LMFDB - Embedded newform 2808.2.q.g.1873.11

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			1873.11
Character	$\chi$	$=$	2808.1873
Dual form			2808.2.q.g.937.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.89006 - 3.27368i) q^{5} +(-0.659907 - 1.14299i) q^{7} +(2.20538 + 3.81984i) q^{11} +(-0.500000 + 0.866025i) q^{13} +3.06205 q^{17} +7.13223 q^{19} +(1.08498 - 1.87925i) q^{23} +(-4.64465 - 8.04477i) q^{25} +(4.17231 + 7.22665i) q^{29} +(-0.669313 + 1.15928i) q^{31} -4.98905 q^{35} +1.24930 q^{37} +(1.62494 - 2.81448i) q^{41} +(5.93250 + 10.2754i) q^{43} +(0.203201 + 0.351955i) q^{47} +(2.62905 - 4.55364i) q^{49} -2.17588 q^{53} +16.6732 q^{55} +(-4.05223 + 7.01866i) q^{59} +(-4.39869 - 7.61875i) q^{61} +(1.89006 + 3.27368i) q^{65} +(4.69012 - 8.12353i) q^{67} +9.45879 q^{71} -12.2720 q^{73} +(2.91070 - 5.04147i) q^{77} +(-3.64033 - 6.30524i) q^{79} +(6.04966 + 10.4783i) q^{83} +(5.78746 - 10.0242i) q^{85} +5.47746 q^{89} +1.31981 q^{91} +(13.4803 - 23.3486i) q^{95} +(-6.70181 - 11.6079i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

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

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
$5$	1.89006	−	3.27368i	0.845260	−	1.46403i	−0.0401347	−	0.999194i	$-0.512779\pi$
$5$	1.89006	−	3.27368i	0.845260	−	1.46403i	0.885395	−	0.464839i	$-0.153888\pi$
$6$		0			0
$7$	−0.659907	−	1.14299i	−0.249421	−	0.432011i	0.713944	−	0.700203i	$-0.246907\pi$
$7$	−0.659907	−	1.14299i	−0.249421	−	0.432011i	−0.963365	+	0.268192i	$0.913574\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.20538	+	3.81984i	0.664948	+	1.15172i	0.979299	+	0.202417i	$0.0648796\pi$
$11$	2.20538	+	3.81984i	0.664948	+	1.15172i	−0.314351	+	0.949307i	$0.601787\pi$
$12$		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
$17$	3.06205			0.742656			0.371328	−	0.928502i	$-0.378902\pi$
$17$	3.06205			0.742656			0.371328	+	0.928502i	$0.378902\pi$
$18$		0			0
$19$	7.13223			1.63624			0.818122	−	0.575044i	$-0.195015\pi$
$19$	7.13223			1.63624			0.818122	+	0.575044i	$0.195015\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.08498	−	1.87925i	0.226235	−	0.391850i	−0.730454	−	0.682961i	$-0.760692\pi$
$23$	1.08498	−	1.87925i	0.226235	−	0.391850i	0.956689	+	0.291111i	$0.0940250\pi$
$24$		0			0
$25$	−4.64465	−	8.04477i	−0.928930	−	1.60895i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.17231	+	7.22665i	0.774778	+	1.34195i	0.934919	+	0.354861i	$0.115472\pi$
$29$	4.17231	+	7.22665i	0.774778	+	1.34195i	−0.160141	+	0.987094i	$0.551195\pi$
$30$		0			0
$31$	−0.669313	+	1.15928i	−0.120212	+	0.208213i	−0.919851	−	0.392267i	$-0.871691\pi$
$31$	−0.669313	+	1.15928i	−0.120212	+	0.208213i	0.799639	+	0.600481i	$0.205024\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.98905			−0.843304
$36$		0			0
$37$	1.24930			0.205384			0.102692	−	0.994713i	$-0.467254\pi$
$37$	1.24930			0.205384			0.102692	+	0.994713i	$0.467254\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.62494	−	2.81448i	0.253773	−	0.439548i	−0.710788	−	0.703406i	$-0.751662\pi$
$41$	1.62494	−	2.81448i	0.253773	−	0.439548i	0.964562	+	0.263858i	$0.0849950\pi$
$42$		0			0
$43$	5.93250	+	10.2754i	0.904698	+	1.56698i	0.821322	+	0.570465i	$0.193237\pi$
$43$	5.93250	+	10.2754i	0.904698	+	1.56698i	0.0833760	+	0.996518i	$0.473430\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.203201	+	0.351955i	0.0296400	+	0.0513380i	0.880465	−	0.474111i	$-0.157231\pi$
$47$	0.203201	+	0.351955i	0.0296400	+	0.0513380i	−0.850825	+	0.525449i	$0.823897\pi$
$48$		0			0
$49$	2.62905	−	4.55364i	0.375578	−	0.650520i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.17588			−0.298881			−0.149440	−	0.988771i	$-0.547747\pi$
$53$	−2.17588			−0.298881			−0.149440	+	0.988771i	$0.547747\pi$
$54$		0			0
$55$	16.6732			2.24822
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.05223	+	7.01866i	−0.527555	+	0.913752i	0.471929	+	0.881636i	$0.343558\pi$
$59$	−4.05223	+	7.01866i	−0.527555	+	0.913752i	−0.999484	+	0.0321154i	$0.989776\pi$
$60$		0			0
$61$	−4.39869	−	7.61875i	−0.563194	−	0.975481i	−0.997215	−	0.0745784i	$-0.976239\pi$
$61$	−4.39869	−	7.61875i	−0.563194	−	0.975481i	0.434021	−	0.900903i	$-0.357094\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.89006	+	3.27368i	0.234433	+	0.406050i
$66$		0			0
$67$	4.69012	−	8.12353i	0.572990	−	0.992448i	−0.423267	−	0.906005i	$-0.639117\pi$
$67$	4.69012	−	8.12353i	0.572990	−	0.992448i	0.996257	−	0.0864426i	$-0.0275499\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.45879			1.12255			0.561276	−	0.827629i	$-0.310311\pi$
$71$	9.45879			1.12255			0.561276	+	0.827629i	$0.310311\pi$
$72$		0			0
$73$	−12.2720			−1.43633			−0.718165	−	0.695872i	$-0.755018\pi$
$73$	−12.2720			−1.43633			−0.718165	+	0.695872i	$0.755018\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.91070	−	5.04147i	0.331705	−	0.574529i
$78$		0			0
$79$	−3.64033	−	6.30524i	−0.409569	−	0.709395i	0.585272	−	0.810837i	$-0.300988\pi$
$79$	−3.64033	−	6.30524i	−0.409569	−	0.709395i	−0.994841	+	0.101442i	$0.967654\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.04966	+	10.4783i	0.664036	+	1.15014i	0.979546	+	0.201222i	$0.0644912\pi$
$83$	6.04966	+	10.4783i	0.664036	+	1.15014i	−0.315510	+	0.948922i	$0.602175\pi$
$84$		0			0
$85$	5.78746	−	10.0242i	0.627738	−	1.08727i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.47746			0.580609			0.290305	−	0.956934i	$-0.406243\pi$
$89$	5.47746			0.580609			0.290305	+	0.956934i	$0.406243\pi$
$90$		0			0
$91$	1.31981			0.138354
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.4803	−	23.3486i	1.38305	−	2.39552i
$96$		0			0
$97$	−6.70181	−	11.6079i	−0.680465	−	1.17860i	−0.974839	−	0.222910i	$-0.928444\pi$
$97$	−6.70181	−	11.6079i	−0.680465	−	1.17860i	0.294374	−	0.955690i	$-0.404889\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.65426	−	11.5255i	−0.662124	−	1.14683i	−0.980057	−	0.198719i	$-0.936322\pi$
$101$	−6.65426	−	11.5255i	−0.662124	−	1.14683i	0.317933	−	0.948113i	$-0.397011\pi$
$102$		0			0
$103$	0.675053	−	1.16923i	0.0665149	−	0.115207i	−0.830850	−	0.556496i	$-0.812145\pi$
$103$	0.675053	−	1.16923i	0.0665149	−	0.115207i	0.897365	+	0.441289i	$0.145479\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.74705			−0.555588			−0.277794	−	0.960641i	$-0.589603\pi$
$107$	−5.74705			−0.555588			−0.277794	+	0.960641i	$0.589603\pi$
$108$		0			0
$109$	−16.1839			−1.55014			−0.775069	−	0.631877i	$-0.782285\pi$
$109$	−16.1839			−1.55014			−0.775069	+	0.631877i	$0.782285\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.55790	+	4.43040i	−0.240627	+	0.416777i	−0.960893	−	0.276920i	$-0.910686\pi$
$113$	−2.55790	+	4.43040i	−0.240627	+	0.416777i	0.720266	+	0.693698i	$0.244020\pi$
$114$		0			0
$115$	−4.10137	−	7.10378i	−0.382455	−	0.662431i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.02067	−	3.49990i	−0.185234	−	0.320835i
$120$		0			0
$121$	−4.22743	+	7.32212i	−0.384312	+	0.665647i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−16.2141			−1.45023
$126$		0			0
$127$	−17.4520			−1.54861			−0.774305	−	0.632812i	$-0.781901\pi$
$127$	−17.4520			−1.54861			−0.774305	+	0.632812i	$0.781901\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.53713	+	2.66239i	−0.134300	+	0.232614i	−0.925330	−	0.379163i	$-0.876212\pi$
$131$	−1.53713	+	2.66239i	−0.134300	+	0.232614i	0.791030	+	0.611777i	$0.209545\pi$
$132$		0			0
$133$	−4.70661	−	8.15208i	−0.408115	−	0.706875i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.40458	+	11.0931i	0.547181	+	0.947745i	0.998466	+	0.0553650i	$0.0176322\pi$
$137$	6.40458	+	11.0931i	0.547181	+	0.947745i	−0.451286	+	0.892380i	$0.649034\pi$
$138$		0			0
$139$	9.62910	−	16.6781i	0.816730	−	1.41462i	−0.0913499	−	0.995819i	$-0.529118\pi$
$139$	9.62910	−	16.6781i	0.816730	−	1.41462i	0.908079	−	0.418798i	$-0.137549\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.41077			−0.368847
$144$		0			0
$145$	31.5436			2.61956
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.33391	−	9.23860i	0.436971	−	0.756856i	−0.560483	−	0.828166i	$-0.689385\pi$
$149$	5.33391	−	9.23860i	0.436971	−	0.756856i	0.997454	+	0.0713099i	$0.0227179\pi$
$150$		0			0
$151$	−4.99869	−	8.65799i	−0.406788	−	0.704577i	0.587740	−	0.809050i	$-0.300018\pi$
$151$	−4.99869	−	8.65799i	−0.406788	−	0.704577i	−0.994528	+	0.104473i	$0.966684\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.53008	+	4.38223i	0.203221	+	0.351989i
$156$		0			0
$157$	−6.75613	+	11.7020i	−0.539198	+	0.933919i	0.459749	+	0.888049i	$0.347939\pi$
$157$	−6.75613	+	11.7020i	−0.539198	+	0.933919i	−0.998947	+	0.0458699i	$0.985394\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.86395			−0.225711
$162$		0			0
$163$	−5.22981			−0.409631			−0.204815	−	0.978801i	$-0.565659\pi$
$163$	−5.22981			−0.409631			−0.204815	+	0.978801i	$0.565659\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.85695	+	8.41248i	−0.375842	+	0.650977i	−0.990453	−	0.137853i	$-0.955980\pi$
$167$	−4.85695	+	8.41248i	−0.375842	+	0.650977i	0.614611	+	0.788831i	$0.289313\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.2600	−	21.2350i	−0.932113	−	1.61447i	−0.779703	−	0.626149i	$-0.784630\pi$
$173$	−12.2600	−	21.2350i	−0.932113	−	1.61447i	−0.152410	−	0.988317i	$-0.548703\pi$
$174$		0			0
$175$	−6.13007	+	10.6176i	−0.463390	+	0.802615i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.4506			1.45381			0.726903	−	0.686740i	$-0.240959\pi$
$179$	19.4506			1.45381			0.726903	+	0.686740i	$0.240959\pi$
$180$		0			0
$181$	15.4476			1.14821			0.574104	−	0.818782i	$-0.305350\pi$
$181$	15.4476			1.14821			0.574104	+	0.818782i	$0.305350\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.36125	−	4.08981i	0.173603	−	0.300689i
$186$		0			0
$187$	6.75300	+	11.6965i	0.493828	+	0.855335i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.31759	−	7.47829i	−0.312410	−	0.541110i	0.666474	−	0.745529i	$-0.267803\pi$
$191$	−4.31759	−	7.47829i	−0.312410	−	0.541110i	−0.978884	+	0.204419i	$0.934470\pi$
$192$		0			0
$193$	−9.17262	+	15.8874i	−0.660260	+	1.14360i	0.320287	+	0.947320i	$0.396220\pi$
$193$	−9.17262	+	15.8874i	−0.660260	+	1.14360i	−0.980547	+	0.196283i	$0.937113\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.26698			0.232762			0.116381	−	0.993205i	$-0.462871\pi$
$197$	3.26698			0.232762			0.116381	+	0.993205i	$0.462871\pi$
$198$		0			0
$199$	18.4899			1.31071			0.655357	−	0.755320i	$-0.272518\pi$
$199$	18.4899			1.31071			0.655357	+	0.755320i	$0.272518\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.50667	−	9.53783i	0.386493	−	0.669425i
$204$		0			0
$205$	−6.14247	−	10.6391i	−0.429009	−	0.743065i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.7293	+	27.2439i	1.08802	+	1.88450i
$210$		0			0
$211$	−1.54759	+	2.68050i	−0.106540	+	0.184533i	−0.914367	−	0.404888i	$-0.867311\pi$
$211$	−1.54759	+	2.68050i	−0.106540	+	0.184533i	0.807826	+	0.589421i	$0.200644\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	44.8511			3.05882
$216$		0			0
$217$	1.76674			0.119934
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.53103	+	2.65181i	−0.102988	+	0.178380i
$222$		0			0
$223$	−1.28975	−	2.23391i	−0.0863679	−	0.149594i	0.819605	−	0.572928i	$-0.194193\pi$
$223$	−1.28975	−	2.23391i	−0.0863679	−	0.149594i	−0.905973	+	0.423335i	$0.860859\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.97484	−	8.61667i	−0.330192	−	0.571909i	0.652358	−	0.757911i	$-0.273780\pi$
$227$	−4.97484	−	8.61667i	−0.330192	−	0.571909i	−0.982549	+	0.186003i	$0.940447\pi$
$228$		0			0
$229$	−2.34074	+	4.05428i	−0.154680	+	0.267914i	−0.932943	−	0.360025i	$-0.882768\pi$
$229$	−2.34074	+	4.05428i	−0.154680	+	0.267914i	0.778262	+	0.627939i	$0.216101\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.8323			1.03721			0.518604	−	0.855015i	$-0.326452\pi$
$233$	15.8323			1.03721			0.518604	+	0.855015i	$0.326452\pi$
$234$		0			0
$235$	1.53625			0.100214
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.53305	+	11.3156i	−0.422588	+	0.731944i	−0.996192	−	0.0871889i	$-0.972212\pi$
$239$	−6.53305	+	11.3156i	−0.422588	+	0.731944i	0.573604	+	0.819133i	$0.305545\pi$
$240$		0			0
$241$	−8.30511	−	14.3849i	−0.534979	−	0.926611i	−0.999164	−	0.0408729i	$-0.986986\pi$
$241$	−8.30511	−	14.3849i	−0.534979	−	0.926611i	0.464185	−	0.885738i	$-0.346347\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.93810	−	17.2133i	−0.634922	−	1.09972i
$246$		0			0
$247$	−3.56611	+	6.17669i	−0.226906	+	0.393013i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.1047			1.26900			0.634500	−	0.772923i	$-0.281206\pi$
$251$	20.1047			1.26900			0.634500	+	0.772923i	$0.281206\pi$
$252$		0			0
$253$	9.57122			0.601737
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.56913	+	14.8422i	−0.534527	+	0.925829i	0.464659	+	0.885490i	$0.346177\pi$
$257$	−8.56913	+	14.8422i	−0.534527	+	0.925829i	−0.999186	+	0.0403387i	$0.987156\pi$
$258$		0			0
$259$	−0.824422	−	1.42794i	−0.0512271	−	0.0887279i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.0441	−	17.3970i	−0.619348	−	1.07274i	−0.989605	−	0.143813i	$-0.954064\pi$
$263$	−10.0441	−	17.3970i	−0.619348	−	1.07274i	0.370256	−	0.928930i	$-0.379270\pi$
$264$		0			0
$265$	−4.11255	+	7.12314i	−0.252632	+	0.437571i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.10780			0.250457			0.125228	−	0.992128i	$-0.460034\pi$
$269$	4.10780			0.250457			0.125228	+	0.992128i	$0.460034\pi$
$270$		0			0
$271$	13.1958			0.801585			0.400793	−	0.916169i	$-0.368735\pi$
$271$	13.1958			0.801585			0.400793	+	0.916169i	$0.368735\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	20.4865	−	35.4836i	1.23538	−	2.13974i
$276$		0			0
$277$	11.3036	+	19.5785i	0.679170	+	1.17636i	0.975231	+	0.221188i	$0.0709935\pi$
$277$	11.3036	+	19.5785i	0.679170	+	1.17636i	−0.296061	+	0.955169i	$0.595673\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.61926	+	13.1969i	0.454527	+	0.787263i	0.998661	−	0.0517348i	$-0.0164750\pi$
$281$	7.61926	+	13.1969i	0.454527	+	0.787263i	−0.544134	+	0.838998i	$0.683142\pi$
$282$		0			0
$283$	−4.59195	+	7.95348i	−0.272963	+	0.472786i	−0.969619	−	0.244620i	$-0.921337\pi$
$283$	−4.59195	+	7.95348i	−0.272963	+	0.472786i	0.696656	+	0.717405i	$0.254670\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.28924			−0.253186
$288$		0			0
$289$	−7.62384			−0.448461
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.232689	+	0.403030i	−0.0135939	+	0.0235453i	−0.872742	−	0.488181i	$-0.837661\pi$
$293$	−0.232689	+	0.403030i	−0.0135939	+	0.0235453i	0.859148	+	0.511726i	$0.170994\pi$
$294$		0			0
$295$	15.3179	+	26.5314i	0.891842	+	1.54472i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.08498	+	1.87925i	0.0627462	+	0.108680i
$300$		0			0
$301$	7.82980	−	13.5616i	0.451302	−	0.781678i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.2551			−1.90418
$306$		0			0
$307$	−13.8035			−0.787810			−0.393905	−	0.919151i	$-0.628876\pi$
$307$	−13.8035			−0.787810			−0.393905	+	0.919151i	$0.628876\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.59547	−	6.22754i	0.203880	−	0.353131i	−0.745895	−	0.666063i	$-0.767978\pi$
$311$	3.59547	−	6.22754i	0.203880	−	0.353131i	0.949775	+	0.312932i	$0.101311\pi$
$312$		0			0
$313$	4.14223	+	7.17456i	0.234133	+	0.405530i	0.959020	−	0.283337i	$-0.0914416\pi$
$313$	4.14223	+	7.17456i	0.234133	+	0.405530i	−0.724888	+	0.688867i	$0.758108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.21178	+	3.83091i	0.124226	+	0.215165i	0.921430	−	0.388544i	$-0.127022\pi$
$317$	2.21178	+	3.83091i	0.124226	+	0.215165i	−0.797204	+	0.603710i	$0.793689\pi$
$318$		0			0
$319$	−18.4031	+	31.8751i	−1.03037	+	1.78466i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	21.8392			1.21517
$324$		0			0
$325$	9.28930			0.515278
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.268188	−	0.464516i	0.0147857	−	0.0256096i
$330$		0			0
$331$	11.7011	+	20.2668i	0.643148	+	1.11397i	0.984726	+	0.174112i	$0.0557055\pi$
$331$	11.7011	+	20.2668i	0.643148	+	1.11397i	−0.341577	+	0.939854i	$0.610961\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.7292	−	30.7079i	−0.968651	−	1.67775i
$336$		0			0
$337$	15.8641	−	27.4774i	0.864171	−	1.49679i	−0.00369700	−	0.999993i	$-0.501177\pi$
$337$	15.8641	−	27.4774i	0.864171	−	1.49679i	0.867868	−	0.496795i	$-0.165490\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.90436			−0.319739
$342$		0			0
$343$	−16.1784			−0.873552
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.6119	+	27.0407i	−0.838092	+	1.45162i	0.0533955	+	0.998573i	$0.482996\pi$
$347$	−15.6119	+	27.0407i	−0.838092	+	1.45162i	−0.891488	+	0.453045i	$0.850338\pi$
$348$		0			0
$349$	−13.8233	−	23.9426i	−0.739944	−	1.28162i	−0.952520	−	0.304475i	$-0.901519\pi$
$349$	−13.8233	−	23.9426i	−0.739944	−	1.28162i	0.212577	−	0.977144i	$-0.431814\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.49914	+	4.32864i	0.133016	+	0.230390i	0.924838	−	0.380362i	$-0.124201\pi$
$353$	2.49914	+	4.32864i	0.133016	+	0.230390i	−0.791822	+	0.610752i	$0.790867\pi$
$354$		0			0
$355$	17.8777	−	30.9650i	0.948848	−	1.64345i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.1761			0.537075			0.268537	−	0.963269i	$-0.413460\pi$
$359$	10.1761			0.537075			0.268537	+	0.963269i	$0.413460\pi$
$360$		0			0
$361$	31.8686			1.67730
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−23.1948	+	40.1746i	−1.21407	+	2.10284i
$366$		0			0
$367$	3.85242	+	6.67258i	0.201094	+	0.348306i	0.948881	−	0.315633i	$-0.102217\pi$
$367$	3.85242	+	6.67258i	0.201094	+	0.348306i	−0.747787	+	0.663939i	$0.768884\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.43588	+	2.48702i	0.0745472	+	0.129120i
$372$		0			0
$373$	−14.6275	+	25.3356i	−0.757384	+	1.31183i	0.186797	+	0.982399i	$0.440189\pi$
$373$	−14.6275	+	25.3356i	−0.757384	+	1.31183i	−0.944180	+	0.329429i	$0.893144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.34461			−0.429769
$378$		0			0
$379$	−20.9948			−1.07843			−0.539215	−	0.842168i	$-0.681279\pi$
$379$	−20.9948			−1.07843			−0.539215	+	0.842168i	$0.681279\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.49637	+	14.7161i	−0.434144	+	0.751960i	−0.997225	−	0.0744417i	$-0.976283\pi$
$383$	−8.49637	+	14.7161i	−0.434144	+	0.751960i	0.563081	+	0.826402i	$0.309616\pi$
$384$		0			0
$385$	−11.0028	−	19.0574i	−0.560753	−	0.971253i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.5122	+	19.9398i	0.583694	+	1.01099i	0.995037	+	0.0995069i	$0.0317265\pi$
$389$	11.5122	+	19.9398i	0.583694	+	1.01099i	−0.411343	+	0.911481i	$0.634940\pi$
$390$		0			0
$391$	3.32228	−	5.75435i	0.168015	−	0.291010i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−27.5218			−1.38477
$396$		0			0
$397$	−10.2161			−0.512731			−0.256365	−	0.966580i	$-0.582525\pi$
$397$	−10.2161			−0.512731			−0.256365	+	0.966580i	$0.582525\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.0998	−	29.6177i	0.853921	−	1.47904i	−0.0237206	−	0.999719i	$-0.507551\pi$
$401$	17.0998	−	29.6177i	0.853921	−	1.47904i	0.877642	−	0.479317i	$-0.159115\pi$
$402$		0			0
$403$	−0.669313	−	1.15928i	−0.0333408	−	0.0577480i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.75518	+	4.77212i	0.136569	+	0.236545i
$408$		0			0
$409$	−14.9088	+	25.8229i	−0.737195	+	1.27686i	0.216558	+	0.976270i	$0.430517\pi$
$409$	−14.9088	+	25.8229i	−0.737195	+	1.27686i	−0.953754	+	0.300590i	$0.902817\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.6964			0.526334
$414$		0			0
$415$	45.7368			2.24513
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.45668	−	12.9153i	0.364283	−	0.630956i	−0.624378	−	0.781122i	$-0.714648\pi$
$419$	7.45668	−	12.9153i	0.364283	−	0.630956i	0.988661	+	0.150166i	$0.0479809\pi$
$420$		0			0
$421$	−6.46463	−	11.1971i	−0.315067	−	0.545712i	0.664385	−	0.747391i	$-0.268694\pi$
$421$	−6.46463	−	11.1971i	−0.315067	−	0.545712i	−0.979452	+	0.201679i	$0.935360\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.2222	−	24.6335i	−0.689876	−	1.19490i
$426$		0			0
$427$	−5.80545	+	10.0553i	−0.280946	+	0.486612i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.00134			−0.433579			−0.216790	−	0.976218i	$-0.569559\pi$
$431$	−9.00134			−0.433579			−0.216790	+	0.976218i	$0.569559\pi$
$432$		0			0
$433$	11.7638			0.565333			0.282667	−	0.959218i	$-0.408781\pi$
$433$	11.7638			0.565333			0.282667	+	0.959218i	$0.408781\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.73835	−	13.4032i	0.370175	−	0.641163i
$438$		0			0
$439$	−15.8633	−	27.4760i	−0.757114	−	1.31136i	−0.944316	−	0.329039i	$-0.893275\pi$
$439$	−15.8633	−	27.4760i	−0.757114	−	1.31136i	0.187202	−	0.982321i	$-0.440058\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.83382	−	3.17626i	−0.0871272	−	0.150909i	0.819168	−	0.573553i	$-0.194435\pi$
$443$	−1.83382	−	3.17626i	−0.0871272	−	0.150909i	−0.906296	+	0.422644i	$0.861102\pi$
$444$		0			0
$445$	10.3527	−	17.9314i	0.490766	−	0.850031i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.4850			−1.86341			−0.931706	−	0.363213i	$-0.881680\pi$
$449$	−39.4850			−1.86341			−0.931706	+	0.363213i	$0.881680\pi$
$450$		0			0
$451$	14.3345			0.674984
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.49453	−	4.32065i	0.116945	−	0.202555i
$456$		0			0
$457$	−12.1028	−	20.9626i	−0.566144	−	0.980590i	−0.996942	−	0.0781417i	$-0.975101\pi$
$457$	−12.1028	−	20.9626i	−0.566144	−	0.980590i	0.430798	−	0.902448i	$-0.358232\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.62828	−	8.01641i	−0.215560	−	0.373362i	0.737885	−	0.674926i	$-0.235824\pi$
$461$	−4.62828	−	8.01641i	−0.215560	−	0.373362i	−0.953446	+	0.301564i	$0.902491\pi$
$462$		0			0
$463$	−4.06304	+	7.03739i	−0.188826	+	0.327055i	−0.944859	−	0.327478i	$-0.893801\pi$
$463$	−4.06304	+	7.03739i	−0.188826	+	0.327055i	0.756033	+	0.654533i	$0.227135\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.39825			−0.157252			−0.0786261	−	0.996904i	$-0.525053\pi$
$467$	−3.39825			−0.157252			−0.0786261	+	0.996904i	$0.525053\pi$
$468$		0			0
$469$	−12.3802			−0.571664
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−26.1669	+	45.3224i	−1.20315	+	2.08392i
$474$		0			0
$475$	−33.1267	−	57.3771i	−1.51996	−	2.63264i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.5576	+	26.9466i	0.710847	+	1.23122i	0.964540	+	0.263938i	$0.0850215\pi$
$479$	15.5576	+	26.9466i	0.710847	+	1.23122i	−0.253693	+	0.967285i	$0.581645\pi$
$480$		0			0
$481$	−0.624650	+	1.08193i	−0.0284816	+	0.0493315i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−50.6672			−2.30068
$486$		0			0
$487$	5.20054			0.235659			0.117829	−	0.993034i	$-0.462406\pi$
$487$	5.20054			0.235659			0.117829	+	0.993034i	$0.462406\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.2623	+	29.8992i	−0.779037	+	1.34933i	0.153460	+	0.988155i	$0.450958\pi$
$491$	−17.2623	+	29.8992i	−0.779037	+	1.34933i	−0.932497	+	0.361177i	$0.882375\pi$
$492$		0			0
$493$	12.7758	+	22.1284i	0.575394	+	0.996611i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.24192	−	10.8113i	−0.279988	−	0.484954i
$498$		0			0
$499$	−4.65602	+	8.06446i	−0.208432	+	0.361015i	−0.951221	−	0.308511i	$-0.900169\pi$
$499$	−4.65602	+	8.06446i	−0.208432	+	0.361015i	0.742789	+	0.669526i	$0.233503\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.3420			0.728656			0.364328	−	0.931271i	$-0.381299\pi$
$503$	16.3420			0.728656			0.364328	+	0.931271i	$0.381299\pi$
$504$		0			0
$505$	−50.3078			−2.23867
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.7266	−	34.1675i	0.874368	−	1.51445i	0.0169335	−	0.999857i	$-0.494610\pi$
$509$	19.7266	−	34.1675i	0.874368	−	1.51445i	0.857435	−	0.514593i	$-0.172057\pi$
$510$		0			0
$511$	8.09839	+	14.0268i	0.358252	+	0.620510i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.55178	−	4.41981i	−0.112445	−	0.194760i
$516$		0			0
$517$	−0.896274	+	1.55239i	−0.0394181	+	0.0682741i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.7139			1.08274			0.541368	−	0.840786i	$-0.317907\pi$
$521$	24.7139			1.08274			0.541368	+	0.840786i	$0.317907\pi$
$522$		0			0
$523$	25.3771			1.10966			0.554832	−	0.831963i	$-0.312783\pi$
$523$	25.3771			1.10966			0.554832	+	0.831963i	$0.312783\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.04947	+	3.54978i	−0.0892763	+	0.154631i
$528$		0			0
$529$	9.14562	+	15.8407i	0.397636	+	0.688725i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.62494	+	2.81448i	0.0703840	+	0.121909i
$534$		0			0
$535$	−10.8623	+	18.8140i	−0.469616	+	0.813400i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	23.1922			0.998959
$540$		0			0
$541$	−22.0818			−0.949370			−0.474685	−	0.880156i	$-0.657438\pi$
$541$	−22.0818			−0.949370			−0.474685	+	0.880156i	$0.657438\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−30.5886	+	52.9809i	−1.31027	+	2.26945i
$546$		0			0
$547$	−3.02667	−	5.24235i	−0.129411	−	0.224147i	0.794037	−	0.607869i	$-0.207975\pi$
$547$	−3.02667	−	5.24235i	−0.129411	−	0.224147i	−0.923449	+	0.383722i	$0.874642\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	29.7578	+	51.5421i	1.26773	+	2.19577i
$552$		0			0
$553$	−4.80456	+	8.32174i	−0.204311	+	0.353876i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.8981			−1.18208			−0.591039	−	0.806643i	$-0.701282\pi$
$557$	−27.8981			−1.18208			−0.591039	+	0.806643i	$0.701282\pi$
$558$		0			0
$559$	−11.8650			−0.501836
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.9423	+	18.9526i	−0.461162	+	0.798755i	−0.999019	−	0.0442804i	$-0.985900\pi$
$563$	−10.9423	+	18.9526i	−0.461162	+	0.798755i	0.537858	+	0.843036i	$0.319234\pi$
$564$		0			0
$565$	9.66915	+	16.7475i	0.406784	+	0.704571i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.0845017	−	0.146361i	−0.00354249	−	0.00613578i	0.864249	−	0.503065i	$-0.167794\pi$
$569$	−0.0845017	−	0.146361i	−0.00354249	−	0.00613578i	−0.867791	+	0.496929i	$0.834461\pi$
$570$		0			0
$571$	−12.7297	+	22.0485i	−0.532722	+	0.922701i	0.466548	+	0.884496i	$0.345497\pi$
$571$	−12.7297	+	22.0485i	−0.532722	+	0.922701i	−0.999270	+	0.0382052i	$0.987836\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.1575			−0.840625
$576$		0			0
$577$	−2.82205			−0.117484			−0.0587418	−	0.998273i	$-0.518709\pi$
$577$	−2.82205			−0.117484			−0.0587418	+	0.998273i	$0.518709\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.98442	−	13.8294i	0.331250	−	0.573741i
$582$		0			0
$583$	−4.79866	−	8.31151i	−0.198740	−	0.344228i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.4040	−	24.9485i	−0.594518	−	1.02973i	−0.993615	−	0.112826i	$-0.964010\pi$
$587$	−14.4040	−	24.9485i	−0.594518	−	1.02973i	0.399097	−	0.916909i	$-0.369324\pi$
$588$		0			0
$589$	−4.77369	+	8.26827i	−0.196696	+	0.340688i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.6059			−1.13364			−0.566819	−	0.823842i	$-0.691826\pi$
$593$	−27.6059			−1.13364			−0.566819	+	0.823842i	$0.691826\pi$
$594$		0			0
$595$	−15.2767			−0.626285
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.9533	+	36.2921i	−0.856127	+	1.48285i	0.0194691	+	0.999810i	$0.493802\pi$
$599$	−20.9533	+	36.2921i	−0.856127	+	1.48285i	−0.875596	+	0.483044i	$0.839531\pi$
$600$		0			0
$601$	10.1877	+	17.6457i	0.415566	+	0.719781i	0.995488	−	0.0948909i	$-0.0302502\pi$
$601$	10.1877	+	17.6457i	0.415566	+	0.719781i	−0.579922	+	0.814672i	$0.696917\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.9802	+	27.6785i	0.649687	+	1.12529i
$606$		0			0
$607$	−21.4089	+	37.0812i	−0.868959	+	1.50508i	−0.00589722	+	0.999983i	$0.501877\pi$
$607$	−21.4089	+	37.0812i	−0.868959	+	1.50508i	−0.863062	+	0.505098i	$0.831456\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.406403			−0.0164413
$612$		0			0
$613$	26.9648			1.08910			0.544549	−	0.838729i	$-0.316701\pi$
$613$	26.9648			1.08910			0.544549	+	0.838729i	$0.316701\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12.2801	+	21.2697i	−0.494378	+	0.856287i	−0.999979	−	0.00647995i	$-0.997937\pi$
$617$	−12.2801	+	21.2697i	−0.494378	+	0.856287i	0.505601	+	0.862767i	$0.331271\pi$
$618$		0			0
$619$	8.80180	+	15.2452i	0.353774	+	0.612755i	0.986907	−	0.161288i	$-0.0515648\pi$
$619$	8.80180	+	15.2452i	0.353774	+	0.612755i	−0.633133	+	0.774043i	$0.718231\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.61461	−	6.26069i	−0.144816	−	0.250829i
$624$		0			0
$625$	−7.42228	+	12.8558i	−0.296891	+	0.514231i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.82542			0.152529
$630$		0			0
$631$	11.6431			0.463503			0.231752	−	0.972775i	$-0.425554\pi$
$631$	11.6431			0.463503			0.231752	+	0.972775i	$0.425554\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−32.9852	+	57.1321i	−1.30898	+	2.26722i
$636$		0			0
$637$	2.62905	+	4.55364i	0.104167	+	0.180422i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−16.3997	−	28.4051i	−0.647750	−	1.12194i	−0.983659	−	0.180041i	$-0.942377\pi$
$641$	−16.3997	−	28.4051i	−0.647750	−	1.12194i	0.335909	−	0.941894i	$-0.390956\pi$
$642$		0			0
$643$	3.44097	−	5.95994i	0.135699	−	0.235037i	−0.790165	−	0.612894i	$-0.790005\pi$
$643$	3.44097	−	5.95994i	0.135699	−	0.235037i	0.925864	+	0.377857i	$0.123339\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.22182			−0.323233			−0.161617	−	0.986854i	$-0.551671\pi$
$647$	−8.22182			−0.323233			−0.161617	+	0.986854i	$0.551671\pi$
$648$		0			0
$649$	−35.7468			−1.40319
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.1786	−	34.9503i	0.789649	−	1.36771i	−0.136533	−	0.990635i	$-0.543596\pi$
$653$	20.1786	−	34.9503i	0.789649	−	1.36771i	0.926182	−	0.377077i	$-0.123071\pi$
$654$		0			0
$655$	5.81054	+	10.0641i	0.227037	+	0.393239i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.0788685	−	0.136604i	−0.00307228	−	0.00532135i	0.864485	−	0.502658i	$-0.167645\pi$
$659$	−0.0788685	−	0.136604i	−0.00307228	−	0.00532135i	−0.867557	+	0.497337i	$0.834311\pi$
$660$		0			0
$661$	12.8347	−	22.2304i	0.499214	−	0.864663i	−0.500786	−	0.865571i	$-0.666956\pi$
$661$	12.8347	−	22.2304i	0.499214	−	0.864663i	1.00000		0.000907872i	$0.000288985\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−35.5831			−1.37985
$666$		0			0
$667$	18.1075			0.701127
$668$		0			0
$669$		0			0
$670$		0			0
$671$	19.4016	−	33.6045i	0.748990	−	1.29729i
$672$		0			0
$673$	−8.87413	−	15.3704i	−0.342072	−	0.592487i	0.642745	−	0.766080i	$-0.277796\pi$
$673$	−8.87413	−	15.3704i	−0.342072	−	0.592487i	−0.984817	+	0.173593i	$0.944462\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.6184	−	32.2480i	−0.715564	−	1.23939i	−0.962742	−	0.270422i	$-0.912837\pi$
$677$	−18.6184	−	32.2480i	−0.715564	−	1.23939i	0.247178	−	0.968970i	$-0.420497\pi$
$678$		0			0
$679$	−8.84514	+	15.3202i	−0.339445	+	0.587937i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13.1276			−0.502315			−0.251158	−	0.967946i	$-0.580811\pi$
$683$	−13.1276			−0.502315			−0.251158	+	0.967946i	$0.580811\pi$
$684$		0			0
$685$	48.4202			1.85004
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.08794	−	1.88437i	0.0414473	−	0.0717888i
$690$		0			0
$691$	−2.44120	−	4.22828i	−0.0928676	−	0.160851i	0.815849	−	0.578265i	$-0.196270\pi$
$691$	−2.44120	−	4.22828i	−0.0928676	−	0.160851i	−0.908717	+	0.417414i	$0.862937\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−36.3991	−	63.0452i	−1.38070	−	2.39144i
$696$		0			0
$697$	4.97565	−	8.61808i	0.188466	−	0.326433i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.88397			0.0711565			0.0355782	−	0.999367i	$-0.488673\pi$
$701$	1.88397			0.0711565			0.0355782	+	0.999367i	$0.488673\pi$
$702$		0			0
$703$	8.91029			0.336058
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.78239	+	15.2115i	−0.330296	+	0.572089i
$708$		0			0
$709$	−10.2918	−	17.8259i	−0.386516	−	0.669466i	0.605462	−	0.795874i	$-0.292988\pi$
$709$	−10.2918	−	17.8259i	−0.386516	−	0.669466i	−0.991978	+	0.126408i	$0.959655\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.45239	+	2.51561i	0.0543923	+	0.0942102i
$714$		0			0
$715$	−8.33661	+	14.4394i	−0.311772	+	0.540004i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.09546			−0.301910			−0.150955	−	0.988541i	$-0.548235\pi$
$719$	−8.09546			−0.301910			−0.150955	+	0.988541i	$0.548235\pi$
$720$		0			0
$721$	−1.78189			−0.0663610
$722$		0			0
$723$		0			0
$724$		0			0
$725$	38.7578	−	67.1305i	1.43943	−	2.49316i
$726$		0			0
$727$	−16.8348	−	29.1587i	−0.624367	−	1.08144i	−0.988663	−	0.150152i	$-0.952024\pi$
$727$	−16.8348	−	29.1587i	−0.624367	−	1.08144i	0.364296	−	0.931283i	$-0.381310\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.1656	+	31.4638i	0.671880	+	1.16373i
$732$		0			0
$733$	19.7069	−	34.1334i	0.727891	−	1.26074i	−0.229882	−	0.973219i	$-0.573834\pi$
$733$	19.7069	−	34.1334i	0.727891	−	1.26074i	0.957773	−	0.287526i	$-0.0928327\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	41.3741			1.52403
$738$		0			0
$739$	13.3865			0.492429			0.246215	−	0.969215i	$-0.420813\pi$
$739$	13.3865			0.492429			0.246215	+	0.969215i	$0.420813\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.03777	+	15.6539i	−0.331564	+	0.574285i	−0.982819	−	0.184574i	$-0.940909\pi$
$743$	−9.03777	+	15.6539i	−0.331564	+	0.574285i	0.651255	+	0.758859i	$0.274243\pi$
$744$		0			0
$745$	−20.1628	−	34.9230i	−0.738708	−	1.27948i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.79252	+	6.56883i	0.138576	+	0.240020i
$750$		0			0
$751$	−10.4782	+	18.1488i	−0.382356	+	0.662260i	−0.991399	−	0.130877i	$-0.958221\pi$
$751$	−10.4782	+	18.1488i	−0.382356	+	0.662260i	0.609042	+	0.793138i	$0.291554\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−37.7913			−1.37537
$756$		0			0
$757$	40.0649			1.45618			0.728092	−	0.685479i	$-0.240407\pi$
$757$	40.0649			1.45618			0.728092	+	0.685479i	$0.240407\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.455916	−	0.789670i	0.0165269	−	0.0286255i	−0.857644	−	0.514244i	$-0.828072\pi$
$761$	0.455916	−	0.789670i	0.0165269	−	0.0286255i	0.874171	+	0.485619i	$0.161406\pi$
$762$		0			0
$763$	10.6799	+	18.4981i	0.386638	+	0.669676i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.05223	−	7.01866i	−0.146317	−	0.253429i
$768$		0			0
$769$	−12.3273	+	21.3514i	−0.444532	+	0.769952i	−0.998019	−	0.0629054i	$-0.979963\pi$
$769$	−12.3273	+	21.3514i	−0.444532	+	0.769952i	0.553487	+	0.832858i	$0.313297\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	41.8304			1.50454			0.752268	−	0.658858i	$-0.228960\pi$
$773$	41.8304			1.50454			0.752268	+	0.658858i	$0.228960\pi$
$774$		0			0
$775$	12.4349			0.446674
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.5894	−	20.0735i	0.415235	−	0.719208i
$780$		0			0
$781$	20.8602	+	36.1310i	0.746438	+	1.29287i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25.5390	+	44.2348i	0.911526	+	1.57881i
$786$		0			0
$787$	8.80612	−	15.2526i	0.313904	−	0.543698i	−0.665300	−	0.746576i	$-0.731696\pi$
$787$	8.80612	−	15.2526i	0.313904	−	0.543698i	0.979204	+	0.202878i	$0.0650296\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.75189			0.240070
$792$		0			0
$793$	8.79738			0.312404
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.39125	+	2.40971i	−0.0492805	+	0.0853563i	−0.889613	−	0.456714i	$-0.849026\pi$
$797$	−1.39125	+	2.40971i	−0.0492805	+	0.0853563i	0.840333	+	0.542071i	$0.182359\pi$
$798$		0			0
$799$	0.622213	+	1.07771i	0.0220123	+	0.0381265i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.0645	−	46.8771i	−0.955085	−	1.65426i
$804$		0			0
$805$	−5.41304	+	9.37567i	−0.190785	+	0.330449i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.0046			−0.703324			−0.351662	−	0.936127i	$-0.614383\pi$
$809$	−20.0046			−0.703324			−0.351662	+	0.936127i	$0.614383\pi$
$810$		0			0
$811$	21.7172			0.762593			0.381297	−	0.924453i	$-0.375478\pi$
$811$	21.7172			0.762593			0.381297	+	0.924453i	$0.375478\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.88466	+	17.1207i	−0.346244	+	0.599713i
$816$		0			0
$817$	42.3119	+	73.2864i	1.48031	+	2.56397i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.6483	+	44.4242i	0.895133	+	1.55042i	0.833640	+	0.552309i	$0.186253\pi$
$821$	25.6483	+	44.4242i	0.895133	+	1.55042i	0.0614936	+	0.998107i	$0.480414\pi$
$822$		0			0
$823$	−7.54360	+	13.0659i	−0.262953	+	0.455449i	−0.967025	−	0.254680i	$-0.918030\pi$
$823$	−7.54360	+	13.0659i	−0.262953	+	0.455449i	0.704072	+	0.710129i	$0.251363\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.8406			−0.863792			−0.431896	−	0.901923i	$-0.642155\pi$
$827$	−24.8406			−0.863792			−0.431896	+	0.901923i	$0.642155\pi$
$828$		0			0
$829$	−23.4924			−0.815924			−0.407962	−	0.912999i	$-0.633760\pi$
$829$	−23.4924			−0.815924			−0.407962	+	0.912999i	$0.633760\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.05027	−	13.9435i	0.278925	−	0.483113i
$834$		0			0
$835$	18.3598	+	31.8002i	0.635368	+	1.10049i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.7569	−	39.4161i	−0.785656	−	1.36080i	−0.928606	−	0.371066i	$-0.878992\pi$
$839$	−22.7569	−	39.4161i	−0.785656	−	1.36080i	0.142950	−	0.989730i	$-0.454341\pi$
$840$		0			0
$841$	−20.3163	+	35.1889i	−0.700562	+	1.21341i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.78012			−0.130040
$846$		0			0
$847$	11.1588			0.383422
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.35547	−	2.34774i	0.0464649	−	0.0804796i
$852$		0			0
$853$	8.22123	+	14.2396i	0.281489	+	0.487554i	0.971752	−	0.236005i	$-0.0758382\pi$
$853$	8.22123	+	14.2396i	0.281489	+	0.487554i	−0.690262	+	0.723559i	$0.742505\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.08343	+	12.2689i	0.241965	+	0.419096i	0.961274	−	0.275594i	$-0.0888747\pi$
$857$	7.08343	+	12.2689i	0.241965	+	0.419096i	−0.719309	+	0.694690i	$0.755541\pi$
$858$		0			0
$859$	−26.9365	+	46.6554i	−0.919062	+	1.59186i	−0.118218	+	0.992988i	$0.537718\pi$
$859$	−26.9365	+	46.6554i	−0.919062	+	1.59186i	−0.800844	+	0.598874i	$0.795615\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	23.9687			0.815905			0.407952	−	0.913003i	$-0.366243\pi$
$863$	23.9687			0.815905			0.407952	+	0.913003i	$0.366243\pi$
$864$		0			0
$865$	−92.6887			−3.15151
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.0566	−	27.8109i	0.544684	−	0.943421i
$870$		0			0
$871$	4.69012	+	8.12353i	0.158919	+	0.275255i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.6998	+	18.5325i	0.361718	+	0.626514i
$876$		0			0
$877$	7.89845	−	13.6805i	0.266712	−	0.461958i	−0.701299	−	0.712867i	$-0.747396\pi$
$877$	7.89845	−	13.6805i	0.266712	−	0.461958i	0.968011	+	0.250909i	$0.0807295\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47.9966			1.61705			0.808523	−	0.588464i	$-0.200267\pi$
$881$	47.9966			1.61705			0.808523	+	0.588464i	$0.200267\pi$
$882$		0			0
$883$	−30.6008			−1.02980			−0.514899	−	0.857251i	$-0.672171\pi$
$883$	−30.6008			−1.02980			−0.514899	+	0.857251i	$0.672171\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	17.1661	−	29.7326i	0.576382	−	0.998323i	−0.419508	−	0.907752i	$-0.637797\pi$
$887$	17.1661	−	29.7326i	0.576382	−	0.998323i	0.995890	−	0.0905710i	$-0.0288692\pi$
$888$		0			0
$889$	11.5167	+	19.9475i	0.386257	+	0.669016i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.44928	+	2.51022i	0.0484983	+	0.0840015i
$894$		0			0
$895$	36.7628	−	63.6750i	1.22884	−	2.12842i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.1703			−0.372551
$900$		0			0
$901$	−6.66266			−0.221966
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29.1968	−	50.5704i	0.970535	−	1.68102i
$906$		0			0
$907$	−8.32595	−	14.4210i	−0.276459	−	0.478840i	0.694044	−	0.719933i	$-0.255828\pi$
$907$	−8.32595	−	14.4210i	−0.276459	−	0.478840i	−0.970502	+	0.241093i	$0.922494\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.00817	−	6.94235i	−0.132797	−	0.230010i	0.791957	−	0.610577i	$-0.209062\pi$
$911$	−4.00817	−	6.94235i	−0.132797	−	0.230010i	−0.924754	+	0.380566i	$0.875729\pi$
$912$		0			0
$913$	−26.6836	+	46.2174i	−0.883099	+	1.52957i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.05746			0.133989
$918$		0			0
$919$	3.98767			0.131541			0.0657705	−	0.997835i	$-0.479049\pi$
$919$	3.98767			0.131541			0.0657705	+	0.997835i	$0.479049\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.72939	+	8.19155i	−0.155670	+	0.269628i
$924$		0			0
$925$	−5.80256	−	10.0503i	−0.190787	−	0.330453i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.22140	−	5.57962i	−0.105691	−	0.183062i	0.808330	−	0.588730i	$-0.200372\pi$
$929$	−3.22140	−	5.57962i	−0.105691	−	0.183062i	−0.914020	+	0.405669i	$0.867039\pi$
$930$		0			0
$931$	18.7509	−	32.4776i	0.614537	−	1.06441i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	51.0542			1.66965
$936$		0			0
$937$	12.2388			0.399824			0.199912	−	0.979814i	$-0.435934\pi$
$937$	12.2388			0.399824			0.199912	+	0.979814i	$0.435934\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.99773	−	17.3166i	0.325917	−	0.564504i	−0.655781	−	0.754951i	$-0.727660\pi$
$941$	9.99773	−	17.3166i	0.325917	−	0.564504i	0.981697	+	0.190447i	$0.0609938\pi$
$942$		0			0
$943$	−3.52607	−	6.10733i	−0.114825	−	0.198882i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.6942	+	30.6472i	0.574983	+	0.995899i	0.996044	+	0.0888669i	$0.0283246\pi$
$947$	17.6942	+	30.6472i	0.574983	+	0.995899i	−0.421061	+	0.907032i	$0.638342\pi$
$948$		0			0
$949$	6.13601	−	10.6279i	0.199183	−	0.344996i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.3690			−0.368278			−0.184139	−	0.982900i	$-0.558950\pi$
$953$	−11.3690			−0.368278			−0.184139	+	0.982900i	$0.558950\pi$
$954$		0			0
$955$	−32.6420			−1.05627
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.45286	−	14.6408i	0.272957	−	0.472776i
$960$		0			0
$961$	14.6040	+	25.2949i	0.471098	+	0.815966i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	34.6736	+	60.0564i	1.11618	+	1.93329i
$966$		0			0
$967$	−29.3632	+	50.8586i	−0.944258	+	1.63550i	−0.187029	+	0.982354i	$0.559886\pi$
$967$	−29.3632	+	50.8586i	−0.944258	+	1.63550i	−0.757229	+	0.653149i	$0.773447\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.1168			−0.485121			−0.242560	−	0.970136i	$-0.577987\pi$
$971$	−15.1168			−0.485121			−0.242560	+	0.970136i	$0.577987\pi$
$972$		0			0
$973$	−25.4172			−0.814839
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.1415	−	47.0105i	0.868335	−	1.50400i	0.00463697	−	0.999989i	$-0.498524\pi$
$977$	27.1415	−	47.0105i	0.868335	−	1.50400i	0.863698	−	0.504010i	$-0.168143\pi$
$978$		0			0
$979$	12.0799	+	20.9230i	0.386075	+	0.668701i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.7221	+	27.2316i	0.501459	+	0.868552i	0.999999	+	0.00168512i	$0.000536391\pi$
$983$	15.7221	+	27.2316i	0.501459	+	0.868552i	−0.498540	+	0.866867i	$0.666130\pi$
$984$		0			0
$985$	6.17478	−	10.6950i	0.196745	−	0.340772i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25.7467			0.818697
$990$		0			0
$991$	−42.6265			−1.35408			−0.677038	−	0.735948i	$-0.736737\pi$
$991$	−42.6265			−1.35408			−0.677038	+	0.735948i	$0.736737\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	34.9470	−	60.5300i	1.10789	−	1.91893i
$996$		0			0
$997$	5.60659	+	9.71091i	0.177563	+	0.307548i	0.941045	−	0.338281i	$-0.109845\pi$
$997$	5.60659	+	9.71091i	0.177563	+	0.307548i	−0.763483	+	0.645829i	$0.776512\pi$
$998$		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.1873.11		22
3.2	odd	2		936.2.q.g.625.4	yes	22
9.2	odd	6		936.2.q.g.313.4	✓	22
9.4	even	3		8424.2.a.be.1.1		11
9.5	odd	6		8424.2.a.bf.1.11		11
9.7	even	3	inner	2808.2.q.g.937.11		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.q.g.313.4	✓	22	9.2	odd	6
936.2.q.g.625.4	yes	22	3.2	odd	2
2808.2.q.g.937.11		22	9.7	even	3	inner
2808.2.q.g.1873.11		22	1.1	even	1	trivial
8424.2.a.be.1.1		11	9.4	even	3
8424.2.a.bf.1.11		11	9.5	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)

\(f(q)\)	\(=\)	\(q+(1.89006 - 3.27368i) q^{5} +(-0.659907 - 1.14299i) q^{7} +(2.20538 + 3.81984i) q^{11} +(-0.500000 + 0.866025i) q^{13} +3.06205 q^{17} +7.13223 q^{19} +(1.08498 - 1.87925i) q^{23} +(-4.64465 - 8.04477i) q^{25} +(4.17231 + 7.22665i) q^{29} +(-0.669313 + 1.15928i) q^{31} -4.98905 q^{35} +1.24930 q^{37} +(1.62494 - 2.81448i) q^{41} +(5.93250 + 10.2754i) q^{43} +(0.203201 + 0.351955i) q^{47} +(2.62905 - 4.55364i) q^{49} -2.17588 q^{53} +16.6732 q^{55} +(-4.05223 + 7.01866i) q^{59} +(-4.39869 - 7.61875i) q^{61} +(1.89006 + 3.27368i) q^{65} +(4.69012 - 8.12353i) q^{67} +9.45879 q^{71} -12.2720 q^{73} +(2.91070 - 5.04147i) q^{77} +(-3.64033 - 6.30524i) q^{79} +(6.04966 + 10.4783i) q^{83} +(5.78746 - 10.0242i) q^{85} +5.47746 q^{89} +1.31981 q^{91} +(13.4803 - 23.3486i) q^{95} +(-6.70181 - 11.6079i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{5} - 4 q^{7} - 5 q^{11} - 11 q^{13} - 8 q^{17} + 10 q^{19} - 9 q^{23} - 24 q^{25} + 16 q^{29} - q^{31} + 18 q^{37} + 6 q^{41} - 7 q^{43} - 21 q^{47} - 27 q^{49} - 32 q^{53} + 34 q^{55} - 11 q^{59}+ \cdots - 10 q^{97}+O(q^{100}) \) 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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