LMFDB - Embedded newform 668.2.b.a.667.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(667,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("668.667");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.b (of order $2$, degree $1$, 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:	$5.33400685502$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			667.6
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.07158 + 0.922888i) q^{2} +3.45525i q^{3} +(0.296557 - 1.97789i) q^{4} +(-3.18880 - 3.70256i) q^{6} -4.00920i q^{7} +(1.50759 + 2.39315i) q^{8} -8.93873 q^{9} +O(q^{10})$	\(q+(-1.07158 + 0.922888i) q^{2} +3.45525i q^{3} +(0.296557 - 1.97789i) q^{4} +(-3.18880 - 3.70256i) q^{6} -4.00920i q^{7} +(1.50759 + 2.39315i) q^{8} -8.93873 q^{9} -2.44151i q^{11} +(6.83410 + 1.02468i) q^{12} +(3.70004 + 4.29617i) q^{14} +(-3.82411 - 1.17311i) q^{16} +(9.57854 - 8.24944i) q^{18} -8.71732i q^{19} +13.8528 q^{21} +(2.25324 + 2.61627i) q^{22} +(-8.26893 + 5.20909i) q^{24} +5.00000 q^{25} -20.5198i q^{27} +(-7.92976 - 1.18896i) q^{28} -10.3616 q^{29} -2.77092i q^{31} +(5.18048 - 2.27214i) q^{32} +8.43601 q^{33} +(-2.65084 + 17.6798i) q^{36} +(8.04510 + 9.34128i) q^{38} +(-14.8443 + 12.7845i) q^{42} +(-4.82904 - 0.724046i) q^{44} +7.00673i q^{47} +(4.05340 - 13.2132i) q^{48} -9.07366 q^{49} +(-5.35789 + 4.61444i) q^{50} +(18.9374 + 21.9885i) q^{54} +(9.59462 - 6.04422i) q^{56} +30.1205 q^{57} +(11.1032 - 9.56256i) q^{58} -14.5538 q^{61} +(2.55725 + 2.96926i) q^{62} +35.8371i q^{63} +(-3.45436 + 7.21577i) q^{64} +(-9.03984 + 7.78549i) q^{66} +(-13.4759 - 21.3917i) q^{72} +17.2762i q^{75} +(-17.2419 - 2.58518i) q^{76} -9.78849 q^{77} +44.0847 q^{81} +(4.10813 - 27.3993i) q^{84} -35.8018i q^{87} +(5.84290 - 3.68079i) q^{88} +14.0588 q^{89} +9.57422 q^{93} +(-6.46642 - 7.50825i) q^{94} +(7.85080 + 17.8998i) q^{96} -19.6890 q^{97} +(9.72313 - 8.37397i) q^{98} +21.8240i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 66 q^{9}+O(q^{10}) $ 22 * q - 66 * q^9	$ 22 q - 66 q^{9} + 110 q^{25} - 11 q^{42} - 33 q^{44} + 55 q^{48} - 154 q^{49} + 77 q^{54} - 99 q^{62} - 121 q^{72} + 198 q^{81} + 143 q^{84} + 165 q^{98}+O(q^{100}) $ 22 * q - 66 * q^9 + 110 * q^25 - 11 * q^42 - 33 * q^44 + 55 * q^48 - 154 * q^49 + 77 * q^54 - 99 * q^62 - 121 * q^72 + 198 * q^81 + 143 * q^84 + 165 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/668\mathbb{Z}\right)^\times$.

$n$	$5$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.07158	+	0.922888i	−0.757720	+	0.652580i
$2$	−1.07158	+	0.922888i	−0.757720	+	0.652580i
$3$			3.45525i			1.99489i	0.0714572	+	0.997444i	$0.477235\pi$
$3$			3.45525i			1.99489i	−0.0714572	+	0.997444i	$0.522765\pi$
$4$	0.296557	−	1.97789i	0.148278	−	0.988946i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−3.18880	−	3.70256i	−1.30182	−	1.51157i
$7$		−	4.00920i		−	1.51533i	−0.652641	−	0.757667i	$-0.726339\pi$
$7$		−	4.00920i		−	1.51533i	0.652641	−	0.757667i	$-0.273661\pi$
$8$	1.50759	+	2.39315i	0.533013	+	0.846107i
$9$	−8.93873			−2.97958
$10$		0			0
$11$		−	2.44151i		−	0.736142i	−0.929798	−	0.368071i	$-0.880018\pi$
$11$		−	2.44151i		−	0.736142i	0.929798	−	0.368071i	$-0.119982\pi$
$12$	6.83410	+	1.02468i	1.97284	+	0.295799i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.70004	+	4.29617i	0.988877	+	1.14820i
$15$		0			0
$16$	−3.82411	−	1.17311i	−0.956027	−	0.293279i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	9.57854	−	8.24944i	2.25768	−	1.94441i
$19$		−	8.71732i		−	1.99989i	−0.0105100	−	0.999945i	$-0.503346\pi$
$19$		−	8.71732i		−	1.99989i	0.0105100	−	0.999945i	$-0.496654\pi$
$20$		0			0
$21$	13.8528			3.02292
$22$	2.25324	+	2.61627i	0.480392	+	0.557790i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−8.26893	+	5.20909i	−1.68789	+	1.06330i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	20.5198i		−	3.94903i
$28$	−7.92976	−	1.18896i	−1.49858	−	0.224691i
$29$	−10.3616			−1.92409			−0.962047	−	0.272883i	$-0.912023\pi$
$29$	−10.3616			−1.92409			−0.962047	+	0.272883i	$0.912023\pi$
$30$		0			0
$31$		−	2.77092i		−	0.497672i	−0.968546	−	0.248836i	$-0.919952\pi$
$31$		−	2.77092i		−	0.497672i	0.968546	−	0.248836i	$-0.0800481\pi$
$32$	5.18048	−	2.27214i	0.915788	−	0.401661i
$33$	8.43601			1.46852
$34$		0			0
$35$		0			0
$36$	−2.65084	+	17.6798i	−0.441807	+	2.94664i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	8.04510	+	9.34128i	1.30509	+	1.51536i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−14.8443	+	12.7845i	−2.29053	+	1.97270i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−4.82904	−	0.724046i	−0.728005	−	0.109154i
$45$		0			0
$46$		0			0
$47$			7.00673i			1.02204i	0.859570	+	0.511018i	$0.170732\pi$
$47$			7.00673i			1.02204i	−0.859570	+	0.511018i	$0.829268\pi$
$48$	4.05340	−	13.2132i	0.585058	−	1.90717i
$49$	−9.07366			−1.29624
$50$	−5.35789	+	4.61444i	−0.757720	+	0.652580i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	18.9374	+	21.9885i	2.57706	+	2.99226i
$55$		0			0
$56$	9.59462	−	6.04422i	1.28214	−	0.807692i
$57$	30.1205			3.98955
$58$	11.1032	−	9.56256i	1.45792	−	1.25563i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−14.5538			−1.86343			−0.931713	−	0.363195i	$-0.881686\pi$
$61$	−14.5538			−1.86343			−0.931713	+	0.363195i	$0.881686\pi$
$62$	2.55725	+	2.96926i	0.324771	+	0.377096i
$63$			35.8371i			4.51505i
$64$	−3.45436	+	7.21577i	−0.431795	+	0.901972i
$65$		0			0
$66$	−9.03984	+	7.78549i	−1.11273	+	0.958328i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−13.4759	−	21.3917i	−1.58815	−	2.52104i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			17.2762i			1.99489i
$76$	−17.2419	−	2.58518i	−1.97778	−	0.296541i
$77$	−9.78849			−1.11550
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	44.0847			4.89829
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	4.10813	−	27.3993i	0.448234	−	2.98950i
$85$		0			0
$86$		0			0
$87$		−	35.8018i		−	3.83835i
$88$	5.84290	−	3.68079i	0.622855	−	0.392373i
$89$	14.0588			1.49023			0.745113	−	0.666939i	$-0.232396\pi$
$89$	14.0588			1.49023			0.745113	+	0.666939i	$0.232396\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	9.57422			0.992800
$94$	−6.46642	−	7.50825i	−0.666961	−	0.774417i
$95$		0			0
$96$	7.85080	+	17.8998i	0.801269	+	1.82689i
$97$	−19.6890			−1.99911			−0.999556	−	0.0298024i	$-0.990512\pi$
$97$	−19.6890			−1.99911			−0.999556	+	0.0298024i	$0.990512\pi$
$98$	9.72313	−	8.37397i	0.982185	−	0.845899i
$99$			21.8240i			2.19339i
$100$	1.48278	−	9.88946i	0.148278	−	0.988946i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.9057i		−	1.73101i	−0.500904	−	0.865503i	$-0.666999\pi$
$107$		−	17.9057i		−	1.73101i	0.500904	−	0.865503i	$-0.333001\pi$
$108$	−40.5859	−	6.08528i	−3.90538	−	0.585556i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−4.70325	+	15.3316i	−0.444415	+	1.44870i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−32.2764	+	27.7978i	−3.02296	+	2.60350i
$115$		0			0
$116$	−3.07279	+	20.4941i	−0.285302	+	1.90282i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.03904			0.458094
$122$	15.5956	−	13.4315i	1.41196	−	1.21604i
$123$		0			0
$124$	−5.48058	−	0.821736i	−0.492171	−	0.0737941i
$125$		0			0
$126$	−33.0736	−	38.4023i	−2.94643	−	3.42114i
$127$		−	8.49861i		−	0.754129i	−0.926187	−	0.377065i	$-0.876933\pi$
$127$		−	8.49861i		−	0.754129i	0.926187	−	0.377065i	$-0.123067\pi$
$128$	−2.95774	−	10.9202i	−0.261429	−	0.965223i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	2.50176	−	16.6855i	0.217750	−	1.45229i
$133$	−34.9494			−3.03050
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.72626			0.830971			0.415485	−	0.909600i	$-0.363612\pi$
$137$	9.72626			0.830971			0.415485	+	0.909600i	$0.363612\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−24.2100			−2.03885
$142$		0			0
$143$		0			0
$144$	34.1827	+	10.4862i	2.84855	+	0.873846i
$145$		0			0
$146$		0			0
$147$		−	31.3517i		−	2.58585i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−15.9440	−	18.5128i	−1.30182	−	1.51157i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	20.8619	−	13.1421i	1.69212	−	1.06597i
$153$		0			0
$154$	10.4891	−	9.03368i	0.845238	−	0.727954i
$155$		0			0
$156$		0			0
$157$	23.6442			1.88701			0.943506	−	0.331355i	$-0.107506\pi$
$157$	23.6442			1.88701			0.943506	+	0.331355i	$0.107506\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−47.2401	+	40.6852i	−3.71153	+	3.19653i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.9228i		−	1.00000i
$167$		−	12.9228i		−	1.00000i
$168$	20.8843	+	33.1518i	1.61126	+	2.55772i
$169$	13.0000			1.00000
$170$		0			0
$171$			77.9217i			5.95882i
$172$		0			0
$173$	2.60890			0.198351			0.0991754	−	0.995070i	$-0.468380\pi$
$173$	2.60890			0.198351			0.0991754	+	0.995070i	$0.468380\pi$
$174$	33.0410	+	38.3644i	2.50483	+	2.90840i
$175$		−	20.0460i		−	1.51533i
$176$	−2.86417	+	9.33659i	−0.215895	+	0.703772i
$177$		0			0
$178$	−15.0650	+	12.9747i	−1.12917	+	0.972491i
$179$		−	12.1385i		−	0.907274i	−0.891187	−	0.453637i	$-0.850126\pi$
$179$		−	12.1385i		−	0.907274i	0.891187	−	0.453637i	$-0.149874\pi$
$180$		0			0
$181$	−17.9994			−1.33789			−0.668943	−	0.743313i	$-0.733253\pi$
$181$	−17.9994			−1.33789			−0.668943	+	0.743313i	$0.733253\pi$
$182$		0			0
$183$		−	50.2871i		−	3.71733i
$184$		0			0
$185$		0			0
$186$	−10.2595	+	8.83593i	−0.752264	+	0.647882i
$187$		0			0
$188$	13.8585	+	2.07789i	1.01074	+	0.151546i
$189$	−82.2678			−5.98410
$190$		0			0
$191$			23.0790i			1.66994i	0.550299	+	0.834968i	$0.314514\pi$
$191$			23.0790i			1.66994i	−0.550299	+	0.834968i	$0.685486\pi$
$192$	−24.9323	−	11.9357i	−1.79933	−	0.861382i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	21.0983	−	18.1707i	1.51477	−	1.30458i
$195$		0			0
$196$	−2.69086	+	17.9467i	−0.192204	+	1.28191i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−20.1411	−	23.3861i	−1.43136	−	1.66198i
$199$			23.1477i			1.64089i	0.571723	+	0.820447i	$0.306275\pi$
$199$			23.1477i			1.64089i	−0.571723	+	0.820447i	$0.693725\pi$
$200$	7.53794	+	11.9658i	0.533013	+	0.846107i
$201$		0			0
$202$		0			0
$203$			41.5416i			2.91565i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.2834			−1.47220
$210$		0			0
$211$		−	24.9202i		−	1.71558i	−0.514001	−	0.857790i	$-0.671837\pi$
$211$		−	24.9202i		−	1.71558i	0.514001	−	0.857790i	$-0.328163\pi$
$212$		0			0
$213$		0			0
$214$	16.5249	+	19.1873i	1.12962	+	1.31162i
$215$		0			0
$216$	49.1069	−	30.9353i	3.34130	−	2.10488i
$217$	−11.1092			−0.754140
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	0.980822i		−	0.0656807i	−0.999461	−	0.0328403i	$-0.989545\pi$
$223$		−	0.980822i		−	0.0656807i	0.999461	−	0.0328403i	$-0.0104553\pi$
$224$	−9.10945	−	20.7696i	−0.608651	−	1.38773i
$225$	−44.6936			−2.97958
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	8.93243	−	59.5750i	0.591565	−	3.94545i
$229$	17.2709			1.14129			0.570646	−	0.821196i	$-0.306693\pi$
$229$	17.2709			1.14129			0.570646	+	0.821196i	$0.306693\pi$
$230$		0			0
$231$		−	33.8216i		−	2.22530i
$232$	−15.6210	−	24.7968i	−1.02557	−	1.62799i
$233$	−28.0695			−1.83889			−0.919447	−	0.393215i	$-0.871363\pi$
$233$	−28.0695			−1.83889			−0.919447	+	0.393215i	$0.871363\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5.92967i		−	0.383559i	−0.981438	−	0.191779i	$-0.938574\pi$
$239$		−	5.92967i		−	0.383559i	0.981438	−	0.191779i	$-0.0614258\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−5.39972	+	4.65046i	−0.347107	+	0.298943i
$243$			90.7640i			5.82252i
$244$	−4.31604	+	28.7859i	−0.276306	+	1.84283i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	6.63124	−	4.17741i	0.421084	−	0.265266i
$249$		0			0
$250$		0			0
$251$		−	6.14952i		−	0.388154i	−0.980986	−	0.194077i	$-0.937829\pi$
$251$		−	6.14952i		−	0.388154i	0.980986	−	0.194077i	$-0.0621712\pi$
$252$	70.8819	+	10.6277i	4.46514	+	0.669485i
$253$		0			0
$254$	7.84326	+	9.10691i	0.492130	+	0.571419i
$255$		0			0
$256$	13.2476	+	8.97223i	0.827975	+	0.560765i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	92.6192			5.73299
$262$		0			0
$263$		−	14.8018i		−	0.912719i	−0.889796	−	0.456359i	$-0.849153\pi$
$263$		−	14.8018i		−	0.912719i	0.889796	−	0.456359i	$-0.150847\pi$
$264$	12.7180	+	20.1887i	0.782741	+	1.24253i
$265$		0			0
$266$	37.4510	−	32.2544i	2.29627	−	1.97764i
$267$			48.5765i			2.97283i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−10.4224	+	8.97625i	−0.629643	+	0.542275i
$275$		−	12.2075i		−	0.736142i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$			24.7685i			1.48285i
$280$		0			0
$281$	17.0292			1.01588			0.507938	−	0.861394i	$-0.330408\pi$
$281$	17.0292			1.01588			0.507938	+	0.861394i	$0.330408\pi$
$282$	25.9429	−	22.3431i	1.54487	−	1.33051i
$283$		−	24.3217i		−	1.44578i	−0.690965	−	0.722889i	$-0.742814\pi$
$283$		−	24.3217i		−	1.44578i	0.690965	−	0.722889i	$-0.257186\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−46.3069	+	20.3100i	−2.72866	+	1.19678i
$289$	17.0000			1.00000
$290$		0			0
$291$		−	68.0302i		−	3.98800i
$292$		0			0
$293$	−2.35163			−0.137384			−0.0686918	−	0.997638i	$-0.521883\pi$
$293$	−2.35163			−0.137384			−0.0686918	+	0.997638i	$0.521883\pi$
$294$	28.9341	+	33.5958i	1.68747	+	1.95935i
$295$		0			0
$296$		0			0
$297$	−50.0992			−2.90705
$298$		0			0
$299$		0			0
$300$	34.1705	+	5.12339i	1.97284	+	0.295799i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−10.2264	+	33.3360i	−0.586525	+	1.91195i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−2.90284	+	19.3606i	−0.165405	+	1.10317i
$309$		0			0
$310$		0			0
$311$		−	25.8457i		−	1.46557i	−0.680458	−	0.732787i	$-0.738219\pi$
$311$		−	25.8457i		−	1.46557i	0.680458	−	0.732787i	$-0.261781\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−25.3366	+	21.8209i	−1.42983	+	1.23143i
$315$		0			0
$316$		0			0
$317$	−12.5735			−0.706201			−0.353100	−	0.935585i	$-0.614873\pi$
$317$	−12.5735			−0.706201			−0.353100	+	0.935585i	$0.614873\pi$
$318$		0			0
$319$			25.2979i			1.41641i
$320$		0			0
$321$	61.8685			3.45316
$322$		0			0
$323$		0			0
$324$	13.0736	−	87.1946i	0.726312	−	4.84415i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	28.0914			1.54873
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	11.9263	+	13.8478i	0.652580	+	0.757720i
$335$		0			0
$336$	−52.9745	−	16.2509i	−2.88999	−	0.886558i
$337$	17.4497			0.950545			0.475272	−	0.879839i	$-0.342350\pi$
$337$	17.4497			0.950545			0.475272	+	0.879839i	$0.342350\pi$
$338$	−13.9305	+	11.9975i	−0.757720	+	0.652580i
$339$		0			0
$340$		0			0
$341$	−6.76523			−0.366358
$342$	−71.9130	−	83.4992i	−3.88861	−	4.51512i
$343$			8.31373i			0.448899i
$344$		0			0
$345$		0			0
$346$	−2.79564	+	2.40772i	−0.150294	+	0.129440i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−70.8120	−	10.6173i	−3.79592	−	0.569145i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	18.5002	+	21.4808i	0.988877	+	1.14820i
$351$		0			0
$352$	−5.54745	−	12.6482i	−0.295680	−	0.674151i
$353$	31.0823			1.65434			0.827172	−	0.561950i	$-0.189948\pi$
$353$	31.0823			1.65434			0.827172	+	0.561950i	$0.189948\pi$
$354$		0			0
$355$		0			0
$356$	4.16922	−	27.8067i	0.220968	−	1.47375i
$357$		0			0
$358$	11.2025	+	13.0073i	0.592069	+	0.687459i
$359$			37.8452i			1.99739i	0.0510543	+	0.998696i	$0.483742\pi$
$359$			37.8452i			1.99739i	−0.0510543	+	0.998696i	$0.516258\pi$
$360$		0			0
$361$	−56.9916			−2.99956
$362$	19.2878	−	16.6114i	1.01374	−	0.873078i
$363$			17.4111i			0.913846i
$364$		0			0
$365$		0			0
$366$	46.4093	+	53.8865i	2.42585	+	2.81669i
$367$			13.8491i			0.722916i	0.932388	+	0.361458i	$0.117721\pi$
$367$			13.8491i			0.722916i	−0.932388	+	0.361458i	$0.882279\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2.83930	−	18.9368i	0.147211	−	0.981826i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−16.7682	+	10.5633i	−0.864752	+	0.544758i
$377$		0			0
$378$	88.1563	−	75.9239i	4.53427	−	3.90510i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	29.3648			1.50440
$382$	−21.2993	−	24.7309i	−1.08977	−	1.26534i
$383$			23.7837i			1.21529i	0.794208	+	0.607646i	$0.207886\pi$
$383$			23.7837i			1.21529i	−0.794208	+	0.607646i	$0.792114\pi$
$384$	37.7321	−	10.2197i	1.92551	−	0.521522i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−5.83890	+	38.9426i	−0.296425	+	1.97701i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−13.6793	−	21.7147i	−0.690911	−	1.09676i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	43.1655	+	6.47205i	2.16915	+	0.325233i
$397$	−17.5242			−0.879517			−0.439758	−	0.898116i	$-0.644936\pi$
$397$	−17.5242			−0.879517			−0.439758	+	0.898116i	$0.644936\pi$
$398$	−21.3627	−	24.8045i	−1.07081	−	1.24334i
$399$		−	120.759i		−	6.04551i
$400$	−19.1205	−	5.86557i	−0.956027	−	0.293279i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−38.3382	−	44.5150i	−1.90269	−	2.20924i
$407$		0			0
$408$		0			0
$409$	−24.4339			−1.20818			−0.604089	−	0.796917i	$-0.706463\pi$
$409$	−24.4339			−1.20818			−0.604089	+	0.796917i	$0.706463\pi$
$410$		0			0
$411$			33.6066i			1.65769i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	22.8068	−	19.6422i	1.11552	−	0.960731i
$419$			0.929667i			0.0454172i	0.999742	+	0.0227086i	$0.00722899\pi$
$419$			0.929667i			0.0454172i	−0.999742	+	0.0227086i	$0.992771\pi$
$420$		0			0
$421$	3.45708			0.168488			0.0842438	−	0.996445i	$-0.473153\pi$
$421$	3.45708			0.168488			0.0842438	+	0.996445i	$0.473153\pi$
$422$	22.9986	+	26.7040i	1.11955	+	1.29993i
$423$		−	62.6312i		−	3.04523i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			58.3492i			2.82371i
$428$	−35.4155	−	5.31005i	−1.71187	−	0.256671i
$429$		0			0
$430$		0			0
$431$			26.6611i			1.28422i	0.766612	+	0.642111i	$0.221941\pi$
$431$			26.6611i			1.28422i	−0.766612	+	0.642111i	$0.778059\pi$
$432$	−24.0720	+	78.4698i	−1.15817	+	3.77538i
$433$	18.1827			0.873804			0.436902	−	0.899509i	$-0.356076\pi$
$433$	18.1827			0.873804			0.436902	+	0.899509i	$0.356076\pi$
$434$	11.9043	−	10.2525i	0.571427	−	0.492137i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	81.1070			3.86224
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	0.905188	+	1.05103i	0.0428619	+	0.0497676i
$447$		0			0
$448$	28.9295	+	13.8492i	1.36679	+	0.654313i
$449$	−29.3271			−1.38403			−0.692017	−	0.721882i	$-0.743277\pi$
$449$	−29.3271			−1.38403			−0.692017	+	0.721882i	$0.743277\pi$
$450$	47.8927	−	41.2472i	2.25768	−	1.94441i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	45.4093	+	72.0829i	2.12648	+	3.37559i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−18.5071	+	15.9391i	−0.864780	+	0.744785i
$459$		0			0
$460$		0			0
$461$	−41.3878			−1.92762			−0.963812	−	0.266583i	$-0.914105\pi$
$461$	−41.3878			−1.92762			−0.963812	+	0.266583i	$0.914105\pi$
$462$	31.2136	+	36.2425i	1.45219	+	1.68615i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	39.6237	+	12.1553i	1.83949	+	0.564296i
$465$		0			0
$466$	30.0786	−	25.9050i	1.39337	−	1.20003i
$467$			41.9609i			1.94172i	0.239654	+	0.970858i	$0.422966\pi$
$467$			41.9609i			1.94172i	−0.239654	+	0.970858i	$0.577034\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			81.6965i			3.76438i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	43.5866i		−	1.99989i
$476$		0			0
$477$		0			0
$478$	5.47242	+	6.35410i	0.250303	+	0.290630i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.49436	−	9.96667i	0.0679255	−	0.453030i
$485$		0			0
$486$	−83.7650	−	97.2607i	−3.79966	−	4.41184i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−21.9412	−	34.8295i	−0.993230	−	1.57666i
$489$		0			0
$490$		0			0
$491$			25.8457i			1.16640i	0.812329	+	0.583200i	$0.198200\pi$
$491$			25.8457i			1.16640i	−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−3.25061	+	10.5963i	−0.145957	+	0.475788i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	44.6516			1.99489
$502$	5.67531	+	6.58968i	0.253302	+	0.294112i
$503$			35.5333i			1.58435i	0.610293	+	0.792175i	$0.291052\pi$
$503$			35.5333i			1.58435i	−0.610293	+	0.792175i	$0.708948\pi$
$504$	−85.7637	+	54.0276i	−3.82022	+	2.40658i
$505$		0			0
$506$		0			0
$507$			44.9182i			1.99489i
$508$	−16.8093	−	2.52032i	−0.745793	−	0.111821i
$509$	27.2792			1.20913			0.604565	−	0.796556i	$-0.293347\pi$
$509$	27.2792			1.20913			0.604565	+	0.796556i	$0.293347\pi$
$510$		0			0
$511$		0			0
$512$	−22.4762	+	2.61161i	−0.993317	+	0.115418i
$513$	−178.877			−7.89762
$514$		0			0
$515$		0			0
$516$		0			0
$517$	17.1070			0.752364
$518$		0			0
$519$			9.01438i			0.395687i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−99.2487	+	85.4771i	−4.34400	+	3.74123i
$523$		−	36.0039i		−	1.57434i	−0.616735	−	0.787171i	$-0.711545\pi$
$523$		−	36.0039i		−	1.57434i	0.616735	−	0.787171i	$-0.288455\pi$
$524$		0			0
$525$	69.2638			3.02292
$526$	13.6604	+	15.8613i	0.595622	+	0.691585i
$527$		0			0
$528$	−32.2602	−	9.89641i	−1.40395	−	0.430686i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	−10.3645	+	69.1262i	−0.449358	+	2.99700i
$533$		0			0
$534$	−44.8306	−	52.0535i	−1.94001	−	2.25257i
$535$		0			0
$536$		0			0
$537$	41.9415			1.80991
$538$		0			0
$539$			22.1534i			0.954216i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	62.1925i		−	2.66893i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	2.88439	−	19.2375i	0.123215	−	0.821785i
$549$	130.093			5.55222
$550$	11.2662	+	13.0813i	0.480392	+	0.557790i
$551$			90.3251i			3.84798i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−46.7317			−1.98009			−0.990044	−	0.140761i	$-0.955045\pi$
$557$	−46.7317			−1.98009			−0.990044	+	0.140761i	$0.955045\pi$
$558$	−22.8586	−	26.5414i	−0.967680	−	1.12359i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−18.2481	+	15.7160i	−0.769749	+	0.662940i
$563$		−	1.87496i		−	0.0790203i	−0.999219	−	0.0395102i	$-0.987420\pi$
$563$		−	1.87496i		−	0.0790203i	0.999219	−	0.0395102i	$-0.0125797\pi$
$564$	−7.17963	+	47.8847i	−0.302317	+	2.01631i
$565$		0			0
$566$	22.4462	+	26.0626i	0.943485	+	1.09549i
$567$		−	176.744i		−	7.42255i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	−79.7435			−3.33133
$574$		0			0
$575$		0			0
$576$	30.8776	−	64.4998i	1.28657	−	2.68749i
$577$	41.0674			1.70966			0.854829	−	0.518909i	$-0.173662\pi$
$577$	41.0674			1.70966			0.854829	+	0.518909i	$0.173662\pi$
$578$	−18.2168	+	15.6891i	−0.757720	+	0.652580i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	62.7843	+	72.8997i	2.60249	+	3.02179i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	2.51995	−	2.17029i	0.104098	−	0.0896539i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−62.0103	−	9.29758i	−2.55726	−	0.383426i
$589$	−24.1550			−0.995290
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	53.6851	−	46.2359i	2.20273	−	1.89708i
$595$		0			0
$596$		0			0
$597$	−79.9808			−3.27340
$598$		0			0
$599$		−	47.8389i		−	1.95465i	−0.211755	−	0.977323i	$-0.567918\pi$
$599$		−	47.8389i		−	1.95465i	0.211755	−	0.977323i	$-0.432082\pi$
$600$	−41.3447	+	26.0454i	−1.68789	+	1.06330i
$601$	3.44983			0.140721			0.0703607	−	0.997522i	$-0.477585\pi$
$601$	3.44983			0.140721			0.0703607	+	0.997522i	$0.477585\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−19.8070	−	45.1599i	−0.803278	−	1.83148i
$609$	−143.536			−5.81639
$610$		0			0
$611$		0			0
$612$		0			0
$613$	15.3495			0.619962			0.309981	−	0.950743i	$-0.399677\pi$
$613$	15.3495			0.619962			0.309981	+	0.950743i	$0.399677\pi$
$614$		0			0
$615$		0			0
$616$	−14.7570	−	23.4253i	−0.594577	−	0.943834i
$617$	31.0871			1.25152			0.625760	−	0.780016i	$-0.284789\pi$
$617$	31.0871			1.25152			0.625760	+	0.780016i	$0.284789\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	23.8527	+	27.6957i	0.956405	+	1.11050i
$623$		−	56.3643i		−	2.25819i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	73.5394i		−	2.93688i
$628$	7.01185	−	46.7657i	0.279803	−	1.86615i
$629$		0			0
$630$		0			0
$631$		−	42.4438i		−	1.68966i	−0.535035	−	0.844830i	$-0.679702\pi$
$631$		−	42.4438i		−	1.68966i	0.535035	−	0.844830i	$-0.320298\pi$
$632$		0			0
$633$	86.1055			3.42239
$634$	13.4735	−	11.6040i	0.535102	−	0.460853i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−23.3471	−	27.1086i	−0.924320	−	1.07324i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−66.2969	+	57.0977i	−2.61653	+	2.25346i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	66.4615	+	105.501i	2.61085	+	4.14448i
$649$		0			0
$650$		0			0
$651$		−	38.3849i		−	1.50442i
$652$		0			0
$653$	14.7936			0.578919			0.289459	−	0.957190i	$-0.406524\pi$
$653$	14.7936			0.578919			0.289459	+	0.957190i	$0.406524\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−30.1021	+	25.9252i	−1.17350	+	1.01067i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−25.5600	−	3.83236i	−0.988946	−	0.148278i
$669$	3.38898			0.131026
$670$		0			0
$671$			35.5333i			1.37175i
$672$	71.7640	−	31.4754i	2.76836	−	1.21419i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−18.6987	+	16.1041i	−0.720247	+	0.620307i
$675$		−	102.599i		−	3.94903i
$676$	3.85524	−	25.7126i	0.148278	−	0.988946i
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$			78.9370i			3.02932i
$680$		0			0
$681$		0			0
$682$	7.24947	−	6.24355i	0.277597	−	0.239078i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	154.121	+	23.1082i	5.89295	+	0.883565i
$685$		0			0
$686$	−7.67263	−	8.90880i	−0.292943	−	0.340140i
$687$			59.6752i			2.27675i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	0.773686	−	5.16011i	0.0294111	−	0.196158i
$693$	87.4966			3.32372
$694$		0			0
$695$		0			0
$696$	85.6791	−	53.9743i	3.24766	−	2.04589i
$697$		0			0
$698$		0			0
$699$		−	96.9870i		−	3.66839i
$700$	−39.6488	−	5.94478i	−1.49858	−	0.224691i
$701$	−18.5144			−0.699279			−0.349639	−	0.936884i	$-0.613696\pi$
$701$	−18.5144			−0.699279			−0.349639	+	0.936884i	$0.613696\pi$
$702$		0			0
$703$		0			0
$704$	17.6174	+	8.43385i	0.663980	+	0.317863i
$705$		0			0
$706$	−33.3071	+	28.6855i	−1.25353	+	1.07959i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	21.1948	+	33.6448i	0.794309	+	1.26089i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−24.0086	−	3.59975i	−0.897244	−	0.134529i
$717$	20.4885			0.765156
$718$	−34.9268	−	40.5540i	−1.30346	−	1.51346i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	61.0709	−	52.5968i	2.27282	−	1.95745i
$723$		0			0
$724$	−5.33785	+	35.6009i	−0.198380	+	1.32310i
$725$	−51.8078			−1.92409
$726$	−16.0685	−	18.6574i	−0.596358	−	0.692439i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−181.358			−6.71697
$730$		0			0
$731$		0			0
$732$	−99.4623	−	14.9130i	−3.67623	−	0.551199i
$733$	53.2653			1.96740			0.983699	−	0.179822i	$-0.0575521\pi$
$733$	53.2653			1.96740			0.983699	+	0.179822i	$0.0575521\pi$
$734$	−12.7811	−	14.8404i	−0.471761	−	0.547768i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.8457i		−	0.948187i	−0.880475	−	0.474093i	$-0.842776\pi$
$743$		−	25.8457i		−	0.948187i	0.880475	−	0.474093i	$-0.157224\pi$
$744$	14.4340	+	22.9126i	0.529175	+	0.840016i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−71.7874			−2.62305
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	8.21970	−	26.7945i	0.299741	−	0.977094i
$753$	21.2481			0.774324
$754$		0			0
$755$		0			0
$756$	−24.3971	+	162.717i	−0.887313	+	5.91795i
$757$	−26.8063			−0.974293			−0.487146	−	0.873320i	$-0.661962\pi$
$757$	−26.8063			−0.974293			−0.487146	+	0.873320i	$0.661962\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11.0883			0.401952			0.200976	−	0.979596i	$-0.435589\pi$
$761$	11.0883			0.401952			0.200976	+	0.979596i	$0.435589\pi$
$762$	−31.4666	+	27.1004i	−1.13992	+	0.981743i
$763$		0			0
$764$	45.6477	+	6.84423i	1.65148	+	0.247615i
$765$		0			0
$766$	−21.9497	−	25.4861i	−0.793076	−	0.920851i
$767$		0			0
$768$	−31.0013	+	45.7737i	−1.11866	+	1.65172i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	13.8546i		−	0.497672i
$776$	−29.6828	−	47.1187i	−1.06555	−	1.69146i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			212.617i			7.59831i
$784$	34.6987	+	10.6444i	1.23924	+	0.380159i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	51.1439			1.82077
$790$		0			0
$791$		0			0
$792$	−52.2281	+	32.9016i	−1.85584	+	1.16911i
$793$		0			0
$794$	18.7786	−	16.1729i	0.666427	−	0.573955i
$795$		0			0
$796$	45.7835	+	6.86460i	1.62275	+	0.243309i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	111.447	+	129.403i	3.94518	+	4.58080i
$799$		0			0
$800$	25.9024	−	11.3607i	0.915788	−	0.401661i
$801$	−125.667			−4.44024
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.6144			−1.39277			−0.696384	−	0.717670i	$-0.745209\pi$
$809$	−39.6144			−1.39277			−0.696384	+	0.717670i	$0.745209\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	82.1647	+	12.3194i	2.88342	+	0.432327i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	26.1828	−	22.5497i	0.915460	−	0.788433i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−31.0151	−	36.0121i	−1.08178	−	1.25607i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	42.1801			1.46852
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−6.31174	+	42.0963i	−0.218296	+	1.45593i
$837$	−56.8587			−1.96532
$838$	−0.857978	−	0.996210i	−0.0296384	−	0.0344135i
$839$		−	47.3926i		−	1.63618i	−0.575094	−	0.818088i	$-0.695034\pi$
$839$		−	47.3926i		−	1.63618i	0.575094	−	0.818088i	$-0.304966\pi$
$840$		0			0
$841$	78.3621			2.70214
$842$	−3.70453	+	3.19049i	−0.127666	+	0.109952i
$843$			58.8400i			2.02656i
$844$	−49.2895	−	7.39026i	−1.69661	−	0.254383i
$845$		0			0
$846$	57.8016	+	67.1142i	1.98726	+	2.30743i
$847$		−	20.2025i		−	0.694166i
$848$		0			0
$849$	84.0376			2.88416
$850$		0			0
$851$		0			0
$852$		0			0
$853$	56.6944			1.94118			0.970590	−	0.240737i	$-0.0773891\pi$
$853$	56.6944			1.94118			0.970590	+	0.240737i	$0.0773891\pi$
$854$	−53.8497	−	62.5257i	−1.84270	−	2.13958i
$855$		0			0
$856$	42.8510	−	26.9944i	1.46462	−	0.922648i
$857$	41.9376			1.43256			0.716280	−	0.697813i	$-0.245843\pi$
$857$	41.9376			1.43256			0.716280	+	0.697813i	$0.245843\pi$
$858$		0			0
$859$			37.6802i			1.28563i	0.766021	+	0.642816i	$0.222234\pi$
$859$			37.6802i			1.28563i	−0.766021	+	0.642816i	$0.777766\pi$
$860$		0			0
$861$		0			0
$862$	−24.6052	−	28.5695i	−0.838058	−	0.973081i
$863$		−	30.3788i		−	1.03411i	−0.855953	−	0.517054i	$-0.827029\pi$
$863$		−	30.3788i		−	1.03411i	0.855953	−	0.517054i	$-0.172971\pi$
$864$	−46.6237	−	106.302i	−1.58617	−	3.61648i
$865$		0			0
$866$	−19.4841	+	16.7806i	−0.662098	+	0.570227i
$867$			58.7392i			1.99489i
$868$	−3.29450	+	21.9727i	−0.111823	+	0.745803i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	175.994			5.95650
$874$		0			0
$875$		0			0
$876$		0			0
$877$	53.6318			1.81102			0.905508	−	0.424328i	$-0.139490\pi$
$877$	53.6318			1.81102			0.905508	+	0.424328i	$0.139490\pi$
$878$		0			0
$879$		−	8.12546i		−	0.274065i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−86.9124	+	74.8526i	−2.92649	+	2.52042i
$883$		−	45.6986i		−	1.53788i	−0.639321	−	0.768940i	$-0.720784\pi$
$883$		−	45.6986i		−	1.53788i	0.639321	−	0.768940i	$-0.279216\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−34.0726			−1.14276
$890$		0			0
$891$		−	107.633i		−	3.60584i
$892$	−1.93996	−	0.290869i	−0.0649546	−	0.00973903i
$893$	61.0799			2.04396
$894$		0			0
$895$		0			0
$896$	−43.7814	+	11.8581i	−1.46263	+	0.396153i
$897$		0			0
$898$	31.4263	−	27.0657i	1.04871	−	0.903192i
$899$			28.7111i			0.957569i
$900$	−13.2542	+	88.3991i	−0.441807	+	2.94664i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			12.3274i			0.409324i	0.978833	+	0.204662i	$0.0656095\pi$
$907$			12.3274i			0.409324i	−0.978833	+	0.204662i	$0.934390\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	59.3106i		−	1.96505i	−0.186134	−	0.982524i	$-0.559596\pi$
$911$		−	59.3106i		−	1.96505i	0.186134	−	0.982524i	$-0.440404\pi$
$912$	−115.184	−	35.3348i	−3.81412	−	1.17005i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	5.12180	−	34.1599i	0.169229	−	1.12868i
$917$		0			0
$918$		0			0
$919$			48.7183i			1.60707i	0.595258	+	0.803535i	$0.297050\pi$
$919$			48.7183i			1.60707i	−0.595258	+	0.803535i	$0.702950\pi$
$920$		0			0
$921$		0			0
$922$	44.3503	−	38.1963i	1.46060	−	1.25793i
$923$		0			0
$924$	−66.8955	−	10.0300i	−2.20070	−	0.329964i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−53.6779	+	23.5429i	−1.76206	+	0.772834i
$929$	23.3662			0.766621			0.383311	−	0.923620i	$-0.374784\pi$
$929$	23.3662			0.766621			0.383311	+	0.923620i	$0.374784\pi$
$930$		0			0
$931$			79.0980i			2.59233i
$932$	−8.32420	+	55.5184i	−0.272668	+	1.81857i
$933$	89.3032			2.92366
$934$	−38.7251	−	44.9643i	−1.26713	−	1.47128i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$	−75.3967	−	87.5442i	−2.45656	−	2.85234i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	19.2379i		−	0.625147i	−0.949894	−	0.312573i	$-0.898809\pi$
$947$		−	19.2379i		−	0.625147i	0.949894	−	0.312573i	$-0.101191\pi$
$948$		0			0
$949$		0			0
$950$	40.2255	+	46.7064i	1.30509	+	1.51536i
$951$		−	43.4447i		−	1.40879i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−11.7282	−	1.75848i	−0.379319	−	0.0568735i
$957$	−87.4103			−2.82557
$958$		0			0
$959$		−	38.9945i		−	1.25920i
$960$		0			0
$961$	23.3220			0.752322
$962$		0			0
$963$			160.054i			5.15766i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	47.0695i		−	1.51365i	−0.653615	−	0.756827i	$-0.726749\pi$
$967$		−	47.0695i		−	1.51365i	0.653615	−	0.756827i	$-0.273251\pi$
$968$	7.59679	+	12.0592i	0.244170	+	0.387597i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	179.521	+	26.9167i	5.75815	+	0.863354i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	55.6554	+	17.0733i	1.78149	+	0.546503i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		−	34.3246i		−	1.09702i
$980$		0			0
$981$		0			0
$982$	−23.8527	−	27.6957i	−0.761169	−	0.883804i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			97.0626i			3.08954i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−6.29592	−	14.3547i	−0.199896	−	0.455763i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	58.7123			1.85944			0.929719	−	0.368270i	$-0.120050\pi$
$997$	58.7123			1.85944			0.929719	+	0.368270i	$0.120050\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.b.a.667.6	yes	22
4.3	odd	2	inner	668.2.b.a.667.5	✓	22
167.166	odd	2	CM	668.2.b.a.667.6	yes	22
668.667	even	2	inner	668.2.b.a.667.5	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.5	✓	22	4.3	odd	2	inner
668.2.b.a.667.5	✓	22	668.667	even	2	inner
668.2.b.a.667.6	yes	22	1.1	even	1	trivial
668.2.b.a.667.6	yes	22	167.166	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.07158 + 0.922888i) q^{2} +3.45525i q^{3} +(0.296557 - 1.97789i) q^{4} +(-3.18880 - 3.70256i) q^{6} -4.00920i q^{7} +(1.50759 + 2.39315i) q^{8} -8.93873 q^{9} +O(q^{10})\)	\(q+(-1.07158 + 0.922888i) q^{2} +3.45525i q^{3} +(0.296557 - 1.97789i) q^{4} +(-3.18880 - 3.70256i) q^{6} -4.00920i q^{7} +(1.50759 + 2.39315i) q^{8} -8.93873 q^{9} -2.44151i q^{11} +(6.83410 + 1.02468i) q^{12} +(3.70004 + 4.29617i) q^{14} +(-3.82411 - 1.17311i) q^{16} +(9.57854 - 8.24944i) q^{18} -8.71732i q^{19} +13.8528 q^{21} +(2.25324 + 2.61627i) q^{22} +(-8.26893 + 5.20909i) q^{24} +5.00000 q^{25} -20.5198i q^{27} +(-7.92976 - 1.18896i) q^{28} -10.3616 q^{29} -2.77092i q^{31} +(5.18048 - 2.27214i) q^{32} +8.43601 q^{33} +(-2.65084 + 17.6798i) q^{36} +(8.04510 + 9.34128i) q^{38} +(-14.8443 + 12.7845i) q^{42} +(-4.82904 - 0.724046i) q^{44} +7.00673i q^{47} +(4.05340 - 13.2132i) q^{48} -9.07366 q^{49} +(-5.35789 + 4.61444i) q^{50} +(18.9374 + 21.9885i) q^{54} +(9.59462 - 6.04422i) q^{56} +30.1205 q^{57} +(11.1032 - 9.56256i) q^{58} -14.5538 q^{61} +(2.55725 + 2.96926i) q^{62} +35.8371i q^{63} +(-3.45436 + 7.21577i) q^{64} +(-9.03984 + 7.78549i) q^{66} +(-13.4759 - 21.3917i) q^{72} +17.2762i q^{75} +(-17.2419 - 2.58518i) q^{76} -9.78849 q^{77} +44.0847 q^{81} +(4.10813 - 27.3993i) q^{84} -35.8018i q^{87} +(5.84290 - 3.68079i) q^{88} +14.0588 q^{89} +9.57422 q^{93} +(-6.46642 - 7.50825i) q^{94} +(7.85080 + 17.8998i) q^{96} -19.6890 q^{97} +(9.72313 - 8.37397i) q^{98} +21.8240i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 22 * q - 66 * q^9	\( 22 q - 66 q^{9} + 110 q^{25} - 11 q^{42} - 33 q^{44} + 55 q^{48} - 154 q^{49} + 77 q^{54} - 99 q^{62} - 121 q^{72} + 198 q^{81} + 143 q^{84} + 165 q^{98}+O(q^{100}) \) 22 * q - 66 * q^9 + 110 * q^25 - 11 * q^42 - 33 * q^44 + 55 * q^48 - 154 * q^49 + 77 * q^54 - 99 * q^62 - 121 * q^72 + 198 * q^81 + 143 * q^84 + 165 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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