LMFDB - Embedded newform 2808.2.q.g.937.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(937,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.937"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [22,0,0,0,3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			937.10
Character	$\chi$	$=$	2808.937
Dual form			2808.2.q.g.1873.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.80041 + 3.11840i) q^{5} +(-2.42567 + 4.20138i) q^{7} +(-2.96999 + 5.14417i) q^{11} +(-0.500000 - 0.866025i) q^{13} -3.14211 q^{17} +4.53444 q^{19} +(-2.40410 - 4.16402i) q^{23} +(-3.98293 + 6.89863i) q^{25} +(-0.967295 + 1.67540i) q^{29} +(2.30944 + 4.00007i) q^{31} -17.4688 q^{35} +10.7700 q^{37} +(-0.389576 - 0.674766i) q^{41} +(-0.164787 + 0.285420i) q^{43} +(-2.01335 + 3.48722i) q^{47} +(-8.26774 - 14.3202i) q^{49} +6.03880 q^{53} -21.3888 q^{55} +(1.75266 + 3.03569i) q^{59} +(2.77604 - 4.80825i) q^{61} +(1.80041 - 3.11840i) q^{65} +(-2.91273 - 5.04500i) q^{67} -3.18402 q^{71} +11.7545 q^{73} +(-14.4084 - 24.9561i) q^{77} +(1.97184 - 3.41532i) q^{79} +(1.28229 - 2.22100i) q^{83} +(-5.65708 - 9.79835i) q^{85} -11.4418 q^{89} +4.85134 q^{91} +(8.16383 + 14.1402i) q^{95} +(2.58047 - 4.46950i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{5} - 4 q^{7} - 5 q^{11} - 11 q^{13} - 8 q^{17} + 10 q^{19} - 9 q^{23} - 24 q^{25} + 16 q^{29} - q^{31} + 18 q^{37} + 6 q^{41} - 7 q^{43} - 21 q^{47} - 27 q^{49} - 32 q^{53} + 34 q^{55} - 11 q^{59}+ \cdots - 10 q^{97}+O(q^{100}) $ 22 * q + 3 * q^5 - 4 * q^7 - 5 * q^11 - 11 * q^13 - 8 * q^17 + 10 * q^19 - 9 * q^23 - 24 * q^25 + 16 * q^29 - q^31 + 18 * q^37 + 6 * q^41 - 7 * q^43 - 21 * q^47 - 27 * q^49 - 32 * q^53 + 34 * q^55 - 11 * q^59 - 2 * q^61 + 3 * q^65 - 18 * q^67 + 30 * q^71 + 18 * q^73 + 9 * q^77 - 30 * q^79 + 16 * q^83 - 32 * q^85 - 26 * q^89 + 8 * q^91 - 20 * q^95 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	1.80041	+	3.11840i	0.805166	+	1.39459i	0.916179	+	0.400770i	$0.131257\pi$
$5$	1.80041	+	3.11840i	0.805166	+	1.39459i	−0.111013	+	0.993819i	$0.535409\pi$
$6$		0			0
$7$	−2.42567	+	4.20138i	−0.916817	+	1.58797i	−0.112598	+	0.993641i	$0.535917\pi$
$7$	−2.42567	+	4.20138i	−0.916817	+	1.58797i	−0.804219	+	0.594333i	$0.797416\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.96999	+	5.14417i	−0.895486	+	1.55103i	−0.0622837	+	0.998058i	$0.519838\pi$
$11$	−2.96999	+	5.14417i	−0.895486	+	1.55103i	−0.833202	+	0.552969i	$0.813495\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.14211			−0.762074			−0.381037	−	0.924560i	$-0.624433\pi$
$17$	−3.14211			−0.762074			−0.381037	+	0.924560i	$0.624433\pi$
$18$		0			0
$19$	4.53444			1.04027			0.520136	−	0.854084i	$-0.325881\pi$
$19$	4.53444			1.04027			0.520136	+	0.854084i	$0.325881\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.40410	−	4.16402i	−0.501289	−	0.868258i	−0.999999	−	0.00148934i	$-0.999526\pi$
$23$	−2.40410	−	4.16402i	−0.501289	−	0.868258i	0.498710	−	0.866769i	$-0.333807\pi$
$24$		0			0
$25$	−3.98293	+	6.89863i	−0.796585	+	1.37973i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.967295	+	1.67540i	−0.179622	+	0.311115i	−0.941751	−	0.336310i	$-0.890821\pi$
$29$	−0.967295	+	1.67540i	−0.179622	+	0.311115i	0.762129	+	0.647425i	$0.224154\pi$
$30$		0			0
$31$	2.30944	+	4.00007i	0.414787	+	0.718433i	0.995406	−	0.0957427i	$-0.0305226\pi$
$31$	2.30944	+	4.00007i	0.414787	+	0.718433i	−0.580619	+	0.814176i	$0.697189\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.4688			−2.95276
$36$		0			0
$37$	10.7700			1.77058			0.885290	−	0.465039i	$-0.153960\pi$
$37$	10.7700			1.77058			0.885290	+	0.465039i	$0.153960\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.389576	−	0.674766i	−0.0608416	−	0.105381i	0.834000	−	0.551764i	$-0.186045\pi$
$41$	−0.389576	−	0.674766i	−0.0608416	−	0.105381i	−0.894842	+	0.446383i	$0.852712\pi$
$42$		0			0
$43$	−0.164787	+	0.285420i	−0.0251298	+	0.0435261i	−0.878317	−	0.478079i	$-0.841333\pi$
$43$	−0.164787	+	0.285420i	−0.0251298	+	0.0435261i	0.853187	+	0.521605i	$0.174667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.01335	+	3.48722i	−0.293677	+	0.508663i	−0.974676	−	0.223621i	$-0.928212\pi$
$47$	−2.01335	+	3.48722i	−0.293677	+	0.508663i	0.680999	+	0.732284i	$0.261546\pi$
$48$		0			0
$49$	−8.26774	−	14.3202i	−1.18111	−	2.04574i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.03880			0.829493			0.414747	−	0.909937i	$-0.363870\pi$
$53$	6.03880			0.829493			0.414747	+	0.909937i	$0.363870\pi$
$54$		0			0
$55$	−21.3888			−2.88406
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.75266	+	3.03569i	0.228176	+	0.395213i	0.957268	−	0.289203i	$-0.0933903\pi$
$59$	1.75266	+	3.03569i	0.228176	+	0.395213i	−0.729091	+	0.684417i	$0.760057\pi$
$60$		0			0
$61$	2.77604	−	4.80825i	0.355436	−	0.615633i	−0.631757	−	0.775167i	$-0.717666\pi$
$61$	2.77604	−	4.80825i	0.355436	−	0.615633i	0.987192	+	0.159534i	$0.0509991\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.80041	−	3.11840i	0.223313	−	0.386789i
$66$		0			0
$67$	−2.91273	−	5.04500i	−0.355847	−	0.616345i	0.631415	−	0.775445i	$-0.282474\pi$
$67$	−2.91273	−	5.04500i	−0.355847	−	0.616345i	−0.987263	+	0.159099i	$0.949141\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.18402			−0.377873			−0.188937	−	0.981989i	$-0.560504\pi$
$71$	−3.18402			−0.377873			−0.188937	+	0.981989i	$0.560504\pi$
$72$		0			0
$73$	11.7545			1.37576			0.687879	−	0.725826i	$-0.258542\pi$
$73$	11.7545			1.37576			0.687879	+	0.725826i	$0.258542\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.4084	−	24.9561i	−1.64199	−	2.84402i
$78$		0			0
$79$	1.97184	−	3.41532i	0.221849	−	0.384254i	−0.733520	−	0.679667i	$-0.762124\pi$
$79$	1.97184	−	3.41532i	0.221849	−	0.384254i	0.955369	+	0.295414i	$0.0954575\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.28229	−	2.22100i	0.140750	−	0.243786i	−0.787029	−	0.616916i	$-0.788382\pi$
$83$	1.28229	−	2.22100i	0.140750	−	0.243786i	0.927779	+	0.373130i	$0.121715\pi$
$84$		0			0
$85$	−5.65708	−	9.79835i	−0.613596	−	1.06278i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.4418			−1.21282			−0.606412	−	0.795150i	$-0.707392\pi$
$89$	−11.4418			−1.21282			−0.606412	+	0.795150i	$0.707392\pi$
$90$		0			0
$91$	4.85134			0.508559
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.16383	+	14.1402i	0.837591	+	1.45075i
$96$		0			0
$97$	2.58047	−	4.46950i	0.262007	−	0.453809i	−0.704768	−	0.709437i	$-0.748949\pi$
$97$	2.58047	−	4.46950i	0.262007	−	0.453809i	0.966775	+	0.255629i	$0.0822824\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.09235	−	8.82021i	0.506708	−	0.877644i	−0.493262	−	0.869881i	$-0.664196\pi$
$101$	5.09235	−	8.82021i	0.506708	−	0.877644i	0.999970	−	0.00776297i	$-0.00247106\pi$
$102$		0			0
$103$	−1.59094	−	2.75559i	−0.156760	−	0.271516i	0.776938	−	0.629577i	$-0.216772\pi$
$103$	−1.59094	−	2.75559i	−0.156760	−	0.271516i	−0.933698	+	0.358060i	$0.883438\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.4926			−1.59440			−0.797201	−	0.603714i	$-0.793687\pi$
$107$	−16.4926			−1.59440			−0.797201	+	0.603714i	$0.793687\pi$
$108$		0			0
$109$	−6.31192			−0.604572			−0.302286	−	0.953217i	$-0.597750\pi$
$109$	−6.31192			−0.604572			−0.302286	+	0.953217i	$0.597750\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.06899	+	7.04770i	0.382779	+	0.662992i	0.991458	−	0.130424i	$-0.0416339\pi$
$113$	4.06899	+	7.04770i	0.382779	+	0.662992i	−0.608680	+	0.793416i	$0.708301\pi$
$114$		0			0
$115$	8.65671	−	14.9939i	0.807242	−	1.39818i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.62173	−	13.2012i	0.698682	−	1.21015i
$120$		0			0
$121$	−12.1417	−	21.0300i	−1.10379	−	1.91182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.6795			−0.955201
$126$		0			0
$127$	9.41029			0.835028			0.417514	−	0.908671i	$-0.362901\pi$
$127$	9.41029			0.835028			0.417514	+	0.908671i	$0.362901\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.18147	+	15.9028i	0.802189	+	1.38943i	0.918173	+	0.396180i	$0.129664\pi$
$131$	9.18147	+	15.9028i	0.802189	+	1.38943i	−0.115984	+	0.993251i	$0.537002\pi$
$132$		0			0
$133$	−10.9990	+	19.0509i	−0.953738	+	1.65192i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.72459	+	6.45117i	−0.318213	+	0.551161i	−0.980115	−	0.198429i	$-0.936416\pi$
$137$	−3.72459	+	6.45117i	−0.318213	+	0.551161i	0.661902	+	0.749590i	$0.269749\pi$
$138$		0			0
$139$	11.5014	+	19.9210i	0.975537	+	1.68968i	0.678151	+	0.734923i	$0.262782\pi$
$139$	11.5014	+	19.9210i	0.975537	+	1.68968i	0.297387	+	0.954757i	$0.403885\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.93998			0.496726
$144$		0			0
$145$	−6.96610			−0.578503
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.82412	+	4.89152i	0.231361	+	0.400729i	0.958209	−	0.286069i	$-0.0923488\pi$
$149$	2.82412	+	4.89152i	0.231361	+	0.400729i	−0.726848	+	0.686799i	$0.759015\pi$
$150$		0			0
$151$	−5.96305	+	10.3283i	−0.485266	+	0.840506i	−0.999857	−	0.0169303i	$-0.994611\pi$
$151$	−5.96305	+	10.3283i	−0.485266	+	0.840506i	0.514590	+	0.857436i	$0.327944\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.31586	+	14.4035i	−0.667946	+	1.15692i
$156$		0			0
$157$	−4.04029	−	6.99798i	−0.322450	−	0.558500i	0.658543	−	0.752543i	$-0.271173\pi$
$157$	−4.04029	−	6.99798i	−0.322450	−	0.558500i	−0.980993	+	0.194043i	$0.937840\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	23.3262			1.83836
$162$		0			0
$163$	12.7054			0.995160			0.497580	−	0.867418i	$-0.334222\pi$
$163$	12.7054			0.995160			0.497580	+	0.867418i	$0.334222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.13446	−	10.6252i	−0.474699	−	0.822202i	0.524881	−	0.851175i	$-0.324110\pi$
$167$	−6.13446	−	10.6252i	−0.474699	−	0.822202i	−0.999580	+	0.0289730i	$0.990776\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.38764	−	11.0637i	0.485643	−	0.841159i	−0.514221	−	0.857658i	$-0.671919\pi$
$173$	6.38764	−	11.0637i	0.485643	−	0.841159i	0.999864	+	0.0164990i	$0.00525204\pi$
$174$		0			0
$175$	−19.3225	−	33.4676i	−1.46065	−	2.52991i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.65209			0.422457			0.211229	−	0.977437i	$-0.432254\pi$
$179$	5.65209			0.422457			0.211229	+	0.977437i	$0.432254\pi$
$180$		0			0
$181$	−11.9835			−0.890723			−0.445362	−	0.895351i	$-0.646925\pi$
$181$	−11.9835			−0.890723			−0.445362	+	0.895351i	$0.646925\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.3904	+	33.5852i	1.42561	+	2.46923i
$186$		0			0
$187$	9.33204	−	16.1636i	0.682427	−	1.18200i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.69583	+	11.5975i	−0.484493	+	0.839167i	−0.999841	−	0.0178139i	$-0.994329\pi$
$191$	−6.69583	+	11.5975i	−0.484493	+	0.839167i	0.515348	+	0.856981i	$0.327663\pi$
$192$		0			0
$193$	−0.00315107	−	0.00545781i	−0.000226819	−	0.000392862i	0.865912	−	0.500196i	$-0.166739\pi$
$193$	−0.00315107	−	0.00545781i	−0.000226819	−	0.000392862i	−0.866139	+	0.499804i	$0.833406\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.67863			−0.689574			−0.344787	−	0.938681i	$-0.612049\pi$
$197$	−9.67863			−0.689574			−0.344787	+	0.938681i	$0.612049\pi$
$198$		0			0
$199$	−26.2710			−1.86230			−0.931149	−	0.364638i	$-0.881193\pi$
$199$	−26.2710			−1.86230			−0.931149	+	0.364638i	$0.881193\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.69268	−	8.12795i	−0.329361	−	0.570470i
$204$		0			0
$205$	1.40279	−	2.42971i	0.0979752	−	0.169698i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.4672	+	23.3259i	−0.931548	+	1.61349i
$210$		0			0
$211$	−0.764674	−	1.32445i	−0.0526424	−	0.0911792i	0.838503	−	0.544896i	$-0.183431\pi$
$211$	−0.764674	−	1.32445i	−0.0526424	−	0.0911792i	−0.891146	+	0.453717i	$0.850098\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.18674			−0.0809347
$216$		0			0
$217$	−22.4077			−1.52114
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.57106	+	2.72115i	0.105681	+	0.183044i
$222$		0			0
$223$	−6.79374	+	11.7671i	−0.454943	+	0.787984i	−0.998685	−	0.0512684i	$-0.983674\pi$
$223$	−6.79374	+	11.7671i	−0.454943	+	0.787984i	0.543742	+	0.839252i	$0.317007\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.46880	+	12.9363i	−0.495722	+	0.858615i	−0.999988	−	0.00493317i	$-0.998430\pi$
$227$	−7.46880	+	12.9363i	−0.495722	+	0.858615i	0.504266	+	0.863548i	$0.331763\pi$
$228$		0			0
$229$	1.41828	+	2.45653i	0.0937225	+	0.162332i	0.909075	−	0.416633i	$-0.136790\pi$
$229$	1.41828	+	2.45653i	0.0937225	+	0.162332i	−0.815352	+	0.578965i	$0.803457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.7200			0.833312			0.416656	−	0.909064i	$-0.363202\pi$
$233$	12.7200			0.833312			0.416656	+	0.909064i	$0.363202\pi$
$234$		0			0
$235$	−14.4994			−0.945834
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.06154	+	1.83865i	0.0686655	+	0.118932i	0.898314	−	0.439354i	$-0.144793\pi$
$239$	1.06154	+	1.83865i	0.0686655	+	0.118932i	−0.829649	+	0.558286i	$0.811459\pi$
$240$		0			0
$241$	1.66094	−	2.87684i	0.106991	−	0.185313i	−0.807559	−	0.589787i	$-0.799212\pi$
$241$	1.66094	−	2.87684i	0.106991	−	0.185313i	0.914550	+	0.404473i	$0.132545\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	29.7706	−	51.5642i	1.90197	−	3.29432i
$246$		0			0
$247$	−2.26722	−	3.92694i	−0.144260	−	0.249865i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.9517			−0.943746			−0.471873	−	0.881667i	$-0.656422\pi$
$251$	−14.9517			−0.943746			−0.471873	+	0.881667i	$0.656422\pi$
$252$		0			0
$253$	28.5606			1.79559
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.01395	−	8.68441i	−0.312761	−	0.541719i	0.666198	−	0.745775i	$-0.267921\pi$
$257$	−5.01395	−	8.68441i	−0.312761	−	0.541719i	−0.978959	+	0.204057i	$0.934587\pi$
$258$		0			0
$259$	−26.1245	+	45.2490i	−1.62330	+	2.81163i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.40234	+	4.16098i	−0.148135	+	0.256577i	−0.930538	−	0.366195i	$-0.880660\pi$
$263$	−2.40234	+	4.16098i	−0.148135	+	0.256577i	0.782403	+	0.622772i	$0.213994\pi$
$264$		0			0
$265$	10.8723	+	18.8314i	0.667880	+	1.15680i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.4507			0.637190			0.318595	−	0.947891i	$-0.396789\pi$
$269$	10.4507			0.637190			0.318595	+	0.947891i	$0.396789\pi$
$270$		0			0
$271$	12.4552			0.756596			0.378298	−	0.925684i	$-0.376509\pi$
$271$	12.4552			0.756596			0.378298	+	0.925684i	$0.376509\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.6585	−	40.9777i	−1.42666	−	2.47105i
$276$		0			0
$277$	6.23063	−	10.7918i	0.374362	−	0.648414i	−0.615869	−	0.787848i	$-0.711195\pi$
$277$	6.23063	−	10.7918i	0.374362	−	0.648414i	0.990231	+	0.139434i	$0.0445283\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.0408	+	26.0515i	−0.897261	+	1.55410i	−0.0662788	+	0.997801i	$0.521113\pi$
$281$	−15.0408	+	26.0515i	−0.897261	+	1.55410i	−0.830982	+	0.556300i	$0.812221\pi$
$282$		0			0
$283$	10.8709	+	18.8289i	0.646207	+	1.11926i	0.984021	+	0.178050i	$0.0569790\pi$
$283$	10.8709	+	18.8289i	0.646207	+	1.11926i	−0.337814	+	0.941213i	$0.609688\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.77993			0.223122
$288$		0			0
$289$	−7.12713			−0.419243
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.200069	−	0.346530i	−0.0116882	−	0.0202445i	0.860122	−	0.510088i	$-0.170387\pi$
$293$	−0.200069	−	0.346530i	−0.0116882	−	0.0202445i	−0.871810	+	0.489844i	$0.837054\pi$
$294$		0			0
$295$	−6.31099	+	10.9310i	−0.367440	+	0.636425i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.40410	+	4.16402i	−0.139033	+	0.240812i
$300$		0			0
$301$	−0.799438	−	1.38467i	−0.0460789	−	0.0798109i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.9920			1.14474
$306$		0			0
$307$	−23.5084			−1.34169			−0.670847	−	0.741596i	$-0.734069\pi$
$307$	−23.5084			−1.34169			−0.670847	+	0.741596i	$0.734069\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.79013	−	15.2249i	−0.498442	−	0.863328i	0.501556	−	0.865125i	$-0.332761\pi$
$311$	−8.79013	−	15.2249i	−0.498442	−	0.863328i	−0.999998	+	0.00179754i	$0.999428\pi$
$312$		0			0
$313$	−15.4108	+	26.6924i	−0.871073	+	1.50874i	−0.0101841	+	0.999948i	$0.503242\pi$
$313$	−15.4108	+	26.6924i	−0.871073	+	1.50874i	−0.860888	+	0.508794i	$0.830092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.02914	+	1.78252i	−0.0578021	+	0.100116i	−0.893479	−	0.449106i	$-0.851743\pi$
$317$	−1.02914	+	1.78252i	−0.0578021	+	0.100116i	0.835676	+	0.549222i	$0.185076\pi$
$318$		0			0
$319$	−5.74571	−	9.95187i	−0.321698	−	0.557198i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.2477			−0.792764
$324$		0			0
$325$	7.96585			0.441866
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.76743	−	16.9177i	−0.538496	−	0.932702i
$330$		0			0
$331$	−2.56919	+	4.44997i	−0.141216	+	0.244593i	−0.927955	−	0.372693i	$-0.878434\pi$
$331$	−2.56919	+	4.44997i	−0.141216	+	0.244593i	0.786739	+	0.617286i	$0.211768\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.4882	−	18.1661i	0.573032	−	0.992521i
$336$		0			0
$337$	10.6912	+	18.5177i	0.582388	+	1.00873i	0.995196	+	0.0979071i	$0.0312148\pi$
$337$	10.6912	+	18.5177i	0.582388	+	1.00873i	−0.412808	+	0.910818i	$0.635452\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−27.4360			−1.48575
$342$		0			0
$343$	46.2599			2.49780
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.80180	+	13.5131i	0.418822	+	0.725422i	0.995821	−	0.0913233i	$-0.0291097\pi$
$347$	7.80180	+	13.5131i	0.418822	+	0.725422i	−0.576999	+	0.816745i	$0.695776\pi$
$348$		0			0
$349$	−11.5993	+	20.0905i	−0.620895	+	1.07542i	0.368425	+	0.929658i	$0.379897\pi$
$349$	−11.5993	+	20.0905i	−0.620895	+	1.07542i	−0.989319	+	0.145764i	$0.953436\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.36520	−	14.4889i	0.445235	−	0.771169i	−0.552834	−	0.833291i	$-0.686454\pi$
$353$	8.36520	−	14.4889i	0.445235	−	0.771169i	0.998069	+	0.0621226i	$0.0197870\pi$
$354$		0			0
$355$	−5.73252	−	9.92902i	−0.304251	−	0.526978i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.8608			0.731543			0.365772	−	0.930705i	$-0.380805\pi$
$359$	13.8608			0.731543			0.365772	+	0.930705i	$0.380805\pi$
$360$		0			0
$361$	1.56113			0.0821647
$362$		0			0
$363$		0			0
$364$		0			0
$365$	21.1628	+	36.6551i	1.10771	+	1.91862i
$366$		0			0
$367$	3.39093	−	5.87326i	0.177005	−	0.306582i	−0.763848	−	0.645396i	$-0.776692\pi$
$367$	3.39093	−	5.87326i	0.177005	−	0.306582i	0.940853	+	0.338814i	$0.110026\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.6481	+	25.3713i	−0.760494	+	1.31721i
$372$		0			0
$373$	−14.4619	−	25.0488i	−0.748811	−	1.29698i	−0.948393	−	0.317097i	$-0.897292\pi$
$373$	−14.4619	−	25.0488i	−0.748811	−	1.29698i	0.199582	−	0.979881i	$-0.436041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.93459			0.0996364
$378$		0			0
$379$	4.07463			0.209300			0.104650	−	0.994509i	$-0.466628\pi$
$379$	4.07463			0.209300			0.104650	+	0.994509i	$0.466628\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.9208	−	31.0398i	−0.915711	−	1.58606i	−0.805857	−	0.592110i	$-0.798295\pi$
$383$	−17.9208	−	31.0398i	−0.915711	−	1.58606i	−0.109853	−	0.993948i	$-0.535038\pi$
$384$		0			0
$385$	51.8821	−	89.8624i	2.64415	−	4.57981i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.2149	+	22.8888i	−0.670020	+	1.16051i	0.307878	+	0.951426i	$0.400381\pi$
$389$	−13.2149	+	22.8888i	−0.670020	+	1.16051i	−0.977898	+	0.209083i	$0.932952\pi$
$390$		0			0
$391$	7.55395	+	13.0838i	0.382020	+	0.661677i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.2004			0.714501
$396$		0			0
$397$	12.0428			0.604413			0.302206	−	0.953243i	$-0.402277\pi$
$397$	12.0428			0.604413			0.302206	+	0.953243i	$0.402277\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.35012	+	12.7308i	0.367047	+	0.635744i	0.989102	−	0.147229i	$-0.0470354\pi$
$401$	7.35012	+	12.7308i	0.367047	+	0.635744i	−0.622055	+	0.782973i	$0.713702\pi$
$402$		0			0
$403$	2.30944	−	4.00007i	0.115041	−	0.199257i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−31.9869	+	55.4029i	−1.58553	+	2.74622i
$408$		0			0
$409$	5.45923	+	9.45566i	0.269941	+	0.467552i	0.968846	−	0.247662i	$-0.0796623\pi$
$409$	5.45923	+	9.45566i	0.269941	+	0.467552i	−0.698905	+	0.715214i	$0.746329\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.0055			−0.836784
$414$		0			0
$415$	9.23459			0.453308
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.9845	−	24.2219i	−0.683187	−	1.18332i	−0.974003	−	0.226536i	$-0.927260\pi$
$419$	−13.9845	−	24.2219i	−0.683187	−	1.18332i	0.290815	−	0.956779i	$-0.406073\pi$
$420$		0			0
$421$	4.71722	−	8.17046i	0.229903	−	0.398204i	−0.727876	−	0.685709i	$-0.759492\pi$
$421$	4.71722	−	8.17046i	0.229903	−	0.398204i	0.957779	+	0.287505i	$0.0928257\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.5148	−	21.6763i	0.607057	−	1.05145i
$426$		0			0
$427$	13.4675	+	23.3264i	0.651739	+	1.12885i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.51693			−0.410246			−0.205123	−	0.978736i	$-0.565759\pi$
$431$	−8.51693			−0.410246			−0.205123	+	0.978736i	$0.565759\pi$
$432$		0			0
$433$	21.6386			1.03989			0.519943	−	0.854201i	$-0.325953\pi$
$433$	21.6386			1.03989			0.519943	+	0.854201i	$0.325953\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.9012	−	18.8815i	−0.521477	−	0.903224i
$438$		0			0
$439$	17.4828	−	30.2811i	0.834409	−	1.44524i	−0.0601023	−	0.998192i	$-0.519143\pi$
$439$	17.4828	−	30.2811i	0.834409	−	1.44524i	0.894511	−	0.447046i	$-0.147524\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.7585	−	29.0265i	0.796219	−	1.37909i	−0.125844	−	0.992050i	$-0.540164\pi$
$443$	16.7585	−	29.0265i	0.796219	−	1.37909i	0.922062	−	0.387041i	$-0.126503\pi$
$444$		0			0
$445$	−20.5998	−	35.6799i	−0.976525	−	1.69139i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.27418			0.296097			0.148048	−	0.988980i	$-0.452701\pi$
$449$	6.27418			0.296097			0.148048	+	0.988980i	$0.452701\pi$
$450$		0			0
$451$	4.62815			0.217931
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.73438	+	15.1284i	0.409474	+	0.709230i
$456$		0			0
$457$	−8.38237	+	14.5187i	−0.392111	+	0.679155i	−0.992728	−	0.120381i	$-0.961588\pi$
$457$	−8.38237	+	14.5187i	−0.392111	+	0.679155i	0.600617	+	0.799537i	$0.294922\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.3348	+	30.0248i	−0.807362	+	1.39839i	0.107323	+	0.994224i	$0.465772\pi$
$461$	−17.3348	+	30.0248i	−0.807362	+	1.39839i	−0.914685	+	0.404168i	$0.867561\pi$
$462$		0			0
$463$	16.1627	+	27.9946i	0.751143	+	1.30102i	0.947269	+	0.320439i	$0.103830\pi$
$463$	16.1627	+	27.9946i	0.751143	+	1.30102i	−0.196126	+	0.980579i	$0.562836\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.51171			0.393875			0.196937	−	0.980416i	$-0.436900\pi$
$467$	8.51171			0.393875			0.196937	+	0.980416i	$0.436900\pi$
$468$		0			0
$469$	28.2613			1.30499
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.978833	−	1.69539i	−0.0450068	−	0.0779540i
$474$		0			0
$475$	−18.0603	+	31.2814i	−0.828665	+	1.43529i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.650570	+	1.12682i	−0.0297253	+	0.0514857i	−0.880505	−	0.474036i	$-0.842797\pi$
$479$	−0.650570	+	1.12682i	−0.0297253	+	0.0514857i	0.850780	+	0.525522i	$0.176130\pi$
$480$		0			0
$481$	−5.38501	−	9.32711i	−0.245535	−	0.425280i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.5835			0.843835
$486$		0			0
$487$	−26.7805			−1.21354			−0.606771	−	0.794877i	$-0.707536\pi$
$487$	−26.7805			−1.21354			−0.606771	+	0.794877i	$0.707536\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.9518	−	27.6293i	−0.719895	−	1.24690i	−0.961041	−	0.276406i	$-0.910857\pi$
$491$	−15.9518	−	27.6293i	−0.719895	−	1.24690i	0.241146	−	0.970489i	$-0.422477\pi$
$492$		0			0
$493$	3.03935	−	5.26431i	0.136885	−	0.237092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.72337	−	13.3773i	0.346440	−	0.600052i
$498$		0			0
$499$	−3.26459	−	5.65444i	−0.146143	−	0.253127i	0.783656	−	0.621195i	$-0.213353\pi$
$499$	−3.26459	−	5.65444i	−0.146143	−	0.253127i	−0.929799	+	0.368068i	$0.880019\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.4117			1.62352			0.811758	−	0.583994i	$-0.198511\pi$
$503$	36.4117			1.62352			0.811758	+	0.583994i	$0.198511\pi$
$504$		0			0
$505$	36.6732			1.63194
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.36812	−	2.36966i	−0.0606410	−	0.105033i	0.834111	−	0.551596i	$-0.185981\pi$
$509$	−1.36812	−	2.36966i	−0.0606410	−	0.105033i	−0.894752	+	0.446563i	$0.852648\pi$
$510$		0			0
$511$	−28.5125	+	49.3851i	−1.26132	+	2.18467i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.72868	−	9.92236i	0.252436	−	0.437231i
$516$		0			0
$517$	−11.9592	−	20.7140i	−0.525967	−	0.911001i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	33.7588			1.47900			0.739500	−	0.673157i	$-0.235062\pi$
$521$	33.7588			1.47900			0.739500	+	0.673157i	$0.235062\pi$
$522$		0			0
$523$	18.0617			0.789782			0.394891	−	0.918728i	$-0.370782\pi$
$523$	18.0617			0.789782			0.394891	+	0.918728i	$0.370782\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.25652	−	12.5687i	−0.316099	−	0.547499i
$528$		0			0
$529$	−0.0593818	+	0.102852i	−0.00258182	+	0.00447184i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.389576	+	0.674766i	−0.0168744	+	0.0292274i
$534$		0			0
$535$	−29.6934	−	51.4305i	−1.28376	−	2.22354i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	98.2205			4.23066
$540$		0			0
$541$	9.82851			0.422561			0.211280	−	0.977426i	$-0.432237\pi$
$541$	9.82851			0.422561			0.211280	+	0.977426i	$0.432237\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.3640	−	19.6831i	−0.486781	−	0.843129i
$546$		0			0
$547$	−6.09142	+	10.5506i	−0.260450	+	0.451113i	−0.966362	−	0.257187i	$-0.917204\pi$
$547$	−6.09142	+	10.5506i	−0.260450	+	0.451113i	0.705911	+	0.708300i	$0.250538\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.38614	+	7.59702i	−0.186856	+	0.323644i
$552$		0			0
$553$	9.56605	+	16.5689i	0.406790	+	0.704581i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.07479			−0.384511			−0.192256	−	0.981345i	$-0.561580\pi$
$557$	−9.07479			−0.384511			−0.192256	+	0.981345i	$0.561580\pi$
$558$		0			0
$559$	0.329574			0.0139395
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.3055	+	28.2419i	0.687193	+	1.19025i	0.972742	+	0.231889i	$0.0744907\pi$
$563$	16.3055	+	28.2419i	0.687193	+	1.19025i	−0.285549	+	0.958364i	$0.592176\pi$
$564$		0			0
$565$	−14.6517	+	25.3775i	−0.616401	+	1.06764i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.2251	+	17.7105i	−0.428660	+	0.742461i	−0.996754	−	0.0805022i	$-0.974348\pi$
$569$	−10.2251	+	17.7105i	−0.428660	+	0.742461i	0.568094	+	0.822964i	$0.307681\pi$
$570$		0			0
$571$	12.7365	+	22.0603i	0.533007	+	0.923195i	0.999257	+	0.0385419i	$0.0122713\pi$
$571$	12.7365	+	22.0603i	0.533007	+	0.923195i	−0.466250	+	0.884653i	$0.654395\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	38.3014			1.59728
$576$		0			0
$577$	42.2188			1.75759			0.878795	−	0.477199i	$-0.158348\pi$
$577$	42.2188			1.75759			0.878795	+	0.477199i	$0.158348\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.22083	+	10.7748i	0.258084	+	0.447014i
$582$		0			0
$583$	−17.9352	+	31.0647i	−0.742800	+	1.28657i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.3873	−	23.1875i	0.552555	−	0.957053i	−0.445535	−	0.895265i	$-0.646986\pi$
$587$	13.3873	−	23.1875i	0.552555	−	0.957053i	0.998089	−	0.0617879i	$-0.0196802\pi$
$588$		0			0
$589$	10.4720	+	18.1380i	0.431492	+	0.747365i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.74780			−0.194969			−0.0974844	−	0.995237i	$-0.531080\pi$
$593$	−4.74780			−0.194969			−0.0974844	+	0.995237i	$0.531080\pi$
$594$		0			0
$595$	54.8888			2.25022
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.9972	−	31.1721i	−0.735347	−	1.27366i	−0.954571	−	0.297984i	$-0.903686\pi$
$599$	−17.9972	−	31.1721i	−0.735347	−	1.27366i	0.219224	−	0.975675i	$-0.429647\pi$
$600$		0			0
$601$	2.62645	−	4.54915i	0.107135	−	0.185564i	−0.807473	−	0.589904i	$-0.799166\pi$
$601$	2.62645	−	4.54915i	0.107135	−	0.185564i	0.914609	+	0.404340i	$0.132499\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	43.7199	−	75.7252i	1.77747	−	3.07867i
$606$		0			0
$607$	0.624003	+	1.08081i	0.0253275	+	0.0438685i	0.878411	−	0.477905i	$-0.158604\pi$
$607$	0.624003	+	1.08081i	0.0253275	+	0.0438685i	−0.853084	+	0.521774i	$0.825270\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.02669			0.162903
$612$		0			0
$613$	−18.4731			−0.746120			−0.373060	−	0.927807i	$-0.621692\pi$
$613$	−18.4731			−0.746120			−0.373060	+	0.927807i	$0.621692\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0419	+	31.2495i	0.726340	+	1.25806i	0.958420	+	0.285361i	$0.0921136\pi$
$617$	18.0419	+	31.2495i	0.726340	+	1.25806i	−0.232080	+	0.972697i	$0.574553\pi$
$618$		0			0
$619$	22.2098	−	38.4685i	0.892688	−	1.54618i	0.0560473	−	0.998428i	$-0.482150\pi$
$619$	22.2098	−	38.4685i	0.892688	−	1.54618i	0.836640	−	0.547752i	$-0.184516\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.7539	−	48.0712i	1.11194	−	1.92593i
$624$		0			0
$625$	0.687235	+	1.19033i	0.0274894	+	0.0476131i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.8406			−1.34931
$630$		0			0
$631$	27.2686			1.08555			0.542773	−	0.839880i	$-0.317375\pi$
$631$	27.2686			1.08555			0.542773	+	0.839880i	$0.317375\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.9423	+	29.3450i	0.672336	+	1.16452i
$636$		0			0
$637$	−8.26774	+	14.3202i	−0.327580	+	0.567385i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.1339	−	17.5524i	0.400265	−	0.693280i	−0.593492	−	0.804840i	$-0.702251\pi$
$641$	10.1339	−	17.5524i	0.400265	−	0.693280i	0.993758	+	0.111560i	$0.0355846\pi$
$642$		0			0
$643$	−10.0914	−	17.4789i	−0.397967	−	0.689299i	0.595508	−	0.803349i	$-0.296951\pi$
$643$	−10.0914	−	17.4789i	−0.397967	−	0.689299i	−0.993475	+	0.114051i	$0.963617\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.41833			0.370273			0.185136	−	0.982713i	$-0.440727\pi$
$647$	9.41833			0.370273			0.185136	+	0.982713i	$0.440727\pi$
$648$		0			0
$649$	−20.8215			−0.817315
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.8178	+	41.2537i	0.932063	+	1.61438i	0.779790	+	0.626042i	$0.215326\pi$
$653$	23.8178	+	41.2537i	0.932063	+	1.61438i	0.152273	+	0.988338i	$0.451341\pi$
$654$		0			0
$655$	−33.0607	+	57.2629i	−1.29179	+	2.23745i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.5791	−	18.3235i	0.412102	−	0.713781i	−0.583017	−	0.812460i	$-0.698128\pi$
$659$	10.5791	−	18.3235i	0.412102	−	0.713781i	0.995119	+	0.0986782i	$0.0314614\pi$
$660$		0			0
$661$	−11.3505	−	19.6596i	−0.441482	−	0.764669i	0.556318	−	0.830969i	$-0.312214\pi$
$661$	−11.3505	−	19.6596i	−0.441482	−	0.764669i	−0.997800	+	0.0663009i	$0.978880\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−79.2110			−3.07167
$666$		0			0
$667$	9.30189			0.360171
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.4896	+	28.5609i	0.636576	+	1.10258i
$672$		0			0
$673$	4.80850	−	8.32857i	0.185354	−	0.321043i	−0.758342	−	0.651857i	$-0.773990\pi$
$673$	4.80850	−	8.32857i	0.185354	−	0.321043i	0.943696	+	0.330815i	$0.107323\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.82985	+	4.90145i	−0.108760	+	0.188378i	−0.915268	−	0.402845i	$-0.868021\pi$
$677$	−2.82985	+	4.90145i	−0.108760	+	0.188378i	0.806508	+	0.591223i	$0.201355\pi$
$678$		0			0
$679$	12.5187	+	21.6830i	0.480424	+	0.832119i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−33.6477			−1.28749			−0.643747	−	0.765239i	$-0.722621\pi$
$683$	−33.6477			−1.28749			−0.643747	+	0.765239i	$0.722621\pi$
$684$		0			0
$685$	−26.8231			−1.02486
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.01940	−	5.22976i	−0.115030	−	0.199238i
$690$		0			0
$691$	−4.26302	+	7.38378i	−0.162173	+	0.280892i	−0.935648	−	0.352935i	$-0.885184\pi$
$691$	−4.26302	+	7.38378i	−0.162173	+	0.280892i	0.773475	+	0.633827i	$0.218517\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−41.4144	+	71.7319i	−1.57094	+	2.72095i
$696$		0			0
$697$	1.22409	+	2.12019i	0.0463658	+	0.0803079i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.9968			−0.528651			−0.264326	−	0.964434i	$-0.585149\pi$
$701$	−13.9968			−0.528651			−0.264326	+	0.964434i	$0.585149\pi$
$702$		0			0
$703$	48.8360			1.84188
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.7047	+	42.7898i	0.929117	+	1.60928i
$708$		0			0
$709$	−8.16793	+	14.1473i	−0.306753	+	0.531312i	−0.977650	−	0.210238i	$-0.932576\pi$
$709$	−8.16793	+	14.1473i	−0.306753	+	0.531312i	0.670897	+	0.741551i	$0.265909\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.1042	−	19.2331i	0.415857	−	0.720285i
$714$		0			0
$715$	10.6944	+	18.5232i	0.399947	+	0.692729i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.1133			−0.638219			−0.319110	−	0.947718i	$-0.603384\pi$
$719$	−17.1133			−0.638219			−0.319110	+	0.947718i	$0.603384\pi$
$720$		0			0
$721$	15.4364			0.574881
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.70533	−	13.3460i	−0.286169	−	0.495659i
$726$		0			0
$727$	−3.23624	+	5.60534i	−0.120026	+	0.207890i	−0.919778	−	0.392440i	$-0.871631\pi$
$727$	−3.23624	+	5.60534i	−0.120026	+	0.207890i	0.799752	+	0.600331i	$0.204964\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.517780	−	0.896821i	0.0191508	−	0.0331701i
$732$		0			0
$733$	24.1525	+	41.8334i	0.892093	+	1.54515i	0.837361	+	0.546650i	$0.184097\pi$
$733$	24.1525	+	41.8334i	0.892093	+	1.54515i	0.0547322	+	0.998501i	$0.482569\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	34.6032			1.27462
$738$		0			0
$739$	−18.1182			−0.666488			−0.333244	−	0.942841i	$-0.608143\pi$
$739$	−18.1182			−0.666488			−0.333244	+	0.942841i	$0.608143\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.4567	+	31.9680i	0.677112	+	1.17279i	0.975847	+	0.218457i	$0.0701022\pi$
$743$	18.4567	+	31.9680i	0.677112	+	1.17279i	−0.298734	+	0.954336i	$0.596564\pi$
$744$		0			0
$745$	−10.1691	+	17.6135i	−0.372568	+	0.645307i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	40.0057	−	69.2918i	1.46177	−	2.53187i
$750$		0			0
$751$	−22.5544	−	39.0654i	−0.823023	−	1.42552i	−0.903421	−	0.428755i	$-0.858952\pi$
$751$	−22.5544	−	39.0654i	−0.823023	−	1.42552i	0.0803976	−	0.996763i	$-0.474381\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−42.9437			−1.56288
$756$		0			0
$757$	−14.8646			−0.540263			−0.270132	−	0.962823i	$-0.587067\pi$
$757$	−14.8646			−0.540263			−0.270132	+	0.962823i	$0.587067\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.24075	−	3.88109i	−0.0812271	−	0.140689i	0.822550	−	0.568693i	$-0.192551\pi$
$761$	−2.24075	−	3.88109i	−0.0812271	−	0.140689i	−0.903777	+	0.428003i	$0.859217\pi$
$762$		0			0
$763$	15.3106	−	26.5188i	0.554282	−	0.960044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.75266	−	3.03569i	0.0632848	−	0.109612i
$768$		0			0
$769$	2.08442	+	3.61031i	0.0751659	+	0.130191i	0.901158	−	0.433490i	$-0.142718\pi$
$769$	2.08442	+	3.61031i	0.0751659	+	0.130191i	−0.825992	+	0.563681i	$0.809385\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.3000			−1.19772			−0.598858	−	0.800855i	$-0.704379\pi$
$773$	−33.3000			−1.19772			−0.598858	+	0.800855i	$0.704379\pi$
$774$		0			0
$775$	−36.7933			−1.32165
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.76651	−	3.05968i	−0.0632918	−	0.109625i
$780$		0			0
$781$	9.45650	−	16.3791i	0.338380	−	0.586091i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	14.5483	−	25.1984i	0.519252	−	0.899370i
$786$		0			0
$787$	−6.82197	−	11.8160i	−0.243177	−	0.421195i	0.718441	−	0.695588i	$-0.244856\pi$
$787$	−6.82197	−	11.8160i	−0.243177	−	0.421195i	−0.961617	+	0.274394i	$0.911523\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−39.4801			−1.40375
$792$		0			0
$793$	−5.55209			−0.197160
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.4935	−	28.5676i	−0.584230	−	1.01192i	−0.994971	−	0.100164i	$-0.968063\pi$
$797$	−16.4935	−	28.5676i	−0.584230	−	1.01192i	0.410741	−	0.911752i	$-0.365270\pi$
$798$		0			0
$799$	6.32616	−	10.9572i	0.223803	−	0.387639i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−34.9107	+	60.4671i	−1.23197	+	2.13384i
$804$		0			0
$805$	41.9966	+	72.7403i	1.48019	+	2.56376i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.2406			1.06320			0.531601	−	0.846995i	$-0.321590\pi$
$809$	30.2406			1.06320			0.531601	+	0.846995i	$0.321590\pi$
$810$		0			0
$811$	−38.3497			−1.34664			−0.673320	−	0.739351i	$-0.735132\pi$
$811$	−38.3497			−1.34664			−0.673320	+	0.739351i	$0.735132\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	22.8748	+	39.6203i	0.801270	+	1.38784i
$816$		0			0
$817$	−0.747217	+	1.29422i	−0.0261418	+	0.0452790i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.5824	+	26.9896i	−0.543831	+	0.941942i	0.454849	+	0.890569i	$0.349693\pi$
$821$	−15.5824	+	26.9896i	−0.543831	+	0.941942i	−0.998680	+	0.0513735i	$0.983640\pi$
$822$		0			0
$823$	1.44096	+	2.49581i	0.0502287	+	0.0869986i	0.890047	−	0.455870i	$-0.150672\pi$
$823$	1.44096	+	2.49581i	0.0502287	+	0.0869986i	−0.839818	+	0.542868i	$0.817338\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.9312			0.797396			0.398698	−	0.917082i	$-0.369462\pi$
$827$	22.9312			0.797396			0.398698	+	0.917082i	$0.369462\pi$
$828$		0			0
$829$	−44.0256			−1.52907			−0.764536	−	0.644581i	$-0.777032\pi$
$829$	−44.0256			−1.52907			−0.764536	+	0.644581i	$0.777032\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	25.9782	+	44.9955i	0.900091	+	1.55900i
$834$		0			0
$835$	22.0890	−	38.2593i	0.764423	−	1.32402i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.75434	−	6.50270i	0.129614	−	0.224498i	−0.793913	−	0.608031i	$-0.791960\pi$
$839$	3.75434	−	6.50270i	0.129614	−	0.224498i	0.923527	+	0.383533i	$0.125293\pi$
$840$		0			0
$841$	12.6287	+	21.8735i	0.435472	+	0.754259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.60081			−0.123872
$846$		0			0
$847$	117.807			4.04789
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.8922	−	44.8466i	−0.887573	−	1.53732i
$852$		0			0
$853$	−17.4967	+	30.3052i	−0.599076	+	1.03763i	0.393882	+	0.919161i	$0.371132\pi$
$853$	−17.4967	+	30.3052i	−0.599076	+	1.03763i	−0.992958	+	0.118469i	$0.962201\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.94529	+	3.36935i	−0.0664500	+	0.115095i	−0.897336	−	0.441347i	$-0.854501\pi$
$857$	−1.94529	+	3.36935i	−0.0664500	+	0.115095i	0.830886	+	0.556442i	$0.187834\pi$
$858$		0			0
$859$	−7.39306	−	12.8052i	−0.252248	−	0.436906i	0.711896	−	0.702285i	$-0.247837\pi$
$859$	−7.39306	−	12.8052i	−0.252248	−	0.436906i	−0.964144	+	0.265378i	$0.914503\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	39.4680			1.34351			0.671753	−	0.740775i	$-0.265541\pi$
$863$	39.4680			1.34351			0.671753	+	0.740775i	$0.265541\pi$
$864$		0			0
$865$	46.0014			1.56409
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.7127	+	20.2869i	0.397325	+	0.688188i
$870$		0			0
$871$	−2.91273	+	5.04500i	−0.0986942	+	0.170943i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	25.9049	−	44.8686i	0.875745	−	1.51683i
$876$		0			0
$877$	−15.9102	−	27.5573i	−0.537249	−	0.930543i	−0.999051	−	0.0435599i	$-0.986130\pi$
$877$	−15.9102	−	27.5573i	−0.537249	−	0.930543i	0.461801	−	0.886983i	$-0.347203\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.104015			0.00350437			0.00175218	−	0.999998i	$-0.499442\pi$
$881$	0.104015			0.00350437			0.00175218	+	0.999998i	$0.499442\pi$
$882$		0			0
$883$	−37.7445			−1.27020			−0.635102	−	0.772428i	$-0.719042\pi$
$883$	−37.7445			−1.27020			−0.635102	+	0.772428i	$0.719042\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.0832	−	24.3928i	−0.472867	−	0.819031i	0.526650	−	0.850082i	$-0.323448\pi$
$887$	−14.0832	−	24.3928i	−0.472867	−	0.819031i	−0.999518	+	0.0310515i	$0.990114\pi$
$888$		0			0
$889$	−22.8262	+	39.5362i	−0.765568	+	1.32600i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.12939	+	15.8126i	−0.305504	+	0.529148i
$894$		0			0
$895$	10.1761	+	17.6255i	0.340148	+	0.589154i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.93563			−0.298020
$900$		0			0
$901$	−18.9746			−0.632136
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−21.5751	−	37.3691i	−0.717180	−	1.24219i
$906$		0			0
$907$	18.1360	−	31.4125i	0.602197	−	1.04304i	−0.390291	−	0.920692i	$-0.627626\pi$
$907$	18.1360	−	31.4125i	0.602197	−	1.04304i	0.992488	−	0.122344i	$-0.0390411\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.43178	−	7.67607i	0.146831	−	0.254319i	−0.783223	−	0.621740i	$-0.786426\pi$
$911$	4.43178	−	7.67607i	0.146831	−	0.254319i	0.930055	+	0.367421i	$0.119759\pi$
$912$		0			0
$913$	7.61679	+	13.1927i	0.252079	+	0.436614i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−89.0848			−2.94184
$918$		0			0
$919$	17.7678			0.586106			0.293053	−	0.956096i	$-0.405329\pi$
$919$	17.7678			0.586106			0.293053	+	0.956096i	$0.405329\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.59201	+	2.75744i	0.0524016	+	0.0907622i
$924$		0			0
$925$	−42.8962	+	74.2984i	−1.41042	+	2.44292i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.7193	−	30.6908i	0.581352	−	1.00693i	−0.413967	−	0.910292i	$-0.635857\pi$
$929$	17.7193	−	30.6908i	0.581352	−	1.00693i	0.995319	−	0.0966396i	$-0.0308094\pi$
$930$		0			0
$931$	−37.4896	−	64.9339i	−1.22867	−	2.12812i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	67.2059			2.19787
$936$		0			0
$937$	26.0877			0.852248			0.426124	−	0.904665i	$-0.359879\pi$
$937$	26.0877			0.852248			0.426124	+	0.904665i	$0.359879\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.2832	+	17.8111i	0.335223	+	0.580624i	0.983528	−	0.180757i	$-0.0578549\pi$
$941$	10.2832	+	17.8111i	0.335223	+	0.580624i	−0.648304	+	0.761381i	$0.724522\pi$
$942$		0			0
$943$	−1.87316	+	3.24441i	−0.0609985	+	0.105652i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.2111	−	19.4182i	0.364311	−	0.631006i	−0.624354	−	0.781141i	$-0.714638\pi$
$947$	11.2111	−	19.4182i	0.364311	−	0.631006i	0.988665	+	0.150136i	$0.0479711\pi$
$948$		0			0
$949$	−5.87724	−	10.1797i	−0.190783	−	0.330446i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.5106			0.502438			0.251219	−	0.967930i	$-0.419168\pi$
$953$	15.5106			0.502438			0.251219	+	0.967930i	$0.419168\pi$
$954$		0			0
$955$	−48.2209			−1.56039
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0692	−	31.2968i	−0.583486	−	1.01063i
$960$		0			0
$961$	4.83299	−	8.37098i	0.155903	−	0.270031i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.0113464	−	0.0196526i	0.000365254	−	0.000632638i
$966$		0			0
$967$	−6.61051	−	11.4497i	−0.212580	−	0.368199i	0.739942	−	0.672671i	$-0.234853\pi$
$967$	−6.61051	−	11.4497i	−0.212580	−	0.368199i	−0.952521	+	0.304473i	$0.901520\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.0317			−0.707031			−0.353515	−	0.935429i	$-0.615014\pi$
$971$	−22.0317			−0.707031			−0.353515	+	0.935429i	$0.615014\pi$
$972$		0			0
$973$	−111.595			−3.57756
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.8551	+	41.3182i	0.763192	+	1.32189i	0.941197	+	0.337858i	$0.109702\pi$
$977$	23.8551	+	41.3182i	0.763192	+	1.32189i	−0.178005	+	0.984030i	$0.556964\pi$
$978$		0			0
$979$	33.9819	−	58.8584i	1.08607	−	1.88112i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.5800	−	33.9135i	0.624504	−	1.08167i	−0.364132	−	0.931347i	$-0.618634\pi$
$983$	19.5800	−	33.9135i	0.624504	−	1.08167i	0.988637	−	0.150326i	$-0.0480322\pi$
$984$		0			0
$985$	−17.4255	−	30.1818i	−0.555221	−	0.961672i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.58466			0.0503892
$990$		0			0
$991$	−46.7781			−1.48596			−0.742978	−	0.669316i	$-0.766587\pi$
$991$	−46.7781			−1.48596			−0.742978	+	0.669316i	$0.766587\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−47.2984	−	81.9232i	−1.49946	−	2.59714i
$996$		0			0
$997$	11.2351	−	19.4598i	0.355819	−	0.616297i	−0.631439	−	0.775426i	$-0.717535\pi$
$997$	11.2351	−	19.4598i	0.355819	−	0.616297i	0.987258	+	0.159129i	$0.0508686\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.q.g.937.10		22
3.2	odd	2		936.2.q.g.313.6	✓	22
9.2	odd	6		8424.2.a.bf.1.10		11
9.4	even	3	inner	2808.2.q.g.1873.10		22
9.5	odd	6		936.2.q.g.625.6	yes	22
9.7	even	3		8424.2.a.be.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.q.g.313.6	✓	22	3.2	odd	2
936.2.q.g.625.6	yes	22	9.5	odd	6
2808.2.q.g.937.10		22	1.1	even	1	trivial
2808.2.q.g.1873.10		22	9.4	even	3	inner
8424.2.a.be.1.2		11	9.7	even	3
8424.2.a.bf.1.10		11	9.2	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.80041 + 3.11840i) q^{5} +(-2.42567 + 4.20138i) q^{7} +(-2.96999 + 5.14417i) q^{11} +(-0.500000 - 0.866025i) q^{13} -3.14211 q^{17} +4.53444 q^{19} +(-2.40410 - 4.16402i) q^{23} +(-3.98293 + 6.89863i) q^{25} +(-0.967295 + 1.67540i) q^{29} +(2.30944 + 4.00007i) q^{31} -17.4688 q^{35} +10.7700 q^{37} +(-0.389576 - 0.674766i) q^{41} +(-0.164787 + 0.285420i) q^{43} +(-2.01335 + 3.48722i) q^{47} +(-8.26774 - 14.3202i) q^{49} +6.03880 q^{53} -21.3888 q^{55} +(1.75266 + 3.03569i) q^{59} +(2.77604 - 4.80825i) q^{61} +(1.80041 - 3.11840i) q^{65} +(-2.91273 - 5.04500i) q^{67} -3.18402 q^{71} +11.7545 q^{73} +(-14.4084 - 24.9561i) q^{77} +(1.97184 - 3.41532i) q^{79} +(1.28229 - 2.22100i) q^{83} +(-5.65708 - 9.79835i) q^{85} -11.4418 q^{89} +4.85134 q^{91} +(8.16383 + 14.1402i) q^{95} +(2.58047 - 4.46950i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{5} - 4 q^{7} - 5 q^{11} - 11 q^{13} - 8 q^{17} + 10 q^{19} - 9 q^{23} - 24 q^{25} + 16 q^{29} - q^{31} + 18 q^{37} + 6 q^{41} - 7 q^{43} - 21 q^{47} - 27 q^{49} - 32 q^{53} + 34 q^{55} - 11 q^{59}+ \cdots - 10 q^{97}+O(q^{100}) \) 22 * q + 3 * q^5 - 4 * q^7 - 5 * q^11 - 11 * q^13 - 8 * q^17 + 10 * q^19 - 9 * q^23 - 24 * q^25 + 16 * q^29 - q^31 + 18 * q^37 + 6 * q^41 - 7 * q^43 - 21 * q^47 - 27 * q^49 - 32 * q^53 + 34 * q^55 - 11 * q^59 - 2 * q^61 + 3 * q^65 - 18 * q^67 + 30 * q^71 + 18 * q^73 + 9 * q^77 - 30 * q^79 + 16 * q^83 - 32 * q^85 - 26 * q^89 + 8 * q^91 - 20 * q^95 - 10 * q^97

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