LMFDB - Embedded newform 804.2.s.a.137.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(5,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 5]))

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

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

chi := DirichletCharacter("804.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.s (of order $22$, degree $10$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			137.2
Root			$-0.888835 + 0.458227i$ of defining polynomial
Character	$\chi$	$=$	804.137
Dual form			804.2.s.a.581.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})$	$q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +(4.59805 + 3.98424i) q^{13} +(-4.38235 + 1.28677i) q^{19} +(0.540809 - 1.18421i) q^{21} +(3.27430 - 3.77875i) q^{25} +(4.72659 + 2.15856i) q^{27} +(6.01248 - 5.20984i) q^{31} -1.31262 q^{37} +(6.90090 + 7.96406i) q^{39} +(0.410625 + 0.638945i) q^{43} +(5.41352 + 3.47906i) q^{49} +(-7.83038 + 1.12584i) q^{57} +(-13.5021 - 6.16621i) q^{61} +(1.21908 - 1.89692i) q^{63} +(-6.54788 + 4.91174i) q^{67} +(-3.51029 + 7.68646i) q^{73} +(6.54498 - 5.67126i) q^{75} +(-6.28748 - 5.44813i) q^{79} +(7.57128 + 4.86577i) q^{81} +(3.84702 - 2.47233i) q^{91} +(11.5921 - 7.44981i) q^{93} +11.8855i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^9	$ 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) $ 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{13}{22}\right)$	$1$

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
$3$	1.71442	+	0.246497i	0.989821	+	0.142315i
$3$	1.71442	+	0.246497i	0.989821	+	0.142315i
$4$		0			0
$5$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$5$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$6$		0			0
$7$	0.211757	−	0.721178i	0.0800367	−	0.272580i	−0.909746	−	0.415166i	$-0.863724\pi$
$7$	0.211757	−	0.721178i	0.0800367	−	0.272580i	0.989782	+	0.142586i	$0.0455417\pi$
$8$		0			0
$9$	2.87848	+	0.845198i	0.959493	+	0.281733i
$10$		0			0
$11$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$11$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$12$		0			0
$13$	4.59805	+	3.98424i	1.27527	+	1.10503i	0.989162	+	0.146831i	$0.0469074\pi$
$13$	4.59805	+	3.98424i	1.27527	+	1.10503i	0.286109	+	0.958197i	$0.407638\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$17$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$18$		0			0
$19$	−4.38235	+	1.28677i	−1.00538	+	0.295206i	−0.742662	−	0.669666i	$-0.766437\pi$
$19$	−4.38235	+	1.28677i	−1.00538	+	0.295206i	−0.262718	+	0.964873i	$0.584619\pi$
$20$		0			0
$21$	0.540809	−	1.18421i	0.118014	−	0.258415i
$22$		0			0
$23$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$23$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$24$		0			0
$25$	3.27430	−	3.77875i	0.654861	−	0.755750i
$26$		0			0
$27$	4.72659	+	2.15856i	0.909632	+	0.415415i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	6.01248	−	5.20984i	1.07987	−	0.935715i	0.0817313	−	0.996654i	$-0.473955\pi$
$31$	6.01248	−	5.20984i	1.07987	−	0.935715i	0.998141	+	0.0609393i	$0.0194096\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.31262			−0.215794			−0.107897	−	0.994162i	$-0.534412\pi$
$37$	−1.31262			−0.215794			−0.107897	+	0.994162i	$0.534412\pi$
$38$		0			0
$39$	6.90090	+	7.96406i	1.10503	+	1.27527i
$40$		0			0
$41$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$41$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$42$		0			0
$43$	0.410625	+	0.638945i	0.0626198	+	0.0974383i	0.871152	−	0.491014i	$-0.163374\pi$
$43$	0.410625	+	0.638945i	0.0626198	+	0.0974383i	−0.808532	+	0.588453i	$0.799737\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$47$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$48$		0			0
$49$	5.41352	+	3.47906i	0.773360	+	0.497008i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$53$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.83038	+	1.12584i	−1.03716	+	0.149121i
$58$		0			0
$59$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$59$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$60$		0			0
$61$	−13.5021	−	6.16621i	−1.72877	−	0.789502i	−0.993772	−	0.111430i	$-0.964457\pi$
$61$	−13.5021	−	6.16621i	−1.72877	−	0.789502i	−0.734996	−	0.678072i	$-0.762816\pi$
$62$		0			0
$63$	1.21908	−	1.89692i	0.153589	−	0.238989i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.54788	+	4.91174i	−0.799951	+	0.600065i
$67$	−6.54788	+	4.91174i	−0.799951	+	0.600065i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$71$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$72$		0			0
$73$	−3.51029	+	7.68646i	−0.410848	+	0.899633i	0.585206	+	0.810885i	$0.301014\pi$
$73$	−3.51029	+	7.68646i	−0.410848	+	0.899633i	−0.996054	+	0.0887477i	$0.971714\pi$
$74$		0			0
$75$	6.54498	−	5.67126i	0.755750	−	0.654861i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.28748	−	5.44813i	−0.707396	−	0.612962i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	−6.28748	−	5.44813i	−0.707396	−	0.612962i	−0.932414	+	0.361392i	$0.882301\pi$
$80$		0			0
$81$	7.57128	+	4.86577i	0.841254	+	0.540641i
$82$		0			0
$83$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$83$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$89$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$90$		0			0
$91$	3.84702	−	2.47233i	0.403277	−	0.259170i
$92$		0			0
$93$	11.5921	−	7.44981i	1.20205	−	0.772509i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			11.8855i			1.20679i	0.797442	+	0.603396i	$0.206186\pi$
$97$			11.8855i			1.20679i	−0.797442	+	0.603396i	$0.793814\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$101$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$102$		0			0
$103$	−11.9276	−	13.7652i	−1.17526	−	1.35632i	−0.921180	−	0.389138i	$-0.872773\pi$
$103$	−11.9276	−	13.7652i	−1.17526	−	1.35632i	−0.254079	−	0.967183i	$-0.581772\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$107$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$108$		0			0
$109$	−5.15041	−	4.46286i	−0.493320	−	0.427464i	0.372340	−	0.928096i	$-0.378556\pi$
$109$	−5.15041	−	4.46286i	−0.493320	−	0.427464i	−0.865660	+	0.500632i	$0.833101\pi$
$110$		0			0
$111$	−2.25038	−	0.323556i	−0.213597	−	0.0307106i
$112$		0			0
$113$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$113$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.86793	+	15.3548i	0.912291	+	1.41955i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.20347	−	8.31325i	0.654861	−	0.755750i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−20.1025	−	5.90263i	−1.78381	−	0.523774i	−0.788038	−	0.615627i	$-0.788903\pi$
$127$	−20.1025	−	5.90263i	−1.78381	−	0.523774i	−0.995773	+	0.0918526i	$0.970721\pi$
$128$		0			0
$129$	0.546487	+	1.19664i	0.0481155	+	0.105358i
$130$		0			0
$131$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$131$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$132$		0			0
$133$			3.43294i			0.297674i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$137$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$138$		0			0
$139$	−16.6482	+	7.60297i	−1.41208	+	0.644876i	−0.967963	−	0.251093i	$-0.919210\pi$
$139$	−16.6482	+	7.60297i	−1.41208	+	0.644876i	−0.444118	+	0.895969i	$0.646483\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	8.42347	+	7.29898i	0.694756	+	0.602010i
$148$		0			0
$149$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$149$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$150$		0			0
$151$	17.1580	−	11.0268i	1.39630	−	0.897346i	0.396511	−	0.918030i	$-0.370221\pi$
$151$	17.1580	−	11.0268i	1.39630	−	0.897346i	0.999786	+	0.0206838i	$0.00658434\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.74505	−	19.0923i	0.219079	−	1.52373i	−0.522369	−	0.852719i	$-0.674952\pi$
$157$	2.74505	−	19.0923i	0.219079	−	1.52373i	0.741448	−	0.671010i	$-0.234139\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.37981			−0.499706			−0.249853	−	0.968284i	$-0.580382\pi$
$163$	−6.37981			−0.499706			−0.249853	+	0.968284i	$0.580382\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$167$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$168$		0			0
$169$	3.41787	+	23.7718i	0.262913	+	1.82860i
$170$		0			0
$171$	−13.7021			−1.04782
$172$		0			0
$173$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$173$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$174$		0			0
$175$	−2.03179	−	3.16153i	−0.153589	−	0.238989i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$179$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$180$		0			0
$181$	−0.449359	+	3.12536i	−0.0334006	+	0.232306i	−0.999683	−	0.0251785i	$-0.991985\pi$
$181$	−0.449359	+	3.12536i	−0.0334006	+	0.232306i	0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$	−21.6284	−	13.8997i	−1.59881	−	1.02750i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.55759	−	2.95162i	0.186038	−	0.214699i
$190$		0			0
$191$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$191$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$192$		0			0
$193$	6.36956	+	7.35086i	0.458491	+	0.529127i	0.937175	−	0.348861i	$-0.113431\pi$
$193$	6.36956	+	7.35086i	0.458491	+	0.529127i	−0.478684	+	0.877987i	$0.658886\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$197$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$198$		0			0
$199$	−24.3979	−	7.16388i	−1.72952	−	0.507834i	−0.742698	−	0.669626i	$-0.766454\pi$
$199$	−24.3979	−	7.16388i	−1.72952	−	0.507834i	−0.986825	+	0.161793i	$0.948272\pi$
$200$		0			0
$201$	−12.4366	+	6.80677i	−0.877207	+	0.480112i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−4.08501	−	28.4119i	−0.281224	−	1.95595i	−0.293279	−	0.956027i	$-0.594746\pi$
$211$	−4.08501	−	28.4119i	−0.281224	−	1.95595i	0.0120548	−	0.999927i	$-0.496163\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.48404	−	5.43929i	−0.168628	−	0.369243i
$218$		0			0
$219$	−7.91280	+	12.3126i	−0.534698	+	0.832006i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.03102	−	14.1261i	−0.136007	−	0.945952i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−2.03102	−	14.1261i	−0.136007	−	0.945952i	0.801502	−	0.597992i	$-0.204035\pi$
$224$		0			0
$225$	12.6188	−	8.10961i	0.841254	−	0.540641i
$226$		0			0
$227$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$227$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$228$		0			0
$229$	−22.8520	+	19.8014i	−1.51010	+	1.30851i	−0.726900	+	0.686743i	$0.759040\pi$
$229$	−22.8520	+	19.8014i	−1.51010	+	1.30851i	−0.783202	+	0.621768i	$0.786415\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$233$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−9.43644	−	10.8902i	−0.612962	−	0.707396i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−12.5090	+	27.3908i	−0.805774	+	1.76440i	−0.181179	+	0.983450i	$0.557991\pi$
$241$	−12.5090	+	27.3908i	−0.805774	+	1.76440i	−0.624595	+	0.780949i	$0.714736\pi$
$242$		0			0
$243$	11.7810	+	10.2083i	0.755750	+	0.654861i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−25.2771	−	11.5437i	−1.60834	−	0.734506i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$251$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$257$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$258$		0			0
$259$	−0.277957	+	0.946634i	−0.0172714	+	0.0588210i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$263$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	8.64153	+	1.24246i	0.524936	+	0.0754744i	0.399688	−	0.916651i	$-0.369118\pi$
$271$	8.64153	+	1.24246i	0.524936	+	0.0754744i	0.125247	+	0.992126i	$0.460028\pi$
$272$		0			0
$273$	7.20482	−	3.29033i	0.436056	−	0.199140i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	10.2048	+	2.99639i	0.613145	+	0.180036i	0.573537	−	0.819180i	$-0.305571\pi$
$277$	10.2048	+	2.99639i	0.613145	+	0.180036i	0.0396081	+	0.999215i	$0.487389\pi$
$278$		0			0
$279$	21.7101	−	9.91469i	1.29975	−	0.593577i
$280$		0			0
$281$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$281$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$282$		0			0
$283$	−3.68456	+	1.08188i	−0.219024	+	0.0643113i	−0.389404	−	0.921067i	$-0.627319\pi$
$283$	−3.68456	+	1.08188i	−0.219024	+	0.0643113i	0.170379	+	0.985379i	$0.445501\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.06206	−	15.4637i	0.415415	−	0.909632i
$290$		0			0
$291$	−2.92974	+	20.3768i	−0.171744	+	1.19451i
$292$		0			0
$293$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$293$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.547746	−	0.160833i	0.0315716	−	0.00927025i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.22355	−	2.56611i	−0.126905	−	0.146456i	0.688741	−	0.725007i	$-0.258164\pi$
$307$	−2.22355	−	2.56611i	−0.126905	−	0.146456i	−0.815646	+	0.578551i	$0.803618\pi$
$308$		0			0
$309$	−17.0558	−	26.5394i	−0.970272	−	1.50977i
$310$		0			0
$311$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$311$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$312$		0			0
$313$	32.4546	−	4.66627i	1.83444	−	0.263753i	0.863724	−	0.503966i	$-0.168126\pi$
$313$	32.4546	−	4.66627i	1.83444	−	0.263753i	0.970720	+	0.240212i	$0.0772171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$317$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	30.1108	−	4.32929i	1.67025	−	0.240146i
$326$		0			0
$327$	−7.72990	−	8.92078i	−0.427464	−	0.493320i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.7918	+	24.5724i	−0.867993	+	1.35062i	0.0676621	+	0.997708i	$0.478446\pi$
$331$	−15.7918	+	24.5724i	−0.867993	+	1.35062i	−0.935655	+	0.352915i	$0.885190\pi$
$332$		0			0
$333$	−3.77835	−	1.10942i	−0.207052	−	0.0607961i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−9.79570	+	33.3611i	−0.533606	+	1.81729i	0.0413966	+	0.999143i	$0.486819\pi$
$337$	−9.79570	+	33.3611i	−0.533606	+	1.81729i	−0.575002	+	0.818152i	$0.694999\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	7.63165	−	6.61286i	0.412070	−	0.357061i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$347$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$348$		0			0
$349$	19.5107	+	12.5388i	1.04438	+	0.671184i	0.946067	−	0.323971i	$-0.105018\pi$
$349$	19.5107	+	12.5388i	1.04438	+	0.671184i	0.0983163	+	0.995155i	$0.468654\pi$
$350$		0			0
$351$	13.1329	+	28.7570i	0.700982	+	1.53494i
$352$		0			0
$353$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$353$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$359$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$360$		0			0
$361$	1.56539	−	1.00602i	0.0823890	−	0.0529482i
$362$		0			0
$363$	14.3990	−	12.4768i	0.755750	−	0.654861i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	12.0668	−	1.73495i	0.629883	−	0.0905635i	0.180023	−	0.983662i	$-0.442383\pi$
$367$	12.0668	−	1.73495i	0.629883	−	0.0905635i	0.449860	+	0.893099i	$0.351474\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			4.05827i			0.210130i	0.994465	+	0.105065i	$0.0335050\pi$
$373$			4.05827i			0.210130i	−0.994465	+	0.105065i	$0.966495\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	33.9585	+	4.88250i	1.74433	+	0.250797i	0.939462	−	0.342653i	$-0.111326\pi$
$379$	33.9585	+	4.88250i	1.74433	+	0.250797i	0.804871	+	0.593450i	$0.202235\pi$
$380$		0			0
$381$	−33.0092	−	15.0748i	−1.69111	−	0.772305i
$382$		0			0
$383$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$383$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.641941	+	2.18625i	0.0326317	+	0.111133i
$388$		0			0
$389$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$389$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.5500	+	36.2394i	0.830620	+	1.81880i	0.435778	+	0.900054i	$0.356473\pi$
$397$	16.5500	+	36.2394i	0.830620	+	1.81880i	0.394842	+	0.918749i	$0.370799\pi$
$398$		0			0
$399$	−0.846208	+	5.88550i	−0.0423634	+	0.294644i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	48.4029			2.41112
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8.11667	−	27.6429i	0.401344	−	1.36685i	−0.472794	−	0.881173i	$-0.656755\pi$
$409$	8.11667	−	27.6429i	0.401344	−	1.36685i	0.874138	−	0.485678i	$-0.161427\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−30.4161	+	8.93097i	−1.48948	+	0.437352i
$418$		0			0
$419$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$419$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$420$		0			0
$421$	38.9428	−	11.4346i	1.89796	−	0.557290i	0.907384	−	0.420303i	$-0.138076\pi$
$421$	38.9428	−	11.4346i	1.89796	−	0.557290i	0.990573	−	0.136988i	$-0.0437421\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.30610	+	8.43169i	−0.353567	+	0.408038i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	19.4307	−	16.8368i	0.933782	−	0.809126i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	19.4307	−	16.8368i	0.933782	−	0.809126i	0.981839	+	0.189718i	$0.0607574\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	40.4106			1.92869			0.964347	−	0.264642i	$-0.0852539\pi$
$439$	40.4106			1.92869			0.964347	+	0.264642i	$0.0852539\pi$
$440$		0			0
$441$	12.6422	+	14.5899i	0.602010	+	0.694756i
$442$		0			0
$443$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$443$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$449$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	32.1341	−	14.6751i	1.50979	−	0.689498i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.38599	−	1.59952i	0.0648341	−	0.0748225i	−0.722406	−	0.691469i	$-0.756964\pi$
$457$	1.38599	−	1.59952i	0.0648341	−	0.0748225i	0.787240	+	0.616647i	$0.211509\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$461$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$462$		0			0
$463$	28.1321	+	12.8475i	1.30741	+	0.597075i	0.942574	−	0.333997i	$-0.108397\pi$
$463$	28.1321	+	12.8475i	1.30741	+	0.597075i	0.364837	+	0.931071i	$0.381125\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$467$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$468$		0			0
$469$	2.15568	+	5.76229i	0.0995402	+	0.266078i
$470$		0			0
$471$	9.41236	−	32.0555i	0.433699	−	1.47704i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.48675	+	20.7731i	−0.435282	+	0.953135i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$479$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$480$		0			0
$481$	−6.03550	−	5.22979i	−0.275195	−	0.238458i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.2853	+	17.5603i	−0.511388	+	0.795735i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−11.2853	+	17.5603i	−0.511388	+	0.795735i	0.485528	+	0.874221i	$0.338628\pi$
$488$		0			0
$489$	−10.9377	−	1.57260i	−0.494619	−	0.0711155i
$490$		0			0
$491$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$491$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	44.6005i		−	1.99659i	−0.0583392	−	0.998297i	$-0.518580\pi$
$499$		−	44.6005i		−	1.99659i	0.0583392	−	0.998297i	$-0.481420\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$503$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			41.5973i			1.84740i
$508$		0			0
$509$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$509$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$510$		0			0
$511$	4.79998	+	4.15921i	0.212339	+	0.183993i
$512$		0			0
$513$	−23.4911	−	3.37752i	−1.03716	−	0.149121i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$521$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$522$		0			0
$523$	−13.3351	+	15.3896i	−0.583104	+	0.672938i	−0.968269	−	0.249910i	$-0.919599\pi$
$523$	−13.3351	+	15.3896i	−0.583104	+	0.672938i	0.385165	+	0.922848i	$0.374145\pi$
$524$		0			0
$525$	−2.70404	−	5.92103i	−0.118014	−	0.258415i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.0683	−	6.47985i	−0.959493	−	0.281733i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−27.9237	+	12.7523i	−1.20054	+	0.548266i	−0.912391	−	0.409320i	$-0.865766\pi$
$541$	−27.9237	+	12.7523i	−1.20054	+	0.548266i	−0.288145	+	0.957587i	$0.593039\pi$
$542$		0			0
$543$	−1.54078	+	5.24742i	−0.0661212	+	0.225188i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.11550	−	0.509431i	0.0476953	−	0.0217817i	−0.391425	−	0.920210i	$-0.628018\pi$
$547$	1.11550	−	0.509431i	0.0476953	−	0.0217817i	0.439120	+	0.898428i	$0.355290\pi$
$548$		0			0
$549$	−33.6539	−	29.1612i	−1.43631	−	1.24457i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−5.26049	+	3.38071i	−0.223699	+	0.143762i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$557$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$558$		0			0
$559$	−0.657632	+	4.57393i	−0.0278149	+	0.193457i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$563$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.11236	−	4.42988i	0.214699	−	0.186038i
$568$		0			0
$569$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$569$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$570$		0			0
$571$	6.01962	+	41.8674i	0.251913	+	1.75210i	0.586703	+	0.809802i	$0.300426\pi$
$571$	6.01962	+	41.8674i	0.251913	+	1.75210i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−19.5101	−	30.3582i	−0.812214	−	1.26383i	−0.961437	−	0.275027i	$-0.911313\pi$
$577$	−19.5101	−	30.3582i	−0.812214	−	1.26383i	0.149222	−	0.988804i	$-0.452323\pi$
$578$		0			0
$579$	9.10815	+	14.1726i	0.378522	+	0.588991i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$587$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$588$		0			0
$589$	−19.6449	+	30.5681i	−0.809454	+	1.25953i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$593$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−40.0624	−	18.2959i	−1.63965	−	0.748802i
$598$		0			0
$599$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$599$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$600$		0			0
$601$	−25.9177	−	7.61013i	−1.05721	−	0.310424i	−0.293481	−	0.955965i	$-0.594814\pi$
$601$	−25.9177	−	7.61013i	−1.05721	−	0.310424i	−0.763725	+	0.645541i	$0.776632\pi$
$602$		0			0
$603$	−22.9993	+	8.60410i	−0.936605	+	0.350386i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−26.8168	−	17.2341i	−1.08846	−	0.699511i	−0.131964	−	0.991254i	$-0.542128\pi$
$607$	−26.8168	−	17.2341i	−1.08846	−	0.699511i	−0.956497	+	0.291743i	$0.905765\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.53422	−	17.6259i	−0.102356	−	0.711903i	−0.974782	−	0.223157i	$-0.928364\pi$
$613$	−2.53422	−	17.6259i	−0.102356	−	0.711903i	0.872426	−	0.488746i	$-0.162545\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$617$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$618$		0			0
$619$	19.9212	+	43.6213i	0.800700	+	1.75329i	0.643094	+	0.765787i	$0.277650\pi$
$619$	19.9212	+	43.6213i	0.800700	+	1.75329i	0.157606	+	0.987502i	$0.449622\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.55787	−	24.7455i	−0.142315	−	0.989821i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	35.5226	−	30.7805i	1.41413	−	1.22535i	0.475828	−	0.879538i	$-0.342149\pi$
$631$	35.5226	−	30.7805i	1.41413	−	1.22535i	0.938303	−	0.345813i	$-0.112397\pi$
$632$		0			0
$633$		−	49.7169i		−	1.97607i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	11.0303	+	37.5656i	0.437035	+	1.48840i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	20.9808	−	45.9416i	0.827402	−	1.81176i	0.331158	−	0.943575i	$-0.392561\pi$
$643$	20.9808	−	45.9416i	0.827402	−	1.81176i	0.496245	−	0.868183i	$-0.334712\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$647$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.91792	−	9.93754i	−0.114362	−	0.389483i
$652$		0			0
$653$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$653$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−16.6009	+	19.1584i	−0.647662	+	0.747442i
$658$		0			0
$659$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$659$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$660$		0			0
$661$	−2.42220	+	8.24927i	−0.0942128	+	0.320859i	−0.993092	−	0.117337i	$-0.962564\pi$
$661$	−2.42220	+	8.24927i	−0.0942128	+	0.320859i	0.898879	+	0.438196i	$0.144382\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	24.7187i		−	0.955680i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	50.7162	+	7.29189i	1.95497	+	0.281082i	0.999884	−	0.0152551i	$-0.00485604\pi$
$673$	50.7162	+	7.29189i	1.95497	+	0.281082i	0.955083	+	0.296337i	$0.0957651\pi$
$674$		0			0
$675$	23.6329	−	10.7928i	0.909632	−	0.415415i
$676$		0			0
$677$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$677$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$678$		0			0
$679$	8.57159	+	2.51684i	0.328947	+	0.0965876i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$683$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−44.0589	+	28.3149i	−1.68095	+	1.08028i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−14.1370	+	30.9557i	−0.537797	+	1.17761i	0.424455	+	0.905449i	$0.360466\pi$
$691$	−14.1370	+	30.9557i	−0.537797	+	1.17761i	−0.962252	+	0.272161i	$0.912262\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$701$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$702$		0			0
$703$	5.75236	−	1.68905i	0.216955	−	0.0637036i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−29.3855	−	33.9126i	−1.10359	−	1.27362i	−0.958778	−	0.284158i	$-0.908286\pi$
$709$	−29.3855	−	33.9126i	−1.10359	−	1.27362i	−0.144817	−	0.989458i	$-0.546259\pi$
$710$		0			0
$711$	−13.4936	−	20.9965i	−0.506050	−	0.787430i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$719$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$720$		0			0
$721$	−12.4529	+	5.68704i	−0.463769	+	0.211796i
$722$		0			0
$723$	−28.1974	+	43.8760i	−1.04867	+	1.63177i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−2.41466	+	0.347176i	−0.0895549	+	0.0128760i	−0.186947	−	0.982370i	$-0.559859\pi$
$727$	−2.41466	+	0.347176i	−0.0895549	+	0.0128760i	0.0973917	+	0.995246i	$0.468950\pi$
$728$		0			0
$729$	17.6812	+	20.4052i	0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	18.3976	−	28.6273i	0.679532	−	1.05737i	−0.314602	−	0.949224i	$-0.601871\pi$
$733$	18.3976	−	28.6273i	0.679532	−	1.05737i	0.994135	−	0.108149i	$-0.0344924\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−14.7524	+	50.2420i	−0.542675	+	1.84818i	−0.0131073	+	0.999914i	$0.504172\pi$
$739$	−14.7524	+	50.2420i	−0.542675	+	1.84818i	−0.529568	+	0.848268i	$0.677646\pi$
$740$		0			0
$741$	−40.4901	−	26.0214i	−1.48744	−	0.955921i
$742$		0			0
$743$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$743$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−44.6784	−	28.7131i	−1.63034	−	1.04776i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−44.6784	−	28.7131i	−1.63034	−	1.04776i	−0.681586	−	0.731738i	$-0.738709\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.46821	−	0.498653i	−0.126054	−	0.0181239i	0.0789989	−	0.996875i	$-0.474828\pi$
$757$	−3.46821	−	0.498653i	−0.126054	−	0.0181239i	−0.205053	+	0.978751i	$0.565737\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$761$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$762$		0			0
$763$	−4.30915	+	2.76933i	−0.156002	+	0.100256i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−37.4227	+	5.38058i	−1.34950	+	0.194029i	−0.778873	−	0.627182i	$-0.784208\pi$
$769$	−37.4227	+	5.38058i	−1.34950	+	0.194029i	−0.570626	+	0.821210i	$0.693299\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$773$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$774$		0			0
$775$		−	39.7782i		−	1.42888i
$776$		0			0
$777$	−0.709877	+	1.55441i	−0.0254667	+	0.0557643i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	21.6545	+	33.6951i	0.771901	+	1.20110i	0.975057	+	0.221955i	$0.0712438\pi$
$787$	21.6545	+	33.6951i	0.771901	+	1.20110i	−0.203156	+	0.979146i	$0.565120\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−37.5158	−	82.1482i	−1.33223	−	2.91717i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$797$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$809$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$810$		0			0
$811$	−10.0061	+	34.0778i	−0.351363	+	1.19663i	0.574414	+	0.818565i	$0.305230\pi$
$811$	−10.0061	+	34.0778i	−0.351363	+	1.19663i	−0.925778	+	0.378069i	$0.876588\pi$
$812$		0			0
$813$	14.5090	+	4.26022i	0.508851	+	0.149412i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.62168	−	2.27170i	−0.0917210	−	0.0794767i
$818$		0			0
$819$	13.1632	−	3.86505i	0.459958	−	0.135056i
$820$		0			0
$821$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$821$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$822$		0			0
$823$	54.4275	−	15.9814i	1.89722	−	0.557075i	0.906303	−	0.422628i	$-0.138892\pi$
$823$	54.4275	−	15.9814i	1.89722	−	0.557075i	0.990921	−	0.134447i	$-0.0429257\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$827$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$828$		0			0
$829$	−37.6060	+	43.3997i	−1.30611	+	1.50733i	−0.596984	+	0.802253i	$0.703635\pi$
$829$	−37.6060	+	43.3997i	−1.30611	+	1.50733i	−0.709127	+	0.705080i	$0.750911\pi$
$830$		0			0
$831$	16.7567	+	7.65251i	0.581282	+	0.265463i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	39.6642	−	11.6465i	1.37100	−	0.402561i
$838$		0			0
$839$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$839$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−4.46995	−	6.95537i	−0.153589	−	0.238989i
$848$		0			0
$849$	−6.58356	+	0.946573i	−0.225947	+	0.0324863i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	36.3572	+	23.3654i	1.24485	+	0.800015i	0.986136	−	0.165940i	$-0.0530657\pi$
$853$	36.3572	+	23.3654i	1.24485	+	0.800015i	0.258712	+	0.965955i	$0.416702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$857$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$858$		0			0
$859$	15.4431	−	17.8223i	0.526911	−	0.608088i	−0.428437	−	0.903572i	$-0.640935\pi$
$859$	15.4431	−	17.8223i	0.526911	−	0.608088i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$863$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	15.9191	−	24.7706i	0.540641	−	0.841254i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−49.6771	−	3.50385i	−1.68324	−	0.118723i
$872$		0			0
$873$	−10.0456	+	34.2122i	−0.339993	+	1.15791i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.87604	+	17.2461i	−0.265955	+	0.582360i	−0.994746	−	0.102376i	$-0.967355\pi$
$877$	−7.87604	+	17.2461i	−0.265955	+	0.582360i	0.728791	+	0.684737i	$0.240083\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$881$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$882$		0			0
$883$	0.349391	+	0.302749i	0.0117579	+	0.0101883i	0.660720	−	0.750633i	$-0.270251\pi$
$883$	0.349391	+	0.302749i	0.0117579	+	0.0101883i	−0.648962	+	0.760821i	$0.724797\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$887$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$888$		0			0
$889$	−8.51370	+	13.2476i	−0.285540	+	0.444310i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.978713	−	0.140718i	0.0325695	−	0.00468279i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	37.7071	+	43.5163i	1.25204	+	1.44494i	0.847827	+	0.530273i	$0.177910\pi$
$907$	37.7071	+	43.5163i	1.25204	+	1.44494i	0.404217	+	0.914663i	$0.367544\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$911$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−9.67737	−	32.9581i	−0.319227	−	1.08719i	−0.950267	−	0.311437i	$-0.899190\pi$
$919$	−9.67737	−	32.9581i	−0.319227	−	1.08719i	0.631040	−	0.775751i	$-0.282629\pi$
$920$		0			0
$921$	−3.17956	−	4.94750i	−0.104770	−	0.163026i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.29792	+	4.96006i	−0.141315	+	0.163086i
$926$		0			0
$927$	−22.6990	−	49.7039i	−0.745533	−	1.63249i
$928$		0			0
$929$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$929$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$930$		0			0
$931$	−28.2007	−	8.28047i	−0.924240	−	0.271381i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	60.6837i		−	1.98245i	−0.132190	−	0.991224i	$-0.542201\pi$
$937$		−	60.6837i		−	1.98245i	0.132190	−	0.991224i	$-0.457799\pi$
$938$		0			0
$939$	56.7911			1.85331
$940$		0			0
$941$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$941$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$947$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$948$		0			0
$949$	−46.7652	+	21.3569i	−1.51806	+	0.693276i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$953$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.59568	−	31.9636i	0.148248	−	1.03109i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−55.4440			−1.78296			−0.891480	−	0.453061i	$-0.850332\pi$
$967$	−55.4440			−1.78296			−0.891480	+	0.453061i	$0.850332\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$971$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$972$		0			0
$973$	1.95773	+	13.6163i	0.0627619	+	0.436518i
$974$		0			0
$975$	52.6898			1.68742
$976$		0			0
$977$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$977$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−11.0534	−	17.1994i	−0.352907	−	0.549133i
$982$		0			0
$983$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$983$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−31.7526	+	49.4079i	−1.00865	+	1.56949i	−0.201211	+	0.979548i	$0.564488\pi$
$991$	−31.7526	+	49.4079i	−1.00865	+	1.56949i	−0.807442	+	0.589947i	$0.799149\pi$
$992$		0			0
$993$	−33.1307	+	38.2349i	−1.05137	+	1.21335i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	41.3494	+	47.7198i	1.30955	+	1.51130i	0.667033	+	0.745028i	$0.267564\pi$
$997$	41.3494	+	47.7198i	1.30955	+	1.51130i	0.642516	+	0.766272i	$0.277891\pi$
$998$		0			0
$999$	−6.20422	−	2.83337i	−0.196293	−	0.0896439i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.137.2	✓	20
3.2	odd	2	CM	804.2.s.a.137.2	✓	20
67.45	odd	22	inner	804.2.s.a.581.2	yes	20
201.179	even	22	inner	804.2.s.a.581.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.137.2	✓	20	1.1	even	1	trivial
804.2.s.a.137.2	✓	20	3.2	odd	2	CM
804.2.s.a.581.2	yes	20	67.45	odd	22	inner
804.2.s.a.581.2	yes	20	201.179	even	22	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.s (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})\)	\(q+(1.71442 + 0.246497i) q^{3} +(0.211757 - 0.721178i) q^{7} +(2.87848 + 0.845198i) q^{9} +(4.59805 + 3.98424i) q^{13} +(-4.38235 + 1.28677i) q^{19} +(0.540809 - 1.18421i) q^{21} +(3.27430 - 3.77875i) q^{25} +(4.72659 + 2.15856i) q^{27} +(6.01248 - 5.20984i) q^{31} -1.31262 q^{37} +(6.90090 + 7.96406i) q^{39} +(0.410625 + 0.638945i) q^{43} +(5.41352 + 3.47906i) q^{49} +(-7.83038 + 1.12584i) q^{57} +(-13.5021 - 6.16621i) q^{61} +(1.21908 - 1.89692i) q^{63} +(-6.54788 + 4.91174i) q^{67} +(-3.51029 + 7.68646i) q^{73} +(6.54498 - 5.67126i) q^{75} +(-6.28748 - 5.44813i) q^{79} +(7.57128 + 4.86577i) q^{81} +(3.84702 - 2.47233i) q^{91} +(11.5921 - 7.44981i) q^{93} +11.8855i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^9	\( 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) \) 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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