LMFDB - Embedded newform 804.2.ba.a.353.1

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			353.1
Root			$-0.888835 + 0.458227i$ of defining polynomial
Character	$\chi$	$=$	804.353
Dual form			804.2.ba.a.41.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.936417 - 1.45709i) q^{3} +(-0.502989 + 5.26754i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})$	$q+(0.936417 - 1.45709i) q^{3} +(-0.502989 + 5.26754i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(4.62817 - 4.85388i) q^{13} +(4.83520 - 0.461706i) q^{19} +(7.20430 + 5.66552i) q^{21} +(4.79746 - 1.40866i) q^{25} +(-5.14326 - 0.739490i) q^{27} +(7.68371 + 8.05844i) q^{31} +(3.82151 + 6.61906i) q^{37} +(-2.73867 - 11.2889i) q^{39} +(4.02392 - 3.48675i) q^{43} +(-20.6205 - 3.97428i) q^{49} +(3.85502 - 7.47769i) q^{57} +(-4.67299 - 11.6726i) q^{61} +(15.0014 - 5.19204i) q^{63} +(3.56930 - 7.36615i) q^{67} +(-15.2494 + 6.10492i) q^{73} +(2.43988 - 8.30945i) q^{75} +(-14.8646 - 3.60611i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(23.2401 + 26.8205i) q^{91} +(18.9371 - 3.64982i) q^{93} +(-11.5396 + 6.66241i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{13}{66}\right)$	$1$

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
$3$	0.936417	−	1.45709i	0.540641	−	0.841254i
$3$	0.936417	−	1.45709i	0.540641	−	0.841254i
$4$		0			0
$5$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$5$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$6$		0			0
$7$	−0.502989	+	5.26754i	−0.190112	+	1.99094i	−0.0947833	+	0.995498i	$0.530216\pi$
$7$	−0.502989	+	5.26754i	−0.190112	+	1.99094i	−0.0953288	+	0.995446i	$0.530390\pi$
$8$		0			0
$9$	−1.24625	−	2.72890i	−0.415415	−	0.909632i
$10$		0			0
$11$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$11$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$12$		0			0
$13$	4.62817	−	4.85388i	1.28362	−	1.34622i	0.375896	−	0.926662i	$-0.377335\pi$
$13$	4.62817	−	4.85388i	1.28362	−	1.34622i	0.907726	−	0.419563i	$-0.137817\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$17$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$18$		0			0
$19$	4.83520	−	0.461706i	1.10927	−	0.105923i	0.475700	−	0.879607i	$-0.342195\pi$
$19$	4.83520	−	0.461706i	1.10927	−	0.105923i	0.633571	+	0.773685i	$0.281589\pi$
$20$		0			0
$21$	7.20430	+	5.66552i	1.57211	+	1.23632i
$22$		0			0
$23$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$23$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$24$		0			0
$25$	4.79746	−	1.40866i	0.959493	−	0.281733i
$26$		0			0
$27$	−5.14326	−	0.739490i	−0.989821	−	0.142315i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	7.68371	+	8.05844i	1.38003	+	1.44734i	0.699699	+	0.714438i	$0.253318\pi$
$31$	7.68371	+	8.05844i	1.38003	+	1.44734i	0.680336	+	0.732900i	$0.261834\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.82151	+	6.61906i	0.628253	+	1.08817i	0.987902	+	0.155079i	$0.0495631\pi$
$37$	3.82151	+	6.61906i	0.628253	+	1.08817i	−0.359649	+	0.933088i	$0.617104\pi$
$38$		0			0
$39$	−2.73867	−	11.2889i	−0.438537	−	1.80768i
$40$		0			0
$41$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$41$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$42$		0			0
$43$	4.02392	−	3.48675i	0.613643	−	0.531724i	−0.291643	−	0.956527i	$-0.594202\pi$
$43$	4.02392	−	3.48675i	0.613643	−	0.531724i	0.905286	+	0.424803i	$0.139657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$47$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$48$		0			0
$49$	−20.6205	−	3.97428i	−2.94579	−	0.567754i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$53$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.85502	−	7.47769i	0.510609	−	0.990444i
$58$		0			0
$59$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$59$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$60$		0			0
$61$	−4.67299	−	11.6726i	−0.598315	−	1.49452i	−0.850039	−	0.526719i	$-0.823422\pi$
$61$	−4.67299	−	11.6726i	−0.598315	−	1.49452i	0.251725	−	0.967799i	$-0.419002\pi$
$62$		0			0
$63$	15.0014	−	5.19204i	1.89000	−	0.654136i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.56930	−	7.36615i	0.436059	−	0.899918i
$67$	3.56930	−	7.36615i	0.436059	−	0.899918i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$71$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$72$		0			0
$73$	−15.2494	+	6.10492i	−1.78480	+	0.714527i	−0.789953	+	0.613167i	$0.789895\pi$
$73$	−15.2494	+	6.10492i	−1.78480	+	0.714527i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	2.43988	−	8.30945i	0.281733	−	0.959493i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−14.8646	−	3.60611i	−1.67240	−	0.405719i	−0.716073	−	0.698026i	$-0.754062\pi$
$79$	−14.8646	−	3.60611i	−1.67240	−	0.405719i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	−5.89375	+	6.80175i	−0.654861	+	0.755750i
$82$		0			0
$83$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$83$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$89$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$90$		0			0
$91$	23.2401	+	26.8205i	2.43623	+	2.81155i
$92$		0			0
$93$	18.9371	−	3.64982i	1.96368	−	0.378469i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−11.5396	+	6.66241i	−1.17167	+	0.676466i	−0.954074	−	0.299573i	$-0.903156\pi$
$97$	−11.5396	+	6.66241i	−1.17167	+	0.676466i	−0.217599	+	0.976038i	$0.569823\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$101$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$102$		0			0
$103$	−3.73247	+	3.55890i	−0.367771	+	0.350669i	−0.851319	−	0.524649i	$-0.824197\pi$
$103$	−3.73247	+	3.55890i	−0.367771	+	0.350669i	0.483547	+	0.875318i	$0.339348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$107$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$108$		0			0
$109$	4.95120	+	16.8622i	0.474239	+	1.61511i	0.755271	+	0.655412i	$0.227505\pi$
$109$	4.95120	+	16.8622i	0.474239	+	1.61511i	−0.281032	+	0.959698i	$0.590677\pi$
$110$		0			0
$111$	13.2231	+	0.629895i	1.25508	+	0.0597870i
$112$		0			0
$113$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$113$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−19.0136	−	6.58066i	−1.75780	−	0.608382i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.96107	−	7.59087i	−0.723734	−	0.690079i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	14.7475	+	1.40821i	1.30863	+	0.124959i	0.725951	−	0.687747i	$-0.241400\pi$
$127$	14.7475	+	1.40821i	1.30863	+	0.124959i	0.582675	+	0.812705i	$0.302006\pi$
$128$		0			0
$129$	−1.31245	−	9.12829i	−0.115555	−	0.803701i
$130$		0			0
$131$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$131$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$132$		0			0
$133$			25.7019i			2.22863i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$137$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$138$		0			0
$139$	−16.6350	+	2.39175i	−1.41096	+	0.202865i	−0.805304	−	0.592862i	$-0.797998\pi$
$139$	−16.6350	+	2.39175i	−1.41096	+	0.202865i	−0.605655	+	0.795727i	$0.707089\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−25.1003	+	26.3244i	−2.07024	+	2.17120i
$148$		0			0
$149$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$149$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$150$		0			0
$151$	−1.73309	+	5.00743i	−0.141037	+	0.407499i	−0.993293	−	0.115626i	$-0.963113\pi$
$151$	−1.73309	+	5.00743i	−0.141037	+	0.407499i	0.852256	+	0.523125i	$0.175234\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.17633	−	24.6942i	0.0938812	−	1.97081i	−0.116506	−	0.993190i	$-0.537169\pi$
$157$	1.17633	−	24.6942i	0.0938812	−	1.97081i	0.210387	−	0.977618i	$-0.432527\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.1493	+	17.5790i	−0.794951	+	1.37690i	0.127919	+	0.991785i	$0.459170\pi$
$163$	−10.1493	+	17.5790i	−0.794951	+	1.37690i	−0.922870	+	0.385111i	$0.874163\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$167$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$168$		0			0
$169$	−1.52167	−	31.9439i	−0.117052	−	2.45722i
$170$		0			0
$171$	−7.28579	−	12.6194i	−0.557158	−	0.965026i
$172$		0			0
$173$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$173$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$174$		0			0
$175$	5.00712	+	25.9794i	0.378502	+	1.96386i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$179$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$180$		0			0
$181$	5.05145	−	2.60420i	0.375472	−	0.193569i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	5.05145	−	2.60420i	0.375472	−	0.193569i	0.635624	+	0.771998i	$0.280743\pi$
$182$		0			0
$183$	−21.3839	−	4.12141i	−1.58074	−	0.304663i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	6.48230	−	26.7204i	0.471518	−	1.94362i
$190$		0			0
$191$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$191$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$192$		0			0
$193$	20.5467	+	6.03304i	1.47898	+	0.434268i	0.919007	−	0.394242i	$-0.128993\pi$
$193$	20.5467	+	6.03304i	1.47898	+	0.434268i	0.559974	+	0.828510i	$0.310811\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$197$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$198$		0			0
$199$	16.3284	−	22.9300i	1.15749	−	1.62546i	0.523958	−	0.851744i	$-0.324455\pi$
$199$	16.3284	−	22.9300i	1.15749	−	1.62546i	0.633529	−	0.773719i	$-0.281606\pi$
$200$		0			0
$201$	−7.39081	−	12.0986i	−0.521308	−	0.853369i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−2.14450	−	1.10557i	−0.147634	−	0.0761104i	0.382827	−	0.923820i	$-0.374951\pi$
$211$	−2.14450	−	1.10557i	−0.147634	−	0.0761104i	−0.530460	+	0.847710i	$0.677981\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−46.3130	+	36.4209i	−3.14393	+	2.47241i
$218$		0			0
$219$	−5.38432	+	27.9365i	−0.363839	+	1.88777i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.85346	+	6.33243i	−0.659836	+	0.424051i	−0.827249	−	0.561836i	$-0.810095\pi$
$223$	−9.85346	+	6.33243i	−0.659836	+	0.424051i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$	−9.82291	−	11.3362i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$227$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$228$		0			0
$229$	0.135285	−	0.0328197i	0.00893986	−	0.00216879i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	0.135285	−	0.0328197i	0.00893986	−	0.00216879i	0.240226	+	0.970717i	$0.422778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$233$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.1739	+	18.2823i	−1.24548	+	1.18756i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	1.27966	−	8.90023i	0.0824301	−	0.573314i	−0.906189	−	0.422873i	$-0.861022\pi$
$241$	1.27966	−	8.90023i	0.0824301	−	0.573314i	0.988619	−	0.150441i	$-0.0480694\pi$
$242$		0			0
$243$	4.39178	+	14.9570i	0.281733	+	0.959493i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	20.1370	−	25.6063i	1.28129	−	1.62929i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$251$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$257$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$258$		0			0
$259$	−36.7883	+	16.8007i	−2.28592	+	1.04394i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$263$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$264$		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
$271$	−17.7628	+	27.6394i	−1.07901	+	1.67897i	−0.485012	+	0.874508i	$0.661185\pi$
$271$	−17.7628	+	27.6394i	−1.07901	+	1.67897i	−0.593999	+	0.804466i	$0.702452\pi$
$272$		0			0
$273$	60.8424	−	8.74782i	3.68235	−	0.529442i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.62677	−	10.1312i	−0.277996	−	0.608726i	0.718203	−	0.695833i	$-0.244965\pi$
$277$	−4.62677	−	10.1312i	−0.277996	−	0.608726i	−0.996199	+	0.0871078i	$0.972238\pi$
$278$		0			0
$279$	12.4149	−	31.0108i	0.743258	−	1.85657i
$280$		0			0
$281$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$281$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$282$		0			0
$283$	2.38135	−	5.21442i	0.141556	−	0.309965i	−0.825554	−	0.564324i	$-0.809137\pi$
$283$	2.38135	−	5.21442i	0.141556	−	0.309965i	0.967110	+	0.254358i	$0.0818643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.3629	−	10.5087i	−0.786053	−	0.618159i
$290$		0			0
$291$	−1.09816	+	23.0531i	−0.0643751	+	1.35140i
$292$		0			0
$293$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$293$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	16.3426	+	22.9500i	0.941972	+	1.32282i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.31293	−	9.53404i	−0.132006	−	0.544137i	−0.998954	−	0.0457370i	$-0.985436\pi$
$307$	−2.31293	−	9.53404i	−0.132006	−	0.544137i	0.866947	−	0.498400i	$-0.166079\pi$
$308$		0			0
$309$	1.69051	+	8.77118i	0.0961696	+	0.498975i
$310$		0			0
$311$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$311$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$312$		0			0
$313$	−17.0433	−	26.5199i	−0.963343	−	1.49899i	−0.863724	−	0.503966i	$-0.831874\pi$
$313$	−17.0433	−	26.5199i	−0.963343	−	1.49899i	−0.0996196	−	0.995026i	$-0.531763\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$317$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	15.3660	−	29.8059i	0.852351	−	1.65333i
$326$		0			0
$327$	29.2063	+	8.57574i	1.61511	+	0.474239i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.691583	+	0.239359i	−0.0380128	+	0.0131564i	−0.346008	−	0.938231i	$-0.612463\pi$
$331$	−0.691583	+	0.239359i	−0.0380128	+	0.0131564i	0.307995	+	0.951388i	$0.400342\pi$
$332$		0			0
$333$	13.3002	−	18.6775i	0.728845	−	1.02352i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	29.5759	+	21.0609i	1.61110	+	1.14726i	0.891656	+	0.452715i	$0.149544\pi$
$337$	29.5759	+	21.0609i	1.61110	+	1.14726i	0.719448	+	0.694547i	$0.244395\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	20.8710	−	71.0802i	1.12693	−	3.83797i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$347$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$348$		0			0
$349$	−11.2829	+	13.0211i	−0.603959	+	0.697006i	−0.972579	−	0.232574i	$-0.925285\pi$
$349$	−11.2829	+	13.0211i	−0.603959	+	0.697006i	0.368620	+	0.929580i	$0.379831\pi$
$350$		0			0
$351$	−27.3933	+	21.5423i	−1.46214	+	1.14984i
$352$		0			0
$353$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$353$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$359$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$360$		0			0
$361$	4.50934	−	0.869104i	0.237334	−	0.0457423i
$362$		0			0
$363$	−18.5155	+	4.49181i	−0.971812	+	0.235759i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.46198	−	0.0696427i	0.0763147	−	0.00363532i	−0.00938982	−	0.999956i	$-0.502989\pi$
$367$	1.46198	−	0.0696427i	0.0763147	−	0.00363532i	0.0857046	+	0.996321i	$0.472686\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−30.2872	−	17.4863i	−1.56821	−	0.905407i	−0.996378	−	0.0850377i	$-0.972899\pi$
$373$	−30.2872	−	17.4863i	−1.56821	−	0.905407i	−0.571834	−	0.820370i	$-0.693768\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	28.5873	+	1.36178i	1.46843	+	0.0699501i	0.766477	−	0.642271i	$-0.222008\pi$
$379$	28.5873	+	1.36178i	1.46843	+	0.0699501i	0.701955	+	0.712221i	$0.252311\pi$
$380$		0			0
$381$	15.8617	−	20.1698i	0.812618	−	1.03333i
$382$		0			0
$383$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$383$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−14.5298	−	6.63552i	−0.738590	−	0.337303i
$388$		0			0
$389$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$389$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.31106	−	36.9392i	−0.266554	−	1.85393i	−0.480387	−	0.877057i	$-0.659504\pi$
$397$	−5.31106	−	36.9392i	−0.266554	−	1.85393i	0.213832	−	0.976870i	$-0.431405\pi$
$398$		0			0
$399$	37.4500	+	24.0677i	1.87485	+	1.20489i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	74.6762			3.71989
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.38792	+	35.4799i	−0.167522	+	1.75437i	0.386892	+	0.922125i	$0.373549\pi$
$409$	−3.38792	+	35.4799i	−0.167522	+	1.75437i	−0.554414	+	0.832241i	$0.687058\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.0923	+	26.4784i	−0.592161	+	1.29665i
$418$		0			0
$419$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$419$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$420$		0			0
$421$	−38.6753	+	3.69304i	−1.88492	+	0.179988i	−0.973066	−	0.230527i	$-0.925955\pi$
$421$	−38.6753	+	3.69304i	−1.88492	+	0.179988i	−0.911854	+	0.410515i	$0.865349\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	63.8362	−	18.7440i	3.08925	−	0.907085i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	27.1352	+	28.4586i	1.30404	+	1.36763i	0.889053	+	0.457804i	$0.151364\pi$
$433$	27.1352	+	28.4586i	1.30404	+	1.36763i	0.414982	+	0.909829i	$0.363788\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	19.3101	+	33.4460i	0.921619	+	1.59629i	0.796911	+	0.604097i	$0.206466\pi$
$439$	19.3101	+	33.4460i	0.921619	+	1.59629i	0.124708	+	0.992194i	$0.460201\pi$
$440$		0			0
$441$	14.8528	+	61.2241i	0.707277	+	2.91543i
$442$		0			0
$443$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$443$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$449$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	5.67340	+	7.21432i	0.266560	+	0.338958i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.41533	−	5.83409i	0.0662066	−	0.272907i	−0.929081	−	0.369877i	$-0.879400\pi$
$457$	1.41533	−	5.83409i	0.0662066	−	0.272907i	0.995287	+	0.0969697i	$0.0309150\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$461$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$462$		0			0
$463$	0.878313	+	2.19392i	0.0408187	+	0.101960i	0.947376	−	0.320122i	$-0.103724\pi$
$463$	0.878313	+	2.19392i	0.0408187	+	0.101960i	−0.906558	+	0.422082i	$0.861300\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$467$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$468$		0			0
$469$	37.0062	+	22.5065i	1.70879	+	1.03925i
$470$		0			0
$471$	−34.8802	−	24.8381i	−1.60719	−	1.14448i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	22.5463	−	9.02618i	1.03450	−	0.414150i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$479$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$480$		0			0
$481$	49.8147	+	12.0849i	2.27136	+	0.551025i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	7.65465	−	39.7161i	0.346865	−	1.79971i	−0.219564	−	0.975598i	$-0.570463\pi$
$487$	7.65465	−	39.7161i	0.346865	−	1.79971i	0.566429	−	0.824110i	$-0.308325\pi$
$488$		0			0
$489$	16.1103	+	31.2497i	0.728535	+	1.41316i
$490$		0			0
$491$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$491$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−33.7139	+	19.4647i	−1.50924	+	0.871361i	−0.509300	+	0.860589i	$0.670096\pi$
$499$	−33.7139	+	19.4647i	−1.50924	+	0.871361i	−0.999942	+	0.0107722i	$0.996571\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$503$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−47.9701	−	27.6956i	−2.13043	−	1.23000i
$508$		0			0
$509$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$509$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$510$		0			0
$511$	−24.4877	−	83.3974i	−1.08327	−	3.68928i
$512$		0			0
$513$	−25.2101	−	1.20091i	−1.11305	−	0.0530213i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$521$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$522$		0			0
$523$	−32.0521	−	30.5616i	−1.40154	−	1.33636i	−0.873227	−	0.487314i	$-0.837977\pi$
$523$	−32.0521	−	30.5616i	−1.40154	−	1.33636i	−0.528312	−	0.849050i	$-0.677175\pi$
$524$		0			0
$525$	42.5432	+	17.0317i	1.85674	+	0.743325i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.8959	−	2.18629i	−0.995472	−	0.0950560i
$530$		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
$541$	−45.2705	+	6.50892i	−1.94633	+	0.279840i	−0.999163	−	0.0408986i	$-0.986978\pi$
$541$	−45.2705	+	6.50892i	−1.94633	+	0.279840i	−0.947168	+	0.320739i	$0.896069\pi$
$542$		0			0
$543$	0.935697	−	9.79906i	0.0401546	−	0.420518i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−11.1012	+	27.7295i	−0.474653	+	1.18563i	0.477124	+	0.878836i	$0.341679\pi$
$547$	−11.1012	+	27.7295i	−0.474653	+	1.18563i	−0.951777	+	0.306791i	$0.900745\pi$
$548$		0			0
$549$	−26.0295	+	27.2990i	−1.11091	+	1.16509i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	26.4721	−	76.4860i	1.12571	−	3.25252i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$557$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$558$		0			0
$559$	1.69912	−	35.6689i	0.0718651	−	1.50863i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$563$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−32.8640	−	34.4668i	−1.38016	−	1.44747i
$568$		0			0
$569$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$569$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$570$		0			0
$571$	−0.305195	−	6.40683i	−0.0127720	−	0.268117i	−0.996216	−	0.0869073i	$-0.972302\pi$
$571$	−0.305195	−	6.40683i	−0.0127720	−	0.268117i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.44272	−	38.6165i	−0.309844	−	1.60763i	−0.718898	−	0.695115i	$-0.755353\pi$
$577$	−7.44272	−	38.6165i	−0.309844	−	1.60763i	0.409054	−	0.912510i	$-0.365859\pi$
$578$		0			0
$579$	28.0310	−	24.2890i	1.16493	−	1.00941i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$587$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$588$		0			0
$589$	40.8729	+	35.4166i	1.68414	+	1.45931i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$593$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−18.1210	−	45.2640i	−0.741642	−	1.85253i
$598$		0			0
$599$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$599$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$600$		0			0
$601$	−5.03945	+	7.07691i	−0.205563	+	0.288673i	−0.904468	−	0.426542i	$-0.859732\pi$
$601$	−5.03945	+	7.07691i	−0.205563	+	0.288673i	0.698905	+	0.715215i	$0.253671\pi$
$602$		0			0
$603$	−24.5497	−	0.560221i	−0.999740	−	0.0228140i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.20349	+	23.7024i	0.332970	+	0.962052i	0.979898	+	0.199501i	$0.0639322\pi$
$607$	8.20349	+	23.7024i	0.332970	+	0.962052i	−0.646928	+	0.762551i	$0.723947\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.569823	+	0.293764i	0.0230149	+	0.0118650i	0.469696	−	0.882828i	$-0.344364\pi$
$613$	0.569823	+	0.293764i	0.0230149	+	0.0118650i	−0.446681	+	0.894693i	$0.647394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$617$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$	−34.4842	+	27.1186i	−1.38604	+	1.08999i	−0.401297	+	0.915948i	$0.631441\pi$
$619$	−34.4842	+	27.1186i	−1.38604	+	1.08999i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.0313	−	13.5160i	0.841254	−	0.540641i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−3.29335	+	0.798959i	−0.131106	+	0.0318061i	−0.300775	−	0.953695i	$-0.597245\pi$
$631$	−3.29335	+	0.798959i	−0.131106	+	0.0318061i	0.169669	+	0.985501i	$0.445730\pi$
$632$		0			0
$633$	−3.61906	+	2.08947i	−0.143845	+	0.0830489i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−114.726	+	81.6959i	−4.54560	+	3.23691i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	0.374642	−	2.60569i	0.0147745	−	0.102759i	−0.981100	−	0.193502i	$-0.938015\pi$
$643$	0.374642	−	2.60569i	0.0147745	−	0.102759i	0.995874	+	0.0907437i	$0.0289244\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$647$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	9.70044	+	101.588i	0.380190	+	3.98153i
$652$		0			0
$653$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$653$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	35.6642	+	34.0057i	1.39139	+	1.32669i
$658$		0			0
$659$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$659$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$660$		0			0
$661$	22.3588	−	10.2109i	0.869657	−	0.397159i	0.0699507	−	0.997550i	$-0.477716\pi$
$661$	22.3588	−	10.2109i	0.869657	−	0.397159i	0.799706	+	0.600391i	$0.204989\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			20.2872i			0.784349i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	17.0015	−	26.4549i	0.655360	−	1.01976i	−0.341446	−	0.939901i	$-0.610917\pi$
$673$	17.0015	−	26.4549i	0.655360	−	1.01976i	0.996806	−	0.0798590i	$-0.0254470\pi$
$674$		0			0
$675$	−25.7163	+	3.69745i	−0.989821	+	0.142315i
$676$		0			0
$677$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$677$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$678$		0			0
$679$	−29.2902	−	64.1367i	−1.12406	−	2.46134i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$683$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.0788615	−	0.227855i	0.00300875	−	0.00869323i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.47353	+	4.30443i	0.208223	+	0.163748i	0.716824	−	0.697254i	$-0.245595\pi$
$691$	5.47353	+	4.30443i	0.208223	+	0.163748i	−0.508601	+	0.861002i	$0.669837\pi$
$692$		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		−0.690079	−	0.723734i	$-0.742424\pi$
$701$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$702$		0			0
$703$	21.5338	+	30.2401i	0.812164	+	1.14053i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.5449	+	43.4665i	0.396020	+	1.63242i	0.725477	+	0.688247i	$0.241619\pi$
$709$	10.5449	+	43.4665i	0.396020	+	1.63242i	−0.329456	+	0.944171i	$0.606865\pi$
$710$		0			0
$711$	8.68422	+	45.0580i	0.325684	+	1.68981i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$719$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$720$		0			0
$721$	−16.8693	−	21.4510i	−0.628245	−	0.798879i
$722$		0			0
$723$	−11.7702	−	10.1989i	−0.437738	−	0.379302i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−10.2122	+	19.8090i	−0.378751	+	0.734675i	−0.998753	−	0.0499257i	$-0.984102\pi$
$727$	−10.2122	+	19.8090i	−0.378751	+	0.734675i	0.620002	+	0.784600i	$0.287132\pi$
$728$		0			0
$729$	25.9063	+	7.60678i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	51.1395	−	17.6996i	1.88888	−	0.653748i	0.933179	−	0.359412i	$-0.117023\pi$
$733$	51.1395	−	17.6996i	1.88888	−	0.653748i	0.955702	−	0.294336i	$-0.0950985\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−44.0283	−	31.3524i	−1.61961	−	1.15332i	−0.872505	−	0.488606i	$-0.837506\pi$
$739$	−44.0283	−	31.3524i	−1.61961	−	1.15332i	−0.747101	−	0.664710i	$-0.768555\pi$
$740$		0			0
$741$	−18.4542	−	53.3198i	−0.677930	−	1.95875i
$742$		0			0
$743$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$743$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−13.7277	+	15.8426i	−0.500931	+	0.578105i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−13.7277	+	15.8426i	−0.500931	+	0.578105i	0.447822	+	0.894123i	$0.352200\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	16.2030	+	31.4294i	0.588908	+	1.14232i	0.975485	+	0.220065i	$0.0706269\pi$
$757$	16.2030	+	31.4294i	0.588908	+	1.14232i	−0.386577	+	0.922257i	$0.626343\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$761$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$762$		0			0
$763$	−91.3130	+	17.5991i	−3.30575	+	0.637132i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.9937	+	0.714238i	−0.540687	+	0.0257561i	−0.316150	−	0.948709i	$-0.602390\pi$
$769$	−14.9937	+	0.714238i	−0.540687	+	0.0257561i	−0.224537	+	0.974465i	$0.572087\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$773$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$774$		0			0
$775$	48.2139	+	27.8363i	1.73190	+	0.999911i
$776$		0			0
$777$	−9.96909	+	69.3365i	−0.357639	+	2.48743i
$778$		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
$787$	−11.7683	−	4.07306i	−0.419495	−	0.145189i	0.109162	−	0.994024i	$-0.465183\pi$
$787$	−11.7683	−	4.07306i	−0.419495	−	0.145189i	−0.528658	+	0.848835i	$0.677304\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−78.2846	−	31.3404i	−2.77997	−	1.11293i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$797$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$798$		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		0.989821	−	0.142315i	$-0.0454545\pi$
$809$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$810$		0			0
$811$	−4.38833	+	45.9566i	−0.154095	+	1.61376i	0.505972	+	0.862550i	$0.331134\pi$
$811$	−4.38833	+	45.9566i	−0.154095	+	1.61376i	−0.660067	+	0.751206i	$0.729472\pi$
$812$		0			0
$813$	23.6398	+	51.7640i	0.829085	+	1.81544i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	17.8466	−	18.7170i	0.624374	−	0.654825i
$818$		0			0
$819$	44.2275	−	96.8448i	1.54543	−	3.38403i
$820$		0			0
$821$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$821$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$822$		0			0
$823$	−10.3634	+	0.989585i	−0.361246	+	0.0344948i	−0.274101	−	0.961701i	$-0.588380\pi$
$823$	−10.3634	+	0.989585i	−0.361246	+	0.0344948i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$827$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$828$		0			0
$829$	45.8455	−	13.4614i	1.59228	−	0.467535i	0.638894	−	0.769295i	$-0.279392\pi$
$829$	45.8455	−	13.4614i	1.59228	−	0.467535i	0.953384	+	0.301759i	$0.0975739\pi$
$830$		0			0
$831$	−19.0947	−	2.74541i	−0.662388	−	0.0952371i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−33.5602	−	47.1287i	−1.16001	−	1.62901i
$838$		0			0
$839$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$839$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$840$		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
$847$	43.9896	−	38.1172i	1.51150	−	1.30972i
$848$		0			0
$849$	−5.36797	−	8.35272i	−0.184228	−	0.286665i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−20.0381	−	3.86204i	−0.686093	−	0.132234i	−0.165720	−	0.986173i	$-0.552995\pi$
$853$	−20.0381	−	3.86204i	−0.686093	−	0.132234i	−0.520373	+	0.853939i	$0.674207\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$857$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$858$		0			0
$859$	−0.855751	+	3.52745i	−0.0291978	+	0.120355i	−0.984546	−	0.175129i	$-0.943966\pi$
$859$	−0.855751	+	3.52745i	−0.0291978	+	0.120355i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$863$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−27.8254	+	9.63047i	−0.945001	+	0.327068i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−19.2351	−	51.4167i	−0.651756	−	1.74219i
$872$		0			0
$873$	32.5623	+	23.1875i	1.10207	+	0.784777i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−27.1418	+	10.8660i	−0.916515	+	0.366917i	−0.781485	−	0.623924i	$-0.785537\pi$
$877$	−27.1418	+	10.8660i	−0.916515	+	0.366917i	−0.135030	+	0.990842i	$0.543113\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$881$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$882$		0			0
$883$	52.3479	+	12.6995i	1.76165	+	0.427371i	0.980427	−	0.196884i	$-0.0630822\pi$
$883$	52.3479	+	12.6995i	1.76165	+	0.427371i	0.781221	+	0.624255i	$0.214597\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$887$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$888$		0			0
$889$	−14.8356	+	76.9745i	−0.497571	+	2.58164i
$890$		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
$903$	48.7438	−	2.32195i	1.62209	−	0.0772697i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	36.3803	−	34.6886i	1.20799	−	1.15182i	0.223811	−	0.974633i	$-0.428150\pi$
$907$	36.3803	−	34.6886i	1.20799	−	1.15182i	0.984178	−	0.177183i	$-0.0566983\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$911$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.47084	+	46.8208i	0.147480	+	1.54448i	0.701922	+	0.712254i	$0.252325\pi$
$919$	4.47084	+	46.8208i	0.147480	+	1.54448i	−0.554442	+	0.832222i	$0.687068\pi$
$920$		0			0
$921$	−16.0579	−	5.55768i	−0.529125	−	0.183132i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	27.6576	+	26.3715i	0.909376	+	0.867089i
$926$		0			0
$927$	14.3635	+	5.75026i	0.471758	+	0.188863i
$928$		0			0
$929$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$929$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$930$		0			0
$931$	−101.539	−	9.69582i	−3.32781	−	0.317768i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	59.3445i		−	1.93870i	−0.245680	−	0.969351i	$-0.579011\pi$
$937$		−	59.3445i		−	1.93870i	0.245680	−	0.969351i	$-0.420989\pi$
$938$		0			0
$939$	−54.6016			−1.78185
$940$		0			0
$941$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$941$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$947$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$948$		0			0
$949$	−40.9440	+	102.273i	−1.32910	+	3.31993i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$953$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−4.42406	+	92.8725i	−0.142712	+	2.99589i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−28.9357	+	50.1181i	−0.930509	+	1.61169i	−0.148057	+	0.988979i	$0.547302\pi$
$967$	−28.9357	+	50.1181i	−0.930509	+	1.61169i	−0.782452	+	0.622710i	$0.786032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$971$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$972$		0			0
$973$	−4.23142	−	88.8284i	−0.135653	−	2.84771i
$974$		0			0
$975$	−29.0409	−	50.3004i	−0.930055	−	1.61090i
$976$		0			0
$977$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$977$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	39.8449	−	34.5258i	1.27215	−	1.10232i
$982$		0			0
$983$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$983$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	37.7580	+	32.7175i	1.19942	+	1.03931i	0.998213	+	0.0597587i	$0.0190331\pi$
$991$	37.7580	+	32.7175i	1.19942	+	1.03931i	0.201211	+	0.979548i	$0.435512\pi$
$992$		0			0
$993$	−0.298841	+	1.23184i	−0.00948344	+	0.0390913i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	24.2737	+	7.12741i	0.768757	+	0.225727i	0.642516	−	0.766272i	$-0.277891\pi$
$997$	24.2737	+	7.12741i	0.768757	+	0.225727i	0.126241	+	0.992000i	$0.459709\pi$
$998$		0			0
$999$	−14.7603	−	36.8695i	−0.466996	−	1.16650i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.353.1	yes	20
3.2	odd	2	CM	804.2.ba.a.353.1	yes	20
67.41	odd	66	inner	804.2.ba.a.41.1	✓	20
201.41	even	66	inner	804.2.ba.a.41.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.41.1	✓	20	67.41	odd	66	inner
804.2.ba.a.41.1	✓	20	201.41	even	66	inner
804.2.ba.a.353.1	yes	20	1.1	even	1	trivial
804.2.ba.a.353.1	yes	20	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.936417 - 1.45709i) q^{3} +(-0.502989 + 5.26754i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})\)	\(q+(0.936417 - 1.45709i) q^{3} +(-0.502989 + 5.26754i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(4.62817 - 4.85388i) q^{13} +(4.83520 - 0.461706i) q^{19} +(7.20430 + 5.66552i) q^{21} +(4.79746 - 1.40866i) q^{25} +(-5.14326 - 0.739490i) q^{27} +(7.68371 + 8.05844i) q^{31} +(3.82151 + 6.61906i) q^{37} +(-2.73867 - 11.2889i) q^{39} +(4.02392 - 3.48675i) q^{43} +(-20.6205 - 3.97428i) q^{49} +(3.85502 - 7.47769i) q^{57} +(-4.67299 - 11.6726i) q^{61} +(15.0014 - 5.19204i) q^{63} +(3.56930 - 7.36615i) q^{67} +(-15.2494 + 6.10492i) q^{73} +(2.43988 - 8.30945i) q^{75} +(-14.8646 - 3.60611i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(23.2401 + 26.8205i) q^{91} +(18.9371 - 3.64982i) q^{93} +(-11.5396 + 6.66241i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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