LMFDB - Embedded newform 804.2.ba.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(41,804)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 53]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("804.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.ba (of order $66$, degree $20$, 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:	$20$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			113.1
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	804.113
Dual form			804.2.ba.a.185.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.57553 + 0.719520i) q^{3} +(3.27565 + 1.13371i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})$	$q+(1.57553 + 0.719520i) q^{3} +(3.27565 + 1.13371i) q^{7} +(1.96458 + 2.26725i) q^{9} +(-0.172289 - 0.334195i) q^{13} +(0.0563415 + 0.162788i) q^{19} +(4.34515 + 4.14309i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(1.46393 + 4.98567i) q^{27} +(0.658548 - 1.27741i) q^{31} +(-2.33343 + 4.04162i) q^{37} +(-0.0309871 - 0.650499i) q^{39} +(8.39906 - 1.20760i) q^{43} +(3.94220 + 3.10018i) q^{49} +(-0.0283616 + 0.297016i) q^{57} +(5.40915 + 1.31225i) q^{61} +(3.86487 + 9.65398i) q^{63} +(-1.34943 - 8.07335i) q^{67} +(-0.548063 + 2.25915i) q^{73} +(-4.68209 - 7.28547i) q^{75} +(-17.2085 - 0.819741i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-0.185479 - 1.29003i) q^{91} +(1.95668 - 1.53875i) q^{93} +(-2.91553 - 1.68328i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^7 + 6 * q^9	$ 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) $ 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{29}{66}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.57553	+	0.719520i	0.909632	+	0.415415i
$3$	1.57553	+	0.719520i	0.909632	+	0.415415i
$4$		0			0
$5$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$5$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$6$		0			0
$7$	3.27565	+	1.13371i	1.23808	+	0.428503i	0.866162	−	0.499763i	$-0.166579\pi$
$7$	3.27565	+	1.13371i	1.23808	+	0.428503i	0.371917	+	0.928266i	$0.378701\pi$
$8$		0			0
$9$	1.96458	+	2.26725i	0.654861	+	0.755750i
$10$		0			0
$11$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$11$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$12$		0			0
$13$	−0.172289	−	0.334195i	−0.0477845	−	0.0926889i	0.863734	−	0.503948i	$-0.168120\pi$
$13$	−0.172289	−	0.334195i	−0.0477845	−	0.0926889i	−0.911519	+	0.411259i	$0.865089\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$17$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$18$		0			0
$19$	0.0563415	+	0.162788i	0.0129256	+	0.0373462i	0.951279	−	0.308331i	$-0.0997704\pi$
$19$	0.0563415	+	0.162788i	0.0129256	+	0.0373462i	−0.938353	+	0.345677i	$0.887649\pi$
$20$		0			0
$21$	4.34515	+	4.14309i	0.948190	+	0.904097i
$22$		0			0
$23$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$23$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$24$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	0.658548	−	1.27741i	0.118279	−	0.229429i	−0.822262	−	0.569109i	$-0.807288\pi$
$31$	0.658548	−	1.27741i	0.118279	−	0.229429i	0.940541	+	0.339680i	$0.110319\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.33343	+	4.04162i	−0.383614	+	0.664439i	−0.991576	−	0.129527i	$-0.958654\pi$
$37$	−2.33343	+	4.04162i	−0.383614	+	0.664439i	0.607962	+	0.793966i	$0.291987\pi$
$38$		0			0
$39$	−0.0309871	−	0.650499i	−0.00496191	−	0.104163i
$40$		0			0
$41$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$41$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$42$		0			0
$43$	8.39906	−	1.20760i	1.28084	−	0.184158i	0.531909	−	0.846801i	$-0.321475\pi$
$43$	8.39906	−	1.20760i	1.28084	−	0.184158i	0.748935	+	0.662644i	$0.230566\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$47$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$48$		0			0
$49$	3.94220	+	3.10018i	0.563172	+	0.442883i
$50$		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
$57$	−0.0283616	+	0.297016i	−0.00375659	+	0.0393408i
$58$		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
$61$	5.40915	+	1.31225i	0.692571	+	0.168016i	0.566569	−	0.824014i	$-0.308270\pi$
$61$	5.40915	+	1.31225i	0.692571	+	0.168016i	0.126001	+	0.992030i	$0.459786\pi$
$62$		0			0
$63$	3.86487	+	9.65398i	0.486928	+	1.21629i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.34943	−	8.07335i	−0.164860	−	0.986317i
$67$	−1.34943	−	8.07335i	−0.164860	−	0.986317i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$71$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$72$		0			0
$73$	−0.548063	+	2.25915i	−0.0641459	+	0.264413i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−0.548063	+	2.25915i	−0.0641459	+	0.264413i	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
$79$	−17.2085	−	0.819741i	−1.93610	−	0.0922280i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.2085	−	0.819741i	−1.93610	−	0.0922280i	−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		0.723734	−	0.690079i	$-0.242424\pi$
$83$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$84$		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
$91$	−0.185479	−	1.29003i	−0.0194434	−	0.135232i
$92$		0			0
$93$	1.95668	−	1.53875i	0.202898	−	0.159561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.91553	−	1.68328i	−0.296027	−	0.170911i	0.344630	−	0.938739i	$-0.388004\pi$
$97$	−2.91553	−	1.68328i	−0.296027	−	0.170911i	−0.640657	+	0.767828i	$0.721338\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$101$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$102$		0			0
$103$	−13.9682	−	7.20113i	−1.37633	−	0.709548i	−0.398153	−	0.917319i	$-0.630349\pi$
$103$	−13.9682	−	7.20113i	−1.37633	−	0.709548i	−0.978178	+	0.207771i	$0.933379\pi$
$104$		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
$109$	7.38644	−	11.4935i	0.707492	−	1.10088i	−0.282433	−	0.959287i	$-0.591141\pi$
$109$	7.38644	−	11.4935i	0.707492	−	1.10088i	0.989925	−	0.141592i	$-0.0452222\pi$
$110$		0			0
$111$	−6.58442	+	4.68874i	−0.624965	+	0.445036i
$112$		0			0
$113$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$113$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.419226	−	1.04718i	0.0387575	−	0.0968114i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.77719	−	5.04049i	0.888835	−	0.458227i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.08389	+	8.91031i	−0.273651	+	0.790663i	0.721510	+	0.692404i	$0.243448\pi$
$127$	−3.08389	+	8.91031i	−0.273651	+	0.790663i	−0.995161	+	0.0982585i	$0.968673\pi$
$128$		0			0
$129$	14.1018	+	4.14068i	1.24160	+	0.364566i
$130$		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
$139$	6.50241	−	22.1452i	0.551528	−	1.87833i	0.0793066	−	0.996850i	$-0.474729\pi$
$139$	6.50241	−	22.1452i	0.551528	−	1.87833i	0.472221	−	0.881480i	$-0.343452\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.98041	+	7.72092i	0.328299	+	0.636811i
$148$		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
$151$	−22.7526	−	9.10878i	−1.85158	−	0.741262i	−0.953460	−	0.301518i	$-0.902507\pi$
$151$	−22.7526	−	9.10878i	−1.85158	−	0.741262i	−0.898123	−	0.439744i	$-0.855069\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.79352	+	5.32725i	0.302756	+	0.425161i	0.937726	−	0.347375i	$-0.112927\pi$
$157$	3.79352	+	5.32725i	0.302756	+	0.425161i	−0.634970	+	0.772537i	$0.718988\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.792696	+	1.37299i	0.0620887	+	0.107541i	0.895399	−	0.445265i	$-0.146890\pi$
$163$	0.792696	+	1.37299i	0.0620887	+	0.107541i	−0.833310	+	0.552806i	$0.813557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$167$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$168$		0			0
$169$	7.45874	−	10.4743i	0.573749	−	0.805718i
$170$		0			0
$171$	−0.258394	+	0.447551i	−0.0197599	+	0.0342251i
$172$		0			0
$173$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$173$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$174$		0			0
$175$	−10.7136	−	13.6235i	−0.809872	−	1.02984i
$176$		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
$181$	−9.39520	+	0.897132i	−0.698339	+	0.0666833i	−0.438186	−	0.898884i	$-0.644379\pi$
$181$	−9.39520	+	0.897132i	−0.698339	+	0.0666833i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	7.57808	+	5.95947i	0.560188	+	0.440537i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.857016	+	17.9910i	−0.0623388	+	1.30865i
$190$		0			0
$191$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$191$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$192$		0			0
$193$	6.06421	−	3.89723i	0.436511	−	0.280529i	−0.303873	−	0.952713i	$-0.598280\pi$
$193$	6.06421	−	3.89723i	0.436511	−	0.280529i	0.740384	+	0.672184i	$0.234643\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$197$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$198$		0			0
$199$	−19.5091	+	3.76007i	−1.38296	+	0.266544i	−0.825848	−	0.563892i	$-0.809303\pi$
$199$	−19.5091	+	3.76007i	−1.38296	+	0.266544i	−0.557114	+	0.830436i	$0.688091\pi$
$200$		0			0
$201$	3.68287	−	13.6907i	0.259769	−	0.965671i
$202$		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
$211$	−28.1851	−	2.69135i	−1.94034	−	0.185280i	−0.948875	−	0.315652i	$-0.897777\pi$
$211$	−28.1851	−	2.69135i	−1.94034	−	0.185280i	−0.991465	+	0.130372i	$0.958383\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.60538	−	3.43773i	0.244749	−	0.233368i
$218$		0			0
$219$	−2.48899	+	3.16501i	−0.168190	+	0.213871i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.2636	−	22.4742i	−0.687303	−	1.50498i	−0.854715	−	0.519097i	$-0.826268\pi$
$223$	−10.2636	−	22.4742i	−0.687303	−	1.50498i	0.167412	−	0.985887i	$-0.446459\pi$
$224$		0			0
$225$	−2.13472	−	14.8473i	−0.142315	−	0.989821i
$226$		0			0
$227$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$227$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$228$		0			0
$229$	−8.38361	+	0.399360i	−0.554004	+	0.0263905i	−0.322718	−	0.946495i	$-0.604597\pi$
$229$	−8.38361	+	0.399360i	−0.554004	+	0.0263905i	−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$233$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−26.5226	−	13.6734i	−1.72283	−	0.888180i
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−0.244718	+	0.0718558i	−0.0157637	+	0.00462864i	−0.289605	−	0.957146i	$-0.593524\pi$
$241$	−0.244718	+	0.0718558i	−0.0157637	+	0.00462864i	0.273841	+	0.961775i	$0.411706\pi$
$242$		0			0
$243$	−8.42776	+	13.1138i	−0.540641	+	0.841254i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.0446959	−	0.0468757i	0.00284393	−	0.00298263i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$251$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$257$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$258$		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
$271$	−10.6504	−	4.86387i	−0.646966	−	0.295459i	0.0647769	−	0.997900i	$-0.479366\pi$
$271$	−10.6504	−	4.86387i	−0.646966	−	0.295459i	−0.711742	+	0.702440i	$0.752094\pi$
$272$		0			0
$273$	0.635977	−	2.16594i	0.0384911	−	0.131088i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	10.7480	+	12.4039i	0.645787	+	0.745278i	0.980387	−	0.197082i	$-0.0631466\pi$
$277$	10.7480	+	12.4039i	0.645787	+	0.745278i	−0.334600	+	0.942360i	$0.608601\pi$
$278$		0			0
$279$	4.18997	−	1.01648i	0.250847	−	0.0608548i
$280$		0			0
$281$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$281$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$282$		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
$289$	12.3035	+	11.7313i	0.723734	+	0.690079i
$290$		0			0
$291$	−3.38234	−	4.74983i	−0.198276	−	0.278440i
$292$		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
$301$	28.8814	+	5.56644i	1.66470	+	0.320844i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.51544	+	31.8131i	0.0864909	+	1.81567i	0.459867	+	0.887988i	$0.347897\pi$
$307$	1.51544	+	31.8131i	0.0864909	+	1.81567i	−0.373376	+	0.927680i	$0.621800\pi$
$308$		0			0
$309$	−16.8260	−	21.3960i	−0.957198	−	1.21718i
$310$		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		−0.786053	−	0.618159i	$-0.787879\pi$
$317$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.178701	+	1.87145i	−0.00991257	+	0.103809i
$326$		0			0
$327$	19.9074	−	12.7937i	1.10088	−	0.707492i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.53875	+	18.8309i	0.414367	+	1.03504i	0.977924	+	0.208962i	$0.0670085\pi$
$331$	7.53875	+	18.8309i	0.414367	+	1.03504i	−0.563556	+	0.826078i	$0.690567\pi$
$332$		0			0
$333$	−13.7476	+	2.64963i	−0.753363	+	0.145199i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.30248	+	32.7004i	0.343318	+	1.78130i	0.585089	+	0.810969i	$0.301059\pi$
$337$	6.30248	+	32.7004i	0.343318	+	1.78130i	−0.241771	+	0.970333i	$0.577728\pi$
$338$		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		−0.998867	−	0.0475819i	$-0.984848\pi$
$347$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$348$		0			0
$349$	−5.31044	+	36.9349i	−0.284261	+	1.97708i	−0.0917948	+	0.995778i	$0.529260\pi$
$349$	−5.31044	+	36.9349i	−0.284261	+	1.97708i	−0.192467	+	0.981304i	$0.561649\pi$
$350$		0			0
$351$	1.41397	−	1.34821i	0.0754719	−	0.0719624i
$352$		0			0
$353$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$353$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$354$		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
$361$	14.9117	−	11.7267i	0.784825	−	0.617194i
$362$		0			0
$363$	19.0310	−	0.906557i	0.998867	−	0.0475819i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	30.7002	+	21.8615i	1.60254	+	1.14116i	0.905691	+	0.423938i	$0.139353\pi$
$367$	30.7002	+	21.8615i	1.60254	+	1.14116i	0.696844	+	0.717222i	$0.254587\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.336490	−	0.194273i	0.0174228	−	0.0100591i	−0.491263	−	0.871011i	$-0.663465\pi$
$373$	0.336490	−	0.194273i	0.0174228	−	0.0100591i	0.508686	+	0.860952i	$0.330131\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	29.2453	−	20.8255i	1.50223	−	1.06973i	0.526653	−	0.850081i	$-0.323447\pi$
$379$	29.2453	−	20.8255i	1.50223	−	1.06973i	0.975578	−	0.219653i	$-0.0704926\pi$
$380$		0			0
$381$	−11.2699	+	11.8195i	−0.577375	+	0.605533i
$382$		0			0
$383$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$383$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	19.2386	+	16.6703i	0.977951	+	0.847400i
$388$		0			0
$389$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$389$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.3155	+	3.90978i	0.668284	+	0.196226i	0.598239	−	0.801318i	$-0.295867\pi$
$397$	13.3155	+	3.90978i	0.668284	+	0.196226i	0.0700455	+	0.997544i	$0.477686\pi$
$398$		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
$403$	−0.540363			−0.0269174
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.6809	+	7.84995i	1.12150	+	0.388155i	0.824013	−	0.566571i	$-0.191730\pi$
$409$	22.6809	+	7.84995i	1.12150	+	0.388155i	0.297487	+	0.954726i	$0.403852\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	26.1786	−	30.2118i	1.28197	−	1.47948i
$418$		0			0
$419$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$419$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$420$		0			0
$421$	10.9748	+	31.7095i	0.534877	+	1.54543i	0.811444	+	0.584431i	$0.198682\pi$
$421$	10.9748	+	31.7095i	0.534877	+	1.54543i	−0.276567	+	0.960995i	$0.589197\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	16.2308	+	10.4309i	0.785462	+	0.504786i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	15.5288	−	30.1217i	0.746268	−	1.44756i	−0.142786	−	0.989754i	$-0.545606\pi$
$433$	15.5288	−	30.1217i	0.746268	−	1.44756i	0.889053	−	0.457804i	$-0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−18.7908	+	32.5466i	−0.896836	+	1.55337i	−0.0653196	+	0.997864i	$0.520807\pi$
$439$	−18.7908	+	32.5466i	−0.896836	+	1.55337i	−0.831516	+	0.555501i	$0.812527\pi$
$440$		0			0
$441$	0.715896	+	15.0285i	0.0340903	+	0.715644i
$442$		0			0
$443$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$443$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$449$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−29.2935	−	30.7221i	−1.37633	−	1.44345i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.71353	+	35.9715i	−0.0801557	+	1.68267i	0.498334	+	0.866985i	$0.333945\pi$
$457$	−1.71353	+	35.9715i	−0.0801557	+	1.68267i	−0.578490	+	0.815690i	$0.696358\pi$
$458$		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
$463$	−38.9391	−	9.44652i	−1.80965	−	0.439017i	−0.818812	−	0.574061i	$-0.805367\pi$
$463$	−38.9391	−	9.44652i	−1.80965	−	0.439017i	−0.990840	+	0.135044i	$0.956882\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$467$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$468$		0			0
$469$	4.73260	−	27.9753i	0.218531	−	1.29178i
$470$		0			0
$471$	2.14374	+	11.1228i	0.0987781	+	0.512509i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.203062	−	0.837033i	0.00931713	−	0.0384057i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$479$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$480$		0			0
$481$	1.75272	+	0.0834921i	0.0799169	+	0.00380691i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.12087	+	10.3265i	−0.367992	+	0.467940i	−0.934421	−	0.356171i	$-0.884082\pi$
$487$	−8.12087	+	10.3265i	−0.367992	+	0.467940i	0.566429	+	0.824110i	$0.308325\pi$
$488$		0			0
$489$	0.261022	+	2.73355i	0.0118038	+	0.123615i
$490$		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
$499$	37.6965	+	21.7641i	1.68753	+	0.974295i	0.956402	+	0.292052i	$0.0943381\pi$
$499$	37.6965	+	21.7641i	1.68753	+	0.974295i	0.731126	+	0.682243i	$0.238995\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$503$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	19.2879	−	11.1359i	0.856608	−	0.494563i
$508$		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
$511$	−4.35649	+	6.77882i	−0.192720	+	0.299878i
$512$		0			0
$513$	−0.729128	+	0.519210i	−0.0321918	+	0.0229237i
$514$		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
$523$	25.5940	−	13.1946i	1.11915	−	0.576962i	0.203554	−	0.979064i	$-0.434751\pi$
$523$	25.5940	−	13.1946i	1.11915	−	0.576962i	0.915595	+	0.402102i	$0.131720\pi$
$524$		0			0
$525$	−7.07724	−	29.1728i	−0.308876	−	1.27320i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.52256	+	21.7350i	−0.327068	+	0.945001i
$530$		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
$541$	−6.62871	+	22.5753i	−0.284991	+	0.970589i	0.685222	+	0.728334i	$0.259705\pi$
$541$	−6.62871	+	22.5753i	−0.284991	+	0.970589i	−0.970213	+	0.242255i	$0.922113\pi$
$542$		0			0
$543$	−15.4479	−	5.34657i	−0.662933	−	0.229443i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.42972	−	1.80243i	0.317672	−	0.0770664i	−0.0737527	−	0.997277i	$-0.523498\pi$
$547$	7.42972	−	1.80243i	0.317672	−	0.0770664i	0.391425	+	0.920210i	$0.371982\pi$
$548$		0			0
$549$	7.65153	+	14.8419i	0.326559	+	0.633437i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−55.4396	−	22.1947i	−2.35753	−	0.943813i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$557$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$558$		0			0
$559$	−1.85064	−	2.59886i	−0.0782738	−	0.109920i
$560$		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
$567$	−14.2951	+	27.7287i	−0.600339	+	1.16449i
$568$		0			0
$569$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$569$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$570$		0			0
$571$	11.7219	−	16.4611i	0.490547	−	0.688878i	−0.492897	−	0.870088i	$-0.664062\pi$
$571$	11.7219	−	16.4611i	0.490547	−	0.688878i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.4235	−	17.0694i	−0.558827	−	0.710607i	0.421756	−	0.906709i	$-0.361414\pi$
$577$	−13.4235	−	17.0694i	−0.558827	−	0.710607i	−0.980583	+	0.196102i	$0.937171\pi$
$578$		0			0
$579$	12.3585	−	1.77688i	0.513601	−	0.0738446i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$587$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$588$		0			0
$589$	0.245050	+	0.0352329i	0.0100971	+	0.00145175i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$593$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−33.4426	−	8.11308i	−1.36871	−	0.332046i
$598$		0			0
$599$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$599$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$600$		0			0
$601$	−45.6864	+	8.80533i	−1.86359	+	0.359177i	−0.991493	−	0.130163i	$-0.958450\pi$
$601$	−45.6864	+	8.80533i	−1.86359	+	0.359177i	−0.872093	+	0.489339i	$0.837238\pi$
$602$		0			0
$603$	15.6532	−	18.9203i	0.637449	−	0.770493i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	26.1439	−	10.4665i	1.06115	−	0.424820i	0.225591	−	0.974222i	$-0.427569\pi$
$607$	26.1439	−	10.4665i	1.06115	−	0.424820i	0.835558	+	0.549402i	$0.185144\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−37.6683	−	3.59689i	−1.52141	−	0.145277i	−0.699391	−	0.714739i	$-0.746545\pi$
$613$	−37.6683	−	3.59689i	−1.52141	−	0.145277i	−0.822017	+	0.569462i	$0.807151\pi$
$614$		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
$619$	−21.3405	+	20.3482i	−0.857749	+	0.817862i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−21.3405	+	20.3482i	−0.857749	+	0.817862i	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
$631$	29.9166	−	1.42510i	1.19096	−	0.0567325i	0.557279	−	0.830326i	$-0.311846\pi$
$631$	29.9166	−	1.42510i	1.19096	−	0.0567325i	0.633683	+	0.773593i	$0.281543\pi$
$632$		0			0
$633$	−42.4699	−	24.5200i	−1.68803	−	0.974583i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.356865	−	1.85159i	0.0141395	−	0.0733627i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		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		0.814576	−	0.580057i	$-0.196970\pi$
$647$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.15390	−	2.82209i	0.319577	−	0.110607i
$652$		0			0
$653$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$653$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.19876	+	3.19568i	−0.241837	+	0.124675i
$658$		0			0
$659$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$659$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$660$		0			0
$661$	13.6310	−	11.8114i	0.530186	−	0.459408i	−0.348160	−	0.937435i	$-0.613193\pi$
$661$	13.6310	−	11.8114i	0.530186	−	0.459408i	0.878345	+	0.478027i	$0.158648\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	42.7936i		−	1.65450i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−41.2280	−	18.8282i	−1.58922	−	0.725773i	−0.592416	−	0.805632i	$-0.701826\pi$
$673$	−41.2280	−	18.8282i	−1.58922	−	0.725773i	−0.996806	+	0.0798590i	$0.974553\pi$
$674$		0			0
$675$	7.31963	−	24.9284i	0.281733	−	0.959493i
$676$		0			0
$677$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$677$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$678$		0			0
$679$	−7.64188	−	8.81920i	−0.293269	−	0.338450i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$683$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−13.4960	−	5.40297i	−0.514903	−	0.206136i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	15.4608	+	14.7418i	0.588155	+	0.560805i	0.924620	−	0.380891i	$-0.124383\pi$
$691$	15.4608	+	14.7418i	0.588155	+	0.560805i	−0.336465	+	0.941696i	$0.609231\pi$
$692$		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		0.458227	−	0.888835i	$-0.348485\pi$
$701$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$702$		0			0
$703$	−0.789398	−	0.152144i	−0.0297727	−	0.00573822i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.53392	+	53.1936i	0.0951635	+	1.99773i	0.0501074	+	0.998744i	$0.484044\pi$
$709$	2.53392	+	53.1936i	0.0951635	+	1.99773i	0.0450561	+	0.998984i	$0.485653\pi$
$710$		0			0
$711$	−31.9489	−	40.6263i	−1.19818	−	1.52361i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$719$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$720$		0			0
$721$	−37.5910	−	39.4243i	−1.39996	−	1.46824i
$722$		0			0
$723$	−0.437262	−	0.0628688i	−0.0162620	−	0.00233812i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−4.91494	+	51.4716i	−0.182285	+	1.90898i	0.191555	+	0.981482i	$0.438647\pi$
$727$	−4.91494	+	51.4716i	−0.182285	+	1.90898i	−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
$733$	−8.98548	−	22.4446i	−0.331886	−	0.829012i	−0.996637	−	0.0819464i	$-0.973886\pi$
$733$	−8.98548	−	22.4446i	−0.331886	−	0.829012i	0.664750	−	0.747065i	$-0.268538\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−5.85808	−	30.3946i	−0.215493	−	1.11808i	−0.914999	−	0.403456i	$-0.867809\pi$
$739$	−5.85808	−	30.3946i	−0.215493	−	1.11808i	0.699506	−	0.714627i	$-0.253403\pi$
$740$		0			0
$741$	0.104148	−	0.0416944i	0.00382596	−	0.00153168i
$742$		0			0
$743$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$743$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$744$		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
$757$	−3.48863	−	36.5346i	−0.126797	−	1.32787i	−0.805121	−	0.593111i	$-0.797900\pi$
$757$	−3.48863	−	36.5346i	−0.126797	−	1.32787i	0.678324	−	0.734763i	$-0.262706\pi$
$758$		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
$763$	37.2257	−	29.2746i	1.34766	−	1.05981i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.520940	−	0.370960i	−0.0187856	−	0.0133771i	0.570626	−	0.821210i	$-0.306701\pi$
$769$	−0.520940	−	0.370960i	−0.0187856	−	0.0133771i	−0.589411	+	0.807833i	$0.700640\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$773$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$774$		0			0
$775$	−6.22312	+	3.59292i	−0.223541	+	0.129061i
$776$		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
$787$	−10.1693	+	25.4017i	−0.362497	+	0.905474i	0.629243	+	0.777209i	$0.283365\pi$
$787$	−10.1693	+	25.4017i	−0.362497	+	0.905474i	−0.991740	+	0.128265i	$0.959059\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.493393	−	2.03380i	−0.0175209	−	0.0722222i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$797$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$798$		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
$811$	32.9784	+	11.4139i	1.15803	+	0.400797i	0.837498	−	0.546441i	$-0.184018\pi$
$811$	32.9784	+	11.4139i	1.15803	+	0.400797i	0.320530	+	0.947238i	$0.396139\pi$
$812$		0			0
$813$	−13.2804	−	15.3263i	−0.465762	−	0.537518i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.669799	+	1.29923i	0.0234333	+	0.0454543i
$818$		0			0
$819$	2.56043	−	2.95490i	0.0894688	−	0.103253i
$820$		0			0
$821$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$821$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$822$		0			0
$823$	−18.2011	−	52.5887i	−0.634452	−	1.83313i	−0.547307	−	0.836932i	$-0.684347\pi$
$823$	−18.2011	−	52.5887i	−0.634452	−	1.83313i	−0.0871445	−	0.996196i	$-0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$827$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$828$		0			0
$829$	46.7566	+	30.0486i	1.62392	+	1.04363i	0.953384	+	0.301759i	$0.0975739\pi$
$829$	46.7566	+	30.0486i	1.62392	+	1.04363i	0.670540	+	0.741873i	$0.266062\pi$
$830$		0			0
$831$	8.00899	+	27.2761i	0.277829	+	0.946198i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.33279	+	1.41328i	0.253458	+	0.0488501i
$838$		0			0
$839$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$839$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		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
$853$	−3.27046	−	2.57192i	−0.111978	−	0.0880607i	0.560600	−	0.828087i	$-0.310571\pi$
$853$	−3.27046	−	2.57192i	−0.111978	−	0.0880607i	−0.672578	+	0.740026i	$0.734813\pi$
$854$		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
$859$	0.949996	−	19.9429i	0.0324134	−	0.680442i	−0.922934	−	0.384958i	$-0.874216\pi$
$859$	0.949996	−	19.9429i	0.0324134	−	0.680442i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$		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
$867$	10.9436	+	27.3357i	0.371662	+	0.928368i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2.46558	+	1.84193i	−0.0835430	+	0.0624113i
$872$		0			0
$873$	−1.91138	−	9.91716i	−0.0646903	−	0.335645i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.18314	−	33.7314i	0.276325	−	1.13903i	−0.649284	−	0.760546i	$-0.724931\pi$
$877$	8.18314	−	33.7314i	0.276325	−	1.13903i	0.925609	−	0.378481i	$-0.123554\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$881$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$882$		0			0
$883$	−31.7048	−	1.51028i	−1.06695	−	0.0508251i	−0.493248	−	0.869889i	$-0.664191\pi$
$883$	−31.7048	−	1.51028i	−1.06695	−	0.0508251i	−0.573703	+	0.819064i	$0.694494\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$887$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$888$		0			0
$889$	−20.2035	+	25.6908i	−0.677603	+	0.861642i
$890$		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
$903$	41.4984	+	29.5509i	1.38098	+	0.983391i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.8363	−	11.7729i	−0.758266	−	0.390914i	0.0353161	−	0.999376i	$-0.488756\pi$
$907$	−22.8363	−	11.7729i	−0.758266	−	0.390914i	−0.793582	+	0.608463i	$0.791787\pi$
$908$		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
$919$	−42.1131	+	14.5755i	−1.38918	+	0.480801i	−0.916360	−	0.400356i	$-0.868887\pi$
$919$	−42.1131	+	14.5755i	−1.38918	+	0.480801i	−0.472824	+	0.881157i	$0.656765\pi$
$920$		0			0
$921$	−20.5025	+	51.2128i	−0.675581	+	1.68752i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	20.7404	−	10.6924i	0.681939	−	0.351564i
$926$		0			0
$927$	−11.1150	−	45.8167i	−0.365065	−	1.50482i
$928$		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
$931$	−0.282563	+	0.816413i	−0.00926063	+	0.0267568i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	27.4954i		−	0.898234i	−0.893473	−	0.449117i	$-0.851739\pi$
$937$		−	27.4954i		−	0.898234i	0.893473	−	0.449117i	$-0.148261\pi$
$938$		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
$949$	0.849420	−	0.206067i	0.0275733	−	0.00668922i
$950$		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
$961$	16.7837	+	23.5694i	0.541409	+	0.760303i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−18.1845	−	31.4965i	−0.584774	−	1.01286i	−0.994904	−	0.100831i	$-0.967850\pi$
$967$	−18.1845	−	31.4965i	−0.584774	−	1.01286i	0.410129	−	0.912027i	$-0.365484\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$971$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$972$		0			0
$973$	46.4059	−	65.1680i	1.48771	−	2.08919i
$974$		0			0
$975$	−1.62809	+	2.81994i	−0.0521407	+	0.0903103i
$976$		0			0
$977$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$977$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	40.5699	−	5.83307i	1.29530	−	0.186236i
$982$		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
$993$	−1.67168	+	35.0929i	−0.0530492	+	1.11364i
$994$		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
$999$	−23.5662	−	5.71709i	−0.745601	−	0.180881i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.113.1	✓	20
3.2	odd	2	CM	804.2.ba.a.113.1	✓	20
67.51	odd	66	inner	804.2.ba.a.185.1	yes	20
201.185	even	66	inner	804.2.ba.a.185.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.113.1	✓	20	1.1	even	1	trivial
804.2.ba.a.113.1	✓	20	3.2	odd	2	CM
804.2.ba.a.185.1	yes	20	67.51	odd	66	inner
804.2.ba.a.185.1	yes	20	201.185	even	66	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.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(1.57553 + 0.719520i) q^{3} +(3.27565 + 1.13371i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\)	\(q+(1.57553 + 0.719520i) q^{3} +(3.27565 + 1.13371i) q^{7} +(1.96458 + 2.26725i) q^{9} +(-0.172289 - 0.334195i) q^{13} +(0.0563415 + 0.162788i) q^{19} +(4.34515 + 4.14309i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(1.46393 + 4.98567i) q^{27} +(0.658548 - 1.27741i) q^{31} +(-2.33343 + 4.04162i) q^{37} +(-0.0309871 - 0.650499i) q^{39} +(8.39906 - 1.20760i) q^{43} +(3.94220 + 3.10018i) q^{49} +(-0.0283616 + 0.297016i) q^{57} +(5.40915 + 1.31225i) q^{61} +(3.86487 + 9.65398i) q^{63} +(-1.34943 - 8.07335i) q^{67} +(-0.548063 + 2.25915i) q^{73} +(-4.68209 - 7.28547i) q^{75} +(-17.2085 - 0.819741i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-0.185479 - 1.29003i) q^{91} +(1.95668 - 1.53875i) q^{93} +(-2.91553 - 1.68328i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^7 + 6 * q^9	\( 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) \) 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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