LMFDB - Embedded newform 804.2.s.a.713.1

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			713.1
Root			$0.981929 - 0.189251i$ of defining polynomial
Character	$\chi$	$=$	804.713
Dual form			804.2.s.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.936417 - 1.45709i) q^{3} +(0.686312 + 0.313428i) q^{7} +(-1.24625 + 2.72890i) q^{9} +O(q^{10})$	$q+(-0.936417 - 1.45709i) q^{3} +(0.686312 + 0.313428i) q^{7} +(-1.24625 + 2.72890i) q^{9} +(1.32339 - 4.50707i) q^{13} +(-3.57376 - 7.82544i) q^{19} +(-0.185980 - 1.29352i) q^{21} +(4.79746 + 1.40866i) q^{25} +(5.14326 - 0.739490i) q^{27} +(-0.698783 - 2.37984i) q^{31} -11.5468 q^{37} +(-7.80647 + 2.29219i) q^{39} +(-9.23919 - 8.00581i) q^{43} +(-4.21124 - 4.86003i) q^{49} +(-8.05587 + 12.5352i) q^{57} +(13.2610 - 1.90664i) q^{61} +(-1.71063 + 1.48227i) q^{63} +(-7.18797 - 3.91575i) q^{67} +(2.15444 + 14.9844i) q^{73} +(-2.43988 - 8.30945i) q^{75} +(4.99958 - 17.0270i) q^{79} +(-5.89375 - 6.80175i) q^{81} +(2.32090 - 2.67847i) q^{91} +(-2.81329 + 3.24671i) q^{93} +4.08778i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{3}{22}\right)$	$1$

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
$3$	−0.936417	−	1.45709i	−0.540641	−	0.841254i
$3$	−0.936417	−	1.45709i	−0.540641	−	0.841254i
$4$		0			0
$5$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$5$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$6$		0			0
$7$	0.686312	+	0.313428i	0.259402	+	0.118465i	0.540871	−	0.841105i	$-0.318095\pi$
$7$	0.686312	+	0.313428i	0.259402	+	0.118465i	−0.281470	+	0.959570i	$0.590822\pi$
$8$		0			0
$9$	−1.24625	+	2.72890i	−0.415415	+	0.909632i
$10$		0			0
$11$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$11$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$12$		0			0
$13$	1.32339	−	4.50707i	0.367044	−	1.25004i	−0.544475	−	0.838777i	$-0.683271\pi$
$13$	1.32339	−	4.50707i	0.367044	−	1.25004i	0.911519	−	0.411259i	$-0.134911\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$17$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$18$		0			0
$19$	−3.57376	−	7.82544i	−0.819876	−	1.79528i	−0.557158	−	0.830406i	$-0.688108\pi$
$19$	−3.57376	−	7.82544i	−0.819876	−	1.79528i	−0.262718	−	0.964873i	$-0.584619\pi$
$20$		0			0
$21$	−0.185980	−	1.29352i	−0.0405842	−	0.282269i
$22$		0			0
$23$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$23$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$24$		0			0
$25$	4.79746	+	1.40866i	0.959493	+	0.281733i
$26$		0			0
$27$	5.14326	−	0.739490i	0.989821	−	0.142315i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.698783	−	2.37984i	−0.125505	−	0.427431i	0.872636	−	0.488371i	$-0.162409\pi$
$31$	−0.698783	−	2.37984i	−0.125505	−	0.427431i	−0.998141	+	0.0609393i	$0.980590\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.5468			−1.89829			−0.949144	−	0.314844i	$-0.898048\pi$
$37$	−11.5468			−1.89829			−0.949144	+	0.314844i	$0.898048\pi$
$38$		0			0
$39$	−7.80647	+	2.29219i	−1.25004	+	0.367044i
$40$		0			0
$41$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$41$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$42$		0			0
$43$	−9.23919	−	8.00581i	−1.40896	−	1.22087i	−0.941567	−	0.336826i	$-0.890646\pi$
$43$	−9.23919	−	8.00581i	−1.40896	−	1.22087i	−0.467397	−	0.884048i	$-0.654808\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$47$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$48$		0			0
$49$	−4.21124	−	4.86003i	−0.601605	−	0.694290i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$53$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.05587	+	12.5352i	−1.06703	+	1.66032i
$58$		0			0
$59$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$59$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$60$		0			0
$61$	13.2610	−	1.90664i	1.69789	−	0.244120i	0.775771	−	0.631015i	$-0.217361\pi$
$61$	13.2610	−	1.90664i	1.69789	−	0.244120i	0.922124	+	0.386895i	$0.126452\pi$
$62$		0			0
$63$	−1.71063	+	1.48227i	−0.215519	+	0.186748i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.18797	−	3.91575i	−0.878150	−	0.478385i
$67$	−7.18797	−	3.91575i	−0.878150	−	0.478385i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$71$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$72$		0			0
$73$	2.15444	+	14.9844i	0.252158	+	1.75379i	0.585206	+	0.810885i	$0.301014\pi$
$73$	2.15444	+	14.9844i	0.252158	+	1.75379i	−0.333048	+	0.942910i	$0.608077\pi$
$74$		0			0
$75$	−2.43988	−	8.30945i	−0.281733	−	0.959493i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.99958	−	17.0270i	0.562497	−	1.91569i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.99958	−	17.0270i	0.562497	−	1.91569i	0.337479	−	0.941333i	$-0.390426\pi$
$80$		0			0
$81$	−5.89375	−	6.80175i	−0.654861	−	0.755750i
$82$		0			0
$83$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$83$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$89$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$	2.32090	−	2.67847i	0.243297	−	0.280780i
$92$		0			0
$93$	−2.81329	+	3.24671i	−0.291725	+	0.336668i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			4.08778i			0.415051i	0.978230	+	0.207526i	$0.0665410\pi$
$97$			4.08778i			0.415051i	−0.978230	+	0.207526i	$0.933459\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$101$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$102$		0			0
$103$	10.9952	−	3.22847i	1.08339	−	0.318111i	0.309154	−	0.951012i	$-0.399954\pi$
$103$	10.9952	−	3.22847i	1.08339	−	0.318111i	0.774233	+	0.632901i	$0.218136\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$107$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$108$		0			0
$109$	−1.39108	+	4.73759i	−0.133242	+	0.453779i	−0.998902	−	0.0468528i	$-0.985081\pi$
$109$	−1.39108	+	4.73759i	−0.133242	+	0.453779i	0.865660	+	0.500632i	$0.166899\pi$
$110$		0			0
$111$	10.8127	+	16.8248i	1.02629	+	1.59694i
$112$		0			0
$113$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$113$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.6500	+	9.22832i	0.984597	+	0.853158i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5544	+	3.09906i	0.959493	+	0.281733i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.75548	−	14.7924i	0.599452	−	1.31262i	−0.330107	−	0.943944i	$-0.607085\pi$
$127$	6.75548	−	14.7924i	0.599452	−	1.31262i	0.929559	−	0.368674i	$-0.120188\pi$
$128$		0			0
$129$	−3.01347	+	20.9591i	−0.265321	+	1.84535i
$130$		0			0
$131$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$131$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$132$		0			0
$133$		−	6.49081i		−	0.562825i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$137$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$138$		0			0
$139$	22.9848	+	3.30472i	1.94955	+	0.280303i	0.999495	−	0.0317847i	$-0.0101191\pi$
$139$	22.9848	+	3.30472i	1.94955	+	0.280303i	0.950053	+	0.312087i	$0.101028\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.13804	+	10.6872i	−0.258821	+	0.881464i
$148$		0			0
$149$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$149$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$150$		0			0
$151$	−16.0907	+	18.5696i	−1.30944	+	1.51118i	−0.639089	+	0.769133i	$0.720688\pi$
$151$	−16.0907	+	18.5696i	−1.30944	+	1.51118i	−0.670352	+	0.742043i	$0.733857\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.0125	+	7.07728i	−0.878890	+	0.564828i	−0.900460	−	0.434939i	$-0.856770\pi$
$157$	−11.0125	+	7.07728i	−0.878890	+	0.564828i	0.0215699	+	0.999767i	$0.493134\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	13.0871			1.02506			0.512529	−	0.858670i	$-0.328709\pi$
$163$	13.0871			1.02506			0.512529	+	0.858670i	$0.328709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$167$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$168$		0			0
$169$	−7.62599	−	4.90093i	−0.586615	−	0.376995i
$170$		0			0
$171$	25.8086			1.97363
$172$		0			0
$173$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$173$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$174$		0			0
$175$	2.85104	+	2.47044i	0.215519	+	0.186748i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$179$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$180$		0			0
$181$	15.2493	−	9.80015i	1.13347	−	0.728440i	0.167192	−	0.985924i	$-0.446530\pi$
$181$	15.2493	−	9.80015i	1.13347	−	0.728440i	0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$	−15.1960	−	17.5371i	−1.12332	−	1.29638i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3.76166	+	1.10452i	0.273621	+	0.0803422i
$190$		0			0
$191$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$191$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$192$		0			0
$193$	23.3902	−	6.86797i	1.68366	−	0.494367i	0.706651	−	0.707562i	$-0.250205\pi$
$193$	23.3902	−	6.86797i	1.68366	−	0.494367i	0.977010	+	0.213195i	$0.0683868\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$197$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$198$		0			0
$199$	−8.70465	+	19.0605i	−0.617056	+	1.35116i	0.300586	+	0.953755i	$0.402818\pi$
$199$	−8.70465	+	19.0605i	−0.617056	+	1.35116i	−0.917642	+	0.397409i	$0.869909\pi$
$200$		0			0
$201$	1.02532	+	14.1403i	0.0723206	+	0.997381i
$202$		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
$211$	18.2761	+	11.7453i	1.25818	+	0.808583i	0.988034	−	0.154237i	$-0.0492918\pi$
$211$	18.2761	+	11.7453i	1.25818	+	0.808583i	0.270146	+	0.962819i	$0.412928\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.266325	−	1.85233i	0.0180793	−	0.125744i
$218$		0			0
$219$	19.8163	−	17.1709i	1.33906	−	1.16030i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−24.5424	−	15.7724i	−1.64348	−	1.05620i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−24.5424	−	15.7724i	−1.64348	−	1.05620i	−0.705971	−	0.708241i	$-0.749489\pi$
$224$		0			0
$225$	−9.82291	+	11.3362i	−0.654861	+	0.755750i
$226$		0			0
$227$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$227$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$228$		0			0
$229$	3.87229	+	13.1878i	0.255888	+	0.871476i	0.982789	+	0.184733i	$0.0591420\pi$
$229$	3.87229	+	13.1878i	0.255888	+	0.871476i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$233$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−29.4916	+	8.65953i	−1.91569	+	0.562497i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.800564	+	5.56804i	0.0515688	+	0.358669i	0.999225	+	0.0393750i	$0.0125367\pi$
$241$	0.800564	+	5.56804i	0.0515688	+	0.358669i	−0.947656	+	0.319294i	$0.896554\pi$
$242$		0			0
$243$	−4.39178	+	14.9570i	−0.281733	+	0.959493i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−39.9993	+	5.75103i	−2.54509	+	0.365929i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$251$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$257$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$258$		0			0
$259$	−7.92473	−	3.61910i	−0.492419	−	0.224880i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$263$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$264$		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
$271$	17.6600	+	27.4795i	1.07277	+	1.66926i	0.641748	+	0.766915i	$0.278209\pi$
$271$	17.6600	+	27.4795i	1.07277	+	1.66926i	0.431019	+	0.902343i	$0.358154\pi$
$272$		0			0
$273$	−6.07611	−	0.873613i	−0.367743	−	0.0528735i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.547691	−	1.19928i	0.0329076	−	0.0720575i	−0.892464	−	0.451118i	$-0.851025\pi$
$277$	0.547691	−	1.19928i	0.0329076	−	0.0720575i	0.925372	+	0.379060i	$0.123753\pi$
$278$		0			0
$279$	7.36518	+	1.05895i	0.440942	+	0.0633979i
$280$		0			0
$281$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$281$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$282$		0			0
$283$	−6.16501	−	13.4995i	−0.366472	−	0.802461i	−0.999596	−	0.0284112i	$-0.990955\pi$
$283$	−6.16501	−	13.4995i	−0.366472	−	0.802461i	0.633125	−	0.774050i	$-0.281772\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.41935	−	16.8270i	−0.142315	−	0.989821i
$290$		0			0
$291$	5.95628	−	3.82787i	0.349163	−	0.224394i
$292$		0			0
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.83172	−	8.39030i	−0.220857	−	0.483609i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.5542	−	3.68624i	0.716504	−	0.210385i	0.0968945	−	0.995295i	$-0.469109\pi$
$307$	12.5542	−	3.68624i	0.716504	−	0.210385i	0.619610	+	0.784910i	$0.287291\pi$
$308$		0			0
$309$	−15.0003	−	12.9978i	−0.853335	−	0.739419i
$310$		0			0
$311$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$311$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$312$		0			0
$313$	−4.59522	+	7.15031i	−0.259737	+	0.404159i	−0.946491	−	0.322731i	$-0.895399\pi$
$313$	−4.59522	+	7.15031i	−0.259737	+	0.404159i	0.686753	+	0.726891i	$0.259035\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$317$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	12.6979	−	19.7583i	0.704352	−	1.09599i
$326$		0			0
$327$	8.20575	−	2.40943i	0.453779	−	0.133242i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.06287	+	1.78749i	−0.113386	+	0.0982491i	−0.709708	−	0.704496i	$-0.751173\pi$
$331$	−2.06287	+	1.78749i	−0.113386	+	0.0982491i	0.596323	+	0.802745i	$0.296628\pi$
$332$		0			0
$333$	14.3902	−	31.5101i	0.788577	−	1.72674i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.38039	−	1.54377i	−0.184142	−	0.0840947i	0.321211	−	0.947008i	$-0.395910\pi$
$337$	−3.38039	−	1.54377i	−0.184142	−	0.0840947i	−0.505352	+	0.862913i	$0.668637\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.85491	−	9.72294i	−0.154151	−	0.524989i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$347$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$348$		0			0
$349$	−16.8265	−	19.4188i	−0.900702	−	1.03947i	−0.999018	−	0.0443098i	$-0.985891\pi$
$349$	−16.8265	−	19.4188i	−0.900702	−	1.03947i	0.0983163	−	0.995155i	$-0.468654\pi$
$350$		0			0
$351$	3.47364	−	24.1597i	0.185409	−	1.28955i
$352$		0			0
$353$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$353$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$359$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	−36.0234	+	41.5732i	−1.89597	+	2.18806i
$362$		0			0
$363$	−5.36773	−	18.2808i	−0.281733	−	0.959493i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	5.07241	−	7.89282i	0.264777	−	0.412002i	−0.683253	−	0.730182i	$-0.739435\pi$
$367$	5.07241	−	7.89282i	0.264777	−	0.412002i	0.948030	+	0.318180i	$0.103072\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			37.4441i			1.93878i	0.245523	+	0.969391i	$0.421040\pi$
$373$			37.4441i			1.93878i	−0.245523	+	0.969391i	$0.578960\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12.4923	+	19.4384i	0.641687	+	0.998485i	0.997944	+	0.0640964i	$0.0204165\pi$
$379$	12.4923	+	19.4384i	0.641687	+	0.998485i	−0.356257	+	0.934388i	$0.615947\pi$
$380$		0			0
$381$	−27.8799	+	4.00853i	−1.42833	+	0.205363i
$382$		0			0
$383$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$383$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	33.3613	−	15.2356i	1.69585	−	0.774469i
$388$		0			0
$389$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$389$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.40415	−	37.5867i	0.271227	−	1.88642i	−0.164551	−	0.986369i	$-0.552617\pi$
$397$	5.40415	−	37.5867i	0.271227	−	1.88642i	0.435778	−	0.900054i	$-0.356473\pi$
$398$		0			0
$399$	−9.45772	+	6.07811i	−0.473478	+	0.304286i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−11.6509			−0.580370
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	36.0080	+	16.4443i	1.78048	+	0.813117i	0.975560	+	0.219732i	$0.0705182\pi$
$409$	36.0080	+	16.4443i	1.78048	+	0.813117i	0.804918	+	0.593386i	$0.202209\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−16.7081	−	36.5857i	−0.818200	−	1.79161i
$418$		0			0
$419$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$419$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$420$		0			0
$421$	−4.71469	−	10.3237i	−0.229780	−	0.503148i	0.759262	−	0.650785i	$-0.225560\pi$
$421$	−4.71469	−	10.3237i	−0.229780	−	0.503148i	−0.989041	+	0.147638i	$0.952833\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9.69877	+	2.84781i	0.469356	+	0.137815i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−11.3958	−	38.8104i	−0.547645	−	1.86511i	−0.499588	−	0.866263i	$-0.666515\pi$
$433$	−11.3958	−	38.8104i	−0.547645	−	1.86511i	−0.0480569	−	0.998845i	$-0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−41.8980			−1.99968			−0.999842	−	0.0177654i	$-0.994345\pi$
$439$	−41.8980			−1.99968			−0.999842	+	0.0177654i	$0.994345\pi$
$440$		0			0
$441$	18.5107	−	5.43525i	0.881464	−	0.258821i
$442$		0			0
$443$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$443$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$449$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	42.1253	+	6.05670i	1.97922	+	0.284569i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	40.2669	+	11.8234i	1.88361	+	0.553077i	0.995653	+	0.0931420i	$0.0296911\pi$
$457$	40.2669	+	11.8234i	1.88361	+	0.553077i	0.887953	+	0.459935i	$0.152127\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$461$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$	−33.6758	+	4.84185i	−1.56505	+	0.225020i	−0.869673	−	0.493629i	$-0.835670\pi$
$463$	−33.6758	+	4.84185i	−1.56505	+	0.225020i	−0.695374	+	0.718648i	$0.744761\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$467$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$468$		0			0
$469$	−3.70588	−	4.94034i	−0.171122	−	0.228124i
$470$		0			0
$471$	20.6245	+	9.41890i	0.950328	+	0.434000i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−6.12158	−	42.5765i	−0.280877	−	1.95354i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$479$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$480$		0			0
$481$	−15.2810	+	52.0424i	−0.696754	+	2.37293i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−23.4165	+	20.2905i	−1.06110	+	0.919451i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−23.4165	+	20.2905i	−1.06110	+	0.919451i	−0.0641880	+	0.997938i	$0.520446\pi$
$488$		0			0
$489$	−12.2550	−	19.0691i	−0.554188	−	0.862334i
$490$		0			0
$491$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$491$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	20.8986i		−	0.935549i	−0.883848	−	0.467775i	$-0.845056\pi$
$499$		−	20.8986i		−	0.935549i	0.883848	−	0.467775i	$-0.154944\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$503$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			15.7011i			0.697311i
$508$		0			0
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$	−3.21793	+	10.9593i	−0.142353	+	0.484809i
$512$		0			0
$513$	−24.1676	−	37.6055i	−1.06703	−	1.66032i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$521$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$522$		0			0
$523$	−36.1157	−	10.6045i	−1.57923	−	0.463704i	−0.629559	−	0.776952i	$-0.716765\pi$
$523$	−36.1157	−	10.6045i	−1.57923	−	0.463704i	−0.949671	+	0.313249i	$0.898583\pi$
$524$		0			0
$525$	0.929901	−	6.46760i	0.0405842	−	0.282269i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.55455	−	20.9215i	0.415415	−	0.909632i
$530$		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
$541$	38.9015	+	5.59319i	1.67250	+	0.240470i	0.912391	−	0.409320i	$-0.134234\pi$
$541$	38.9015	+	5.59319i	1.67250	+	0.240470i	0.760114	+	0.649790i	$0.225143\pi$
$542$		0			0
$543$	−28.5595	−	13.0427i	−1.22560	−	0.559715i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.2044	−	2.04228i	−0.607336	−	0.0873217i	−0.168216	−	0.985750i	$-0.553801\pi$
$547$	−14.2044	−	2.04228i	−0.607336	−	0.0873217i	−0.439120	+	0.898428i	$0.644710\pi$
$548$		0			0
$549$	−11.3234	+	38.5640i	−0.483271	+	1.64587i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	8.76802	−	10.1188i	0.372854	−	0.430296i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$557$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$558$		0			0
$559$	−48.3098	+	31.0468i	−2.04329	+	1.31314i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$563$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.91309	−	6.51539i	−0.0803422	−	0.273621i
$568$		0			0
$569$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$569$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$570$		0			0
$571$	9.15855	+	5.88585i	0.383274	+	0.246315i	0.718063	−	0.695978i	$-0.245029\pi$
$571$	9.15855	+	5.88585i	0.383274	+	0.246315i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	35.9009	+	31.1083i	1.49458	+	1.29506i	0.845008	+	0.534754i	$0.179596\pi$
$577$	35.9009	+	31.1083i	1.49458	+	1.29506i	0.649568	+	0.760303i	$0.274950\pi$
$578$		0			0
$579$	−31.9102	−	27.6504i	−1.32614	−	1.14911i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$587$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$588$		0			0
$589$	−16.1260	+	13.9732i	−0.664460	+	0.575758i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$593$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	35.9241	−	5.16511i	1.47028	−	0.211394i
$598$		0			0
$599$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$599$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$600$		0			0
$601$	18.6298	−	40.7936i	0.759926	−	1.66401i	0.0122663	−	0.999925i	$-0.496095\pi$
$601$	18.6298	−	40.7936i	0.759926	−	1.66401i	0.747660	−	0.664082i	$-0.231177\pi$
$602$		0			0
$603$	19.6436	−	14.7352i	0.799951	−	0.600065i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	21.3848	+	24.6793i	0.867981	+	1.00170i	0.999945	+	0.0104491i	$0.00332612\pi$
$607$	21.3848	+	24.6793i	0.867981	+	1.00170i	−0.131964	+	0.991254i	$0.542128\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−25.3245	−	16.2751i	−1.02285	−	0.657344i	−0.0821600	−	0.996619i	$-0.526182\pi$
$613$	−25.3245	−	16.2751i	−1.02285	−	0.657344i	−0.940687	+	0.339275i	$0.889818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$617$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$618$		0			0
$619$	−4.71961	+	32.8256i	−0.189697	+	1.31937i	0.643094	+	0.765787i	$0.277650\pi$
$619$	−4.71961	+	32.8256i	−0.189697	+	1.31937i	−0.832792	+	0.553587i	$0.813259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.0313	+	13.5160i	0.841254	+	0.540641i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	12.4491	+	42.3976i	0.495589	+	1.68782i	0.704348	+	0.709855i	$0.251239\pi$
$631$	12.4491	+	42.3976i	0.495589	+	1.68782i	−0.208759	+	0.977967i	$0.566942\pi$
$632$		0			0
$633$		−	37.6286i		−	1.49560i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−27.4776	+	12.5486i	−1.08870	+	0.497194i
$638$		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
$643$	7.08107	+	49.2499i	0.279250	+	1.94223i	0.331158	+	0.943575i	$0.392561\pi$
$643$	7.08107	+	49.2499i	0.279250	+	1.94223i	−0.0519076	+	0.998652i	$0.516530\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$647$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.94841	+	1.34649i	−0.115557	+	0.0527732i
$652$		0			0
$653$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$653$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−43.5759	−	12.7950i	−1.70006	−	0.499182i
$658$		0			0
$659$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$659$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$660$		0			0
$661$	−5.48821	−	2.50638i	−0.213467	−	0.0974870i	0.305812	−	0.952092i	$-0.401072\pi$
$661$	−5.48821	−	2.50638i	−0.213467	−	0.0974870i	−0.519279	+	0.854605i	$0.673799\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			50.5302i			1.95361i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−25.6908	−	39.9756i	−0.990306	−	1.54095i	−0.832908	−	0.553411i	$-0.813326\pi$
$673$	−25.6908	−	39.9756i	−0.990306	−	1.54095i	−0.157398	−	0.987535i	$-0.550311\pi$
$674$		0			0
$675$	25.7163	+	3.69745i	0.989821	+	0.142315i
$676$		0			0
$677$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$677$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$678$		0			0
$679$	−1.28123	+	2.80549i	−0.0491689	+	0.107665i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$683$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	15.5898	−	17.9916i	0.594788	−	0.686422i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−3.17578	−	22.0881i	−0.120813	−	0.840269i	−0.956639	−	0.291276i	$-0.905920\pi$
$691$	−3.17578	−	22.0881i	−0.120813	−	0.840269i	0.835827	−	0.548994i	$-0.184989\pi$
$692$		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		−0.281733	−	0.959493i	$-0.590909\pi$
$701$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$702$		0			0
$703$	41.2656	+	90.3590i	1.55636	+	3.40795i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.39971	−	2.17275i	0.277902	−	0.0815994i	−0.139812	−	0.990178i	$-0.544650\pi$
$709$	7.39971	−	2.17275i	0.277902	−	0.0815994i	0.417714	+	0.908579i	$0.362832\pi$
$710$		0			0
$711$	40.2342	+	34.8632i	1.50890	+	1.30747i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$719$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$720$		0			0
$721$	8.55801	+	1.23046i	0.318717	+	0.0458246i
$722$		0			0
$723$	7.36350	−	6.38051i	0.273851	−	0.237294i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	6.93991	−	10.7987i	0.257387	−	0.400502i	−0.688379	−	0.725351i	$-0.741677\pi$
$727$	6.93991	−	10.7987i	0.257387	−	0.400502i	0.945766	+	0.324850i	$0.105314\pi$
$728$		0			0
$729$	25.9063	−	7.60678i	0.959493	−	0.281733i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−27.8474	+	24.1299i	−1.02857	+	0.891258i	−0.994135	−	0.108149i	$-0.965508\pi$
$733$	−27.8474	+	24.1299i	−1.02857	+	0.891258i	−0.0344317	+	0.999407i	$0.510962\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	41.9518	+	19.1588i	1.54322	+	0.704766i	0.991602	−	0.129329i	$-0.0412822\pi$
$739$	41.9518	+	19.1588i	1.54322	+	0.704766i	0.551621	+	0.834095i	$0.314010\pi$
$740$		0			0
$741$	45.8358	+	52.8973i	1.68382	+	1.94323i
$742$		0			0
$743$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$743$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−30.8719	−	35.6281i	−1.12653	−	1.30009i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−30.8719	−	35.6281i	−1.12653	−	1.30009i	−0.177779	−	0.984070i	$-0.556891\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	17.6450	+	27.4562i	0.641320	+	0.997913i	0.997971	+	0.0636752i	$0.0202822\pi$
$757$	17.6450	+	27.4562i	0.641320	+	0.997913i	−0.356651	+	0.934238i	$0.616081\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$761$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$	−2.43961	+	2.81546i	−0.0883199	+	0.101927i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	28.4469	−	44.2643i	1.02582	−	1.59621i	0.246949	−	0.969028i	$-0.420572\pi$
$769$	28.4469	−	44.2643i	1.02582	−	1.59621i	0.778873	−	0.627182i	$-0.215792\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$773$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$774$		0			0
$775$		−	12.4015i		−	0.445476i
$776$		0			0
$777$	2.14748	+	14.9361i	0.0770405	+	0.535828i
$778$		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
$787$	14.4354	+	12.5084i	0.514567	+	0.445875i	0.873028	−	0.487670i	$-0.162153\pi$
$787$	14.4354	+	12.5084i	0.514567	+	0.445875i	−0.358461	+	0.933545i	$0.616698\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8.95615	−	62.2914i	0.318042	−	2.21203i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		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		−0.989821	−	0.142315i	$-0.954545\pi$
$809$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$810$		0			0
$811$	23.4215	+	10.6963i	0.822441	+	0.375596i	0.781763	−	0.623576i	$-0.214321\pi$
$811$	23.4215	+	10.6963i	0.822441	+	0.375596i	0.0406786	+	0.999172i	$0.487048\pi$
$812$		0			0
$813$	23.5030	−	51.4645i	0.824288	−	1.80494i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−29.6303	+	100.912i	−1.03663	+	3.53045i
$818$		0			0
$819$	4.41684	+	9.67153i	0.154337	+	0.337951i
$820$		0			0
$821$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$821$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$822$		0			0
$823$	−0.189356	−	0.414631i	−0.00660053	−	0.0144531i	0.906303	−	0.422628i	$-0.138892\pi$
$823$	−0.189356	−	0.414631i	−0.00660053	−	0.0144531i	−0.912904	+	0.408175i	$0.866165\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$827$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$828$		0			0
$829$	54.0228	+	15.8625i	1.87629	+	0.550929i	0.997244	+	0.0741933i	$0.0236382\pi$
$829$	54.0228	+	15.8625i	1.87629	+	0.550929i	0.879047	+	0.476735i	$0.158180\pi$
$830$		0			0
$831$	−2.26033	+	0.324986i	−0.0784098	+	0.0112736i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.35389	−	11.7234i	−0.185058	−	0.405219i
$838$		0			0
$839$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$839$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.27230	+	5.43497i	0.215519	+	0.186748i
$848$		0			0
$849$	−13.8970	+	21.6241i	−0.476944	+	0.742139i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−35.1652	−	40.5828i	−1.20403	−	1.38953i	−0.899440	−	0.437045i	$-0.856025\pi$
$853$	−35.1652	−	40.5828i	−1.20403	−	1.38953i	−0.304592	−	0.952483i	$-0.598520\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$857$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$858$		0			0
$859$	56.2372	+	16.5127i	1.91879	+	0.563407i	0.963440	+	0.267923i	$0.0863373\pi$
$859$	56.2372	+	16.5127i	1.91879	+	0.563407i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$863$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−22.2529	+	19.2823i	−0.755750	+	0.654861i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−27.1611	+	27.2146i	−0.920318	+	0.922131i
$872$		0			0
$873$	−11.1551	−	5.09437i	−0.377544	−	0.172418i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−8.28826	−	57.6461i	−0.279875	−	1.94657i	−0.320108	−	0.947381i	$-0.603719\pi$
$877$	−8.28826	−	57.6461i	−0.279875	−	1.94657i	0.0402330	−	0.999190i	$-0.487190\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$881$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$882$		0			0
$883$	12.7388	−	43.3845i	0.428696	−	1.46000i	−0.408326	−	0.912836i	$-0.633887\pi$
$883$	12.7388	−	43.3845i	0.428696	−	1.46000i	0.837022	−	0.547169i	$-0.184294\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$887$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$888$		0			0
$889$	9.27274	−	8.03487i	0.310998	−	0.269481i
$890$		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
$903$	−8.63737	+	13.4400i	−0.287434	+	0.447255i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	37.3073	−	10.9544i	1.23877	−	0.363736i	0.404217	−	0.914663i	$-0.367544\pi$
$907$	37.3073	−	10.9544i	1.23877	−	0.363736i	0.834553	+	0.550927i	$0.185726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$911$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−50.8554	+	23.2249i	−1.67756	+	0.766118i	−0.678048	+	0.735017i	$0.737174\pi$
$919$	−50.8554	+	23.2249i	−1.67756	+	0.766118i	−0.999516	+	0.0311006i	$0.990099\pi$
$920$		0			0
$921$	−17.1271	−	14.8407i	−0.564358	−	0.489019i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−55.3955	−	16.2656i	−1.82139	−	0.534809i
$926$		0			0
$927$	−4.89251	+	34.0282i	−0.160691	+	1.11763i
$928$		0			0
$929$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$929$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$930$		0			0
$931$	−22.9819	+	50.3234i	−0.753202	+	1.64928i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			55.9456i			1.82766i	0.406095	+	0.913831i	$0.366890\pi$
$937$			55.9456i			1.82766i	−0.406095	+	0.913831i	$0.633110\pi$
$938$		0			0
$939$	14.7217			0.480425
$940$		0			0
$941$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$941$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$947$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$948$		0			0
$949$	70.3870	+	10.1201i	2.28486	+	0.328513i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$953$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	20.9035	−	13.4339i	0.674308	−	0.433351i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.0131			0.482789			0.241395	−	0.970427i	$-0.422395\pi$
$967$	15.0131			0.482789			0.241395	+	0.970427i	$0.422395\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$971$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$972$		0			0
$973$	14.7390	+	9.47216i	0.472510	+	0.303664i
$974$		0			0
$975$	−40.6802			−1.30281
$976$		0			0
$977$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$977$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−11.1948	−	9.70032i	−0.357422	−	0.309707i
$982$		0			0
$983$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$983$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	28.0710	−	24.3236i	0.891704	−	0.772666i	−0.0829100	−	0.996557i	$-0.526421\pi$
$991$	28.0710	−	24.3236i	0.891704	−	0.772666i	0.974614	+	0.223891i	$0.0718760\pi$
$992$		0			0
$993$	4.53624	+	1.33196i	0.143953	+	0.0422685i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−40.4173	+	11.8676i	−1.28003	+	0.375850i	−0.849912	−	0.526924i	$-0.823345\pi$
$997$	−40.4173	+	11.8676i	−1.28003	+	0.375850i	−0.430115	+	0.902774i	$0.641527\pi$
$998$		0			0
$999$	−59.3884	+	8.53876i	−1.87897	+	0.270154i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.713.1	yes	20
3.2	odd	2	CM	804.2.s.a.713.1	yes	20
67.53	odd	22	inner	804.2.s.a.53.1	✓	20
201.53	even	22	inner	804.2.s.a.53.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.53.1	✓	20	67.53	odd	22	inner
804.2.s.a.53.1	✓	20	201.53	even	22	inner
804.2.s.a.713.1	yes	20	1.1	even	1	trivial
804.2.s.a.713.1	yes	20	3.2	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(-0.936417 - 1.45709i) q^{3} +(0.686312 + 0.313428i) q^{7} +(-1.24625 + 2.72890i) q^{9} +O(q^{10})\)	\(q+(-0.936417 - 1.45709i) q^{3} +(0.686312 + 0.313428i) q^{7} +(-1.24625 + 2.72890i) q^{9} +(1.32339 - 4.50707i) q^{13} +(-3.57376 - 7.82544i) q^{19} +(-0.185980 - 1.29352i) q^{21} +(4.79746 + 1.40866i) q^{25} +(5.14326 - 0.739490i) q^{27} +(-0.698783 - 2.37984i) q^{31} -11.5468 q^{37} +(-7.80647 + 2.29219i) q^{39} +(-9.23919 - 8.00581i) q^{43} +(-4.21124 - 4.86003i) q^{49} +(-8.05587 + 12.5352i) q^{57} +(13.2610 - 1.90664i) q^{61} +(-1.71063 + 1.48227i) q^{63} +(-7.18797 - 3.91575i) q^{67} +(2.15444 + 14.9844i) q^{73} +(-2.43988 - 8.30945i) q^{75} +(4.99958 - 17.0270i) q^{79} +(-5.89375 - 6.80175i) q^{81} +(2.32090 - 2.67847i) q^{91} +(-2.81329 + 3.24671i) q^{93} +4.08778i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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