LMFDB - Embedded newform 2268.2.x.i.1889.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(377,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("2268.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.x (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:	$18.1100711784$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1889.1
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2268.1889
Dual form			2268.2.x.i.377.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 - 3.00000i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(3.67423 + 2.12132i) q^{11} +(4.24264 - 2.44949i) q^{13} +3.46410 q^{17} -4.89898i q^{19} +(3.67423 - 2.12132i) q^{23} +(-3.50000 + 6.06218i) q^{25} +(3.67423 + 2.12132i) q^{29} +(-3.46410 + 8.48528i) q^{35} -8.00000 q^{37} +(1.73205 + 3.00000i) q^{41} +(1.00000 - 1.73205i) q^{43} +(3.46410 - 6.00000i) q^{47} +(-1.74264 + 6.77962i) q^{49} -12.7279i q^{53} -14.6969i q^{55} +(6.92820 + 12.0000i) q^{59} +(-8.48528 - 4.89898i) q^{61} +(-14.6969 - 8.48528i) q^{65} +(-4.00000 - 6.92820i) q^{67} +4.24264i q^{71} -4.89898i q^{73} +(-1.52192 - 11.1213i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-3.46410 + 6.00000i) q^{83} +(-6.00000 - 10.3923i) q^{85} -10.3923 q^{89} +(-12.0000 - 4.89898i) q^{91} +(-14.6969 + 8.48528i) q^{95} +(-4.24264 - 2.44949i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} - 28 q^{25} - 64 q^{37} + 8 q^{43} + 20 q^{49} - 32 q^{67} + 16 q^{79} - 48 q^{85} - 96 q^{91}+O(q^{100}) $ 8 * q + 4 * q^7 - 28 * q^25 - 64 * q^37 + 8 * q^43 + 20 * q^49 - 32 * q^67 + 16 * q^79 - 48 * q^85 - 96 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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	$-0.884620\pi$
$5$	−1.73205	−	3.00000i	−0.774597	−	1.34164i	0.160424	−	0.987048i	$-0.448714\pi$
$6$		0			0
$7$	−1.62132	−	2.09077i	−0.612801	−	0.790237i
$7$	−1.62132	−	2.09077i	−0.612801	−	0.790237i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.67423	+	2.12132i	1.10782	+	0.639602i	0.938265	−	0.345918i	$-0.112432\pi$
$11$	3.67423	+	2.12132i	1.10782	+	0.639602i	0.169559	+	0.985520i	$0.445766\pi$
$12$		0			0
$13$	4.24264	−	2.44949i	1.17670	−	0.679366i	0.221449	−	0.975172i	$-0.428921\pi$
$13$	4.24264	−	2.44949i	1.17670	−	0.679366i	0.955248	+	0.295806i	$0.0955881\pi$
$14$		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
$19$		−	4.89898i		−	1.12390i	−0.827170	−	0.561951i	$-0.810051\pi$
$19$		−	4.89898i		−	1.12390i	0.827170	−	0.561951i	$-0.189949\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.67423	−	2.12132i	0.766131	−	0.442326i	−0.0653618	−	0.997862i	$-0.520820\pi$
$23$	3.67423	−	2.12132i	0.766131	−	0.442326i	0.831493	+	0.555536i	$0.187487\pi$
$24$		0			0
$25$	−3.50000	+	6.06218i	−0.700000	+	1.21244i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.67423	+	2.12132i	0.682288	+	0.393919i	0.800717	−	0.599043i	$-0.204452\pi$
$29$	3.67423	+	2.12132i	0.682288	+	0.393919i	−0.118428	+	0.992963i	$0.537786\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.46410	+	8.48528i	−0.585540	+	1.43427i
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.73205	+	3.00000i	0.270501	+	0.468521i	0.968990	−	0.247099i	$-0.0794774\pi$
$41$	1.73205	+	3.00000i	0.270501	+	0.468521i	−0.698489	+	0.715621i	$0.746144\pi$
$42$		0			0
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	$-0.784601\pi$
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	0.932145	+	0.362084i	$0.117935\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.46410	−	6.00000i	0.505291	−	0.875190i	−0.494690	−	0.869069i	$-0.664718\pi$
$47$	3.46410	−	6.00000i	0.505291	−	0.875190i	0.999981	−	0.00612051i	$-0.00194823\pi$
$48$		0			0
$49$	−1.74264	+	6.77962i	−0.248949	+	0.968517i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	12.7279i		−	1.74831i	−0.485643	−	0.874157i	$-0.661414\pi$
$53$		−	12.7279i		−	1.74831i	0.485643	−	0.874157i	$-0.338586\pi$
$54$		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	0.824927	+	0.565240i	$0.191216\pi$
$59$	6.92820	+	12.0000i	0.901975	+	1.56227i	0.0770484	+	0.997027i	$0.475450\pi$
$60$		0			0
$61$	−8.48528	−	4.89898i	−1.08643	−	0.627250i	−0.153806	−	0.988101i	$-0.549153\pi$
$61$	−8.48528	−	4.89898i	−1.08643	−	0.627250i	−0.932623	+	0.360851i	$0.882486\pi$
$62$		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	0.511237	−	0.859440i	$-0.329187\pi$
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	−0.999915	+	0.0130248i	$0.995854\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.24264i			0.503509i	0.967791	+	0.251754i	$0.0810075\pi$
$71$			4.24264i			0.503509i	−0.967791	+	0.251754i	$0.918992\pi$
$72$		0			0
$73$		−	4.89898i		−	0.573382i	−0.958023	−	0.286691i	$-0.907445\pi$
$73$		−	4.89898i		−	0.573382i	0.958023	−	0.286691i	$-0.0925553\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.52192	−	11.1213i	−0.173439	−	1.26739i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.46410	+	6.00000i	−0.380235	+	0.658586i	−0.991096	−	0.133152i	$-0.957490\pi$
$83$	−3.46410	+	6.00000i	−0.380235	+	0.658586i	0.610861	+	0.791738i	$0.290823\pi$
$84$		0			0
$85$	−6.00000	−	10.3923i	−0.650791	−	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	−12.0000	−	4.89898i	−1.25794	−	0.513553i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.6969	+	8.48528i	−1.50787	+	0.870572i
$96$		0			0
$97$	−4.24264	−	2.44949i	−0.430775	−	0.248708i	0.268902	−	0.963168i	$-0.413339\pi$
$97$	−4.24264	−	2.44949i	−0.430775	−	0.248708i	−0.699677	+	0.714460i	$0.746673\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.19615	+	9.00000i	−0.517036	+	0.895533i	0.482768	+	0.875748i	$0.339632\pi$
$101$	−5.19615	+	9.00000i	−0.517036	+	0.895533i	−0.999804	+	0.0197851i	$0.993702\pi$
$102$		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
$107$		−	4.24264i		−	0.410152i	−0.978746	−	0.205076i	$-0.934256\pi$
$107$		−	4.24264i		−	0.410152i	0.978746	−	0.205076i	$-0.0657441\pi$
$108$		0			0
$109$	4.00000			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.0227	+	6.36396i	−1.03693	+	0.598671i	−0.918962	−	0.394346i	$-0.870971\pi$
$113$	−11.0227	+	6.36396i	−1.03693	+	0.598671i	−0.117967	+	0.993018i	$0.537638\pi$
$114$		0			0
$115$	−12.7279	−	7.34847i	−1.18688	−	0.685248i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.61642	−	7.24264i	−0.514856	−	0.663932i
$120$		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	$-0.804291\pi$
$131$	−10.3923	−	18.0000i	−0.907980	−	1.57267i	−0.0911134	−	0.995841i	$-0.529043\pi$
$132$		0			0
$133$	−10.2426	+	7.94282i	−0.888150	+	0.688729i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.67423	+	2.12132i	0.313911	+	0.181237i	0.648675	−	0.761065i	$-0.275323\pi$
$137$	3.67423	+	2.12132i	0.313911	+	0.181237i	−0.334764	+	0.942302i	$0.608657\pi$
$138$		0			0
$139$	8.48528	−	4.89898i	0.719712	−	0.415526i	−0.0949346	−	0.995484i	$-0.530264\pi$
$139$	8.48528	−	4.89898i	0.719712	−	0.415526i	0.814647	+	0.579957i	$0.196931\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.7846			1.73810
$144$		0			0
$145$		−	14.6969i		−	1.22051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.0227	−	6.36396i	0.903015	−	0.521356i	0.0248379	−	0.999691i	$-0.492093\pi$
$149$	11.0227	−	6.36396i	0.903015	−	0.521356i	0.878177	+	0.478335i	$0.158760\pi$
$150$		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	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.3923	−	4.24264i	−0.819028	−	0.334367i
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.46410	+	6.00000i	0.268060	+	0.464294i	0.968361	−	0.249554i	$-0.0802840\pi$
$167$	3.46410	+	6.00000i	0.268060	+	0.464294i	−0.700301	+	0.713848i	$0.746951\pi$
$168$		0			0
$169$	5.50000	−	9.52628i	0.423077	−	0.732791i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.19615	+	9.00000i	−0.395056	+	0.684257i	−0.993108	−	0.117200i	$-0.962608\pi$
$173$	−5.19615	+	9.00000i	−0.395056	+	0.684257i	0.598052	+	0.801457i	$0.295942\pi$
$174$		0			0
$175$	18.3492	−	2.51104i	1.38707	−	0.189817i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.24264i			0.317110i	0.987350	+	0.158555i	$0.0506835\pi$
$179$			4.24264i			0.317110i	−0.987350	+	0.158555i	$0.949317\pi$
$180$		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	0.985190	−	0.171466i	$-0.0548503\pi$
$191$	18.3712	+	10.6066i	1.32929	+	0.767467i	0.344101	+	0.938933i	$0.388184\pi$
$192$		0			0
$193$	10.0000	+	17.3205i	0.719816	+	1.24676i	0.961073	+	0.276296i	$0.0891071\pi$
$193$	10.0000	+	17.3205i	0.719816	+	1.24676i	−0.241257	+	0.970461i	$0.577560\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.24264i		−	0.302276i	−0.988513	−	0.151138i	$-0.951706\pi$
$197$		−	4.24264i		−	0.302276i	0.988513	−	0.151138i	$-0.0482937\pi$
$198$		0			0
$199$		−	24.4949i		−	1.73640i	−0.496217	−	0.868199i	$-0.665278\pi$
$199$		−	24.4949i		−	1.73640i	0.496217	−	0.868199i	$-0.334722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.52192	−	11.1213i	−0.106818	−	0.780564i
$204$		0			0
$205$	6.00000	−	10.3923i	0.419058	−	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.3923	−	18.0000i	0.718851	−	1.24509i
$210$		0			0
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	0.640996	−	0.767544i	$-0.278521\pi$
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	−0.985211	+	0.171347i	$0.945188\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.92820			−0.472500
$216$		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
$223$	−12.7279	−	7.34847i	−0.852325	−	0.492090i	0.00910984	−	0.999959i	$-0.497100\pi$
$223$	−12.7279	−	7.34847i	−0.852325	−	0.492090i	−0.861435	+	0.507869i	$0.830434\pi$
$224$		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
$233$			21.2132i			1.38972i	0.719144	+	0.694862i	$0.244534\pi$
$233$			21.2132i			1.38972i	−0.719144	+	0.694862i	$0.755466\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.7196	−	14.8492i	1.66367	−	0.960518i	0.692723	−	0.721204i	$-0.256411\pi$
$239$	25.7196	−	14.8492i	1.66367	−	0.960518i	0.970942	−	0.239314i	$-0.0769225\pi$
$240$		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
$245$	23.3572	−	6.51472i	1.49224	−	0.416210i
$246$		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	$-0.570166\pi$
$251$	−6.92820			−0.437304			−0.218652	+	0.975803i	$0.570166\pi$
$252$		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	0.981335	−	0.192304i	$-0.0615961\pi$
$257$	5.19615	+	9.00000i	0.324127	+	0.561405i	−0.657208	+	0.753709i	$0.728263\pi$
$258$		0			0
$259$	12.9706	+	16.7262i	0.805952	+	1.03931i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.67423	−	2.12132i	−0.226563	−	0.130806i	0.382422	−	0.923988i	$-0.375090\pi$
$263$	−3.67423	−	2.12132i	−0.226563	−	0.130806i	−0.608985	+	0.793181i	$0.708423\pi$
$264$		0			0
$265$	−38.1838	+	22.0454i	−2.34561	+	1.35424i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.46410			−0.211210			−0.105605	−	0.994408i	$-0.533678\pi$
$269$	−3.46410			−0.211210			−0.105605	+	0.994408i	$0.533678\pi$
$270$		0			0
$271$			19.5959i			1.19037i	0.803590	+	0.595184i	$0.202921\pi$
$271$			19.5959i			1.19037i	−0.803590	+	0.595184i	$0.797079\pi$
$272$		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
$281$	−3.67423	−	2.12132i	−0.219186	−	0.126547i	0.386387	−	0.922337i	$-0.373723\pi$
$281$	−3.67423	−	2.12132i	−0.219186	−	0.126547i	−0.605574	+	0.795789i	$0.707056\pi$
$282$		0			0
$283$	12.7279	−	7.34847i	0.756596	−	0.436821i	−0.0714760	−	0.997442i	$-0.522771\pi$
$283$	12.7279	−	7.34847i	0.756596	−	0.436821i	0.828072	+	0.560621i	$0.189438\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.46410	−	8.48528i	0.204479	−	0.500870i
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.66025	−	15.0000i	−0.505937	−	0.876309i	−0.999976	−	0.00686959i	$-0.997813\pi$
$293$	−8.66025	−	15.0000i	−0.505937	−	0.876309i	0.494039	−	0.869440i	$-0.335520\pi$
$294$		0			0
$295$	24.0000	−	41.5692i	1.39733	−	2.42025i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3923	−	18.0000i	0.601003	−	1.04097i
$300$		0			0
$301$	−5.24264	+	0.717439i	−0.302181	+	0.0413525i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			33.9411i			1.94346i
$306$		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	$-0.295182\pi$
$311$	−6.92820	−	12.0000i	−0.392862	−	0.680458i	−0.992826	+	0.119570i	$0.961848\pi$
$312$		0			0
$313$	−16.9706	−	9.79796i	−0.959233	−	0.553813i	−0.0632961	−	0.997995i	$-0.520161\pi$
$313$	−16.9706	−	9.79796i	−0.959233	−	0.553813i	−0.895937	+	0.444181i	$0.853495\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.0227	+	6.36396i	0.619097	+	0.357436i	0.776517	−	0.630096i	$-0.216984\pi$
$317$	11.0227	+	6.36396i	0.619097	+	0.357436i	−0.157421	+	0.987532i	$0.550318\pi$
$318$		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
$325$			34.2929i			1.90223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.1610	+	2.48528i	−1.00125	+	0.137018i
$330$		0			0
$331$	−11.0000	+	19.0526i	−0.604615	+	1.04722i	0.387498	+	0.921871i	$0.373340\pi$
$331$	−11.0000	+	19.0526i	−0.604615	+	1.04722i	−0.992112	+	0.125353i	$0.959994\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.8564	+	24.0000i	−0.757056	+	1.31126i
$336$		0			0
$337$	−8.00000	−	13.8564i	−0.435788	−	0.754807i	0.561572	−	0.827428i	$-0.310197\pi$
$337$	−8.00000	−	13.8564i	−0.435788	−	0.754807i	−0.997360	+	0.0726214i	$0.976864\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.3712	−	10.6066i	0.986216	−	0.569392i	0.0820751	−	0.996626i	$-0.473845\pi$
$347$	18.3712	−	10.6066i	0.986216	−	0.569392i	0.904141	+	0.427234i	$0.140512\pi$
$348$		0			0
$349$	16.9706	+	9.79796i	0.908413	+	0.524473i	0.879920	−	0.475121i	$-0.157596\pi$
$349$	16.9706	+	9.79796i	0.908413	+	0.524473i	0.0284931	+	0.999594i	$0.490929\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.73205	+	3.00000i	−0.0921878	+	0.159674i	−0.908431	−	0.418034i	$-0.862719\pi$
$353$	−1.73205	+	3.00000i	−0.0921878	+	0.159674i	0.816244	+	0.577708i	$0.196053\pi$
$354$		0			0
$355$	12.7279	−	7.34847i	0.675528	−	0.390016i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.2132i		−	1.11959i	−0.828631	−	0.559795i	$-0.810880\pi$
$359$		−	21.2132i		−	1.11959i	0.828631	−	0.559795i	$-0.189120\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.6969	+	8.48528i	−0.769273	+	0.444140i
$366$		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
$371$	−26.6112	+	20.6360i	−1.38158	+	1.07137i
$372$		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	0.763847	−	0.645398i	$-0.223308\pi$
$383$	−3.46410	−	6.00000i	−0.177007	−	0.306586i	−0.940854	+	0.338812i	$0.889975\pi$
$384$		0			0
$385$	−30.7279	+	23.8284i	−1.56604	+	1.21441i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.3712	−	10.6066i	−0.931455	−	0.537776i	−0.0441839	−	0.999023i	$-0.514069\pi$
$389$	−18.3712	−	10.6066i	−0.931455	−	0.537776i	−0.887272	+	0.461247i	$0.847402\pi$
$390$		0			0
$391$	12.7279	−	7.34847i	0.643679	−	0.371628i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.8564			−0.697191
$396$		0			0
$397$			29.3939i			1.47524i	0.675218	+	0.737618i	$0.264050\pi$
$397$			29.3939i			1.47524i	−0.675218	+	0.737618i	$0.735950\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.3712	+	10.6066i	−0.917413	+	0.529668i	−0.882809	−	0.469733i	$-0.844350\pi$
$401$	−18.3712	+	10.6066i	−0.917413	+	0.529668i	−0.0346039	+	0.999401i	$0.511017\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−29.3939	−	16.9706i	−1.45700	−	0.841200i
$408$		0			0
$409$	−4.24264	+	2.44949i	−0.209785	+	0.121119i	−0.601212	−	0.799090i	$-0.705315\pi$
$409$	−4.24264	+	2.44949i	−0.209785	+	0.121119i	0.391426	+	0.920209i	$0.371982\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.8564	−	33.9411i	0.681829	−	1.67013i
$414$		0			0
$415$	24.0000			1.17811
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.8564	+	24.0000i	0.676930	+	1.17248i	0.975901	+	0.218215i	$0.0700233\pi$
$419$	13.8564	+	24.0000i	0.676930	+	1.17248i	−0.298971	+	0.954262i	$0.596643\pi$
$420$		0			0
$421$	17.0000	−	29.4449i	0.828529	−	1.43505i	−0.0706626	−	0.997500i	$-0.522511\pi$
$421$	17.0000	−	29.4449i	0.828529	−	1.43505i	0.899192	−	0.437555i	$-0.144155\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.1244	+	21.0000i	−0.588118	+	1.01865i
$426$		0			0
$427$	3.51472	+	25.6836i	0.170089	+	1.24292i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.24264i		−	0.204361i	−0.994766	−	0.102180i	$-0.967418\pi$
$431$		−	4.24264i		−	0.204361i	0.994766	−	0.102180i	$-0.0325819\pi$
$432$		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	$-0.219269\pi$
$439$	12.7279	+	7.34847i	0.607471	+	0.350723i	−0.164504	+	0.986376i	$0.552602\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.0227	+	6.36396i	0.523704	+	0.302361i	0.738449	−	0.674309i	$-0.235559\pi$
$443$	11.0227	+	6.36396i	0.523704	+	0.302361i	−0.214745	+	0.976670i	$0.568892\pi$
$444$		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	$0.0319199\pi$
$449$			4.24264i			0.200223i	−0.994976	+	0.100111i	$0.968080\pi$
$450$		0			0
$451$			14.6969i			0.692052i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.08767	+	44.4853i	0.285394	+	2.08550i
$456$		0			0
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	−0.757174	−	0.653213i	$-0.773421\pi$
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	0.944286	+	0.329125i	$0.106754\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.73205	−	3.00000i	0.0806696	−	0.139724i	−0.822868	−	0.568232i	$-0.807627\pi$
$461$	1.73205	−	3.00000i	0.0806696	−	0.139724i	0.903538	+	0.428508i	$0.140961\pi$
$462$		0			0
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	0.534450	−	0.845200i	$-0.320519\pi$
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	−8.00000	+	19.5959i	−0.369406	+	0.904855i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.34847	−	4.24264i	0.337883	−	0.195077i
$474$		0			0
$475$	29.6985	+	17.1464i	1.36266	+	0.786732i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.92820	−	12.0000i	0.316558	−	0.548294i	−0.663210	−	0.748434i	$-0.730806\pi$
$479$	6.92820	−	12.0000i	0.316558	−	0.548294i	0.979767	+	0.200140i	$0.0641396\pi$
$480$		0			0
$481$	−33.9411	+	19.5959i	−1.54758	+	0.893497i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			16.9706i			0.770594i
$486$		0			0
$487$	−14.0000			−0.634401			−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000			−0.634401			−0.317200	+	0.948359i	$0.602743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.0227	−	6.36396i	0.497448	−	0.287202i	−0.230211	−	0.973141i	$-0.573942\pi$
$491$	11.0227	−	6.36396i	0.497448	−	0.287202i	0.727659	+	0.685939i	$0.240608\pi$
$492$		0			0
$493$	12.7279	+	7.34847i	0.573237	+	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.87039	−	6.87868i	0.397891	−	0.308551i
$498$		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	$-0.780885\pi$
$503$	−34.6410			−1.54457			−0.772283	+	0.635278i	$0.780885\pi$
$504$		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	$0.0139278\pi$
$509$	12.1244	+	21.0000i	0.537403	+	0.930809i	−0.461640	+	0.887067i	$0.652739\pi$
$510$		0			0
$511$	−10.2426	+	7.94282i	−0.453108	+	0.351369i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−29.3939	−	16.9706i	−1.29525	−	0.747812i
$516$		0			0
$517$	25.4558	−	14.6969i	1.11955	−	0.646371i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.46410			−0.151765			−0.0758825	−	0.997117i	$-0.524177\pi$
$521$	−3.46410			−0.151765			−0.0758825	+	0.997117i	$0.524177\pi$
$522$		0			0
$523$		−	39.1918i		−	1.71374i	−0.515533	−	0.856870i	$-0.672406\pi$
$523$		−	39.1918i		−	1.71374i	0.515533	−	0.856870i	$-0.327594\pi$
$524$		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
$533$	14.6969	+	8.48528i	0.636595	+	0.367538i
$534$		0			0
$535$	−12.7279	+	7.34847i	−0.550276	+	0.317702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.7846	+	21.2132i	−0.895257	+	0.913717i
$540$		0			0
$541$	26.0000			1.11783			0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000			1.11783			0.558914	+	0.829226i	$0.311218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.92820	−	12.0000i	−0.296772	−	0.514024i
$546$		0			0
$547$	20.0000	−	34.6410i	0.855138	−	1.48114i	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	20.0000	−	34.6410i	0.855138	−	1.48114i	0.876517	−	0.481371i	$-0.159861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.3923	−	18.0000i	0.442727	−	0.766826i
$552$		0			0
$553$	−10.4853	+	1.43488i	−0.445880	+	0.0610172i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4.24264i		−	0.179766i	−0.995952	−	0.0898832i	$-0.971351\pi$
$557$		−	4.24264i		−	0.179766i	0.995952	−	0.0898832i	$-0.0286494\pi$
$558$		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	$-0.310974\pi$
$563$	−10.3923	−	18.0000i	−0.437983	−	0.758610i	−0.997534	+	0.0701867i	$0.977640\pi$
$564$		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	0.421003	−	0.907059i	$-0.361678\pi$
$569$	−3.67423	−	2.12132i	−0.154032	−	0.0889304i	−0.575035	+	0.818129i	$0.695012\pi$
$570$		0			0
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			29.6985i			1.23851i
$576$		0			0
$577$			9.79796i			0.407894i	0.978982	+	0.203947i	$0.0653771\pi$
$577$			9.79796i			0.407894i	−0.978982	+	0.203947i	$0.934623\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.1610	−	2.48528i	0.753447	−	0.103107i
$582$		0			0
$583$	27.0000	−	46.7654i	1.11823	−	1.93682i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.3923	−	18.0000i	0.428936	−	0.742940i	−0.567843	−	0.823137i	$-0.692222\pi$
$587$	10.3923	−	18.0000i	0.428936	−	0.742940i	0.996779	+	0.0801976i	$0.0255551\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.3205			0.711268			0.355634	−	0.934625i	$-0.384265\pi$
$593$	17.3205			0.711268			0.355634	+	0.934625i	$0.384265\pi$
$594$		0			0
$595$	−12.0000	+	29.3939i	−0.491952	+	1.20503i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.67423	+	2.12132i	−0.150125	+	0.0866748i	−0.573181	−	0.819429i	$-0.694291\pi$
$599$	−3.67423	+	2.12132i	−0.150125	+	0.0866748i	0.423056	+	0.906104i	$0.360957\pi$
$600$		0			0
$601$	−25.4558	−	14.6969i	−1.03837	−	0.599501i	−0.118996	−	0.992895i	$-0.537967\pi$
$601$	−25.4558	−	14.6969i	−1.03837	−	0.599501i	−0.919370	+	0.393394i	$0.871301\pi$
$602$		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
$611$		−	33.9411i		−	1.37311i
$612$		0			0
$613$	14.0000			0.565455			0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000			0.565455			0.282727	+	0.959200i	$0.408761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.0227	+	6.36396i	−0.443757	+	0.256203i	−0.705190	−	0.709018i	$-0.749138\pi$
$617$	−11.0227	+	6.36396i	−0.443757	+	0.256203i	0.261433	+	0.965222i	$0.415805\pi$
$618$		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
$623$	16.8493	+	21.7279i	0.675051	+	0.870511i
$624$		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
$637$	9.21320	+	33.0321i	0.365040	+	1.30878i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.7196	+	14.8492i	1.01586	+	0.586510i	0.912903	−	0.408176i	$-0.133835\pi$
$641$	25.7196	+	14.8492i	1.01586	+	0.586510i	0.102961	+	0.994685i	$0.467168\pi$
$642$		0			0
$643$	−12.7279	+	7.34847i	−0.501940	+	0.289795i	−0.729514	−	0.683965i	$-0.760254\pi$
$643$	−12.7279	+	7.34847i	−0.501940	+	0.289795i	0.227574	+	0.973761i	$0.426921\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−41.5692			−1.63425			−0.817127	−	0.576457i	$-0.804435\pi$
$647$	−41.5692			−1.63425			−0.817127	+	0.576457i	$0.804435\pi$
$648$		0			0
$649$			58.7878i			2.30762i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	40.4166	−	23.3345i	1.58162	−	0.913150i	0.587000	−	0.809587i	$-0.300309\pi$
$653$	40.4166	−	23.3345i	1.58162	−	0.913150i	0.994623	−	0.103564i	$-0.0330246\pi$
$654$		0			0
$655$	−36.0000	+	62.3538i	−1.40664	+	2.43637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.4166	+	23.3345i	1.57441	+	0.908984i	0.995619	+	0.0935057i	$0.0298073\pi$
$659$	40.4166	+	23.3345i	1.57441	+	0.908984i	0.578788	+	0.815478i	$0.303526\pi$
$660$		0			0
$661$	−33.9411	+	19.5959i	−1.32016	+	0.762193i	−0.983753	−	0.179525i	$-0.942544\pi$
$661$	−33.9411	+	19.5959i	−1.32016	+	0.762193i	−0.336403	+	0.941718i	$0.609211\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	41.5692	+	16.9706i	1.61199	+	0.658090i
$666$		0			0
$667$	18.0000			0.696963
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20.7846	−	36.0000i	−0.802381	−	1.38976i
$672$		0			0
$673$	10.0000	−	17.3205i	0.385472	−	0.667657i	−0.606363	−	0.795188i	$-0.707372\pi$
$673$	10.0000	−	17.3205i	0.385472	−	0.667657i	0.991835	+	0.127532i	$0.0407054\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.5885	+	27.0000i	−0.599113	+	1.03769i	0.393839	+	0.919179i	$0.371147\pi$
$677$	−15.5885	+	27.0000i	−0.599113	+	1.03769i	−0.992952	+	0.118515i	$0.962187\pi$
$678$		0			0
$679$	1.75736	+	12.8418i	0.0674413	+	0.492823i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			12.7279i			0.487020i	0.969898	+	0.243510i	$0.0782989\pi$
$683$			12.7279i			0.487020i	−0.969898	+	0.243510i	$0.921701\pi$
$684$		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	$-0.522186\pi$
$691$	−25.4558	−	14.6969i	−0.968386	−	0.559098i	−0.898744	+	0.438474i	$0.855519\pi$
$692$		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	$0.0772666\pi$
$701$			12.7279i			0.480727i	−0.970683	+	0.240363i	$0.922733\pi$
$702$		0			0
$703$			39.1918i			1.47815i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.2416	−	3.72792i	1.02452	−	0.140203i
$708$		0			0
$709$	−2.00000	+	3.46410i	−0.0751116	+	0.130097i	−0.901135	−	0.433539i	$-0.857265\pi$
$709$	−2.00000	+	3.46410i	−0.0751116	+	0.130097i	0.826023	+	0.563636i	$0.190598\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−36.0000	−	62.3538i	−1.34632	−	2.33190i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.7128			1.03351			0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128			1.03351			0.516757	+	0.856132i	$0.327139\pi$
$720$		0			0
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.7196	+	14.8492i	−0.955204	+	0.551487i
$726$		0			0
$727$	25.4558	+	14.6969i	0.944105	+	0.545079i	0.891245	−	0.453523i	$-0.149833\pi$
$727$	25.4558	+	14.6969i	0.944105	+	0.545079i	0.0528602	+	0.998602i	$0.483166\pi$
$728$		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
$737$		−	33.9411i		−	1.25024i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.67423	+	2.12132i	−0.134795	+	0.0778237i	−0.565881	−	0.824487i	$-0.691464\pi$
$743$	−3.67423	+	2.12132i	−0.134795	+	0.0778237i	0.431086	+	0.902311i	$0.358130\pi$
$744$		0			0
$745$	−38.1838	−	22.0454i	−1.39894	−	0.807681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.87039	+	6.87868i	−0.324117	+	0.251341i
$750$		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	$0.137272\pi$
$761$	22.5167	+	39.0000i	0.816228	+	1.41375i	−0.0922143	+	0.995739i	$0.529394\pi$
$762$		0			0
$763$	−6.48528	−	8.36308i	−0.234783	−	0.302764i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	58.7878	+	33.9411i	2.12270	+	1.22554i
$768$		0			0
$769$	8.48528	−	4.89898i	0.305987	−	0.176662i	−0.339142	−	0.940735i	$-0.610137\pi$
$769$	8.48528	−	4.89898i	0.305987	−	0.176662i	0.645129	+	0.764073i	$0.276803\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.3205			−0.622975			−0.311488	−	0.950250i	$-0.600827\pi$
$773$	−17.3205			−0.622975			−0.311488	+	0.950250i	$0.600827\pi$
$774$		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	$-0.553100\pi$
$787$	16.9706	−	9.79796i	0.604935	−	0.349260i	0.770981	+	0.636859i	$0.219766\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	31.1769	+	12.7279i	1.10852	+	0.452553i
$792$		0			0
$793$	−48.0000			−1.70453
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.19615	+	9.00000i	0.184057	+	0.318796i	0.943258	−	0.332060i	$-0.107744\pi$
$797$	5.19615	+	9.00000i	0.184057	+	0.318796i	−0.759201	+	0.650856i	$0.774410\pi$
$798$		0			0
$799$	12.0000	−	20.7846i	0.424529	−	0.735307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.3923	−	18.0000i	0.366736	−	0.635206i
$804$		0			0
$805$	5.27208	+	38.5254i	0.185816	+	1.35784i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	38.1838i		−	1.34247i	−0.741245	−	0.671235i	$-0.765764\pi$
$809$		−	38.1838i		−	1.34247i	0.741245	−	0.671235i	$-0.234236\pi$
$810$		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	0.295164	−	0.955447i	$-0.404626\pi$
$821$	−11.0227	−	6.36396i	−0.384695	−	0.222104i	−0.679859	+	0.733343i	$0.737959\pi$
$822$		0			0
$823$	−10.0000	−	17.3205i	−0.348578	−	0.603755i	0.637419	−	0.770517i	$-0.280002\pi$
$823$	−10.0000	−	17.3205i	−0.348578	−	0.603755i	−0.985997	+	0.166762i	$0.946669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	12.7279i		−	0.442593i	−0.975207	−	0.221297i	$-0.928971\pi$
$827$		−	12.7279i		−	0.442593i	0.975207	−	0.221297i	$-0.0710289\pi$
$828$		0			0
$829$		−	24.4949i		−	0.850743i	−0.905019	−	0.425371i	$-0.860143\pi$
$829$		−	24.4949i		−	0.850743i	0.905019	−	0.425371i	$-0.139857\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.03668	+	23.4853i	−0.209159	+	0.813717i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.3205	+	30.0000i	−0.597970	+	1.03572i	0.395150	+	0.918617i	$0.370693\pi$
$839$	−17.3205	+	30.0000i	−0.597970	+	1.03572i	−0.993120	+	0.117098i	$0.962641\pi$
$840$		0			0
$841$	−5.50000	−	9.52628i	−0.189655	−	0.328492i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−38.1051			−1.31086
$846$		0			0
$847$	7.00000	−	17.1464i	0.240523	−	0.589158i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−29.3939	+	16.9706i	−1.00761	+	0.581743i
$852$		0			0
$853$	8.48528	+	4.89898i	0.290531	+	0.167738i	0.638181	−	0.769886i	$-0.279687\pi$
$853$	8.48528	+	4.89898i	0.290531	+	0.167738i	−0.347651	+	0.937624i	$0.613020\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.1244	+	21.0000i	−0.414160	+	0.717346i	−0.995340	−	0.0964289i	$-0.969258\pi$
$857$	−12.1244	+	21.0000i	−0.414160	+	0.717346i	0.581180	+	0.813775i	$0.302591\pi$
$858$		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
$863$			55.1543i			1.87748i	0.344633	+	0.938738i	$0.388003\pi$
$863$			55.1543i			1.87748i	−0.344633	+	0.938738i	$0.611997\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.6969	−	8.48528i	0.498559	−	0.287843i
$870$		0			0
$871$	−33.9411	−	19.5959i	−1.15005	−	0.663982i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−11.2328	−	14.4853i	−0.379739	−	0.489692i
$876$		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	$-0.676005\pi$
$881$	−31.1769			−1.05038			−0.525188	+	0.850986i	$0.676005\pi$
$882$		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	0.918304	−	0.395876i	$-0.129559\pi$
$887$	3.46410	+	6.00000i	0.116313	+	0.201460i	−0.801991	+	0.597336i	$0.796226\pi$
$888$		0			0
$889$	6.48528	+	8.36308i	0.217509	+	0.280489i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−29.3939	−	16.9706i	−0.983629	−	0.567898i
$894$		0			0
$895$	12.7279	−	7.34847i	0.425448	−	0.245632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	44.0908i		−	1.46888i
$902$		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
$911$	−18.3712	−	10.6066i	−0.608664	−	0.351412i	0.163778	−	0.986497i	$-0.447632\pi$
$911$	−18.3712	−	10.6066i	−0.608664	−	0.351412i	−0.772442	+	0.635085i	$0.780965\pi$
$912$		0			0
$913$	−25.4558	+	14.6969i	−0.842465	+	0.486398i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.7846	+	50.9117i	−0.686368	+	1.68125i
$918$		0			0
$919$	−14.0000			−0.461817			−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000			−0.461817			−0.230909	+	0.972975i	$0.574170\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.3923	+	18.0000i	0.342067	+	0.592477i
$924$		0			0
$925$	28.0000	−	48.4974i	0.920634	−	1.59459i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22.5167	+	39.0000i	−0.738748	+	1.27955i	0.214312	+	0.976765i	$0.431249\pi$
$929$	−22.5167	+	39.0000i	−0.738748	+	1.27955i	−0.953059	+	0.302783i	$0.902084\pi$
$930$		0			0
$931$	33.2132	+	8.53716i	1.08852	+	0.279794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	50.9117i		−	1.66499i
$936$		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	0.955151	+	0.296119i	$0.0956925\pi$
$941$	22.5167	+	39.0000i	0.734022	+	1.27136i	−0.221129	+	0.975245i	$0.570974\pi$
$942$		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	0.0200502	−	0.999799i	$-0.493617\pi$
$947$	−25.7196	−	14.8492i	−0.835776	−	0.482536i	−0.855826	+	0.517263i	$0.826951\pi$
$948$		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	$-0.787762\pi$
$953$		−	38.1838i		−	1.23689i	0.785827	−	0.618447i	$-0.212238\pi$
$954$		0			0
$955$		−	73.4847i		−	2.37791i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.52192	−	11.1213i	−0.0491453	−	0.359126i
$960$		0			0
$961$	−15.5000	+	26.8468i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	34.6410	−	60.0000i	1.11513	−	1.93147i
$966$		0			0
$967$	14.0000	+	24.2487i	0.450210	+	0.779786i	0.998399	−	0.0565684i	$-0.0180159\pi$
$967$	14.0000	+	24.2487i	0.450210	+	0.779786i	−0.548189	+	0.836354i	$0.684683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.7128			0.889346			0.444673	−	0.895693i	$-0.353320\pi$
$971$	27.7128			0.889346			0.444673	+	0.895693i	$0.353320\pi$
$972$		0			0
$973$	−24.0000	−	9.79796i	−0.769405	−	0.314108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.67423	−	2.12132i	0.117549	−	0.0678671i	−0.440073	−	0.897962i	$-0.645047\pi$
$977$	3.67423	−	2.12132i	0.117549	−	0.0678671i	0.557622	+	0.830095i	$0.311714\pi$
$978$		0			0
$979$	−38.1838	−	22.0454i	−1.22036	−	0.704574i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.3205	−	30.0000i	0.552438	−	0.956851i	−0.445659	−	0.895203i	$-0.647031\pi$
$983$	17.3205	−	30.0000i	0.552438	−	0.956851i	0.998098	−	0.0616488i	$-0.0196359\pi$
$984$		0			0
$985$	−12.7279	+	7.34847i	−0.405545	+	0.234142i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	8.48528i		−	0.269816i
$990$		0			0
$991$	−26.0000			−0.825917			−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000			−0.825917			−0.412959	+	0.910750i	$0.635505\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−73.4847	+	42.4264i	−2.32962	+	1.34501i
$996$		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	2268.2.x.i.1889.1		8
3.2	odd	2	inner	2268.2.x.i.1889.3		8
7.6	odd	2	inner	2268.2.x.i.1889.4		8
9.2	odd	6		252.2.f.a.125.2	yes	4
9.4	even	3	inner	2268.2.x.i.377.2		8
9.5	odd	6	inner	2268.2.x.i.377.4		8
9.7	even	3		252.2.f.a.125.4	yes	4
21.20	even	2	inner	2268.2.x.i.1889.2		8
36.7	odd	6		1008.2.k.b.881.3		4
36.11	even	6		1008.2.k.b.881.1		4
45.2	even	12		6300.2.f.b.3149.2		8
45.7	odd	12		6300.2.f.b.3149.1		8
45.29	odd	6		6300.2.d.c.3401.2		4
45.34	even	6		6300.2.d.c.3401.1		4
45.38	even	12		6300.2.f.b.3149.8		8
45.43	odd	12		6300.2.f.b.3149.7		8
63.2	odd	6		1764.2.t.b.521.3		8
63.11	odd	6		1764.2.t.b.1097.4		8
63.13	odd	6	inner	2268.2.x.i.377.3		8
63.16	even	3		1764.2.t.b.521.2		8
63.20	even	6		252.2.f.a.125.3	yes	4
63.25	even	3		1764.2.t.b.1097.1		8
63.34	odd	6		252.2.f.a.125.1	✓	4
63.38	even	6		1764.2.t.b.1097.2		8
63.41	even	6	inner	2268.2.x.i.377.1		8
63.47	even	6		1764.2.t.b.521.1		8
63.52	odd	6		1764.2.t.b.1097.3		8
63.61	odd	6		1764.2.t.b.521.4		8
72.11	even	6		4032.2.k.d.3905.3		4
72.29	odd	6		4032.2.k.a.3905.4		4
72.43	odd	6		4032.2.k.d.3905.1		4
72.61	even	6		4032.2.k.a.3905.2		4
252.83	odd	6		1008.2.k.b.881.4		4
252.223	even	6		1008.2.k.b.881.2		4
315.34	odd	6		6300.2.d.c.3401.3		4
315.83	odd	12		6300.2.f.b.3149.4		8
315.97	even	12		6300.2.f.b.3149.5		8
315.209	even	6		6300.2.d.c.3401.4		4
315.223	even	12		6300.2.f.b.3149.3		8
315.272	odd	12		6300.2.f.b.3149.6		8
504.83	odd	6		4032.2.k.d.3905.2		4
504.349	odd	6		4032.2.k.a.3905.3		4
504.461	even	6		4032.2.k.a.3905.1		4
504.475	even	6		4032.2.k.d.3905.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.f.a.125.1	✓	4	63.34	odd	6
252.2.f.a.125.2	yes	4	9.2	odd	6
252.2.f.a.125.3	yes	4	63.20	even	6
252.2.f.a.125.4	yes	4	9.7	even	3
1008.2.k.b.881.1		4	36.11	even	6
1008.2.k.b.881.2		4	252.223	even	6
1008.2.k.b.881.3		4	36.7	odd	6
1008.2.k.b.881.4		4	252.83	odd	6
1764.2.t.b.521.1		8	63.47	even	6
1764.2.t.b.521.2		8	63.16	even	3
1764.2.t.b.521.3		8	63.2	odd	6
1764.2.t.b.521.4		8	63.61	odd	6
1764.2.t.b.1097.1		8	63.25	even	3
1764.2.t.b.1097.2		8	63.38	even	6
1764.2.t.b.1097.3		8	63.52	odd	6
1764.2.t.b.1097.4		8	63.11	odd	6
2268.2.x.i.377.1		8	63.41	even	6	inner
2268.2.x.i.377.2		8	9.4	even	3	inner
2268.2.x.i.377.3		8	63.13	odd	6	inner
2268.2.x.i.377.4		8	9.5	odd	6	inner
2268.2.x.i.1889.1		8	1.1	even	1	trivial
2268.2.x.i.1889.2		8	21.20	even	2	inner
2268.2.x.i.1889.3		8	3.2	odd	2	inner
2268.2.x.i.1889.4		8	7.6	odd	2	inner
4032.2.k.a.3905.1		4	504.461	even	6
4032.2.k.a.3905.2		4	72.61	even	6
4032.2.k.a.3905.3		4	504.349	odd	6
4032.2.k.a.3905.4		4	72.29	odd	6
4032.2.k.d.3905.1		4	72.43	odd	6
4032.2.k.d.3905.2		4	504.83	odd	6
4032.2.k.d.3905.3		4	72.11	even	6
4032.2.k.d.3905.4		4	504.475	even	6
6300.2.d.c.3401.1		4	45.34	even	6
6300.2.d.c.3401.2		4	45.29	odd	6
6300.2.d.c.3401.3		4	315.34	odd	6
6300.2.d.c.3401.4		4	315.209	even	6
6300.2.f.b.3149.1		8	45.7	odd	12
6300.2.f.b.3149.2		8	45.2	even	12
6300.2.f.b.3149.3		8	315.223	even	12
6300.2.f.b.3149.4		8	315.83	odd	12
6300.2.f.b.3149.5		8	315.97	even	12
6300.2.f.b.3149.6		8	315.272	odd	12
6300.2.f.b.3149.7		8	45.43	odd	12
6300.2.f.b.3149.8		8	45.38	even	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 - 3.00000i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(3.67423 + 2.12132i) q^{11} +(4.24264 - 2.44949i) q^{13} +3.46410 q^{17} -4.89898i q^{19} +(3.67423 - 2.12132i) q^{23} +(-3.50000 + 6.06218i) q^{25} +(3.67423 + 2.12132i) q^{29} +(-3.46410 + 8.48528i) q^{35} -8.00000 q^{37} +(1.73205 + 3.00000i) q^{41} +(1.00000 - 1.73205i) q^{43} +(3.46410 - 6.00000i) q^{47} +(-1.74264 + 6.77962i) q^{49} -12.7279i q^{53} -14.6969i q^{55} +(6.92820 + 12.0000i) q^{59} +(-8.48528 - 4.89898i) q^{61} +(-14.6969 - 8.48528i) q^{65} +(-4.00000 - 6.92820i) q^{67} +4.24264i q^{71} -4.89898i q^{73} +(-1.52192 - 11.1213i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-3.46410 + 6.00000i) q^{83} +(-6.00000 - 10.3923i) q^{85} -10.3923 q^{89} +(-12.0000 - 4.89898i) q^{91} +(-14.6969 + 8.48528i) q^{95} +(-4.24264 - 2.44949i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} - 28 q^{25} - 64 q^{37} + 8 q^{43} + 20 q^{49} - 32 q^{67} + 16 q^{79} - 48 q^{85} - 96 q^{91}+O(q^{100}) \) 8 * q + 4 * q^7 - 28 * q^25 - 64 * q^37 + 8 * q^43 + 20 * q^49 - 32 * q^67 + 16 * q^79 - 48 * q^85 - 96 * q^91

Introduction

L-functions

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