LMFDB - Embedded newform 2304.2.k.l.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(577,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2304.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.k (of order $4$, degree $2$, 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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 768)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.577
Dual form			2304.2.k.l.1729.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.732051 + 0.732051i) q^{5} -2.44949i q^{7} +O(q^{10})$	$q+(-0.732051 + 0.732051i) q^{5} -2.44949i q^{7} +(-3.86370 + 3.86370i) q^{11} +(-3.00000 - 3.00000i) q^{13} +3.46410 q^{17} +(-0.378937 - 0.378937i) q^{19} +2.82843i q^{23} +3.92820i q^{25} +(4.19615 + 4.19615i) q^{29} +7.34847 q^{31} +(1.79315 + 1.79315i) q^{35} +(6.46410 - 6.46410i) q^{37} -11.4641i q^{41} +(2.44949 - 2.44949i) q^{43} +2.82843 q^{47} +1.00000 q^{49} +(6.73205 - 6.73205i) q^{53} -5.65685i q^{55} +(-9.79796 + 9.79796i) q^{59} +(6.46410 + 6.46410i) q^{61} +4.39230 q^{65} +(0.757875 + 0.757875i) q^{67} -16.2127i q^{71} +4.00000i q^{73} +(9.46410 + 9.46410i) q^{77} +2.44949 q^{79} +(-1.03528 - 1.03528i) q^{83} +(-2.53590 + 2.53590i) q^{85} +8.92820i q^{89} +(-7.34847 + 7.34847i) q^{91} +0.554803 q^{95} +14.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} - 24 q^{13} - 8 q^{29} + 24 q^{37} + 8 q^{49} + 40 q^{53} + 24 q^{61} - 48 q^{65} + 48 q^{77} - 48 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 24 * q^13 - 8 * q^29 + 24 * q^37 + 8 * q^49 + 40 * q^53 + 24 * q^61 - 48 * q^65 + 48 * q^77 - 48 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−0.732051	+	0.732051i	−0.327383	+	0.327383i	−0.851591	−	0.524207i	$-0.824362\pi$
$5$	−0.732051	+	0.732051i	−0.327383	+	0.327383i	0.524207	+	0.851591i	$0.324362\pi$
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.86370	+	3.86370i	−1.16495	+	1.16495i	−0.181573	+	0.983377i	$0.558119\pi$
$11$	−3.86370	+	3.86370i	−1.16495	+	1.16495i	−0.983377	+	0.181573i	$0.941881\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		0			0
$19$	−0.378937	−	0.378937i	−0.0869342	−	0.0869342i	0.662302	−	0.749237i	$-0.269579\pi$
$19$	−0.378937	−	0.378937i	−0.0869342	−	0.0869342i	−0.749237	+	0.662302i	$0.769579\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$			3.92820i			0.785641i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.19615	+	4.19615i	0.779206	+	0.779206i	0.979696	−	0.200490i	$-0.0642534\pi$
$29$	4.19615	+	4.19615i	0.779206	+	0.779206i	−0.200490	+	0.979696i	$0.564253\pi$
$30$		0			0
$31$	7.34847			1.31982			0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847			1.31982			0.659912	+	0.751343i	$0.270594\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.79315	+	1.79315i	0.303098	+	0.303098i
$36$		0			0
$37$	6.46410	−	6.46410i	1.06269	−	1.06269i	0.0647930	−	0.997899i	$-0.479361\pi$
$37$	6.46410	−	6.46410i	1.06269	−	1.06269i	0.997899	−	0.0647930i	$-0.0206387\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.4641i		−	1.79039i	−0.445673	−	0.895196i	$-0.647036\pi$
$41$		−	11.4641i		−	1.79039i	0.445673	−	0.895196i	$-0.352964\pi$
$42$		0			0
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	−0.495222	−	0.868766i	$-0.664913\pi$
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	0.868766	+	0.495222i	$0.164913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.82843			0.412568			0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843			0.412568			0.206284	+	0.978492i	$0.433863\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.73205	−	6.73205i	0.924718	−	0.924718i	−0.0726399	−	0.997358i	$-0.523142\pi$
$53$	6.73205	−	6.73205i	0.924718	−	0.924718i	0.997358	+	0.0726399i	$0.0231424\pi$
$54$		0			0
$55$		−	5.65685i		−	0.762770i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.79796	+	9.79796i	−1.27559	+	1.27559i	−0.332473	+	0.943113i	$0.607883\pi$
$59$	−9.79796	+	9.79796i	−1.27559	+	1.27559i	−0.943113	+	0.332473i	$0.892117\pi$
$60$		0			0
$61$	6.46410	+	6.46410i	0.827643	+	0.827643i	0.987190	−	0.159547i	$-0.0510033\pi$
$61$	6.46410	+	6.46410i	0.827643	+	0.827643i	−0.159547	+	0.987190i	$0.551003\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.39230			0.544798
$66$		0			0
$67$	0.757875	+	0.757875i	0.0925891	+	0.0925891i	0.751884	−	0.659295i	$-0.229145\pi$
$67$	0.757875	+	0.757875i	0.0925891	+	0.0925891i	−0.659295	+	0.751884i	$0.729145\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	16.2127i		−	1.92409i	−0.272887	−	0.962046i	$-0.587979\pi$
$71$		−	16.2127i		−	1.92409i	0.272887	−	0.962046i	$-0.412021\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.46410	+	9.46410i	1.07853	+	1.07853i
$78$		0			0
$79$	2.44949			0.275589			0.137795	−	0.990461i	$-0.455999\pi$
$79$	2.44949			0.275589			0.137795	+	0.990461i	$0.455999\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.03528	−	1.03528i	−0.113636	−	0.113636i	0.648002	−	0.761638i	$-0.275605\pi$
$83$	−1.03528	−	1.03528i	−0.113636	−	0.113636i	−0.761638	+	0.648002i	$0.775605\pi$
$84$		0			0
$85$	−2.53590	+	2.53590i	−0.275057	+	0.275057i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			8.92820i			0.946388i	0.880958	+	0.473194i	$0.156899\pi$
$89$			8.92820i			0.946388i	−0.880958	+	0.473194i	$0.843101\pi$
$90$		0			0
$91$	−7.34847	+	7.34847i	−0.770329	+	0.770329i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.554803			0.0569216
$96$		0			0
$97$	14.9282			1.51573			0.757865	−	0.652412i	$-0.226243\pi$
$97$	14.9282			1.51573			0.757865	+	0.652412i	$0.226243\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.196152	−	0.196152i	0.0195179	−	0.0195179i	−0.697280	−	0.716798i	$-0.745607\pi$
$101$	0.196152	−	0.196152i	0.0195179	−	0.0195179i	0.716798	+	0.697280i	$0.245607\pi$
$102$		0			0
$103$		−	8.10634i		−	0.798742i	−0.916789	−	0.399371i	$-0.869229\pi$
$103$		−	8.10634i		−	0.798742i	0.916789	−	0.399371i	$-0.130771\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	−0.557047	−	0.830481i	$-0.688066\pi$
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	0.830481	+	0.557047i	$0.188066\pi$
$108$		0			0
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	0.657592	−	0.753374i	$-0.271575\pi$
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	−0.753374	+	0.657592i	$0.771575\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$	−2.07055	−	2.07055i	−0.193080	−	0.193080i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	8.48528i		−	0.777844i
$120$		0			0
$121$		−	18.8564i		−	1.71422i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.53590	−	6.53590i	−0.584589	−	0.584589i
$126$		0			0
$127$	−19.4201			−1.72325			−0.861625	−	0.507545i	$-0.830553\pi$
$127$	−19.4201			−1.72325			−0.861625	+	0.507545i	$0.830553\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.757875	+	0.757875i	0.0662158	+	0.0662158i	0.739439	−	0.673223i	$-0.235091\pi$
$131$	0.757875	+	0.757875i	0.0662158	+	0.0662158i	−0.673223	+	0.739439i	$0.735091\pi$
$132$		0			0
$133$	−0.928203	+	0.928203i	−0.0804854	+	0.0804854i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.39230i			0.546131i	0.961995	+	0.273066i	$0.0880377\pi$
$137$			6.39230i			0.546131i	−0.961995	+	0.273066i	$0.911962\pi$
$138$		0			0
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	−0.883658	−	0.468132i	$-0.844927\pi$
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	0.468132	+	0.883658i	$0.344927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	23.1822			1.93859
$144$		0			0
$145$	−6.14359			−0.510198
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.1962	−	14.1962i	1.16299	−	1.16299i	0.179177	−	0.983817i	$-0.442656\pi$
$149$	14.1962	−	14.1962i	1.16299	−	1.16299i	0.983817	−	0.179177i	$-0.0573436\pi$
$150$		0			0
$151$			17.9043i			1.45703i	0.685029	+	0.728516i	$0.259789\pi$
$151$			17.9043i			1.45703i	−0.685029	+	0.728516i	$0.740211\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.37945	+	5.37945i	−0.432088	+	0.432088i
$156$		0			0
$157$	8.46410	+	8.46410i	0.675509	+	0.675509i	0.958981	−	0.283472i	$-0.0914862\pi$
$157$	8.46410	+	8.46410i	0.675509	+	0.675509i	−0.283472	+	0.958981i	$0.591486\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.92820			0.546019
$162$		0			0
$163$	8.10634	+	8.10634i	0.634938	+	0.634938i	0.949302	−	0.314364i	$-0.101791\pi$
$163$	8.10634	+	8.10634i	0.634938	+	0.634938i	−0.314364	+	0.949302i	$0.601791\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.07055i		−	0.160224i	−0.996786	−	0.0801121i	$-0.974472\pi$
$167$		−	2.07055i		−	0.160224i	0.996786	−	0.0801121i	$-0.0255278\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.80385	−	1.80385i	−0.137144	−	0.137144i	0.635202	−	0.772346i	$-0.280917\pi$
$173$	−1.80385	−	1.80385i	−0.137144	−	0.137144i	−0.772346	+	0.635202i	$0.780917\pi$
$174$		0			0
$175$	9.62209			0.727362
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.58630	−	3.58630i	−0.268053	−	0.268053i	0.560262	−	0.828315i	$-0.310700\pi$
$179$	−3.58630	−	3.58630i	−0.268053	−	0.268053i	−0.828315	+	0.560262i	$0.810700\pi$
$180$		0			0
$181$	5.00000	−	5.00000i	0.371647	−	0.371647i	−0.496430	−	0.868077i	$-0.665356\pi$
$181$	5.00000	−	5.00000i	0.371647	−	0.371647i	0.868077	+	0.496430i	$0.165356\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			9.46410i			0.695815i
$186$		0			0
$187$	−13.3843	+	13.3843i	−0.978754	+	0.978754i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.07055			0.149820			0.0749100	−	0.997190i	$-0.476133\pi$
$191$	2.07055			0.149820			0.0749100	+	0.997190i	$0.476133\pi$
$192$		0			0
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.19615	−	4.19615i	0.298963	−	0.298963i	−0.541644	−	0.840608i	$-0.682198\pi$
$197$	4.19615	−	4.19615i	0.298963	−	0.298963i	0.840608	+	0.541644i	$0.182198\pi$
$198$		0			0
$199$			3.96524i			0.281088i	0.990074	+	0.140544i	$0.0448852\pi$
$199$			3.96524i			0.281088i	−0.990074	+	0.140544i	$0.955115\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.2784	−	10.2784i	0.721405	−	0.721405i
$204$		0			0
$205$	8.39230	+	8.39230i	0.586144	+	0.586144i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.92820			0.202548
$210$		0			0
$211$	4.89898	+	4.89898i	0.337260	+	0.337260i	0.855335	−	0.518075i	$-0.173351\pi$
$211$	4.89898	+	4.89898i	0.337260	+	0.337260i	−0.518075	+	0.855335i	$0.673351\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.58630i			0.244584i
$216$		0			0
$217$		−	18.0000i		−	1.22192i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.3923	−	10.3923i	−0.699062	−	0.699062i
$222$		0			0
$223$	19.4201			1.30046			0.650231	−	0.759736i	$-0.274672\pi$
$223$	19.4201			1.30046			0.650231	+	0.759736i	$0.274672\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.4901	+	16.4901i	1.09449	+	1.09449i	0.995043	+	0.0994423i	$0.0317059\pi$
$227$	16.4901	+	16.4901i	1.09449	+	1.09449i	0.0994423	+	0.995043i	$0.468294\pi$
$228$		0			0
$229$	−1.92820	+	1.92820i	−0.127419	+	0.127419i	−0.767940	−	0.640521i	$-0.778718\pi$
$229$	−1.92820	+	1.92820i	−0.127419	+	0.127419i	0.640521	+	0.767940i	$0.278718\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.928203i			0.0608086i	0.999538	+	0.0304043i	$0.00967948\pi$
$233$			0.928203i			0.0608086i	−0.999538	+	0.0304043i	$0.990321\pi$
$234$		0			0
$235$	−2.07055	+	2.07055i	−0.135068	+	0.135068i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.07055			0.133933			0.0669664	−	0.997755i	$-0.478668\pi$
$239$	2.07055			0.133933			0.0669664	+	0.997755i	$0.478668\pi$
$240$		0			0
$241$	−14.7846			−0.952360			−0.476180	−	0.879348i	$-0.657979\pi$
$241$	−14.7846			−0.952360			−0.476180	+	0.879348i	$0.657979\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.732051	+	0.732051i	−0.0467690	+	0.0467690i
$246$		0			0
$247$			2.27362i			0.144667i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.52056	+	9.52056i	−0.600932	+	0.600932i	−0.940560	−	0.339628i	$-0.889699\pi$
$251$	−9.52056	+	9.52056i	−0.600932	+	0.600932i	0.339628	+	0.940560i	$0.389699\pi$
$252$		0			0
$253$	−10.9282	−	10.9282i	−0.687050	−	0.687050i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.7846			−1.17175			−0.585876	−	0.810401i	$-0.699249\pi$
$257$	−18.7846			−1.17175			−0.585876	+	0.810401i	$0.699249\pi$
$258$		0			0
$259$	−15.8338	−	15.8338i	−0.983861	−	0.983861i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.6274i		−	1.39527i	−0.716455	−	0.697633i	$-0.754237\pi$
$263$		−	22.6274i		−	1.39527i	0.716455	−	0.697633i	$-0.245763\pi$
$264$		0			0
$265$			9.85641i			0.605474i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.7321	+	10.7321i	0.654345	+	0.654345i	0.954036	−	0.299691i	$-0.0968837\pi$
$269$	10.7321	+	10.7321i	0.654345	+	0.654345i	−0.299691	+	0.954036i	$0.596884\pi$
$270$		0			0
$271$	−8.10634			−0.492425			−0.246213	−	0.969216i	$-0.579186\pi$
$271$	−8.10634			−0.492425			−0.246213	+	0.969216i	$0.579186\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.1774	−	15.1774i	−0.915232	−	0.915232i
$276$		0			0
$277$	−18.8564	+	18.8564i	−1.13297	+	1.13297i	−0.143291	+	0.989681i	$0.545769\pi$
$277$	−18.8564	+	18.8564i	−1.13297	+	1.13297i	−0.989681	+	0.143291i	$0.954231\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	12.9282i		−	0.771232i	−0.922659	−	0.385616i	$-0.873989\pi$
$281$		−	12.9282i		−	0.771232i	0.922659	−	0.385616i	$-0.126011\pi$
$282$		0			0
$283$	12.6264	−	12.6264i	0.750561	−	0.750561i	−0.224023	−	0.974584i	$-0.571919\pi$
$283$	12.6264	−	12.6264i	0.750561	−	0.750561i	0.974584	+	0.224023i	$0.0719191\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−28.0812			−1.65758
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.66025	+	9.66025i	−0.564358	+	0.564358i	−0.930542	−	0.366184i	$-0.880664\pi$
$293$	−9.66025	+	9.66025i	−0.564358	+	0.564358i	0.366184	+	0.930542i	$0.380664\pi$
$294$		0			0
$295$		−	14.3452i		−	0.835210i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.48528	−	8.48528i	0.490716	−	0.490716i
$300$		0			0
$301$	−6.00000	−	6.00000i	−0.345834	−	0.345834i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.46410			−0.541913
$306$		0			0
$307$	−6.41473	−	6.41473i	−0.366108	−	0.366108i	0.499948	−	0.866056i	$-0.333353\pi$
$307$	−6.41473	−	6.41473i	−0.366108	−	0.366108i	−0.866056	+	0.499948i	$0.833353\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			12.8295i			0.727492i	0.931498	+	0.363746i	$0.118502\pi$
$311$			12.8295i			0.727492i	−0.931498	+	0.363746i	$0.881498\pi$
$312$		0			0
$313$			23.8564i			1.34844i	0.738529	+	0.674222i	$0.235521\pi$
$313$			23.8564i			1.34844i	−0.738529	+	0.674222i	$0.764479\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.26795	−	1.26795i	−0.0712151	−	0.0712151i	0.670602	−	0.741817i	$-0.266036\pi$
$317$	−1.26795	−	1.26795i	−0.0712151	−	0.0712151i	−0.741817	+	0.670602i	$0.766036\pi$
$318$		0			0
$319$	−32.4254			−1.81547
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.31268	−	1.31268i	−0.0730393	−	0.0730393i
$324$		0			0
$325$	11.7846	−	11.7846i	0.653693	−	0.653693i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	6.92820i		−	0.381964i
$330$		0			0
$331$	10.5558	−	10.5558i	0.580201	−	0.580201i	−0.354757	−	0.934958i	$-0.615437\pi$
$331$	10.5558	−	10.5558i	0.580201	−	0.580201i	0.934958	+	0.354757i	$0.115437\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.10961			−0.0606242
$336$		0			0
$337$	17.8564			0.972700			0.486350	−	0.873764i	$-0.338328\pi$
$337$	17.8564			0.972700			0.486350	+	0.873764i	$0.338328\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−28.3923	+	28.3923i	−1.53753	+	1.53753i
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.69213	−	6.69213i	0.359252	−	0.359252i	−0.504285	−	0.863537i	$-0.668244\pi$
$347$	6.69213	−	6.69213i	0.359252	−	0.359252i	0.863537	+	0.504285i	$0.168244\pi$
$348$		0			0
$349$	−1.39230	−	1.39230i	−0.0745284	−	0.0745284i	0.668860	−	0.743388i	$-0.266783\pi$
$349$	−1.39230	−	1.39230i	−0.0745284	−	0.0745284i	−0.743388	+	0.668860i	$0.766783\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.9282			−0.688099			−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282			−0.688099			−0.344049	+	0.938952i	$0.611799\pi$
$354$		0			0
$355$	11.8685	+	11.8685i	0.629915	+	0.629915i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			30.1518i			1.59135i	0.605725	+	0.795674i	$0.292883\pi$
$359$			30.1518i			1.59135i	−0.605725	+	0.795674i	$0.707117\pi$
$360$		0			0
$361$		−	18.7128i		−	0.984885i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.92820	−	2.92820i	−0.153269	−	0.153269i
$366$		0			0
$367$	27.7023			1.44605			0.723023	−	0.690824i	$-0.242752\pi$
$367$	27.7023			1.44605			0.723023	+	0.690824i	$0.242752\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−16.4901	−	16.4901i	−0.856123	−	0.856123i
$372$		0			0
$373$	11.3923	−	11.3923i	0.589871	−	0.589871i	−0.347725	−	0.937596i	$-0.613046\pi$
$373$	11.3923	−	11.3923i	0.589871	−	0.589871i	0.937596	+	0.347725i	$0.113046\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	25.1769i		−	1.29668i
$378$		0			0
$379$	15.0759	−	15.0759i	0.774396	−	0.774396i	−0.204476	−	0.978872i	$-0.565549\pi$
$379$	15.0759	−	15.0759i	0.774396	−	0.774396i	0.978872	+	0.204476i	$0.0655490\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.554803			−0.0283491			−0.0141746	−	0.999900i	$-0.504512\pi$
$383$	−0.554803			−0.0283491			−0.0141746	+	0.999900i	$0.504512\pi$
$384$		0			0
$385$	−13.8564			−0.706188
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.26795	−	3.26795i	0.165692	−	0.165692i	−0.619391	−	0.785083i	$-0.712620\pi$
$389$	3.26795	−	3.26795i	0.165692	−	0.165692i	0.785083	+	0.619391i	$0.212620\pi$
$390$		0			0
$391$			9.79796i			0.495504i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.79315	+	1.79315i	−0.0902232	+	0.0902232i
$396$		0			0
$397$	−18.3205	−	18.3205i	−0.919480	−	0.919480i	0.0775115	−	0.996991i	$-0.475303\pi$
$397$	−18.3205	−	18.3205i	−0.919480	−	0.919480i	−0.996991	+	0.0775115i	$0.975303\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.39230			0.119466			0.0597330	−	0.998214i	$-0.480975\pi$
$401$	2.39230			0.119466			0.0597330	+	0.998214i	$0.480975\pi$
$402$		0			0
$403$	−22.0454	−	22.0454i	−1.09816	−	1.09816i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			49.9507i			2.47597i
$408$		0			0
$409$			22.9282i			1.13373i	0.823812	+	0.566863i	$0.191843\pi$
$409$			22.9282i			1.13373i	−0.823812	+	0.566863i	$0.808157\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.0000	+	24.0000i	1.18096	+	1.18096i
$414$		0			0
$415$	1.51575			0.0744052
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.4195	+	14.4195i	0.704440	+	0.704440i	0.965360	−	0.260920i	$-0.0840259\pi$
$419$	14.4195	+	14.4195i	0.704440	+	0.704440i	−0.260920	+	0.965360i	$0.584026\pi$
$420$		0			0
$421$	−0.0717968	+	0.0717968i	−0.00349916	+	0.00349916i	−0.708854	−	0.705355i	$-0.750788\pi$
$421$	−0.0717968	+	0.0717968i	−0.00349916	+	0.00349916i	0.705355	+	0.708854i	$0.250788\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			13.6077i			0.660070i
$426$		0			0
$427$	15.8338	−	15.8338i	0.766249	−	0.766249i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.9401			−1.15315			−0.576577	−	0.817043i	$-0.695612\pi$
$431$	−23.9401			−1.15315			−0.576577	+	0.817043i	$0.695612\pi$
$432$		0			0
$433$	2.14359			0.103015			0.0515073	−	0.998673i	$-0.483597\pi$
$433$	2.14359			0.103015			0.0515073	+	0.998673i	$0.483597\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.07180	−	1.07180i	0.0512710	−	0.0512710i
$438$		0			0
$439$			28.4601i			1.35833i	0.733986	+	0.679164i	$0.237658\pi$
$439$			28.4601i			1.35833i	−0.733986	+	0.679164i	$0.762342\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.480473	+	0.480473i	−0.0228280	+	0.0228280i	−0.718429	−	0.695601i	$-0.755138\pi$
$443$	−0.480473	+	0.480473i	−0.0228280	+	0.0228280i	0.695601	+	0.718429i	$0.255138\pi$
$444$		0			0
$445$	−6.53590	−	6.53590i	−0.309831	−	0.309831i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.3923			−0.679215			−0.339607	−	0.940567i	$-0.610294\pi$
$449$	−14.3923			−0.679215			−0.339607	+	0.940567i	$0.610294\pi$
$450$		0			0
$451$	44.2939	+	44.2939i	2.08572	+	2.08572i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	10.7589i		−	0.504385i
$456$		0			0
$457$			20.7846i			0.972263i	0.873886	+	0.486132i	$0.161592\pi$
$457$			20.7846i			0.972263i	−0.873886	+	0.486132i	$0.838408\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16.0526	−	16.0526i	−0.747642	−	0.747642i	0.226394	−	0.974036i	$-0.427306\pi$
$461$	−16.0526	−	16.0526i	−0.747642	−	0.747642i	−0.974036	+	0.226394i	$0.927306\pi$
$462$		0			0
$463$	−4.72311			−0.219502			−0.109751	−	0.993959i	$-0.535005\pi$
$463$	−4.72311			−0.219502			−0.109751	+	0.993959i	$0.535005\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.9038	+	12.9038i	0.597116	+	0.597116i	0.939544	−	0.342428i	$-0.111249\pi$
$467$	12.9038	+	12.9038i	0.597116	+	0.597116i	−0.342428	+	0.939544i	$0.611249\pi$
$468$		0			0
$469$	1.85641	−	1.85641i	0.0857209	−	0.0857209i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			18.9282i			0.870320i
$474$		0			0
$475$	1.48854	−	1.48854i	0.0682990	−	0.0682990i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.8401			−1.77465			−0.887325	−	0.461145i	$-0.847439\pi$
$479$	−38.8401			−1.77465			−0.887325	+	0.461145i	$0.847439\pi$
$480$		0			0
$481$	−38.7846			−1.76843
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.9282	+	10.9282i	−0.496224	+	0.496224i
$486$		0			0
$487$		−	22.8033i		−	1.03332i	−0.856192	−	0.516658i	$-0.827176\pi$
$487$		−	22.8033i		−	1.03332i	0.856192	−	0.516658i	$-0.172824\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.4548	−	15.4548i	0.697466	−	0.697466i	−0.266397	−	0.963863i	$-0.585833\pi$
$491$	15.4548	−	15.4548i	0.697466	−	0.697466i	0.963863	+	0.266397i	$0.0858333\pi$
$492$		0			0
$493$	14.5359	+	14.5359i	0.654664	+	0.654664i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−39.7128			−1.78136
$498$		0			0
$499$	−2.82843	−	2.82843i	−0.126618	−	0.126618i	0.640958	−	0.767576i	$-0.278537\pi$
$499$	−2.82843	−	2.82843i	−0.126618	−	0.126618i	−0.767576	+	0.640958i	$0.778537\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			33.1833i			1.47957i	0.672844	+	0.739784i	$0.265072\pi$
$503$			33.1833i			1.47957i	−0.672844	+	0.739784i	$0.734928\pi$
$504$		0			0
$505$			0.287187i			0.0127797i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.1244	−	19.1244i	−0.847672	−	0.847672i	0.142170	−	0.989842i	$-0.454592\pi$
$509$	−19.1244	−	19.1244i	−0.847672	−	0.847672i	−0.989842	+	0.142170i	$0.954592\pi$
$510$		0			0
$511$	9.79796			0.433436
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.93426	+	5.93426i	0.261495	+	0.261495i
$516$		0			0
$517$	−10.9282	+	10.9282i	−0.480622	+	0.480622i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.3205i		−	0.583582i	−0.956482	−	0.291791i	$-0.905749\pi$
$521$		−	13.3205i		−	0.583582i	0.956482	−	0.291791i	$-0.0942512\pi$
$522$		0			0
$523$	17.3495	−	17.3495i	0.758641	−	0.758641i	−0.217434	−	0.976075i	$-0.569769\pi$
$523$	17.3495	−	17.3495i	0.758641	−	0.758641i	0.976075	+	0.217434i	$0.0697688\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.4558			1.10887
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.3923	+	34.3923i	−1.48970	+	1.48970i
$534$		0			0
$535$			4.14110i			0.179036i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.86370	+	3.86370i	−0.166421	+	0.166421i
$540$		0			0
$541$	−6.07180	−	6.07180i	−0.261047	−	0.261047i	0.564432	−	0.825479i	$-0.309095\pi$
$541$	−6.07180	−	6.07180i	−0.261047	−	0.261047i	−0.825479	+	0.564432i	$0.809095\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.46410			0.0627152
$546$		0			0
$547$	−27.7023	−	27.7023i	−1.18446	−	1.18446i	−0.978576	−	0.205888i	$-0.933992\pi$
$547$	−27.7023	−	27.7023i	−1.18446	−	1.18446i	−0.205888	−	0.978576i	$-0.566008\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3.18016i		−	0.135479i
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−30.1962	−	30.1962i	−1.27945	−	1.27945i	−0.940974	−	0.338478i	$-0.890088\pi$
$557$	−30.1962	−	30.1962i	−1.27945	−	1.27945i	−0.338478	−	0.940974i	$-0.609912\pi$
$558$		0			0
$559$	−14.6969			−0.621614
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.86370	−	3.86370i	−0.162836	−	0.162836i	0.620986	−	0.783822i	$-0.286732\pi$
$563$	−3.86370	−	3.86370i	−0.162836	−	0.162836i	−0.783822	+	0.620986i	$0.786732\pi$
$564$		0			0
$565$	−13.1769	+	13.1769i	−0.554357	+	0.554357i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.46410i			0.145223i	0.997360	+	0.0726113i	$0.0231333\pi$
$569$			3.46410i			0.145223i	−0.997360	+	0.0726113i	$0.976867\pi$
$570$		0			0
$571$	−16.7675	+	16.7675i	−0.701698	+	0.701698i	−0.964775	−	0.263077i	$-0.915263\pi$
$571$	−16.7675	+	16.7675i	−0.701698	+	0.701698i	0.263077	+	0.964775i	$0.415263\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.1106			−0.463346
$576$		0			0
$577$	5.85641			0.243805			0.121903	−	0.992542i	$-0.461100\pi$
$577$	5.85641			0.243805			0.121903	+	0.992542i	$0.461100\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.53590	+	2.53590i	−0.105207	+	0.105207i
$582$		0			0
$583$			52.0213i			2.15450i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.5569	−	20.5569i	0.848473	−	0.848473i	−0.141470	−	0.989943i	$-0.545183\pi$
$587$	20.5569	−	20.5569i	0.848473	−	0.848473i	0.989943	+	0.141470i	$0.0451829\pi$
$588$		0			0
$589$	−2.78461	−	2.78461i	−0.114738	−	0.114738i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.7128			−0.727378			−0.363689	−	0.931520i	$-0.618483\pi$
$593$	−17.7128			−0.727378			−0.363689	+	0.931520i	$0.618483\pi$
$594$		0			0
$595$	6.21166	+	6.21166i	0.254653	+	0.254653i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	12.6264i		−	0.515900i	−0.966158	−	0.257950i	$-0.916953\pi$
$599$		−	12.6264i		−	0.515900i	0.966158	−	0.257950i	$-0.0830470\pi$
$600$		0			0
$601$			35.7128i			1.45676i	0.685176	+	0.728378i	$0.259725\pi$
$601$			35.7128i			1.45676i	−0.685176	+	0.728378i	$0.740275\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.8038	+	13.8038i	0.561206	+	0.561206i
$606$		0			0
$607$	−31.8434			−1.29248			−0.646241	−	0.763133i	$-0.723660\pi$
$607$	−31.8434			−1.29248			−0.646241	+	0.763133i	$0.723660\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.48528	−	8.48528i	−0.343278	−	0.343278i
$612$		0			0
$613$	−30.3205	+	30.3205i	−1.22463	+	1.22463i	−0.258667	+	0.965966i	$0.583283\pi$
$613$	−30.3205	+	30.3205i	−1.22463	+	1.22463i	−0.965966	+	0.258667i	$0.916717\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.1436i			0.488883i	0.969664	+	0.244441i	$0.0786046\pi$
$617$			12.1436i			0.488883i	−0.969664	+	0.244441i	$0.921395\pi$
$618$		0			0
$619$	−8.48528	+	8.48528i	−0.341052	+	0.341052i	−0.856763	−	0.515711i	$-0.827528\pi$
$619$	−8.48528	+	8.48528i	−0.341052	+	0.341052i	0.515711	+	0.856763i	$0.327528\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.8695			0.876185
$624$		0			0
$625$	−10.0718			−0.402872
$626$		0			0
$627$		0			0
$628$		0			0
$629$	22.3923	−	22.3923i	0.892840	−	0.892840i
$630$		0			0
$631$		−	0.933740i		−	0.0371716i	−0.999827	−	0.0185858i	$-0.994084\pi$
$631$		−	0.933740i		−	0.0371716i	0.999827	−	0.0185858i	$-0.00591639\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.2165	−	14.2165i	0.564163	−	0.564163i
$636$		0			0
$637$	−3.00000	−	3.00000i	−0.118864	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	46.3923			1.83239			0.916193	−	0.400737i	$-0.131246\pi$
$641$	46.3923			1.83239			0.916193	+	0.400737i	$0.131246\pi$
$642$		0			0
$643$	−25.0769	−	25.0769i	−0.988937	−	0.988937i	0.0110028	−	0.999939i	$-0.496498\pi$
$643$	−25.0769	−	25.0769i	−0.988937	−	0.988937i	−0.999939	+	0.0110028i	$0.996498\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21.3147i		−	0.837969i	−0.907993	−	0.418984i	$-0.862386\pi$
$647$		−	21.3147i		−	0.837969i	0.907993	−	0.418984i	$-0.137614\pi$
$648$		0			0
$649$		−	75.7128i		−	2.97199i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.339746	−	0.339746i	−0.0132953	−	0.0132953i	0.700428	−	0.713723i	$-0.252993\pi$
$653$	−0.339746	−	0.339746i	−0.0132953	−	0.0132953i	−0.713723	+	0.700428i	$0.752993\pi$
$654$		0			0
$655$	−1.10961			−0.0433559
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.9706	−	16.9706i	−0.661079	−	0.661079i	0.294555	−	0.955634i	$-0.404829\pi$
$659$	−16.9706	−	16.9706i	−0.661079	−	0.661079i	−0.955634	+	0.294555i	$0.904829\pi$
$660$		0			0
$661$	25.3923	−	25.3923i	0.987646	−	0.987646i	−0.0122784	−	0.999925i	$-0.503908\pi$
$661$	25.3923	−	25.3923i	0.987646	−	0.987646i	0.999925	+	0.0122784i	$0.00390844\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.35898i		−	0.0526991i
$666$		0			0
$667$	−11.8685	+	11.8685i	−0.459551	+	0.459551i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−49.9507			−1.92833
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.6603	−	19.6603i	0.755605	−	0.755605i	−0.219914	−	0.975519i	$-0.570578\pi$
$677$	19.6603	−	19.6603i	0.755605	−	0.755605i	0.975519	+	0.219914i	$0.0705778\pi$
$678$		0			0
$679$		−	36.5665i		−	1.40329i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.2880	−	26.2880i	1.00588	−	1.00588i	0.00590163	−	0.999983i	$-0.498121\pi$
$683$	26.2880	−	26.2880i	1.00588	−	1.00588i	0.999983	−	0.00590163i	$-0.00187856\pi$
$684$		0			0
$685$	−4.67949	−	4.67949i	−0.178794	−	0.178794i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−40.3923			−1.53882
$690$		0			0
$691$	−3.96524	−	3.96524i	−0.150845	−	0.150845i	0.627650	−	0.778495i	$-0.284017\pi$
$691$	−3.96524	−	3.96524i	−0.150845	−	0.150845i	−0.778495	+	0.627650i	$0.784017\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.17260i		−	0.272072i
$696$		0			0
$697$		−	39.7128i		−	1.50423i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.5885	−	20.5885i	−0.777615	−	0.777615i	0.201809	−	0.979425i	$-0.435318\pi$
$701$	−20.5885	−	20.5885i	−0.777615	−	0.777615i	−0.979425	+	0.201809i	$0.935318\pi$
$702$		0			0
$703$	−4.89898			−0.184769
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.480473	−	0.480473i	−0.0180701	−	0.0180701i
$708$		0			0
$709$	−20.7128	+	20.7128i	−0.777886	+	0.777886i	−0.979471	−	0.201585i	$-0.935391\pi$
$709$	−20.7128	+	20.7128i	−0.777886	+	0.777886i	0.201585	+	0.979471i	$0.435391\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			20.7846i			0.778390i
$714$		0			0
$715$	−16.9706	+	16.9706i	−0.634663	+	0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	31.6675			1.18100			0.590499	−	0.807038i	$-0.298931\pi$
$719$	31.6675			1.18100			0.590499	+	0.807038i	$0.298931\pi$
$720$		0			0
$721$	−19.8564			−0.739491
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.4833	+	16.4833i	−0.612176	+	0.612176i
$726$		0			0
$727$			3.20736i			0.118955i	0.998230	+	0.0594773i	$0.0189434\pi$
$727$			3.20736i			0.118955i	−0.998230	+	0.0594773i	$0.981057\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.48528	−	8.48528i	0.313839	−	0.313839i
$732$		0			0
$733$	5.00000	+	5.00000i	0.184679	+	0.184679i	0.793391	−	0.608712i	$-0.208314\pi$
$733$	5.00000	+	5.00000i	0.184679	+	0.184679i	−0.608712	+	0.793391i	$0.708314\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.85641			−0.215724
$738$		0			0
$739$	25.4558	+	25.4558i	0.936408	+	0.936408i	0.998096	−	0.0616872i	$-0.0196481\pi$
$739$	25.4558	+	25.4558i	0.936408	+	0.936408i	−0.0616872	+	0.998096i	$0.519648\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			21.6665i			0.794866i	0.917631	+	0.397433i	$0.130099\pi$
$743$			21.6665i			0.794866i	−0.917631	+	0.397433i	$0.869901\pi$
$744$		0			0
$745$			20.7846i			0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.92820	−	6.92820i	−0.253151	−	0.253151i
$750$		0			0
$751$	21.6937			0.791614			0.395807	−	0.918334i	$-0.370465\pi$
$751$	21.6937			0.791614			0.395807	+	0.918334i	$0.370465\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13.1069	−	13.1069i	−0.477007	−	0.477007i
$756$		0			0
$757$	17.7846	−	17.7846i	0.646393	−	0.646393i	−0.305727	−	0.952119i	$-0.598899\pi$
$757$	17.7846	−	17.7846i	0.646393	−	0.646393i	0.952119	+	0.305727i	$0.0988994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3.75129i		−	0.135984i	−0.997686	−	0.0679921i	$-0.978341\pi$
$761$		−	3.75129i		−	0.135984i	0.997686	−	0.0679921i	$-0.0216593\pi$
$762$		0			0
$763$	−2.44949	+	2.44949i	−0.0886775	+	0.0886775i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	58.7878			2.12270
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.41154	−	3.41154i	0.122705	−	0.122705i	−0.643088	−	0.765792i	$-0.722347\pi$
$773$	3.41154	−	3.41154i	0.122705	−	0.122705i	0.765792	+	0.643088i	$0.222347\pi$
$774$		0			0
$775$			28.8663i			1.03691i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.34418	+	4.34418i	−0.155646	+	0.155646i
$780$		0			0
$781$	62.6410	+	62.6410i	2.24147	+	2.24147i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.3923			−0.442300
$786$		0			0
$787$	11.6926	+	11.6926i	0.416798	+	0.416798i	0.884098	−	0.467301i	$-0.154773\pi$
$787$	11.6926	+	11.6926i	0.416798	+	0.416798i	−0.467301	+	0.884098i	$0.654773\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	44.0908i		−	1.56769i
$792$		0			0
$793$		−	38.7846i		−	1.37728i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.5167	+	29.5167i	1.04553	+	1.04553i	0.998913	+	0.0466211i	$0.0148453\pi$
$797$	29.5167	+	29.5167i	1.04553	+	1.04553i	0.0466211	+	0.998913i	$0.485155\pi$
$798$		0			0
$799$	9.79796			0.346627
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.4548	−	15.4548i	−0.545389	−	0.545389i
$804$		0			0
$805$	−5.07180	+	5.07180i	−0.178757	+	0.178757i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			41.3205i			1.45275i	0.687298	+	0.726376i	$0.258797\pi$
$809$			41.3205i			1.45275i	−0.687298	+	0.726376i	$0.741203\pi$
$810$		0			0
$811$	23.7642	−	23.7642i	0.834475	−	0.834475i	−0.153650	−	0.988125i	$-0.549103\pi$
$811$	23.7642	−	23.7642i	0.834475	−	0.834475i	0.988125	+	0.153650i	$0.0491029\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−11.8685			−0.415736
$816$		0			0
$817$	−1.85641			−0.0649474
$818$		0			0
$819$		0			0
$820$		0			0
$821$	28.1962	−	28.1962i	0.984053	−	0.984053i	−0.0158223	−	0.999875i	$-0.505037\pi$
$821$	28.1962	−	28.1962i	0.984053	−	0.984053i	0.999875	+	0.0158223i	$0.00503661\pi$
$822$		0			0
$823$			43.9149i			1.53078i	0.643567	+	0.765389i	$0.277454\pi$
$823$			43.9149i			1.53078i	−0.643567	+	0.765389i	$0.722546\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.27362	−	2.27362i	0.0790617	−	0.0790617i	−0.666470	−	0.745532i	$-0.732196\pi$
$827$	2.27362	−	2.27362i	0.0790617	−	0.0790617i	0.745532	+	0.666470i	$0.232196\pi$
$828$		0			0
$829$	−28.8564	−	28.8564i	−1.00222	−	1.00222i	−0.999998	−	0.00222690i	$-0.999291\pi$
$829$	−28.8564	−	28.8564i	−1.00222	−	1.00222i	−0.00222690	−	0.999998i	$-0.500709\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.46410			0.120024
$834$		0			0
$835$	1.51575	+	1.51575i	0.0524547	+	0.0524547i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	3.38323i		−	0.116802i	−0.998293	−	0.0584010i	$-0.981400\pi$
$839$		−	3.38323i		−	0.116802i	0.998293	−	0.0584010i	$-0.0186002\pi$
$840$		0			0
$841$			6.21539i			0.214324i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.66025	−	3.66025i	−0.125917	−	0.125917i
$846$		0			0
$847$	−46.1886			−1.58706
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.2832	+	18.2832i	0.626741	+	0.626741i
$852$		0			0
$853$	−15.2487	+	15.2487i	−0.522106	+	0.522106i	−0.918207	−	0.396101i	$-0.870363\pi$
$853$	−15.2487	+	15.2487i	−0.522106	+	0.522106i	0.396101	+	0.918207i	$0.370363\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.1769i		−	0.655071i	−0.944839	−	0.327535i	$-0.893782\pi$
$857$		−	19.1769i		−	0.655071i	0.944839	−	0.327535i	$-0.106218\pi$
$858$		0			0
$859$	−11.4896	+	11.4896i	−0.392019	+	0.392019i	−0.875407	−	0.483387i	$-0.839406\pi$
$859$	−11.4896	+	11.4896i	−0.392019	+	0.392019i	0.483387	+	0.875407i	$0.339406\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31.4644			1.07106			0.535531	−	0.844516i	$-0.320112\pi$
$863$	31.4644			1.07106			0.535531	+	0.844516i	$0.320112\pi$
$864$		0			0
$865$	2.64102			0.0897972
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.46410	+	9.46410i	−0.321048	+	0.321048i
$870$		0			0
$871$		−	4.54725i		−	0.154078i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.0096	+	16.0096i	−0.541224	+	0.541224i
$876$		0			0
$877$	−23.2487	−	23.2487i	−0.785053	−	0.785053i	0.195625	−	0.980679i	$-0.437326\pi$
$877$	−23.2487	−	23.2487i	−0.785053	−	0.785053i	−0.980679	+	0.195625i	$0.937326\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	32.6410			1.09970			0.549852	−	0.835262i	$-0.314684\pi$
$881$	32.6410			1.09970			0.549852	+	0.835262i	$0.314684\pi$
$882$		0			0
$883$	18.6622	+	18.6622i	0.628032	+	0.628032i	0.947573	−	0.319540i	$-0.103529\pi$
$883$	18.6622	+	18.6622i	0.628032	+	0.628032i	−0.319540	+	0.947573i	$0.603529\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.62536i			0.0881508i	0.999028	+	0.0440754i	$0.0140342\pi$
$887$			2.62536i			0.0881508i	−0.999028	+	0.0440754i	$0.985966\pi$
$888$		0			0
$889$			47.5692i			1.59542i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.07180	−	1.07180i	−0.0358663	−	0.0358663i
$894$		0			0
$895$	5.25071			0.175512
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.8353	+	30.8353i	1.02841	+	1.02841i
$900$		0			0
$901$	23.3205	−	23.3205i	0.776919	−	0.776919i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			7.32051i			0.243342i
$906$		0			0
$907$	−15.2789	+	15.2789i	−0.507329	+	0.507329i	−0.913706	−	0.406377i	$-0.866792\pi$
$907$	−15.2789	+	15.2789i	−0.507329	+	0.507329i	0.406377	+	0.913706i	$0.366792\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−51.4665			−1.70516			−0.852580	−	0.522596i	$-0.824964\pi$
$911$	−51.4665			−1.70516			−0.852580	+	0.522596i	$0.824964\pi$
$912$		0			0
$913$	8.00000			0.264761
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.85641	−	1.85641i	0.0613039	−	0.0613039i
$918$		0			0
$919$		−	44.6728i		−	1.47362i	−0.676100	−	0.736810i	$-0.736331\pi$
$919$		−	44.6728i		−	1.47362i	0.676100	−	0.736810i	$-0.263669\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48.6381	+	48.6381i	−1.60094	+	1.60094i
$924$		0			0
$925$	25.3923	+	25.3923i	0.834894	+	0.834894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.6795			0.350383			0.175191	−	0.984534i	$-0.443946\pi$
$929$	10.6795			0.350383			0.175191	+	0.984534i	$0.443946\pi$
$930$		0			0
$931$	−0.378937	−	0.378937i	−0.0124192	−	0.0124192i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	19.5959i		−	0.640855i
$936$		0			0
$937$			2.00000i			0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$			2.00000i			0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.33975	−	8.33975i	−0.271868	−	0.271868i	0.557984	−	0.829852i	$-0.311575\pi$
$941$	−8.33975	−	8.33975i	−0.271868	−	0.271868i	−0.829852	+	0.557984i	$0.811575\pi$
$942$		0			0
$943$	32.4254			1.05592
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.3548	+	30.3548i	0.986399	+	0.986399i	0.999909	−	0.0135095i	$-0.00430034\pi$
$947$	30.3548	+	30.3548i	0.986399	+	0.986399i	−0.0135095	+	0.999909i	$0.504300\pi$
$948$		0			0
$949$	12.0000	−	12.0000i	0.389536	−	0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			25.6077i			0.829515i	0.909932	+	0.414757i	$0.136134\pi$
$953$			25.6077i			0.829515i	−0.909932	+	0.414757i	$0.863866\pi$
$954$		0			0
$955$	−1.51575	+	1.51575i	−0.0490485	+	0.0490485i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.6579			0.505619
$960$		0			0
$961$	23.0000			0.741935
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.39230	+	4.39230i	−0.141393	+	0.141393i
$966$		0			0
$967$		−	34.8749i		−	1.12150i	−0.827985	−	0.560750i	$-0.810513\pi$
$967$		−	34.8749i		−	1.12150i	0.827985	−	0.560750i	$-0.189487\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.5606	−	18.5606i	0.595639	−	0.595639i	−0.343510	−	0.939149i	$-0.611616\pi$
$971$	18.5606	−	18.5606i	0.595639	−	0.595639i	0.939149	+	0.343510i	$0.111616\pi$
$972$		0			0
$973$	12.0000	+	12.0000i	0.384702	+	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.1769			0.869467			0.434733	−	0.900559i	$-0.356843\pi$
$977$	27.1769			0.869467			0.434733	+	0.900559i	$0.356843\pi$
$978$		0			0
$979$	−34.4959	−	34.4959i	−1.10249	−	1.10249i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			35.0507i			1.11794i	0.829186	+	0.558972i	$0.188804\pi$
$983$			35.0507i			1.11794i	−0.829186	+	0.558972i	$0.811196\pi$
$984$		0			0
$985$			6.14359i			0.195751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.92820	+	6.92820i	0.220304	+	0.220304i
$990$		0			0
$991$	12.2474			0.389053			0.194527	−	0.980897i	$-0.437683\pi$
$991$	12.2474			0.389053			0.194527	+	0.980897i	$0.437683\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.90276	−	2.90276i	−0.0920236	−	0.0920236i
$996$		0			0
$997$	−5.53590	+	5.53590i	−0.175324	+	0.175324i	−0.789314	−	0.613990i	$-0.789563\pi$
$997$	−5.53590	+	5.53590i	−0.175324	+	0.175324i	0.613990	+	0.789314i	$0.289563\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.l.577.1		8
3.2	odd	2		768.2.j.e.577.4	yes	8
4.3	odd	2	inner	2304.2.k.l.577.2		8
8.3	odd	2		2304.2.k.e.577.4		8
8.5	even	2		2304.2.k.e.577.3		8
12.11	even	2		768.2.j.e.577.2	yes	8
16.3	odd	4		2304.2.k.e.1729.3		8
16.5	even	4	inner	2304.2.k.l.1729.2		8
16.11	odd	4	inner	2304.2.k.l.1729.1		8
16.13	even	4		2304.2.k.e.1729.4		8
24.5	odd	2		768.2.j.f.577.1	yes	8
24.11	even	2		768.2.j.f.577.3	yes	8
32.5	even	8		9216.2.a.bi.1.2		4
32.11	odd	8		9216.2.a.bi.1.3		4
32.21	even	8		9216.2.a.bc.1.3		4
32.27	odd	8		9216.2.a.bc.1.2		4
48.5	odd	4		768.2.j.e.193.4	yes	8
48.11	even	4		768.2.j.e.193.2	✓	8
48.29	odd	4		768.2.j.f.193.1	yes	8
48.35	even	4		768.2.j.f.193.3	yes	8
96.5	odd	8		3072.2.a.l.1.3		4
96.11	even	8		3072.2.a.l.1.2		4
96.29	odd	8		3072.2.d.h.1537.7		8
96.35	even	8		3072.2.d.h.1537.3		8
96.53	odd	8		3072.2.a.r.1.2		4
96.59	even	8		3072.2.a.r.1.3		4
96.77	odd	8		3072.2.d.h.1537.2		8
96.83	even	8		3072.2.d.h.1537.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.e.193.2	✓	8	48.11	even	4
768.2.j.e.193.4	yes	8	48.5	odd	4
768.2.j.e.577.2	yes	8	12.11	even	2
768.2.j.e.577.4	yes	8	3.2	odd	2
768.2.j.f.193.1	yes	8	48.29	odd	4
768.2.j.f.193.3	yes	8	48.35	even	4
768.2.j.f.577.1	yes	8	24.5	odd	2
768.2.j.f.577.3	yes	8	24.11	even	2
2304.2.k.e.577.3		8	8.5	even	2
2304.2.k.e.577.4		8	8.3	odd	2
2304.2.k.e.1729.3		8	16.3	odd	4
2304.2.k.e.1729.4		8	16.13	even	4
2304.2.k.l.577.1		8	1.1	even	1	trivial
2304.2.k.l.577.2		8	4.3	odd	2	inner
2304.2.k.l.1729.1		8	16.11	odd	4	inner
2304.2.k.l.1729.2		8	16.5	even	4	inner
3072.2.a.l.1.2		4	96.11	even	8
3072.2.a.l.1.3		4	96.5	odd	8
3072.2.a.r.1.2		4	96.53	odd	8
3072.2.a.r.1.3		4	96.59	even	8
3072.2.d.h.1537.2		8	96.77	odd	8
3072.2.d.h.1537.3		8	96.35	even	8
3072.2.d.h.1537.6		8	96.83	even	8
3072.2.d.h.1537.7		8	96.29	odd	8
9216.2.a.bc.1.2		4	32.27	odd	8
9216.2.a.bc.1.3		4	32.21	even	8
9216.2.a.bi.1.2		4	32.5	even	8
9216.2.a.bi.1.3		4	32.11	odd	8

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.732051 + 0.732051i) q^{5} -2.44949i q^{7} +O(q^{10})\)	\(q+(-0.732051 + 0.732051i) q^{5} -2.44949i q^{7} +(-3.86370 + 3.86370i) q^{11} +(-3.00000 - 3.00000i) q^{13} +3.46410 q^{17} +(-0.378937 - 0.378937i) q^{19} +2.82843i q^{23} +3.92820i q^{25} +(4.19615 + 4.19615i) q^{29} +7.34847 q^{31} +(1.79315 + 1.79315i) q^{35} +(6.46410 - 6.46410i) q^{37} -11.4641i q^{41} +(2.44949 - 2.44949i) q^{43} +2.82843 q^{47} +1.00000 q^{49} +(6.73205 - 6.73205i) q^{53} -5.65685i q^{55} +(-9.79796 + 9.79796i) q^{59} +(6.46410 + 6.46410i) q^{61} +4.39230 q^{65} +(0.757875 + 0.757875i) q^{67} -16.2127i q^{71} +4.00000i q^{73} +(9.46410 + 9.46410i) q^{77} +2.44949 q^{79} +(-1.03528 - 1.03528i) q^{83} +(-2.53590 + 2.53590i) q^{85} +8.92820i q^{89} +(-7.34847 + 7.34847i) q^{91} +0.554803 q^{95} +14.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} - 24 q^{13} - 8 q^{29} + 24 q^{37} + 8 q^{49} + 40 q^{53} + 24 q^{61} - 48 q^{65} + 48 q^{77} - 48 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 24 * q^13 - 8 * q^29 + 24 * q^37 + 8 * q^49 + 40 * q^53 + 24 * q^61 - 48 * q^65 + 48 * q^77 - 48 * q^85 + 64 * q^97

Introduction

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