LMFDB - Embedded newform 668.2.b.a.667.2

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			667.2
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.40042 + 0.197044i) q^{2} -2.07563i q^{3} +(1.92235 - 0.551888i) q^{4} +(0.408990 + 2.90675i) q^{6} +4.81975i q^{7} +(-2.58335 + 1.15166i) q^{8} -1.30824 q^{9} +O(q^{10})$	\(q+(-1.40042 + 0.197044i) q^{2} -2.07563i q^{3} +(1.92235 - 0.551888i) q^{4} +(0.408990 + 2.90675i) q^{6} +4.81975i q^{7} +(-2.58335 + 1.15166i) q^{8} -1.30824 q^{9} -6.62447i q^{11} +(-1.14552 - 3.99008i) q^{12} +(-0.949702 - 6.74967i) q^{14} +(3.39084 - 2.12184i) q^{16} +(1.83209 - 0.257781i) q^{18} +1.14991i q^{19} +10.0040 q^{21} +(1.30531 + 9.27703i) q^{22} +(2.39042 + 5.36207i) q^{24} +5.00000 q^{25} -3.51146i q^{27} +(2.65996 + 9.26523i) q^{28} +10.7699 q^{29} -8.16200i q^{31} +(-4.33050 + 3.63961i) q^{32} -13.7499 q^{33} +(-2.51490 + 0.722004i) q^{36} +(-0.226583 - 1.61036i) q^{38} +(-14.0098 + 1.97123i) q^{42} +(-3.65596 - 12.7345i) q^{44} -12.6630i q^{47} +(-4.40416 - 7.03813i) q^{48} -16.2300 q^{49} +(-7.00210 + 0.985220i) q^{50} +(0.691912 + 4.91752i) q^{54} +(-5.55072 - 12.4511i) q^{56} +2.38679 q^{57} +(-15.0823 + 2.12214i) q^{58} +5.24315 q^{61} +(1.60827 + 11.4302i) q^{62} -6.30540i q^{63} +(5.34735 - 5.95028i) q^{64} +(19.2557 - 2.70934i) q^{66} +(3.37964 - 1.50665i) q^{72} -10.3782i q^{75} +(0.634623 + 2.21053i) q^{76} +31.9283 q^{77} -11.2132 q^{81} +(19.2312 - 5.52110i) q^{84} -22.3543i q^{87} +(7.62914 + 17.1133i) q^{88} +0.303653 q^{89} -16.9413 q^{93} +(2.49517 + 17.7335i) q^{94} +(7.55449 + 8.98852i) q^{96} -8.71308 q^{97} +(22.7287 - 3.19801i) q^{98} +8.66641i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.40042	+	0.197044i	−0.990246	+	0.139331i
$2$	−1.40042	+	0.197044i	−0.990246	+	0.139331i
$3$		−	2.07563i		−	1.19837i	−0.800612	−	0.599183i	$-0.795492\pi$
$3$		−	2.07563i		−	1.19837i	0.800612	−	0.599183i	$-0.204508\pi$
$4$	1.92235	−	0.551888i	0.961174	−	0.275944i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	0.408990	+	2.90675i	0.166970	+	1.18668i
$7$			4.81975i			1.82169i	0.412745	+	0.910847i	$0.364570\pi$
$7$			4.81975i			1.82169i	−0.412745	+	0.910847i	$0.635430\pi$
$8$	−2.58335	+	1.15166i	−0.913351	+	0.407174i
$9$	−1.30824			−0.436081
$10$		0			0
$11$		−	6.62447i		−	1.99735i	−0.0514425	−	0.998676i	$-0.516382\pi$
$11$		−	6.62447i		−	1.99735i	0.0514425	−	0.998676i	$-0.483618\pi$
$12$	−1.14552	−	3.99008i	−0.330682	−	1.15184i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.949702	−	6.74967i	−0.253818	−	1.80392i
$15$		0			0
$16$	3.39084	−	2.12184i	0.847710	−	0.530460i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	1.83209	−	0.257781i	0.431827	−	0.0607596i
$19$			1.14991i			0.263808i	0.991263	+	0.131904i	$0.0421090\pi$
$19$			1.14991i			0.263808i	−0.991263	+	0.131904i	$0.957891\pi$
$20$		0			0
$21$	10.0040			2.18305
$22$	1.30531	+	9.27703i	0.278293	+	1.97787i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	2.39042	+	5.36207i	0.487943	+	1.09453i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	3.51146i		−	0.675781i
$28$	2.65996	+	9.26523i	0.502685	+	1.75096i
$29$	10.7699			1.99992			0.999958	−	0.00921094i	$-0.00293197\pi$
$29$	10.7699			1.99992			0.999958	+	0.00921094i	$0.00293197\pi$
$30$		0			0
$31$		−	8.16200i		−	1.46594i	−0.680261	−	0.732970i	$-0.738134\pi$
$31$		−	8.16200i		−	1.46594i	0.680261	−	0.732970i	$-0.261866\pi$
$32$	−4.33050	+	3.63961i	−0.765531	+	0.643398i
$33$	−13.7499			−2.39356
$34$		0			0
$35$		0			0
$36$	−2.51490	+	0.722004i	−0.419150	+	0.120334i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.226583	−	1.61036i	−0.0367566	−	0.261235i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−14.0098	+	1.97123i	−2.16176	+	0.304167i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−3.65596	−	12.7345i	−0.551157	−	1.91980i
$45$		0			0
$46$		0			0
$47$		−	12.6630i		−	1.84709i	−0.383487	−	0.923546i	$-0.625277\pi$
$47$		−	12.6630i		−	1.84709i	0.383487	−	0.923546i	$-0.374723\pi$
$48$	−4.40416	−	7.03813i	−0.635686	−	1.01587i
$49$	−16.2300			−2.31857
$50$	−7.00210	+	0.985220i	−0.990246	+	0.139331i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.691912	+	4.91752i	0.0941574	+	0.669190i
$55$		0			0
$56$	−5.55072	−	12.4511i	−0.741746	−	1.66384i
$57$	2.38679			0.316138
$58$	−15.0823	+	2.12214i	−1.98041	+	0.278650i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	5.24315			0.671317			0.335658	−	0.941984i	$-0.391041\pi$
$61$	5.24315			0.671317			0.335658	+	0.941984i	$0.391041\pi$
$62$	1.60827	+	11.4302i	0.204251	+	1.45164i
$63$		−	6.30540i		−	0.794406i
$64$	5.34735	−	5.95028i	0.668419	−	0.743785i
$65$		0			0
$66$	19.2557	−	2.70934i	2.37021	−	0.333497i
$67$		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	$0.5\pi$
$72$	3.37964	−	1.50665i	0.398295	−	0.177561i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	10.3782i		−	1.19837i
$76$	0.634623	+	2.21053i	0.0727962	+	0.253565i
$77$	31.9283			3.63856
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−11.2132			−1.24591
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	19.2312	−	5.52110i	2.09829	−	0.602401i
$85$		0			0
$86$		0			0
$87$		−	22.3543i		−	2.39663i
$88$	7.62914	+	17.1133i	0.813269	+	1.82428i
$89$	0.303653			0.0321871			0.0160936	−	0.999870i	$-0.494877\pi$
$89$	0.303653			0.0321871			0.0160936	+	0.999870i	$0.494877\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−16.9413			−1.75673
$94$	2.49517	+	17.7335i	0.257357	+	1.82908i
$95$		0			0
$96$	7.55449	+	8.98852i	0.771027	+	0.917387i
$97$	−8.71308			−0.884679			−0.442340	−	0.896848i	$-0.645851\pi$
$97$	−8.71308			−0.884679			−0.442340	+	0.896848i	$0.645851\pi$
$98$	22.7287	−	3.19801i	2.29595	−	0.323048i
$99$			8.66641i			0.871007i
$100$	9.61174	−	2.75944i	0.961174	−	0.275944i
$101$		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
$107$			14.2608i			1.37865i	0.724454	+	0.689323i	$0.242092\pi$
$107$			14.2608i			1.37865i	−0.724454	+	0.689323i	$0.757908\pi$
$108$	−1.93793	−	6.75025i	−0.186478	−	0.649543i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	10.2267	+	16.3430i	0.966336	+	1.54427i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−3.34251	+	0.470303i	−0.313055	+	0.0440479i
$115$		0			0
$116$	20.7034	−	5.94376i	1.92227	−	0.551865i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−32.8836			−2.98941
$122$	−7.34261	+	1.03313i	−0.664769	+	0.0935353i
$123$		0			0
$124$	−4.50451	−	15.6902i	−0.404517	−	1.40902i
$125$		0			0
$126$	1.24244	+	8.83020i	0.110685	+	0.786657i
$127$			15.4583i			1.37170i	0.727742	+	0.685851i	$0.240570\pi$
$127$			15.4583i			1.37170i	−0.727742	+	0.685851i	$0.759430\pi$
$128$	−6.31607	+	9.38655i	−0.558267	+	0.829662i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−26.4322	+	7.58843i	−2.30063	+	0.660488i
$133$	−5.54228			−0.480577
$134$		0			0
$135$		0			0
$136$		0			0
$137$	23.4094			2.00000			1.00000		7.73045e-5i	$-2.46068e-5\pi$
$137$	23.4094			2.00000			1.00000		7.73045e-5i	$2.46068e-5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−26.2838			−2.21349
$142$		0			0
$143$		0			0
$144$	−4.43604	+	2.77588i	−0.369670	+	0.231324i
$145$		0			0
$146$		0			0
$147$			33.6874i			2.77849i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	2.04495	+	14.5338i	0.166970	+	1.18668i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−1.32431	−	2.97062i	−0.107416	−	0.240949i
$153$		0			0
$154$	−44.7129	+	6.29127i	−3.60307	+	0.506965i
$155$		0			0
$156$		0			0
$157$	24.3801			1.94574			0.972872	−	0.231345i	$-0.0743125\pi$
$157$	24.3801			1.94574			0.972872	+	0.231345i	$0.0743125\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	15.7032	−	2.20950i	1.23376	−	0.173595i
$163$		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
$168$	−25.8438	+	11.5212i	−1.99389	+	0.888883i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	1.50436i		−	0.115042i
$172$		0			0
$173$	−22.7269			−1.72790			−0.863949	−	0.503580i	$-0.832016\pi$
$173$	−22.7269			−1.72790			−0.863949	+	0.503580i	$0.832016\pi$
$174$	4.40477	+	31.3054i	0.333925	+	2.37325i
$175$			24.0987i			1.82169i
$176$	−14.0561	−	22.4625i	−1.05952	−	1.69317i
$177$		0			0
$178$	−0.425241	+	0.0598329i	−0.0318732	+	0.00448466i
$179$		−	21.8763i		−	1.63511i	−0.575848	−	0.817556i	$-0.695328\pi$
$179$		−	21.8763i		−	1.63511i	0.575848	−	0.817556i	$-0.304672\pi$
$180$		0			0
$181$	22.3585			1.66190			0.830948	−	0.556350i	$-0.187799\pi$
$181$	22.3585			1.66190			0.830948	+	0.556350i	$0.187799\pi$
$182$		0			0
$183$		−	10.8828i		−	0.804483i
$184$		0			0
$185$		0			0
$186$	23.7249	−	3.33818i	1.73960	−	0.244767i
$187$		0			0
$188$	−6.98857	−	24.3427i	−0.509694	−	1.77538i
$189$	16.9244			1.23107
$190$		0			0
$191$			27.6387i			1.99987i	0.0115231	+	0.999934i	$0.496332\pi$
$191$			27.6387i			1.99987i	−0.0115231	+	0.999934i	$0.503668\pi$
$192$	−12.3506	−	11.0991i	−0.891327	−	0.801011i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	12.2020	−	1.71686i	0.876050	−	0.123263i
$195$		0			0
$196$	−31.1996	+	8.95712i	−2.22854	+	0.639794i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−1.70766	−	12.1366i	−0.121358	−	0.862511i
$199$			24.2885i			1.72177i	0.508803	+	0.860883i	$0.330088\pi$
$199$			24.2885i			1.72177i	−0.508803	+	0.860883i	$0.669912\pi$
$200$	−12.9167	+	5.75831i	−0.913351	+	0.407174i
$201$		0			0
$202$		0			0
$203$			51.9081i			3.64323i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.61755			0.526917
$210$		0			0
$211$		−	12.8911i		−	0.887457i	−0.896161	−	0.443729i	$-0.853655\pi$
$211$		−	12.8911i		−	0.887457i	0.896161	−	0.443729i	$-0.146345\pi$
$212$		0			0
$213$		0			0
$214$	−2.81001	−	19.9711i	−0.192088	−	1.36520i
$215$		0			0
$216$	4.04402	+	9.07132i	0.275161	+	0.617225i
$217$	39.3388			2.67049
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			23.2016i			1.55370i	0.629688	+	0.776848i	$0.283183\pi$
$223$			23.2016i			1.55370i	−0.629688	+	0.776848i	$0.716817\pi$
$224$	−17.5420	−	20.8719i	−1.17207	−	1.39456i
$225$	−6.54121			−0.436081
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	4.58824	−	1.31724i	0.303864	−	0.0872365i
$229$	−23.5734			−1.55778			−0.778889	−	0.627162i	$-0.784216\pi$
$229$	−23.5734			−1.55778			−0.778889	+	0.627162i	$0.784216\pi$
$230$		0			0
$231$		−	66.2713i		−	4.36033i
$232$	−27.8223	+	12.4032i	−1.82662	+	0.814313i
$233$	−17.1235			−1.12180			−0.560900	−	0.827884i	$-0.689545\pi$
$233$	−17.1235			−1.12180			−0.560900	+	0.827884i	$0.689545\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.0504i		−	1.23227i	−0.787642	−	0.616133i	$-0.788698\pi$
$239$		−	19.0504i		−	1.23227i	0.787642	−	0.616133i	$-0.211302\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	46.0508	−	6.47951i	2.96026	−	0.416518i
$243$			12.7401i			0.817280i
$244$	10.0792	−	2.89363i	0.645252	−	0.185246i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	9.39987	+	21.0853i	0.596892	+	1.33892i
$249$		0			0
$250$		0			0
$251$			14.6577i			0.925183i	0.886572	+	0.462591i	$0.153080\pi$
$251$			14.6577i			0.925183i	−0.886572	+	0.462591i	$0.846920\pi$
$252$	−3.47987	−	12.1212i	−0.219211	−	0.763562i
$253$		0			0
$254$	−3.04596	−	21.6481i	−0.191121	−	1.35832i
$255$		0			0
$256$	6.99558	−	14.3896i	0.437224	−	0.899353i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−14.0896			−0.872125
$262$		0			0
$263$			31.5041i			1.94263i	0.237799	+	0.971314i	$0.423574\pi$
$263$			31.5041i			1.94263i	−0.237799	+	0.971314i	$0.576426\pi$
$264$	35.5209	−	15.8353i	2.18616	−	0.974594i
$265$		0			0
$266$	7.76152	−	1.09207i	0.475889	−	0.0669593i
$267$		−	0.630271i		−	0.0385719i
$268$		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
$274$	−32.7830	+	4.61268i	−1.98049	+	0.278662i
$275$		−	33.1223i		−	1.99735i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$			10.6779i			0.639268i
$280$		0			0
$281$	−32.9772			−1.96725			−0.983626	−	0.180219i	$-0.942319\pi$
$281$	−32.9772			−1.96725			−0.983626	+	0.180219i	$0.942319\pi$
$282$	36.8083	−	5.17906i	2.19190	−	0.308408i
$283$			11.0432i			0.656450i	0.944600	+	0.328225i	$0.106450\pi$
$283$			11.0432i			0.656450i	−0.944600	+	0.328225i	$0.893550\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	5.66535	−	4.76150i	0.333834	−	0.280574i
$289$	17.0000			1.00000
$290$		0			0
$291$			18.0851i			1.06017i
$292$		0			0
$293$	16.4865			0.963153			0.481577	−	0.876404i	$-0.340064\pi$
$293$	16.4865			0.963153			0.481577	+	0.876404i	$0.340064\pi$
$294$	−6.63790	−	47.1765i	−0.387130	−	2.75139i
$295$		0			0
$296$		0			0
$297$	−23.2616			−1.34977
$298$		0			0
$299$		0			0
$300$	−5.72758	−	19.9504i	−0.330682	−	1.15184i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2.43993	+	3.89917i	0.139940	+	0.223633i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	61.3772	−	17.6208i	3.49729	−	1.00404i
$309$		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			$\pi$
$314$	−34.1424	+	4.80395i	−1.92676	+	0.271103i
$315$		0			0
$316$		0			0
$317$	−16.9441			−0.951673			−0.475837	−	0.879534i	$-0.657855\pi$
$317$	−16.9441			−0.951673			−0.475837	+	0.879534i	$0.657855\pi$
$318$		0			0
$319$		−	71.3447i		−	3.99453i
$320$		0			0
$321$	29.6002			1.65212
$322$		0			0
$323$		0			0
$324$	−21.5557	+	6.18845i	−1.19754	+	0.343803i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	61.0326			3.36484
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	2.54637	+	18.0974i	0.139331	+	0.990246i
$335$		0			0
$336$	33.9220	−	21.2269i	1.85060	−	1.15802i
$337$	29.4912			1.60649			0.803245	−	0.595649i	$-0.203105\pi$
$337$	29.4912			1.60649			0.803245	+	0.595649i	$0.203105\pi$
$338$	−18.2054	+	2.56157i	−0.990246	+	0.139331i
$339$		0			0
$340$		0			0
$341$	−54.0689			−2.92800
$342$	0.296426	+	2.10674i	0.0160289	+	0.113919i
$343$		−	44.4861i		−	2.40202i
$344$		0			0
$345$		0			0
$346$	31.8272	−	4.47820i	1.71104	−	0.240750i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−12.3371	−	42.9727i	−0.661336	−	2.30358i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−4.74851	−	33.7483i	−0.253818	−	1.80392i
$351$		0			0
$352$	24.1105	+	28.6873i	1.28509	+	1.52904i
$353$	−23.8741			−1.27069			−0.635346	−	0.772228i	$-0.719142\pi$
$353$	−23.8741			−1.27069			−0.635346	+	0.772228i	$0.719142\pi$
$354$		0			0
$355$		0			0
$356$	0.583726	−	0.167582i	0.0309374	−	0.00888184i
$357$		0			0
$358$	4.31060	+	30.6360i	0.227822	+	1.61916i
$359$			30.7914i			1.62511i	0.582885	+	0.812554i	$0.301924\pi$
$359$			30.7914i			1.62511i	−0.582885	+	0.812554i	$0.698076\pi$
$360$		0			0
$361$	17.6777			0.930405
$362$	−31.3113	+	4.40561i	−1.64569	+	0.231554i
$363$			68.2541i			3.58241i
$364$		0			0
$365$		0			0
$366$	2.14440	+	15.2405i	0.112089	+	0.796636i
$367$			33.3894i			1.74291i	0.490472	+	0.871457i	$0.336825\pi$
$367$			33.3894i			1.74291i	−0.490472	+	0.871457i	$0.663175\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−32.5671	+	9.34971i	−1.68852	+	0.484760i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	14.5835	+	32.7130i	0.752088	+	1.68704i
$377$		0			0
$378$	−23.7012	+	3.33484i	−1.21906	+	0.171526i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	32.0857			1.64380
$382$	−5.44604	−	38.7058i	−0.278644	−	1.98036i
$383$		−	14.0624i		−	0.718556i	−0.933231	−	0.359278i	$-0.883023\pi$
$383$		−	14.0624i		−	0.718556i	0.933231	−	0.359278i	$-0.116977\pi$
$384$	19.4830	+	13.1098i	0.994238	+	0.669008i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−16.7496	+	4.80865i	−0.850331	+	0.244122i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	41.9276	−	18.6914i	2.11766	−	0.944059i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	4.78289	+	16.6599i	0.240349	+	0.837189i
$397$	38.5240			1.93346			0.966731	−	0.255794i	$-0.0823368\pi$
$397$	38.5240			1.93346			0.966731	+	0.255794i	$0.0823368\pi$
$398$	−4.78590	−	34.0141i	−0.239896	−	1.70497i
$399$			11.5037i			0.575907i
$400$	16.9542	−	10.6092i	0.847710	−	0.530460i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−10.2282	−	72.6930i	−0.507615	−	3.60769i
$407$		0			0
$408$		0			0
$409$	−39.4706			−1.95170			−0.975849	−	0.218447i	$-0.929901\pi$
$409$	−39.4706			−1.95170			−0.975849	+	0.218447i	$0.929901\pi$
$410$		0			0
$411$		−	48.5893i		−	2.39673i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−10.6678	+	1.50099i	−0.521778	+	0.0734159i
$419$			22.9097i			1.11921i	0.828759	+	0.559605i	$0.189047\pi$
$419$			22.9097i			1.11921i	−0.828759	+	0.559605i	$0.810953\pi$
$420$		0			0
$421$	−14.8373			−0.723125			−0.361562	−	0.932348i	$-0.617757\pi$
$421$	−14.8373			−0.723125			−0.361562	+	0.932348i	$0.617757\pi$
$422$	2.54010	+	18.0529i	0.123650	+	0.878801i
$423$			16.5663i			0.805482i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			25.2707i			1.22293i
$428$	7.87038	+	27.4143i	0.380429	+	1.32512i
$429$		0			0
$430$		0			0
$431$			6.59658i			0.317746i	0.987299	+	0.158873i	$0.0507860\pi$
$431$			6.59658i			0.317746i	−0.987299	+	0.158873i	$0.949214\pi$
$432$	−7.45077	−	11.9068i	−0.358475	−	0.572867i
$433$	−39.6418			−1.90506			−0.952531	−	0.304441i	$-0.901530\pi$
$433$	−39.6418			−1.90506			−0.952531	+	0.304441i	$0.901530\pi$
$434$	−55.0908	+	7.75147i	−2.64444	+	0.372082i
$435$		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
$441$	21.2327			1.01108
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	−4.57174	−	32.4920i	−0.216478	−	1.53854i
$447$		0			0
$448$	28.6788	+	25.7729i	1.35495	+	1.21765i
$449$	−26.1077			−1.23210			−0.616050	−	0.787707i	$-0.711268\pi$
$449$	−26.1077			−1.23210			−0.616050	+	0.787707i	$0.711268\pi$
$450$	9.16044	−	1.28891i	0.431827	−	0.0607596i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−6.16591	+	2.74878i	−0.288745	+	0.128723i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	33.0127	−	4.64500i	1.54258	−	0.217047i
$459$		0			0
$460$		0			0
$461$	−5.44095			−0.253410			−0.126705	−	0.991940i	$-0.540440\pi$
$461$	−5.44095			−0.253410			−0.126705	+	0.991940i	$0.540440\pi$
$462$	13.0583	+	92.8075i	0.607529	+	4.31780i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	36.5189	−	22.8520i	1.69535	−	1.06088i
$465$		0			0
$466$	23.9801	−	3.37409i	1.11086	−	0.156302i
$467$		−	43.1793i		−	1.99810i	−0.0435759	−	0.999050i	$-0.513875\pi$
$467$		−	43.1793i		−	1.99810i	0.0435759	−	0.999050i	$-0.486125\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	50.6041i		−	2.33171i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			5.74956i			0.263808i
$476$		0			0
$477$		0			0
$478$	3.75376	+	26.6785i	0.171693	+	1.22025i
$479$		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
$484$	−63.2136	+	18.1480i	−2.87335	+	0.824911i
$485$		0			0
$486$	−2.51037	−	17.8415i	−0.113872	−	0.809308i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−13.5449	+	6.03833i	−0.613148	+	0.273343i
$489$		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
$496$	−17.3185	−	27.6760i	−0.777623	−	1.24269i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−26.8231			−1.19837
$502$	−2.88820	−	20.5269i	−0.128907	−	0.916158i
$503$		−	43.9579i		−	1.95999i	−0.199029	−	0.979994i	$-0.563779\pi$
$503$		−	43.9579i		−	1.95999i	0.199029	−	0.979994i	$-0.436221\pi$
$504$	7.26168	+	16.2890i	0.323461	+	0.725571i
$505$		0			0
$506$		0			0
$507$		−	26.9832i		−	1.19837i
$508$	8.53125	+	29.7162i	0.378513	+	1.31844i
$509$	−21.3620			−0.946855			−0.473427	−	0.880833i	$-0.656983\pi$
$509$	−21.3620			−0.946855			−0.473427	+	0.880833i	$0.656983\pi$
$510$		0			0
$511$		0			0
$512$	−6.96135	+	21.5300i	−0.307651	+	0.951499i
$513$	4.03787			0.178276
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−83.8858			−3.68929
$518$		0			0
$519$			47.1727i			2.07065i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	19.7314	−	2.77627i	0.863618	−	0.121514i
$523$		−	45.5391i		−	1.99128i	−0.0932541	−	0.995642i	$-0.529727\pi$
$523$		−	45.5391i		−	1.99128i	0.0932541	−	0.995642i	$-0.470273\pi$
$524$		0			0
$525$	50.0201			2.18305
$526$	−6.20770	−	44.1190i	−0.270669	−	1.92368i
$527$		0			0
$528$	−46.6239	+	29.1752i	−2.02904	+	1.26969i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	−10.6542	+	3.05872i	−0.461918	+	0.132612i
$533$		0			0
$534$	0.124191	+	0.882643i	0.00537427	+	0.0381957i
$535$		0			0
$536$		0			0
$537$	−45.4072			−1.95946
$538$		0			0
$539$			107.515i			4.63099i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	46.4080i		−	1.99156i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	45.0010	−	12.9194i	1.92235	−	0.551888i
$549$	−6.85931			−0.292748
$550$	6.52655	+	46.3852i	0.278293	+	1.97787i
$551$			12.3844i			0.527593i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.4568			−1.07864			−0.539319	−	0.842101i	$-0.681318\pi$
$557$	−25.4568			−1.07864			−0.539319	+	0.842101i	$0.681318\pi$
$558$	−2.10401	−	14.9535i	−0.0890699	−	0.633033i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	46.1819	−	6.49795i	1.94806	−	0.274099i
$563$			47.2024i			1.98934i	0.103096	+	0.994671i	$0.467125\pi$
$563$			47.2024i			1.98934i	−0.103096	+	0.994671i	$0.532875\pi$
$564$	−50.5265	+	14.5057i	−2.12755	+	0.610800i
$565$		0			0
$566$	−2.17599	−	15.4651i	−0.0914638	−	0.650046i
$567$		−	54.0449i		−	2.26967i
$568$		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
$573$	57.3678			2.39657
$574$		0			0
$575$		0			0
$576$	−6.99563	+	7.78441i	−0.291485	+	0.324350i
$577$	21.0703			0.877169			0.438584	−	0.898690i	$-0.355480\pi$
$577$	21.0703			0.877169			0.438584	+	0.898690i	$0.355480\pi$
$578$	−23.8071	+	3.34975i	−0.990246	+	0.139331i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−3.56357	−	25.3268i	−0.147715	−	1.04983i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−23.0880	+	3.24857i	−0.953758	+	0.134197i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	18.5917	+	64.7589i	0.766708	+	2.67061i
$589$	9.38559			0.386726
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	32.5760	−	4.58355i	1.33661	−	0.188065i
$595$		0			0
$596$		0			0
$597$	50.4140			2.06331
$598$		0			0
$599$			42.9809i			1.75615i	0.478521	+	0.878076i	$0.341173\pi$
$599$			42.9809i			1.75615i	−0.478521	+	0.878076i	$0.658827\pi$
$600$	11.9521	+	26.8104i	0.487943	+	1.09453i
$601$	10.4692			0.427047			0.213524	−	0.976938i	$-0.431506\pi$
$601$	10.4692			0.427047			0.213524	+	0.976938i	$0.431506\pi$
$602$		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	$0.5\pi$
$608$	−4.18523	−	4.97969i	−0.169734	−	0.201953i
$609$	107.742			4.36592
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.5397			−0.506475			−0.253238	−	0.967404i	$-0.581495\pi$
$613$	−12.5397			−0.506475			−0.253238	+	0.967404i	$0.581495\pi$
$614$		0			0
$615$		0			0
$616$	−82.4817	+	36.7705i	−3.32328	+	1.48153i
$617$	−40.7451			−1.64034			−0.820168	−	0.572123i	$-0.806120\pi$
$617$	−40.7451			−1.64034			−0.820168	+	0.572123i	$0.806120\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	5.09274	+	36.1948i	0.204200	+	1.45128i
$623$			1.46353i			0.0586350i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	15.8112i		−	0.631440i
$628$	46.8670	−	13.4551i	1.87020	−	0.536916i
$629$		0			0
$630$		0			0
$631$			48.1092i			1.91520i	0.288107	+	0.957598i	$0.406974\pi$
$631$			48.1092i			1.91520i	−0.288107	+	0.957598i	$0.593026\pi$
$632$		0			0
$633$	−26.7571			−1.06350
$634$	23.7288	−	3.33872i	0.942391	−	0.132598i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	14.0580	+	99.9124i	0.556563	+	3.95557i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−41.4527	+	5.83254i	−1.63601	+	0.230192i
$643$		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	$0.5\pi$
$648$	28.9676	−	12.9138i	1.13796	−	0.507304i
$649$		0			0
$650$		0			0
$651$		−	81.6528i		−	3.20023i
$652$		0			0
$653$	38.8932			1.52201			0.761005	−	0.648746i	$-0.224706\pi$
$653$	38.8932			1.52201			0.761005	+	0.648746i	$0.224706\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−85.4712	+	12.0261i	−3.33201	+	0.468826i
$659$		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
$668$	−7.13197	−	24.8422i	−0.275944	−	0.961174i
$669$	48.1580			1.86190
$670$		0			0
$671$		−	34.7331i		−	1.34086i
$672$	−43.3224	+	36.4107i	−1.67120	+	1.40457i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−41.3001	+	5.81107i	−1.59082	+	0.223834i
$675$		−	17.5573i		−	0.675781i
$676$	24.9905	−	7.17455i	0.961174	−	0.275944i
$677$	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
$679$		−	41.9949i		−	1.61161i
$680$		0			0
$681$		0			0
$682$	75.7192	−	10.6540i	2.89944	−	0.407961i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−0.830240	−	2.89191i	−0.0317450	−	0.110575i
$685$		0			0
$686$	8.76571	+	62.2991i	0.334676	+	2.37859i
$687$			48.9298i			1.86679i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−43.6891	+	12.5427i	−1.66081	+	0.476803i
$693$	−41.7699			−1.58671
$694$		0			0
$695$		0			0
$696$	25.7446	+	57.7488i	0.975845	+	2.18896i
$697$		0			0
$698$		0			0
$699$			35.5421i			1.34433i
$700$	13.2998	+	46.3261i	0.502685	+	1.75096i
$701$	49.6176			1.87403			0.937015	−	0.349289i	$-0.113577\pi$
$701$	49.6176			1.87403			0.937015	+	0.349289i	$0.113577\pi$
$702$		0			0
$703$		0			0
$704$	−39.4174	−	35.4234i	−1.48560	−	1.33507i
$705$		0			0
$706$	33.4338	−	4.70425i	1.25830	−	0.177047i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	−0.784440	+	0.349705i	−0.0293981	+	0.0131057i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−12.0733	−	42.0539i	−0.451200	−	1.57163i
$717$	−39.5415			−1.47670
$718$	−6.06726	−	43.1209i	−0.226428	−	1.60926i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−24.7562	+	3.48328i	−0.921330	+	0.129634i
$723$		0			0
$724$	42.9809	−	12.3394i	1.59737	−	0.458590i
$725$	53.8494			1.99992
$726$	−13.4491	−	95.5844i	−0.499141	−	3.54747i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−7.19588			−0.266514
$730$		0			0
$731$		0			0
$732$	−6.00611	−	20.9206i	−0.221992	−	0.773248i
$733$	2.05744			0.0759932			0.0379966	−	0.999278i	$-0.487902\pi$
$733$	2.05744			0.0759932			0.0379966	+	0.999278i	$0.487902\pi$
$734$	−6.57918	−	46.7592i	−0.242842	−	1.72591i
$735$		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	$-0.157224\pi$
$744$	43.7653	−	19.5106i	1.60451	−	0.715295i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−68.7336			−2.51147
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−26.8689	−	42.9383i	−0.979809	−	1.56580i
$753$	30.4239			1.10871
$754$		0			0
$755$		0			0
$756$	32.5345	−	9.34036i	1.18327	−	0.339705i
$757$	−54.8494			−1.99354			−0.996768	−	0.0803335i	$-0.974401\pi$
$757$	−54.8494			−1.99354			−0.996768	+	0.0803335i	$0.974401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.5845			1.21744			0.608718	−	0.793387i	$-0.291684\pi$
$761$	33.5845			1.21744			0.608718	+	0.793387i	$0.291684\pi$
$762$	−44.9334	+	6.32229i	−1.62777	+	0.229033i
$763$		0			0
$764$	15.2535	+	53.1312i	0.551851	+	1.92222i
$765$		0			0
$766$	2.77091	+	19.6933i	0.100117	+	0.711547i
$767$		0			0
$768$	−29.8676	−	14.5202i	−1.07775	−	0.523954i
$769$		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
$775$		−	40.8100i		−	1.46594i
$776$	22.5089	−	10.0345i	0.808023	−	0.360218i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	37.8180i		−	1.35151i
$784$	−55.0332	+	34.4374i	−1.96547	+	1.22991i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	65.3910			2.32798
$790$		0			0
$791$		0			0
$792$	−9.98077	−	22.3883i	−0.354651	−	0.795535i
$793$		0			0
$794$	−53.9497	+	7.59091i	−1.91460	+	0.269391i
$795$		0			0
$796$	13.4045	+	46.6909i	0.475111	+	1.65492i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	−2.26674	−	16.1100i	−0.0802418	−	0.570290i
$799$		0			0
$800$	−21.6525	+	18.1981i	−0.765531	+	0.643398i
$801$	−0.397251			−0.0140362
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.6796			0.727054			0.363527	−	0.931584i	$-0.381572\pi$
$809$	20.6796			0.727054			0.363527	+	0.931584i	$0.381572\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	28.6474	+	99.7853i	1.00533	+	3.50178i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	55.2754	−	7.77745i	1.93266	−	0.271932i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	9.57422	+	68.0453i	0.333939	+	2.37335i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−68.7497			−2.39356
$826$		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
$836$	14.6436	−	4.20404i	0.506459	−	0.145400i
$837$	−28.6606			−0.990654
$838$	−4.51421	−	32.0831i	−0.155941	−	1.10829i
$839$			5.85721i			0.202213i	0.994876	+	0.101107i	$0.0322383\pi$
$839$			5.85721i			0.202213i	−0.994876	+	0.101107i	$0.967762\pi$
$840$		0			0
$841$	86.9902			2.99966
$842$	20.7784	−	2.92360i	0.716071	−	0.100754i
$843$			68.4484i			2.35749i
$844$	−7.11442	−	24.7811i	−0.244888	−	0.853000i
$845$		0			0
$846$	−3.26429	−	23.1998i	−0.112229	−	0.797625i
$847$		−	158.490i		−	5.44580i
$848$		0			0
$849$	22.9216			0.786667
$850$		0			0
$851$		0			0
$852$		0			0
$853$	10.7605			0.368431			0.184216	−	0.982886i	$-0.441026\pi$
$853$	10.7605			0.368431			0.184216	+	0.982886i	$0.441026\pi$
$854$	−4.97943	−	35.3895i	−0.170393	−	1.21100i
$855$		0			0
$856$	−16.4236	−	36.8407i	−0.561349	−	1.25919i
$857$	−46.4088			−1.58529			−0.792647	−	0.609680i	$-0.791298\pi$
$857$	−46.4088			−1.58529			−0.792647	+	0.609680i	$0.791298\pi$
$858$		0			0
$859$		−	23.5035i		−	0.801929i	−0.916094	−	0.400965i	$-0.868675\pi$
$859$		−	23.5035i		−	0.801929i	0.916094	−	0.400965i	$-0.131325\pi$
$860$		0			0
$861$		0			0
$862$	−1.29982	−	9.23797i	−0.0442719	−	0.314647i
$863$			33.1260i			1.12762i	0.825904	+	0.563810i	$0.190665\pi$
$863$			33.1260i			1.12762i	−0.825904	+	0.563810i	$0.809335\pi$
$864$	12.7804	+	15.2064i	0.434797	+	0.517332i
$865$		0			0
$866$	55.5151	−	7.81117i	1.88648	−	0.265434i
$867$		−	35.2857i		−	1.19837i
$868$	75.6228	−	21.7106i	2.56681	−	0.736906i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	11.3988			0.385792
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−32.5091			−1.09775			−0.548877	−	0.835903i	$-0.684944\pi$
$877$	−32.5091			−1.09775			−0.548877	+	0.835903i	$0.684944\pi$
$878$		0			0
$879$		−	34.2199i		−	1.15421i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−29.7347	+	4.18378i	−1.00122	+	0.140875i
$883$			33.1430i			1.11535i	0.830059	+	0.557675i	$0.188307\pi$
$883$			33.1430i			1.11535i	−0.830059	+	0.557675i	$0.811693\pi$
$884$		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
$889$	−74.5051			−2.49882
$890$		0			0
$891$			74.2817i			2.48853i
$892$	12.8047	+	44.6016i	0.428733	+	1.49337i
$893$	14.5614			0.487277
$894$		0			0
$895$		0			0
$896$	−45.2408	−	30.4418i	−1.51139	−	1.01699i
$897$		0			0
$898$	36.5617	−	5.14437i	1.22008	−	0.171670i
$899$		−	87.9037i		−	2.93175i
$900$	−12.5745	+	3.61002i	−0.419150	+	0.120334i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	52.6301i		−	1.74756i	−0.486326	−	0.873778i	$-0.661663\pi$
$907$		−	52.6301i		−	1.74756i	0.486326	−	0.873778i	$-0.338337\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			19.5625i			0.648134i	0.946034	+	0.324067i	$0.105050\pi$
$911$			19.5625i			0.648134i	−0.946034	+	0.324067i	$0.894950\pi$
$912$	8.09323	−	5.06439i	0.267994	−	0.167699i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−45.3164	+	13.0099i	−1.49729	+	0.429859i
$917$		0			0
$918$		0			0
$919$			28.7898i			0.949687i	0.880070	+	0.474844i	$0.157495\pi$
$919$			28.7898i			0.949687i	−0.880070	+	0.474844i	$0.842505\pi$
$920$		0			0
$921$		0			0
$922$	7.61961	−	1.07211i	0.250939	−	0.0353079i
$923$		0			0
$924$	−36.5743	−	127.396i	−1.20321	−	4.19103i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−46.6389	+	39.1982i	−1.53100	+	1.28674i
$929$	50.0966			1.64362			0.821808	−	0.569765i	$-0.192966\pi$
$929$	50.0966			1.64362			0.821808	+	0.569765i	$0.192966\pi$
$930$		0			0
$931$		−	18.6630i		−	0.611656i
$932$	−32.9174	+	9.45027i	−1.07824	+	0.309554i
$933$	−53.6461			−1.75630
$934$	8.50822	+	60.4691i	0.278397	+	1.97861i
$935$		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			$\pi$
$942$	9.97123	+	70.8669i	0.324880	+	2.30897i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			56.7814i			1.84515i	0.385821	+	0.922574i	$0.373918\pi$
$947$			56.7814i			1.84515i	−0.385821	+	0.922574i	$0.626082\pi$
$948$		0			0
$949$		0			0
$950$	−1.13292	−	8.05179i	−0.0367566	−	0.261235i
$951$			35.1696i			1.14045i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−10.5137	−	36.6214i	−0.340036	−	1.18442i
$957$	−148.085			−4.78691
$958$		0			0
$959$			112.827i			3.64339i
$960$		0			0
$961$	−35.6183			−1.14898
$962$		0			0
$963$		−	18.6566i		−	0.601201i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	17.6201i		−	0.566625i	−0.959028	−	0.283312i	$-0.908567\pi$
$967$		−	17.6201i		−	0.566625i	0.959028	−	0.283312i	$-0.0914333\pi$
$968$	84.9496	−	37.8707i	2.73038	−	1.21721i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	7.03113	+	24.4910i	0.225524	+	0.785548i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	17.7787	−	11.1251i	0.569082	−	0.356107i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		−	2.01154i		−	0.0642890i
$980$		0			0
$981$		0			0
$982$	−5.09274	−	36.1948i	−0.162516	−	1.15502i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	126.681i		−	4.03230i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	29.7065	+	35.3456i	0.943183	+	1.12222i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−49.7820			−1.57661			−0.788305	−	0.615284i	$-0.789041\pi$
$997$	−49.7820			−1.57661			−0.788305	+	0.615284i	$0.789041\pi$
$998$		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.667.2	yes	22
4.3	odd	2	inner	668.2.b.a.667.1	✓	22
167.166	odd	2	CM	668.2.b.a.667.2	yes	22
668.667	even	2	inner	668.2.b.a.667.1	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.1	✓	22	4.3	odd	2	inner
668.2.b.a.667.1	✓	22	668.667	even	2	inner
668.2.b.a.667.2	yes	22	1.1	even	1	trivial
668.2.b.a.667.2	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)

\(f(q)\)	\(=\)	\(q+(-1.40042 + 0.197044i) q^{2} -2.07563i q^{3} +(1.92235 - 0.551888i) q^{4} +(0.408990 + 2.90675i) q^{6} +4.81975i q^{7} +(-2.58335 + 1.15166i) q^{8} -1.30824 q^{9} +O(q^{10})\)	\(q+(-1.40042 + 0.197044i) q^{2} -2.07563i q^{3} +(1.92235 - 0.551888i) q^{4} +(0.408990 + 2.90675i) q^{6} +4.81975i q^{7} +(-2.58335 + 1.15166i) q^{8} -1.30824 q^{9} -6.62447i q^{11} +(-1.14552 - 3.99008i) q^{12} +(-0.949702 - 6.74967i) q^{14} +(3.39084 - 2.12184i) q^{16} +(1.83209 - 0.257781i) q^{18} +1.14991i q^{19} +10.0040 q^{21} +(1.30531 + 9.27703i) q^{22} +(2.39042 + 5.36207i) q^{24} +5.00000 q^{25} -3.51146i q^{27} +(2.65996 + 9.26523i) q^{28} +10.7699 q^{29} -8.16200i q^{31} +(-4.33050 + 3.63961i) q^{32} -13.7499 q^{33} +(-2.51490 + 0.722004i) q^{36} +(-0.226583 - 1.61036i) q^{38} +(-14.0098 + 1.97123i) q^{42} +(-3.65596 - 12.7345i) q^{44} -12.6630i q^{47} +(-4.40416 - 7.03813i) q^{48} -16.2300 q^{49} +(-7.00210 + 0.985220i) q^{50} +(0.691912 + 4.91752i) q^{54} +(-5.55072 - 12.4511i) q^{56} +2.38679 q^{57} +(-15.0823 + 2.12214i) q^{58} +5.24315 q^{61} +(1.60827 + 11.4302i) q^{62} -6.30540i q^{63} +(5.34735 - 5.95028i) q^{64} +(19.2557 - 2.70934i) q^{66} +(3.37964 - 1.50665i) q^{72} -10.3782i q^{75} +(0.634623 + 2.21053i) q^{76} +31.9283 q^{77} -11.2132 q^{81} +(19.2312 - 5.52110i) q^{84} -22.3543i q^{87} +(7.62914 + 17.1133i) q^{88} +0.303653 q^{89} -16.9413 q^{93} +(2.49517 + 17.7335i) q^{94} +(7.55449 + 8.98852i) q^{96} -8.71308 q^{97} +(22.7287 - 3.19801i) q^{98} +8.66641i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 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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