LMFDB - Embedded newform 804.2.s.a.161.1

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			161.1
Root			$-0.786053 - 0.618159i$ of defining polynomial
Character	$\chi$	$=$	804.161
Dual form			804.2.s.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57553 - 0.719520i) q^{3} +(-3.36481 + 2.91562i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})$	$q+(-1.57553 - 0.719520i) q^{3} +(-3.36481 + 2.91562i) q^{7} +(1.96458 + 2.26725i) q^{9} +(1.63571 - 2.54522i) q^{13} +(4.23982 - 4.89301i) q^{19} +(7.39920 - 2.17260i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(-1.46393 - 4.98567i) q^{27} +(-6.00018 - 9.33646i) q^{31} +12.1582 q^{37} +(-4.40845 + 2.83314i) q^{39} +(-12.9347 + 1.85973i) q^{43} +(1.82487 - 12.6923i) q^{49} +(-10.2006 + 4.65844i) q^{57} +(-4.39867 - 14.9805i) q^{61} +(-13.2209 - 1.90088i) q^{63} +(5.79361 + 5.78222i) q^{67} +(5.46060 - 1.60338i) q^{73} +(4.68209 + 7.28547i) q^{75} +(9.04681 - 14.0771i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(1.91704 + 13.3333i) q^{91} +(2.73569 + 19.0271i) q^{93} -6.97869i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{17}{22}\right)$	$1$

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
$3$	−1.57553	−	0.719520i	−0.909632	−	0.415415i
$3$	−1.57553	−	0.719520i	−0.909632	−	0.415415i
$4$		0			0
$5$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$5$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$6$		0			0
$7$	−3.36481	+	2.91562i	−1.27178	+	1.10200i	−0.281995	+	0.959416i	$0.590996\pi$
$7$	−3.36481	+	2.91562i	−1.27178	+	1.10200i	−0.989782	+	0.142586i	$0.954458\pi$
$8$		0			0
$9$	1.96458	+	2.26725i	0.654861	+	0.755750i
$10$		0			0
$11$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$11$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$12$		0			0
$13$	1.63571	−	2.54522i	0.453665	−	0.705917i	−0.536795	−	0.843712i	$-0.680365\pi$
$13$	1.63571	−	2.54522i	0.453665	−	0.705917i	0.990461	+	0.137795i	$0.0440016\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$17$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$18$		0			0
$19$	4.23982	−	4.89301i	0.972680	−	1.12253i	−0.0197599	−	0.999805i	$-0.506290\pi$
$19$	4.23982	−	4.89301i	0.972680	−	1.12253i	0.992440	−	0.122728i	$-0.0391644\pi$
$20$		0			0
$21$	7.39920	−	2.17260i	1.61464	−	0.474100i
$22$		0			0
$23$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$23$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$24$		0			0
$25$	−4.20627	−	2.70320i	−0.841254	−	0.540641i
$26$		0			0
$27$	−1.46393	−	4.98567i	−0.281733	−	0.959493i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−6.00018	−	9.33646i	−1.07766	−	1.67688i	−0.607589	−	0.794252i	$-0.707863\pi$
$31$	−6.00018	−	9.33646i	−1.07766	−	1.67688i	−0.470075	−	0.882626i	$-0.655773\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	12.1582			1.99880			0.999398	−	0.0346855i	$-0.0110430\pi$
$37$	12.1582			1.99880			0.999398	+	0.0346855i	$0.0110430\pi$
$38$		0			0
$39$	−4.40845	+	2.83314i	−0.705917	+	0.453665i
$40$		0			0
$41$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$41$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$42$		0			0
$43$	−12.9347	+	1.85973i	−1.97252	+	0.283606i	−0.974198	+	0.225693i	$0.927535\pi$
$43$	−12.9347	+	1.85973i	−1.97252	+	0.283606i	−0.998322	+	0.0579125i	$0.981556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$47$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$48$		0			0
$49$	1.82487	−	12.6923i	0.260696	−	1.81318i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$53$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−10.2006	+	4.65844i	−1.35110	+	0.617026i
$58$		0			0
$59$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$59$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$60$		0			0
$61$	−4.39867	−	14.9805i	−0.563192	−	1.91806i	−0.311467	−	0.950257i	$-0.600820\pi$
$61$	−4.39867	−	14.9805i	−0.563192	−	1.91806i	−0.251725	−	0.967799i	$-0.580998\pi$
$62$		0			0
$63$	−13.2209	−	1.90088i	−1.66567	−	0.239488i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.79361	+	5.78222i	0.707802	+	0.706411i
$67$	5.79361	+	5.78222i	0.707802	+	0.706411i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$71$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$72$		0			0
$73$	5.46060	−	1.60338i	0.639115	−	0.187661i	0.0539089	−	0.998546i	$-0.482832\pi$
$73$	5.46060	−	1.60338i	0.639115	−	0.187661i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	4.68209	+	7.28547i	0.540641	+	0.841254i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.04681	−	14.0771i	1.01785	−	1.58380i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	9.04681	−	14.0771i	1.01785	−	1.58380i	0.792829	−	0.609445i	$-0.208608\pi$
$80$		0			0
$81$	−1.28083	+	8.90839i	−0.142315	+	0.989821i
$82$		0			0
$83$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$83$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$89$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$	1.91704	+	13.3333i	0.200960	+	1.39771i
$92$		0			0
$93$	2.73569	+	19.0271i	0.283677	+	1.97302i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	6.97869i		−	0.708578i	−0.935136	−	0.354289i	$-0.884723\pi$
$97$		−	6.97869i		−	0.708578i	0.935136	−	0.354289i	$-0.115277\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$101$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$102$		0			0
$103$	8.81568	−	5.66549i	0.868635	−	0.558238i	−0.0286999	−	0.999588i	$-0.509137\pi$
$103$	8.81568	−	5.66549i	0.868635	−	0.558238i	0.897335	+	0.441350i	$0.145500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$107$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$108$		0			0
$109$	10.4772	−	16.3028i	1.00353	−	1.56153i	0.188534	−	0.982067i	$-0.439626\pi$
$109$	10.4772	−	16.3028i	1.00353	−	1.56153i	0.814999	−	0.579462i	$-0.196737\pi$
$110$		0			0
$111$	−19.1556	−	8.74807i	−1.81817	−	0.830330i
$112$		0			0
$113$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$113$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	8.98414	−	1.29172i	0.830584	−	0.119420i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.25379	−	5.94705i	−0.841254	−	0.540641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.33874	+	8.46936i	0.651208	+	0.751534i	0.981315	−	0.192410i	$-0.0616302\pi$
$127$	7.33874	+	8.46936i	0.651208	+	0.751534i	−0.330107	+	0.943944i	$0.607085\pi$
$128$		0			0
$129$	21.7171	+	6.37671i	1.91208	+	0.561438i
$130$		0			0
$131$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$131$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$132$		0			0
$133$			28.8257i			2.49951i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$137$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$138$		0			0
$139$	−6.12749	+	20.8683i	−0.519727	+	1.77003i	0.110782	+	0.993845i	$0.464665\pi$
$139$	−6.12749	+	20.8683i	−0.519727	+	1.77003i	−0.630509	+	0.776182i	$0.717154\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−12.0075	+	18.6840i	−0.990360	+	1.54103i
$148$		0			0
$149$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$149$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$150$		0			0
$151$	0.426045	+	2.96321i	0.0346711	+	0.241142i	0.999786	−	0.0206838i	$-0.00658434\pi$
$151$	0.426045	+	2.96321i	0.0346711	+	0.241142i	−0.965115	+	0.261826i	$0.915675\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.3339	+	22.6280i	−0.824732	+	1.80591i	−0.302363	+	0.953193i	$0.597776\pi$
$157$	−10.3339	+	22.6280i	−0.824732	+	1.80591i	−0.522369	+	0.852719i	$0.674952\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−25.1404			−1.96915			−0.984575	−	0.174966i	$-0.944019\pi$
$163$	−25.1404			−1.96915			−0.984575	+	0.174966i	$0.944019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$167$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$168$		0			0
$169$	1.59781	+	3.49871i	0.122908	+	0.269132i
$170$		0			0
$171$	19.4231			1.48532
$172$		0			0
$173$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$173$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$174$		0			0
$175$	22.0348	−	3.16813i	1.66567	−	0.239488i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$179$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$180$		0			0
$181$	1.86882	−	4.09214i	0.138908	−	0.304166i	−0.827374	−	0.561651i	$-0.810166\pi$
$181$	1.86882	−	4.09214i	0.138908	−	0.304166i	0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$	−3.84853	+	26.7671i	−0.284492	+	1.97868i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	19.4622	+	12.5076i	1.41566	+	0.909792i
$190$		0			0
$191$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$191$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$192$		0			0
$193$	−18.7210	+	12.0312i	−1.34757	+	0.866028i	−0.997498	−	0.0706968i	$-0.977478\pi$
$193$	−18.7210	+	12.0312i	−1.34757	+	0.866028i	−0.350068	+	0.936724i	$0.613841\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$197$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$198$		0			0
$199$	0.364085	+	0.420177i	0.0258093	+	0.0297855i	0.768507	−	0.639841i	$-0.221000\pi$
$199$	0.364085	+	0.420177i	0.0258093	+	0.0297855i	−0.742698	+	0.669626i	$0.766454\pi$
$200$		0			0
$201$	−4.96758	−	13.2787i	−0.350386	−	0.936605i
$202$		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
$211$	−9.21538	−	20.1789i	−0.634413	−	1.38917i	−0.904558	−	0.426350i	$-0.859799\pi$
$211$	−9.21538	−	20.1789i	−0.634413	−	1.38917i	0.270146	−	0.962819i	$-0.412928\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	47.4111	+	13.9211i	3.21847	+	0.945029i
$218$		0			0
$219$	−9.75699	−	1.40284i	−0.659316	−	0.0947954i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.75894	−	19.1794i	−0.586542	−	1.28435i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−8.75894	−	19.1794i	−0.586542	−	1.28435i	0.350967	−	0.936388i	$-0.385853\pi$
$224$		0			0
$225$	−2.13472	−	14.8473i	−0.142315	−	0.989821i
$226$		0			0
$227$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$227$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$228$		0			0
$229$	3.02274	+	4.70347i	0.199748	+	0.310814i	0.926649	−	0.375929i	$-0.122676\pi$
$229$	3.02274	+	4.70347i	0.199748	+	0.310814i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$233$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−24.3823	+	15.6695i	−1.58380	+	1.01785i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	3.07530	−	0.902989i	0.198097	−	0.0581666i	−0.181179	−	0.983450i	$-0.557991\pi$
$241$	3.07530	−	0.902989i	0.198097	−	0.0581666i	0.379276	+	0.925284i	$0.376173\pi$
$242$		0			0
$243$	8.42776	−	13.1138i	0.540641	−	0.841254i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.51866	−	18.7948i	−0.351144	−	1.19589i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$251$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$257$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$258$		0			0
$259$	−40.9100	+	35.4487i	−2.54203	+	2.20268i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$		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
$271$	8.93119	+	4.07874i	0.542531	+	0.247766i	0.667779	−	0.744360i	$-0.267245\pi$
$271$	8.93119	+	4.07874i	0.542531	+	0.247766i	−0.125247	+	0.992126i	$0.539972\pi$
$272$		0			0
$273$	6.57322	−	22.3863i	0.397830	−	1.35488i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.8956	−	18.3445i	−0.955073	−	1.10221i	−0.994682	−	0.102998i	$-0.967156\pi$
$277$	−15.8956	−	18.3445i	−0.955073	−	1.10221i	0.0396081	−	0.999215i	$-0.487389\pi$
$278$		0			0
$279$	9.38024	−	31.9461i	0.561580	−	1.91257i
$280$		0			0
$281$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$281$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$282$		0			0
$283$	18.1894	−	20.9916i	1.08125	−	1.24782i	0.114135	−	0.993465i	$-0.463590\pi$
$283$	18.1894	−	20.9916i	1.08125	−	1.24782i	0.967110	−	0.254358i	$-0.0818643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.3114	+	4.78945i	−0.959493	+	0.281733i
$290$		0			0
$291$	−5.02131	+	10.9951i	−0.294354	+	0.644546i
$292$		0			0
$293$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$293$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	38.1005	−	43.9703i	2.19607	−	2.53440i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	27.8763	−	17.9150i	1.59098	−	1.02246i	0.619610	−	0.784910i	$-0.287291\pi$
$307$	27.8763	−	17.9150i	1.59098	−	1.02246i	0.971374	−	0.237553i	$-0.0763455\pi$
$308$		0			0
$309$	−17.9658	+	2.58309i	−1.02204	+	0.146947i
$310$		0			0
$311$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$311$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$312$		0			0
$313$	−31.6320	+	14.4459i	−1.78795	+	0.816528i	−0.817227	+	0.576316i	$0.804490\pi$
$313$	−31.6320	+	14.4459i	−1.78795	+	0.816528i	−0.970720	+	0.240212i	$0.922783\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$317$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−13.7605	+	6.28421i	−0.763295	+	0.348585i
$326$		0			0
$327$	−28.2373	+	18.1470i	−1.56153	+	1.00353i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	21.6900	+	3.11855i	1.19219	+	0.171411i	0.709708	−	0.704496i	$-0.248827\pi$
$331$	21.6900	+	3.11855i	1.19219	+	0.171411i	0.482481	+	0.875907i	$0.339736\pi$
$332$		0			0
$333$	23.8858	+	27.5657i	1.30893	+	1.51059i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.08244	+	4.40396i	−0.276858	+	0.239899i	−0.782211	−	0.623014i	$-0.785908\pi$
$337$	−5.08244	+	4.40396i	−0.276858	+	0.239899i	0.505352	+	0.862913i	$0.331363\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	14.0159	+	21.8092i	0.756789	+	1.17759i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$347$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$348$		0			0
$349$	−5.03054	+	34.9882i	−0.269279	+	1.87287i	0.186034	+	0.982543i	$0.440437\pi$
$349$	−5.03054	+	34.9882i	−0.269279	+	1.87287i	−0.455313	+	0.890332i	$0.650472\pi$
$350$		0			0
$351$	−15.0842	−	4.42912i	−0.805135	−	0.236409i
$352$		0			0
$353$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$353$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$359$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$360$		0			0
$361$	−3.26151	−	22.6843i	−0.171658	−	1.19391i
$362$		0			0
$363$	10.3006	+	16.0280i	0.540641	+	0.841254i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	32.2325	−	14.7201i	1.68252	−	0.768382i	0.683253	−	0.730182i	$-0.260565\pi$
$367$	32.2325	−	14.7201i	1.68252	−	0.768382i	0.999270	−	0.0382006i	$-0.0121626\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			24.1815i			1.25207i	0.779795	+	0.626035i	$0.215323\pi$
$373$			24.1815i			1.25207i	−0.779795	+	0.626035i	$0.784677\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−17.1989	−	7.85448i	−0.883449	−	0.403458i	−0.0785782	−	0.996908i	$-0.525038\pi$
$379$	−17.1989	−	7.85448i	−0.883449	−	0.403458i	−0.804871	+	0.593450i	$0.797765\pi$
$380$		0			0
$381$	−5.46853	−	18.6241i	−0.280161	−	0.954141i
$382$		0			0
$383$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$383$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−29.6277	−	25.6726i	−1.50606	−	1.30501i
$388$		0			0
$389$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$389$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−15.0970	−	4.43288i	−0.757697	−	0.222480i	−0.120007	−	0.992773i	$-0.538292\pi$
$397$	−15.0970	−	4.43288i	−0.757697	−	0.222480i	−0.637690	+	0.770293i	$0.720110\pi$
$398$		0			0
$399$	20.7407	−	45.4158i	1.03833	−	2.27363i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−33.5779			−1.67264
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.95685	−	1.69562i	0.0967601	−	0.0838431i	−0.605137	−	0.796121i	$-0.706882\pi$
$409$	1.95685	−	1.69562i	0.0967601	−	0.0838431i	0.701898	+	0.712278i	$0.252336\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.6692	−	28.4698i	1.20806	−	1.39417i
$418$		0			0
$419$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$419$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$420$		0			0
$421$	0.144573	−	0.166847i	0.00704607	−	0.00813160i	−0.752216	−	0.658917i	$-0.771015\pi$
$421$	0.144573	−	0.166847i	0.00704607	−	0.00813160i	0.759262	+	0.650785i	$0.225560\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	58.4781	+	37.5816i	2.82996	+	1.81870i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−19.4909	−	30.3285i	−0.936674	−	1.45749i	−0.888617	−	0.458649i	$-0.848333\pi$
$433$	−19.4909	−	30.3285i	−0.936674	−	1.45749i	−0.0480569	−	0.998845i	$-0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	26.8748			1.28266			0.641331	−	0.767264i	$-0.278382\pi$
$439$	26.8748			1.28266			0.641331	+	0.767264i	$0.278382\pi$
$440$		0			0
$441$	32.3616	−	20.7976i	1.54103	−	0.990360i
$442$		0			0
$443$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$443$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$449$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.46084	−	4.97517i	0.0686363	−	0.233754i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−20.9197	−	13.4443i	−0.978583	−	0.628897i	−0.0495022	−	0.998774i	$-0.515763\pi$
$457$	−20.9197	−	13.4443i	−0.978583	−	0.628897i	−0.929081	+	0.369877i	$0.879400\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$461$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$462$		0			0
$463$	−2.77184	−	9.44002i	−0.128818	−	0.438715i	0.869673	−	0.493629i	$-0.164330\pi$
$463$	−2.77184	−	9.44002i	−0.128818	−	0.438715i	−0.998491	+	0.0549137i	$0.982512\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$467$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$468$		0			0
$469$	−36.3532	−	2.56408i	−1.67863	−	0.118398i
$470$		0			0
$471$	32.5626	−	28.2157i	1.50041	−	1.30011i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−31.0606	+	9.12022i	−1.42516	+	0.418464i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$479$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$480$		0			0
$481$	19.8873	−	30.9453i	0.906785	−	1.41098i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−43.5968	−	6.26828i	−1.97556	−	0.284043i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−43.5968	−	6.26828i	−1.97556	−	0.284043i	−0.978645	−	0.205556i	$-0.934100\pi$
$488$		0			0
$489$	39.6094	+	18.0890i	1.79120	+	0.818014i
$490$		0			0
$491$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$491$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			8.92718i			0.399636i	0.979833	+	0.199818i	$0.0640350\pi$
$499$			8.92718i			0.399636i	−0.979833	+	0.199818i	$0.935965\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$503$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	6.66198i		−	0.295869i
$508$		0			0
$509$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$509$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$	−13.6990	+	21.3161i	−0.606009	+	0.942969i
$512$		0			0
$513$	−30.6017	−	13.9753i	−1.35110	−	0.617026i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$521$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$522$		0			0
$523$	33.0384	+	21.2325i	1.44467	+	0.928431i	0.999455	+	0.0330064i	$0.0105082\pi$
$523$	33.0384	+	21.2325i	1.44467	+	0.928431i	0.445212	+	0.895425i	$0.353128\pi$
$524$		0			0
$525$	−36.9960	−	10.8630i	−1.61464	−	0.474100i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−15.0618	−	17.3822i	−0.654861	−	0.755750i
$530$		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
$541$	12.5500	−	42.7414i	0.539567	−	1.83760i	−0.00668980	−	0.999978i	$-0.502129\pi$
$541$	12.5500	−	42.7414i	0.539567	−	1.83760i	0.546257	−	0.837618i	$-0.316052\pi$
$542$		0			0
$543$	−5.88875	+	5.10263i	−0.252710	+	0.218975i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.1268	+	41.3002i	−0.518506	+	1.76587i	0.116316	+	0.993212i	$0.462891\pi$
$547$	−12.1268	+	41.3002i	−0.518506	+	1.76587i	−0.634822	+	0.772658i	$0.718927\pi$
$548$		0			0
$549$	25.3230	−	39.4033i	1.08076	−	1.68169i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	10.6028	+	73.7439i	0.450876	+	3.13591i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$557$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$558$		0			0
$559$	−16.4240	+	35.9636i	−0.694662	+	1.52110i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$563$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−21.6638	−	33.7095i	−0.909792	−	1.41566i
$568$		0			0
$569$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$569$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$570$		0			0
$571$	14.2558	+	31.2159i	0.596588	+	1.30635i	0.931378	+	0.364054i	$0.118607\pi$
$571$	14.2558	+	31.2159i	0.596588	+	1.30635i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.332013	+	0.0477362i	−0.0138219	+	0.00198729i	−0.149222	−	0.988804i	$-0.547677\pi$
$577$	−0.332013	+	0.0477362i	−0.0138219	+	0.00198729i	0.135400	+	0.990791i	$0.456768\pi$
$578$		0			0
$579$	38.1522	−	5.48545i	1.58555	−	0.227968i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$587$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$588$		0			0
$589$	−71.1230	−	10.2259i	−2.93057	−	0.421353i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$593$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.271301	−	0.923967i	−0.0111036	−	0.0378155i
$598$		0			0
$599$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$599$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$600$		0			0
$601$	−8.66735	−	10.0027i	−0.353549	−	0.408017i	0.550919	−	0.834559i	$-0.314277\pi$
$601$	−8.66735	−	10.0027i	−0.353549	−	0.408017i	−0.904468	+	0.426542i	$0.859732\pi$
$602$		0			0
$603$	−1.72771	+	24.4952i	−0.0703577	+	0.997522i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.70746	−	46.6514i	0.272247	−	1.89352i	−0.152650	−	0.988280i	$-0.548781\pi$
$607$	6.70746	−	46.6514i	0.272247	−	1.89352i	0.424897	−	0.905242i	$-0.360310\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	20.5327	+	44.9603i	0.829308	+	1.81593i	0.469696	+	0.882828i	$0.344364\pi$
$613$	20.5327	+	44.9603i	0.829308	+	1.81593i	0.359612	+	0.933102i	$0.382909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$617$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	39.7606	+	11.6748i	1.59812	+	0.469249i	0.955021	−	0.296538i	$-0.0958321\pi$
$619$	39.7606	+	11.6748i	1.59812	+	0.469249i	0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	10.3854	+	22.7408i	0.415415	+	0.909632i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−27.0845	−	42.1444i	−1.07822	−	1.67774i	−0.602390	−	0.798202i	$-0.705785\pi$
$631$	−27.0845	−	42.1444i	−1.07822	−	1.67774i	−0.475828	−	0.879538i	$-0.657851\pi$
$632$		0			0
$633$			38.4230i			1.52718i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−29.3197	−	25.4056i	−1.16169	−	1.00661i
$638$		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
$643$	−24.1475	+	7.09036i	−0.952286	+	0.279616i	−0.720738	−	0.693207i	$-0.756197\pi$
$643$	−24.1475	+	7.09036i	−0.952286	+	0.279616i	−0.231548	+	0.972824i	$0.574379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$647$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−64.6809	−	56.0464i	−2.53505	−	2.19663i
$652$		0			0
$653$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$653$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	14.3630	+	9.23057i	0.560356	+	0.360119i
$658$		0			0
$659$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$659$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$660$		0			0
$661$	32.1520	−	27.8598i	1.25057	−	1.08362i	0.257474	−	0.966285i	$-0.417110\pi$
$661$	32.1520	−	27.8598i	1.25057	−	1.08362i	0.993092	−	0.117337i	$-0.0374357\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			36.5199i			1.41194i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	24.9073	+	11.3748i	0.960107	+	0.438466i	0.832908	−	0.553411i	$-0.186674\pi$
$673$	24.9073	+	11.3748i	0.960107	+	0.438466i	0.127198	+	0.991877i	$0.459401\pi$
$674$		0			0
$675$	−7.31963	+	24.9284i	−0.281733	+	0.959493i
$676$		0			0
$677$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$677$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$678$		0			0
$679$	20.3472	+	23.4819i	0.780855	+	0.901154i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$683$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.37817	−	9.58538i	−0.0525804	−	0.365705i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	33.4120	−	9.81065i	1.27105	−	0.373215i	0.424455	−	0.905449i	$-0.360466\pi$
$691$	33.4120	−	9.81065i	1.27105	−	0.373215i	0.846597	+	0.532234i	$0.178648\pi$
$692$		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		−0.540641	−	0.841254i	$-0.681818\pi$
$701$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$702$		0			0
$703$	51.5485	−	59.4902i	1.94419	−	2.24372i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−29.4234	+	18.9093i	−1.10502	+	0.710153i	−0.960202	−	0.279306i	$-0.909896\pi$
$709$	−29.4234	+	18.9093i	−1.10502	+	0.710153i	−0.144817	+	0.989458i	$0.546259\pi$
$710$		0			0
$711$	49.6895	−	7.14428i	1.86350	−	0.267931i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$719$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$720$		0			0
$721$	−13.1446	+	44.7665i	−0.489532	+	1.66719i
$722$		0			0
$723$	−5.49494	−	0.790053i	−0.204359	−	0.0293824i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−43.6524	+	19.9354i	−1.61898	+	0.739362i	−0.998976	−	0.0452379i	$-0.985595\pi$
$727$	−43.6524	+	19.9354i	−1.61898	+	0.739362i	−0.620002	+	0.784600i	$0.712868\pi$
$728$		0			0
$729$	−22.7138	+	14.5973i	−0.841254	+	0.540641i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	50.8753	+	7.31477i	1.87912	+	0.270177i	0.984334	−	0.176312i	$-0.0564167\pi$
$733$	50.8753	+	7.31477i	1.87912	+	0.270177i	0.894789	+	0.446489i	$0.147326\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.38022	+	5.52850i	−0.234700	+	0.203369i	−0.764268	−	0.644899i	$-0.776900\pi$
$739$	−6.38022	+	5.52850i	−0.234700	+	0.203369i	0.529568	+	0.848268i	$0.322354\pi$
$740$		0			0
$741$	−4.82845	+	33.5826i	−0.177377	+	1.23369i
$742$		0			0
$743$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$743$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.38669	−	9.64466i	0.0506011	−	0.351939i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	1.38669	−	9.64466i	0.0506011	−	0.351939i	0.999355	−	0.0359215i	$-0.0114366\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	35.6648	+	16.2875i	1.29626	+	0.591981i	0.939606	−	0.342257i	$-0.111191\pi$
$757$	35.6648	+	16.2875i	1.29626	+	0.591981i	0.356651	+	0.934238i	$0.383919\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$761$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$	12.2792	+	85.4035i	0.444536	+	3.09181i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−41.4298	+	18.9204i	−1.49400	+	0.682286i	−0.984046	−	0.177916i	$-0.943064\pi$
$769$	−41.4298	+	18.9204i	−1.49400	+	0.682286i	−0.509953	+	0.860202i	$0.670337\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$773$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$774$		0			0
$775$			55.4914i			1.99331i
$776$		0			0
$777$	89.9610	−	26.4149i	3.22733	−	0.947630i
$778$		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
$787$	32.8522	−	4.72344i	1.17106	−	0.168372i	0.470784	−	0.882248i	$-0.343971\pi$
$787$	32.8522	−	4.72344i	1.17106	−	0.168372i	0.700272	+	0.713876i	$0.253062\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−45.3236	−	13.3082i	−1.60949	−	0.472589i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$797$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$798$		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		0.281733	−	0.959493i	$-0.409091\pi$
$809$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$810$		0			0
$811$	−29.4882	+	25.5517i	−1.03547	+	0.897240i	−0.994791	−	0.101932i	$-0.967497\pi$
$811$	−29.4882	+	25.5517i	−1.03547	+	0.897240i	−0.0406786	+	0.999172i	$0.512952\pi$
$812$		0			0
$813$	−11.1366	−	12.8523i	−0.390578	−	0.450751i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−45.7410	+	71.1744i	−1.60028	+	2.49008i
$818$		0			0
$819$	−26.4637	+	30.5408i	−0.924717	+	1.06718i
$820$		0			0
$821$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$821$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$822$		0			0
$823$	34.3008	−	39.5852i	1.19565	−	1.37985i	0.289346	−	0.957224i	$-0.406562\pi$
$823$	34.3008	−	39.5852i	1.19565	−	1.37985i	0.906303	−	0.422628i	$-0.138892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$827$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$828$		0			0
$829$	23.3380	+	14.9984i	0.810562	+	0.520917i	0.879047	−	0.476735i	$-0.158180\pi$
$829$	23.3380	+	14.9984i	0.810562	+	0.520917i	−0.0684845	+	0.997652i	$0.521816\pi$
$830$		0			0
$831$	11.8447	+	40.3395i	0.410889	+	1.39936i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−37.7647	+	43.5828i	−1.30534	+	1.50644i
$838$		0			0
$839$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$839$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	48.4766	−	6.96988i	1.66567	−	0.239488i
$848$		0			0
$849$	−43.7618	+	19.9853i	−1.50190	+	0.685895i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	6.41086	−	44.5885i	0.219504	−	1.52668i	−0.520373	−	0.853939i	$-0.674207\pi$
$853$	6.41086	−	44.5885i	0.219504	−	1.52668i	0.739877	−	0.672743i	$-0.234884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$857$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$858$		0			0
$859$	47.5093	+	30.5324i	1.62100	+	1.04175i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	47.5093	+	30.5324i	1.62100	+	1.04175i	0.665648	+	0.746266i	$0.268155\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$863$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.1452	+	4.19044i	0.989821	+	0.142315i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	24.1937	−	5.28796i	0.819773	−	0.179176i
$872$		0			0
$873$	15.8224	−	13.7102i	0.535508	−	0.464020i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−13.8039	+	4.05319i	−0.466124	+	0.136866i	−0.506357	−	0.862324i	$-0.669008\pi$
$877$	−13.8039	+	4.05319i	−0.466124	+	0.136866i	0.0402330	+	0.999190i	$0.487190\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$881$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$882$		0			0
$883$	−9.29179	+	14.4583i	−0.312694	+	0.486561i	−0.961656	−	0.274260i	$-0.911567\pi$
$883$	−9.29179	+	14.4583i	−0.312694	+	0.486561i	0.648962	+	0.760821i	$0.275203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$887$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$888$		0			0
$889$	−49.3869	−	7.10076i	−1.65638	−	0.238152i
$890$		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
$903$	−91.6658	+	41.8624i	−3.05045	+	1.39309i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	42.9604	−	27.6089i	1.42648	−	0.916740i	0.426551	−	0.904464i	$-0.359729\pi$
$907$	42.9604	−	27.6089i	1.42648	−	0.916740i	0.999925	−	0.0122762i	$-0.00390773\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$911$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−11.5357	−	9.99575i	−0.380528	−	0.329729i	0.443504	−	0.896272i	$-0.353735\pi$
$919$	−11.5357	−	9.99575i	−0.380528	−	0.329729i	−0.824032	+	0.566543i	$0.808281\pi$
$920$		0			0
$921$	−56.8101	+	8.16806i	−1.87196	+	0.269147i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−51.1407	−	32.8661i	−1.68149	−	1.08063i
$926$		0			0
$927$	30.1642	+	8.85701i	0.990723	+	0.290902i
$928$		0			0
$929$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$929$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$930$		0			0
$931$	−54.3663	−	62.7420i	−1.78178	−	2.05629i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	17.8475i		−	0.583051i	−0.956563	−	0.291526i	$-0.905837\pi$
$937$		−	17.8475i		−	0.583051i	0.956563	−	0.291526i	$-0.0941629\pi$
$938$		0			0
$939$	60.2312			1.96557
$940$		0			0
$941$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$941$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$947$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$948$		0			0
$949$	4.85103	−	16.5211i	0.157471	−	0.536297i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$953$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−38.2895	+	83.8423i	−1.23515	+	2.70459i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	45.2597			1.45545			0.727727	−	0.685867i	$-0.240577\pi$
$967$	45.2597			1.45545			0.727727	+	0.685867i	$0.240577\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$971$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$972$		0			0
$973$	−40.2263	−	88.0833i	−1.28960	−	2.82382i
$974$		0			0
$975$	26.2017			0.839125
$976$		0			0
$977$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$977$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	57.5459	−	8.27385i	1.83730	−	0.264164i
$982$		0			0
$983$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$983$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	55.0394	+	7.91347i	1.74838	+	0.251380i	0.940942	−	0.338567i	$-0.109942\pi$
$991$	55.0394	+	7.91347i	1.74838	+	0.251380i	0.807442	+	0.589947i	$0.200851\pi$
$992$		0			0
$993$	−31.9293	−	20.5197i	−1.01325	−	0.651174i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	51.2098	−	32.9105i	1.62183	−	1.04229i	0.667033	−	0.745028i	$-0.267564\pi$
$997$	51.2098	−	32.9105i	1.62183	−	1.04229i	0.954797	−	0.297259i	$-0.0960725\pi$
$998$		0			0
$999$	−17.7987	−	60.6168i	−0.563126	−	1.91783i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.161.1	yes	20
3.2	odd	2	CM	804.2.s.a.161.1	yes	20
67.5	odd	22	inner	804.2.s.a.5.1	✓	20
201.5	even	22	inner	804.2.s.a.5.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.5.1	✓	20	67.5	odd	22	inner
804.2.s.a.5.1	✓	20	201.5	even	22	inner
804.2.s.a.161.1	yes	20	1.1	even	1	trivial
804.2.s.a.161.1	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)

\(f(q)\)	\(=\)	\(q+(-1.57553 - 0.719520i) q^{3} +(-3.36481 + 2.91562i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\)	\(q+(-1.57553 - 0.719520i) q^{3} +(-3.36481 + 2.91562i) q^{7} +(1.96458 + 2.26725i) q^{9} +(1.63571 - 2.54522i) q^{13} +(4.23982 - 4.89301i) q^{19} +(7.39920 - 2.17260i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(-1.46393 - 4.98567i) q^{27} +(-6.00018 - 9.33646i) q^{31} +12.1582 q^{37} +(-4.40845 + 2.83314i) q^{39} +(-12.9347 + 1.85973i) q^{43} +(1.82487 - 12.6923i) q^{49} +(-10.2006 + 4.65844i) q^{57} +(-4.39867 - 14.9805i) q^{61} +(-13.2209 - 1.90088i) q^{63} +(5.79361 + 5.78222i) q^{67} +(5.46060 - 1.60338i) q^{73} +(4.68209 + 7.28547i) q^{75} +(9.04681 - 14.0771i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(1.91704 + 13.3333i) q^{91} +(2.73569 + 19.0271i) q^{93} -6.97869i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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