LMFDB - Embedded newform 804.2.s.a.125.2

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			125.2
Root			$0.580057 - 0.814576i$ of defining polynomial
Character	$\chi$	$=$	804.125
Dual form			804.2.s.a.521.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})$	$q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(-6.83933 + 0.983347i) q^{13} +(7.27844 - 4.67757i) q^{19} +(-5.75924 - 6.64652i) q^{21} +(0.711574 + 4.94911i) q^{25} +(-3.92699 + 3.40276i) q^{27} +(-9.72847 - 1.39874i) q^{31} +5.43454 q^{37} +(-1.70321 + 11.8461i) q^{39} +(5.85766 + 2.67510i) q^{43} +(-7.80220 - 17.0844i) q^{49} +(-4.22191 - 14.3785i) q^{57} +(8.00476 - 6.93617i) q^{61} +(-13.8562 + 6.32789i) q^{63} +(-2.85294 - 7.67208i) q^{67} +(-2.56926 - 2.96508i) q^{73} +(8.57211 + 1.23248i) q^{75} +(1.47911 - 0.212664i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-14.5745 + 31.9138i) q^{91} +(-7.07181 + 15.4851i) q^{93} +9.35087i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{15}{22}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.487975	−	1.66189i	0.281733	−	0.959493i
$3$	0.487975	−	1.66189i	0.281733	−	0.959493i
$4$		0			0
$5$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$5$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$6$		0			0
$7$	2.74514	−	4.27152i	1.03757	−	1.61448i	0.281995	−	0.959416i	$-0.409004\pi$
$7$	2.74514	−	4.27152i	1.03757	−	1.61448i	0.755570	−	0.655068i	$-0.227360\pi$
$8$		0			0
$9$	−2.52376	−	1.62192i	−0.841254	−	0.540641i
$10$		0			0
$11$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$11$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$12$		0			0
$13$	−6.83933	+	0.983347i	−1.89689	+	0.272731i	−0.989162	−	0.146831i	$-0.953093\pi$
$13$	−6.83933	+	0.983347i	−1.89689	+	0.272731i	−0.907726	+	0.419563i	$0.862183\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$17$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$18$		0			0
$19$	7.27844	−	4.67757i	1.66979	−	1.07311i	0.768542	−	0.639799i	$-0.220983\pi$
$19$	7.27844	−	4.67757i	1.66979	−	1.07311i	0.901246	−	0.433308i	$-0.142654\pi$
$20$		0			0
$21$	−5.75924	−	6.64652i	−1.25677	−	1.45039i
$22$		0			0
$23$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$23$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$24$		0			0
$25$	0.711574	+	4.94911i	0.142315	+	0.989821i
$26$		0			0
$27$	−3.92699	+	3.40276i	−0.755750	+	0.654861i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−9.72847	−	1.39874i	−1.74728	−	0.251222i	−0.806744	−	0.590901i	$-0.798772\pi$
$31$	−9.72847	−	1.39874i	−1.74728	−	0.251222i	−0.940541	+	0.339680i	$0.889681\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.43454			0.893432			0.446716	−	0.894676i	$-0.352593\pi$
$37$	5.43454			0.893432			0.446716	+	0.894676i	$0.352593\pi$
$38$		0			0
$39$	−1.70321	+	11.8461i	−0.272731	+	1.89689i
$40$		0			0
$41$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$41$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$42$		0			0
$43$	5.85766	+	2.67510i	0.893285	+	0.407949i	0.808532	−	0.588453i	$-0.200263\pi$
$43$	5.85766	+	2.67510i	0.893285	+	0.407949i	0.0847529	+	0.996402i	$0.472990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$47$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$48$		0			0
$49$	−7.80220	−	17.0844i	−1.11460	−	2.44063i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$53$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.22191	−	14.3785i	−0.559205	−	1.90448i
$58$		0			0
$59$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$59$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$60$		0			0
$61$	8.00476	−	6.93617i	1.02490	−	0.888085i	0.0311325	−	0.999515i	$-0.490089\pi$
$61$	8.00476	−	6.93617i	1.02490	−	0.888085i	0.993772	+	0.111430i	$0.0355432\pi$
$62$		0			0
$63$	−13.8562	+	6.32789i	−1.74571	+	0.797240i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.85294	−	7.67208i	−0.348542	−	0.937293i
$67$	−2.85294	−	7.67208i	−0.348542	−	0.937293i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$71$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$72$		0			0
$73$	−2.56926	−	2.96508i	−0.300709	−	0.347037i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	−2.56926	−	2.96508i	−0.300709	−	0.347037i	−0.885915	+	0.463848i	$0.846468\pi$
$74$		0			0
$75$	8.57211	+	1.23248i	0.989821	+	0.142315i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.47911	−	0.212664i	0.166413	−	0.0239266i	−0.0586047	−	0.998281i	$-0.518665\pi$
$79$	1.47911	−	0.212664i	0.166413	−	0.0239266i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	3.73874	+	8.18669i	0.415415	+	0.909632i
$82$		0			0
$83$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$83$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$89$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$90$		0			0
$91$	−14.5745	+	31.9138i	−1.52783	+	3.34547i
$92$		0			0
$93$	−7.07181	+	15.4851i	−0.733312	+	1.60573i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			9.35087i			0.949437i	0.880138	+	0.474718i	$0.157450\pi$
$97$			9.35087i			0.949437i	−0.880138	+	0.474718i	$0.842550\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$101$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$102$		0			0
$103$	2.66099	−	18.5076i	0.262195	−	1.82361i	−0.254079	−	0.967183i	$-0.581772\pi$
$103$	2.66099	−	18.5076i	0.262195	−	1.82361i	0.516274	−	0.856423i	$-0.327319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$		0			0
$109$	11.9764	−	1.72194i	1.14713	−	0.164932i	0.457578	−	0.889170i	$-0.348717\pi$
$109$	11.9764	−	1.72194i	1.14713	−	0.164932i	0.689551	+	0.724238i	$0.257808\pi$
$110$		0			0
$111$	2.65192	−	9.03160i	0.251709	−	0.857242i
$112$		0			0
$113$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$113$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	18.8557	+	8.61113i	1.74321	+	0.796099i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.56546	+	10.8880i	0.142315	+	0.989821i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	11.0480	+	7.10014i	0.980354	+	0.630036i	0.929559	−	0.368674i	$-0.120188\pi$
$127$	11.0480	+	7.10014i	0.980354	+	0.630036i	0.0507955	+	0.998709i	$0.483824\pi$
$128$		0			0
$129$	7.30412	−	8.42940i	0.643092	−	0.742168i
$130$		0			0
$131$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$131$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$132$		0			0
$133$		−	43.9306i		−	3.80926i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$137$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$138$		0			0
$139$	15.9664	+	13.8350i	1.35426	+	1.17347i	0.967963	+	0.251093i	$0.0807900\pi$
$139$	15.9664	+	13.8350i	1.35426	+	1.17347i	0.386293	+	0.922376i	$0.373755\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−32.1997	+	4.62962i	−2.65579	+	0.381845i
$148$		0			0
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$	−6.52472	+	14.2871i	−0.530974	+	1.16267i	0.434141	+	0.900845i	$0.357052\pi$
$151$	−6.52472	+	14.2871i	−0.530974	+	1.16267i	−0.965115	+	0.261826i	$0.915675\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.518645	−	0.152288i	−0.0413924	−	0.0121539i	0.260971	−	0.965347i	$-0.415957\pi$
$157$	−0.518645	−	0.152288i	−0.0413924	−	0.0121539i	−0.302363	+	0.953193i	$0.597776\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	25.3807			1.98797			0.993986	−	0.109509i	$-0.0349277\pi$
$163$	25.3807			1.98797			0.993986	+	0.109509i	$0.0349277\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$167$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$168$		0			0
$169$	33.3360	−	9.78834i	2.56431	−	0.752949i
$170$		0			0
$171$	−25.9557			−1.98488
$172$		0			0
$173$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$173$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$174$		0			0
$175$	23.0936	+	10.5465i	1.74571	+	0.797240i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$179$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$180$		0			0
$181$	22.0635	+	6.47842i	1.63996	+	0.481537i	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	22.0635	+	6.47842i	1.63996	+	0.481537i	0.673682	+	0.739022i	$0.264712\pi$
$182$		0			0
$183$	−7.62102	−	16.6877i	−0.563362	−	1.23359i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3.75481	+	26.1153i	0.273122	+	1.89961i
$190$		0			0
$191$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$191$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$192$		0			0
$193$	1.89281	−	13.1648i	0.136248	−	0.947623i	−0.800927	−	0.598762i	$-0.795660\pi$
$193$	1.89281	−	13.1648i	0.136248	−	0.947623i	0.937175	−	0.348861i	$-0.113431\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$197$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$198$		0			0
$199$	7.13430	+	4.58494i	0.505737	+	0.325018i	0.768507	−	0.639841i	$-0.221000\pi$
$199$	7.13430	+	4.58494i	0.505737	+	0.325018i	−0.262770	+	0.964858i	$0.584636\pi$
$200$		0			0
$201$	−14.1423	+	0.997492i	−0.997522	+	0.0703577i
$202$		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
$211$	27.6370	−	8.11495i	1.90261	−	0.558656i	0.914574	−	0.404419i	$-0.132526\pi$
$211$	27.6370	−	8.11495i	1.90261	−	0.558656i	0.988034	−	0.154237i	$-0.0492918\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−32.6808	+	37.7156i	−2.21852	+	2.56030i
$218$		0			0
$219$	−6.18138	+	2.82294i	−0.417699	+	0.190757i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−28.5868	+	8.39384i	−1.91431	+	0.562093i	−0.937509	+	0.347960i	$0.886874\pi$
$223$	−28.5868	+	8.39384i	−1.91431	+	0.562093i	−0.976804	+	0.214133i	$0.931307\pi$
$224$		0			0
$225$	6.23123	−	13.6445i	0.415415	−	0.909632i
$226$		0			0
$227$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$227$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$228$		0			0
$229$	−24.4823	−	3.52002i	−1.61783	−	0.232609i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−24.4823	−	3.52002i	−1.61783	−	0.232609i	−0.890934	+	0.454133i	$0.849949\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$233$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.368345	−	2.56189i	0.0239266	−	0.166413i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	19.2681	+	22.2365i	1.24116	+	1.43238i	0.861889	+	0.507097i	$0.169281\pi$
$241$	19.2681	+	22.2365i	1.24116	+	1.43238i	0.379276	+	0.925284i	$0.376173\pi$
$242$		0			0
$243$	15.4298	−	2.21847i	0.989821	−	0.142315i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−45.1799	+	39.1486i	−2.87473	+	2.49097i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$251$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$257$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$258$		0			0
$259$	14.9186	−	23.2137i	0.926994	−	1.44243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		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
$271$	−8.36996	+	28.5055i	−0.508439	+	1.73158i	0.159340	+	0.987224i	$0.449063\pi$
$271$	−8.36996	+	28.5055i	−0.508439	+	1.73158i	−0.667779	+	0.744360i	$0.732755\pi$
$272$		0			0
$273$	45.9252	+	39.7944i	2.77952	+	2.40847i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−24.9912	−	16.0609i	−1.50158	−	0.965005i	−0.994682	−	0.102998i	$-0.967156\pi$
$277$	−24.9912	−	16.0609i	−1.50158	−	0.965005i	−0.506896	−	0.862007i	$-0.669207\pi$
$278$		0			0
$279$	22.2837	+	19.3089i	1.33409	+	1.15599i
$280$		0			0
$281$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$281$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$282$		0			0
$283$	−11.0217	+	7.08324i	−0.655174	+	0.421055i	−0.825554	−	0.564324i	$-0.809137\pi$
$283$	−11.0217	+	7.08324i	−0.655174	+	0.421055i	0.170379	+	0.985379i	$0.445501\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−11.1326	−	12.8477i	−0.654861	−	0.755750i
$290$		0			0
$291$	15.5401	+	4.56299i	0.910978	+	0.267487i
$292$		0			0
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	27.5069	−	17.6776i	1.58547	−	1.01892i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	4.98190	−	34.6498i	0.284332	−	1.97757i	0.0968945	−	0.995295i	$-0.469109\pi$
$307$	4.98190	−	34.6498i	0.284332	−	1.97757i	0.187437	−	0.982277i	$-0.439982\pi$
$308$		0			0
$309$	−29.4591	−	13.4535i	−1.67587	−	0.765344i
$310$		0			0
$311$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$311$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$312$		0			0
$313$	−3.21722	−	10.9568i	−0.181848	−	0.619317i	−0.999075	−	0.0430013i	$-0.986308\pi$
$313$	−3.21722	−	10.9568i	−0.181848	−	0.619317i	0.817227	−	0.576316i	$-0.195510\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$317$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−9.73338	−	33.1488i	−0.539911	−	1.83877i
$326$		0			0
$327$	2.98249	−	20.7437i	0.164932	−	1.14713i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.91635	+	3.15859i	−0.380157	+	0.173612i	−0.596323	−	0.802745i	$-0.703372\pi$
$331$	−6.91635	+	3.15859i	−0.380157	+	0.173612i	0.216166	+	0.976357i	$0.430645\pi$
$332$		0			0
$333$	−13.7155	−	8.81440i	−0.751603	−	0.483026i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−19.2608	+	29.9704i	−1.04920	+	1.63259i	−0.321211	+	0.947008i	$0.604090\pi$
$337$	−19.2608	+	29.9704i	−1.04920	+	1.63259i	−0.727993	+	0.685585i	$0.759547\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−59.2134	−	8.51361i	−3.19723	−	0.459692i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$347$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$348$		0			0
$349$	2.88747	+	6.32267i	0.154562	+	0.338445i	0.971034	−	0.238941i	$-0.0768002\pi$
$349$	2.88747	+	6.32267i	0.154562	+	0.338445i	−0.816472	+	0.577386i	$0.804073\pi$
$350$		0			0
$351$	23.5119	−	27.1342i	1.25497	−	1.44831i
$352$		0			0
$353$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$353$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	23.2031	−	50.8077i	1.22122	−	2.67409i
$362$		0			0
$363$	18.8586	+	2.71146i	0.989821	+	0.142315i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6.17855	−	21.0422i	−0.322518	−	1.09839i	−0.948030	−	0.318180i	$-0.896928\pi$
$367$	−6.17855	−	21.0422i	−0.322518	−	1.09839i	0.625513	−	0.780214i	$-0.284890\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	26.3727i		−	1.36553i	−0.730639	−	0.682764i	$-0.760778\pi$
$373$		−	26.3727i		−	1.36553i	0.730639	−	0.682764i	$-0.239222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−10.2498	+	34.9075i	−0.526495	+	1.79308i	0.0785782	+	0.996908i	$0.474962\pi$
$379$	−10.2498	+	34.9075i	−0.526495	+	1.79308i	−0.605073	+	0.796170i	$0.706856\pi$
$380$		0			0
$381$	17.1908	−	14.8959i	0.880712	−	0.763142i
$382$		0			0
$383$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$383$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.4445	−	16.2520i	−0.530925	−	0.826135i
$388$		0			0
$389$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$389$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	3.13170	−	3.61417i	0.157175	−	0.181390i	−0.671700	−	0.740823i	$-0.734436\pi$
$397$	3.13170	−	3.61417i	0.157175	−	0.181390i	0.828876	+	0.559433i	$0.188981\pi$
$398$		0			0
$399$	−73.0078	−	21.4370i	−3.65496	−	1.07319i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	67.9116			3.38292
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.6206	+	16.5260i	−0.525155	+	0.817157i	−0.997949	−	0.0640160i	$-0.979609\pi$
$409$	−10.6206	+	16.5260i	−0.525155	+	0.817157i	0.472794	+	0.881173i	$0.343245\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	30.7835	−	19.7833i	1.50747	−	0.968794i
$418$		0			0
$419$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$419$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$	−31.4792	+	20.2304i	−1.53420	+	0.985971i	−0.545159	+	0.838333i	$0.683531\pi$
$421$	−31.4792	+	20.2304i	−1.53420	+	0.985971i	−0.989041	+	0.147638i	$0.952833\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.65379	−	53.2333i	−0.370393	−	2.57614i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	3.89621	+	0.560190i	0.187240	+	0.0269210i	0.235297	−	0.971924i	$-0.424394\pi$
$433$	3.89621	+	0.560190i	0.187240	+	0.0269210i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−16.7279			−0.798379			−0.399190	−	0.916868i	$-0.630708\pi$
$439$	−16.7279			−0.798379			−0.399190	+	0.916868i	$0.630708\pi$
$440$		0			0
$441$	−8.01874	+	55.7716i	−0.381845	+	2.65579i
$442$		0			0
$443$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$443$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$449$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	20.5598	+	17.8151i	0.965982	+	0.837028i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.39561	+	30.5722i	0.205618	+	1.43011i	0.787240	+	0.616647i	$0.211509\pi$
$457$	4.39561	+	30.5722i	0.205618	+	1.43011i	−0.581622	+	0.813459i	$0.697582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$461$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$	32.3981	−	28.0731i	1.50567	−	1.30467i	0.695374	−	0.718648i	$-0.255239\pi$
$463$	32.3981	−	28.0731i	1.50567	−	1.30467i	0.810296	−	0.586021i	$-0.199307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$467$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$468$		0			0
$469$	−40.6032	−	8.87453i	−1.87488	−	0.409788i
$470$		0			0
$471$	−0.506172	+	0.787619i	−0.0233232	+	0.0362916i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	28.3289	+	32.6933i	1.29982	+	1.50007i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$479$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$480$		0			0
$481$	−37.1686	+	5.34403i	−1.69474	+	0.243667i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−37.7159	+	17.2243i	−1.70907	+	0.780507i	−0.712157	+	0.702020i	$0.752282\pi$
$487$	−37.7159	+	17.2243i	−1.70907	+	0.780507i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	12.3852	−	42.1800i	0.560076	−	1.90744i
$490$		0			0
$491$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$491$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			36.1112i			1.61656i	0.588798	+	0.808280i	$0.299601\pi$
$499$			36.1112i			1.61656i	−0.588798	+	0.808280i	$0.700399\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$503$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	60.1773i		−	2.67257i
$508$		0			0
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$	−19.7184	+	2.83508i	−0.872290	+	0.125416i
$512$		0			0
$513$	−12.6657	+	43.1355i	−0.559205	+	1.90448i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$521$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$522$		0			0
$523$	−2.50714	−	17.4375i	−0.109629	−	0.762489i	−0.968269	−	0.249910i	$-0.919599\pi$
$523$	−2.50714	−	17.4375i	−0.109629	−	0.762489i	0.858640	−	0.512579i	$-0.171310\pi$
$524$		0			0
$525$	28.7962	−	33.2326i	1.25677	−	1.45039i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.3488	+	12.4347i	0.841254	+	0.540641i
$530$		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
$541$	−29.4478	−	25.5166i	−1.26606	−	1.09705i	−0.990751	−	0.135690i	$-0.956675\pi$
$541$	−29.4478	−	25.5166i	−1.26606	−	1.09705i	−0.275307	−	0.961356i	$-0.588780\pi$
$542$		0			0
$543$	21.5328	−	33.5058i	0.924063	−	1.43787i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	35.1110	+	30.4239i	1.50124	+	1.30083i	0.826791	+	0.562510i	$0.190164\pi$
$547$	35.1110	+	30.4239i	1.50124	+	1.30083i	0.674449	+	0.738321i	$0.264381\pi$
$548$		0			0
$549$	−31.4520	+	4.52212i	−1.34234	+	0.192999i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	3.15197	−	6.90185i	0.134035	−	0.293496i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$557$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$558$		0			0
$559$	−42.6930	−	12.5358i	−1.80572	−	0.530208i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$563$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	45.2330	+	6.50352i	1.89961	+	0.273122i
$568$		0			0
$569$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$569$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$570$		0			0
$571$	−2.55660	+	0.750686i	−0.106990	+	0.0314152i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−2.55660	+	0.750686i	−0.106990	+	0.0314152i	0.227799	+	0.973708i	$0.426847\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−23.3689	−	10.6722i	−0.972859	−	0.444290i	−0.135400	−	0.990791i	$-0.543232\pi$
$577$	−23.3689	−	10.6722i	−0.972859	−	0.444290i	−0.837458	+	0.546501i	$0.815959\pi$
$578$		0			0
$579$	−20.9548	−	9.56974i	−0.870852	−	0.397705i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$587$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$588$		0			0
$589$	−77.3508	+	35.3249i	−3.18718	+	1.45554i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$593$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	11.1010	−	9.61910i	0.454335	−	0.393683i
$598$		0			0
$599$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$599$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$600$		0			0
$601$	−12.1053	−	7.77957i	−0.493784	−	0.317335i	0.269942	−	0.962877i	$-0.412995\pi$
$601$	−12.1053	−	7.77957i	−0.493784	−	0.317335i	−0.763725	+	0.645541i	$0.776632\pi$
$602$		0			0
$603$	−5.24337	+	23.9897i	−0.213527	+	0.976937i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.12466	−	6.84205i	−0.126826	−	0.277710i	0.835558	−	0.549402i	$-0.185144\pi$
$607$	−3.12466	−	6.84205i	−0.126826	−	0.277710i	−0.962384	+	0.271691i	$0.912417\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	46.3137	−	13.5989i	1.87059	−	0.549256i	0.872426	−	0.488746i	$-0.162545\pi$
$613$	46.3137	−	13.5989i	1.87059	−	0.549256i	0.998168	−	0.0605099i	$-0.0192727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$617$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$618$		0			0
$619$	32.5816	−	37.6012i	1.30956	−	1.51132i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.5816	−	37.6012i	1.30956	−	1.51132i	0.666471	−	0.745531i	$-0.267804\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.9873	+	7.04331i	−0.959493	+	0.281733i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	35.2997	+	5.07533i	1.40526	+	0.202046i	0.802869	−	0.596156i	$-0.203306\pi$
$631$	35.2997	+	5.07533i	1.40526	+	0.202046i	0.602390	+	0.798202i	$0.294215\pi$
$632$		0			0
$633$		−	49.8895i		−	1.98293i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	70.1617	+	109.174i	2.77991	+	4.32562i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	7.68998	+	8.87471i	0.303263	+	0.349984i	0.886843	−	0.462072i	$-0.152894\pi$
$643$	7.68998	+	8.87471i	0.303263	+	0.349984i	−0.583579	+	0.812056i	$0.698348\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$647$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	46.7318	+	72.7162i	1.83157	+	2.84997i
$652$		0			0
$653$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$653$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.67506	+	11.6503i	0.0653503	+	0.454521i
$658$		0			0
$659$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$659$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$660$		0			0
$661$	−26.4678	+	41.1848i	−1.02948	+	1.60190i	−0.257474	+	0.966285i	$0.582890\pi$
$661$	−26.4678	+	41.1848i	−1.02948	+	1.60190i	−0.772005	+	0.635616i	$0.780746\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			51.6041i			1.99513i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−13.2024	+	44.9633i	−0.508915	+	1.73321i	0.157398	+	0.987535i	$0.449689\pi$
$673$	−13.2024	+	44.9633i	−0.508915	+	1.73321i	−0.666314	+	0.745672i	$0.732129\pi$
$674$		0			0
$675$	−19.6349	−	17.0138i	−0.755750	−	0.654861i
$676$		0			0
$677$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$677$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$678$		0			0
$679$	39.9424	+	25.6694i	1.53285	+	0.985103i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$683$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−17.7966	+	38.9692i	−0.678984	+	1.48677i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	32.9356	+	38.0098i	1.25293	+	1.44596i	0.846597	+	0.532234i	$0.178648\pi$
$691$	32.9356	+	38.0098i	1.25293	+	1.44596i	0.406334	+	0.913725i	$0.366807\pi$
$692$		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		−0.989821	−	0.142315i	$-0.954545\pi$
$701$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$702$		0			0
$703$	39.5549	−	25.4204i	1.49184	−	0.958749i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.05961	−	7.36976i	0.0397945	−	0.276777i	−0.960202	−	0.279306i	$-0.909896\pi$
$709$	1.05961	−	7.36976i	0.0397945	−	0.276777i	0.999997	+	0.00252851i	$0.000804851\pi$
$710$		0			0
$711$	−4.07785	−	1.86229i	−0.152931	−	0.0698413i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$719$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$720$		0			0
$721$	−71.7508	−	62.1724i	−2.67214	−	2.31542i
$722$		0			0
$723$	46.3570	−	21.1705i	1.72404	−	0.787341i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−14.9248	−	50.8293i	−0.553531	−	1.88515i	−0.456140	−	0.889908i	$-0.650768\pi$
$727$	−14.9248	−	50.8293i	−0.553531	−	1.88515i	−0.0973917	−	0.995246i	$-0.531050\pi$
$728$		0			0
$729$	3.84250	−	26.7252i	0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−8.68418	+	3.96593i	−0.320757	+	0.146485i	−0.569286	−	0.822140i	$-0.692780\pi$
$733$	−8.68418	+	3.96593i	−0.320757	+	0.146485i	0.248529	+	0.968625i	$0.420053\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−3.80151	+	5.91526i	−0.139841	+	0.217596i	−0.904109	−	0.427302i	$-0.859464\pi$
$739$	−3.80151	+	5.91526i	−0.139841	+	0.217596i	0.764268	+	0.644899i	$0.223100\pi$
$740$		0			0
$741$	43.0141	+	94.1877i	1.58016	+	3.46007i
$742$		0			0
$743$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$743$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.42864	−	5.31797i	−0.0886222	−	0.194056i	0.860131	−	0.510073i	$-0.170382\pi$
$751$	−2.42864	−	5.31797i	−0.0886222	−	0.194056i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.41166	+	11.6190i	−0.123999	+	0.422301i	−0.997971	−	0.0636752i	$-0.979718\pi$
$757$	−3.41166	+	11.6190i	−0.123999	+	0.422301i	0.873972	+	0.485977i	$0.161536\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$761$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$762$		0			0
$763$	25.5215	−	55.8843i	0.923940	−	2.02315i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−2.78000	−	9.46783i	−0.100250	−	0.341419i	0.894060	−	0.447947i	$-0.147845\pi$
$769$	−2.78000	−	9.46783i	−0.100250	−	0.341419i	−0.994310	+	0.106528i	$0.966026\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$773$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$774$		0			0
$775$		−	49.1426i		−	1.76525i
$776$		0			0
$777$	−31.2988	−	36.1207i	−1.12284	−	1.29582i
$778$		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
$787$	−49.9725	−	22.8217i	−1.78133	−	0.813504i	−0.975057	−	0.221955i	$-0.928756\pi$
$787$	−49.9725	−	22.8217i	−1.78133	−	0.813504i	−0.806269	−	0.591549i	$-0.798517\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−47.9265	+	55.3102i	−1.70192	+	1.96412i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$797$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$798$		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		−0.755750	−	0.654861i	$-0.772727\pi$
$809$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$810$		0			0
$811$	−29.1681	+	45.3865i	−1.02423	+	1.59374i	−0.242469	+	0.970159i	$0.577957\pi$
$811$	−29.1681	+	45.3865i	−1.02423	+	1.59374i	−0.781763	+	0.623576i	$0.785679\pi$
$812$		0			0
$813$	43.2886	+	27.8199i	1.51820	+	0.975687i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	55.1476	−	7.92902i	1.92937	−	0.277401i
$818$		0			0
$819$	88.5443	−	56.9039i	3.09399	−	1.98838i
$820$		0			0
$821$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$821$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$822$		0			0
$823$	25.7721	−	16.5627i	0.898359	−	0.577340i	−0.00794450	−	0.999968i	$-0.502529\pi$
$823$	25.7721	−	16.5627i	0.898359	−	0.577340i	0.906303	+	0.422628i	$0.138892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$827$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$828$		0			0
$829$	1.76492	+	12.2753i	0.0612980	+	0.426337i	0.997244	+	0.0741933i	$0.0236382\pi$
$829$	1.76492	+	12.2753i	0.0612980	+	0.426337i	−0.935946	+	0.352144i	$0.885453\pi$
$830$		0			0
$831$	−38.8865	+	33.6954i	−1.34896	+	1.16888i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	42.9632	−	27.6108i	1.48503	−	0.954368i
$838$		0			0
$839$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$839$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	50.8059	+	23.2023i	1.74571	+	0.797240i
$848$		0			0
$849$	6.39324	+	21.7734i	0.219415	+	0.747260i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	6.27773	+	13.7463i	0.214945	+	0.470665i	0.986136	−	0.165940i	$-0.0530657\pi$
$853$	6.27773	+	13.7463i	0.214945	+	0.470665i	−0.771191	+	0.636604i	$0.780338\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$857$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$858$		0			0
$859$	−1.30568	−	9.08120i	−0.0445492	−	0.309846i	−0.999897	−	0.0143623i	$-0.995428\pi$
$859$	−1.30568	−	9.08120i	−0.0445492	−	0.309846i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−26.7840	+	12.2318i	−0.909632	+	0.415415i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	27.0565	+	49.6664i	0.916774	+	1.68288i
$872$		0			0
$873$	15.1664	−	23.5994i	0.513304	−	0.798717i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−35.9010	−	41.4319i	−1.21229	−	1.39906i	−0.892182	−	0.451676i	$-0.850826\pi$
$877$	−35.9010	−	41.4319i	−1.21229	−	1.39906i	−0.320108	−	0.947381i	$-0.603719\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$881$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$882$		0			0
$883$	53.6982	−	7.72064i	1.80709	−	0.259820i	0.845434	−	0.534080i	$-0.179342\pi$
$883$	53.6982	−	7.72064i	1.80709	−	0.259820i	0.961656	+	0.274260i	$0.0884329\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$887$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$888$		0			0
$889$	60.6568	−	27.7010i	2.03436	−	0.929063i
$890$		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
$903$	−15.9555	−	54.3396i	−0.530967	−	1.80831i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−8.57139	+	59.6153i	−0.284608	+	1.97949i	−0.130153	+	0.991494i	$0.541547\pi$
$907$	−8.57139	+	59.6153i	−0.284608	+	1.97949i	−0.154455	+	0.988000i	$0.549362\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	10.2086	+	15.8849i	0.336751	+	0.523995i	0.967791	−	0.251755i	$-0.0810079\pi$
$919$	10.2086	+	15.8849i	0.336751	+	0.523995i	−0.631040	+	0.775751i	$0.717371\pi$
$920$		0			0
$921$	−55.1532	−	25.1876i	−1.81736	−	0.829961i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.86708	+	26.8961i	0.127149	+	0.884338i
$926$		0			0
$927$	−36.7336	+	42.3928i	−1.20649	+	1.39236i
$928$		0			0
$929$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$929$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$930$		0			0
$931$	−136.701	−	87.8526i	−4.48021	−	2.87925i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	0.625795i		−	0.0204438i	−0.999948	−	0.0102219i	$-0.996746\pi$
$937$		−	0.625795i		−	0.0204438i	0.999948	−	0.0102219i	$-0.00325379\pi$
$938$		0			0
$939$	−19.7790			−0.645463
$940$		0			0
$941$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$941$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$947$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$948$		0			0
$949$	20.4877	+	17.7527i	0.665059	+	0.576277i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$953$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	62.9424	+	18.4815i	2.03040	+	0.596179i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.59871			0.0835688			0.0417844	−	0.999127i	$-0.486696\pi$
$967$	2.59871			0.0835688			0.0417844	+	0.999127i	$0.486696\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$971$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$972$		0			0
$973$	102.927	−	30.2220i	3.29968	−	0.968872i
$974$		0			0
$975$	−59.8394			−1.91639
$976$		0			0
$977$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$977$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−33.0183	−	15.0790i	−1.05419	−	0.481435i
$982$		0			0
$983$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$983$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−57.0735	+	26.0646i	−1.81300	+	0.827970i	−0.872058	+	0.489402i	$0.837215\pi$
$991$	−57.0735	+	26.0646i	−1.81300	+	0.827970i	−0.940942	+	0.338567i	$0.890058\pi$
$992$		0			0
$993$	1.87423	+	13.0355i	0.0594768	+	0.413670i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.86556	−	26.8856i	0.122424	−	0.851474i	−0.832373	−	0.554215i	$-0.813018\pi$
$997$	3.86556	−	26.8856i	0.122424	−	0.851474i	0.954797	−	0.297259i	$-0.0960725\pi$
$998$		0			0
$999$	−21.3414	+	18.4924i	−0.675211	+	0.585074i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.s.a.125.2	✓	20
3.2	odd	2	CM	804.2.s.a.125.2	✓	20
67.52	odd	22	inner	804.2.s.a.521.2	yes	20
201.119	even	22	inner	804.2.s.a.521.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.s.a.125.2	✓	20	1.1	even	1	trivial
804.2.s.a.125.2	✓	20	3.2	odd	2	CM
804.2.s.a.521.2	yes	20	67.52	odd	22	inner
804.2.s.a.521.2	yes	20	201.119	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)

\(f(q)\)	\(=\)	\(q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)	\(q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(-6.83933 + 0.983347i) q^{13} +(7.27844 - 4.67757i) q^{19} +(-5.75924 - 6.64652i) q^{21} +(0.711574 + 4.94911i) q^{25} +(-3.92699 + 3.40276i) q^{27} +(-9.72847 - 1.39874i) q^{31} +5.43454 q^{37} +(-1.70321 + 11.8461i) q^{39} +(5.85766 + 2.67510i) q^{43} +(-7.80220 - 17.0844i) q^{49} +(-4.22191 - 14.3785i) q^{57} +(8.00476 - 6.93617i) q^{61} +(-13.8562 + 6.32789i) q^{63} +(-2.85294 - 7.67208i) q^{67} +(-2.56926 - 2.96508i) q^{73} +(8.57211 + 1.23248i) q^{75} +(1.47911 - 0.212664i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-14.5745 + 31.9138i) q^{91} +(-7.07181 + 15.4851i) q^{93} +9.35087i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{9}+O(q^{10}) \) 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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