LMFDB - Embedded newform 804.2.ba.a.197.1

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			197.1
Root			$0.0475819 - 0.998867i$ of defining polynomial
Character	$\chi$	$=$	804.197
Dual form			804.2.ba.a.653.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.936417 + 1.45709i) q^{3} +(-0.614593 - 0.437650i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})$	$q+(-0.936417 + 1.45709i) q^{3} +(-0.614593 - 0.437650i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(-2.32224 - 0.563370i) q^{13} +(-4.99015 - 7.00769i) q^{19} +(1.21321 - 0.485697i) q^{21} +(4.79746 - 1.40866i) q^{25} +(5.14326 + 0.739490i) q^{27} +(-10.3416 + 2.50884i) q^{31} +(5.77342 - 9.99985i) q^{37} +(2.99547 - 2.85617i) q^{39} +(1.51198 - 1.31014i) q^{43} +(-2.10329 - 6.07705i) q^{49} +(14.8837 - 0.708999i) q^{57} +(8.44064 - 10.7332i) q^{61} +(-0.428367 + 2.22258i) q^{63} +(6.98512 + 4.26709i) q^{67} +(-11.3456 - 8.92225i) q^{73} +(-2.43988 + 8.30945i) q^{75} +(-5.50042 + 5.76867i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(1.18067 + 1.36257i) q^{91} +(6.02841 - 17.4180i) q^{93} +(-16.2217 - 9.36562i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{35}{66}\right)$	$1$

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
$7$	−0.614593	−	0.437650i	−0.232294	−	0.165416i	0.457983	−	0.888961i	$-0.348572\pi$
$7$	−0.614593	−	0.437650i	−0.232294	−	0.165416i	−0.690277	+	0.723545i	$0.742511\pi$
$8$		0			0
$9$	−1.24625	−	2.72890i	−0.415415	−	0.909632i
$10$		0			0
$11$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$11$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$12$		0			0
$13$	−2.32224	−	0.563370i	−0.644074	−	0.156251i	−0.0995988	−	0.995028i	$-0.531756\pi$
$13$	−2.32224	−	0.563370i	−0.644074	−	0.156251i	−0.544475	+	0.838777i	$0.683271\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$17$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$18$		0			0
$19$	−4.99015	−	7.00769i	−1.14482	−	1.60767i	−0.704245	−	0.709957i	$-0.748714\pi$
$19$	−4.99015	−	7.00769i	−1.14482	−	1.60767i	−0.440574	−	0.897716i	$-0.645225\pi$
$20$		0			0
$21$	1.21321	−	0.485697i	0.264745	−	0.105988i
$22$		0			0
$23$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$23$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$24$		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	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−10.3416	+	2.50884i	−1.85740	+	0.450601i	−0.998141	−	0.0609393i	$-0.980590\pi$
$31$	−10.3416	+	2.50884i	−1.85740	+	0.450601i	−0.859260	+	0.511540i	$0.829075\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.77342	−	9.99985i	0.949144	−	1.64396i	0.201909	−	0.979404i	$-0.435285\pi$
$37$	5.77342	−	9.99985i	0.949144	−	1.64396i	0.747234	−	0.664561i	$-0.231381\pi$
$38$		0			0
$39$	2.99547	−	2.85617i	0.479659	−	0.457354i
$40$		0			0
$41$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$41$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$42$		0			0
$43$	1.51198	−	1.31014i	0.230574	−	0.199794i	−0.531909	−	0.846801i	$-0.678525\pi$
$43$	1.51198	−	1.31014i	0.230574	−	0.199794i	0.762484	+	0.647008i	$0.223980\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$47$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$48$		0			0
$49$	−2.10329	−	6.07705i	−0.300470	−	0.868150i
$50$		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
$57$	14.8837	−	0.708999i	1.97140	−	0.0939092i
$58$		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
$61$	8.44064	−	10.7332i	1.08071	−	1.37424i	0.158590	−	0.987345i	$-0.449305\pi$
$61$	8.44064	−	10.7332i	1.08071	−	1.37424i	0.922124	−	0.386895i	$-0.126452\pi$
$62$		0			0
$63$	−0.428367	+	2.22258i	−0.0539692	+	0.280019i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.98512	+	4.26709i	0.853369	+	0.521308i
$67$	6.98512	+	4.26709i	0.853369	+	0.521308i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$71$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$72$		0			0
$73$	−11.3456	−	8.92225i	−1.32790	−	1.04427i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−11.3456	−	8.92225i	−1.32790	−	1.04427i	−0.333048	−	0.942910i	$-0.608077\pi$
$74$		0			0
$75$	−2.43988	+	8.30945i	−0.281733	+	0.959493i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.50042	+	5.76867i	−0.618846	+	0.649027i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−5.50042	+	5.76867i	−0.618846	+	0.649027i	0.337479	+	0.941333i	$0.390426\pi$
$80$		0			0
$81$	−5.89375	+	6.80175i	−0.654861	+	0.755750i
$82$		0			0
$83$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$83$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$84$		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
$91$	1.18067	+	1.36257i	0.123768	+	0.142836i
$92$		0			0
$93$	6.02841	−	17.4180i	0.625117	−	1.80616i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.2217	−	9.36562i	−1.64707	−	0.950934i	−0.978230	−	0.207526i	$-0.933459\pi$
$97$	−16.2217	−	9.36562i	−1.64707	−	0.950934i	−0.668837	−	0.743409i	$-0.733208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$101$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$102$		0			0
$103$	−2.70165	−	11.1363i	−0.266201	−	1.09730i	−0.935225	−	0.354055i	$-0.884803\pi$
$103$	−2.70165	−	11.1363i	−0.266201	−	1.09730i	0.669024	−	0.743241i	$-0.266712\pi$
$104$		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
$109$	5.64566	+	19.2274i	0.540757	+	1.84165i	0.540027	+	0.841648i	$0.318414\pi$
$109$	5.64566	+	19.2274i	0.540757	+	1.84165i	0.000729916		1.00000i	$0.499768\pi$
$110$		0			0
$111$	9.16439	+	17.7764i	0.869846	+	1.68726i
$112$		0			0
$113$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$113$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.35670	+	7.03925i	0.125427	+	0.650779i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.59335	+	10.6899i	−0.235759	+	0.971812i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−12.5560	+	17.6325i	−1.11417	+	1.56463i	−0.330107	+	0.943944i	$0.607085\pi$
$127$	−12.5560	+	17.6325i	−1.11417	+	1.56463i	−0.784060	+	0.620685i	$0.786855\pi$
$128$		0			0
$129$	0.493149	+	3.42993i	0.0434193	+	0.301988i
$130$		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
$133$			6.49081i			0.562825i
$134$		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
$139$	−7.98138	+	1.14755i	−0.676972	+	0.0973339i	−0.472221	−	0.881480i	$-0.656548\pi$
$139$	−7.98138	+	1.14755i	−0.676972	+	0.0973339i	−0.204751	+	0.978814i	$0.565638\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.8244	+	2.62597i	0.892781	+	0.216586i
$148$		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
$151$	−11.6313	+	2.24175i	−0.946541	+	0.182431i	−0.639089	−	0.769133i	$-0.720688\pi$
$151$	−11.6313	+	2.24175i	−0.946541	+	0.182431i	−0.307452	+	0.951564i	$0.599476\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.6353	−	5.99843i	0.928601	−	0.478727i	0.0735617	−	0.997291i	$-0.476563\pi$
$157$	11.6353	−	5.99843i	0.928601	−	0.478727i	0.855039	+	0.518564i	$0.173533\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.22226	−	10.7773i	−0.487365	−	0.844142i	0.512529	−	0.858670i	$-0.328709\pi$
$163$	−6.22226	−	10.7773i	−0.487365	−	0.844142i	−0.999894	+	0.0145283i	$0.995375\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$167$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$168$		0			0
$169$	−6.47944	−	3.34038i	−0.498419	−	0.256953i
$170$		0			0
$171$	−12.9043	+	22.3509i	−0.986816	+	1.70922i
$172$		0			0
$173$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$173$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$174$		0			0
$175$	−3.56499	−	1.23385i	−0.269488	−	0.0932706i
$176$		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
$181$	−1.25066	+	26.2546i	−0.0929610	+	1.95149i	0.167192	+	0.985924i	$0.446530\pi$
$181$	−1.25066	+	26.2546i	−0.0929610	+	1.95149i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	7.73524	+	22.3495i	0.571806	+	1.65212i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.83738	−	2.70543i	−0.206389	−	0.196791i
$190$		0			0
$191$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$191$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$192$		0			0
$193$	−0.617267	−	0.181246i	−0.0444319	−	0.0130464i	0.259441	−	0.965759i	$-0.416462\pi$
$193$	−0.617267	−	0.181246i	−0.0444319	−	0.0130464i	−0.303873	+	0.952713i	$0.598280\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$197$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$198$		0			0
$199$	5.85768	+	0.559341i	0.415240	+	0.0396506i	0.300586	−	0.953755i	$-0.402818\pi$
$199$	5.85768	+	0.559341i	0.415240	+	0.0396506i	0.114654	+	0.993405i	$0.463424\pi$
$200$		0			0
$201$	−12.7585	+	6.18221i	−0.899918	+	0.436059i
$202$		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
$211$	0.277952	+	5.83493i	0.0191350	+	0.401693i	0.988034	+	0.154237i	$0.0492918\pi$
$211$	0.277952	+	5.83493i	0.0191350	+	0.401693i	−0.968899	+	0.247457i	$0.920405\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.45385	+	2.98407i	0.506000	+	0.202572i
$218$		0			0
$219$	23.6247	−	8.17660i	1.59641	−	0.552524i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.9305	−	10.8806i	1.13375	−	0.728618i	0.167412	−	0.985887i	$-0.446459\pi$
$223$	16.9305	−	10.8806i	1.13375	−	0.728618i	0.966340	+	0.257269i	$0.0828225\pi$
$224$		0			0
$225$	−9.82291	−	11.3362i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$227$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$228$		0			0
$229$	11.3723	+	11.9269i	0.751502	+	0.788153i	0.982789	−	0.184733i	$-0.0591420\pi$
$229$	11.3723	+	11.9269i	0.751502	+	0.788153i	−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$233$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3.25481	−	13.4165i	−0.211423	−	0.871496i
$238$		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
$241$	3.36304	−	23.3905i	0.216633	−	1.50671i	−0.533712	−	0.845666i	$-0.679203\pi$
$241$	3.36304	−	23.3905i	0.216633	−	1.50671i	0.750345	−	0.661047i	$-0.229888\pi$
$242$		0			0
$243$	−4.39178	−	14.9570i	−0.281733	−	0.959493i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.64042	+	19.0848i	0.486148	+	1.21434i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$251$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$257$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$258$		0			0
$259$	−7.92473	+	3.61910i	−0.492419	+	0.224880i
$260$		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
$271$	−10.7607	+	16.7440i	−0.653667	+	1.01713i	0.343294	+	0.939228i	$0.388457\pi$
$271$	−10.7607	+	16.7440i	−0.653667	+	1.01713i	−0.996961	+	0.0778984i	$0.975179\pi$
$272$		0			0
$273$	−3.09100	+	0.444418i	−0.187076	+	0.0268974i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−12.2396	−	26.8011i	−0.735409	−	1.61032i	−0.790962	−	0.611866i	$-0.790419\pi$
$277$	−12.2396	−	26.8011i	−0.735409	−	1.61032i	0.0555527	−	0.998456i	$-0.482308\pi$
$278$		0			0
$279$	19.7345	+	25.0944i	1.18147	+	1.50237i
$280$		0			0
$281$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$281$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$282$		0			0
$283$	−6.16501	+	13.4995i	−0.366472	+	0.802461i	0.633125	+	0.774050i	$0.281772\pi$
$283$	−6.16501	+	13.4995i	−0.366472	+	0.802461i	−0.999596	+	0.0284112i	$0.990955\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.7823	−	6.31826i	0.928368	−	0.371662i
$290$		0			0
$291$	28.8369	−	14.8665i	1.69045	−	0.871487i
$292$		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
$301$	−1.50263	+	0.143484i	−0.0866102	+	0.00827027i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	25.1102	−	23.9425i	1.43311	−	1.36647i	0.619610	−	0.784910i	$-0.287291\pi$
$307$	25.1102	−	23.9425i	1.43311	−	1.36647i	0.813503	−	0.581560i	$-0.197558\pi$
$308$		0			0
$309$	18.7566	+	6.49171i	1.06702	+	0.369300i
$310$		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
$313$	−4.59522	−	7.15031i	−0.259737	−	0.404159i	0.686753	−	0.726891i	$-0.259035\pi$
$313$	−4.59522	−	7.15031i	−0.259737	−	0.404159i	−0.946491	+	0.322731i	$0.895399\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$317$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−11.9345	+	0.568509i	−0.662005	+	0.0315352i
$326$		0			0
$327$	−33.3028	−	9.77858i	−1.84165	−	0.540757i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.20515	−	32.1954i	0.341066	−	1.76962i	−0.255257	−	0.966873i	$-0.582160\pi$
$331$	6.20515	−	32.1954i	0.341066	−	1.76962i	0.596323	−	0.802745i	$-0.296628\pi$
$332$		0			0
$333$	−34.4836	−	3.29279i	−1.88969	−	0.180444i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.18353	+	33.3394i	−0.173418	+	1.81611i	0.321211	+	0.947008i	$0.395910\pi$
$337$	−3.18353	+	33.3394i	−0.173418	+	1.81611i	−0.494628	+	0.869105i	$0.664696\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.85491	+	9.72294i	−0.154151	+	0.524989i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$347$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$348$		0			0
$349$	23.7967	−	27.4628i	1.27381	−	1.47005i	0.461136	−	0.887330i	$-0.347442\pi$
$349$	23.7967	−	27.4628i	1.27381	−	1.47005i	0.812672	−	0.582722i	$-0.198012\pi$
$350$		0			0
$351$	−11.5273	−	4.61483i	−0.615281	−	0.246321i
$352$		0			0
$353$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$353$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$354$		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
$361$	−17.9918	+	51.9838i	−0.946935	+	2.73599i
$362$		0			0
$363$	−13.1478	−	13.7890i	−0.690079	−	0.723734i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.29918	−	8.33924i	0.224415	−	0.435305i	−0.749567	−	0.661928i	$-0.769738\pi$
$367$	4.29918	−	8.33924i	0.224415	−	0.435305i	0.973982	+	0.226623i	$0.0727687\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	9.10103	−	5.25448i	0.471233	−	0.272067i	−0.245523	−	0.969391i	$-0.578960\pi$
$373$	9.10103	−	5.25448i	0.471233	−	0.272067i	0.716756	+	0.697324i	$0.245626\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−17.7302	−	34.3917i	−0.910738	−	1.76658i	−0.554481	−	0.832196i	$-0.687083\pi$
$379$	−17.7302	−	34.3917i	−0.910738	−	1.76658i	−0.356257	−	0.934388i	$-0.615947\pi$
$380$		0			0
$381$	−13.9345	−	34.8067i	−0.713885	−	1.78320i
$382$		0			0
$383$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$383$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.45952	−	2.49328i	−0.277523	−	0.126740i
$388$		0			0
$389$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$389$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.21269	−	8.43443i	−0.0608631	−	0.423312i	−0.997359	−	0.0726328i	$-0.976860\pi$
$397$	−1.21269	−	8.43443i	−0.0608631	−	0.423312i	0.936496	−	0.350679i	$-0.114049\pi$
$398$		0			0
$399$	−9.45772	−	6.07811i	−0.473478	−	0.304286i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	25.4290			1.26671
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−32.2451	−	22.9617i	−1.59442	−	1.13538i	−0.916346	−	0.400387i	$-0.868876\pi$
$409$	−32.2451	−	22.9617i	−1.59442	−	1.13538i	−0.678073	−	0.734994i	$-0.737185\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	5.80182	−	12.7042i	0.284116	−	0.622128i
$418$		0			0
$419$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$419$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$420$		0			0
$421$	23.1020	+	32.4423i	1.12592	+	1.58114i	0.759262	+	0.650785i	$0.225560\pi$
$421$	23.1020	+	32.4423i	1.12592	+	1.58114i	0.366663	+	0.930354i	$0.380500\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−9.88492	+	2.90247i	−0.478365	+	0.140461i
$428$		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
$433$	39.3086	−	9.53618i	1.88905	−	0.458279i	0.889053	−	0.457804i	$-0.151364\pi$
$433$	39.3086	−	9.53618i	1.88905	−	0.458279i	1.00000		0.000475561i	$-0.000151376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−10.7969	+	18.7007i	−0.515306	+	0.892537i	0.484536	+	0.874771i	$0.338989\pi$
$439$	−10.7969	+	18.7007i	−0.515306	+	0.892537i	−0.999842	+	0.0177654i	$0.994345\pi$
$440$		0			0
$441$	−13.9624	+	13.3132i	−0.664878	+	0.633960i
$442$		0			0
$443$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$443$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$449$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	7.62530	−	19.0471i	0.358268	−	0.894910i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.3085	+	19.3641i	0.949991	+	0.905815i	0.995653	−	0.0931420i	$-0.0296911\pi$
$457$	20.3085	+	19.3641i	0.949991	+	0.905815i	−0.0456614	+	0.998957i	$0.514540\pi$
$458$		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
$463$	21.0311	−	26.7432i	0.977396	−	1.24286i	0.00734134	−	0.999973i	$-0.497663\pi$
$463$	21.0311	−	26.7432i	0.977396	−	1.24286i	0.970055	−	0.242887i	$-0.0780944\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$467$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$468$		0			0
$469$	−2.42552	−	5.67956i	−0.112000	−	0.262258i
$470$		0			0
$471$	−2.15525	+	22.5708i	−0.0993087	+	1.04001i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−33.8115	−	26.5897i	−1.55138	−	1.22002i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$479$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$480$		0			0
$481$	−19.0409	+	19.9695i	−0.868189	+	0.910530i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	11.0835	−	3.83603i	0.502241	−	0.173827i	−0.0641880	−	0.997938i	$-0.520446\pi$
$487$	11.0835	−	3.83603i	0.502241	−	0.173827i	0.566429	+	0.824110i	$0.308325\pi$
$488$		0			0
$489$	21.5301	+	1.02561i	0.973627	+	0.0463796i
$490$		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
$499$	−18.0987	−	10.4493i	−0.810209	−	0.467775i	0.0368191	−	0.999322i	$-0.488277\pi$
$499$	−18.0987	−	10.4493i	−0.810209	−	0.467775i	−0.847029	+	0.531547i	$0.821611\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$503$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	10.9347	−	6.31316i	0.485628	−	0.280377i
$508$		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
$511$	3.06808	+	10.4489i	0.135724	+	0.462234i
$512$		0			0
$513$	−20.4835	−	39.7325i	−0.904371	−	1.75423i
$514$		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
$523$	8.87407	−	36.5794i	0.388036	−	1.59951i	−0.358081	−	0.933691i	$-0.616569\pi$
$523$	8.87407	−	36.5794i	0.388036	−	1.59951i	0.746117	−	0.665815i	$-0.231916\pi$
$524$		0			0
$525$	5.13616	−	4.03912i	0.224160	−	0.176282i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.3413	−	18.7352i	0.580057	−	0.814576i
$530$		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
$541$	1.88319	−	0.270762i	0.0809647	−	0.0116410i	−0.101713	−	0.994814i	$-0.532432\pi$
$541$	1.88319	−	0.270762i	0.0809647	−	0.0116410i	0.182678	+	0.983173i	$0.441523\pi$
$542$		0			0
$543$	−37.0843	−	26.4076i	−1.59144	−	1.13326i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	19.3982	+	24.6668i	0.829406	+	1.05468i	0.997622	+	0.0689253i	$0.0219570\pi$
$547$	19.3982	+	24.6668i	0.829406	+	1.05468i	−0.168216	+	0.985750i	$0.553801\pi$
$548$		0			0
$549$	−39.8088	−	9.65750i	−1.69900	−	0.412172i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.90518	−	1.13813i	0.251114	−	0.0483982i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$557$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$558$		0			0
$559$	−4.24927	+	2.19065i	−0.179725	+	0.0926546i
$560$		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
$567$	6.59904	−	1.60091i	0.277133	−	0.0672318i
$568$		0			0
$569$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$569$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$570$		0			0
$571$	40.6586	+	20.9609i	1.70151	+	0.877188i	0.983444	+	0.181210i	$0.0580014\pi$
$571$	40.6586	+	20.9609i	1.70151	+	0.877188i	0.718063	+	0.695978i	$0.245029\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	28.3125	+	9.79906i	1.17867	+	0.407940i	0.845008	−	0.534754i	$-0.179596\pi$
$577$	28.3125	+	9.79906i	1.17867	+	0.407940i	0.333658	+	0.942694i	$0.391717\pi$
$578$		0			0
$579$	0.842112	−	0.729694i	0.0349970	−	0.0303251i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$587$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$588$		0			0
$589$	69.1871	+	59.9510i	2.85081	+	2.47024i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$593$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−6.30024	+	8.01141i	−0.257852	+	0.327885i
$598$		0			0
$599$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$599$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$600$		0			0
$601$	5.23519	+	0.499900i	0.213548	+	0.0203914i	0.201282	−	0.979533i	$-0.435489\pi$
$601$	5.23519	+	0.499900i	0.213548	+	0.0203914i	0.0122663	+	0.999925i	$0.496095\pi$
$602$		0			0
$603$	2.93926	−	24.3795i	0.119696	−	0.992811i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−15.3463	−	2.95776i	−0.622888	−	0.120052i	−0.131964	−	0.991254i	$-0.542128\pi$
$607$	−15.3463	−	2.95776i	−0.622888	−	0.120052i	−0.490923	+	0.871203i	$0.663341\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.33631	+	49.0452i	0.0943626	+	1.98092i	0.176523	+	0.984297i	$0.443515\pi$
$613$	2.33631	+	49.0452i	0.0943626	+	1.98092i	−0.0821600	+	0.996619i	$0.526182\pi$
$614$		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
$619$	−45.2196	−	18.1032i	−1.81753	−	0.727629i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−45.2196	−	18.1032i	−1.81753	−	0.727629i	−0.832792	−	0.553587i	$-0.813259\pi$
$620$		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
$631$	30.4929	+	31.9800i	1.21390	+	1.27310i	0.951324	+	0.308193i	$0.0997243\pi$
$631$	30.4929	+	31.9800i	1.21390	+	1.27310i	0.262578	+	0.964911i	$0.415427\pi$
$632$		0			0
$633$	−8.76233	−	5.05893i	−0.348271	−	0.201074i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.46072	+	15.2973i	0.0578757	+	0.606102i
$638$		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
$643$	7.08107	−	49.2499i	0.279250	−	1.94223i	−0.0519076	−	0.998652i	$-0.516530\pi$
$643$	7.08107	−	49.2499i	0.279250	−	1.94223i	0.331158	−	0.943575i	$-0.392561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$647$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−11.3280	+	8.06662i	−0.443979	+	0.316156i
$652$		0			0
$653$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$653$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−10.2085	+	42.0802i	−0.398273	+	1.64170i
$658$		0			0
$659$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$659$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$660$		0			0
$661$	−37.4828	+	17.1178i	−1.45791	+	0.665806i	−0.977442	−	0.211205i	$-0.932261\pi$
$661$	−37.4828	+	17.1178i	−1.45791	+	0.665806i	−0.480470	+	0.877011i	$0.659534\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			34.8582i			1.34769i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	22.5987	−	35.1642i	0.871115	−	1.35548i	−0.0628142	−	0.998025i	$-0.520008\pi$
$673$	22.5987	−	35.1642i	0.871115	−	1.35548i	0.933930	−	0.357457i	$-0.116356\pi$
$674$		0			0
$675$	25.7163	−	3.69745i	0.989821	−	0.142315i
$676$		0			0
$677$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$677$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$678$		0			0
$679$	5.87090	+	12.8555i	0.225304	+	0.493348i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$683$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−28.0279	+	5.40193i	−1.06933	+	0.206096i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	20.7167	−	8.29372i	0.788101	−	0.315508i	0.0575291	−	0.998344i	$-0.481678\pi$
$691$	20.7167	−	8.29372i	0.788101	−	0.315508i	0.730572	+	0.682836i	$0.239254\pi$
$692$		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		0.971812	−	0.235759i	$-0.0757576\pi$
$701$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$702$		0			0
$703$	−98.8860	+	9.44247i	−3.72956	+	0.356129i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.58152	+	5.32196i	−0.209618	+	0.199871i	−0.787614	−	0.616169i	$-0.788684\pi$
$709$	−5.58152	+	5.32196i	−0.209618	+	0.199871i	0.577995	+	0.816040i	$0.303835\pi$
$710$		0			0
$711$	22.5970	+	7.82089i	0.847453	+	0.293306i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$719$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$720$		0			0
$721$	−3.21340	+	8.02669i	−0.119673	+	0.298929i
$722$		0			0
$723$	30.9329	+	26.8035i	1.15041	+	0.996834i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	51.7184	−	2.46365i	1.91813	−	0.0913717i	0.945766	−	0.324850i	$-0.105314\pi$
$727$	51.7184	−	2.46365i	1.91813	−	0.0913717i	0.972362	+	0.233478i	$0.0750106\pi$
$728$		0			0
$729$	25.9063	+	7.60678i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−6.97341	+	36.1815i	−0.257569	+	1.33639i	0.590727	+	0.806871i	$0.298841\pi$
$733$	−6.97341	+	36.1815i	−0.257569	+	1.33639i	−0.848296	+	0.529522i	$0.822371\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.178224	+	1.86645i	−0.00655608	+	0.0686584i	−0.998158	−	0.0606703i	$-0.980676\pi$
$739$	−0.178224	+	1.86645i	−0.00655608	+	0.0686584i	0.991602	+	0.129329i	$0.0412822\pi$
$740$		0			0
$741$	−34.9630	−	6.73857i	−1.28440	−	0.247547i
$742$		0			0
$743$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$743$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−30.8719	+	35.6281i	−1.12653	+	1.30009i	−0.177779	+	0.984070i	$0.556891\pi$
$751$	−30.8719	+	35.6281i	−1.12653	+	1.30009i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−22.0244	−	1.04915i	−0.800492	−	0.0381321i	−0.356651	−	0.934238i	$-0.616081\pi$
$757$	−22.0244	−	1.04915i	−0.800492	−	0.0381321i	−0.443841	+	0.896106i	$0.646384\pi$
$758$		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
$763$	4.94506	−	14.2878i	0.179023	−	0.517254i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−19.0135	+	36.8810i	−0.685643	+	1.32996i	0.246949	+	0.969028i	$0.420572\pi$
$769$	−19.0135	+	36.8810i	−0.685643	+	1.32996i	−0.932592	+	0.360933i	$0.882458\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$773$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$774$		0			0
$775$	−46.0792	+	26.6038i	−1.65521	+	0.955638i
$776$		0			0
$777$	2.14748	−	14.9361i	0.0770405	−	0.535828i
$778$		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
$787$	3.61485	+	18.7556i	0.128856	+	0.668566i	0.987704	+	0.156336i	$0.0499683\pi$
$787$	3.61485	+	18.7556i	0.128856	+	0.668566i	−0.858848	+	0.512230i	$0.828820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−25.6479	+	20.1698i	−0.910785	+	0.716249i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$797$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$798$		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
$811$	46.3262	+	32.9887i	1.62673	+	1.15839i	0.844969	+	0.534815i	$0.179619\pi$
$811$	46.3262	+	32.9887i	1.62673	+	1.15839i	0.781763	+	0.623576i	$0.214321\pi$
$812$		0			0
$813$	−14.3211	−	31.3588i	−0.502262	−	1.09980i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−16.7260	−	4.05769i	−0.585169	−	0.141960i
$818$		0			0
$819$	2.24690	−	4.92003i	0.0785132	−	0.171920i
$820$		0			0
$821$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$821$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$822$		0			0
$823$	−28.6894	−	40.2886i	−1.00005	−	1.40437i	−0.912904	−	0.408175i	$-0.866165\pi$
$823$	−28.6894	−	40.2886i	−1.00005	−	1.40437i	−0.0871445	−	0.996196i	$-0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$827$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$828$		0			0
$829$	−37.0488	+	10.8785i	−1.28676	+	0.377826i	−0.852388	−	0.522909i	$-0.824847\pi$
$829$	−37.0488	+	10.8785i	−1.28676	+	0.377826i	−0.434369	+	0.900735i	$0.643028\pi$
$830$		0			0
$831$	50.5131	+	7.26269i	1.75228	+	0.251940i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−55.0447	+	5.25613i	−1.90262	+	0.181678i
$838$		0			0
$839$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$839$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$840$		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
$847$	6.27230	−	5.43497i	0.215519	−	0.186748i
$848$		0			0
$849$	−13.8970	−	21.6241i	−0.476944	−	0.742139i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	15.2929	+	44.1860i	0.523619	+	1.51290i	0.828212	+	0.560415i	$0.189359\pi$
$853$	15.2929	+	44.1860i	0.523619	+	1.51290i	−0.304592	+	0.952483i	$0.598520\pi$
$854$		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
$859$	20.6818	+	19.7201i	0.705656	+	0.672841i	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	20.6818	+	19.7201i	0.705656	+	0.672841i	−0.249692	+	0.968325i	$0.580329\pi$
$860$		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
$867$	−5.57248	+	28.9128i	−0.189251	+	0.981929i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−13.8172	−	13.8444i	−0.468178	−	0.469100i
$872$		0			0
$873$	−5.34155	+	55.9393i	−0.180784	+	1.89326i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15.5504	+	12.2290i	0.525100	+	0.412943i	0.845208	−	0.534438i	$-0.179477\pi$
$877$	15.5504	+	12.2290i	0.525100	+	0.412943i	−0.320108	+	0.947381i	$0.603719\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$881$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$882$		0			0
$883$	31.2027	−	32.7244i	1.05005	−	1.10126i	0.0553507	−	0.998467i	$-0.482372\pi$
$883$	31.2027	−	32.7244i	1.05005	−	1.10126i	0.994702	−	0.102797i	$-0.0327792\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$887$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$888$		0			0
$889$	15.4337	−	5.34165i	0.517629	−	0.179153i
$890$		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
$903$	1.19802	−	2.32383i	0.0398676	−	0.0773324i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.16686	−	37.7863i	−0.304381	−	1.25467i	−0.894394	−	0.447280i	$-0.852393\pi$
$907$	−9.16686	−	37.7863i	−0.304381	−	1.25467i	0.590013	−	0.807394i	$-0.299123\pi$
$908$		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
$919$	45.5410	−	32.4296i	1.50226	−	1.06975i	0.526692	−	0.850056i	$-0.323432\pi$
$919$	45.5410	−	32.4296i	1.50226	−	1.06975i	0.975568	−	0.219698i	$-0.0705073\pi$
$920$		0			0
$921$	11.3729	+	59.0080i	0.374749	+	1.94438i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	13.6113	−	56.1067i	0.447538	−	1.84478i
$926$		0			0
$927$	−27.0230	+	21.2511i	−0.887551	+	0.697978i
$928$		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
$931$	−32.0904	+	45.0646i	−1.05172	+	1.47693i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			6.44207i			0.210453i	0.994448	+	0.105227i	$0.0335568\pi$
$937$			6.44207i			0.210453i	−0.994448	+	0.105227i	$0.966443\pi$
$938$		0			0
$939$	14.7217			0.480425
$940$		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
$949$	21.3206	+	27.1114i	0.692096	+	0.880072i
$950$		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
$961$	73.0999	−	37.6856i	2.35806	−	1.21567i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−22.3808	−	38.7646i	−0.719717	−	1.24659i	−0.961112	−	0.276160i	$-0.910938\pi$
$967$	−22.3808	−	38.7646i	−0.719717	−	1.24659i	0.241395	−	0.970427i	$-0.422395\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$971$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$972$		0			0
$973$	5.40753	+	2.78777i	0.173357	+	0.0893719i
$974$		0			0
$975$	10.3473	−	17.9220i	0.331378	−	0.573963i
$976$		0			0
$977$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$977$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	45.4336	−	39.3684i	1.45058	−	1.25694i
$982$		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
$991$	28.0710	+	24.3236i	0.891704	+	0.772666i	0.974614	−	0.223891i	$-0.0718760\pi$
$991$	28.0710	+	24.3236i	0.891704	+	0.772666i	−0.0829100	+	0.996557i	$0.526421\pi$
$992$		0			0
$993$	41.1011	+	39.1898i	1.30430	+	1.24365i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−40.4173	−	11.8676i	−1.28003	−	0.375850i	−0.430115	−	0.902774i	$-0.641527\pi$
$997$	−40.4173	−	11.8676i	−1.28003	−	0.375850i	−0.849912	+	0.526924i	$0.823345\pi$
$998$		0			0
$999$	37.0890	−	47.1625i	1.17344	−	1.49215i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.197.1	✓	20
3.2	odd	2	CM	804.2.ba.a.197.1	✓	20
67.50	odd	66	inner	804.2.ba.a.653.1	yes	20
201.50	even	66	inner	804.2.ba.a.653.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.197.1	✓	20	1.1	even	1	trivial
804.2.ba.a.197.1	✓	20	3.2	odd	2	CM
804.2.ba.a.653.1	yes	20	67.50	odd	66	inner
804.2.ba.a.653.1	yes	20	201.50	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)

\(f(q)\)	\(=\)	\(q+(-0.936417 + 1.45709i) q^{3} +(-0.614593 - 0.437650i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})\)	\(q+(-0.936417 + 1.45709i) q^{3} +(-0.614593 - 0.437650i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(-2.32224 - 0.563370i) q^{13} +(-4.99015 - 7.00769i) q^{19} +(1.21321 - 0.485697i) q^{21} +(4.79746 - 1.40866i) q^{25} +(5.14326 + 0.739490i) q^{27} +(-10.3416 + 2.50884i) q^{31} +(5.77342 - 9.99985i) q^{37} +(2.99547 - 2.85617i) q^{39} +(1.51198 - 1.31014i) q^{43} +(-2.10329 - 6.07705i) q^{49} +(14.8837 - 0.708999i) q^{57} +(8.44064 - 10.7332i) q^{61} +(-0.428367 + 2.22258i) q^{63} +(6.98512 + 4.26709i) q^{67} +(-11.3456 - 8.92225i) q^{73} +(-2.43988 + 8.30945i) q^{75} +(-5.50042 + 5.76867i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(1.18067 + 1.36257i) q^{91} +(6.02841 - 17.4180i) q^{93} +(-16.2217 - 9.36562i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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