LMFDB - Embedded newform 4032.2.v.d.1583.7

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			1583.7
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.871498 - 0.871498i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-0.871498 - 0.871498i) q^{5} +1.00000 q^{7} +(-0.987453 + 0.987453i) q^{11} +(0.526623 + 0.526623i) q^{13} +5.93491i q^{17} +(-2.78433 + 2.78433i) q^{19} -8.59031i q^{23} -3.48098i q^{25} +(-5.07293 + 5.07293i) q^{29} -7.72929i q^{31} +(-0.871498 - 0.871498i) q^{35} +(3.36904 - 3.36904i) q^{37} -4.55399 q^{41} +(-1.26872 - 1.26872i) q^{43} -5.02553 q^{47} +1.00000 q^{49} +(2.07649 + 2.07649i) q^{53} +1.72113 q^{55} +(6.52331 - 6.52331i) q^{59} +(8.74207 + 8.74207i) q^{61} -0.917901i q^{65} +(6.20569 - 6.20569i) q^{67} -1.72584i q^{71} -10.5554i q^{73} +(-0.987453 + 0.987453i) q^{77} -5.03627i q^{79} +(-10.2866 - 10.2866i) q^{83} +(5.17226 - 5.17226i) q^{85} -2.97331 q^{89} +(0.526623 + 0.526623i) q^{91} +4.85308 q^{95} -8.71046 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.871498	−	0.871498i	−0.389746	−	0.389746i	0.484851	−	0.874597i	$-0.338874\pi$
$5$	−0.871498	−	0.871498i	−0.389746	−	0.389746i	−0.874597	+	0.484851i	$0.838874\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.987453	+	0.987453i	−0.297728	+	0.297728i	−0.840123	−	0.542395i	$-0.817518\pi$
$11$	−0.987453	+	0.987453i	−0.297728	+	0.297728i	0.542395	+	0.840123i	$0.317518\pi$
$12$		0			0
$13$	0.526623	+	0.526623i	0.146059	+	0.146059i	0.776355	−	0.630296i	$-0.217066\pi$
$13$	0.526623	+	0.526623i	0.146059	+	0.146059i	−0.630296	+	0.776355i	$0.717066\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.93491i			1.43943i	0.694272	+	0.719713i	$0.255727\pi$
$17$			5.93491i			1.43943i	−0.694272	+	0.719713i	$0.744273\pi$
$18$		0			0
$19$	−2.78433	+	2.78433i	−0.638769	+	0.638769i	−0.950252	−	0.311483i	$-0.899174\pi$
$19$	−2.78433	+	2.78433i	−0.638769	+	0.638769i	0.311483	+	0.950252i	$0.399174\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.59031i		−	1.79120i	−0.444858	−	0.895601i	$-0.646746\pi$
$23$		−	8.59031i		−	1.79120i	0.444858	−	0.895601i	$-0.353254\pi$
$24$		0			0
$25$		−	3.48098i		−	0.696197i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.07293	+	5.07293i	−0.942019	+	0.942019i	−0.998409	−	0.0563901i	$-0.982041\pi$
$29$	−5.07293	+	5.07293i	−0.942019	+	0.942019i	0.0563901	+	0.998409i	$0.482041\pi$
$30$		0			0
$31$		−	7.72929i		−	1.38822i	−0.719868	−	0.694111i	$-0.755798\pi$
$31$		−	7.72929i		−	1.38822i	0.719868	−	0.694111i	$-0.244202\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.871498	−	0.871498i	−0.147310	−	0.147310i
$36$		0			0
$37$	3.36904	−	3.36904i	0.553866	−	0.553866i	−0.373688	−	0.927554i	$-0.621907\pi$
$37$	3.36904	−	3.36904i	0.553866	−	0.553866i	0.927554	+	0.373688i	$0.121907\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.55399			−0.711213			−0.355607	−	0.934636i	$-0.615726\pi$
$41$	−4.55399			−0.711213			−0.355607	+	0.934636i	$0.615726\pi$
$42$		0			0
$43$	−1.26872	−	1.26872i	−0.193478	−	0.193478i	0.603719	−	0.797197i	$-0.293685\pi$
$43$	−1.26872	−	1.26872i	−0.193478	−	0.193478i	−0.797197	+	0.603719i	$0.793685\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.02553			−0.733049			−0.366525	−	0.930408i	$-0.619452\pi$
$47$	−5.02553			−0.733049			−0.366525	+	0.930408i	$0.619452\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.07649	+	2.07649i	0.285228	+	0.285228i	0.835190	−	0.549962i	$-0.185358\pi$
$53$	2.07649	+	2.07649i	0.285228	+	0.285228i	−0.549962	+	0.835190i	$0.685358\pi$
$54$		0			0
$55$	1.72113			0.232077
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.52331	−	6.52331i	0.849263	−	0.849263i	−0.140779	−	0.990041i	$-0.544961\pi$
$59$	6.52331	−	6.52331i	0.849263	−	0.849263i	0.990041	+	0.140779i	$0.0449605\pi$
$60$		0			0
$61$	8.74207	+	8.74207i	1.11931	+	1.11931i	0.991843	+	0.127464i	$0.0406836\pi$
$61$	8.74207	+	8.74207i	1.11931	+	1.11931i	0.127464	+	0.991843i	$0.459316\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.917901i		−	0.113852i
$66$		0			0
$67$	6.20569	−	6.20569i	0.758146	−	0.758146i	−0.217839	−	0.975985i	$-0.569901\pi$
$67$	6.20569	−	6.20569i	0.758146	−	0.758146i	0.975985	+	0.217839i	$0.0699007\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.72584i		−	0.204819i	−0.994742	−	0.102410i	$-0.967345\pi$
$71$		−	1.72584i		−	0.204819i	0.994742	−	0.102410i	$-0.0326553\pi$
$72$		0			0
$73$		−	10.5554i		−	1.23541i	−0.786409	−	0.617707i	$-0.788062\pi$
$73$		−	10.5554i		−	1.23541i	0.786409	−	0.617707i	$-0.211938\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.987453	+	0.987453i	−0.112531	+	0.112531i
$78$		0			0
$79$		−	5.03627i		−	0.566625i	−0.959028	−	0.283312i	$-0.908567\pi$
$79$		−	5.03627i		−	0.566625i	0.959028	−	0.283312i	$-0.0914333\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.2866	−	10.2866i	−1.12910	−	1.12910i	−0.990323	−	0.138779i	$-0.955682\pi$
$83$	−10.2866	−	10.2866i	−1.12910	−	1.12910i	−0.138779	−	0.990323i	$-0.544318\pi$
$84$		0			0
$85$	5.17226	−	5.17226i	0.561010	−	0.561010i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.97331			−0.315171			−0.157585	−	0.987505i	$-0.550371\pi$
$89$	−2.97331			−0.315171			−0.157585	+	0.987505i	$0.550371\pi$
$90$		0			0
$91$	0.526623	+	0.526623i	0.0552050	+	0.0552050i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.85308			0.497915
$96$		0			0
$97$	−8.71046			−0.884414			−0.442207	−	0.896913i	$-0.645804\pi$
$97$	−8.71046			−0.884414			−0.442207	+	0.896913i	$0.645804\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.30460	+	6.30460i	0.627331	+	0.627331i	0.947396	−	0.320065i	$-0.103705\pi$
$101$	6.30460	+	6.30460i	0.627331	+	0.627331i	−0.320065	+	0.947396i	$0.603705\pi$
$102$		0			0
$103$	−11.2000			−1.10357			−0.551786	−	0.833986i	$-0.686053\pi$
$103$	−11.2000			−1.10357			−0.551786	+	0.833986i	$0.686053\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.2212	+	11.2212i	−1.08479	+	1.08479i	−0.0887361	+	0.996055i	$0.528283\pi$
$107$	−11.2212	+	11.2212i	−1.08479	+	1.08479i	−0.996055	+	0.0887361i	$0.971717\pi$
$108$		0			0
$109$	−6.91678	−	6.91678i	−0.662508	−	0.662508i	0.293463	−	0.955971i	$-0.405192\pi$
$109$	−6.91678	−	6.91678i	−0.662508	−	0.662508i	−0.955971	+	0.293463i	$0.905192\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.9402i		−	1.21731i	−0.793435	−	0.608655i	$-0.791709\pi$
$113$		−	12.9402i		−	1.21731i	0.793435	−	0.608655i	$-0.208291\pi$
$114$		0			0
$115$	−7.48643	+	7.48643i	−0.698113	+	0.698113i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.93491i			0.544052i
$120$		0			0
$121$			9.04987i			0.822716i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.39116	+	7.39116i	−0.661085	+	0.661085i
$126$		0			0
$127$		−	9.04469i		−	0.802587i	−0.915950	−	0.401293i	$-0.868561\pi$
$127$		−	9.04469i		−	0.802587i	0.915950	−	0.401293i	$-0.131439\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.77649	−	1.77649i	−0.155213	−	0.155213i	0.625229	−	0.780442i	$-0.285006\pi$
$131$	−1.77649	−	1.77649i	−0.155213	−	0.155213i	−0.780442	+	0.625229i	$0.785006\pi$
$132$		0			0
$133$	−2.78433	+	2.78433i	−0.241432	+	0.241432i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.7432			−0.917855			−0.458927	−	0.888474i	$-0.651766\pi$
$137$	−10.7432			−0.917855			−0.458927	+	0.888474i	$0.651766\pi$
$138$		0			0
$139$	−4.31730	−	4.31730i	−0.366188	−	0.366188i	0.499897	−	0.866085i	$-0.333371\pi$
$139$	−4.31730	−	4.31730i	−0.366188	−	0.366188i	−0.866085	+	0.499897i	$0.833371\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.04003			−0.0869717
$144$		0			0
$145$	8.84209			0.734295
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.6365	−	11.6365i	−0.953299	−	0.953299i	0.0456584	−	0.998957i	$-0.485461\pi$
$149$	−11.6365	−	11.6365i	−0.953299	−	0.953299i	−0.998957	+	0.0456584i	$0.985461\pi$
$150$		0			0
$151$	12.4402			1.01237			0.506186	−	0.862424i	$-0.331055\pi$
$151$	12.4402			1.01237			0.506186	+	0.862424i	$0.331055\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.73606	+	6.73606i	−0.541053	+	0.541053i
$156$		0			0
$157$	13.1511	+	13.1511i	1.04957	+	1.04957i	0.998705	+	0.0508693i	$0.0161992\pi$
$157$	13.1511	+	13.1511i	1.04957	+	1.04957i	0.0508693	+	0.998705i	$0.483801\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.59031i		−	0.677011i
$162$		0			0
$163$	−12.5847	+	12.5847i	−0.985710	+	0.985710i	−0.999899	−	0.0141894i	$-0.995483\pi$
$163$	−12.5847	+	12.5847i	−0.985710	+	0.985710i	0.0141894	+	0.999899i	$0.495483\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.48399i		−	0.424364i	−0.977230	−	0.212182i	$-0.931943\pi$
$167$		−	5.48399i		−	0.424364i	0.977230	−	0.212182i	$-0.0680569\pi$
$168$		0			0
$169$		−	12.4453i		−	0.957334i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.58954	+	2.58954i	−0.196879	+	0.196879i	−0.798661	−	0.601782i	$-0.794458\pi$
$173$	−2.58954	+	2.58954i	−0.196879	+	0.196879i	0.601782	+	0.798661i	$0.294458\pi$
$174$		0			0
$175$		−	3.48098i		−	0.263138i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.1871	−	14.1871i	−1.06039	−	1.06039i	−0.998055	−	0.0623366i	$-0.980145\pi$
$179$	−14.1871	−	14.1871i	−1.06039	−	1.06039i	−0.0623366	−	0.998055i	$-0.519855\pi$
$180$		0			0
$181$	6.26495	−	6.26495i	0.465670	−	0.465670i	−0.434838	−	0.900509i	$-0.643194\pi$
$181$	6.26495	−	6.26495i	0.465670	−	0.465670i	0.900509	+	0.434838i	$0.143194\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.87221			−0.431734
$186$		0			0
$187$	−5.86044	−	5.86044i	−0.428558	−	0.428558i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.527338			0.0381568			0.0190784	−	0.999818i	$-0.493927\pi$
$191$	0.527338			0.0381568			0.0190784	+	0.999818i	$0.493927\pi$
$192$		0			0
$193$	−12.8125			−0.922266			−0.461133	−	0.887331i	$-0.652557\pi$
$193$	−12.8125			−0.922266			−0.461133	+	0.887331i	$0.652557\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.4233	−	11.4233i	−0.813876	−	0.813876i	0.171337	−	0.985212i	$-0.445191\pi$
$197$	−11.4233	−	11.4233i	−0.813876	−	0.813876i	−0.985212	+	0.171337i	$0.945191\pi$
$198$		0			0
$199$	3.39879			0.240934			0.120467	−	0.992717i	$-0.461561\pi$
$199$	3.39879			0.240934			0.120467	+	0.992717i	$0.461561\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.07293	+	5.07293i	−0.356050	+	0.356050i
$204$		0			0
$205$	3.96879	+	3.96879i	0.277192	+	0.277192i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	5.49879i		−	0.380359i
$210$		0			0
$211$	−0.345207	+	0.345207i	−0.0237650	+	0.0237650i	−0.718889	−	0.695124i	$-0.755349\pi$
$211$	−0.345207	+	0.345207i	−0.0237650	+	0.0237650i	0.695124	+	0.718889i	$0.255349\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.21138i			0.150815i
$216$		0			0
$217$		−	7.72929i		−	0.524698i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.12546	+	3.12546i	−0.210241	+	0.210241i
$222$		0			0
$223$		−	16.7844i		−	1.12397i	−0.827147	−	0.561985i	$-0.810038\pi$
$223$		−	16.7844i		−	1.12397i	0.827147	−	0.561985i	$-0.189962\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.27544	−	2.27544i	−0.151026	−	0.151026i	0.627550	−	0.778576i	$-0.284058\pi$
$227$	−2.27544	−	2.27544i	−0.151026	−	0.151026i	−0.778576	+	0.627550i	$0.784058\pi$
$228$		0			0
$229$	19.0489	−	19.0489i	1.25879	−	1.25879i	0.307115	−	0.951672i	$-0.400636\pi$
$229$	19.0489	−	19.0489i	1.25879	−	1.25879i	0.951672	−	0.307115i	$-0.0993637\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.6439			0.697307			0.348653	−	0.937252i	$-0.386639\pi$
$233$	10.6439			0.697307			0.348653	+	0.937252i	$0.386639\pi$
$234$		0			0
$235$	4.37974	+	4.37974i	0.285703	+	0.285703i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.40150			−0.220024			−0.110012	−	0.993930i	$-0.535089\pi$
$239$	−3.40150			−0.220024			−0.110012	+	0.993930i	$0.535089\pi$
$240$		0			0
$241$	−4.68045			−0.301494			−0.150747	−	0.988572i	$-0.548168\pi$
$241$	−4.68045			−0.301494			−0.150747	+	0.988572i	$0.548168\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.871498	−	0.871498i	−0.0556779	−	0.0556779i
$246$		0			0
$247$	−2.93258			−0.186596
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.96396	−	7.96396i	0.502681	−	0.502681i	−0.409589	−	0.912270i	$-0.634328\pi$
$251$	7.96396	−	7.96396i	0.502681	−	0.502681i	0.912270	+	0.409589i	$0.134328\pi$
$252$		0			0
$253$	8.48252	+	8.48252i	0.533292	+	0.533292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			10.7921i			0.673192i	0.941649	+	0.336596i	$0.109276\pi$
$257$			10.7921i			0.673192i	−0.941649	+	0.336596i	$0.890724\pi$
$258$		0			0
$259$	3.36904	−	3.36904i	0.209342	−	0.209342i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1.58657i			0.0978322i	0.998803	+	0.0489161i	$0.0155767\pi$
$263$			1.58657i			0.0978322i	−0.998803	+	0.0489161i	$0.984423\pi$
$264$		0			0
$265$		−	3.61932i		−	0.222333i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.09723	+	4.09723i	−0.249812	+	0.249812i	−0.820894	−	0.571081i	$-0.806524\pi$
$269$	−4.09723	+	4.09723i	−0.249812	+	0.249812i	0.571081	+	0.820894i	$0.306524\pi$
$270$		0			0
$271$		−	27.3889i		−	1.66376i	−0.554955	−	0.831880i	$-0.687265\pi$
$271$		−	27.3889i		−	1.66376i	0.554955	−	0.831880i	$-0.312735\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.43731	+	3.43731i	0.207277	+	0.207277i
$276$		0			0
$277$	8.44268	−	8.44268i	0.507272	−	0.507272i	−0.406416	−	0.913688i	$-0.633222\pi$
$277$	8.44268	−	8.44268i	0.507272	−	0.507272i	0.913688	+	0.406416i	$0.133222\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.27651			−0.255115			−0.127557	−	0.991831i	$-0.540714\pi$
$281$	−4.27651			−0.255115			−0.127557	+	0.991831i	$0.540714\pi$
$282$		0			0
$283$	11.9857	+	11.9857i	0.712474	+	0.712474i	0.967052	−	0.254578i	$-0.0819367\pi$
$283$	11.9857	+	11.9857i	0.712474	+	0.712474i	−0.254578	+	0.967052i	$0.581937\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.55399			−0.268813
$288$		0			0
$289$	−18.2231			−1.07195
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.97067	+	7.97067i	0.465652	+	0.465652i	0.900502	−	0.434851i	$-0.143199\pi$
$293$	7.97067	+	7.97067i	0.465652	+	0.465652i	−0.434851	+	0.900502i	$0.643199\pi$
$294$		0			0
$295$	−11.3701			−0.661993
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.52385	−	4.52385i	0.261621	−	0.261621i
$300$		0			0
$301$	−1.26872	−	1.26872i	−0.0731280	−	0.0731280i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	15.2374i		−	0.872490i
$306$		0			0
$307$	−8.28870	+	8.28870i	−0.473061	+	0.473061i	−0.902904	−	0.429843i	$-0.858569\pi$
$307$	−8.28870	+	8.28870i	−0.473061	+	0.473061i	0.429843	+	0.902904i	$0.358569\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.0334i			0.739058i	0.929219	+	0.369529i	$0.120481\pi$
$311$			13.0334i			0.739058i	−0.929219	+	0.369529i	$0.879519\pi$
$312$		0			0
$313$		−	7.66531i		−	0.433269i	−0.976253	−	0.216635i	$-0.930492\pi$
$313$		−	7.66531i		−	0.433269i	0.976253	−	0.216635i	$-0.0695080\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.392592	+	0.392592i	−0.0220502	+	0.0220502i	−0.718046	−	0.695996i	$-0.754963\pi$
$317$	−0.392592	+	0.392592i	−0.0220502	+	0.0220502i	0.695996	+	0.718046i	$0.254963\pi$
$318$		0			0
$319$		−	10.0186i		−	0.560931i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.5247	−	16.5247i	−0.919461	−	0.919461i
$324$		0			0
$325$	1.83316	−	1.83316i	0.101686	−	0.101686i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.02553			−0.277067
$330$		0			0
$331$	−2.74121	−	2.74121i	−0.150671	−	0.150671i	0.627747	−	0.778418i	$-0.283977\pi$
$331$	−2.74121	−	2.74121i	−0.150671	−	0.150671i	−0.778418	+	0.627747i	$0.783977\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.8165			−0.590968
$336$		0			0
$337$	−5.04588			−0.274867			−0.137433	−	0.990511i	$-0.543885\pi$
$337$	−5.04588			−0.274867			−0.137433	+	0.990511i	$0.543885\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.63231	+	7.63231i	0.413313	+	0.413313i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.5419	+	19.5419i	−1.04906	+	1.04906i	−0.0503322	+	0.998733i	$0.516028\pi$
$347$	−19.5419	+	19.5419i	−1.04906	+	1.04906i	−0.998733	+	0.0503322i	$0.983972\pi$
$348$		0			0
$349$	−14.5817	−	14.5817i	−0.780543	−	0.780543i	0.199380	−	0.979922i	$-0.436107\pi$
$349$	−14.5817	−	14.5817i	−0.780543	−	0.780543i	−0.979922	+	0.199380i	$0.936107\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	30.7047i		−	1.63425i	−0.576464	−	0.817123i	$-0.695568\pi$
$353$		−	30.7047i		−	1.63425i	0.576464	−	0.817123i	$-0.304432\pi$
$354$		0			0
$355$	−1.50406	+	1.50406i	−0.0798275	+	0.0798275i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			28.8465i			1.52246i	0.648482	+	0.761230i	$0.275404\pi$
$359$			28.8465i			1.52246i	−0.648482	+	0.761230i	$0.724596\pi$
$360$		0			0
$361$			3.49500i			0.183948i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.19898	+	9.19898i	−0.481497	+	0.481497i
$366$		0			0
$367$			13.7698i			0.718778i	0.933188	+	0.359389i	$0.117015\pi$
$367$			13.7698i			0.718778i	−0.933188	+	0.359389i	$0.882985\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.07649	+	2.07649i	0.107806	+	0.107806i
$372$		0			0
$373$	−15.3320	+	15.3320i	−0.793863	+	0.793863i	−0.982120	−	0.188257i	$-0.939716\pi$
$373$	−15.3320	+	15.3320i	−0.793863	+	0.793863i	0.188257	+	0.982120i	$0.439716\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.34303			−0.275180
$378$		0			0
$379$	5.42375	+	5.42375i	0.278599	+	0.278599i	0.832550	−	0.553950i	$-0.186880\pi$
$379$	5.42375	+	5.42375i	0.278599	+	0.278599i	−0.553950	+	0.832550i	$0.686880\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	29.1472			1.48935			0.744677	−	0.667426i	$-0.232604\pi$
$383$	29.1472			1.48935			0.744677	+	0.667426i	$0.232604\pi$
$384$		0			0
$385$	1.72113			0.0877167
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.4964	−	22.4964i	−1.14061	−	1.14061i	−0.988338	−	0.152273i	$-0.951341\pi$
$389$	−22.4964	−	22.4964i	−1.14061	−	1.14061i	−0.152273	−	0.988338i	$-0.548659\pi$
$390$		0			0
$391$	50.9827			2.57830
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.38910	+	4.38910i	−0.220839	+	0.220839i
$396$		0			0
$397$	11.9044	+	11.9044i	0.597464	+	0.597464i	0.939637	−	0.342173i	$-0.111163\pi$
$397$	11.9044	+	11.9044i	0.597464	+	0.597464i	−0.342173	+	0.939637i	$0.611163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	37.2464i		−	1.86000i	−0.367563	−	0.929999i	$-0.619808\pi$
$401$		−	37.2464i		−	1.86000i	0.367563	−	0.929999i	$-0.380192\pi$
$402$		0			0
$403$	4.07042	−	4.07042i	0.202762	−	0.202762i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.65353i			0.329803i
$408$		0			0
$409$			31.1673i			1.54112i	0.637365	+	0.770562i	$0.280024\pi$
$409$			31.1673i			1.54112i	−0.637365	+	0.770562i	$0.719976\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.52331	−	6.52331i	0.320991	−	0.320991i
$414$		0			0
$415$			17.9295i			0.880125i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.79721	−	8.79721i	−0.429772	−	0.429772i	0.458779	−	0.888551i	$-0.348287\pi$
$419$	−8.79721	−	8.79721i	−0.429772	−	0.429772i	−0.888551	+	0.458779i	$0.848287\pi$
$420$		0			0
$421$	−6.51667	+	6.51667i	−0.317603	+	0.317603i	−0.847846	−	0.530243i	$-0.822101\pi$
$421$	−6.51667	+	6.51667i	−0.317603	+	0.317603i	0.530243	+	0.847846i	$0.322101\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	20.6593			1.00212
$426$		0			0
$427$	8.74207	+	8.74207i	0.423058	+	0.423058i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.0668			−0.533070			−0.266535	−	0.963825i	$-0.585879\pi$
$431$	−11.0668			−0.533070			−0.266535	+	0.963825i	$0.585879\pi$
$432$		0			0
$433$	−19.1187			−0.918788			−0.459394	−	0.888233i	$-0.651933\pi$
$433$	−19.1187			−0.918788			−0.459394	+	0.888233i	$0.651933\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	23.9183	+	23.9183i	1.14417	+	1.14417i
$438$		0			0
$439$	−11.1926			−0.534192			−0.267096	−	0.963670i	$-0.586064\pi$
$439$	−11.1926			−0.534192			−0.267096	+	0.963670i	$0.586064\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.01173	+	7.01173i	−0.333137	+	0.333137i	−0.853777	−	0.520639i	$-0.825694\pi$
$443$	−7.01173	+	7.01173i	−0.333137	+	0.333137i	0.520639	+	0.853777i	$0.325694\pi$
$444$		0			0
$445$	2.59124	+	2.59124i	0.122836	+	0.122836i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			13.1013i			0.618288i	0.951015	+	0.309144i	$0.100042\pi$
$449$			13.1013i			0.618288i	−0.951015	+	0.309144i	$0.899958\pi$
$450$		0			0
$451$	4.49685	−	4.49685i	0.211748	−	0.211748i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.917901i		−	0.0430318i
$456$		0			0
$457$		−	39.7967i		−	1.86161i	−0.365516	−	0.930805i	$-0.619107\pi$
$457$		−	39.7967i		−	1.86161i	0.365516	−	0.930805i	$-0.380893\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.4401	+	20.4401i	−0.951989	+	0.951989i	−0.998899	−	0.0469105i	$-0.985062\pi$
$461$	−20.4401	+	20.4401i	−0.951989	+	0.951989i	0.0469105	+	0.998899i	$0.485062\pi$
$462$		0			0
$463$		−	26.1475i		−	1.21518i	−0.794252	−	0.607588i	$-0.792137\pi$
$463$		−	26.1475i		−	1.21518i	0.794252	−	0.607588i	$-0.207863\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0619	−	12.0619i	−0.558158	−	0.558158i	0.370625	−	0.928783i	$-0.379144\pi$
$467$	−12.0619	−	12.0619i	−0.558158	−	0.558158i	−0.928783	+	0.370625i	$0.879144\pi$
$468$		0			0
$469$	6.20569	−	6.20569i	0.286552	−	0.286552i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.50561			0.115208
$474$		0			0
$475$	9.69221	+	9.69221i	0.444709	+	0.444709i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	41.5551			1.89870			0.949351	−	0.314218i	$-0.101742\pi$
$479$	41.5551			1.89870			0.949351	+	0.314218i	$0.101742\pi$
$480$		0			0
$481$	3.54842			0.161794
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.59115	+	7.59115i	0.344696	+	0.344696i
$486$		0			0
$487$	39.1199			1.77269			0.886345	−	0.463025i	$-0.153236\pi$
$487$	39.1199			1.77269			0.886345	+	0.463025i	$0.153236\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.23385	+	4.23385i	−0.191071	+	0.191071i	−0.796159	−	0.605088i	$-0.793138\pi$
$491$	−4.23385	+	4.23385i	−0.191071	+	0.191071i	0.605088	+	0.796159i	$0.293138\pi$
$492$		0			0
$493$	−30.1073	−	30.1073i	−1.35597	−	1.35597i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1.72584i		−	0.0774145i
$498$		0			0
$499$	16.1500	−	16.1500i	0.722974	−	0.722974i	−0.246236	−	0.969210i	$-0.579194\pi$
$499$	16.1500	−	16.1500i	0.722974	−	0.722974i	0.969210	+	0.246236i	$0.0791938\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.83843i		−	0.215735i	−0.994165	−	0.107867i	$-0.965598\pi$
$503$		−	4.83843i		−	0.215735i	0.994165	−	0.107867i	$-0.0344022\pi$
$504$		0			0
$505$		−	10.9889i		−	0.488999i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.0539	+	12.0539i	−0.534278	+	0.534278i	−0.921842	−	0.387565i	$-0.873316\pi$
$509$	−12.0539	+	12.0539i	−0.534278	+	0.534278i	0.387565	+	0.921842i	$0.373316\pi$
$510$		0			0
$511$		−	10.5554i		−	0.466942i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.76079	+	9.76079i	0.430112	+	0.430112i
$516$		0			0
$517$	4.96248	−	4.96248i	0.218250	−	0.218250i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	44.2568			1.93893			0.969463	−	0.245236i	$-0.0788654\pi$
$521$	44.2568			1.93893			0.969463	+	0.245236i	$0.0788654\pi$
$522$		0			0
$523$	9.45506	+	9.45506i	0.413441	+	0.413441i	0.882935	−	0.469495i	$-0.155564\pi$
$523$	9.45506	+	9.45506i	0.413441	+	0.413441i	−0.469495	+	0.882935i	$0.655564\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	45.8726			1.99824
$528$		0			0
$529$	−50.7934			−2.20841
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.39823	−	2.39823i	−0.103879	−	0.103879i
$534$		0			0
$535$	19.5584			0.845585
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.987453	+	0.987453i	−0.0425326	+	0.0425326i
$540$		0			0
$541$	10.0695	+	10.0695i	0.432922	+	0.432922i	0.889621	−	0.456699i	$-0.150969\pi$
$541$	10.0695	+	10.0695i	0.432922	+	0.432922i	−0.456699	+	0.889621i	$0.650969\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			12.0559i			0.516419i
$546$		0			0
$547$	−16.7984	+	16.7984i	−0.718248	+	0.718248i	−0.968246	−	0.249998i	$-0.919570\pi$
$547$	−16.7984	+	16.7984i	−0.718248	+	0.718248i	0.249998	+	0.968246i	$0.419570\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	28.2494i		−	1.20347i
$552$		0			0
$553$		−	5.03627i		−	0.214164i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.91690	+	4.91690i	−0.208336	+	0.208336i	−0.803560	−	0.595224i	$-0.797063\pi$
$557$	−4.91690	+	4.91690i	−0.208336	+	0.208336i	0.595224	+	0.803560i	$0.297063\pi$
$558$		0			0
$559$		−	1.33628i		−	0.0565184i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.2939	+	19.2939i	0.813141	+	0.813141i	0.985103	−	0.171963i	$-0.0550108\pi$
$563$	19.2939	+	19.2939i	0.813141	+	0.813141i	−0.171963	+	0.985103i	$0.555011\pi$
$564$		0			0
$565$	−11.2773	+	11.2773i	−0.474441	+	0.474441i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.2625			1.22675			0.613375	−	0.789792i	$-0.289812\pi$
$569$	29.2625			1.22675			0.613375	+	0.789792i	$0.289812\pi$
$570$		0			0
$571$	−3.85993	−	3.85993i	−0.161533	−	0.161533i	0.621712	−	0.783246i	$-0.286437\pi$
$571$	−3.85993	−	3.85993i	−0.161533	−	0.161533i	−0.783246	+	0.621712i	$0.786437\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−29.9027			−1.24703
$576$		0			0
$577$	−36.5617			−1.52208			−0.761041	−	0.648704i	$-0.775311\pi$
$577$	−36.5617			−1.52208			−0.761041	+	0.648704i	$0.775311\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.2866	−	10.2866i	−0.426761	−	0.426761i
$582$		0			0
$583$	−4.10088			−0.169841
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.2902	−	27.2902i	1.12639	−	1.12639i	0.135626	−	0.990760i	$-0.456695\pi$
$587$	27.2902	−	27.2902i	1.12639	−	1.12639i	0.990760	−	0.135626i	$-0.0433047\pi$
$588$		0			0
$589$	21.5209	+	21.5209i	0.886753	+	0.886753i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			10.8459i			0.445388i	0.974888	+	0.222694i	$0.0714851\pi$
$593$			10.8459i			0.445388i	−0.974888	+	0.222694i	$0.928515\pi$
$594$		0			0
$595$	5.17226	−	5.17226i	0.212042	−	0.212042i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3.64194i			0.148806i	0.997228	+	0.0744028i	$0.0237051\pi$
$599$			3.64194i			0.148806i	−0.997228	+	0.0744028i	$0.976295\pi$
$600$		0			0
$601$			32.3789i			1.32076i	0.750929	+	0.660382i	$0.229606\pi$
$601$			32.3789i			1.32076i	−0.750929	+	0.660382i	$0.770394\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.88694	−	7.88694i	0.320650	−	0.320650i
$606$		0			0
$607$			18.0772i			0.733732i	0.930274	+	0.366866i	$0.119569\pi$
$607$			18.0772i			0.733732i	−0.930274	+	0.366866i	$0.880431\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.64656	−	2.64656i	−0.107068	−	0.107068i
$612$		0			0
$613$	16.9465	−	16.9465i	0.684462	−	0.684462i	−0.276540	−	0.961002i	$-0.589188\pi$
$613$	16.9465	−	16.9465i	0.684462	−	0.684462i	0.961002	+	0.276540i	$0.0891880\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.4985			−0.905755			−0.452877	−	0.891573i	$-0.649602\pi$
$617$	−22.4985			−0.905755			−0.452877	+	0.891573i	$0.649602\pi$
$618$		0			0
$619$	11.5591	+	11.5591i	0.464600	+	0.464600i	0.900160	−	0.435560i	$-0.143450\pi$
$619$	11.5591	+	11.5591i	0.464600	+	0.464600i	−0.435560	+	0.900160i	$0.643450\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.97331			−0.119123
$624$		0			0
$625$	−4.52217			−0.180887
$626$		0			0
$627$		0			0
$628$		0			0
$629$	19.9949	+	19.9949i	0.797249	+	0.797249i
$630$		0			0
$631$	−31.1111			−1.23851			−0.619257	−	0.785189i	$-0.712566\pi$
$631$	−31.1111			−1.23851			−0.619257	+	0.785189i	$0.712566\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.88243	+	7.88243i	−0.312805	+	0.312805i
$636$		0			0
$637$	0.526623	+	0.526623i	0.0208655	+	0.0208655i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			22.5963i			0.892499i	0.894909	+	0.446249i	$0.147241\pi$
$641$			22.5963i			0.892499i	−0.894909	+	0.446249i	$0.852759\pi$
$642$		0			0
$643$	−11.9048	+	11.9048i	−0.469479	+	0.469479i	−0.901746	−	0.432266i	$-0.857714\pi$
$643$	−11.9048	+	11.9048i	−0.469479	+	0.469479i	0.432266	+	0.901746i	$0.357714\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			19.3086i			0.759098i	0.925172	+	0.379549i	$0.123921\pi$
$647$			19.3086i			0.759098i	−0.925172	+	0.379549i	$0.876079\pi$
$648$		0			0
$649$			12.8829i			0.505699i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.2618	−	22.2618i	0.871172	−	0.871172i	−0.121429	−	0.992600i	$-0.538748\pi$
$653$	22.2618	−	22.2618i	0.871172	−	0.871172i	0.992600	+	0.121429i	$0.0387476\pi$
$654$		0			0
$655$			3.09642i			0.120987i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.80332	+	8.80332i	0.342929	+	0.342929i	0.857467	−	0.514539i	$-0.172037\pi$
$659$	8.80332	+	8.80332i	0.342929	+	0.342929i	−0.514539	+	0.857467i	$0.672037\pi$
$660$		0			0
$661$	−32.2956	+	32.2956i	−1.25615	+	1.25615i	−0.303237	+	0.952915i	$0.598067\pi$
$661$	−32.2956	+	32.2956i	−1.25615	+	1.25615i	−0.952915	+	0.303237i	$0.901933\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.85308			0.188194
$666$		0			0
$667$	43.5780	+	43.5780i	1.68735	+	1.68735i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.2648			−0.666498
$672$		0			0
$673$	36.4035			1.40325			0.701627	−	0.712545i	$-0.252457\pi$
$673$	36.4035			1.40325			0.701627	+	0.712545i	$0.252457\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.3085	−	12.3085i	−0.473054	−	0.473054i	0.429848	−	0.902901i	$-0.358567\pi$
$677$	−12.3085	−	12.3085i	−0.473054	−	0.473054i	−0.902901	+	0.429848i	$0.858567\pi$
$678$		0			0
$679$	−8.71046			−0.334277
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.66290	+	2.66290i	−0.101893	+	0.101893i	−0.756216	−	0.654323i	$-0.772954\pi$
$683$	−2.66290	+	2.66290i	−0.101893	+	0.101893i	0.654323	+	0.756216i	$0.272954\pi$
$684$		0			0
$685$	9.36268	+	9.36268i	0.357730	+	0.357730i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2.18706i			0.0833203i
$690$		0			0
$691$	−36.4998	+	36.4998i	−1.38852	+	1.38852i	−0.560075	+	0.828442i	$0.689228\pi$
$691$	−36.4998	+	36.4998i	−1.38852	+	1.38852i	−0.828442	+	0.560075i	$0.810772\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.52502i			0.285440i
$696$		0			0
$697$		−	27.0275i		−	1.02374i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.2621	−	18.2621i	0.689751	−	0.689751i	−0.272426	−	0.962177i	$-0.587826\pi$
$701$	18.2621	−	18.2621i	0.689751	−	0.689751i	0.962177	+	0.272426i	$0.0878259\pi$
$702$		0			0
$703$			18.7610i			0.707585i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.30460	+	6.30460i	0.237109	+	0.237109i
$708$		0			0
$709$	−29.4673	+	29.4673i	−1.10667	+	1.10667i	−0.113082	+	0.993586i	$0.536072\pi$
$709$	−29.4673	+	29.4673i	−1.10667	+	1.10667i	−0.993586	+	0.113082i	$0.963928\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−66.3970			−2.48659
$714$		0			0
$715$	0.906383	+	0.906383i	0.0338968	+	0.0338968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	28.5122			1.06333			0.531664	−	0.846956i	$-0.321567\pi$
$719$	28.5122			1.06333			0.531664	+	0.846956i	$0.321567\pi$
$720$		0			0
$721$	−11.2000			−0.417111
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17.6588	+	17.6588i	0.655830	+	0.655830i
$726$		0			0
$727$	−46.9459			−1.74113			−0.870564	−	0.492056i	$-0.836246\pi$
$727$	−46.9459			−1.74113			−0.870564	+	0.492056i	$0.836246\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.52975	−	7.52975i	0.278498	−	0.278498i
$732$		0			0
$733$	24.2876	+	24.2876i	0.897081	+	0.897081i	0.995177	−	0.0980958i	$-0.0312751\pi$
$733$	24.2876	+	24.2876i	0.897081	+	0.897081i	−0.0980958	+	0.995177i	$0.531275\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.2557i			0.451443i
$738$		0			0
$739$	−17.4642	+	17.4642i	−0.642433	+	0.642433i	−0.951153	−	0.308720i	$-0.900099\pi$
$739$	−17.4642	+	17.4642i	−0.642433	+	0.642433i	0.308720	+	0.951153i	$0.400099\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40.4442i		−	1.48375i	−0.670536	−	0.741877i	$-0.733936\pi$
$743$		−	40.4442i		−	1.48375i	0.670536	−	0.741877i	$-0.266064\pi$
$744$		0			0
$745$			20.2824i			0.743088i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.2212	+	11.2212i	−0.410013	+	0.410013i
$750$		0			0
$751$		−	28.7651i		−	1.04965i	−0.851209	−	0.524826i	$-0.824130\pi$
$751$		−	28.7651i		−	1.04965i	0.851209	−	0.524826i	$-0.175870\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.8416	−	10.8416i	−0.394567	−	0.394567i
$756$		0			0
$757$	19.0609	−	19.0609i	0.692779	−	0.692779i	−0.270064	−	0.962842i	$-0.587045\pi$
$757$	19.0609	−	19.0609i	0.692779	−	0.692779i	0.962842	+	0.270064i	$0.0870447\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.23558			0.262289			0.131145	−	0.991363i	$-0.458135\pi$
$761$	7.23558			0.262289			0.131145	+	0.991363i	$0.458135\pi$
$762$		0			0
$763$	−6.91678	−	6.91678i	−0.250404	−	0.250404i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.87064			0.248085
$768$		0			0
$769$	27.3510			0.986301			0.493151	−	0.869944i	$-0.335845\pi$
$769$	27.3510			0.986301			0.493151	+	0.869944i	$0.335845\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.9872	+	10.9872i	0.395182	+	0.395182i	0.876530	−	0.481348i	$-0.159853\pi$
$773$	10.9872	+	10.9872i	0.395182	+	0.395182i	−0.481348	+	0.876530i	$0.659853\pi$
$774$		0			0
$775$	−26.9055			−0.966475
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.6798	−	12.6798i	0.454301	−	0.454301i
$780$		0			0
$781$	1.70419	+	1.70419i	0.0609805	+	0.0609805i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	22.9224i		−	0.818134i
$786$		0			0
$787$	−15.8443	+	15.8443i	−0.564789	+	0.564789i	−0.930664	−	0.365875i	$-0.880770\pi$
$787$	−15.8443	+	15.8443i	−0.564789	+	0.564789i	0.365875	+	0.930664i	$0.380770\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	12.9402i		−	0.460100i
$792$		0			0
$793$			9.20754i			0.326969i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	28.7308	−	28.7308i	1.01770	−	1.01770i	0.0178574	−	0.999841i	$-0.494316\pi$
$797$	28.7308	−	28.7308i	1.01770	−	1.01770i	0.999841	−	0.0178574i	$-0.00568448\pi$
$798$		0			0
$799$		−	29.8261i		−	1.05517i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.4229	+	10.4229i	0.367817	+	0.367817i
$804$		0			0
$805$	−7.48643	+	7.48643i	−0.263862	+	0.263862i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.56022			0.160329			0.0801644	−	0.996782i	$-0.474455\pi$
$809$	4.56022			0.160329			0.0801644	+	0.996782i	$0.474455\pi$
$810$		0			0
$811$	−34.1196	−	34.1196i	−1.19810	−	1.19810i	−0.974734	−	0.223368i	$-0.928295\pi$
$811$	−34.1196	−	34.1196i	−1.19810	−	1.19810i	−0.223368	−	0.974734i	$-0.571705\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.9351			0.768352
$816$		0			0
$817$	7.06509			0.247176
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.3174	+	11.3174i	0.394980	+	0.394980i	0.876458	−	0.481478i	$-0.159900\pi$
$821$	11.3174	+	11.3174i	0.394980	+	0.394980i	−0.481478	+	0.876458i	$0.659900\pi$
$822$		0			0
$823$	40.5648			1.41400			0.707000	−	0.707213i	$-0.250048\pi$
$823$	40.5648			1.41400			0.707000	+	0.707213i	$0.250048\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.72273	−	6.72273i	0.233772	−	0.233772i	−0.580493	−	0.814265i	$-0.697140\pi$
$827$	6.72273	−	6.72273i	0.233772	−	0.233772i	0.814265	+	0.580493i	$0.197140\pi$
$828$		0			0
$829$	26.1731	+	26.1731i	0.909030	+	0.909030i	0.996194	−	0.0871639i	$-0.0277804\pi$
$829$	26.1731	+	26.1731i	0.909030	+	0.909030i	−0.0871639	+	0.996194i	$0.527780\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.93491i			0.205632i
$834$		0			0
$835$	−4.77928	+	4.77928i	−0.165394	+	0.165394i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			41.2076i			1.42264i	0.702866	+	0.711322i	$0.251903\pi$
$839$			41.2076i			1.42264i	−0.702866	+	0.711322i	$0.748097\pi$
$840$		0			0
$841$		−	22.4692i		−	0.774799i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.8461	+	10.8461i	−0.373117	+	0.373117i
$846$		0			0
$847$			9.04987i			0.310957i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−28.9410	−	28.9410i	−0.992086	−	0.992086i
$852$		0			0
$853$	33.8433	−	33.8433i	1.15877	−	1.15877i	0.174032	−	0.984740i	$-0.444320\pi$
$853$	33.8433	−	33.8433i	1.15877	−	1.15877i	0.984740	−	0.174032i	$-0.0556797\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	46.4113			1.58538			0.792690	−	0.609624i	$-0.208680\pi$
$857$	46.4113			1.58538			0.792690	+	0.609624i	$0.208680\pi$
$858$		0			0
$859$	−30.4854	−	30.4854i	−1.04015	−	1.04015i	−0.999160	−	0.0409896i	$-0.986949\pi$
$859$	−30.4854	−	30.4854i	−1.04015	−	1.04015i	−0.0409896	−	0.999160i	$-0.513051\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.5455			−0.495134			−0.247567	−	0.968871i	$-0.579631\pi$
$863$	−14.5455			−0.495134			−0.247567	+	0.968871i	$0.579631\pi$
$864$		0			0
$865$	4.51355			0.153465
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.97308	+	4.97308i	0.168700	+	0.168700i
$870$		0			0
$871$	6.53612			0.221468
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.39116	+	7.39116i	−0.249867	+	0.249867i
$876$		0			0
$877$	−6.88085	−	6.88085i	−0.232350	−	0.232350i	0.581323	−	0.813673i	$-0.302535\pi$
$877$	−6.88085	−	6.88085i	−0.232350	−	0.232350i	−0.813673	+	0.581323i	$0.802535\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	22.2972i		−	0.751211i	−0.926780	−	0.375606i	$-0.877435\pi$
$881$		−	22.2972i		−	0.751211i	0.926780	−	0.375606i	$-0.122565\pi$
$882$		0			0
$883$	27.2968	−	27.2968i	0.918610	−	0.918610i	−0.0783181	−	0.996928i	$-0.524955\pi$
$883$	27.2968	−	27.2968i	0.918610	−	0.918610i	0.996928	+	0.0783181i	$0.0249550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	9.32380i		−	0.313063i	−0.987673	−	0.156531i	$-0.949969\pi$
$887$		−	9.32380i		−	0.313063i	0.987673	−	0.156531i	$-0.0500312\pi$
$888$		0			0
$889$		−	9.04469i		−	0.303349i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.9927	−	13.9927i	0.468250	−	0.468250i
$894$		0			0
$895$			24.7280i			0.826566i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	39.2101	+	39.2101i	1.30773	+	1.30773i
$900$		0			0
$901$	−12.3238	+	12.3238i	−0.410565	+	0.410565i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.9198			−0.362986
$906$		0			0
$907$	−35.9987	−	35.9987i	−1.19532	−	1.19532i	−0.975553	−	0.219763i	$-0.929472\pi$
$907$	−35.9987	−	35.9987i	−1.19532	−	1.19532i	−0.219763	−	0.975553i	$-0.570528\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.3185			0.673181			0.336591	−	0.941651i	$-0.390726\pi$
$911$	20.3185			0.673181			0.336591	+	0.941651i	$0.390726\pi$
$912$		0			0
$913$	20.3151			0.672331
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.77649	−	1.77649i	−0.0586649	−	0.0586649i
$918$		0			0
$919$	−13.2950			−0.438561			−0.219281	−	0.975662i	$-0.570371\pi$
$919$	−13.2950			−0.438561			−0.219281	+	0.975662i	$0.570371\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.908866	−	0.908866i	0.0299157	−	0.0299157i
$924$		0			0
$925$	−11.7276	−	11.7276i	−0.385600	−	0.385600i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			23.0184i			0.755211i	0.925967	+	0.377605i	$0.123252\pi$
$929$			23.0184i			0.755211i	−0.925967	+	0.377605i	$0.876748\pi$
$930$		0			0
$931$	−2.78433	+	2.78433i	−0.0912528	+	0.0912528i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			10.2147i			0.334057i
$936$		0			0
$937$			3.47134i			0.113404i	0.998391	+	0.0567019i	$0.0180585\pi$
$937$			3.47134i			0.113404i	−0.998391	+	0.0567019i	$0.981942\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	28.9167	−	28.9167i	0.942658	−	0.942658i	−0.0557851	−	0.998443i	$-0.517766\pi$
$941$	28.9167	−	28.9167i	0.942658	−	0.942658i	0.998443	+	0.0557851i	$0.0177662\pi$
$942$		0			0
$943$			39.1201i			1.27393i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.4415	+	40.4415i	1.31417	+	1.31417i	0.918309	+	0.395864i	$0.129555\pi$
$947$	40.4415	+	40.4415i	1.31417	+	1.31417i	0.395864	+	0.918309i	$0.370445\pi$
$948$		0			0
$949$	5.55870	−	5.55870i	0.180443	−	0.180443i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	42.9018			1.38973			0.694863	−	0.719142i	$-0.255465\pi$
$953$	42.9018			1.38973			0.694863	+	0.719142i	$0.255465\pi$
$954$		0			0
$955$	−0.459573	−	0.459573i	−0.0148714	−	0.0148714i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.7432			−0.346916
$960$		0			0
$961$	−28.7419			−0.927159
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.1661	+	11.1661i	0.359449	+	0.359449i
$966$		0			0
$967$	−21.7065			−0.698033			−0.349016	−	0.937117i	$-0.613484\pi$
$967$	−21.7065			−0.698033			−0.349016	+	0.937117i	$0.613484\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.23767	+	8.23767i	−0.264359	+	0.264359i	−0.826822	−	0.562463i	$-0.809854\pi$
$971$	−8.23767	+	8.23767i	−0.264359	+	0.264359i	0.562463	+	0.826822i	$0.309854\pi$
$972$		0			0
$973$	−4.31730	−	4.31730i	−0.138406	−	0.138406i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.5243i		−	0.816595i	−0.912849	−	0.408298i	$-0.866123\pi$
$977$		−	25.5243i		−	0.816595i	0.912849	−	0.408298i	$-0.133877\pi$
$978$		0			0
$979$	2.93601	−	2.93601i	0.0938352	−	0.0938352i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			14.2862i			0.455660i	0.973701	+	0.227830i	$0.0731631\pi$
$983$			14.2862i			0.455660i	−0.973701	+	0.227830i	$0.926837\pi$
$984$		0			0
$985$			19.9107i			0.634409i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10.8987	+	10.8987i	−0.346559	+	0.346559i
$990$		0			0
$991$		−	0.571733i		−	0.0181617i	−0.999959	−	0.00908086i	$-0.997109\pi$
$991$		−	0.571733i		−	0.0181617i	0.999959	−	0.00908086i	$-0.00289057\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.96204	−	2.96204i	−0.0939030	−	0.0939030i
$996$		0			0
$997$	−6.20041	+	6.20041i	−0.196369	+	0.196369i	−0.798441	−	0.602072i	$-0.794342\pi$
$997$	−6.20041	+	6.20041i	−0.196369	+	0.196369i	0.602072	+	0.798441i	$0.294342\pi$
$998$		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.1583.7		36
3.2	odd	2	inner	4032.2.v.d.1583.12		36
4.3	odd	2		1008.2.v.d.323.9	✓	36
12.11	even	2		1008.2.v.d.323.10	yes	36
16.5	even	4		1008.2.v.d.827.10	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.12		36
48.5	odd	4		1008.2.v.d.827.9	yes	36
48.11	even	4	inner	4032.2.v.d.3599.7		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.9	✓	36	4.3	odd	2
1008.2.v.d.323.10	yes	36	12.11	even	2
1008.2.v.d.827.9	yes	36	48.5	odd	4
1008.2.v.d.827.10	yes	36	16.5	even	4
4032.2.v.d.1583.7		36	1.1	even	1	trivial
4032.2.v.d.1583.12		36	3.2	odd	2	inner
4032.2.v.d.3599.7		36	48.11	even	4	inner
4032.2.v.d.3599.12		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)

\(f(q)\)	\(=\)	\(q+(-0.871498 - 0.871498i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-0.871498 - 0.871498i) q^{5} +1.00000 q^{7} +(-0.987453 + 0.987453i) q^{11} +(0.526623 + 0.526623i) q^{13} +5.93491i q^{17} +(-2.78433 + 2.78433i) q^{19} -8.59031i q^{23} -3.48098i q^{25} +(-5.07293 + 5.07293i) q^{29} -7.72929i q^{31} +(-0.871498 - 0.871498i) q^{35} +(3.36904 - 3.36904i) q^{37} -4.55399 q^{41} +(-1.26872 - 1.26872i) q^{43} -5.02553 q^{47} +1.00000 q^{49} +(2.07649 + 2.07649i) q^{53} +1.72113 q^{55} +(6.52331 - 6.52331i) q^{59} +(8.74207 + 8.74207i) q^{61} -0.917901i q^{65} +(6.20569 - 6.20569i) q^{67} -1.72584i q^{71} -10.5554i q^{73} +(-0.987453 + 0.987453i) q^{77} -5.03627i q^{79} +(-10.2866 - 10.2866i) q^{83} +(5.17226 - 5.17226i) q^{85} -2.97331 q^{89} +(0.526623 + 0.526623i) q^{91} +4.85308 q^{95} -8.71046 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 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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