LMFDB - Embedded newform 804.2.s.a.209.2

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			209.2
Root			$0.235759 + 0.971812i$ of defining polynomial
Character	$\chi$	$=$	804.209
Dual form			804.2.s.a.377.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})$	$q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +(0.903862 - 0.412780i) q^{13} +(-1.11815 - 7.77691i) q^{19} +(-6.48510 + 4.16772i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(-2.80925 - 4.37128i) q^{27} +(-5.98538 - 2.73343i) q^{31} -2.14797 q^{37} +(0.714956 - 1.56553i) q^{39} +(0.833910 + 2.84004i) q^{43} +(12.2900 + 3.60866i) q^{49} +(-10.2846 - 8.91168i) q^{57} +(8.17314 + 12.7177i) q^{61} +(-3.76173 + 12.8113i) q^{63} +(4.89885 + 6.55754i) q^{67} +(13.3792 - 8.59828i) q^{73} +(-7.87764 - 3.59760i) q^{75} +(-3.23526 + 1.47749i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(-4.24334 + 1.24596i) q^{91} +(-10.9352 + 3.21088i) q^{93} -1.50648i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

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

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
$3$	1.30900	−	1.13425i	0.755750	−	0.654861i
$3$	1.30900	−	1.13425i	0.755750	−	0.654861i
$4$		0			0
$5$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$5$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$6$		0			0
$7$	−4.40541	−	0.633402i	−1.66509	−	0.239404i	−0.755570	−	0.655068i	$-0.772640\pi$
$7$	−4.40541	−	0.633402i	−1.66509	−	0.239404i	−0.909518	+	0.415664i	$0.863549\pi$
$8$		0			0
$9$	0.426945	−	2.96946i	0.142315	−	0.989821i
$10$		0			0
$11$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$11$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$12$		0			0
$13$	0.903862	−	0.412780i	0.250686	−	0.114485i	−0.286109	−	0.958197i	$-0.592362\pi$
$13$	0.903862	−	0.412780i	0.250686	−	0.114485i	0.536795	+	0.843712i	$0.319635\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$17$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$18$		0			0
$19$	−1.11815	−	7.77691i	−0.256521	−	1.78414i	−0.557158	−	0.830406i	$-0.688108\pi$
$19$	−1.11815	−	7.77691i	−0.256521	−	1.78414i	0.300637	−	0.953739i	$-0.402801\pi$
$20$		0			0
$21$	−6.48510	+	4.16772i	−1.41517	+	0.909472i
$22$		0			0
$23$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$23$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$24$		0			0
$25$	−2.07708	−	4.54816i	−0.415415	−	0.909632i
$26$		0			0
$27$	−2.80925	−	4.37128i	−0.540641	−	0.841254i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−5.98538	−	2.73343i	−1.07501	−	0.490939i	−0.202369	−	0.979309i	$-0.564864\pi$
$31$	−5.98538	−	2.73343i	−1.07501	−	0.490939i	−0.872636	+	0.488371i	$0.837591\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.14797			−0.353123			−0.176562	−	0.984290i	$-0.556497\pi$
$37$	−2.14797			−0.353123			−0.176562	+	0.984290i	$0.556497\pi$
$38$		0			0
$39$	0.714956	−	1.56553i	0.114485	−	0.250686i
$40$		0			0
$41$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$41$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$42$		0			0
$43$	0.833910	+	2.84004i	0.127170	+	0.433102i	0.998322	−	0.0579125i	$-0.0184445\pi$
$43$	0.833910	+	2.84004i	0.127170	+	0.433102i	−0.871152	+	0.491014i	$0.836626\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$47$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$48$		0			0
$49$	12.2900	+	3.60866i	1.75571	+	0.515524i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$53$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−10.2846	−	8.91168i	−1.36223	−	1.18038i
$58$		0			0
$59$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$59$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$	8.17314	+	12.7177i	1.04646	+	1.62833i	0.734996	+	0.678072i	$0.237184\pi$
$61$	8.17314	+	12.7177i	1.04646	+	1.62833i	0.311467	+	0.950257i	$0.399180\pi$
$62$		0			0
$63$	−3.76173	+	12.8113i	−0.473934	+	1.61407i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.89885	+	6.55754i	0.598490	+	0.801131i
$67$	4.89885	+	6.55754i	0.598490	+	0.801131i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$71$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$72$		0			0
$73$	13.3792	−	8.59828i	1.56592	−	1.00635i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	13.3792	−	8.59828i	1.56592	−	1.00635i	0.980710	−	0.195468i	$-0.0626226\pi$
$74$		0			0
$75$	−7.87764	−	3.59760i	−0.909632	−	0.415415i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.23526	+	1.47749i	−0.363995	+	0.166231i	−0.589013	−	0.808124i	$-0.700483\pi$
$79$	−3.23526	+	1.47749i	−0.363995	+	0.166231i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−8.63544	−	2.53559i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$83$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$89$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$	−4.24334	+	1.24596i	−0.444823	+	0.130612i
$92$		0			0
$93$	−10.9352	+	3.21088i	−1.13393	+	0.332952i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.50648i		−	0.152960i	−0.997071	−	0.0764798i	$-0.975632\pi$
$97$		−	1.50648i		−	0.152960i	0.997071	−	0.0764798i	$-0.0243681\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$101$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$102$		0			0
$103$	−0.241998	+	0.529902i	−0.0238448	+	0.0522128i	−0.921180	−	0.389138i	$-0.872773\pi$
$103$	−0.241998	+	0.529902i	−0.0238448	+	0.0522128i	0.897335	+	0.441350i	$0.145500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$107$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$108$		0			0
$109$	16.8886	−	7.71276i	1.61763	−	0.738749i	0.618733	−	0.785602i	$-0.287646\pi$
$109$	16.8886	−	7.71276i	1.61763	−	0.738749i	0.998902	+	0.0468528i	$0.0149192\pi$
$110$		0			0
$111$	−2.81168	+	2.43633i	−0.266873	+	0.231247i
$112$		0			0
$113$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$113$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.839836	−	2.86022i	−0.0776429	−	0.264427i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.56957	−	10.0060i	−0.415415	−	0.909632i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.52772	−	17.5807i	0.224299	−	1.56003i	−0.497211	−	0.867630i	$-0.665643\pi$
$127$	2.52772	−	17.5807i	0.224299	−	1.56003i	0.721510	−	0.692404i	$-0.243448\pi$
$128$		0			0
$129$	4.31291	+	2.77174i	0.379730	+	0.244038i
$130$		0			0
$131$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$131$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$			34.9687i			3.03217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$137$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$138$		0			0
$139$	12.6697	−	19.7144i	1.07463	−	1.67215i	0.444118	−	0.895969i	$-0.353517\pi$
$139$	12.6697	−	19.7144i	1.07463	−	1.67215i	0.630509	−	0.776182i	$-0.282846\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.1807	−	9.21621i	1.66447	−	0.760140i
$148$		0			0
$149$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$149$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$150$		0			0
$151$	15.8075	−	4.64150i	1.28640	−	0.377720i	0.434141	−	0.900845i	$-0.357052\pi$
$151$	15.8075	−	4.64150i	1.28640	−	0.377720i	0.852256	+	0.523125i	$0.175234\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.7772	+	17.0538i	1.17935	+	1.36104i	0.918381	+	0.395698i	$0.129497\pi$
$157$	14.7772	+	17.0538i	1.17935	+	1.36104i	0.260971	+	0.965347i	$0.415957\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−14.5076			−1.13632			−0.568161	−	0.822917i	$-0.692345\pi$
$163$	−14.5076			−1.13632			−0.568161	+	0.822917i	$0.692345\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$167$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$168$		0			0
$169$	−7.86661	+	9.07855i	−0.605124	+	0.698350i
$170$		0			0
$171$	−23.5706			−1.80249
$172$		0			0
$173$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$173$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$174$		0			0
$175$	6.26955	+	21.3521i	0.473934	+	1.61407i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$179$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$180$		0			0
$181$	7.72107	+	8.91058i	0.573902	+	0.662318i	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	7.72107	+	8.91058i	0.573902	+	0.662318i	−0.392380	+	0.919803i	$0.628348\pi$
$182$		0			0
$183$	25.1236	+	7.37696i	1.85719	+	0.545321i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.60713	+	21.0367i	0.698816	+	1.53019i
$190$		0			0
$191$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$191$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$192$		0			0
$193$	−11.5134	+	25.2108i	−0.828751	+	1.81471i	−0.350068	+	0.936724i	$0.613841\pi$
$193$	−11.5134	+	25.2108i	−0.828751	+	1.81471i	−0.478684	+	0.877987i	$0.658886\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$197$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$198$		0			0
$199$	3.68451	−	25.6263i	0.261188	−	1.81660i	−0.262770	−	0.964858i	$-0.584636\pi$
$199$	3.68451	−	25.6263i	0.261188	−	1.81660i	0.523958	−	0.851744i	$-0.324455\pi$
$200$		0			0
$201$	13.8505	+	3.02726i	0.976937	+	0.213527i
$202$		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
$211$	−0.229341	+	0.264674i	−0.0157885	+	0.0182209i	−0.763589	−	0.645703i	$-0.776565\pi$
$211$	−0.229341	+	0.264674i	−0.0157885	+	0.0182209i	0.747800	+	0.663924i	$0.231110\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.6367	+	15.8330i	1.67245	+	1.07482i
$218$		0			0
$219$	7.76069	−	26.4305i	0.524419	−	1.78601i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−17.1509	+	19.7931i	−1.14851	+	1.32545i	−0.210997	+	0.977487i	$0.567671\pi$
$223$	−17.1509	+	19.7931i	−1.14851	+	1.32545i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		0			0
$225$	−14.3924	+	4.22599i	−0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$227$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$228$		0			0
$229$	−26.0574	−	11.9000i	−1.72192	−	0.786377i	−0.995024	−	0.0996338i	$-0.968233\pi$
$229$	−26.0574	−	11.9000i	−1.72192	−	0.786377i	−0.726900	−	0.686743i	$-0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$233$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.55910	+	5.60364i	−0.166231	+	0.363995i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	26.0993	−	16.7730i	1.68120	−	1.08044i	0.819313	−	0.573346i	$-0.194355\pi$
$241$	26.0993	−	16.7730i	1.68120	−	1.08044i	0.861889	−	0.507097i	$-0.169281\pi$
$242$		0			0
$243$	−14.1798	+	6.47568i	−0.909632	+	0.415415i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.22081	−	6.56770i	−0.268563	−	0.417893i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$251$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$257$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$258$		0			0
$259$	9.46267	+	1.36053i	0.587981	+	0.0845390i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		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
$271$	−19.0827	+	16.5353i	−1.15919	+	1.00445i	−0.159340	+	0.987224i	$0.550937\pi$
$271$	−19.0827	+	16.5353i	−1.15919	+	1.00445i	−0.999852	+	0.0172215i	$0.994518\pi$
$272$		0			0
$273$	−4.14129	+	6.44397i	−0.250642	+	0.390007i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.38366	+	30.4890i	−0.263388	+	1.83191i	0.243508	+	0.969899i	$0.421702\pi$
$277$	−4.38366	+	30.4890i	−0.263388	+	1.83191i	−0.506896	+	0.862007i	$0.669207\pi$
$278$		0			0
$279$	−10.6722	+	16.6063i	−0.638931	+	0.994196i
$280$		0			0
$281$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$281$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$282$		0			0
$283$	−4.63073	−	32.2074i	−0.275268	−	1.91453i	−0.389404	−	0.921067i	$-0.627319\pi$
$283$	−4.63073	−	32.2074i	−0.275268	−	1.91453i	0.114135	−	0.993465i	$-0.463590\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	14.3013	−	9.19089i	0.841254	−	0.540641i
$290$		0			0
$291$	−1.70873	−	1.97197i	−0.100167	−	0.115599i
$292$		0			0
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.87483	−	13.0397i	−0.108063	−	0.751598i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.0262	−	21.9544i	0.572227	−	1.25300i	−0.373376	−	0.927680i	$-0.621800\pi$
$307$	10.0262	−	21.9544i	0.572227	−	1.25300i	0.945603	−	0.325322i	$-0.105473\pi$
$308$		0			0
$309$	0.284268	+	0.968127i	0.0161714	+	0.0550748i
$310$		0			0
$311$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$311$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$312$		0			0
$313$	19.4379	+	16.8430i	1.09869	+	0.952024i	0.999075	−	0.0430013i	$-0.0136920\pi$
$313$	19.4379	+	16.8430i	1.09869	+	0.952024i	0.0996196	+	0.995026i	$0.468237\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$317$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.75478	−	3.25353i	−0.208278	−	0.180474i
$326$		0			0
$327$	13.3589	−	29.2519i	0.738749	−	1.61763i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.61786	+	12.3213i	−0.198856	+	0.677240i	0.798327	+	0.602224i	$0.205719\pi$
$331$	−3.61786	+	12.3213i	−0.198856	+	0.677240i	−0.997182	+	0.0750153i	$0.976099\pi$
$332$		0			0
$333$	−0.917062	+	6.37831i	−0.0502547	+	0.349529i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	36.3103	+	5.22063i	1.97795	+	0.284386i	0.994864	+	0.101218i	$0.0322738\pi$
$337$	36.3103	+	5.22063i	1.97795	+	0.284386i	0.983082	+	0.183168i	$0.0586352\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−23.5171	−	10.7399i	−1.26980	−	0.579900i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$347$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$	−34.8112	−	10.2215i	−1.86340	−	0.547144i	−0.999018	−	0.0443098i	$-0.985891\pi$
$349$	−34.8112	−	10.2215i	−1.86340	−	0.547144i	−0.864383	−	0.502834i	$-0.832291\pi$
$350$		0			0
$351$	−4.34355	−	2.79143i	−0.231842	−	0.148996i
$352$		0			0
$353$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$353$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−40.9997	+	12.0386i	−2.15788	+	0.633610i
$362$		0			0
$363$	−17.3308	−	7.91472i	−0.909632	−	0.415415i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	25.8607	+	22.4084i	1.34992	+	1.16971i	0.969555	+	0.244874i	$0.0787465\pi$
$367$	25.8607	+	22.4084i	1.34992	+	1.16971i	0.380363	+	0.924837i	$0.375799\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	36.6272i		−	1.89649i	−0.317546	−	0.948243i	$-0.602859\pi$
$373$		−	36.6272i		−	1.89649i	0.317546	−	0.948243i	$-0.397141\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.88608	+	1.63430i	−0.0968817	+	0.0839484i	−0.701955	−	0.712221i	$-0.747689\pi$
$379$	−1.88608	+	1.63430i	−0.0968817	+	0.0839484i	0.605073	+	0.796170i	$0.293144\pi$
$380$		0			0
$381$	−16.6322	−	25.8801i	−0.852091	−	1.32588i
$382$		0			0
$383$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$383$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.78943	−	1.26373i	0.446792	−	0.0642389i
$388$		0			0
$389$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$389$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−22.5179	−	14.4714i	−1.13014	−	0.726297i	−0.164551	−	0.986369i	$-0.552617\pi$
$397$	−22.5179	−	14.4714i	−1.13014	−	0.726297i	−0.965589	+	0.260072i	$0.916254\pi$
$398$		0			0
$399$	39.6633	+	45.7739i	1.98565	+	2.29156i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−6.53826			−0.325694
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−31.8733	−	4.58269i	−1.57604	−	0.226600i	−0.701898	−	0.712278i	$-0.747664\pi$
$409$	−31.8733	−	4.58269i	−1.57604	−	0.226600i	−0.874138	+	0.485678i	$0.838573\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.77653	−	40.1766i	−0.282878	−	1.96746i
$418$		0			0
$419$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$419$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$420$		0			0
$421$	−5.78506	−	40.2360i	−0.281946	−	1.96098i	−0.276567	−	0.960995i	$-0.589197\pi$
$421$	−5.78506	−	40.2360i	−0.281946	−	1.96098i	−0.00537983	−	0.999986i	$-0.501712\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−27.9506	−	61.2034i	−1.35263	−	2.96184i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	14.0531	+	6.41784i	0.675349	+	0.308421i	0.723406	−	0.690423i	$-0.242576\pi$
$433$	14.0531	+	6.41784i	0.675349	+	0.308421i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−18.0823			−0.863019			−0.431509	−	0.902108i	$-0.642019\pi$
$439$	−18.0823			−0.863019			−0.431509	+	0.902108i	$0.642019\pi$
$440$		0			0
$441$	15.9629	−	34.9540i	0.760140	−	1.66447i
$442$		0			0
$443$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$443$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$449$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	15.4273	−	24.0054i	0.724840	−	1.12787i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16.5015	−	36.1333i	−0.771908	−	1.69024i	−0.722406	−	0.691469i	$-0.756964\pi$
$457$	−16.5015	−	36.1333i	−0.771908	−	1.69024i	−0.0495022	−	0.998774i	$-0.515763\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	7.77090	+	12.0918i	0.361145	+	0.561952i	0.973516	−	0.228618i	$-0.0734207\pi$
$463$	7.77090	+	12.0918i	0.361145	+	0.561952i	−0.612372	+	0.790570i	$0.709784\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$467$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$468$		0			0
$469$	−17.4279	−	31.9916i	−0.804745	−	1.47723i
$470$		0			0
$471$	38.6867	+	5.56231i	1.78259	+	0.256298i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−33.0481	+	21.2387i	−1.51635	+	0.974501i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$479$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$480$		0			0
$481$	−1.94146	+	0.886637i	−0.0885231	+	0.0404272i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.42884	+	15.0832i	−0.200690	+	0.683487i	0.796225	+	0.605000i	$0.206827\pi$
$487$	−4.42884	+	15.0832i	−0.200690	+	0.683487i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−18.9904	+	16.4553i	−0.858775	+	0.744133i
$490$		0			0
$491$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$491$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	42.0596i		−	1.88285i	−0.337229	−	0.941423i	$-0.609490\pi$
$499$		−	42.0596i		−	1.88285i	0.337229	−	0.941423i	$-0.390510\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$503$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			20.8065i			0.924050i
$508$		0			0
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$	−64.3870	+	29.4046i	−2.84831	+	1.30078i
$512$		0			0
$513$	−30.8539	+	26.7350i	−1.36223	+	1.18038i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$521$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$522$		0			0
$523$	18.9901	+	41.5824i	0.830377	+	1.81827i	0.445212	+	0.895425i	$0.353128\pi$
$523$	18.9901	+	41.5824i	0.830377	+	1.81827i	0.385165	+	0.922848i	$0.374145\pi$
$524$		0			0
$525$	32.4255	+	20.8386i	1.41517	+	0.909472i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.27324	+	22.7659i	−0.142315	+	0.989821i
$530$		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
$541$	3.41260	−	5.31011i	0.146719	−	0.228299i	−0.760114	−	0.649790i	$-0.774857\pi$
$541$	3.41260	−	5.31011i	0.146719	−	0.228299i	0.906833	+	0.421491i	$0.138493\pi$
$542$		0			0
$543$	20.2137	+	2.90629i	0.867453	+	0.124721i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18.6714	+	29.0533i	−0.798333	+	1.24223i	0.168216	+	0.985750i	$0.446199\pi$
$547$	−18.6714	+	29.0533i	−0.798333	+	1.24223i	−0.966549	+	0.256481i	$0.917437\pi$
$548$		0			0
$549$	41.2541	−	18.8401i	1.76068	−	0.804076i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	15.1885	−	4.45975i	0.645881	−	0.189648i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$557$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$558$		0			0
$559$	1.92605	+	2.22278i	0.0814633	+	0.0940136i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$563$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	36.4366	+	16.6400i	1.53019	+	0.698816i
$568$		0			0
$569$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$569$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$570$		0			0
$571$	15.4260	−	17.8025i	0.645558	−	0.745013i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	15.4260	−	17.8025i	0.645558	−	0.745013i	0.980347	+	0.197280i	$0.0632107\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	10.2906	+	35.0467i	0.428404	+	1.45901i	0.837458	+	0.546501i	$0.184041\pi$
$577$	10.2906	+	35.0467i	0.428404	+	1.45901i	−0.409054	+	0.912510i	$0.634141\pi$
$578$		0			0
$579$	13.5244	+	46.0599i	0.562055	+	1.91418i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$587$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$588$		0			0
$589$	−14.5651	+	49.6041i	−0.600144	+	2.04390i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$593$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−24.2437	−	37.7239i	−0.992229	−	1.54394i
$598$		0			0
$599$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$599$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$600$		0			0
$601$	6.31118	−	43.8952i	0.257438	−	1.79052i	−0.293481	−	0.955965i	$-0.594814\pi$
$601$	6.31118	−	43.8952i	0.257438	−	1.79052i	0.550919	−	0.834559i	$-0.314277\pi$
$602$		0			0
$603$	21.5639	−	11.7473i	0.878150	−	0.478385i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	45.5003	+	13.3601i	1.84680	+	0.542270i	0.999945	+	0.0104491i	$0.00332612\pi$
$607$	45.5003	+	13.3601i	1.84680	+	0.542270i	0.846857	+	0.531821i	$0.178492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−15.2309	+	17.5774i	−0.615170	+	0.709944i	−0.974782	−	0.223157i	$-0.928364\pi$
$613$	−15.2309	+	17.5774i	−0.615170	+	0.709944i	0.359612	+	0.933102i	$0.382909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$617$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$	5.31582	+	3.41627i	0.213661	+	0.137312i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	5.31582	+	3.41627i	0.213661	+	0.137312i	−0.429433	+	0.903099i	$0.641287\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−16.3715	+	18.8937i	−0.654861	+	0.755750i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−44.6925	−	20.4104i	−1.77918	−	0.812524i	−0.976312	−	0.216369i	$-0.930579\pi$
$631$	−44.6925	−	20.4104i	−1.77918	−	0.812524i	−0.802869	−	0.596156i	$-0.796694\pi$
$632$		0			0
$633$			0.606587i			0.0241097i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12.5980	−	1.81132i	0.499152	−	0.0717673i
$638$		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
$643$	−24.8979	+	16.0009i	−0.981877	+	0.631014i	−0.929969	−	0.367638i	$-0.880167\pi$
$643$	−24.8979	+	16.0009i	−0.981877	+	0.631014i	−0.0519076	+	0.998652i	$0.516530\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$647$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	50.2080	−	7.21882i	1.96781	−	0.282928i
$652$		0			0
$653$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$653$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−19.8201	−	43.4000i	−0.773257	−	1.69320i
$658$		0			0
$659$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$659$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$660$		0			0
$661$	43.4964	+	6.25383i	1.69181	+	0.243246i	0.919807	−	0.392370i	$-0.128345\pi$
$661$	43.4964	+	6.25383i	1.69181	+	0.243246i	0.772005	+	0.635616i	$0.219254\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			45.3626i			1.75382i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−11.6199	+	10.0687i	−0.447913	+	0.388119i	−0.849403	−	0.527744i	$-0.823038\pi$
$673$	−11.6199	+	10.0687i	−0.447913	+	0.388119i	0.401490	+	0.915863i	$0.368492\pi$
$674$		0			0
$675$	−14.0463	+	21.8564i	−0.540641	+	0.841254i
$676$		0			0
$677$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$677$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$678$		0			0
$679$	−0.954206	+	6.63665i	−0.0366191	+	0.254691i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$683$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−47.6068	+	13.9786i	−1.81631	+	0.533317i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	36.9668	−	23.7571i	1.40628	−	0.903764i	0.406334	−	0.913725i	$-0.366807\pi$
$691$	36.9668	−	23.7571i	1.40628	−	0.903764i	0.999950	+	0.00996078i	$0.00317067\pi$
$692$		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		−0.909632	−	0.415415i	$-0.863636\pi$
$701$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$702$		0			0
$703$	2.40175	+	16.7045i	0.0905837	+	0.630023i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.24089	−	20.2347i	0.347049	−	0.759931i	−0.652948	−	0.757403i	$-0.726468\pi$
$709$	9.24089	−	20.2347i	0.347049	−	0.759931i	0.999997	−	0.00252851i	$-0.000804851\pi$
$710$		0			0
$711$	3.00609	+	10.2378i	0.112737	+	0.383948i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$719$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$720$		0			0
$721$	1.40174	−	2.18115i	0.0522036	−	0.0812304i
$722$		0			0
$723$	15.1391	−	51.5589i	0.563028	−	1.91750i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	31.9760	+	27.7073i	1.18592	+	1.02761i	0.998976	+	0.0452379i	$0.0144046\pi$
$727$	31.9760	+	27.7073i	1.18592	+	1.02761i	0.186947	+	0.982370i	$0.440141\pi$
$728$		0			0
$729$	−11.2162	+	24.5601i	−0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14.7766	−	50.3245i	0.545786	−	1.85878i	0.0344317	−	0.999407i	$-0.489038\pi$
$733$	14.7766	−	50.3245i	0.545786	−	1.85878i	0.511354	−	0.859370i	$-0.329144\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	44.8874	+	6.45384i	1.65121	+	0.237408i	0.904109	−	0.427302i	$-0.140536\pi$
$739$	44.8874	+	6.45384i	1.65121	+	0.237408i	0.747101	+	0.664710i	$0.231445\pi$
$740$		0			0
$741$	−12.9744	−	3.80964i	−0.476628	−	0.139951i
$742$		0			0
$743$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$743$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−43.1905	−	12.6819i	−1.57604	−	0.462768i	−0.627290	−	0.778785i	$-0.715836\pi$
$751$	−43.1905	−	12.6819i	−1.57604	−	0.462768i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	41.4569	−	35.9226i	1.50678	−	1.30563i	0.701654	−	0.712517i	$-0.252445\pi$
$757$	41.4569	−	35.9226i	1.50678	−	1.30563i	0.805121	−	0.593111i	$-0.202100\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$761$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$	−79.2865	+	23.2806i	−2.87036	+	0.842815i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	18.7758	+	16.2693i	0.677071	+	0.586686i	0.924021	−	0.382343i	$-0.124883\pi$
$769$	18.7758	+	16.2693i	0.677071	+	0.586686i	−0.246949	+	0.969028i	$0.579428\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$773$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$774$		0			0
$775$			32.9000i			1.18180i
$776$		0			0
$777$	13.9298	−	8.95212i	0.499728	−	0.321156i
$778$		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
$787$	−13.9459	−	47.4952i	−0.497116	−	1.69302i	−0.700272	−	0.713876i	$-0.746938\pi$
$787$	−13.9459	−	47.4952i	−0.497116	−	1.69302i	0.203156	−	0.979146i	$-0.434880\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.6370	+	8.12129i	0.448752	+	0.288396i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$797$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$798$		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		0.540641	−	0.841254i	$-0.318182\pi$
$809$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$810$		0			0
$811$	46.1477	+	6.63504i	1.62047	+	0.232988i	0.891981	−	0.452074i	$-0.149316\pi$
$811$	46.1477	+	6.63504i	1.62047	+	0.232988i	0.728485	+	0.685062i	$0.240225\pi$
$812$		0			0
$813$	−6.22405	+	43.2892i	−0.218287	+	1.51822i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	21.1543	−	9.66083i	0.740094	−	0.337990i
$818$		0			0
$819$	1.88815	+	13.1324i	0.0659774	+	0.458883i
$820$		0			0
$821$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$821$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$822$		0			0
$823$	4.46901	+	31.0826i	0.155780	+	1.08347i	0.906303	+	0.422628i	$0.138892\pi$
$823$	4.46901	+	31.0826i	0.155780	+	1.08347i	−0.750523	+	0.660844i	$0.770198\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$827$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$828$		0			0
$829$	−14.2808	−	31.2706i	−0.495992	−	1.08607i	−0.977751	−	0.209768i	$-0.932729\pi$
$829$	−14.2808	−	31.2706i	−0.495992	−	1.08607i	0.481759	−	0.876304i	$-0.339998\pi$
$830$		0			0
$831$	28.8440	+	44.8822i	1.00059	+	1.55694i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.86584	+	33.8427i	0.168188	+	1.16977i
$838$		0			0
$839$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$839$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.7930	+	46.9747i	0.473934	+	1.61407i
$848$		0			0
$849$	−42.5929	−	36.9070i	−1.46179	−	1.26664i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	29.1649	+	8.56360i	0.998588	+	0.293212i	0.739877	−	0.672743i	$-0.234884\pi$
$853$	29.1649	+	8.56360i	0.998588	+	0.293212i	0.258712	+	0.965955i	$0.416702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$857$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$858$		0			0
$859$	3.11892	+	6.82947i	0.106416	+	0.233019i	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	3.11892	+	6.82947i	0.106416	+	0.233019i	−0.848932	+	0.528503i	$0.822754\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.29558	−	28.2521i	0.281733	−	0.959493i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	7.13470	+	3.90496i	0.241750	+	0.132315i
$872$		0			0
$873$	−4.47343	−	0.643182i	−0.151403	−	0.0217684i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	36.3128	−	23.3368i	1.22620	−	0.788028i	0.242901	−	0.970051i	$-0.421901\pi$
$877$	36.3128	−	23.3368i	1.22620	−	0.788028i	0.983294	+	0.182023i	$0.0582646\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$881$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$	−29.5799	+	13.5087i	−0.995444	+	0.454604i	−0.845434	−	0.534080i	$-0.820658\pi$
$883$	−29.5799	+	13.5087i	−0.995444	+	0.454604i	−0.150010	+	0.988684i	$0.547931\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$887$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$888$		0			0
$889$	−22.2713	+	75.8491i	−0.746955	+	2.54390i
$890$		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
$903$	−17.2445	−	14.9424i	−0.573861	−	0.497253i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.86473	+	8.46257i	−0.128326	+	0.280995i	−0.962879	−	0.269932i	$-0.912999\pi$
$907$	−3.86473	+	8.46257i	−0.128326	+	0.280995i	0.834553	+	0.550927i	$0.185726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$911$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	53.7879	−	7.73353i	1.77430	−	0.255106i	0.824032	−	0.566543i	$-0.191719\pi$
$919$	53.7879	−	7.73353i	1.77430	−	0.255106i	0.950267	+	0.311437i	$0.100810\pi$
$920$		0			0
$921$	−11.7775	−	40.1105i	−0.388082	−	1.32168i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.46148	+	9.76929i	0.146693	+	0.321212i
$926$		0			0
$927$	1.47020	+	0.944843i	0.0482879	+	0.0310327i
$928$		0			0
$929$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$929$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$930$		0			0
$931$	14.3222	−	99.6131i	0.469391	−	3.26469i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	45.8555i		−	1.49803i	−0.662552	−	0.749016i	$-0.730527\pi$
$937$		−	45.8555i		−	1.49803i	0.662552	−	0.749016i	$-0.269473\pi$
$938$		0			0
$939$	44.5484			1.45378
$940$		0			0
$941$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$941$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$	8.54374	−	13.2943i	0.277342	−	0.431552i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$953$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	8.05243	+	9.29300i	0.259756	+	0.299774i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−31.4087			−1.01003			−0.505017	−	0.863109i	$-0.668514\pi$
$967$	−31.4087			−1.01003			−0.505017	+	0.863109i	$0.668514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$971$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$972$		0			0
$973$	−68.3021	+	78.8249i	−2.18967	+	2.52701i
$974$		0			0
$975$	−8.60532			−0.275591
$976$		0			0
$977$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$977$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−15.6923	−	53.4430i	−0.501016	−	1.70630i
$982$		0			0
$983$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$983$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−3.97137	+	13.5252i	−0.126155	+	0.429644i	−0.998213	−	0.0597587i	$-0.980967\pi$
$991$	−3.97137	+	13.5252i	−0.126155	+	0.429644i	0.872058	+	0.489402i	$0.162785\pi$
$992$		0			0
$993$	9.23969	+	20.2321i	0.293213	+	0.642046i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−22.2964	+	48.8222i	−0.706133	+	1.54621i	0.126241	+	0.992000i	$0.459709\pi$
$997$	−22.2964	+	48.8222i	−0.706133	+	1.54621i	−0.832373	+	0.554215i	$0.813018\pi$
$998$		0			0
$999$	6.03418	+	9.38936i	0.190913	+	0.297066i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.209.2	✓	20
3.2	odd	2	CM	804.2.s.a.209.2	✓	20
67.42	odd	22	inner	804.2.s.a.377.2	yes	20
201.176	even	22	inner	804.2.s.a.377.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.209.2	✓	20	1.1	even	1	trivial
804.2.s.a.209.2	✓	20	3.2	odd	2	CM
804.2.s.a.377.2	yes	20	67.42	odd	22	inner
804.2.s.a.377.2	yes	20	201.176	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)

\(f(q)\)	\(=\)	\(q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\)	\(q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +(0.903862 - 0.412780i) q^{13} +(-1.11815 - 7.77691i) q^{19} +(-6.48510 + 4.16772i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(-2.80925 - 4.37128i) q^{27} +(-5.98538 - 2.73343i) q^{31} -2.14797 q^{37} +(0.714956 - 1.56553i) q^{39} +(0.833910 + 2.84004i) q^{43} +(12.2900 + 3.60866i) q^{49} +(-10.2846 - 8.91168i) q^{57} +(8.17314 + 12.7177i) q^{61} +(-3.76173 + 12.8113i) q^{63} +(4.89885 + 6.55754i) q^{67} +(13.3792 - 8.59828i) q^{73} +(-7.87764 - 3.59760i) q^{75} +(-3.23526 + 1.47749i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(-4.24334 + 1.24596i) q^{91} +(-10.9352 + 3.21088i) q^{93} -1.50648i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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