LMFDB - Embedded newform 804.2.ba.a.413.1

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			413.1
Root			$0.580057 - 0.814576i$ of defining polynomial
Character	$\chi$	$=$	804.413
Dual form			804.2.ba.a.329.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57553 + 0.719520i) q^{3} +(-0.842599 + 4.37182i) q^{7} +(1.96458 - 2.26725i) q^{9} +O(q^{10})$	$q+(-1.57553 + 0.719520i) q^{3} +(-0.842599 + 4.37182i) q^{7} +(1.96458 - 2.26725i) q^{9} +(7.17337 - 0.341710i) q^{13} +(-6.35738 + 1.22528i) q^{19} +(-1.81807 - 7.49420i) q^{21} +(-4.20627 + 2.70320i) q^{25} +(-1.46393 + 4.98567i) q^{27} +(-4.75556 - 0.226535i) q^{31} +(-6.07910 + 10.5293i) q^{37} +(-11.0560 + 5.69976i) q^{39} +(5.51453 + 0.792870i) q^{43} +(-11.9043 - 4.76575i) q^{49} +(9.13462 - 6.50473i) q^{57} +(5.09645 + 5.34500i) q^{61} +(8.25665 + 10.4992i) q^{63} +(-7.90436 - 2.12630i) q^{67} +(-8.03940 + 7.66555i) q^{73} +(4.68209 - 7.28547i) q^{75} +(-1.45319 + 2.81879i) q^{79} +(-1.28083 - 8.90839i) q^{81} +(-4.55038 + 31.6486i) q^{91} +(7.65551 - 3.06480i) q^{93} +(-10.7932 - 6.23144i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{59}{66}\right)$	$1$

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	$-0.590909\pi$
$5$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$6$		0			0
$7$	−0.842599	+	4.37182i	−0.318473	+	1.65239i	0.371408	+	0.928470i	$0.378875\pi$
$7$	−0.842599	+	4.37182i	−0.318473	+	1.65239i	−0.689881	+	0.723923i	$0.742337\pi$
$8$		0			0
$9$	1.96458	−	2.26725i	0.654861	−	0.755750i
$10$		0			0
$11$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$11$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$12$		0			0
$13$	7.17337	−	0.341710i	1.98953	−	0.0947732i	0.990461	−	0.137795i	$-0.0440016\pi$
$13$	7.17337	−	0.341710i	1.98953	−	0.0947732i	0.999074	+	0.0430222i	$0.0136986\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$17$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$18$		0			0
$19$	−6.35738	+	1.22528i	−1.45848	+	0.281099i	−0.855976	−	0.517015i	$-0.827043\pi$
$19$	−6.35738	+	1.22528i	−1.45848	+	0.281099i	−0.602506	+	0.798114i	$0.705831\pi$
$20$		0			0
$21$	−1.81807	−	7.49420i	−0.396736	−	1.63537i
$22$		0			0
$23$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$23$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$24$		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
$31$	−4.75556	−	0.226535i	−0.854123	−	0.0406869i	−0.384048	−	0.923313i	$-0.625470\pi$
$31$	−4.75556	−	0.226535i	−0.854123	−	0.0406869i	−0.470075	+	0.882626i	$0.655773\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.07910	+	10.5293i	−0.999398	+	1.73101i	−0.469661	+	0.882847i	$0.655624\pi$
$37$	−6.07910	+	10.5293i	−0.999398	+	1.73101i	−0.529738	+	0.848162i	$0.677710\pi$
$38$		0			0
$39$	−11.0560	+	5.69976i	−1.77037	+	0.912691i
$40$		0			0
$41$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$41$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$42$		0			0
$43$	5.51453	+	0.792870i	0.840958	+	0.120911i	0.549315	−	0.835616i	$-0.314889\pi$
$43$	5.51453	+	0.792870i	0.840958	+	0.120911i	0.291643	+	0.956527i	$0.405798\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$47$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$48$		0			0
$49$	−11.9043	−	4.76575i	−1.70061	−	0.680821i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$53$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.13462	−	6.50473i	1.20991	−	0.861573i
$58$		0			0
$59$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$59$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$60$		0			0
$61$	5.09645	+	5.34500i	0.652533	+	0.684357i	0.964001	−	0.265900i	$-0.0856690\pi$
$61$	5.09645	+	5.34500i	0.652533	+	0.684357i	−0.311467	+	0.950257i	$0.600820\pi$
$62$		0			0
$63$	8.25665	+	10.4992i	1.04024	+	1.32277i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.90436	−	2.12630i	−0.965671	−	0.259769i
$67$	−7.90436	−	2.12630i	−0.965671	−	0.259769i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$71$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$72$		0			0
$73$	−8.03940	+	7.66555i	−0.940941	+	0.897185i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−8.03940	+	7.66555i	−0.940941	+	0.897185i	0.0539089	+	0.998546i	$0.482832\pi$
$74$		0			0
$75$	4.68209	−	7.28547i	0.540641	−	0.841254i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.45319	+	2.81879i	−0.163496	+	0.317138i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−1.45319	+	2.81879i	−0.163496	+	0.317138i	0.792829	+	0.609445i	$0.208608\pi$
$80$		0			0
$81$	−1.28083	−	8.90839i	−0.142315	−	0.989821i
$82$		0			0
$83$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$83$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$89$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$90$		0			0
$91$	−4.55038	+	31.6486i	−0.477010	+	3.31768i
$92$		0			0
$93$	7.65551	−	3.06480i	0.793840	−	0.317805i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.7932	−	6.23144i	−1.09588	−	0.632707i	−0.160744	−	0.986996i	$-0.551390\pi$
$97$	−10.7932	−	6.23144i	−1.09588	−	0.632707i	−0.935136	+	0.354289i	$0.884723\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$101$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$102$		0			0
$103$	0.498621	−	10.4674i	0.0491306	−	1.03138i	−0.830888	−	0.556440i	$-0.812167\pi$
$103$	0.498621	−	10.4674i	0.0491306	−	1.03138i	0.880019	−	0.474939i	$-0.157530\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$107$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$108$		0			0
$109$	−1.59842	−	2.48719i	−0.153101	−	0.238230i	0.756227	−	0.654309i	$-0.227040\pi$
$109$	−1.59842	−	2.48719i	−0.153101	−	0.238230i	−0.909329	+	0.416079i	$0.863404\pi$
$110$		0			0
$111$	2.00175	−	20.9633i	0.189998	−	1.98975i
$112$		0			0
$113$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$113$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	13.3179	−	16.9351i	1.23124	−	1.56565i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.523401	−	10.9875i	−0.0475819	−	0.998867i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	22.1314	+	4.26548i	1.96385	+	0.378501i	0.982533	+	0.186091i	$0.0595819\pi$
$127$	22.1314	+	4.26548i	1.96385	+	0.378501i	0.981315	+	0.192410i	$0.0616302\pi$
$128$		0			0
$129$	−9.25879	+	2.71862i	−0.815191	+	0.239362i
$130$		0			0
$131$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$131$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$132$		0			0
$133$		−	28.8257i		−	2.49951i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$137$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$138$		0			0
$139$	5.28614	+	18.0029i	0.448365	+	1.52699i	0.805304	+	0.592862i	$0.202002\pi$
$139$	5.28614	+	18.0029i	0.448365	+	1.52699i	−0.356939	+	0.934128i	$0.616180\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	22.1846	−	1.05678i	1.82975	−	0.0871618i
$148$		0			0
$149$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$149$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$150$		0			0
$151$	15.4290	+	12.1335i	1.25560	+	0.987411i	0.999786	+	0.0206838i	$0.00658434\pi$
$151$	15.4290	+	12.1335i	1.25560	+	0.987411i	0.255809	+	0.966727i	$0.417658\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	24.7634	+	2.36462i	1.97633	+	0.188717i	0.999661	−	0.0260253i	$-0.00828506\pi$
$157$	24.7634	+	2.36462i	1.97633	+	0.188717i	0.976671	+	0.214742i	$0.0688911\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.35057	−	7.53540i	−0.340763	−	0.590218i	0.643812	−	0.765184i	$-0.277352\pi$
$163$	−4.35057	−	7.53540i	−0.340763	−	0.590218i	−0.984575	+	0.174966i	$0.944019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$167$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$168$		0			0
$169$	38.3993	−	3.66669i	2.95379	−	0.282053i
$170$		0			0
$171$	−9.71157	+	16.8209i	−0.742662	+	1.28633i
$172$		0			0
$173$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$173$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$174$		0			0
$175$	−8.27372	−	20.6668i	−0.625435	−	1.56226i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$179$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$180$		0			0
$181$	−14.6312	+	20.5466i	−1.08753	+	1.52722i	−0.260153	+	0.965567i	$0.583773\pi$
$181$	−14.6312	+	20.5466i	−1.08753	+	1.52722i	−0.827374	+	0.561651i	$0.810166\pi$
$182$		0			0
$183$	−11.8754	−	4.75421i	−0.877857	−	0.351441i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−20.5630	−	10.6009i	−1.49574	−	0.771105i
$190$		0			0
$191$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$191$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$192$		0			0
$193$	21.4810	+	13.8050i	1.54624	+	0.993705i	0.986261	+	0.165195i	$0.0528253\pi$
$193$	21.4810	+	13.8050i	1.54624	+	0.993705i	0.559974	+	0.828510i	$0.310811\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$197$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$198$		0			0
$199$	−8.08081	−	23.3480i	−0.572834	−	1.65509i	−0.742698	−	0.669626i	$-0.766454\pi$
$199$	−8.08081	−	23.3480i	−0.572834	−	1.65509i	0.169865	−	0.985467i	$-0.445667\pi$
$200$		0			0
$201$	13.9835	−	2.33729i	0.986317	−	0.164860i
$202$		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
$211$	15.8572	+	22.2683i	1.09165	+	1.53301i	0.821509	+	0.570195i	$0.193132\pi$
$211$	15.8572	+	22.2683i	1.09165	+	1.53301i	0.270146	+	0.962819i	$0.412928\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.99740	−	20.5996i	0.339246	−	1.39839i
$218$		0			0
$219$	7.15079	−	17.8618i	0.483206	−	1.20699i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	11.9893	−	26.2529i	0.802864	−	1.75803i	0.167412	−	0.985887i	$-0.446459\pi$
$223$	11.9893	−	26.2529i	0.802864	−	1.75803i	0.635452	−	0.772140i	$-0.280814\pi$
$224$		0			0
$225$	−2.13472	+	14.8473i	−0.142315	+	0.989821i
$226$		0			0
$227$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$227$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$228$		0			0
$229$	10.5227	+	20.4113i	0.695362	+	1.34881i	0.926649	+	0.375929i	$0.122676\pi$
$229$	10.5227	+	20.4113i	0.695362	+	1.34881i	−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$233$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.261363	−	5.48668i	0.0169773	−	0.356398i
$238$		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
$241$	−27.1993	−	7.98643i	−1.75206	−	0.514451i	−0.761103	−	0.648630i	$-0.775342\pi$
$241$	−27.1993	−	7.98643i	−1.75206	−	0.514451i	−0.990957	+	0.134179i	$0.957160\pi$
$242$		0			0
$243$	8.42776	+	13.1138i	0.540641	+	0.841254i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−45.1851	+	10.9618i	−2.87506	+	0.697482i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$251$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$257$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$258$		0			0
$259$	−40.9100	−	35.4487i	−2.54203	−	2.20268i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$263$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−29.2219	+	13.3452i	−1.77511	+	0.810664i	−0.796582	+	0.604530i	$0.793361\pi$
$271$	−29.2219	+	13.3452i	−1.77511	+	0.810664i	−0.978524	+	0.206133i	$0.933912\pi$
$272$		0			0
$273$	−15.6025	−	53.1374i	−0.944309	−	3.21602i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.96986	+	5.73553i	−0.298610	+	0.344614i	−0.885150	−	0.465306i	$-0.845944\pi$
$277$	−4.96986	+	5.73553i	−0.298610	+	0.344614i	0.586540	+	0.809920i	$0.300490\pi$
$278$		0			0
$279$	−9.85629	+	10.3370i	−0.590081	+	0.618859i
$280$		0			0
$281$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$281$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$282$		0			0
$283$	18.1894	+	20.9916i	1.08125	+	1.24782i	0.967110	+	0.254358i	$0.0818643\pi$
$283$	18.1894	+	20.9916i	1.08125	+	1.24782i	0.114135	+	0.993465i	$0.463590\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00790	+	16.5208i	0.235759	+	0.971812i
$290$		0			0
$291$	21.4886	+	2.05191i	1.25968	+	0.120285i
$292$		0			0
$293$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$293$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.11282	+	23.4405i	−0.467615	+	1.35109i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	23.5019	−	12.1160i	1.34132	−	0.691499i	0.369947	−	0.929053i	$-0.379376\pi$
$307$	23.5019	−	12.1160i	1.34132	−	0.691499i	0.971374	+	0.237553i	$0.0763455\pi$
$308$		0			0
$309$	6.74588	+	16.8504i	0.383759	+	0.958585i
$310$		0			0
$311$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$311$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$312$		0			0
$313$	−31.6320	−	14.4459i	−1.78795	−	0.816528i	−0.970720	−	0.240212i	$-0.922783\pi$
$313$	−31.6320	−	14.4459i	−1.78795	−	0.816528i	−0.817227	−	0.576316i	$-0.804490\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$317$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−29.2494	+	20.8284i	−1.62246	+	1.15535i
$326$		0			0
$327$	4.30795	+	2.76855i	0.238230	+	0.153101i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.3238	+	28.3870i	1.22703	+	1.56029i	0.709708	+	0.704496i	$0.248827\pi$
$331$	22.3238	+	28.3870i	1.22703	+	1.56029i	0.517317	+	0.855794i	$0.326931\pi$
$332$		0			0
$333$	11.9297	+	34.4685i	0.653742	+	1.88886i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	26.3615	−	9.12381i	1.43600	−	0.497006i	0.505352	−	0.862913i	$-0.331363\pi$
$337$	26.3615	−	9.12381i	1.43600	−	0.497006i	0.930651	+	0.365907i	$0.119241\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	14.0159	−	21.8092i	0.756789	−	1.17759i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$347$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$348$		0			0
$349$	1.02341	+	7.11795i	0.0547817	+	0.381015i	0.998706	+	0.0508535i	$0.0161942\pi$
$349$	1.02341	+	7.11795i	0.0547817	+	0.381015i	−0.943924	+	0.330162i	$0.892897\pi$
$350$		0			0
$351$	−8.79763	+	36.2643i	−0.469582	+	1.93565i
$352$		0			0
$353$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$353$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$359$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$360$		0			0
$361$	21.2759	−	8.51760i	1.11979	−	0.448295i
$362$		0			0
$363$	8.73039	+	16.9346i	0.458227	+	0.888835i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−3.36828	−	35.2742i	−0.175823	−	1.84130i	−0.466552	−	0.884494i	$-0.654504\pi$
$367$	−3.36828	−	35.2742i	−0.175823	−	1.84130i	0.290730	−	0.956805i	$-0.406102\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	33.0614	−	19.0880i	1.71185	−	0.988340i	0.779795	−	0.626035i	$-0.215323\pi$
$373$	33.0614	−	19.0880i	1.71185	−	0.988340i	0.932060	−	0.362305i	$-0.118010\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.90330	−	19.9323i	0.0977662	−	1.02385i	−0.804871	−	0.593450i	$-0.797765\pi$
$379$	1.90330	−	19.9323i	0.0977662	−	1.02385i	0.902637	−	0.430403i	$-0.141629\pi$
$380$		0			0
$381$	−37.9378	+	9.20361i	−1.94361	+	0.471515i
$382$		0			0
$383$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$383$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	12.6314	−	10.9452i	0.642089	−	0.556373i
$388$		0			0
$389$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$389$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−22.8740	+	6.71641i	−1.14801	+	0.337087i	−0.799763	−	0.600315i	$-0.795042\pi$
$397$	−22.8740	+	6.71641i	−1.14801	+	0.337087i	−0.348249	+	0.937402i	$0.613224\pi$
$398$		0			0
$399$	20.7407	+	45.4158i	1.03833	+	2.27363i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−34.1908			−1.70316
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.490026	−	2.54250i	0.0242302	−	0.125718i	−0.967800	−	0.251722i	$-0.919003\pi$
$409$	0.490026	−	2.54250i	0.0242302	−	0.125718i	0.992030	+	0.126004i	$0.0402152\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−21.2819	−	24.5607i	−1.04218	−	1.20274i
$418$		0			0
$419$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$419$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$420$		0			0
$421$	−34.7876	+	6.70476i	−1.69544	+	0.326770i	−0.943228	−	0.332147i	$-0.892227\pi$
$421$	−34.7876	+	6.70476i	−1.69544	+	0.326770i	−0.752216	+	0.658917i	$0.771015\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−27.6616	+	17.7771i	−1.33864	+	0.860292i
$428$		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
$433$	36.0107	+	1.71540i	1.73056	+	0.0824369i	0.889053	−	0.457804i	$-0.151364\pi$
$433$	36.0107	+	1.71540i	1.73056	+	0.0824369i	0.841511	+	0.540241i	$0.181667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	20.6409	−	35.7511i	0.985136	−	1.70631i	0.343805	−	0.939041i	$-0.388284\pi$
$439$	20.6409	−	35.7511i	0.985136	−	1.70631i	0.641331	−	0.767264i	$-0.278382\pi$
$440$		0			0
$441$	−34.1920	+	17.6272i	−1.62819	+	0.839392i
$442$		0			0
$443$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$443$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$449$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−33.0392	−	8.01521i	−1.55232	−	0.376588i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.7202	+	8.10434i	0.735361	+	0.379105i	0.784863	−	0.619669i	$-0.212733\pi$
$457$	15.7202	+	8.10434i	0.735361	+	0.379105i	−0.0495022	+	0.998774i	$0.515763\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$461$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$462$		0			0
$463$	−6.78938	−	7.12050i	−0.315529	−	0.330918i	0.546802	−	0.837262i	$-0.315845\pi$
$463$	−6.78938	−	7.12050i	−0.315529	−	0.330918i	−0.862331	+	0.506344i	$0.830996\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$467$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$468$		0			0
$469$	15.9560	−	32.7648i	0.736781	−	1.51294i
$470$		0			0
$471$	−40.7168	+	14.0922i	−1.87613	+	0.649335i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	23.4286	−	22.3392i	1.07498	−	1.02499i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$479$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$480$		0			0
$481$	−40.0097	+	77.6080i	−1.82428	+	3.53862i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−9.09682	+	22.7228i	−0.412216	+	1.02967i	0.566429	+	0.824110i	$0.308325\pi$
$487$	−9.09682	+	22.7228i	−0.412216	+	1.02967i	−0.978645	+	0.205556i	$0.934100\pi$
$488$		0			0
$489$	12.2763	+	8.74193i	0.555154	+	0.395324i
$490$		0			0
$491$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$491$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7.73117	+	4.46359i	0.346095	+	0.199818i	0.662964	−	0.748651i	$-0.269298\pi$
$499$	7.73117	+	4.46359i	0.346095	+	0.199818i	−0.316869	+	0.948469i	$0.602632\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$503$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−57.8610	+	33.4061i	−2.56970	+	1.48362i
$508$		0			0
$509$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$509$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$510$		0			0
$511$	−26.7384	−	41.6058i	−1.18284	−	1.84053i
$512$		0			0
$513$	3.19786	−	33.4895i	0.141189	−	1.47860i
$514$		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	$-0.227273\pi$
$521$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$522$		0			0
$523$	1.86868	+	39.2283i	0.0817115	+	1.71533i	0.552855	+	0.833278i	$0.313538\pi$
$523$	1.86868	+	39.2283i	0.0817115	+	1.71533i	−0.471143	+	0.882057i	$0.656158\pi$
$524$		0			0
$525$	27.9056	+	26.6080i	1.21790	+	1.16127i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	22.5844	+	4.35278i	0.981929	+	0.189251i
$530$		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
$541$	−9.54545	−	32.5088i	−0.410391	−	1.39766i	−0.862661	−	0.505782i	$-0.831204\pi$
$541$	−9.54545	−	32.5088i	−0.410391	−	1.39766i	0.452270	−	0.891881i	$-0.350614\pi$
$542$		0			0
$543$	8.26815	−	42.8992i	0.354820	−	1.84098i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.90970	−	4.10038i	0.167167	−	0.175319i	−0.634822	−	0.772658i	$-0.718927\pi$
$547$	3.90970	−	4.10038i	0.167167	−	0.175319i	0.801989	+	0.597339i	$0.203775\pi$
$548$		0			0
$549$	22.1308	−	1.05422i	0.944521	−	0.0449931i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−11.0988	−	8.72818i	−0.471968	−	0.371160i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$557$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$558$		0			0
$559$	39.8287	+	3.80318i	1.68457	+	0.160857i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$563$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	40.0251	+	1.90663i	1.68090	+	0.0800710i
$568$		0			0
$569$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$569$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$570$		0			0
$571$	45.7558	−	4.36915i	1.91482	−	0.182843i	0.931378	−	0.364054i	$-0.118607\pi$
$571$	45.7558	−	4.36915i	1.91482	−	0.182843i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	15.4004	+	38.4684i	0.641128	+	1.60146i	0.790350	+	0.612656i	$0.209899\pi$
$577$	15.4004	+	38.4684i	0.641128	+	1.60146i	−0.149222	+	0.988804i	$0.547677\pi$
$578$		0			0
$579$	−43.7769	−	6.29416i	−1.81930	−	0.261577i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$587$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$588$		0			0
$589$	30.5104	−	4.38674i	1.25716	−	0.180752i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$593$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	29.5309	+	30.9711i	1.20862	+	1.26756i
$598$		0			0
$599$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$599$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$600$		0			0
$601$	15.5368	+	44.8905i	0.633757	+	1.83112i	0.550919	+	0.834559i	$0.314277\pi$
$601$	15.5368	+	44.8905i	0.633757	+	1.83112i	0.0828381	+	0.996563i	$0.473602\pi$
$602$		0			0
$603$	−20.3496	+	13.7438i	−0.828700	+	0.559692i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−28.3098	+	22.2631i	−1.14906	+	0.903631i	−0.996411	−	0.0846490i	$-0.973023\pi$
$607$	−28.3098	+	22.2631i	−1.14906	+	0.903631i	−0.152650	+	0.988280i	$0.548781\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−15.8404	−	22.2447i	−0.639788	−	0.898456i	0.359612	−	0.933102i	$-0.382909\pi$
$613$	−15.8404	−	22.2447i	−0.639788	−	0.898456i	−0.999400	+	0.0346459i	$0.988970\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$617$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$618$		0			0
$619$	−0.739352	+	3.04765i	−0.0297171	+	0.122495i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−0.739352	+	3.04765i	−0.0297171	+	0.122495i	0.955021	+	0.296538i	$0.0958321\pi$
$620$		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
$631$	−22.9558	−	44.5281i	−0.913857	−	1.77263i	−0.523789	−	0.851848i	$-0.675482\pi$
$631$	−22.9558	−	44.5281i	−0.913857	−	1.77263i	−0.390068	−	0.920786i	$-0.627548\pi$
$632$		0			0
$633$	−41.0060	−	23.6748i	−1.62984	−	0.940989i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−87.0222	−	30.1187i	−3.44794	−	1.19335i
$638$		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
$643$	−24.1475	−	7.09036i	−0.952286	−	0.279616i	−0.231548	−	0.972824i	$-0.574379\pi$
$643$	−24.1475	−	7.09036i	−0.952286	−	0.279616i	−0.720738	+	0.693207i	$0.756197\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$647$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	6.94825	+	36.0509i	0.272323	+	1.41295i
$652$		0			0
$653$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$653$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.58565	+	33.2869i	0.0618622	+	1.29865i
$658$		0			0
$659$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$659$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$660$		0			0
$661$	2.82623	+	2.44894i	0.109928	+	0.0952528i	0.708090	−	0.706122i	$-0.249557\pi$
$661$	2.82623	+	2.44894i	0.109928	+	0.0952528i	−0.598163	+	0.801375i	$0.704102\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			49.9888i			1.93268i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	22.2638	−	10.1675i	0.858206	−	0.391930i	0.0628142	−	0.998025i	$-0.479992\pi$
$673$	22.2638	−	10.1675i	0.858206	−	0.391930i	0.795392	+	0.606096i	$0.207265\pi$
$674$		0			0
$675$	−7.31963	−	24.9284i	−0.281733	−	0.959493i
$676$		0			0
$677$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$677$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$678$		0			0
$679$	36.3371	−	41.9352i	1.39449	−	1.60932i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$683$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−31.2652	−	24.5872i	−1.19284	−	0.938061i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−8.20973	−	33.8410i	−0.312313	−	1.28737i	−0.884227	−	0.467057i	$-0.845314\pi$
$691$	−8.20973	−	33.8410i	−0.312313	−	1.28737i	0.571914	−	0.820313i	$-0.306201\pi$
$692$		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		−0.998867	−	0.0475819i	$-0.984848\pi$
$701$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$702$		0			0
$703$	25.7457	−	74.3875i	0.971019	−	2.80558i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	31.0876	−	16.0268i	1.16752	−	0.601898i	0.238215	−	0.971212i	$-0.423438\pi$
$709$	31.0876	−	16.0268i	1.16752	−	0.601898i	0.929305	+	0.369314i	$0.120407\pi$
$710$		0			0
$711$	3.53599	+	8.83247i	0.132610	+	0.331244i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$719$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$720$		0			0
$721$	45.3412	+	10.9997i	1.68860	+	0.409649i
$722$		0			0
$723$	48.5997	−	6.98758i	1.80744	−	0.259871i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−36.8977	+	26.2747i	−1.36846	+	0.974475i	−0.369483	+	0.929237i	$0.620465\pi$
$727$	−36.8977	+	26.2747i	−1.36846	+	0.974475i	−0.998976	+	0.0452379i	$0.985595\pi$
$728$		0			0
$729$	−22.7138	−	14.5973i	−0.841254	−	0.540641i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−31.7724	−	40.4020i	−1.17354	−	1.49228i	−0.834065	−	0.551666i	$-0.813992\pi$
$733$	−31.7724	−	40.4020i	−1.17354	−	1.49228i	−0.339477	−	0.940614i	$-0.610250\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	39.9667	−	13.8326i	1.47020	−	0.508841i	0.529568	−	0.848268i	$-0.322354\pi$
$739$	39.9667	−	13.8326i	1.47020	−	0.508841i	0.940633	+	0.339426i	$0.110233\pi$
$740$		0			0
$741$	63.3032	−	49.7822i	2.32550	−	1.82880i
$742$		0			0
$743$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$743$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.38669	+	9.64466i	0.0506011	+	0.351939i	0.999355	+	0.0359215i	$0.0114366\pi$
$751$	1.38669	+	9.64466i	0.0506011	+	0.351939i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	43.2062	+	30.7670i	1.57035	+	1.11824i	0.939606	+	0.342257i	$0.111191\pi$
$757$	43.2062	+	30.7670i	1.57035	+	1.11824i	0.630748	+	0.775988i	$0.282748\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$761$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$762$		0			0
$763$	12.2204	−	4.89231i	0.442408	−	0.177114i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−4.76998	−	49.9535i	−0.172010	−	1.80137i	−0.509953	−	0.860202i	$-0.670337\pi$
$769$	−4.76998	−	49.9535i	−0.172010	−	1.80137i	0.337943	−	0.941167i	$-0.390269\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$773$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$774$		0			0
$775$	20.6155	−	11.9024i	0.740531	−	0.427546i
$776$		0			0
$777$	89.9610	+	26.4149i	3.22733	+	0.947630i
$778$		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
$787$	−20.5167	+	26.0892i	−0.731343	+	0.929978i	−0.999442	−	0.0334133i	$-0.989362\pi$
$787$	−20.5167	+	26.0892i	−0.731343	+	0.929978i	0.268099	+	0.963391i	$0.413605\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	38.3851	+	36.6002i	1.36310	+	1.29971i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$797$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$798$		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	$-0.590909\pi$
$809$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$810$		0			0
$811$	10.4925	−	54.4402i	0.368441	−	1.91165i	−0.0406786	−	0.999172i	$-0.512952\pi$
$811$	10.4925	−	54.4402i	0.368441	−	1.91165i	0.409120	−	0.912481i	$-0.365836\pi$
$812$		0			0
$813$	36.4379	−	42.0515i	1.27793	−	1.47481i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−36.0294	+	1.71629i	−1.26051	+	0.0600455i
$818$		0			0
$819$	62.8157	+	72.4931i	2.19496	+	2.53312i
$820$		0			0
$821$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$821$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$822$		0			0
$823$	5.80076	−	1.11801i	0.202202	−	0.0389712i	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	5.80076	−	1.11801i	0.202202	−	0.0389712i	0.289346	+	0.957224i	$0.406562\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$827$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$828$		0			0
$829$	−48.4327	+	31.1258i	−1.68214	+	1.08104i	−0.829750	+	0.558135i	$0.811517\pi$
$829$	−48.4327	+	31.1258i	−1.68214	+	1.08104i	−0.852388	+	0.522909i	$0.824847\pi$
$830$		0			0
$831$	3.70333	−	12.6124i	0.128467	−	0.437519i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.09121	−	23.3780i	0.279673	−	0.808062i
$838$		0			0
$839$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$839$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$840$		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
$847$	48.4766	+	6.96988i	1.66567	+	0.239488i
$848$		0			0
$849$	−43.7618	−	19.9853i	−1.50190	−	0.685895i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	50.8069	+	20.3400i	1.73960	+	0.696429i	0.999719	+	0.0236865i	$0.00754034\pi$
$853$	50.8069	+	20.3400i	1.73960	+	0.696429i	0.739877	+	0.672743i	$0.234884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$857$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$858$		0			0
$859$	37.1872	+	19.1713i	1.26881	+	0.654117i	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	37.1872	+	19.1713i	1.26881	+	0.654117i	0.313462	+	0.949601i	$0.398511\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$863$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−18.2016	−	23.1452i	−0.618159	−	0.786053i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−57.4274	−	12.5518i	−1.94585	−	0.425301i
$872$		0			0
$873$	−35.3323	+	12.2286i	−1.19582	+	0.413876i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	30.8049	−	29.3724i	1.04021	−	0.991834i	0.0402330	−	0.999190i	$-0.487190\pi$
$877$	30.8049	−	29.3724i	1.04021	−	0.991834i	0.999973	+	0.00735598i	$0.00234150\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$881$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$882$		0			0
$883$	−7.87537	+	15.2761i	−0.265027	+	0.514081i	−0.983371	−	0.181607i	$-0.941870\pi$
$883$	−7.87537	+	15.2761i	−0.265027	+	0.514081i	0.718344	+	0.695688i	$0.244900\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$887$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$888$		0			0
$889$	−37.2959	+	93.1606i	−1.25086	+	3.12450i
$890$		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
$903$	−4.08389	−	42.7685i	−0.135903	−	1.42325i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.42987	−	51.0092i	0.0806825	−	1.69373i	−0.489331	−	0.872098i	$-0.662759\pi$
$907$	2.42987	−	51.0092i	0.0806825	−	1.69373i	0.570013	−	0.821635i	$-0.306938\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$911$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.88872	−	14.9881i	−0.0952901	−	0.494412i	−0.997947	−	0.0640502i	$-0.979598\pi$
$919$	−2.88872	−	14.9881i	−0.0952901	−	0.494412i	0.902657	−	0.430362i	$-0.141614\pi$
$920$		0			0
$921$	−28.3101	+	35.9992i	−0.932850	+	1.18622i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.89255	−	60.7222i	−0.0951066	−	1.99653i
$926$		0			0
$927$	−22.7525	−	21.6945i	−0.747290	−	0.712540i
$928$		0			0
$929$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$929$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$930$		0			0
$931$	81.5193	+	15.7116i	2.67169	+	0.514925i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			41.7921i			1.36529i	0.730750	+	0.682645i	$0.239170\pi$
$937$			41.7921i			1.36529i	−0.730750	+	0.682645i	$0.760830\pi$
$938$		0			0
$939$	60.2312			1.96557
$940$		0			0
$941$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$941$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$947$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$	−55.0502	+	57.7350i	−1.78701	+	1.87416i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$953$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8.29563	−	0.792137i	−0.267601	−	0.0255528i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−7.15581	−	12.3942i	−0.230115	−	0.398571i	0.727727	−	0.685867i	$-0.240577\pi$
$967$	−7.15581	−	12.3942i	−0.230115	−	0.398571i	−0.957842	+	0.287296i	$0.907244\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$971$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$972$		0			0
$973$	−83.1597	+	7.94079i	−2.66598	+	0.254570i
$974$		0			0
$975$	31.0968	−	53.8613i	0.995895	−	1.72494i
$976$		0			0
$977$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$977$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−8.77932	−	1.26228i	−0.280302	−	0.0403014i
$982$		0			0
$983$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$983$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	55.0394	−	7.91347i	1.74838	−	0.251380i	0.807442	−	0.589947i	$-0.200851\pi$
$991$	55.0394	−	7.91347i	1.74838	−	0.251380i	0.940942	+	0.338567i	$0.109942\pi$
$992$		0			0
$993$	−55.5967	−	28.6621i	−1.76431	−	0.909565i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	51.2098	+	32.9105i	1.62183	+	1.04229i	0.954797	+	0.297259i	$0.0960725\pi$
$997$	51.2098	+	32.9105i	1.62183	+	1.04229i	0.667033	+	0.745028i	$0.267564\pi$
$998$		0			0
$999$	−43.5964	−	45.7225i	−1.37933	−	1.44660i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.413.1	yes	20
3.2	odd	2	CM	804.2.ba.a.413.1	yes	20
67.61	odd	66	inner	804.2.ba.a.329.1	✓	20
201.128	even	66	inner	804.2.ba.a.329.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.329.1	✓	20	67.61	odd	66	inner
804.2.ba.a.329.1	✓	20	201.128	even	66	inner
804.2.ba.a.413.1	yes	20	1.1	even	1	trivial
804.2.ba.a.413.1	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)

\(f(q)\)	\(=\)	\(q+(-1.57553 + 0.719520i) q^{3} +(-0.842599 + 4.37182i) q^{7} +(1.96458 - 2.26725i) q^{9} +O(q^{10})\)	\(q+(-1.57553 + 0.719520i) q^{3} +(-0.842599 + 4.37182i) q^{7} +(1.96458 - 2.26725i) q^{9} +(7.17337 - 0.341710i) q^{13} +(-6.35738 + 1.22528i) q^{19} +(-1.81807 - 7.49420i) q^{21} +(-4.20627 + 2.70320i) q^{25} +(-1.46393 + 4.98567i) q^{27} +(-4.75556 - 0.226535i) q^{31} +(-6.07910 + 10.5293i) q^{37} +(-11.0560 + 5.69976i) q^{39} +(5.51453 + 0.792870i) q^{43} +(-11.9043 - 4.76575i) q^{49} +(9.13462 - 6.50473i) q^{57} +(5.09645 + 5.34500i) q^{61} +(8.25665 + 10.4992i) q^{63} +(-7.90436 - 2.12630i) q^{67} +(-8.03940 + 7.66555i) q^{73} +(4.68209 - 7.28547i) q^{75} +(-1.45319 + 2.81879i) q^{79} +(-1.28083 - 8.90839i) q^{81} +(-4.55038 + 31.6486i) q^{91} +(7.65551 - 3.06480i) q^{93} +(-10.7932 - 6.23144i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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