LMFDB - Embedded newform 804.2.s.a.161.2

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			161.2
Root			$0.928368 - 0.371662i$ of defining polynomial
Character	$\chi$	$=$	804.161
Dual form			804.2.s.a.5.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})$	$q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +(-3.27007 + 5.08833i) q^{13} +(0.112808 - 0.130187i) q^{19} +(-5.76060 + 1.69146i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(1.46393 + 4.98567i) q^{27} +(4.78165 + 7.44039i) q^{31} +4.66687 q^{37} +(-8.81324 + 5.66393i) q^{39} +(4.37247 - 0.628667i) q^{43} +(0.713735 - 4.96414i) q^{49} +(0.271404 - 0.123946i) q^{57} +(-2.77697 - 9.45750i) q^{61} +(-10.2930 - 1.47991i) q^{63} +(7.66645 + 2.86803i) q^{67} +(12.9519 - 3.80303i) q^{73} +(-4.68209 - 7.28547i) q^{75} +(-6.70847 + 10.4386i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-2.98376 - 20.7525i) q^{91} +(2.18012 + 15.1630i) q^{93} -18.4910i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.61965	+	2.26994i	−0.990134	+	0.857956i	−0.989860	−	0.142044i	$-0.954633\pi$
$7$	−2.61965	+	2.26994i	−0.990134	+	0.857956i	−0.000273951		1.00000i	$0.500087\pi$
$8$		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
$13$	−3.27007	+	5.08833i	−0.906954	+	1.41125i	0.00456441	+	0.999990i	$0.498547\pi$
$13$	−3.27007	+	5.08833i	−0.906954	+	1.41125i	−0.911519	+	0.411259i	$0.865089\pi$
$14$		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
$19$	0.112808	−	0.130187i	0.0258799	−	0.0298670i	−0.742662	−	0.669666i	$-0.766437\pi$
$19$	0.112808	−	0.130187i	0.0258799	−	0.0298670i	0.768542	+	0.639799i	$0.220983\pi$
$20$		0			0
$21$	−5.76060	+	1.69146i	−1.25707	+	0.369108i
$22$		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
$31$	4.78165	+	7.44039i	0.858810	+	1.33633i	0.940541	+	0.339680i	$0.110319\pi$
$31$	4.78165	+	7.44039i	0.858810	+	1.33633i	−0.0817313	+	0.996654i	$0.526045\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.66687			0.767228			0.383614	−	0.923494i	$-0.374679\pi$
$37$	4.66687			0.767228			0.383614	+	0.923494i	$0.374679\pi$
$38$		0			0
$39$	−8.81324	+	5.66393i	−1.41125	+	0.906954i
$40$		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
$43$	4.37247	−	0.628667i	0.666796	−	0.0958708i	0.199399	−	0.979918i	$-0.436101\pi$
$43$	4.37247	−	0.628667i	0.666796	−	0.0958708i	0.467397	+	0.884048i	$0.345192\pi$
$44$		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
$49$	0.713735	−	4.96414i	0.101962	−	0.709163i
$50$		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
$57$	0.271404	−	0.123946i	0.0359484	−	0.0164171i
$58$		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
$61$	−2.77697	−	9.45750i	−0.355555	−	1.21091i	−0.922124	−	0.386895i	$-0.873548\pi$
$61$	−2.77697	−	9.45750i	−0.355555	−	1.21091i	0.566569	−	0.824014i	$-0.308270\pi$
$62$		0			0
$63$	−10.2930	−	1.47991i	−1.29680	−	0.186452i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.66645	+	2.86803i	0.936605	+	0.350386i
$67$	7.66645	+	2.86803i	0.936605	+	0.350386i
$68$		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
$73$	12.9519	−	3.80303i	1.51591	−	0.445111i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	12.9519	−	3.80303i	1.51591	−	0.445111i	0.930704	+	0.365774i	$0.119196\pi$
$74$		0			0
$75$	−4.68209	−	7.28547i	−0.540641	−	0.841254i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.70847	+	10.4386i	−0.754762	+	1.17443i	0.225018	+	0.974355i	$0.427756\pi$
$79$	−6.70847	+	10.4386i	−0.754762	+	1.17443i	−0.979780	+	0.200078i	$0.935880\pi$
$80$		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
$91$	−2.98376	−	20.7525i	−0.312783	−	2.17545i
$92$		0			0
$93$	2.18012	+	15.1630i	0.226068	+	1.57233i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	18.4910i		−	1.87748i	−0.344630	−	0.938739i	$-0.611996\pi$
$97$		−	18.4910i		−	1.87748i	0.344630	−	0.938739i	$-0.388004\pi$
$98$		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
$103$	13.2205	−	8.49628i	1.30265	−	0.837164i	0.309154	−	0.951012i	$-0.399954\pi$
$103$	13.2205	−	8.49628i	1.30265	−	0.837164i	0.993498	+	0.113848i	$0.0363178\pi$
$104$		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
$109$	−11.0865	+	17.2509i	−1.06189	+	1.65233i	−0.372340	+	0.928096i	$0.621444\pi$
$109$	−11.0865	+	17.2509i	−1.06189	+	1.65233i	−0.689551	+	0.724238i	$0.742192\pi$
$110$		0			0
$111$	7.35278	+	3.35790i	0.697895	+	0.318718i
$112$		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
$117$	−17.9608	+	2.58238i	−1.66048	+	0.238741i
$118$		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
$127$	14.6974	+	16.9617i	1.30418	+	1.50511i	0.721510	+	0.692404i	$0.243448\pi$
$127$	14.6974	+	16.9617i	1.30418	+	1.50511i	0.582675	+	0.812705i	$0.302006\pi$
$128$		0			0
$129$	7.34130	+	2.15560i	0.646365	+	0.189790i
$130$		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
$133$			0.597112i			0.0517762i
$134$		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
$139$	−2.07325	+	7.06083i	−0.175850	+	0.598891i	0.823644	+	0.567107i	$0.191937\pi$
$139$	−2.07325	+	7.06083i	−0.175850	+	0.598891i	−0.999495	+	0.0317847i	$0.989881\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.69631	−	7.30760i	0.387345	−	0.602720i
$148$		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
$151$	−1.51845	−	10.5610i	−0.123569	−	0.859444i	−0.953460	−	0.301518i	$-0.902507\pi$
$151$	−1.51845	−	10.5610i	−0.123569	−	0.859444i	0.829891	−	0.557926i	$-0.188403\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.71678	−	5.94891i	0.216822	−	0.474775i	−0.769699	−	0.638407i	$-0.779594\pi$
$157$	2.71678	−	5.94891i	0.216822	−	0.474775i	0.986521	+	0.163632i	$0.0523210\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	22.8634			1.79080			0.895399	−	0.445265i	$-0.146890\pi$
$163$	22.8634			1.79080			0.895399	+	0.445265i	$0.146890\pi$
$164$		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
$169$	−9.79733	−	21.4532i	−0.753641	−	1.65024i
$170$		0			0
$171$	0.516787			0.0395197
$172$		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
$175$	17.1551	−	2.46652i	1.29680	−	0.186452i
$176$		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
$181$	7.10480	−	15.5574i	0.528096	−	1.15637i	−0.438186	−	0.898884i	$-0.644379\pi$
$181$	7.10480	−	15.5574i	0.528096	−	1.15637i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	2.42966	−	16.8987i	0.179606	−	1.24918i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−15.1521	−	9.73769i	−1.10216	−	0.708313i
$190$		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
$193$	16.5174	−	10.6151i	1.18895	−	0.764089i	0.211936	−	0.977284i	$-0.432023\pi$
$193$	16.5174	−	10.6151i	1.18895	−	0.764089i	0.977010	+	0.213195i	$0.0683868\pi$
$194$		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
$199$	4.85491	+	5.60287i	0.344156	+	0.397177i	0.901269	−	0.433260i	$-0.142637\pi$
$199$	4.85491	+	5.60287i	0.344156	+	0.397177i	−0.557114	+	0.830436i	$0.688091\pi$
$200$		0			0
$201$	10.0151	+	10.0348i	0.706411	+	0.707802i
$202$		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
$211$	−3.53943	−	7.75028i	−0.243665	−	0.533551i	0.747800	−	0.663924i	$-0.231110\pi$
$211$	−3.53943	−	7.75028i	−0.243665	−	0.533551i	−0.991465	+	0.130372i	$0.958383\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−29.4155	−	8.63717i	−1.99685	−	0.586329i
$218$		0			0
$219$	23.1425	+	3.32739i	1.56383	+	0.224844i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.904960	−	1.98159i	−0.0606006	−	0.132697i	0.876909	−	0.480657i	$-0.159602\pi$
$223$	−0.904960	−	1.98159i	−0.0606006	−	0.132697i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		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
$229$	−15.8836	−	24.7154i	−1.04962	−	1.63324i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−15.8836	−	24.7154i	−1.04962	−	1.63324i	−0.322718	−	0.946495i	$-0.604597\pi$
$230$		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
$237$	−18.0802	+	11.6194i	−1.17443	+	0.754762i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−25.6763	+	7.53923i	−1.65395	+	0.485644i	−0.969842	−	0.243734i	$-0.921628\pi$
$241$	−25.6763	+	7.53923i	−1.65395	+	0.485644i	−0.684111	+	0.729378i	$0.739809\pi$
$242$		0			0
$243$	−8.42776	+	13.1138i	−0.540641	+	0.841254i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.293546	+	0.999725i	0.0186779	+	0.0636110i
$248$		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
$259$	−12.2256	+	10.5935i	−0.759659	+	0.658248i
$260$		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
$271$	29.5662	+	13.5024i	1.79602	+	0.820215i	0.964203	+	0.265167i	$0.0854270\pi$
$271$	29.5662	+	13.5024i	1.79602	+	0.820215i	0.831818	+	0.555048i	$0.187300\pi$
$272$		0			0
$273$	10.2308	−	34.8430i	0.619198	−	2.10880i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	11.0494	+	12.7517i	0.663893	+	0.766173i	0.983408	−	0.181408i	$-0.0580655\pi$
$277$	11.0494	+	12.7517i	0.663893	+	0.766173i	−0.319515	+	0.947581i	$0.603520\pi$
$278$		0			0
$279$	−7.47528	+	25.4585i	−0.447533	+	1.52416i
$280$		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
$283$	22.0240	−	25.4171i	1.30919	−	1.51089i	0.633125	−	0.774050i	$-0.281772\pi$
$283$	22.0240	−	25.4171i	1.30919	−	1.51089i	0.676068	−	0.736839i	$-0.263682\pi$
$284$		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
$291$	13.3046	−	29.1331i	0.779932	−	1.70781i
$292$		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
$301$	−10.0273	+	11.5721i	−0.577965	+	0.667007i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−24.0452	+	15.4529i	−1.37233	+	0.881943i	−0.998954	−	0.0457370i	$-0.985436\pi$
$307$	−24.0452	+	15.4529i	−1.37233	+	0.881943i	−0.373376	+	0.927680i	$0.621800\pi$
$308$		0			0
$309$	26.9425	−	3.87374i	1.53270	−	0.220370i
$310$		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
$313$	1.38404	−	0.632072i	0.0782308	−	0.0357268i	−0.375915	−	0.926654i	$-0.622672\pi$
$313$	1.38404	−	0.632072i	0.0782308	−	0.0357268i	0.454146	+	0.890927i	$0.349944\pi$
$314$		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
$325$	27.5096	−	12.5632i	1.52596	−	0.696881i
$326$		0			0
$327$	−29.8794	+	19.2023i	−1.65233	+	1.06189i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	35.9339	+	5.16652i	1.97511	+	0.283977i	0.997182	+	0.0750153i	$0.0239006\pi$
$331$	35.9339	+	5.16652i	1.97511	+	0.283977i	0.977924	+	0.208962i	$0.0670085\pi$
$332$		0			0
$333$	9.16844	+	10.5809i	0.502428	+	0.579832i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.7016	+	19.6711i	−1.23664	+	1.07155i	−0.241771	+	0.970333i	$0.577728\pi$
$337$	−22.7016	+	19.6711i	−1.23664	+	1.07155i	−0.994864	+	0.101218i	$0.967726\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−3.71958	−	5.78778i	−0.200839	−	0.312511i
$344$		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
$349$	2.42104	−	16.8387i	0.129595	−	0.901356i	−0.816472	−	0.577386i	$-0.804073\pi$
$349$	2.42104	−	16.8387i	0.129595	−	0.901356i	0.946067	−	0.323971i	$-0.105018\pi$
$350$		0			0
$351$	−30.1559	−	8.85456i	−1.60960	−	0.472622i
$352$		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
$361$	2.69976	+	18.7772i	0.142093	+	0.988276i
$362$		0			0
$363$	−10.3006	−	16.0280i	−0.540641	−	0.841254i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−34.2827	+	15.6564i	−1.78954	+	0.817256i	−0.819987	+	0.572383i	$0.806019\pi$
$367$	−34.2827	+	15.6564i	−1.78954	+	0.817256i	−0.969555	+	0.244874i	$0.921254\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			33.2555i			1.72190i	0.508686	+	0.860952i	$0.330131\pi$
$373$			33.2555i			1.72190i	−0.508686	+	0.860952i	$0.669869\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	28.1982	+	12.8777i	1.44844	+	0.661482i	0.975578	−	0.219653i	$-0.0704926\pi$
$379$	28.1982	+	12.8777i	1.44844	+	0.661482i	0.472865	+	0.881135i	$0.343220\pi$
$380$		0			0
$381$	10.9519	+	37.2988i	0.561083	+	1.91087i
$382$		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
$387$	10.0154	+	8.67842i	0.509113	+	0.441149i
$388$		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
$397$	24.3824	+	7.15932i	1.22372	+	0.359316i	0.828876	−	0.559433i	$-0.188981\pi$
$397$	24.3824	+	7.15932i	1.22372	+	0.359316i	0.394842	+	0.918749i	$0.370799\pi$
$398$		0			0
$399$	−0.429634	+	0.940767i	−0.0215086	+	0.0470973i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−53.4955			−2.66480
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−18.1387	+	15.7173i	−0.896902	+	0.777170i	−0.975560	−	0.219732i	$-0.929482\pi$
$409$	−18.1387	+	15.7173i	−0.896902	+	0.777170i	0.0786585	+	0.996902i	$0.474936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−8.34686	+	9.63279i	−0.408748	+	0.471720i
$418$		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
$421$	−24.3843	+	28.1410i	−1.18842	+	1.37151i	−0.276567	+	0.960995i	$0.589197\pi$
$421$	−24.3843	+	28.1410i	−1.18842	+	1.37151i	−0.911854	+	0.410515i	$0.865349\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	28.7426	+	18.4718i	1.39095	+	0.893912i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	18.3218	+	28.5092i	0.880488	+	1.37007i	0.928545	+	0.371221i	$0.121061\pi$
$433$	18.3218	+	28.5092i	0.880488	+	1.37007i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−34.8444			−1.66303			−0.831516	−	0.555501i	$-0.812527\pi$
$439$	−34.8444			−1.66303			−0.831516	+	0.555501i	$0.812527\pi$
$440$		0			0
$441$	12.6571	−	8.13424i	0.602720	−	0.387345i
$442$		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
$453$	5.20651	−	17.7317i	0.244623	−	0.833110i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	31.9378	+	20.5252i	1.49399	+	0.960127i	0.995653	+	0.0931420i	$0.0296911\pi$
$457$	31.9378	+	20.5252i	1.49399	+	0.960127i	0.498334	+	0.866985i	$0.333945\pi$
$458$		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
$463$	11.2886	+	38.4455i	0.524626	+	1.78671i	0.612372	+	0.790570i	$0.290216\pi$
$463$	11.2886	+	38.4455i	0.524626	+	1.78671i	−0.0877454	+	0.996143i	$0.527966\pi$
$464$		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
$469$	−26.5937	+	9.88912i	−1.22798	+	0.456637i
$470$		0			0
$471$	8.56072	−	7.41791i	0.394457	−	0.341799i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.826423	+	0.242660i	−0.0379189	+	0.0111340i
$476$		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
$481$	−15.2610	+	23.7465i	−0.695841	+	1.08275i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−42.6209	−	6.12796i	−1.93134	−	0.277684i	−0.934421	−	0.356171i	$-0.884082\pi$
$487$	−42.6209	−	6.12796i	−1.93134	−	0.277684i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	36.0219	+	16.4507i	1.62897	+	0.743924i
$490$		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
$499$		−	43.5282i		−	1.94859i	−0.225277	−	0.974295i	$-0.572329\pi$
$499$		−	43.5282i		−	1.94859i	0.225277	−	0.974295i	$-0.427671\pi$
$500$		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
$507$		−	40.8495i		−	1.81419i
$508$		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
$511$	−25.2969	+	39.3627i	−1.11907	+	1.74130i
$512$		0			0
$513$	0.814213	+	0.371839i	0.0359484	+	0.0164171i
$514$		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
$523$	−24.2239	−	15.5678i	−1.05924	−	0.680731i	−0.109567	−	0.993979i	$-0.534946\pi$
$523$	−24.2239	−	15.5678i	−1.05924	−	0.680731i	−0.949671	+	0.313249i	$0.898583\pi$
$524$		0			0
$525$	28.8030	+	8.45732i	1.25707	+	0.369108i
$526$		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
$541$	13.1056	−	44.6335i	0.563453	−	1.91894i	0.275307	−	0.961356i	$-0.411220\pi$
$541$	13.1056	−	44.6335i	0.563453	−	1.91894i	0.288145	−	0.957587i	$-0.406961\pi$
$542$		0			0
$543$	22.3876	−	19.3990i	0.960746	−	0.832491i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10.1824	+	34.6779i	−0.435366	+	1.48272i	0.391425	+	0.920210i	$0.371982\pi$
$547$	−10.1824	+	34.6779i	−0.435366	+	1.48272i	−0.826791	+	0.562510i	$0.809836\pi$
$548$		0			0
$549$	15.9869	−	24.8761i	0.682305	−	1.06169i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−6.12111	−	42.5733i	−0.260296	−	1.81040i
$554$		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
$559$	−11.0994	+	24.3044i	−0.469456	+	1.02797i
$560$		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
$567$	−16.8662	−	26.2443i	−0.708313	−	1.10216i
$568$		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
$571$	−19.7781	−	43.3079i	−0.827686	−	1.81238i	−0.492897	−	0.870088i	$-0.664062\pi$
$571$	−19.7781	−	43.3079i	−0.827686	−	1.81238i	−0.334790	−	0.942293i	$-0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	25.9876	−	3.73645i	1.08188	−	0.155550i	0.421756	−	0.906709i	$-0.361414\pi$
$577$	25.9876	−	3.73645i	1.08188	−	0.155550i	0.660121	+	0.751159i	$0.270505\pi$
$578$		0			0
$579$	33.6613	−	4.83977i	1.39892	−	0.201134i
$580$		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
$589$	1.50805	+	0.216825i	0.0621382	+	0.00893413i
$590$		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
$597$	3.61768	+	12.3207i	0.148062	+	0.504252i
$598$		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
$601$	−24.0060	−	27.7044i	−0.979226	−	1.13009i	−0.991493	−	0.130163i	$-0.958450\pi$
$601$	−24.0060	−	27.7044i	−0.979226	−	1.13009i	0.0122663	−	0.999925i	$-0.496095\pi$
$602$		0			0
$603$	8.55882	+	23.0162i	0.348542	+	0.937293i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.97960	+	20.7236i	−0.120938	+	0.841145i	0.835558	+	0.549402i	$0.185144\pi$
$607$	−2.97960	+	20.7236i	−0.120938	+	0.841145i	−0.956497	+	0.291743i	$0.905765\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.3503	−	42.3712i	−0.781551	−	1.71136i	−0.699391	−	0.714739i	$-0.746545\pi$
$613$	−19.3503	−	42.3712i	−0.781551	−	1.71136i	−0.0821600	−	0.996619i	$-0.526182\pi$
$614$		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
$619$	19.1595	+	5.62573i	0.770084	+	0.226117i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	19.1595	+	5.62573i	0.770084	+	0.226117i	0.126990	+	0.991904i	$0.459469\pi$
$620$		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
$631$	−16.1925	−	25.1960i	−0.644612	−	1.00304i	−0.997722	−	0.0674546i	$-0.978512\pi$
$631$	−16.1925	−	25.1960i	−0.644612	−	1.00304i	0.353110	−	0.935582i	$-0.385124\pi$
$632$		0			0
$633$		−	14.7575i		−	0.586557i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.9252	+	19.8648i	0.908329	+	0.787072i
$638$		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
$643$	35.0716	−	10.2979i	1.38309	−	0.406111i	0.496245	−	0.868183i	$-0.334712\pi$
$643$	35.0716	−	10.2979i	1.38309	−	0.406111i	0.886843	+	0.462072i	$0.152894\pi$
$644$		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
$651$	−40.1303	−	34.7731i	−1.57283	−	1.36287i
$652$		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
$657$	34.0676	+	21.8939i	1.32910	+	0.854162i
$658$		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
$661$	24.7004	−	21.4030i	0.960733	−	0.832480i	−0.0251892	−	0.999683i	$-0.508019\pi$
$661$	24.7004	−	21.4030i	0.960733	−	0.832480i	0.985923	+	0.167203i	$0.0534734\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	3.77318i		−	0.145880i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.719977	+	0.328802i	0.0277531	+	0.0126744i	0.429243	−	0.903189i	$-0.358780\pi$
$673$	0.719977	+	0.328802i	0.0277531	+	0.0126744i	−0.401490	+	0.915863i	$0.631508\pi$
$674$		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
$679$	41.9735	+	48.4400i	1.61079	+	1.85895i
$680$		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
$687$	−7.24188	−	50.3684i	−0.276295	−	1.92167i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−20.4972	+	6.01851i	−0.779749	+	0.228955i	−0.647300	−	0.762235i	$-0.724102\pi$
$691$	−20.4972	+	6.01851i	−0.779749	+	0.228955i	−0.132448	+	0.991190i	$0.542284\pi$
$692$		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
$703$	0.526459	−	0.607567i	0.0198558	−	0.0229148i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	44.8001	−	28.7913i	1.68250	−	1.08128i	0.839884	−	0.542766i	$-0.182623\pi$
$709$	44.8001	−	28.7913i	1.68250	−	1.08128i	0.842618	−	0.538512i	$-0.181013\pi$
$710$		0			0
$711$	−36.8462	+	5.29769i	−1.38184	+	0.198679i
$712$		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
$721$	−15.3470	+	52.2670i	−0.571551	+	1.94652i
$722$		0			0
$723$	−45.8783	−	6.59630i	−1.70623	−	0.245319i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−35.5804	+	16.2490i	−1.31961	+	0.602644i	−0.945766	−	0.324850i	$-0.894686\pi$
$727$	−35.5804	+	16.2490i	−1.31961	+	0.602644i	−0.373840	+	0.927493i	$0.621959\pi$
$728$		0			0
$729$	−22.7138	+	14.5973i	−0.841254	+	0.540641i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	23.9304	+	3.44067i	0.883888	+	0.127084i	0.569286	−	0.822140i	$-0.307220\pi$
$733$	23.9304	+	3.44067i	0.883888	+	0.127084i	0.314602	+	0.949224i	$0.398129\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−17.5576	+	15.2138i	−0.645867	+	0.559647i	−0.914999	−	0.403456i	$-0.867809\pi$
$739$	−17.5576	+	15.2138i	−0.645867	+	0.559647i	0.269132	+	0.963103i	$0.413263\pi$
$740$		0			0
$741$	−0.256832	+	1.78631i	−0.00943497	+	0.0656217i
$742$		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
$751$	−3.49306	+	24.2948i	−0.127464	+	0.886528i	0.821290	+	0.570511i	$0.193255\pi$
$751$	−3.49306	+	24.2948i	−0.127464	+	0.886528i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−48.9909	−	22.3734i	−1.78061	−	0.813176i	−0.975485	−	0.220065i	$-0.929373\pi$
$757$	−48.9909	−	22.3734i	−1.78061	−	0.813176i	−0.805121	−	0.593111i	$-0.797900\pi$
$758$		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
$763$	−10.1158	−	70.3568i	−0.366216	−	2.54709i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	43.3970	−	19.8187i	1.56494	−	0.714682i	0.570626	−	0.821210i	$-0.306701\pi$
$769$	43.3970	−	19.8187i	1.56494	−	0.714682i	0.994310	+	0.106528i	$0.0339735\pi$
$770$		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
$775$		−	44.2221i		−	1.58850i
$776$		0			0
$777$	−26.8839	+	7.89384i	−0.964456	+	0.283190i
$778$		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
$787$	27.0832	−	3.89398i	0.965412	−	0.138805i	0.358461	−	0.933545i	$-0.383302\pi$
$787$	27.0832	−	3.89398i	0.965412	−	0.138805i	0.606951	+	0.794739i	$0.292393\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	57.2038	+	16.7965i	2.03137	+	0.596463i
$794$		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
$811$	−16.2738	+	14.1013i	−0.571450	+	0.495165i	−0.891981	−	0.452074i	$-0.850684\pi$
$811$	−16.2738	+	14.1013i	−0.571450	+	0.495165i	0.320530	+	0.947238i	$0.396139\pi$
$812$		0			0
$813$	36.8672	+	42.5470i	1.29299	+	1.49219i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.411405	−	0.640159i	0.0143932	−	0.0223963i
$818$		0			0
$819$	41.1892	−	47.5349i	1.43927	−	1.66100i
$820$		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
$823$	10.2989	−	11.8855i	0.358996	−	0.414303i	−0.547307	−	0.836932i	$-0.684347\pi$
$823$	10.2989	−	11.8855i	0.358996	−	0.414303i	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
$829$	−34.3525	−	22.0770i	−1.19311	−	0.766766i	−0.215361	−	0.976535i	$-0.569093\pi$
$829$	−34.3525	−	22.0770i	−1.19311	−	0.766766i	−0.977751	+	0.209768i	$0.932729\pi$
$830$		0			0
$831$	8.23354	+	28.0409i	0.285618	+	0.972727i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−30.0954	+	34.7319i	−1.04025	+	1.20051i
$838$		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
$847$	37.7411	−	5.42635i	1.29680	−	0.186452i
$848$		0			0
$849$	52.9876	−	24.1986i	1.81853	−	0.830495i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.47699	−	52.0036i	0.256007	−	1.78057i	−0.304592	−	0.952483i	$-0.598520\pi$
$853$	7.47699	−	52.0036i	0.256007	−	1.78057i	0.560600	−	0.828087i	$-0.310571\pi$
$854$		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
$859$	31.7540	+	20.4070i	1.08343	+	0.696279i	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	31.7540	+	20.4070i	1.08343	+	0.696279i	0.128084	+	0.991763i	$0.459117\pi$
$860$		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
$871$	−39.6633	+	29.6307i	−1.34394	+	1.00400i
$872$		0			0
$873$	41.9237	−	36.3271i	1.41890	−	1.22949i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	56.5306	−	16.5989i	1.90890	−	0.560505i	0.925609	−	0.378481i	$-0.123554\pi$
$877$	56.5306	−	16.5989i	1.90890	−	0.560505i	0.983294	−	0.182023i	$-0.0582646\pi$
$878$		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
$883$	17.1603	−	26.7020i	0.577491	−	0.898594i	−0.422478	−	0.906373i	$-0.638840\pi$
$883$	17.1603	−	26.7020i	0.577491	−	0.898594i	0.999970	+	0.00777899i	$0.00247616\pi$
$884$		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
$889$	−77.0041	−	11.0715i	−2.58264	−	0.371327i
$890$		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
$903$	−24.1247	+	11.0174i	−0.802820	+	0.366635i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21.6138	−	13.8903i	0.717674	−	0.461221i	−0.130153	−	0.991494i	$-0.541547\pi$
$907$	21.6138	−	13.8903i	0.717674	−	0.461221i	0.847827	+	0.530273i	$0.177910\pi$
$908$		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
$919$	33.6793	+	29.1833i	1.11098	+	0.962668i	0.999516	−	0.0311006i	$-0.00990124\pi$
$919$	33.6793	+	29.1833i	1.11098	+	0.962668i	0.111462	+	0.993769i	$0.464447\pi$
$920$		0			0
$921$	−49.0025	+	7.04550i	−1.61469	+	0.232157i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−19.6301	−	12.6155i	−0.645433	−	0.414795i
$926$		0			0
$927$	45.2359	+	13.2825i	1.48574	+	0.436253i
$928$		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
$931$	−0.565753	−	0.652913i	−0.0185418	−	0.0213984i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	33.6232i		−	1.09842i	−0.835683	−	0.549212i	$-0.814928\pi$
$937$		−	33.6232i		−	1.09842i	0.835683	−	0.549212i	$-0.185072\pi$
$938$		0			0
$939$	2.63539			0.0860027
$940$		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
$949$	−23.0027	+	78.3399i	−0.746698	+	2.54302i
$950$		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
$961$	−19.6174	+	42.9561i	−0.632820	+	1.38568i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−61.8763			−1.98981			−0.994904	−	0.100831i	$-0.967850\pi$
$967$	−61.8763			−1.98981			−0.994904	+	0.100831i	$0.967850\pi$
$968$		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
$973$	−10.5965	−	23.2030i	−0.339707	−	0.743855i
$974$		0			0
$975$	52.3816			1.67755
$976$		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
$981$	−60.8923	+	8.75499i	−1.94414	+	0.279525i
$982$		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
$991$	−30.4993	−	4.38514i	−0.968842	−	0.139298i	−0.360314	−	0.932831i	$-0.617331\pi$
$991$	−30.4993	−	4.38514i	−0.968842	−	0.139298i	−0.608528	+	0.793533i	$0.708240\pi$
$992$		0			0
$993$	52.8975	+	33.9952i	1.67865	+	1.07880i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−44.2204	+	28.4187i	−1.40047	+	0.900030i	−0.999868	−	0.0162206i	$-0.994837\pi$
$997$	−44.2204	+	28.4187i	−1.40047	+	0.900030i	−0.400606	+	0.916251i	$0.631200\pi$
$998$		0			0
$999$	6.83194	+	23.2675i	0.216153	+	0.736150i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.161.2	yes	20
3.2	odd	2	CM	804.2.s.a.161.2	yes	20
67.5	odd	22	inner	804.2.s.a.5.2	✓	20
201.5	even	22	inner	804.2.s.a.5.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.5.2	✓	20	67.5	odd	22	inner
804.2.s.a.5.2	✓	20	201.5	even	22	inner
804.2.s.a.161.2	yes	20	1.1	even	1	trivial
804.2.s.a.161.2	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)

\(f(q)\)	\(=\)	\(q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\)	\(q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +(-3.27007 + 5.08833i) q^{13} +(0.112808 - 0.130187i) q^{19} +(-5.76060 + 1.69146i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(1.46393 + 4.98567i) q^{27} +(4.78165 + 7.44039i) q^{31} +4.66687 q^{37} +(-8.81324 + 5.66393i) q^{39} +(4.37247 - 0.628667i) q^{43} +(0.713735 - 4.96414i) q^{49} +(0.271404 - 0.123946i) q^{57} +(-2.77697 - 9.45750i) q^{61} +(-10.2930 - 1.47991i) q^{63} +(7.66645 + 2.86803i) q^{67} +(12.9519 - 3.80303i) q^{73} +(-4.68209 - 7.28547i) q^{75} +(-6.70847 + 10.4386i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-2.98376 - 20.7525i) q^{91} +(2.18012 + 15.1630i) q^{93} -18.4910i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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