705170	+	0.709038i	$0.250871\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.1693			0.729758
$378$		0			0
$379$	1.18359	+	1.18359i	0.0607968	+	0.0607968i	0.736851	−	0.676055i	$-0.236312\pi$
$379$	1.18359	+	1.18359i	0.0607968	+	0.0607968i	−0.676055	+	0.736851i	$0.736312\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.0870			1.38408			0.692039	−	0.721860i	$-0.256712\pi$
$383$	27.0870			1.38408			0.692039	+	0.721860i	$0.256712\pi$
$384$		0			0
$385$	−14.8956			−0.759151
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.3801	−	18.3801i	−0.931907	−	0.931907i	0.0659176	−	0.997825i	$-0.479003\pi$
$389$	−18.3801	−	18.3801i	−0.931907	−	0.931907i	−0.997825	+	0.0659176i	$0.979003\pi$
$390$		0			0
$391$	−24.0385			−1.21568
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.00805	−	3.00805i	0.151351	−	0.151351i
$396$		0			0
$397$	9.80810	+	9.80810i	0.492255	+	0.492255i	0.909016	−	0.416761i	$-0.136835\pi$
$397$	9.80810	+	9.80810i	0.492255	+	0.492255i	−0.416761	+	0.909016i	$0.636835\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.3321i		−	0.515958i	−0.966150	−	0.257979i	$-0.916943\pi$
$401$		−	10.3321i		−	0.515958i	0.966150	−	0.257979i	$-0.0830566\pi$
$402$		0			0
$403$	−3.08420	+	3.08420i	−0.153635	+	0.153635i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.8433i			1.23144i
$408$		0			0
$409$		−	30.1466i		−	1.49065i	−0.666699	−	0.745327i	$-0.732293\pi$
$409$		−	30.1466i		−	1.49065i	0.666699	−	0.745327i	$-0.267707\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.29886	−	3.29886i	0.162326	−	0.162326i
$414$		0			0
$415$			23.6331i			1.16010i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.8222	+	16.8222i	0.821817	+	0.821817i	0.986368	−	0.164552i	$-0.0526178\pi$
$419$	16.8222	+	16.8222i	0.821817	+	0.821817i	−0.164552	+	0.986368i	$0.552618\pi$
$420$		0			0
$421$	−12.9752	+	12.9752i	−0.632375	+	0.632375i	−0.948663	−	0.316288i	$-0.897563\pi$
$421$	−12.9752	+	12.9752i	−0.632375	+	0.632375i	0.316288	+	0.948663i	$0.397563\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.7210			−0.908102
$426$		0			0
$427$	5.05790	+	5.05790i	0.244769	+	0.244769i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.3953			−1.17508			−0.587540	−	0.809195i	$-0.699904\pi$
$431$	−24.3953			−1.17508			−0.587540	+	0.809195i	$0.699904\pi$
$432$		0			0
$433$	18.9149			0.908991			0.454496	−	0.890749i	$-0.349820\pi$
$433$	18.9149			0.908991			0.454496	+	0.890749i	$0.349820\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.29351	−	4.29351i	−0.205386	−	0.205386i
$438$		0			0
$439$	4.63471			0.221203			0.110601	−	0.993865i	$-0.464722\pi$
$439$	4.63471			0.221203			0.110601	+	0.993865i	$0.464722\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.736942	+	0.736942i	−0.0350132	+	0.0350132i	−0.724397	−	0.689383i	$-0.757882\pi$
$443$	−0.736942	+	0.736942i	−0.0350132	+	0.0350132i	0.689383	+	0.724397i	$0.257882\pi$
$444$		0			0
$445$	−15.0109	−	15.0109i	−0.711587	−	0.711587i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			24.0699i			1.13593i	0.823052	+	0.567965i	$0.192269\pi$
$449$			24.0699i			1.13593i	−0.823052	+	0.567965i	$0.807731\pi$
$450$		0			0
$451$	32.5456	−	32.5456i	1.53251	−	1.53251i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			10.5601i			0.495066i
$456$		0			0
$457$			16.6529i			0.778990i	0.921029	+	0.389495i	$0.127350\pi$
$457$			16.6529i			0.778990i	−0.921029	+	0.389495i	$0.872650\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.80113	+	6.80113i	−0.316760	+	0.316760i	−0.847521	−	0.530761i	$-0.821906\pi$
$461$	−6.80113	+	6.80113i	−0.316760	+	0.316760i	0.530761	+	0.847521i	$0.321906\pi$
$462$		0			0
$463$			24.3377i			1.13107i	0.824724	+	0.565535i	$0.191330\pi$
$463$			24.3377i			1.13107i	−0.824724	+	0.565535i	$0.808670\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.0585	−	14.0585i	−0.650550	−	0.650550i	0.302575	−	0.953126i	$-0.402154\pi$
$467$	−14.0585	−	14.0585i	−0.650550	−	0.650550i	−0.953126	+	0.302575i	$0.902154\pi$
$468$		0			0
$469$	−2.32750	+	2.32750i	−0.107474	+	0.107474i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	59.1393			2.71923
$474$		0			0
$475$	−3.34376	−	3.34376i	−0.153422	−	0.153422i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.776304			0.0354702			0.0177351	−	0.999843i	$-0.494354\pi$
$479$	0.776304			0.0354702			0.0177351	+	0.999843i	$0.494354\pi$
$480$		0			0
$481$	17.6124			0.803059
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.7505	−	10.7505i	−0.488154	−	0.488154i
$486$		0			0
$487$	12.9158			0.585270			0.292635	−	0.956224i	$-0.405468\pi$
$487$	12.9158			0.585270			0.292635	+	0.956224i	$0.405468\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.9134	−	20.9134i	0.943807	−	0.943807i	−0.0546960	−	0.998503i	$-0.517419\pi$
$491$	20.9134	−	20.9134i	0.943807	−	0.943807i	0.998503	+	0.0546960i	$0.0174190\pi$
$492$		0			0
$493$	16.2558	+	16.2558i	0.732126	+	0.732126i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.79177i			0.125228i
$498$		0			0
$499$	−9.84298	+	9.84298i	−0.440632	+	0.440632i	−0.892224	−	0.451592i	$-0.850856\pi$
$499$	−9.84298	+	9.84298i	−0.440632	+	0.440632i	0.451592	+	0.892224i	$0.350856\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			14.5229i			0.647546i	0.946135	+	0.323773i	$0.104951\pi$
$503$			14.5229i			0.647546i	−0.946135	+	0.323773i	$0.895049\pi$
$504$		0			0
$505$		−	13.6800i		−	0.608750i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.7528	−	17.7528i	0.786881	−	0.786881i	−0.194101	−	0.980982i	$-0.562179\pi$
$509$	17.7528	−	17.7528i	0.786881	−	0.786881i	0.980982	+	0.194101i	$0.0621788\pi$
$510$		0			0
$511$			4.79747i			0.212228i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.5769	+	28.5769i	1.25925	+	1.25925i
$516$		0			0
$517$	13.5460	−	13.5460i	0.595754	−	0.595754i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.2267			−1.19282			−0.596412	−	0.802679i	$-0.703408\pi$
$521$	−27.2267			−1.19282			−0.596412	+	0.802679i	$0.703408\pi$
$522$		0			0
$523$	−4.22218	−	4.22218i	−0.184623	−	0.184623i	0.608744	−	0.793367i	$-0.291674\pi$
$523$	−4.22218	−	4.22218i	−0.184623	−	0.184623i	−0.793367	+	0.608744i	$0.791674\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.07673			−0.308267
$528$		0			0
$529$	7.03175			0.305728
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−23.0729	−	23.0729i	−0.999400	−	0.999400i
$534$		0			0
$535$	50.4416			2.18078
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.69809	+	3.69809i	−0.159288	+	0.159288i
$540$		0			0
$541$	−24.4052	−	24.4052i	−1.04926	−	1.04926i	−0.998722	−	0.0505372i	$-0.983907\pi$
$541$	−24.4052	−	24.4052i	−1.04926	−	1.04926i	−0.0505372	−	0.998722i	$-0.516093\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	20.8880i		−	0.894744i
$546$		0			0
$547$	−23.9757	+	23.9757i	−1.02513	+	1.02513i	−0.0254514	+	0.999676i	$0.508102\pi$
$547$	−23.9757	+	23.9757i	−1.02513	+	1.02513i	−0.999676	+	0.0254514i	$0.991898\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			5.80691i			0.247382i
$552$		0			0
$553$		−	1.49360i		−	0.0635143i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.62057	−	4.62057i	0.195780	−	0.195780i	−0.602408	−	0.798188i	$-0.705792\pi$
$557$	4.62057	−	4.62057i	0.195780	−	0.195780i	0.798188	+	0.602408i	$0.205792\pi$
$558$		0			0
$559$		−	41.9264i		−	1.77330i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.7309	−	10.7309i	−0.452254	−	0.452254i	0.443848	−	0.896102i	$-0.353613\pi$
$563$	−10.7309	−	10.7309i	−0.452254	−	0.452254i	−0.896102	+	0.443848i	$0.853613\pi$
$564$		0			0
$565$	−33.1252	+	33.1252i	−1.39359	+	1.39359i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	39.3131			1.64809			0.824046	−	0.566523i	$-0.191712\pi$
$569$	39.3131			1.64809			0.824046	+	0.566523i	$0.191712\pi$
$570$		0			0
$571$	−10.5965	−	10.5965i	−0.443452	−	0.443452i	0.449719	−	0.893170i	$-0.351524\pi$
$571$	−10.5965	−	10.5965i	−0.443452	−	0.443452i	−0.893170	+	0.449719i	$0.851524\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.4360			−0.518616
$576$		0			0
$577$	14.9143			0.620891			0.310446	−	0.950591i	$-0.399522\pi$
$577$	14.9143			0.620891			0.310446	+	0.950591i	$0.399522\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.86731	+	5.86731i	0.243417	+	0.243417i
$582$		0			0
$583$	69.8488			2.89284
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.39838	+	2.39838i	−0.0989916	+	0.0989916i	−0.754868	−	0.655877i	$-0.772299\pi$
$587$	−2.39838	+	2.39838i	−0.0989916	+	0.0989916i	0.655877	+	0.754868i	$0.272299\pi$
$588$		0			0
$589$	−1.26397	−	1.26397i	−0.0520811	−	0.0520811i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	10.7453i		−	0.441255i	−0.975358	−	0.220627i	$-0.929190\pi$
$593$		−	10.7453i		−	0.441255i	0.975358	−	0.220627i	$-0.0708105\pi$
$594$		0			0
$595$	−12.1152	+	12.1152i	−0.496673	+	0.496673i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			15.1810i			0.620278i	0.950691	+	0.310139i	$0.100376\pi$
$599$			15.1810i			0.620278i	−0.950691	+	0.310139i	$0.899624\pi$
$600$		0			0
$601$		−	6.64451i		−	0.271035i	−0.990775	−	0.135518i	$-0.956730\pi$
$601$		−	6.64451i		−	0.271035i	0.990775	−	0.135518i	$-0.0432697\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	32.9317	−	32.9317i	1.33886	−	1.33886i
$606$		0			0
$607$		−	38.0756i		−	1.54544i	−0.634747	−	0.772720i	$-0.718896\pi$
$607$		−	38.0756i		−	1.54544i	0.634747	−	0.772720i	$-0.281104\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.60335	−	9.60335i	−0.388510	−	0.388510i
$612$		0			0
$613$	27.1964	−	27.1964i	1.09845	−	1.09845i	0.103862	−	0.994592i	$-0.466880\pi$
$613$	27.1964	−	27.1964i	1.09845	−	1.09845i	0.994592	−	0.103862i	$-0.0331199\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.20023			−0.330129			−0.165065	−	0.986283i	$-0.552783\pi$
$617$	−8.20023			−0.330129			−0.165065	+	0.986283i	$0.552783\pi$
$618$		0			0
$619$	12.5626	+	12.5626i	0.504932	+	0.504932i	0.912966	−	0.408035i	$-0.133786\pi$
$619$	12.5626	+	12.5626i	0.504932	+	0.504932i	−0.408035	+	0.912966i	$0.633786\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.45344			−0.298616
$624$		0			0
$625$	30.8754			1.23501
$626$		0			0
$627$		0			0
$628$		0			0
$629$	20.2060	+	20.2060i	0.805665	+	0.805665i
$630$		0			0
$631$	−32.4228			−1.29073			−0.645365	−	0.763874i	$-0.723295\pi$
$631$	−32.4228			−1.29073			−0.645365	+	0.763874i	$0.723295\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	34.8661	−	34.8661i	1.38362	−	1.38362i
$636$		0			0
$637$	2.62173	+	2.62173i	0.103877	+	0.103877i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			7.25637i			0.286609i	0.989679	+	0.143305i	$0.0457729\pi$
$641$			7.25637i			0.286609i	−0.989679	+	0.143305i	$0.954227\pi$
$642$		0			0
$643$	14.2505	−	14.2505i	0.561984	−	0.561984i	−0.367887	−	0.929871i	$-0.619919\pi$
$643$	14.2505	−	14.2505i	0.561984	−	0.561984i	0.929871	+	0.367887i	$0.119919\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	40.6733i		−	1.59903i	−0.600644	−	0.799517i	$-0.705089\pi$
$647$		−	40.6733i		−	1.59903i	0.600644	−	0.799517i	$-0.294911\pi$
$648$		0			0
$649$			24.3990i			0.957743i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.02778	+	5.02778i	−0.196752	+	0.196752i	−0.798606	−	0.601854i	$-0.794429\pi$
$653$	−5.02778	+	5.02778i	−0.196752	+	0.196752i	0.601854	+	0.798606i	$0.294429\pi$
$654$		0			0
$655$			48.9895i			1.91418i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.3313	+	11.3313i	0.441405	+	0.441405i	0.892484	−	0.451079i	$-0.148961\pi$
$659$	11.3313	+	11.3313i	0.441405	+	0.441405i	−0.451079	+	0.892484i	$0.648961\pi$
$660$		0			0
$661$	4.83358	−	4.83358i	0.188005	−	0.188005i	−0.606828	−	0.794833i	$-0.707558\pi$
$661$	4.83358	−	4.83358i	0.188005	−	0.188005i	0.794833	+	0.606828i	$0.207558\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.32777			−0.167824
$666$		0			0
$667$	10.7984	+	10.7984i	0.418116	+	0.418116i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−37.4091			−1.44416
$672$		0			0
$673$	−10.3238			−0.397952			−0.198976	−	0.980004i	$-0.563762\pi$
$673$	−10.3238			−0.397952			−0.198976	+	0.980004i	$0.563762\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.72554	−	9.72554i	−0.373783	−	0.373783i	0.495070	−	0.868853i	$-0.335142\pi$
$677$	−9.72554	−	9.72554i	−0.373783	−	0.373783i	−0.868853	+	0.495070i	$0.835142\pi$
$678$		0			0
$679$	−5.33798			−0.204853
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28.0743	+	28.0743i	−1.07423	+	1.07423i	−0.0772187	+	0.997014i	$0.524604\pi$
$683$	−28.0743	+	28.0743i	−1.07423	+	1.07423i	−0.997014	+	0.0772187i	$0.975396\pi$
$684$		0			0
$685$	28.6841	+	28.6841i	1.09596	+	1.09596i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	49.5188i		−	1.88651i
$690$		0			0
$691$	8.58607	−	8.58607i	0.326630	−	0.326630i	−0.524674	−	0.851303i	$-0.675813\pi$
$691$	8.58607	−	8.58607i	0.326630	−	0.326630i	0.851303	+	0.524674i	$0.175813\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	58.2670i		−	2.21019i
$696$		0			0
$697$		−	52.9411i		−	2.00529i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.04500	−	4.04500i	0.152778	−	0.152778i	−0.626580	−	0.779357i	$-0.715546\pi$
$701$	4.04500	−	4.04500i	0.152778	−	0.152778i	0.779357	+	0.626580i	$0.215546\pi$
$702$		0			0
$703$			7.21797i			0.272231i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.39628	−	3.39628i	−0.127730	−	0.127730i
$708$		0			0
$709$	−8.94418	+	8.94418i	−0.335906	+	0.335906i	−0.854824	−	0.518918i	$-0.826335\pi$
$709$	−8.94418	+	8.94418i	−0.335906	+	0.335906i	0.518918	+	0.854824i	$0.326335\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.70092			−0.176051
$714$		0			0
$715$	−39.0523	−	39.0523i	−1.46047	−	1.46047i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.3665			−0.386605			−0.193302	−	0.981139i	$-0.561920\pi$
$719$	−10.3665			−0.386605			−0.193302	+	0.981139i	$0.561920\pi$
$720$		0			0
$721$	14.1894			0.528440
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8.40974	+	8.40974i	0.312330	+	0.312330i
$726$		0			0
$727$	−25.7450			−0.954827			−0.477414	−	0.878679i	$-0.658426\pi$
$727$	−25.7450			−0.954827			−0.477414	+	0.878679i	$0.658426\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	48.1003	−	48.1003i	1.77905	−	1.77905i
$732$		0			0
$733$	−10.0104	−	10.0104i	−0.369742	−	0.369742i	0.497641	−	0.867383i	$-0.334200\pi$
$733$	−10.0104	−	10.0104i	−0.369742	−	0.369742i	−0.867383	+	0.497641i	$0.834200\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	17.2146i		−	0.634108i
$738$		0			0
$739$	0.288948	−	0.288948i	0.0106291	−	0.0106291i	−0.701772	−	0.712401i	$-0.747608\pi$
$739$	0.288948	−	0.288948i	0.0106291	−	0.0106291i	0.712401	+	0.701772i	$0.247608\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	8.12050i		−	0.297912i	−0.988844	−	0.148956i	$-0.952409\pi$
$743$		−	8.12050i		−	0.297912i	0.988844	−	0.148956i	$-0.0475913\pi$
$744$		0			0
$745$		−	2.53672i		−	0.0929383i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.5230	−	12.5230i	0.457580	−	0.457580i
$750$		0			0
$751$		−	21.7937i		−	0.795264i	−0.917545	−	0.397632i	$-0.869832\pi$
$751$		−	21.7937i		−	0.795264i	0.917545	−	0.397632i	$-0.130168\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.87302	+	4.87302i	0.177347	+	0.177347i
$756$		0			0
$757$	10.1176	−	10.1176i	0.367730	−	0.367730i	−0.498919	−	0.866649i	$-0.666269\pi$
$757$	10.1176	−	10.1176i	0.367730	−	0.367730i	0.866649	+	0.498919i	$0.166269\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26.8920			0.974836			0.487418	−	0.873169i	$-0.337939\pi$
$761$	26.8920			0.974836			0.487418	+	0.873169i	$0.337939\pi$
$762$		0			0
$763$	−5.18580	−	5.18580i	−0.187739	−	0.187739i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	17.2974			0.624575
$768$		0			0
$769$	22.2814			0.803488			0.401744	−	0.915752i	$-0.368404\pi$
$769$	22.2814			0.803488			0.401744	+	0.915752i	$0.368404\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.9647	−	12.9647i	−0.466308	−	0.466308i	0.434408	−	0.900716i	$-0.356958\pi$
$773$	−12.9647	−	12.9647i	−0.466308	−	0.466308i	−0.900716	+	0.434408i	$0.856958\pi$
$774$		0			0
$775$	−3.66105			−0.131509
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.45580	−	9.45580i	0.338789	−	0.338789i
$780$		0			0
$781$	−10.3242	−	10.3242i	−0.369430	−	0.369430i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	0.735618i		−	0.0262553i
$786$		0			0
$787$	11.9548	−	11.9548i	0.426143	−	0.426143i	−0.461169	−	0.887312i	$-0.652570\pi$
$787$	11.9548	−	11.9548i	0.426143	−	0.426143i	0.887312	+	0.461169i	$0.152570\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			16.4478i			0.584817i
$792$		0			0
$793$			26.5209i			0.941785i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.43442	+	7.43442i	−0.263341	+	0.263341i	−0.826410	−	0.563069i	$-0.809620\pi$
$797$	−7.43442	+	7.43442i	−0.263341	+	0.263341i	0.563069	+	0.826410i	$0.309620\pi$
$798$		0			0
$799$		−	22.0350i		−	0.779542i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.7415	−	17.7415i	−0.626083	−	0.626083i
$804$		0			0
$805$	−8.04785	+	8.04785i	−0.283649	+	0.283649i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.8955			−0.769804			−0.384902	−	0.922957i	$-0.625765\pi$
$809$	−21.8955			−0.769804			−0.384902	+	0.922957i	$0.625765\pi$
$810$		0			0
$811$	31.3446	+	31.3446i	1.10066	+	1.10066i	0.994331	+	0.106328i	$0.0339095\pi$
$811$	31.3446	+	31.3446i	1.10066	+	1.10066i	0.106328	+	0.994331i	$0.466091\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	34.1594			1.19655
$816$		0			0
$817$	17.1824			0.601135
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.31382	−	3.31382i	−0.115653	−	0.115653i	0.646912	−	0.762565i	$-0.276060\pi$
$821$	−3.31382	−	3.31382i	−0.115653	−	0.115653i	−0.762565	+	0.646912i	$0.776060\pi$
$822$		0			0
$823$	−12.4162			−0.432801			−0.216401	−	0.976305i	$-0.569432\pi$
$823$	−12.4162			−0.432801			−0.216401	+	0.976305i	$0.569432\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.7481	−	14.7481i	0.512843	−	0.512843i	−0.402554	−	0.915396i	$-0.631877\pi$
$827$	14.7481	−	14.7481i	0.512843	−	0.512843i	0.915396	+	0.402554i	$0.131877\pi$
$828$		0			0
$829$	14.1607	+	14.1607i	0.491820	+	0.491820i	0.908879	−	0.417059i	$-0.136939\pi$
$829$	14.1607	+	14.1607i	0.491820	+	0.491820i	−0.417059	+	0.908879i	$0.636939\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.01559i			0.208428i
$834$		0			0
$835$	−36.3812	+	36.3812i	−1.25902	+	1.25902i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	30.1447i		−	1.04071i	−0.853950	−	0.520356i	$-0.825799\pi$
$839$		−	30.1447i		−	1.04071i	0.853950	−	0.520356i	$-0.174201\pi$
$840$		0			0
$841$			14.3953i			0.496390i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.50429	+	1.50429i	−0.0517493	+	0.0517493i
$846$		0			0
$847$		−	16.3517i		−	0.561851i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	13.4224	+	13.4224i	0.460114	+	0.460114i
$852$		0			0
$853$	−22.7905	+	22.7905i	−0.780333	+	0.780333i	−0.979887	−	0.199554i	$-0.936051\pi$
$853$	−22.7905	+	22.7905i	−0.780333	+	0.780333i	0.199554	+	0.979887i	$0.436051\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.55823			0.189865			0.0949327	−	0.995484i	$-0.469736\pi$
$857$	5.55823			0.189865			0.0949327	+	0.995484i	$0.469736\pi$
$858$		0			0
$859$	−17.0395	−	17.0395i	−0.581380	−	0.581380i	0.353903	−	0.935282i	$-0.384854\pi$
$859$	−17.0395	−	17.0395i	−0.581380	−	0.581380i	−0.935282	+	0.353903i	$0.884854\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.4237			−0.729271			−0.364636	−	0.931150i	$-0.618806\pi$
$863$	−21.4237			−0.729271			−0.364636	+	0.931150i	$0.618806\pi$
$864$		0			0
$865$	−69.1619			−2.35158
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.52346	+	5.52346i	0.187370	+	0.187370i
$870$		0			0
$871$	−12.2041			−0.413522
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.80219	−	3.80219i	0.128538	−	0.128538i
$876$		0			0
$877$	2.28189	+	2.28189i	0.0770538	+	0.0770538i	0.744583	−	0.667530i	$-0.232648\pi$
$877$	2.28189	+	2.28189i	0.0770538	+	0.0770538i	−0.667530	+	0.744583i	$0.732648\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			52.5747i			1.77129i	0.464365	+	0.885644i	$0.346282\pi$
$881$			52.5747i			1.77129i	−0.464365	+	0.885644i	$0.653718\pi$
$882$		0			0
$883$	−15.6588	+	15.6588i	−0.526960	+	0.526960i	−0.919665	−	0.392704i	$-0.871540\pi$
$883$	−15.6588	+	15.6588i	−0.526960	+	0.526960i	0.392704	+	0.919665i	$0.371540\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	11.7979i		−	0.396133i	−0.980189	−	0.198067i	$-0.936534\pi$
$887$		−	11.7979i		−	0.396133i	0.980189	−	0.198067i	$-0.0634663\pi$
$888$		0			0
$889$		−	17.3122i		−	0.580632i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.93567	−	3.93567i	0.131702	−	0.131702i
$894$		0			0
$895$			23.0143i			0.769283i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.17897	+	3.17897i	0.106024	+	0.106024i
$900$		0			0
$901$	56.8107	−	56.8107i	1.89264	−	1.89264i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−69.6295			−2.31456
$906$		0			0
$907$	27.8901	+	27.8901i	0.926077	+	0.926077i	0.997450	−	0.0713727i	$-0.0227380\pi$
$907$	27.8901	+	27.8901i	0.926077	+	0.926077i	−0.0713727	+	0.997450i	$0.522738\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.7274			0.454809			0.227404	−	0.973800i	$-0.426976\pi$
$911$	13.7274			0.454809			0.227404	+	0.973800i	$0.426976\pi$
$912$		0			0
$913$	−43.3956			−1.43619
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.1625	+	12.1625i	0.401641	+	0.401641i
$918$		0			0
$919$	59.5299			1.96371			0.981855	−	0.189632i	$-0.0607295\pi$
$919$	59.5299			1.96371			0.981855	+	0.189632i	$0.0607295\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.31927	+	7.31927i	−0.240917	+	0.240917i
$924$		0			0
$925$	10.4533	+	10.4533i	0.343702	+	0.343702i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			51.7451i			1.69770i	0.528631	+	0.848851i	$0.322705\pi$
$929$			51.7451i			1.69770i	−0.528631	+	0.848851i	$0.677295\pi$
$930$		0			0
$931$	−1.07444	+	1.07444i	−0.0352134	+	0.0352134i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	89.6059i		−	2.93043i
$936$		0			0
$937$		−	12.3020i		−	0.401890i	−0.979603	−	0.200945i	$-0.935599\pi$
$937$		−	12.3020i		−	0.401890i	0.979603	−	0.200945i	$-0.0644012\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.1448	−	22.1448i	0.721899	−	0.721899i	−0.247093	−	0.968992i	$-0.579475\pi$
$941$	22.1448	−	22.1448i	0.721899	−	0.721899i	0.968992	+	0.247093i	$0.0794754\pi$
$942$		0			0
$943$		−	35.1677i		−	1.14522i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.1330	+	26.1330i	0.849207	+	0.849207i	0.990034	−	0.140827i	$-0.0449762\pi$
$947$	26.1330	+	26.1330i	0.849207	+	0.849207i	−0.140827	+	0.990034i	$0.544976\pi$
$948$		0			0
$949$	−12.5777	+	12.5777i	−0.408288	+	0.408288i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.26587			0.138185			0.0690926	−	0.997610i	$-0.477990\pi$
$953$	4.26587			0.138185			0.0690926	+	0.997610i	$0.477990\pi$
$954$		0			0
$955$	7.45503	+	7.45503i	0.241239	+	0.241239i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.2426			0.459918
$960$		0			0
$961$	29.6161			0.955358
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−15.2738	−	15.2738i	−0.491681	−	0.491681i
$966$		0			0
$967$	−27.1336			−0.872557			−0.436278	−	0.899812i	$-0.643704\pi$
$967$	−27.1336			−0.872557			−0.436278	+	0.899812i	$0.643704\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	29.9723	−	29.9723i	0.961858	−	0.961858i	−0.0374407	−	0.999299i	$-0.511921\pi$
$971$	29.9723	−	29.9723i	0.961858	−	0.961858i	0.999299	+	0.0374407i	$0.0119205\pi$
$972$		0			0
$973$	−14.4658	−	14.4658i	−0.463751	−	0.463751i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.00253i		−	0.0640666i	−0.999487	−	0.0320333i	$-0.989802\pi$
$977$		−	2.00253i		−	0.0640666i	0.999487	−	0.0320333i	$-0.0101983\pi$
$978$		0			0
$979$	27.5635	−	27.5635i	0.880933	−	0.880933i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	32.7842i		−	1.04565i	−0.852439	−	0.522827i	$-0.824877\pi$
$983$		−	32.7842i		−	1.04565i	0.852439	−	0.522827i	$-0.175123\pi$
$984$		0			0
$985$			75.7086i			2.41228i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.9520	−	31.9520i	1.01601	−	1.01601i
$990$		0			0
$991$		−	20.5096i		−	0.651508i	−0.945455	−	0.325754i	$-0.894382\pi$
$991$		−	20.5096i		−	0.651508i	0.945455	−	0.325754i	$-0.105618\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.1225	−	17.1225i	−0.542821	−	0.542821i
$996$		0			0
$997$	−4.89947	+	4.89947i	−0.155168	+	0.155168i	−0.780422	−	0.625254i	$-0.784996\pi$
$997$	−4.89947	+	4.89947i	−0.155168	+	0.155168i	0.625254	+	0.780422i	$0.284996\pi$
$998$		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.1583.15		36
3.2	odd	2	inner	4032.2.v.d.1583.4		36
4.3	odd	2		1008.2.v.d.323.15	yes	36
12.11	even	2		1008.2.v.d.323.4	✓	36
16.5	even	4		1008.2.v.d.827.4	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.4		36
48.5	odd	4		1008.2.v.d.827.15	yes	36
48.11	even	4	inner	4032.2.v.d.3599.15		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.4	✓	36	12.11	even	2
1008.2.v.d.323.15	yes	36	4.3	odd	2
1008.2.v.d.827.4	yes	36	16.5	even	4
1008.2.v.d.827.15	yes	36	48.5	odd	4
4032.2.v.d.1583.4		36	3.2	odd	2	inner
4032.2.v.d.1583.15		36	1.1	even	1	trivial
4032.2.v.d.3599.4		36	16.11	odd	4	inner
4032.2.v.d.3599.15		36	48.11	even	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)

\(f(q)\)	\(=\)	\(q+(2.01396 + 2.01396i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(2.01396 + 2.01396i) q^{5} +1.00000 q^{7} +(-3.69809 + 3.69809i) q^{11} +(2.62173 + 2.62173i) q^{13} +6.01559i q^{17} +(-1.07444 + 1.07444i) q^{19} +3.99603i q^{23} +3.11208i q^{25} +(2.70229 - 2.70229i) q^{29} +1.17640i q^{31} +(2.01396 + 2.01396i) q^{35} +(3.35894 - 3.35894i) q^{37} -8.80066 q^{41} +(-7.99594 - 7.99594i) q^{43} -3.66298 q^{47} +1.00000 q^{49} +(-9.44391 - 9.44391i) q^{53} -14.8956 q^{55} +(3.29886 - 3.29886i) q^{59} +(5.05790 + 5.05790i) q^{61} +10.5601i q^{65} +(-2.32750 + 2.32750i) q^{67} +2.79177i q^{71} +4.79747i q^{73} +(-3.69809 + 3.69809i) q^{77} -1.49360i q^{79} +(5.86731 + 5.86731i) q^{83} +(-12.1152 + 12.1152i) q^{85} -7.45344 q^{89} +(2.62173 + 2.62173i) q^{91} -4.32777 q^{95} -5.33798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 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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