LMFDB - Embedded newform 804.2.s.a.137.1

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			137.1
Root			$0.0475819 - 0.998867i$ of defining polynomial
Character	$\chi$	$=$	804.137
Dual form			804.2.s.a.581.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})$	$q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +(-5.44973 - 4.72222i) q^{13} +(2.19755 - 0.645259i) q^{19} +(3.65282 - 7.99857i) q^{21} +(3.27430 - 3.77875i) q^{25} +(-4.72659 - 2.15856i) q^{27} +(-0.512845 + 0.444383i) q^{31} -11.7846 q^{37} +(8.17912 + 9.43921i) q^{39} +(-6.94805 - 10.8114i) q^{43} +(-15.7932 - 10.1497i) q^{49} +(-3.92658 + 0.564557i) q^{57} +(-11.7083 - 5.34702i) q^{61} +(-8.23410 + 12.8125i) q^{63} +(-3.92989 + 7.18025i) q^{67} +(6.96168 - 15.2440i) q^{73} +(-6.54498 + 5.67126i) q^{75} +(10.8567 + 9.40741i) q^{79} +(7.57128 + 4.86577i) q^{81} +(30.7971 - 19.7921i) q^{91} +(0.988771 - 0.635445i) q^{93} +15.8295i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^9	$ 20 q + 6 q^{9} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 20 q^{37} - 24 q^{39} + 10 q^{49} - 66 q^{57} - 132 q^{63} - 16 q^{67} + 90 q^{73} + 44 q^{79} - 18 q^{81} + 48 q^{91} - 36 q^{93}+O(q^{100}) $ 20 * q + 6 * q^9 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 20 * q^37 - 24 * q^39 + 10 * q^49 - 66 * q^57 - 132 * q^63 - 16 * q^67 + 90 * q^73 + 44 * q^79 - 18 * q^81 + 48 * q^91 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.71442	−	0.246497i	−0.989821	−	0.142315i
$3$	−1.71442	−	0.246497i	−0.989821	−	0.142315i
$4$		0			0
$5$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$5$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$6$		0			0
$7$	−1.43029	+	4.87111i	−0.540597	+	1.84111i	0.000273951		1.00000i	$0.499913\pi$
$7$	−1.43029	+	4.87111i	−0.540597	+	1.84111i	−0.540871	+	0.841105i	$0.681905\pi$
$8$		0			0
$9$	2.87848	+	0.845198i	0.959493	+	0.281733i
$10$		0			0
$11$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$11$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$12$		0			0
$13$	−5.44973	−	4.72222i	−1.51148	−	1.30971i	−0.758731	−	0.651404i	$-0.774180\pi$
$13$	−5.44973	−	4.72222i	−1.51148	−	1.30971i	−0.752753	−	0.658303i	$-0.771274\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$17$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$18$		0			0
$19$	2.19755	−	0.645259i	0.504152	−	0.148032i	−0.0197599	−	0.999805i	$-0.506290\pi$
$19$	2.19755	−	0.645259i	0.504152	−	0.148032i	0.523912	+	0.851772i	$0.324472\pi$
$20$		0			0
$21$	3.65282	−	7.99857i	0.797111	−	1.74543i
$22$		0			0
$23$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$23$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$24$		0			0
$25$	3.27430	−	3.77875i	0.654861	−	0.755750i
$26$		0			0
$27$	−4.72659	−	2.15856i	−0.909632	−	0.415415i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.512845	+	0.444383i	−0.0921097	+	0.0798135i	−0.699699	−	0.714438i	$-0.746682\pi$
$31$	−0.512845	+	0.444383i	−0.0921097	+	0.0798135i	0.607589	+	0.794252i	$0.292137\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.7846			−1.93738			−0.968688	−	0.248283i	$-0.920134\pi$
$37$	−11.7846			−1.93738			−0.968688	+	0.248283i	$0.920134\pi$
$38$		0			0
$39$	8.17912	+	9.43921i	1.30971	+	1.51148i
$40$		0			0
$41$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$41$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$42$		0			0
$43$	−6.94805	−	10.8114i	−1.05957	−	1.64872i	−0.697529	−	0.716557i	$-0.745717\pi$
$43$	−6.94805	−	10.8114i	−1.05957	−	1.64872i	−0.362039	−	0.932163i	$-0.617919\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$47$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$48$		0			0
$49$	−15.7932	−	10.1497i	−2.25617	−	1.44995i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$53$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.92658	+	0.564557i	−0.520088	+	0.0747774i
$58$		0			0
$59$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$59$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$60$		0			0
$61$	−11.7083	−	5.34702i	−1.49910	−	0.684616i	−0.514188	−	0.857677i	$-0.671907\pi$
$61$	−11.7083	−	5.34702i	−1.49910	−	0.684616i	−0.984911	+	0.173061i	$0.944634\pi$
$62$		0			0
$63$	−8.23410	+	12.8125i	−1.03740	+	1.61422i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.92989	+	7.18025i	−0.480112	+	0.877207i
$67$	−3.92989	+	7.18025i	−0.480112	+	0.877207i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$71$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$72$		0			0
$73$	6.96168	−	15.2440i	0.814803	−	1.78417i	0.229598	−	0.973286i	$-0.426259\pi$
$73$	6.96168	−	15.2440i	0.814803	−	1.78417i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$	−6.54498	+	5.67126i	−0.755750	+	0.654861i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.8567	+	9.40741i	1.22148	+	1.05842i	0.996461	+	0.0840621i	$0.0267894\pi$
$79$	10.8567	+	9.40741i	1.22148	+	1.05842i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	7.57128	+	4.86577i	0.841254	+	0.540641i
$82$		0			0
$83$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$83$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$89$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$90$		0			0
$91$	30.7971	−	19.7921i	3.22841	−	2.07478i
$92$		0			0
$93$	0.988771	−	0.635445i	0.102531	−	0.0658926i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			15.8295i			1.60724i	0.595143	+	0.803620i	$0.297095\pi$
$97$			15.8295i			1.60724i	−0.595143	+	0.803620i	$0.702905\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$101$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$102$		0			0
$103$	−13.2058	−	15.2403i	−1.30121	−	1.50167i	−0.736644	−	0.676281i	$-0.763591\pi$
$103$	−13.2058	−	15.2403i	−1.30121	−	1.50167i	−0.564562	−	0.825391i	$-0.690955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$107$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$108$		0			0
$109$	7.90023	+	6.84559i	0.756705	+	0.655688i	0.945239	−	0.326378i	$-0.105828\pi$
$109$	7.90023	+	6.84559i	0.756705	+	0.655688i	−0.188534	+	0.982067i	$0.560374\pi$
$110$		0			0
$111$	20.2038	+	2.90486i	1.91766	+	0.275717i
$112$		0			0
$113$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$113$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.6957	−	18.1989i	−1.08127	−	1.68249i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.20347	−	8.31325i	0.654861	−	0.755750i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.03085	−	1.47719i	−0.446415	−	0.131079i	0.0507955	−	0.998709i	$-0.483824\pi$
$127$	−5.03085	−	1.47719i	−0.446415	−	0.131079i	−0.497211	+	0.867630i	$0.665643\pi$
$128$		0			0
$129$	9.24692	+	20.2479i	0.814146	+	1.78273i
$130$		0			0
$131$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$131$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$132$		0			0
$133$			11.6274i			1.00822i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$137$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$138$		0			0
$139$	−12.1638	+	5.55500i	−1.03172	+	0.471169i	−0.858013	−	0.513629i	$-0.828301\pi$
$139$	−12.1638	+	5.55500i	−1.03172	+	0.471169i	−0.173704	+	0.984798i	$0.555574\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	24.5743	+	21.2938i	2.02686	+	1.75628i
$148$		0			0
$149$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$149$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$150$		0			0
$151$	−19.9537	+	12.8235i	−1.62381	+	1.04356i	−0.670352	+	0.742043i	$0.733857\pi$
$151$	−19.9537	+	12.8235i	−1.62381	+	1.04356i	−0.953460	+	0.301518i	$0.902507\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.07835	−	7.50008i	0.0860615	−	0.598571i	−0.900460	−	0.434939i	$-0.856770\pi$
$157$	1.07835	−	7.50008i	0.0860615	−	0.598571i	0.986521	−	0.163632i	$-0.0523210\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.8399			1.55398			0.776989	−	0.629514i	$-0.216746\pi$
$163$	19.8399			1.55398			0.776989	+	0.629514i	$0.216746\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$167$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$168$		0			0
$169$	5.55012	+	38.6020i	0.426933	+	2.96938i
$170$		0			0
$171$	6.87097			0.525436
$172$		0			0
$173$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$173$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$174$		0			0
$175$	13.7235	+	21.3542i	1.03740	+	1.61422i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$179$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$180$		0			0
$181$	1.50254	−	10.4504i	0.111683	−	0.776773i	−0.854599	−	0.519288i	$-0.826197\pi$
$181$	1.50254	−	10.4504i	0.111683	−	0.776773i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	18.7550	+	12.0531i	1.38641	+	0.890991i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	17.2749	−	19.9364i	1.25657	−	1.45016i
$190$		0			0
$191$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$191$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$192$		0			0
$193$	−3.85622	−	4.45032i	−0.277577	−	0.320341i	0.599793	−	0.800155i	$-0.295250\pi$
$193$	−3.85622	−	4.45032i	−0.277577	−	0.320341i	−0.877370	+	0.479814i	$0.840704\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$197$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$198$		0			0
$199$	−20.8040	−	6.10860i	−1.47476	−	0.433027i	−0.557114	−	0.830436i	$-0.688091\pi$
$199$	−20.8040	−	6.10860i	−1.47476	−	0.433027i	−0.917642	+	0.397409i	$0.869909\pi$
$200$		0			0
$201$	8.50739	−	11.3413i	0.600065	−	0.799951i
$202$		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
$211$	2.19318	+	15.2539i	0.150985	+	1.05012i	0.914574	+	0.404419i	$0.132526\pi$
$211$	2.19318	+	15.2539i	0.150985	+	1.05012i	−0.763589	+	0.645703i	$0.776565\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.43112	−	3.13372i	−0.0971508	−	0.212731i
$218$		0			0
$219$	−15.6928	+	24.4185i	−1.06042	+	1.65005i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.896827	+	6.23757i	0.0600560	+	0.417698i	0.997565	+	0.0697384i	$0.0222164\pi$
$223$	0.896827	+	6.23757i	0.0600560	+	0.417698i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		0			0
$225$	12.6188	−	8.10961i	0.841254	−	0.540641i
$226$		0			0
$227$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$227$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$228$		0			0
$229$	−2.27894	+	1.97471i	−0.150596	+	0.130493i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−2.27894	+	1.97471i	−0.150596	+	0.130493i	0.576304	+	0.817235i	$0.304494\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$233$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−16.2941	−	18.8044i	−1.05842	−	1.22148i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	4.89188	−	10.7117i	0.315114	−	0.690003i	−0.684111	−	0.729378i	$-0.739809\pi$
$241$	4.89188	−	10.7117i	0.315114	−	0.690003i	0.999225	+	0.0393750i	$0.0125367\pi$
$242$		0			0
$243$	−11.7810	−	10.2083i	−0.755750	−	0.654861i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−15.0231	−	6.86082i	−0.955897	−	0.436544i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$251$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$257$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$258$		0			0
$259$	16.8553	−	57.4040i	1.04734	−	3.56691i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$263$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−24.2580	−	3.48777i	−1.47357	−	0.211867i	−0.641748	−	0.766915i	$-0.721791\pi$
$271$	−24.2580	−	3.48777i	−1.47357	−	0.211867i	−0.831818	+	0.555048i	$0.812700\pi$
$272$		0			0
$273$	−57.6779	+	26.3406i	−3.49082	+	1.59421i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	31.7684	+	9.32805i	1.90878	+	0.560468i	0.983408	+	0.181408i	$0.0580655\pi$
$277$	31.7684	+	9.32805i	1.90878	+	0.560468i	0.925372	+	0.379060i	$0.123753\pi$
$278$		0			0
$279$	−1.85181	+	0.845691i	−0.110865	+	0.0506302i
$280$		0			0
$281$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$281$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$282$		0			0
$283$	−21.8251	+	6.40842i	−1.29737	+	0.380941i	−0.856274	−	0.516522i	$-0.827227\pi$
$283$	−21.8251	+	6.40842i	−1.29737	+	0.380941i	−0.441091	+	0.897462i	$0.645408\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.06206	−	15.4637i	0.415415	−	0.909632i
$290$		0			0
$291$	3.90191	−	27.1384i	0.228734	−	1.59088i
$292$		0			0
$293$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$293$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	62.6011	−	18.3813i	3.60827	−	1.05948i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	19.8525	+	22.9110i	1.13304	+	1.30760i	0.945603	+	0.325322i	$0.105473\pi$
$307$	19.8525	+	22.9110i	1.13304	+	1.30760i	0.187437	+	0.982277i	$0.439982\pi$
$308$		0			0
$309$	18.8836	+	29.3835i	1.07425	+	1.67157i
$310$		0			0
$311$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$311$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$312$		0			0
$313$	−20.1846	+	2.90210i	−1.14090	+	0.164037i	−0.686753	−	0.726891i	$-0.740965\pi$
$313$	−20.1846	+	2.90210i	−1.14090	+	0.164037i	−0.454146	+	0.890927i	$0.650056\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$317$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−35.6881	+	5.13118i	−1.97962	+	0.284627i
$326$		0			0
$327$	−11.8569	−	13.6836i	−0.655688	−	0.756705i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−18.4571	+	28.7198i	−1.01449	+	1.57858i	−0.216166	+	0.976357i	$0.569355\pi$
$331$	−18.4571	+	28.7198i	−1.01449	+	1.57858i	−0.798327	+	0.602224i	$0.794281\pi$
$332$		0			0
$333$	−33.9217	−	9.96031i	−1.85890	−	0.545822i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.68281	+	15.9482i	−0.255089	+	0.868753i	0.727993	+	0.685585i	$0.240453\pi$
$337$	−4.68281	+	15.9482i	−0.255089	+	0.868753i	−0.983082	+	0.183168i	$0.941365\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	45.1716	−	39.1414i	2.43904	−	2.11344i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$347$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$348$		0			0
$349$	3.09026	+	1.98599i	0.165418	+	0.106308i	0.620730	−	0.784024i	$-0.286836\pi$
$349$	3.09026	+	1.98599i	0.165418	+	0.106308i	−0.455313	+	0.890332i	$0.650472\pi$
$350$		0			0
$351$	15.5654	+	34.0835i	0.830821	+	1.81925i
$352$		0			0
$353$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$353$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$359$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$360$		0			0
$361$	−11.5710	+	7.43620i	−0.608998	+	0.391379i
$362$		0			0
$363$	−14.3990	+	12.4768i	−0.755750	+	0.654861i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−19.2698	+	2.77058i	−1.00588	+	0.144623i	−0.625513	−	0.780214i	$-0.715110\pi$
$367$	−19.2698	+	2.77058i	−1.00588	+	0.144623i	−0.380363	+	0.924837i	$0.624201\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			17.3534i			0.898525i	0.893400	+	0.449262i	$0.148313\pi$
$373$			17.3534i			0.898525i	−0.893400	+	0.449262i	$0.851687\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−38.4204	−	5.52401i	−1.97352	−	0.283750i	−0.997944	−	0.0640964i	$-0.979583\pi$
$379$	−38.4204	−	5.52401i	−1.97352	−	0.283750i	−0.975578	−	0.219653i	$-0.929507\pi$
$380$		0			0
$381$	8.26087	+	3.77261i	0.423217	+	0.193277i
$382$		0			0
$383$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$383$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.8621	−	36.9928i	−0.552150	−	1.88045i
$388$		0			0
$389$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$389$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.21394	+	15.7963i	0.362057	+	0.792794i	0.999747	+	0.0225011i	$0.00716292\pi$
$397$	7.21394	+	15.7963i	0.362057	+	0.792794i	−0.637690	+	0.770293i	$0.720110\pi$
$398$		0			0
$399$	2.86611	−	19.9343i	0.143485	−	0.997961i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	4.89334			0.243755
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	11.3601	−	38.6888i	0.561719	−	1.91304i	0.205387	−	0.978681i	$-0.434155\pi$
$409$	11.3601	−	38.6888i	0.561719	−	1.91304i	0.356332	−	0.934359i	$-0.384027\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	22.2231	−	6.52529i	1.08827	−	0.319545i
$418$		0			0
$419$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$419$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$420$		0			0
$421$	−11.2961	+	3.31684i	−0.550539	+	0.161653i	−0.545159	−	0.838333i	$-0.683531\pi$
$421$	−11.2961	+	3.31684i	−0.550539	+	0.161653i	−0.00537983	+	0.999986i	$0.501712\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	42.7922	−	49.3848i	2.07086	−	2.38990i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−21.7154	+	18.8165i	−1.04357	+	0.904262i	−0.995517	−	0.0945826i	$-0.969848\pi$
$433$	−21.7154	+	18.8165i	−1.04357	+	0.904262i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	6.69959			0.319754			0.159877	−	0.987137i	$-0.448890\pi$
$439$	6.69959			0.319754			0.159877	+	0.987137i	$0.448890\pi$
$440$		0			0
$441$	−36.8819	−	42.5640i	−1.75628	−	2.02686i
$442$		0			0
$443$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$443$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$449$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	37.3701	−	17.0663i	1.75580	−	0.801847i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.9527	+	16.1023i	−0.652679	+	0.753232i	−0.981563	−	0.191139i	$-0.938782\pi$
$457$	−13.9527	+	16.1023i	−0.652679	+	0.753232i	0.328884	+	0.944370i	$0.393327\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$461$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$462$		0			0
$463$	−38.3831	−	17.5290i	−1.78381	−	0.814640i	−0.973516	−	0.228618i	$-0.926579\pi$
$463$	−38.3831	−	17.5290i	−1.78381	−	0.814640i	−0.810296	−	0.586021i	$-0.800693\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$467$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$468$		0			0
$469$	−29.3549	−	29.4127i	−1.35548	−	1.35815i
$470$		0			0
$471$	−3.69749	+	12.5925i	−0.170371	+	0.580231i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	4.75717	−	10.4168i	0.218274	−	0.477954i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$479$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$480$		0			0
$481$	64.2229	+	55.6494i	2.92831	+	2.53740i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−14.4364	+	22.4635i	−0.654176	+	1.01792i	0.342739	+	0.939430i	$0.388645\pi$
$487$	−14.4364	+	22.4635i	−0.654176	+	1.01792i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−34.0139	−	4.89046i	−1.53816	−	0.221154i
$490$		0			0
$491$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$491$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	3.76745i		−	0.168654i	−0.996438	−	0.0843271i	$-0.973126\pi$
$499$		−	3.76745i		−	0.168654i	0.996438	−	0.0843271i	$-0.0268741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$503$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	67.5481i		−	2.99992i
$508$		0			0
$509$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$509$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$510$		0			0
$511$	64.2978	+	55.7143i	2.84437	+	2.46466i
$512$		0			0
$513$	−11.7797	−	1.69367i	−0.520088	−	0.0747774i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$521$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$522$		0			0
$523$	3.28177	−	3.78737i	0.143502	−	0.165610i	−0.679449	−	0.733723i	$-0.737781\pi$
$523$	3.28177	−	3.78737i	0.143502	−	0.165610i	0.822951	+	0.568113i	$0.192326\pi$
$524$		0			0
$525$	−18.2641	−	39.9928i	−0.797111	−	1.74543i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.0683	−	6.47985i	−0.959493	−	0.281733i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.3204	+	7.90996i	−0.744662	+	0.340076i	−0.751352	−	0.659902i	$-0.770598\pi$
$541$	−17.3204	+	7.90996i	−0.744662	+	0.340076i	0.00668980	+	0.999978i	$0.497871\pi$
$542$		0			0
$543$	−5.15198	+	17.5460i	−0.221093	+	0.752972i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	38.2272	−	17.4578i	1.63448	−	0.746441i	0.634822	−	0.772658i	$-0.281073\pi$
$547$	38.2272	−	17.4578i	1.63448	−	0.746441i	0.999656	+	0.0262168i	$0.00834602\pi$
$548$		0			0
$549$	−29.1829	−	25.2871i	−1.24550	−	1.07923i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−61.3528	+	39.4290i	−2.60898	+	1.67669i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$557$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$558$		0			0
$559$	−13.1887	+	91.7293i	−0.557822	+	3.87974i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$563$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−34.5308	+	29.9211i	−1.45016	+	1.25657i
$568$		0			0
$569$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$569$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$570$		0			0
$571$	−6.66773	−	46.3751i	−0.279036	−	1.94074i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−6.66773	−	46.3751i	−0.279036	−	1.94074i	0.0557537	−	0.998445i	$-0.482244\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−25.7341	−	40.0430i	−1.07132	−	1.66701i	−0.649568	−	0.760303i	$-0.725050\pi$
$577$	−25.7341	−	40.0430i	−1.07132	−	1.66701i	−0.421756	−	0.906709i	$-0.638586\pi$
$578$		0			0
$579$	5.51420	+	8.58027i	0.229162	+	0.356584i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$587$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$588$		0			0
$589$	−0.840261	+	1.30747i	−0.0346223	+	0.0538734i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$593$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	34.1610	+	15.6008i	1.39812	+	0.638499i
$598$		0			0
$599$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$599$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$600$		0			0
$601$	46.6443	+	13.6960i	1.90266	+	0.558672i	0.988001	+	0.154446i	$0.0493591\pi$
$601$	46.6443	+	13.6960i	1.90266	+	0.558672i	0.914659	+	0.404226i	$0.132459\pi$
$602$		0			0
$603$	−17.3808	+	17.3467i	−0.707802	+	0.706411i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.47026	−	3.51552i	−0.222031	−	0.142691i	0.424897	−	0.905242i	$-0.360310\pi$
$607$	−5.47026	−	3.51552i	−0.222031	−	0.142691i	−0.646928	+	0.762551i	$0.723947\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.92868	+	34.2797i	0.199067	+	1.38454i	0.806998	+	0.590554i	$0.201091\pi$
$613$	4.92868	+	34.2797i	0.199067	+	1.38454i	−0.607930	+	0.793990i	$0.708000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$617$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$618$		0			0
$619$	−8.87674	−	19.4373i	−0.356786	−	0.781253i	−0.999880	−	0.0154656i	$-0.995077\pi$
$619$	−8.87674	−	19.4373i	−0.356786	−	0.781253i	0.643094	−	0.765787i	$-0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.55787	−	24.7455i	−0.142315	−	0.989821i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	30.3065	−	26.2607i	1.20648	−	1.04542i	0.208759	−	0.977967i	$-0.433058\pi$
$631$	30.3065	−	26.2607i	1.20648	−	1.04542i	0.997722	−	0.0674546i	$-0.0214878\pi$
$632$		0			0
$633$		−	26.6922i		−	1.06092i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	38.1397	+	129.892i	1.51115	+	5.14650i
$638$		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
$643$	6.97674	−	15.2769i	0.275136	−	0.602463i	−0.720738	−	0.693207i	$-0.756197\pi$
$643$	6.97674	−	15.2769i	0.275136	−	0.602463i	0.995874	+	0.0907437i	$0.0289244\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$647$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.68109	+	5.72528i	0.0658873	+	0.224391i
$652$		0			0
$653$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$653$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	32.9232	−	37.9954i	1.28446	−	1.48234i
$658$		0			0
$659$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$659$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$660$		0			0
$661$	13.9982	−	47.6736i	0.544468	−	1.85429i	0.0251892	−	0.999683i	$-0.491981\pi$
$661$	13.9982	−	47.6736i	0.544468	−	1.85429i	0.519279	−	0.854605i	$-0.326201\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	10.9149i		−	0.421994i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	39.3211	+	5.65352i	1.51572	+	0.217927i	0.849403	−	0.527744i	$-0.176962\pi$
$673$	39.3211	+	5.65352i	1.51572	+	0.217927i	0.666314	+	0.745672i	$0.267871\pi$
$674$		0			0
$675$	−23.6329	+	10.7928i	−0.909632	+	0.415415i
$676$		0			0
$677$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$677$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$678$		0			0
$679$	−77.1071	−	22.6407i	−2.95910	−	0.868869i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$683$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.39382	−	2.82373i	0.167635	−	0.107732i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	18.4896	−	40.4866i	0.703378	−	1.54018i	−0.132448	−	0.991190i	$-0.542284\pi$
$691$	18.4896	−	40.4866i	0.703378	−	1.54018i	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		0.755750	−	0.654861i	$-0.227273\pi$
$701$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$702$		0			0
$703$	−25.8972	+	7.60411i	−0.976732	+	0.286794i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	33.4861	+	38.6450i	1.25760	+	1.45134i	0.839884	+	0.542766i	$0.182623\pi$
$709$	33.4861	+	38.6450i	1.25760	+	1.45134i	0.417714	+	0.908579i	$0.362832\pi$
$710$		0			0
$711$	23.2998	+	36.2551i	0.873809	+	1.35967i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$719$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$720$		0			0
$721$	93.1252	−	42.5289i	3.46816	−	1.58386i
$722$		0			0
$723$	−11.0272	+	17.1586i	−0.410104	+	0.638135i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	49.5068	−	7.11800i	1.83611	−	0.263992i	0.864850	−	0.502031i	$-0.167414\pi$
$727$	49.5068	−	7.11800i	1.83611	−	0.263992i	0.971256	+	0.238039i	$0.0765045\pi$
$728$		0			0
$729$	17.6812	+	20.4052i	0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−3.16602	+	4.92643i	−0.116940	+	0.181962i	−0.894789	−	0.446489i	$-0.852674\pi$
$733$	−3.16602	+	4.92643i	−0.116940	+	0.181962i	0.777849	+	0.628451i	$0.216311\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	9.87827	−	33.6423i	0.363378	−	1.23755i	−0.551621	−	0.834095i	$-0.685990\pi$
$739$	9.87827	−	33.6423i	0.363378	−	1.23755i	0.914999	−	0.403456i	$-0.132191\pi$
$740$		0			0
$741$	24.0648	+	15.4655i	0.884041	+	0.568139i
$742$		0			0
$743$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$743$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.9231	−	18.5878i	−1.05542	−	0.678278i	−0.106667	−	0.994295i	$-0.534018\pi$
$751$	−28.9231	−	18.5878i	−1.05542	−	0.678278i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−43.3512	−	6.23296i	−1.57563	−	0.226541i	−0.701654	−	0.712517i	$-0.747555\pi$
$757$	−43.3512	−	6.23296i	−1.57563	−	0.226541i	−0.873972	+	0.485977i	$0.838464\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$761$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$762$		0			0
$763$	−44.6452	+	28.6917i	−1.61626	+	1.03871i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	34.4305	−	4.95036i	1.24160	−	0.178515i	0.509953	−	0.860202i	$-0.329663\pi$
$769$	34.4305	−	4.95036i	1.24160	−	0.178515i	0.731643	+	0.681688i	$0.238754\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$773$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$774$		0			0
$775$			3.39296i			0.121879i
$776$		0			0
$777$	−43.0470	+	94.2599i	−1.54430	+	3.38155i
$778$		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
$787$	24.1074	+	37.5119i	0.859337	+	1.33715i	0.940268	+	0.340435i	$0.110574\pi$
$787$	24.1074	+	37.5119i	0.859337	+	1.33715i	−0.0809308	+	0.996720i	$0.525789\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	38.5575	+	84.4292i	1.36922	+	2.99817i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$797$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$809$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$810$		0			0
$811$	−13.8408	+	47.1375i	−0.486017	+	1.65522i	0.242469	+	0.970159i	$0.422043\pi$
$811$	−13.8408	+	47.1375i	−0.486017	+	1.65522i	−0.728485	+	0.685062i	$0.759775\pi$
$812$		0			0
$813$	40.7286	+	11.9590i	1.42842	+	0.419421i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−22.2448	−	19.2752i	−0.778248	−	0.674355i
$818$		0			0
$819$	105.377	−	30.9415i	3.68217	−	1.08118i
$820$		0			0
$821$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$821$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$822$		0			0
$823$	41.3177	−	12.1320i	1.44024	−	0.422894i	0.533940	−	0.845522i	$-0.320711\pi$
$823$	41.3177	−	12.1320i	1.44024	−	0.422894i	0.906303	+	0.422628i	$0.138892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$827$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$828$		0			0
$829$	−13.0771	+	15.0918i	−0.454187	+	0.524160i	−0.935946	−	0.352144i	$-0.885453\pi$
$829$	−13.0771	+	15.0918i	−0.454187	+	0.524160i	0.481759	+	0.876304i	$0.339998\pi$
$830$		0			0
$831$	−52.1651	−	23.8230i	−1.80959	−	0.826411i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.38323	−	0.993407i	0.116942	−	0.0343372i
$838$		0			0
$839$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$839$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	30.1917	+	46.9792i	1.03740	+	1.61422i
$848$		0			0
$849$	38.9970	−	5.60692i	1.33837	−	0.192429i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	27.5476	+	17.7038i	0.943213	+	0.606166i	0.919304	−	0.393548i	$-0.128752\pi$
$853$	27.5476	+	17.7038i	0.943213	+	0.606166i	0.0239089	+	0.999714i	$0.492389\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$857$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$858$		0			0
$859$	32.5873	−	37.6077i	1.11186	−	1.28316i	0.156516	−	0.987675i	$-0.449974\pi$
$859$	32.5873	−	37.6077i	1.11186	−	1.28316i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$863$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.9191	+	24.7706i	−0.540641	+	0.841254i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	55.3235	−	20.5726i	1.87457	−	0.697077i
$872$		0			0
$873$	−13.3790	+	45.5648i	−0.452812	+	1.54213i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−19.2279	+	42.1033i	−0.649281	+	1.42173i	0.242901	+	0.970051i	$0.421901\pi$
$877$	−19.2279	+	42.1033i	−0.649281	+	1.42173i	−0.892182	+	0.451676i	$0.850826\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$881$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$882$		0			0
$883$	−12.3183	−	10.6739i	−0.414544	−	0.359204i	0.422478	−	0.906373i	$-0.361160\pi$
$883$	−12.3183	−	10.6739i	−0.414544	−	0.359204i	−0.837022	+	0.547169i	$0.815706\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$887$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$888$		0			0
$889$	14.3911	−	22.3930i	0.482662	−	0.751037i
$890$		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
$903$	−111.856	+	16.0824i	−3.72232	+	0.535189i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−15.9440	−	18.4003i	−0.529412	−	0.610974i	0.426551	−	0.904464i	$-0.359729\pi$
$907$	−15.9440	−	18.4003i	−0.529412	−	0.610974i	−0.955962	+	0.293490i	$0.905183\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$911$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−16.9750	−	57.8116i	−0.559954	−	1.90703i	−0.386924	−	0.922112i	$-0.626462\pi$
$919$	−16.9750	−	57.8116i	−0.559954	−	1.90703i	−0.173030	−	0.984917i	$-0.555356\pi$
$920$		0			0
$921$	−28.3880	−	44.1726i	−0.935417	−	1.45554i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−38.5863	+	44.5310i	−1.26871	+	1.46417i
$926$		0			0
$927$	−25.1315	−	55.0304i	−0.825428	−	1.80744i
$928$		0			0
$929$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$929$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$930$		0			0
$931$	−41.2555	−	12.1137i	−1.35209	−	0.397010i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			32.5703i			1.06403i	0.846736	+	0.532013i	$0.178564\pi$
$937$			32.5703i			1.06403i	−0.846736	+	0.532013i	$0.821436\pi$
$938$		0			0
$939$	35.3202			1.15263
$940$		0			0
$941$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$941$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$947$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$948$		0			0
$949$	−109.925	+	50.2009i	−3.56830	+	1.62959i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$953$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−4.34623	+	30.2287i	−0.140201	+	0.975118i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	61.1366			1.96602			0.983010	−	0.183550i	$-0.0587588\pi$
$967$	61.1366			1.96602			0.983010	+	0.183550i	$0.0587588\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$971$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$972$		0			0
$973$	−9.66136	−	67.1962i	−0.309729	−	2.15421i
$974$		0			0
$975$	62.4493			1.99998
$976$		0			0
$977$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$977$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	16.9548	+	26.3821i	0.541324	+	0.842317i
$982$		0			0
$983$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$983$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−11.5245	+	17.9324i	−0.366086	+	0.569642i	−0.974614	−	0.223891i	$-0.928124\pi$
$991$	−11.5245	+	17.9324i	−0.366086	+	0.569642i	0.608528	+	0.793533i	$0.291760\pi$
$992$		0			0
$993$	38.7225	−	44.6882i	1.22882	−	1.41814i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−39.4855	−	45.5687i	−1.25052	−	1.44317i	−0.849912	−	0.526924i	$-0.823345\pi$
$997$	−39.4855	−	45.5687i	−1.25052	−	1.44317i	−0.400606	−	0.916251i	$-0.631200\pi$
$998$		0			0
$999$	55.7009	+	25.4377i	1.76230	+	0.804815i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.137.1	✓	20
3.2	odd	2	CM	804.2.s.a.137.1	✓	20
67.45	odd	22	inner	804.2.s.a.581.1	yes	20
201.179	even	22	inner	804.2.s.a.581.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.137.1	✓	20	1.1	even	1	trivial
804.2.s.a.137.1	✓	20	3.2	odd	2	CM
804.2.s.a.581.1	yes	20	67.45	odd	22	inner
804.2.s.a.581.1	yes	20	201.179	even	22	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})\)	\(q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +(-5.44973 - 4.72222i) q^{13} +(2.19755 - 0.645259i) q^{19} +(3.65282 - 7.99857i) q^{21} +(3.27430 - 3.77875i) q^{25} +(-4.72659 - 2.15856i) q^{27} +(-0.512845 + 0.444383i) q^{31} -11.7846 q^{37} +(8.17912 + 9.43921i) q^{39} +(-6.94805 - 10.8114i) q^{43} +(-15.7932 - 10.1497i) q^{49} +(-3.92658 + 0.564557i) q^{57} +(-11.7083 - 5.34702i) q^{61} +(-8.23410 + 12.8125i) q^{63} +(-3.92989 + 7.18025i) q^{67} +(6.96168 - 15.2440i) q^{73} +(-6.54498 + 5.67126i) q^{75} +(10.8567 + 9.40741i) q^{79} +(7.57128 + 4.86577i) q^{81} +(30.7971 - 19.7921i) q^{91} +(0.988771 - 0.635445i) q^{93} +15.8295i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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