LMFDB - Embedded newform 1764.2.t.b.521.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, 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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.2.t.b.1097.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 + 3.00000i) q^{5} +(-3.67423 - 2.12132i) q^{11} +4.89898i q^{13} +(1.73205 - 3.00000i) q^{17} +(-4.24264 + 2.44949i) q^{19} +(-3.67423 + 2.12132i) q^{23} +(-3.50000 + 6.06218i) q^{25} +4.24264i q^{29} +(4.00000 + 6.92820i) q^{37} +3.46410 q^{41} -2.00000 q^{43} +(-3.46410 - 6.00000i) q^{47} +(-11.0227 - 6.36396i) q^{53} -14.6969i q^{55} +(-6.92820 + 12.0000i) q^{59} +(8.48528 - 4.89898i) q^{61} +(-14.6969 + 8.48528i) q^{65} +(-4.00000 + 6.92820i) q^{67} -4.24264i q^{71} +(4.24264 + 2.44949i) q^{73} +(2.00000 + 3.46410i) q^{79} -6.92820 q^{83} +12.0000 q^{85} +(-5.19615 - 9.00000i) q^{89} +(-14.6969 - 8.48528i) q^{95} +4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 28 q^{25} + 32 q^{37} - 16 q^{43} - 32 q^{67} + 16 q^{79} + 96 q^{85}+O(q^{100}) $ 8 * q - 28 * q^25 + 32 * q^37 - 16 * q^43 - 32 * q^67 + 16 * q^79 + 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.73205	+	3.00000i	0.774597	+	1.34164i	0.935021	+	0.354593i	$0.115380\pi$
$5$	1.73205	+	3.00000i	0.774597	+	1.34164i	−0.160424	+	0.987048i	$0.551286\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.67423	−	2.12132i	−1.10782	−	0.639602i	−0.169559	−	0.985520i	$-0.554234\pi$
$11$	−3.67423	−	2.12132i	−1.10782	−	0.639602i	−0.938265	+	0.345918i	$0.887568\pi$
$12$		0			0
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.73205	−	3.00000i	0.420084	−	0.727607i	−0.575863	−	0.817546i	$-0.695334\pi$
$17$	1.73205	−	3.00000i	0.420084	−	0.727607i	0.995947	+	0.0899392i	$0.0286673\pi$
$18$		0			0
$19$	−4.24264	+	2.44949i	−0.973329	+	0.561951i	−0.900249	−	0.435375i	$-0.856616\pi$
$19$	−4.24264	+	2.44949i	−0.973329	+	0.561951i	−0.0730792	+	0.997326i	$0.523283\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.67423	+	2.12132i	−0.766131	+	0.442326i	−0.831493	−	0.555536i	$-0.812513\pi$
$23$	−3.67423	+	2.12132i	−0.766131	+	0.442326i	0.0653618	+	0.997862i	$0.479180\pi$
$24$		0			0
$25$	−3.50000	+	6.06218i	−0.700000	+	1.21244i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.24264i			0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$			4.24264i			0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	0.981236	+	0.192809i	$0.0617599\pi$
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	−0.323640	+	0.946180i	$0.604907\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.46410			0.541002			0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410			0.541002			0.270501	+	0.962720i	$0.412811\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410	−	6.00000i	−0.505291	−	0.875190i	−0.999981	−	0.00612051i	$-0.998052\pi$
$47$	−3.46410	−	6.00000i	−0.505291	−	0.875190i	0.494690	−	0.869069i	$-0.335282\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.0227	−	6.36396i	−1.51408	−	0.874157i	−0.999864	−	0.0164995i	$-0.994748\pi$
$53$	−11.0227	−	6.36396i	−1.51408	−	0.874157i	−0.514221	−	0.857658i	$-0.671919\pi$
$54$		0			0
$55$		−	14.6969i		−	1.98173i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.92820	+	12.0000i	−0.901975	+	1.56227i	−0.0770484	+	0.997027i	$0.524550\pi$
$59$	−6.92820	+	12.0000i	−0.901975	+	1.56227i	−0.824927	+	0.565240i	$0.808784\pi$
$60$		0			0
$61$	8.48528	−	4.89898i	1.08643	−	0.627250i	0.153806	−	0.988101i	$-0.450847\pi$
$61$	8.48528	−	4.89898i	1.08643	−	0.627250i	0.932623	+	0.360851i	$0.117514\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.6969	+	8.48528i	−1.82293	+	1.05247i
$66$		0			0
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	$-0.995854\pi$
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	0.511237	+	0.859440i	$0.329187\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.24264i		−	0.503509i	−0.967791	−	0.251754i	$-0.918992\pi$
$71$		−	4.24264i		−	0.503509i	0.967791	−	0.251754i	$-0.0810075\pi$
$72$		0			0
$73$	4.24264	+	2.44949i	0.496564	+	0.286691i	0.727293	−	0.686327i	$-0.240778\pi$
$73$	4.24264	+	2.44949i	0.496564	+	0.286691i	−0.230730	+	0.973018i	$0.574111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.92820			−0.760469			−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820			−0.760469			−0.380235	+	0.924890i	$0.624157\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	−0.998218	−	0.0596775i	$-0.980993\pi$
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	0.447427	−	0.894321i	$-0.352341\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.6969	−	8.48528i	−1.50787	−	0.870572i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.19615	−	9.00000i	0.517036	−	0.895533i	−0.482768	−	0.875748i	$-0.660368\pi$
$101$	5.19615	−	9.00000i	0.517036	−	0.895533i	0.999804	−	0.0197851i	$-0.00629819\pi$
$102$		0			0
$103$	8.48528	−	4.89898i	0.836080	−	0.482711i	−0.0198501	−	0.999803i	$-0.506319\pi$
$103$	8.48528	−	4.89898i	0.836080	−	0.482711i	0.855930	+	0.517092i	$0.172986\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.67423	−	2.12132i	0.355202	−	0.205076i	−0.311772	−	0.950157i	$-0.600923\pi$
$107$	3.67423	−	2.12132i	0.355202	−	0.205076i	0.666974	+	0.745081i	$0.267589\pi$
$108$		0			0
$109$	−2.00000	+	3.46410i	−0.191565	+	0.331801i	−0.945769	−	0.324840i	$-0.894690\pi$
$109$	−2.00000	+	3.46410i	−0.191565	+	0.331801i	0.754204	+	0.656640i	$0.228023\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.7279i			1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$113$			12.7279i			1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$114$		0			0
$115$	−12.7279	−	7.34847i	−1.18688	−	0.685248i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.92820			−0.619677
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.3923	+	18.0000i	0.907980	+	1.57267i	0.816866	+	0.576827i	$0.195709\pi$
$131$	10.3923	+	18.0000i	0.907980	+	1.57267i	0.0911134	+	0.995841i	$0.470957\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.67423	−	2.12132i	−0.313911	−	0.181237i	0.334764	−	0.942302i	$-0.391343\pi$
$137$	−3.67423	−	2.12132i	−0.313911	−	0.181237i	−0.648675	+	0.761065i	$0.724677\pi$
$138$		0			0
$139$			9.79796i			0.831052i	0.909581	+	0.415526i	$0.136402\pi$
$139$			9.79796i			0.831052i	−0.909581	+	0.415526i	$0.863598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.3923	−	18.0000i	0.869048	−	1.50524i
$144$		0			0
$145$	−12.7279	+	7.34847i	−1.05700	+	0.610257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.0227	+	6.36396i	−0.903015	+	0.521356i	−0.878177	−	0.478335i	$-0.841240\pi$
$149$	−11.0227	+	6.36396i	−0.903015	+	0.521356i	−0.0248379	+	0.999691i	$0.507907\pi$
$150$		0			0
$151$	−11.0000	+	19.0526i	−0.895167	+	1.55048i	−0.0615699	+	0.998103i	$0.519611\pi$
$151$	−11.0000	+	19.0526i	−0.895167	+	1.55048i	−0.833597	+	0.552372i	$0.813723\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	0.665771	−	0.746156i	$-0.268103\pi$
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.92820			0.536120			0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820			0.536120			0.268060	+	0.963402i	$0.413617\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.19615	+	9.00000i	0.395056	+	0.684257i	0.993108	−	0.117200i	$-0.0373918\pi$
$173$	5.19615	+	9.00000i	0.395056	+	0.684257i	−0.598052	+	0.801457i	$0.704058\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.67423	+	2.12132i	0.274625	+	0.158555i	0.630988	−	0.775793i	$-0.282650\pi$
$179$	3.67423	+	2.12132i	0.274625	+	0.158555i	−0.356362	+	0.934348i	$0.615983\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.8564	+	24.0000i	−1.01874	+	1.76452i
$186$		0			0
$187$	−12.7279	+	7.34847i	−0.930758	+	0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.3712	−	10.6066i	1.32929	−	0.767467i	0.344101	−	0.938933i	$-0.388184\pi$
$191$	18.3712	−	10.6066i	1.32929	−	0.767467i	0.985190	+	0.171466i	$0.0548503\pi$
$192$		0			0
$193$	10.0000	−	17.3205i	0.719816	−	1.24676i	−0.241257	−	0.970461i	$-0.577560\pi$
$193$	10.0000	−	17.3205i	0.719816	−	1.24676i	0.961073	−	0.276296i	$-0.0891071\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.24264i			0.302276i	0.988513	+	0.151138i	$0.0482937\pi$
$197$			4.24264i			0.302276i	−0.988513	+	0.151138i	$0.951706\pi$
$198$		0			0
$199$	21.2132	+	12.2474i	1.50376	+	0.868199i	0.999990	+	0.00436292i	$0.00138876\pi$
$199$	21.2132	+	12.2474i	1.50376	+	0.868199i	0.503774	+	0.863836i	$0.331945\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.7846			1.43770
$210$		0			0
$211$	10.0000			0.688428			0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000			0.688428			0.344214	+	0.938891i	$0.388145\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46410	−	6.00000i	−0.236250	−	0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.6969	+	8.48528i	0.988623	+	0.570782i
$222$		0			0
$223$			14.6969i			0.984180i	0.870544	+	0.492090i	$0.163767\pi$
$223$			14.6969i			0.984180i	−0.870544	+	0.492090i	$0.836233\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−12.7279	+	7.34847i	−0.841085	+	0.485601i	−0.857633	−	0.514263i	$-0.828066\pi$
$229$	−12.7279	+	7.34847i	−0.841085	+	0.485601i	0.0165480	+	0.999863i	$0.494732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.3712	+	10.6066i	−1.20354	+	0.694862i	−0.961340	−	0.275366i	$-0.911201\pi$
$233$	−18.3712	+	10.6066i	−1.20354	+	0.694862i	−0.242196	+	0.970227i	$0.577868\pi$
$234$		0			0
$235$	12.0000	−	20.7846i	0.782794	−	1.35584i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	29.6985i		−	1.92104i	−0.278219	−	0.960518i	$-0.589744\pi$
$239$		−	29.6985i		−	1.92104i	0.278219	−	0.960518i	$-0.410256\pi$
$240$		0			0
$241$	4.24264	+	2.44949i	0.273293	+	0.157786i	0.630383	−	0.776284i	$-0.282898\pi$
$241$	4.24264	+	2.44949i	0.273293	+	0.157786i	−0.357090	+	0.934070i	$0.616231\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.0000	−	20.7846i	−0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.92820			0.437304			0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820			0.437304			0.218652	+	0.975803i	$0.429834\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.19615	−	9.00000i	−0.324127	−	0.561405i	0.657208	−	0.753709i	$-0.271737\pi$
$257$	−5.19615	−	9.00000i	−0.324127	−	0.561405i	−0.981335	+	0.192304i	$0.938404\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.67423	+	2.12132i	0.226563	+	0.130806i	0.608985	−	0.793181i	$-0.291577\pi$
$263$	3.67423	+	2.12132i	0.226563	+	0.130806i	−0.382422	+	0.923988i	$0.624910\pi$
$264$		0			0
$265$		−	44.0908i		−	2.70848i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.73205	+	3.00000i	−0.105605	+	0.182913i	−0.913985	−	0.405747i	$-0.867011\pi$
$269$	−1.73205	+	3.00000i	−0.105605	+	0.182913i	0.808380	+	0.588661i	$0.200345\pi$
$270$		0			0
$271$	16.9706	−	9.79796i	1.03089	−	0.595184i	0.113649	−	0.993521i	$-0.463746\pi$
$271$	16.9706	−	9.79796i	1.03089	−	0.595184i	0.917239	+	0.398337i	$0.130413\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	25.7196	−	14.8492i	1.55095	−	0.895443i
$276$		0			0
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	−0.894503	−	0.447062i	$-0.852470\pi$
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	0.834419	+	0.551131i	$0.185804\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.24264i		−	0.253095i	−0.991961	−	0.126547i	$-0.959610\pi$
$281$		−	4.24264i		−	0.253095i	0.991961	−	0.126547i	$-0.0403896\pi$
$282$		0			0
$283$	−12.7279	−	7.34847i	−0.756596	−	0.436821i	0.0714760	−	0.997442i	$-0.477229\pi$
$283$	−12.7279	−	7.34847i	−0.756596	−	0.436821i	−0.828072	+	0.560621i	$0.810562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.50000	+	4.33013i	0.147059	+	0.254713i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.3205			−1.01187			−0.505937	−	0.862570i	$-0.668853\pi$
$293$	−17.3205			−1.01187			−0.505937	+	0.862570i	$0.668853\pi$
$294$		0			0
$295$	−48.0000			−2.79467
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.3923	−	18.0000i	−0.601003	−	1.04097i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	29.3939	+	16.9706i	1.68309	+	0.971732i
$306$		0			0
$307$			14.6969i			0.838799i	0.907802	+	0.419399i	$0.137759\pi$
$307$			14.6969i			0.838799i	−0.907802	+	0.419399i	$0.862241\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	−0.599963	−	0.800027i	$-0.704818\pi$
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	0.992826	+	0.119570i	$0.0381515\pi$
$312$		0			0
$313$	16.9706	−	9.79796i	0.959233	−	0.553813i	0.0632961	−	0.997995i	$-0.479839\pi$
$313$	16.9706	−	9.79796i	0.959233	−	0.553813i	0.895937	+	0.444181i	$0.146505\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.0227	−	6.36396i	0.619097	−	0.357436i	−0.157421	−	0.987532i	$-0.550318\pi$
$317$	11.0227	−	6.36396i	0.619097	−	0.357436i	0.776517	+	0.630096i	$0.216984\pi$
$318$		0			0
$319$	9.00000	−	15.5885i	0.503903	−	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			16.9706i			0.944267i
$324$		0			0
$325$	−29.6985	−	17.1464i	−1.64738	−	0.951113i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.0000	−	19.0526i	−0.604615	−	1.04722i	−0.992112	−	0.125353i	$-0.959994\pi$
$331$	−11.0000	−	19.0526i	−0.604615	−	1.04722i	0.387498	−	0.921871i	$-0.373340\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−27.7128			−1.51411
$336$		0			0
$337$	16.0000			0.871576			0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000			0.871576			0.435788	+	0.900049i	$0.356470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.3712	+	10.6066i	0.986216	+	0.569392i	0.904141	−	0.427234i	$-0.140512\pi$
$347$	18.3712	+	10.6066i	0.986216	+	0.569392i	0.0820751	+	0.996626i	$0.473845\pi$
$348$		0			0
$349$		−	19.5959i		−	1.04895i	−0.851427	−	0.524473i	$-0.824262\pi$
$349$		−	19.5959i		−	1.04895i	0.851427	−	0.524473i	$-0.175738\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.73205	−	3.00000i	0.0921878	−	0.159674i	−0.816244	−	0.577708i	$-0.803947\pi$
$353$	1.73205	−	3.00000i	0.0921878	−	0.159674i	0.908431	+	0.418034i	$0.137281\pi$
$354$		0			0
$355$	12.7279	−	7.34847i	0.675528	−	0.390016i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.3712	−	10.6066i	0.969593	−	0.559795i	0.0704812	−	0.997513i	$-0.477547\pi$
$359$	18.3712	−	10.6066i	0.969593	−	0.559795i	0.899112	+	0.437718i	$0.144213\pi$
$360$		0			0
$361$	2.50000	−	4.33013i	0.131579	−	0.227901i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			16.9706i			0.888280i
$366$		0			0
$367$	21.2132	+	12.2474i	1.10732	+	0.639312i	0.938134	−	0.346272i	$-0.112553\pi$
$367$	21.2132	+	12.2474i	1.10732	+	0.639312i	0.169186	+	0.985584i	$0.445886\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.00000	−	1.73205i	−0.0517780	−	0.0896822i	0.838975	−	0.544170i	$-0.183156\pi$
$373$	−1.00000	−	1.73205i	−0.0517780	−	0.0896822i	−0.890753	+	0.454488i	$0.849822\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.7846			−1.07046
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.46410	+	6.00000i	0.177007	+	0.306586i	0.940854	−	0.338812i	$-0.110025\pi$
$383$	3.46410	+	6.00000i	0.177007	+	0.306586i	−0.763847	+	0.645398i	$0.776692\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.3712	+	10.6066i	0.931455	+	0.537776i	0.887272	−	0.461247i	$-0.152598\pi$
$389$	18.3712	+	10.6066i	0.931455	+	0.537776i	0.0441839	+	0.999023i	$0.485931\pi$
$390$		0			0
$391$			14.6969i			0.743256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.92820	+	12.0000i	−0.348596	+	0.603786i
$396$		0			0
$397$	25.4558	−	14.6969i	1.27759	−	0.737618i	0.301187	−	0.953565i	$-0.402617\pi$
$397$	25.4558	−	14.6969i	1.27759	−	0.737618i	0.976405	+	0.215947i	$0.0692838\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.3712	−	10.6066i	0.917413	−	0.529668i	0.0346039	−	0.999401i	$-0.488983\pi$
$401$	18.3712	−	10.6066i	0.917413	−	0.529668i	0.882809	+	0.469733i	$0.155650\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	33.9411i		−	1.68240i
$408$		0			0
$409$	4.24264	+	2.44949i	0.209785	+	0.121119i	0.601212	−	0.799090i	$-0.294685\pi$
$409$	4.24264	+	2.44949i	0.209785	+	0.121119i	−0.391426	+	0.920209i	$0.628018\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000	−	20.7846i	−0.589057	−	1.02028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	27.7128			1.35386			0.676930	−	0.736048i	$-0.263310\pi$
$419$	27.7128			1.35386			0.676930	+	0.736048i	$0.263310\pi$
$420$		0			0
$421$	−34.0000			−1.65706			−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000			−1.65706			−0.828529	+	0.559946i	$0.810822\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.1244	+	21.0000i	0.588118	+	1.01865i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.67423	−	2.12132i	−0.176982	−	0.102180i	0.408892	−	0.912583i	$-0.365915\pi$
$431$	−3.67423	−	2.12132i	−0.176982	−	0.102180i	−0.585874	+	0.810402i	$0.699249\pi$
$432$		0			0
$433$			29.3939i			1.41258i	0.707923	+	0.706290i	$0.249632\pi$
$433$			29.3939i			1.41258i	−0.707923	+	0.706290i	$0.750368\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.3923	−	18.0000i	0.497131	−	0.861057i
$438$		0			0
$439$	−12.7279	+	7.34847i	−0.607471	+	0.350723i	−0.771975	−	0.635653i	$-0.780731\pi$
$439$	−12.7279	+	7.34847i	−0.607471	+	0.350723i	0.164504	+	0.986376i	$0.447398\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	−0.214745	−	0.976670i	$-0.568892\pi$
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	0.738449	+	0.674309i	$0.235559\pi$
$444$		0			0
$445$	18.0000	−	31.1769i	0.853282	−	1.47793i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.24264i		−	0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$		−	4.24264i		−	0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$		0			0
$451$	−12.7279	−	7.34847i	−0.599334	−	0.346026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	0.944286	−	0.329125i	$-0.106754\pi$
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	−0.757174	+	0.653213i	$0.773421\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.46410			0.161339			0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410			0.161339			0.0806696	+	0.996741i	$0.474294\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.3923	+	18.0000i	0.480899	+	0.832941i	0.999760	−	0.0219178i	$-0.00697721\pi$
$467$	10.3923	+	18.0000i	0.480899	+	0.832941i	−0.518861	+	0.854858i	$0.673644\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.34847	+	4.24264i	0.337883	+	0.195077i
$474$		0			0
$475$		−	34.2929i		−	1.57346i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.92820	+	12.0000i	−0.316558	+	0.548294i	−0.979767	−	0.200140i	$-0.935860\pi$
$479$	−6.92820	+	12.0000i	−0.316558	+	0.548294i	0.663210	+	0.748434i	$0.269194\pi$
$480$		0			0
$481$	−33.9411	+	19.5959i	−1.54758	+	0.893497i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.6969	+	8.48528i	−0.667354	+	0.385297i
$486$		0			0
$487$	7.00000	−	12.1244i	0.317200	−	0.549407i	−0.662702	−	0.748883i	$-0.730591\pi$
$487$	7.00000	−	12.1244i	0.317200	−	0.549407i	0.979903	+	0.199476i	$0.0639239\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.7279i		−	0.574403i	−0.957870	−	0.287202i	$-0.907275\pi$
$491$		−	12.7279i		−	0.574403i	0.957870	−	0.287202i	$-0.0927249\pi$
$492$		0			0
$493$	12.7279	+	7.34847i	0.573237	+	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−5.00000	−	8.66025i	−0.223831	−	0.387686i	0.732137	−	0.681157i	$-0.238523\pi$
$499$	−5.00000	−	8.66025i	−0.223831	−	0.387686i	−0.955968	+	0.293471i	$0.905190\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.6410			1.54457			0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410			1.54457			0.772283	+	0.635278i	$0.219115\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.1244	−	21.0000i	−0.537403	−	0.930809i	−0.999043	−	0.0437414i	$-0.986072\pi$
$509$	−12.1244	−	21.0000i	−0.537403	−	0.930809i	0.461640	−	0.887067i	$-0.347261\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	29.3939	+	16.9706i	1.29525	+	0.747812i
$516$		0			0
$517$			29.3939i			1.29274i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.73205	+	3.00000i	−0.0758825	+	0.131432i	−0.901470	−	0.432842i	$-0.857511\pi$
$521$	−1.73205	+	3.00000i	−0.0758825	+	0.131432i	0.825587	+	0.564275i	$0.190844\pi$
$522$		0			0
$523$	−33.9411	+	19.5959i	−1.48414	+	0.856870i	−0.999837	−	0.0180299i	$-0.994261\pi$
$523$	−33.9411	+	19.5959i	−1.48414	+	0.856870i	−0.484304	+	0.874900i	$0.660927\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.50000	+	4.33013i	−0.108696	+	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			16.9706i			0.735077i
$534$		0			0
$535$	12.7279	+	7.34847i	0.550276	+	0.317702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.0000	−	22.5167i	−0.558914	−	0.968067i	−0.997587	−	0.0694205i	$-0.977885\pi$
$541$	−13.0000	−	22.5167i	−0.558914	−	0.968067i	0.438674	−	0.898646i	$-0.355448\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.8564			−0.593543
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.3923	−	18.0000i	−0.442727	−	0.766826i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.67423	−	2.12132i	−0.155682	−	0.0898832i	0.420135	−	0.907462i	$-0.361983\pi$
$557$	−3.67423	−	2.12132i	−0.155682	−	0.0898832i	−0.575817	+	0.817578i	$0.695316\pi$
$558$		0			0
$559$		−	9.79796i		−	0.414410i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.3923	−	18.0000i	0.437983	−	0.758610i	−0.559550	−	0.828796i	$-0.689026\pi$
$563$	10.3923	−	18.0000i	0.437983	−	0.758610i	0.997534	+	0.0701867i	$0.0223595\pi$
$564$		0			0
$565$	−38.1838	+	22.0454i	−1.60640	+	0.927457i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.67423	+	2.12132i	−0.154032	+	0.0889304i	−0.575035	−	0.818129i	$-0.695012\pi$
$569$	−3.67423	+	2.12132i	−0.154032	+	0.0889304i	0.421003	+	0.907059i	$0.361678\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	29.6985i		−	1.23851i
$576$		0			0
$577$	−8.48528	−	4.89898i	−0.353247	−	0.203947i	0.312868	−	0.949797i	$-0.398710\pi$
$577$	−8.48528	−	4.89898i	−0.353247	−	0.203947i	−0.666114	+	0.745850i	$0.732044\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	27.0000	+	46.7654i	1.11823	+	1.93682i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.7846			0.857873			0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846			0.857873			0.428936	+	0.903335i	$0.358888\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.66025	+	15.0000i	0.355634	+	0.615976i	0.987226	−	0.159325i	$-0.0509316\pi$
$593$	8.66025	+	15.0000i	0.355634	+	0.615976i	−0.631592	+	0.775301i	$0.717598\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.67423	−	2.12132i	−0.150125	−	0.0866748i	0.423056	−	0.906104i	$-0.360957\pi$
$599$	−3.67423	−	2.12132i	−0.150125	−	0.0866748i	−0.573181	+	0.819429i	$0.694291\pi$
$600$		0			0
$601$			29.3939i			1.19900i	0.800374	+	0.599501i	$0.204634\pi$
$601$			29.3939i			1.19900i	−0.800374	+	0.599501i	$0.795366\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.1244	+	21.0000i	−0.492925	+	0.853771i
$606$		0			0
$607$	4.24264	−	2.44949i	0.172203	−	0.0994217i	−0.411421	−	0.911445i	$-0.634967\pi$
$607$	4.24264	−	2.44949i	0.172203	−	0.0994217i	0.583624	+	0.812024i	$0.301634\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.3939	−	16.9706i	1.18915	−	0.686555i
$612$		0			0
$613$	−7.00000	+	12.1244i	−0.282727	+	0.489698i	−0.972056	−	0.234751i	$-0.924572\pi$
$613$	−7.00000	+	12.1244i	−0.282727	+	0.489698i	0.689328	+	0.724449i	$0.257906\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.7279i			0.512407i	0.966623	+	0.256203i	$0.0824717\pi$
$617$			12.7279i			0.512407i	−0.966623	+	0.256203i	$0.917528\pi$
$618$		0			0
$619$	8.48528	+	4.89898i	0.341052	+	0.196907i	0.660737	−	0.750617i	$-0.270244\pi$
$619$	8.48528	+	4.89898i	0.341052	+	0.196907i	−0.319685	+	0.947524i	$0.603577\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27.7128			1.10498
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.92820	−	12.0000i	−0.274937	−	0.476205i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−25.7196	−	14.8492i	−1.01586	−	0.586510i	−0.102961	−	0.994685i	$-0.532832\pi$
$641$	−25.7196	−	14.8492i	−1.01586	−	0.586510i	−0.912903	+	0.408176i	$0.866165\pi$
$642$		0			0
$643$		−	14.6969i		−	0.579591i	−0.957089	−	0.289795i	$-0.906413\pi$
$643$		−	14.6969i		−	0.579591i	0.957089	−	0.289795i	$-0.0935872\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.7846	+	36.0000i	−0.817127	+	1.41531i	0.0906629	+	0.995882i	$0.471101\pi$
$647$	−20.7846	+	36.0000i	−0.817127	+	1.41531i	−0.907790	+	0.419424i	$0.862232\pi$
$648$		0			0
$649$	50.9117	−	29.3939i	1.99846	−	1.15381i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−40.4166	+	23.3345i	−1.58162	+	0.913150i	−0.587000	+	0.809587i	$0.699691\pi$
$653$	−40.4166	+	23.3345i	−1.58162	+	0.913150i	−0.994623	+	0.103564i	$0.966975\pi$
$654$		0			0
$655$	−36.0000	+	62.3538i	−1.40664	+	2.43637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			46.6690i			1.81797i	0.416831	+	0.908984i	$0.363141\pi$
$659$			46.6690i			1.81797i	−0.416831	+	0.908984i	$0.636859\pi$
$660$		0			0
$661$	33.9411	+	19.5959i	1.32016	+	0.762193i	0.983753	−	0.179525i	$-0.0574561\pi$
$661$	33.9411	+	19.5959i	1.32016	+	0.762193i	0.336403	+	0.941718i	$0.390789\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−9.00000	−	15.5885i	−0.348481	−	0.603587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−41.5692			−1.60476
$672$		0			0
$673$	−20.0000			−0.770943			−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000			−0.770943			−0.385472	+	0.922720i	$0.625961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.5885	+	27.0000i	0.599113	+	1.03769i	0.992952	+	0.118515i	$0.0378134\pi$
$677$	15.5885	+	27.0000i	0.599113	+	1.03769i	−0.393839	+	0.919179i	$0.628853\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.0227	+	6.36396i	0.421772	+	0.243510i	0.695835	−	0.718201i	$-0.255034\pi$
$683$	11.0227	+	6.36396i	0.421772	+	0.243510i	−0.274063	+	0.961712i	$0.588368\pi$
$684$		0			0
$685$		−	14.6969i		−	0.561541i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	31.1769	−	54.0000i	1.18775	−	2.05724i
$690$		0			0
$691$	25.4558	−	14.6969i	0.968386	−	0.559098i	0.0696421	−	0.997572i	$-0.477814\pi$
$691$	25.4558	−	14.6969i	0.968386	−	0.559098i	0.898744	+	0.438474i	$0.144481\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−29.3939	+	16.9706i	−1.11497	+	0.643730i
$696$		0			0
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$	−33.9411	−	19.5959i	−1.28011	−	0.739074i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	0.826023	−	0.563636i	$-0.190598\pi$
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	−0.901135	+	0.433539i	$0.857265\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	72.0000			2.69265
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.8564	+	24.0000i	0.516757	+	0.895049i	0.999811	+	0.0194584i	$0.00619418\pi$
$719$	13.8564	+	24.0000i	0.516757	+	0.895049i	−0.483054	+	0.875591i	$0.660472\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.7196	−	14.8492i	−0.955204	−	0.551487i
$726$		0			0
$727$		−	29.3939i		−	1.09016i	−0.838385	−	0.545079i	$-0.816500\pi$
$727$		−	29.3939i		−	1.09016i	0.838385	−	0.545079i	$-0.183500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.46410	+	6.00000i	−0.128124	+	0.221918i
$732$		0			0
$733$	38.1838	−	22.0454i	1.41035	−	0.814266i	0.414929	−	0.909854i	$-0.363807\pi$
$733$	38.1838	−	22.0454i	1.41035	−	0.814266i	0.995421	+	0.0955883i	$0.0304732\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.3939	−	16.9706i	1.08274	−	0.625119i
$738$		0			0
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.24264i			0.155647i	0.996967	+	0.0778237i	$0.0247971\pi$
$743$			4.24264i			0.155647i	−0.996967	+	0.0778237i	$0.975203\pi$
$744$		0			0
$745$	−38.1838	−	22.0454i	−1.39894	−	0.807681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	0.760263	−	0.649616i	$-0.225070\pi$
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	−0.942715	+	0.333599i	$0.891737\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−76.2102			−2.77357
$756$		0			0
$757$	28.0000			1.01768			0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000			1.01768			0.508839	+	0.860862i	$0.330075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.5167	−	39.0000i	−0.816228	−	1.41375i	−0.908443	−	0.418010i	$-0.862728\pi$
$761$	−22.5167	−	39.0000i	−0.816228	−	1.41375i	0.0922143	−	0.995739i	$-0.470606\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−58.7878	−	33.9411i	−2.12270	−	1.22554i
$768$		0			0
$769$			9.79796i			0.353323i	0.984272	+	0.176662i	$0.0565299\pi$
$769$			9.79796i			0.353323i	−0.984272	+	0.176662i	$0.943470\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.66025	+	15.0000i	−0.311488	+	0.539513i	−0.978685	−	0.205369i	$-0.934161\pi$
$773$	−8.66025	+	15.0000i	−0.311488	+	0.539513i	0.667197	+	0.744881i	$0.267494\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.6969	+	8.48528i	−0.526572	+	0.304017i
$780$		0			0
$781$	−9.00000	+	15.5885i	−0.322045	+	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−16.9706	−	9.79796i	−0.604935	−	0.349260i	0.166045	−	0.986118i	$-0.446900\pi$
$787$	−16.9706	−	9.79796i	−0.604935	−	0.349260i	−0.770981	+	0.636859i	$0.780234\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	24.0000	+	41.5692i	0.852265	+	1.47617i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.3923			0.368114			0.184057	−	0.982916i	$-0.441077\pi$
$797$	10.3923			0.368114			0.184057	+	0.982916i	$0.441077\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.3923	−	18.0000i	−0.366736	−	0.635206i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−33.0681	−	19.0919i	−1.16261	−	0.671235i	−0.210684	−	0.977554i	$-0.567569\pi$
$809$	−33.0681	−	19.0919i	−1.16261	−	0.671235i	−0.951929	+	0.306319i	$0.900902\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.8564	−	24.0000i	0.485369	−	0.840683i
$816$		0			0
$817$	8.48528	−	4.89898i	0.296862	−	0.171394i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.0227	+	6.36396i	−0.384695	+	0.222104i	−0.679859	−	0.733343i	$-0.737959\pi$
$821$	−11.0227	+	6.36396i	−0.384695	+	0.222104i	0.295164	+	0.955447i	$0.404626\pi$
$822$		0			0
$823$	−10.0000	+	17.3205i	−0.348578	+	0.603755i	−0.985997	−	0.166762i	$-0.946669\pi$
$823$	−10.0000	+	17.3205i	−0.348578	+	0.603755i	0.637419	+	0.770517i	$0.280002\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.7279i			0.442593i	0.975207	+	0.221297i	$0.0710289\pi$
$827$			12.7279i			0.442593i	−0.975207	+	0.221297i	$0.928971\pi$
$828$		0			0
$829$	21.2132	+	12.2474i	0.736765	+	0.425371i	0.820892	−	0.571084i	$-0.193477\pi$
$829$	21.2132	+	12.2474i	0.736765	+	0.425371i	−0.0841269	+	0.996455i	$0.526810\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	12.0000	+	20.7846i	0.415277	+	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.6410			−1.19594			−0.597970	−	0.801518i	$-0.704026\pi$
$839$	−34.6410			−1.19594			−0.597970	+	0.801518i	$0.704026\pi$
$840$		0			0
$841$	11.0000			0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.0526	−	33.0000i	−0.655428	−	1.13523i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−29.3939	−	16.9706i	−1.00761	−	0.581743i
$852$		0			0
$853$		−	9.79796i		−	0.335476i	−0.985832	−	0.167738i	$-0.946354\pi$
$853$		−	9.79796i		−	0.335476i	0.985832	−	0.167738i	$-0.0536462\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.1244	−	21.0000i	0.414160	−	0.717346i	−0.581180	−	0.813775i	$-0.697409\pi$
$857$	12.1244	−	21.0000i	0.414160	−	0.717346i	0.995340	+	0.0964289i	$0.0307420\pi$
$858$		0			0
$859$	4.24264	−	2.44949i	0.144757	−	0.0835755i	−0.425872	−	0.904783i	$-0.640033\pi$
$859$	4.24264	−	2.44949i	0.144757	−	0.0835755i	0.570629	+	0.821208i	$0.306699\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−47.7650	+	27.5772i	−1.62594	+	0.938738i	−0.640654	+	0.767830i	$0.721337\pi$
$863$	−47.7650	+	27.5772i	−1.62594	+	0.938738i	−0.985287	+	0.170908i	$0.945330\pi$
$864$		0			0
$865$	−18.0000	+	31.1769i	−0.612018	+	1.06005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	16.9706i		−	0.575687i
$870$		0			0
$871$	−33.9411	−	19.5959i	−1.15005	−	0.663982i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.1769			1.05038			0.525188	−	0.850986i	$-0.323995\pi$
$881$	31.1769			1.05038			0.525188	+	0.850986i	$0.323995\pi$
$882$		0			0
$883$	46.0000			1.54802			0.774012	−	0.633171i	$-0.218247\pi$
$883$	46.0000			1.54802			0.774012	+	0.633171i	$0.218247\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.46410	−	6.00000i	−0.116313	−	0.201460i	0.801991	−	0.597336i	$-0.203774\pi$
$887$	−3.46410	−	6.00000i	−0.116313	−	0.201460i	−0.918304	+	0.395876i	$0.870441\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29.3939	+	16.9706i	0.983629	+	0.567898i
$894$		0			0
$895$			14.6969i			0.491264i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−38.1838	+	22.0454i	−1.27209	+	0.734439i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.6969	+	8.48528i	−0.488543	+	0.282060i
$906$		0			0
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	−0.988816	−	0.149140i	$-0.952349\pi$
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	0.623567	+	0.781770i	$0.285683\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	21.2132i		−	0.702825i	−0.936221	−	0.351412i	$-0.885702\pi$
$911$		−	21.2132i		−	0.702825i	0.936221	−	0.351412i	$-0.114298\pi$
$912$		0			0
$913$	25.4558	+	14.6969i	0.842465	+	0.486398i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	7.00000	+	12.1244i	0.230909	+	0.399946i	0.958076	−	0.286515i	$-0.0924968\pi$
$919$	7.00000	+	12.1244i	0.230909	+	0.399946i	−0.727167	+	0.686461i	$0.759163\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.7846			0.684134
$924$		0			0
$925$	−56.0000			−1.84127
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.5167	+	39.0000i	0.738748	+	1.27955i	0.953059	+	0.302783i	$0.0979158\pi$
$929$	22.5167	+	39.0000i	0.738748	+	1.27955i	−0.214312	+	0.976765i	$0.568751\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−44.0908	−	25.4558i	−1.44192	−	0.832495i
$936$		0			0
$937$		−	19.5959i		−	0.640171i	−0.947389	−	0.320085i	$-0.896288\pi$
$937$		−	19.5959i		−	0.640171i	0.947389	−	0.320085i	$-0.103712\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.5167	+	39.0000i	−0.734022	+	1.27136i	0.221129	+	0.975245i	$0.429026\pi$
$941$	−22.5167	+	39.0000i	−0.734022	+	1.27136i	−0.955151	+	0.296119i	$0.904307\pi$
$942$		0			0
$943$	−12.7279	+	7.34847i	−0.414478	+	0.239299i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.7196	+	14.8492i	−0.835776	+	0.482536i	−0.855826	−	0.517263i	$-0.826951\pi$
$947$	−25.7196	+	14.8492i	−0.835776	+	0.482536i	0.0200502	+	0.999799i	$0.493617\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			38.1838i			1.23689i	0.785827	+	0.618447i	$0.212238\pi$
$953$			38.1838i			1.23689i	−0.785827	+	0.618447i	$0.787762\pi$
$954$		0			0
$955$	63.6396	+	36.7423i	2.05933	+	1.18895i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	69.2820			2.23027
$966$		0			0
$967$	−28.0000			−0.900419			−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000			−0.900419			−0.450210	+	0.892923i	$0.648651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.8564	+	24.0000i	0.444673	+	0.770197i	0.998029	−	0.0627481i	$-0.0199865\pi$
$971$	13.8564	+	24.0000i	0.444673	+	0.770197i	−0.553356	+	0.832945i	$0.686653\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.67423	+	2.12132i	0.117549	+	0.0678671i	0.557622	−	0.830095i	$-0.311714\pi$
$977$	3.67423	+	2.12132i	0.117549	+	0.0678671i	−0.440073	+	0.897962i	$0.645047\pi$
$978$		0			0
$979$			44.0908i			1.40915i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.3205	+	30.0000i	−0.552438	+	0.956851i	0.445659	+	0.895203i	$0.352969\pi$
$983$	−17.3205	+	30.0000i	−0.552438	+	0.956851i	−0.998098	+	0.0616488i	$0.980364\pi$
$984$		0			0
$985$	−12.7279	+	7.34847i	−0.405545	+	0.234142i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.34847	−	4.24264i	0.233668	−	0.134908i
$990$		0			0
$991$	13.0000	−	22.5167i	0.412959	−	0.715265i	−0.582253	−	0.813008i	$-0.697829\pi$
$991$	13.0000	−	22.5167i	0.412959	−	0.715265i	0.995212	+	0.0977423i	$0.0311621\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			84.8528i			2.69002i
$996$		0			0
$997$	25.4558	+	14.6969i	0.806195	+	0.465457i	0.845633	−	0.533765i	$-0.179223\pi$
$997$	25.4558	+	14.6969i	0.806195	+	0.465457i	−0.0394380	+	0.999222i	$0.512557\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.t.b.521.3		8
3.2	odd	2	inner	1764.2.t.b.521.2		8
7.2	even	3	inner	1764.2.t.b.1097.4		8
7.3	odd	6		252.2.f.a.125.3	yes	4
7.4	even	3		252.2.f.a.125.2	yes	4
7.5	odd	6	inner	1764.2.t.b.1097.2		8
7.6	odd	2	inner	1764.2.t.b.521.1		8
21.2	odd	6	inner	1764.2.t.b.1097.1		8
21.5	even	6	inner	1764.2.t.b.1097.3		8
21.11	odd	6		252.2.f.a.125.4	yes	4
21.17	even	6		252.2.f.a.125.1	✓	4
21.20	even	2	inner	1764.2.t.b.521.4		8
28.3	even	6		1008.2.k.b.881.4		4
28.11	odd	6		1008.2.k.b.881.1		4
35.3	even	12		6300.2.f.b.3149.4		8
35.4	even	6		6300.2.d.c.3401.2		4
35.17	even	12		6300.2.f.b.3149.6		8
35.18	odd	12		6300.2.f.b.3149.8		8
35.24	odd	6		6300.2.d.c.3401.4		4
35.32	odd	12		6300.2.f.b.3149.2		8
56.3	even	6		4032.2.k.d.3905.2		4
56.11	odd	6		4032.2.k.d.3905.3		4
56.45	odd	6		4032.2.k.a.3905.1		4
56.53	even	6		4032.2.k.a.3905.4		4
63.4	even	3		2268.2.x.i.1889.3		8
63.11	odd	6		2268.2.x.i.377.2		8
63.25	even	3		2268.2.x.i.377.4		8
63.31	odd	6		2268.2.x.i.1889.2		8
63.32	odd	6		2268.2.x.i.1889.1		8
63.38	even	6		2268.2.x.i.377.3		8
63.52	odd	6		2268.2.x.i.377.1		8
63.59	even	6		2268.2.x.i.1889.4		8
84.11	even	6		1008.2.k.b.881.3		4
84.59	odd	6		1008.2.k.b.881.2		4
105.17	odd	12		6300.2.f.b.3149.5		8
105.32	even	12		6300.2.f.b.3149.1		8
105.38	odd	12		6300.2.f.b.3149.3		8
105.53	even	12		6300.2.f.b.3149.7		8
105.59	even	6		6300.2.d.c.3401.3		4
105.74	odd	6		6300.2.d.c.3401.1		4
168.11	even	6		4032.2.k.d.3905.1		4
168.53	odd	6		4032.2.k.a.3905.2		4
168.59	odd	6		4032.2.k.d.3905.4		4
168.101	even	6		4032.2.k.a.3905.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.f.a.125.1	✓	4	21.17	even	6
252.2.f.a.125.2	yes	4	7.4	even	3
252.2.f.a.125.3	yes	4	7.3	odd	6
252.2.f.a.125.4	yes	4	21.11	odd	6
1008.2.k.b.881.1		4	28.11	odd	6
1008.2.k.b.881.2		4	84.59	odd	6
1008.2.k.b.881.3		4	84.11	even	6
1008.2.k.b.881.4		4	28.3	even	6
1764.2.t.b.521.1		8	7.6	odd	2	inner
1764.2.t.b.521.2		8	3.2	odd	2	inner
1764.2.t.b.521.3		8	1.1	even	1	trivial
1764.2.t.b.521.4		8	21.20	even	2	inner
1764.2.t.b.1097.1		8	21.2	odd	6	inner
1764.2.t.b.1097.2		8	7.5	odd	6	inner
1764.2.t.b.1097.3		8	21.5	even	6	inner
1764.2.t.b.1097.4		8	7.2	even	3	inner
2268.2.x.i.377.1		8	63.52	odd	6
2268.2.x.i.377.2		8	63.11	odd	6
2268.2.x.i.377.3		8	63.38	even	6
2268.2.x.i.377.4		8	63.25	even	3
2268.2.x.i.1889.1		8	63.32	odd	6
2268.2.x.i.1889.2		8	63.31	odd	6
2268.2.x.i.1889.3		8	63.4	even	3
2268.2.x.i.1889.4		8	63.59	even	6
4032.2.k.a.3905.1		4	56.45	odd	6
4032.2.k.a.3905.2		4	168.53	odd	6
4032.2.k.a.3905.3		4	168.101	even	6
4032.2.k.a.3905.4		4	56.53	even	6
4032.2.k.d.3905.1		4	168.11	even	6
4032.2.k.d.3905.2		4	56.3	even	6
4032.2.k.d.3905.3		4	56.11	odd	6
4032.2.k.d.3905.4		4	168.59	odd	6
6300.2.d.c.3401.1		4	105.74	odd	6
6300.2.d.c.3401.2		4	35.4	even	6
6300.2.d.c.3401.3		4	105.59	even	6
6300.2.d.c.3401.4		4	35.24	odd	6
6300.2.f.b.3149.1		8	105.32	even	12
6300.2.f.b.3149.2		8	35.32	odd	12
6300.2.f.b.3149.3		8	105.38	odd	12
6300.2.f.b.3149.4		8	35.3	even	12
6300.2.f.b.3149.5		8	105.17	odd	12
6300.2.f.b.3149.6		8	35.17	even	12
6300.2.f.b.3149.7		8	105.53	even	12
6300.2.f.b.3149.8		8	35.18	odd	12

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 + 3.00000i) q^{5} +(-3.67423 - 2.12132i) q^{11} +4.89898i q^{13} +(1.73205 - 3.00000i) q^{17} +(-4.24264 + 2.44949i) q^{19} +(-3.67423 + 2.12132i) q^{23} +(-3.50000 + 6.06218i) q^{25} +4.24264i q^{29} +(4.00000 + 6.92820i) q^{37} +3.46410 q^{41} -2.00000 q^{43} +(-3.46410 - 6.00000i) q^{47} +(-11.0227 - 6.36396i) q^{53} -14.6969i q^{55} +(-6.92820 + 12.0000i) q^{59} +(8.48528 - 4.89898i) q^{61} +(-14.6969 + 8.48528i) q^{65} +(-4.00000 + 6.92820i) q^{67} -4.24264i q^{71} +(4.24264 + 2.44949i) q^{73} +(2.00000 + 3.46410i) q^{79} -6.92820 q^{83} +12.0000 q^{85} +(-5.19615 - 9.00000i) q^{89} +(-14.6969 - 8.48528i) q^{95} +4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 28 q^{25} + 32 q^{37} - 16 q^{43} - 32 q^{67} + 16 q^{79} + 96 q^{85}+O(q^{100}) \) 8 * q - 28 * q^25 + 32 * q^37 - 16 * q^43 - 32 * q^67 + 16 * q^79 + 96 * q^85

Introduction

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