LMFDB - Embedded newform 668.2.b.a.667.17

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			667.17
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.06600 - 0.929330i) q^{2} -2.77291i q^{3} +(0.272692 - 1.98132i) q^{4} +(-2.57695 - 2.95591i) q^{6} -0.0155204i q^{7} +(-1.55061 - 2.36550i) q^{8} -4.68903 q^{9} +O(q^{10})$	\(q+(1.06600 - 0.929330i) q^{2} -2.77291i q^{3} +(0.272692 - 1.98132i) q^{4} +(-2.57695 - 2.95591i) q^{6} -0.0155204i q^{7} +(-1.55061 - 2.36550i) q^{8} -4.68903 q^{9} -0.605002i q^{11} +(-5.49403 - 0.756151i) q^{12} +(-0.0144236 - 0.0165447i) q^{14} +(-3.85128 - 1.08058i) q^{16} +(-4.99849 + 4.35766i) q^{18} +3.70465i q^{19} -0.0430367 q^{21} +(-0.562247 - 0.644930i) q^{22} +(-6.55932 + 4.29971i) q^{24} +5.00000 q^{25} +4.68354i q^{27} +(-0.0307509 - 0.00423229i) q^{28} +4.56421 q^{29} -11.0698i q^{31} +(-5.10966 + 2.42721i) q^{32} -1.67762 q^{33} +(-1.27866 + 9.29049i) q^{36} +(3.44284 + 3.94914i) q^{38} +(-0.0458769 + 0.0399953i) q^{42} +(-1.19870 - 0.164979i) q^{44} +7.81007i q^{47} +(-2.99636 + 10.6792i) q^{48} +6.99976 q^{49} +(5.32998 - 4.64665i) q^{50} +(4.35255 + 4.99263i) q^{54} +(-0.0367135 + 0.0240662i) q^{56} +10.2727 q^{57} +(4.86542 - 4.24165i) q^{58} -9.17625 q^{61} +(-10.2875 - 11.8004i) q^{62} +0.0727757i q^{63} +(-3.19119 + 7.33596i) q^{64} +(-1.78833 + 1.55906i) q^{66} +(7.27088 + 11.0919i) q^{72} -13.8646i q^{75} +(7.34010 + 1.01023i) q^{76} -0.00938988 q^{77} -1.08007 q^{81} +(-0.0117358 + 0.0852696i) q^{84} -12.6561i q^{87} +(-1.43113 + 0.938125i) q^{88} +5.02368 q^{89} -30.6957 q^{93} +(7.25813 + 8.32549i) q^{94} +(6.73044 + 14.1686i) q^{96} +18.7260 q^{97} +(7.46171 - 6.50508i) q^{98} +2.83688i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.06600	−	0.929330i	0.753773	−	0.657135i
$2$	1.06600	−	0.929330i	0.753773	−	0.657135i
$3$		−	2.77291i		−	1.60094i	−0.599372	−	0.800470i	$-0.704583\pi$
$3$		−	2.77291i		−	1.60094i	0.599372	−	0.800470i	$-0.295417\pi$
$4$	0.272692	−	1.98132i	0.136346	−	0.990661i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−2.57695	−	2.95591i	−1.05203	−	1.20675i
$7$		−	0.0155204i		−	0.00586616i	−0.999996	−	0.00293308i	$-0.999066\pi$
$7$		−	0.0155204i		−	0.00586616i	0.999996	−	0.00293308i	$-0.000933630\pi$
$8$	−1.55061	−	2.36550i	−0.548225	−	0.836331i
$9$	−4.68903			−1.56301
$10$		0			0
$11$		−	0.605002i		−	0.182415i	−0.995832	−	0.0912075i	$-0.970927\pi$
$11$		−	0.605002i		−	0.182415i	0.995832	−	0.0912075i	$-0.0290727\pi$
$12$	−5.49403	−	0.756151i	−1.58599	−	0.218282i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.0144236	−	0.0165447i	−0.00385486	−	0.00442175i
$15$		0			0
$16$	−3.85128	−	1.08058i	−0.962820	−	0.270146i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−4.99849	+	4.35766i	−1.17815	+	1.02711i
$19$			3.70465i			0.849905i	0.905216	+	0.424952i	$0.139709\pi$
$19$			3.70465i			0.849905i	−0.905216	+	0.424952i	$0.860291\pi$
$20$		0			0
$21$	−0.0430367			−0.00939138
$22$	−0.562247	−	0.644930i	−0.119871	−	0.137499i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−6.55932	+	4.29971i	−1.33892	+	0.877675i
$25$	5.00000			1.00000
$26$		0			0
$27$			4.68354i			0.901347i
$28$	−0.0307509	−	0.00423229i	−0.00581138	−	0.000799828i
$29$	4.56421			0.847552			0.423776	−	0.905767i	$-0.360704\pi$
$29$	4.56421			0.847552			0.423776	+	0.905767i	$0.360704\pi$
$30$		0			0
$31$		−	11.0698i		−	1.98820i	−0.108465	−	0.994100i	$-0.534594\pi$
$31$		−	11.0698i		−	1.98820i	0.108465	−	0.994100i	$-0.465406\pi$
$32$	−5.10966	+	2.42721i	−0.903269	+	0.429075i
$33$	−1.67762			−0.292036
$34$		0			0
$35$		0			0
$36$	−1.27866	+	9.29049i	−0.213110	+	1.54841i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	3.44284	+	3.94914i	0.558502	+	0.640635i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.0458769	+	0.0399953i	−0.00707896	+	0.00617141i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−1.19870	−	0.164979i	−0.180712	−	0.0248716i
$45$		0			0
$46$		0			0
$47$			7.81007i			1.13922i	0.821917	+	0.569608i	$0.192905\pi$
$47$			7.81007i			1.13922i	−0.821917	+	0.569608i	$0.807095\pi$
$48$	−2.99636	+	10.6792i	−0.432487	+	1.54142i
$49$	6.99976			0.999966
$50$	5.32998	−	4.64665i	0.753773	−	0.657135i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	4.35255	+	4.99263i	0.592307	+	0.679411i
$55$		0			0
$56$	−0.0367135	+	0.0240662i	−0.00490605	+	0.00321597i
$57$	10.2727			1.36065
$58$	4.86542	−	4.24165i	0.638861	−	0.556956i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−9.17625			−1.17490			−0.587449	−	0.809261i	$-0.699868\pi$
$61$	−9.17625			−1.17490			−0.587449	+	0.809261i	$0.699868\pi$
$62$	−10.2875	−	11.8004i	−1.30652	−	1.49865i
$63$			0.0727757i			0.00916888i
$64$	−3.19119	+	7.33596i	−0.398899	+	0.916995i
$65$		0			0
$66$	−1.78833	+	1.55906i	−0.220128	+	0.191907i
$67$		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	$0.5\pi$
$72$	7.27088	+	11.0919i	0.856881	+	1.30719i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	13.8646i		−	1.60094i
$76$	7.34010	+	1.01023i	0.841968	+	0.115881i
$77$	−0.00938988			−0.00107008
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−1.08007			−0.120007
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−0.0117358	+	0.0852696i	−0.00128048	+	0.00930367i
$85$		0			0
$86$		0			0
$87$		−	12.6561i		−	1.35688i
$88$	−1.43113	+	0.938125i	−0.152559	+	0.100004i
$89$	5.02368			0.532509			0.266254	−	0.963903i	$-0.414214\pi$
$89$	5.02368			0.532509			0.266254	+	0.963903i	$0.414214\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−30.6957			−3.18299
$94$	7.25813	+	8.32549i	0.748619	+	0.858709i
$95$		0			0
$96$	6.73044	+	14.1686i	0.686923	+	1.44608i
$97$	18.7260			1.90134			0.950670	−	0.310203i	$-0.100397\pi$
$97$	18.7260			1.90134			0.950670	+	0.310203i	$0.100397\pi$
$98$	7.46171	−	6.50508i	0.753747	−	0.657113i
$99$			2.83688i			0.285117i
$100$	1.36346	−	9.90661i	0.136346	−	0.990661i
$101$		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
$107$		−	19.5574i		−	1.89068i	−0.326081	−	0.945342i	$-0.605728\pi$
$107$		−	19.5574i		−	1.89068i	0.326081	−	0.945342i	$-0.394272\pi$
$108$	9.27960	+	1.27716i	0.892930	+	0.122895i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.0167711	+	0.0597734i	−0.00158472	+	0.00564805i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	10.9506	−	9.54669i	1.02562	−	0.894129i
$115$		0			0
$116$	1.24462	−	9.04317i	0.115560	−	0.839637i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.6340			0.966725
$122$	−9.78184	+	8.52776i	−0.885606	+	0.772068i
$123$		0			0
$124$	−21.9329	−	3.01866i	−1.96963	−	0.271083i
$125$		0			0
$126$	0.0676326	+	0.0775786i	0.00602519	+	0.00691125i
$127$		−	14.0356i		−	1.24546i	−0.782438	−	0.622728i	$-0.786024\pi$
$127$		−	14.0356i		−	1.24546i	0.782438	−	0.622728i	$-0.213976\pi$
$128$	3.41573	+	10.7858i	0.301910	+	0.953336i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−0.457473	+	3.32390i	−0.0398179	+	0.289308i
$133$	0.0574976			0.00498568
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.33330			−0.284783			−0.142391	−	0.989810i	$-0.545479\pi$
$137$	−3.33330			−0.284783			−0.142391	+	0.989810i	$0.545479\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	21.6566			1.82382
$142$		0			0
$143$		0			0
$144$	18.0588	+	5.06689i	1.50490	+	0.422240i
$145$		0			0
$146$		0			0
$147$		−	19.4097i		−	1.60089i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−12.8847	−	14.7795i	−1.05203	−	1.20675i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	8.76335	−	5.74448i	0.710802	−	0.465939i
$153$		0			0
$154$	−0.0100096	+	0.00872630i	−0.000806594	+	0.000703185i
$155$		0			0
$156$		0			0
$157$	−11.5841			−0.924514			−0.462257	−	0.886746i	$-0.652960\pi$
$157$	−11.5841			−0.924514			−0.462257	+	0.886746i	$0.652960\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.15135	+	1.00374i	−0.0904583	+	0.0788611i
$163$		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
$168$	0.0667333	+	0.101803i	0.00514858	+	0.00785430i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	17.3712i		−	1.32841i
$172$		0			0
$173$	−9.87791			−0.751003			−0.375502	−	0.926822i	$-0.622530\pi$
$173$	−9.87791			−0.751003			−0.375502	+	0.926822i	$0.622530\pi$
$174$	−11.7617	−	13.4914i	−0.891654	−	1.02278i
$175$		−	0.0776020i		−	0.00586616i
$176$	−0.653755	+	2.33003i	−0.0492786	+	0.175633i
$177$		0			0
$178$	5.35522	−	4.66865i	0.401391	−	0.349930i
$179$			26.7341i			1.99820i	0.0424302	+	0.999099i	$0.486490\pi$
$179$			26.7341i			1.99820i	−0.0424302	+	0.999099i	$0.513510\pi$
$180$		0			0
$181$	10.7159			0.796505			0.398252	−	0.917276i	$-0.369617\pi$
$181$	10.7159			0.796505			0.398252	+	0.917276i	$0.369617\pi$
$182$		0			0
$183$			25.4449i			1.88094i
$184$		0			0
$185$		0			0
$186$	−32.7214	+	28.5264i	−2.39925	+	2.09166i
$187$		0			0
$188$	15.4743	+	2.12974i	1.12858	+	0.155328i
$189$	0.0726904			0.00528745
$190$		0			0
$191$			18.3402i			1.32705i	0.748153	+	0.663526i	$0.230941\pi$
$191$			18.3402i			1.32705i	−0.748153	+	0.663526i	$0.769059\pi$
$192$	20.3420	+	8.84890i	1.46805	+	0.638614i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	19.9619	−	17.4027i	1.43318	−	1.24944i
$195$		0			0
$196$	1.90878	−	13.8688i	0.136341	−	0.990627i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	2.63639	+	3.02410i	0.187360	+	0.214913i
$199$			17.6656i			1.25228i	0.779711	+	0.626140i	$0.215366\pi$
$199$			17.6656i			1.25228i	−0.779711	+	0.626140i	$0.784634\pi$
$200$	−7.75307	−	11.8275i	−0.548225	−	0.836331i
$201$		0			0
$202$		0			0
$203$		−	0.0708383i		−	0.00497188i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.24132			0.155035
$210$		0			0
$211$			11.2341i			0.773386i	0.922208	+	0.386693i	$0.126383\pi$
$211$			11.2341i			0.773386i	−0.922208	+	0.386693i	$0.873617\pi$
$212$		0			0
$213$		0			0
$214$	−18.1753	−	20.8481i	−1.24244	−	1.42515i
$215$		0			0
$216$	11.0789	−	7.26236i	0.753825	−	0.494141i
$217$	−0.171808			−0.0116631
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			16.9634i			1.13595i	0.823045	+	0.567976i	$0.192273\pi$
$223$			16.9634i			1.13595i	−0.823045	+	0.567976i	$0.807727\pi$
$224$	0.0376713	+	0.0793040i	0.00251702	+	0.00529872i
$225$	−23.4452			−1.56301
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	2.80127	−	20.3535i	0.185519	−	1.34794i
$229$	7.47330			0.493850			0.246925	−	0.969035i	$-0.420580\pi$
$229$	7.47330			0.493850			0.246925	+	0.969035i	$0.420580\pi$
$230$		0			0
$231$			0.0260373i			0.00171313i
$232$	−7.07732	−	10.7966i	−0.464649	−	0.708834i
$233$	−7.88743			−0.516722			−0.258361	−	0.966048i	$-0.583182\pi$
$233$	−7.88743			−0.516722			−0.258361	+	0.966048i	$0.583182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	11.4176i		−	0.738541i	−0.929322	−	0.369271i	$-0.879608\pi$
$239$		−	11.4176i		−	0.738541i	0.929322	−	0.369271i	$-0.120392\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	11.3358	−	9.88247i	0.728691	−	0.635269i
$243$			17.0455i			1.09347i
$244$	−2.50229	+	18.1811i	−0.160193	+	1.16393i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−26.1857	+	17.1650i	−1.66279	+	1.08998i
$249$		0			0
$250$		0			0
$251$			19.4643i			1.22857i	0.789083	+	0.614287i	$0.210556\pi$
$251$			19.4643i			1.22857i	−0.789083	+	0.614287i	$0.789444\pi$
$252$	0.144192	+	0.0198454i	0.00908325	+	0.00125014i
$253$		0			0
$254$	−13.0437	−	14.9619i	−0.818433	−	0.938790i
$255$		0			0
$256$	13.6647	+	8.32324i	0.854043	+	0.520203i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−21.4017			−1.32473
$262$		0			0
$263$			28.0550i			1.72995i	0.501817	+	0.864974i	$0.332665\pi$
$263$			28.0550i			1.72995i	−0.501817	+	0.864974i	$0.667335\pi$
$264$	2.60134	+	3.96841i	0.160101	+	0.244238i
$265$		0			0
$266$	0.0612922	−	0.0534343i	0.00375807	−	0.00327627i
$267$		−	13.9302i		−	0.852515i
$268$		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
$274$	−3.55328	+	3.09773i	−0.214661	+	0.187141i
$275$		−	3.02501i		−	0.182415i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$			51.9068i			3.10758i
$280$		0			0
$281$	29.9391			1.78602			0.893009	−	0.450039i	$-0.148590\pi$
$281$	29.9391			1.78602			0.893009	+	0.450039i	$0.148590\pi$
$282$	23.0859	−	20.1261i	1.37474	−	1.19849i
$283$		−	29.8862i		−	1.77655i	−0.459314	−	0.888274i	$-0.651905\pi$
$283$		−	29.8862i		−	1.77655i	0.459314	−	0.888274i	$-0.348095\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	23.9594	−	11.3813i	1.41182	−	0.670648i
$289$	17.0000			1.00000
$290$		0			0
$291$		−	51.9256i		−	3.04393i
$292$		0			0
$293$	−33.4713			−1.95542			−0.977708	−	0.209971i	$-0.932663\pi$
$293$	−33.4713			−1.95542			−0.977708	+	0.209971i	$0.932663\pi$
$294$	−18.0380	−	20.6907i	−1.05200	−	1.20670i
$295$		0			0
$296$		0			0
$297$	2.83355			0.164419
$298$		0			0
$299$		0			0
$300$	−27.4702	−	3.78075i	−1.58599	−	0.218282i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	4.00318	−	14.2676i	0.229598	−	0.818305i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−0.00256055	+	0.0186044i	−0.000145901	+	0.00106008i
$309$		0			0
$310$		0			0
$311$			25.8457i			1.46557i	0.680458	+	0.732787i	$0.261781\pi$
$311$			25.8457i			1.46557i	−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−12.3486	+	10.7655i	−0.696873	+	0.607531i
$315$		0			0
$316$		0			0
$317$	7.43404			0.417537			0.208769	−	0.977965i	$-0.433054\pi$
$317$	7.43404			0.417537			0.208769	+	0.977965i	$0.433054\pi$
$318$		0			0
$319$		−	2.76136i		−	0.154606i
$320$		0			0
$321$	−54.2309			−3.02687
$322$		0			0
$323$		0			0
$324$	−0.294526	+	2.13996i	−0.0163625	+	0.118887i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.121215			0.00668282
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	12.0096	+	13.7757i	0.657135	+	0.753773i
$335$		0			0
$336$	0.165746	+	0.0465047i	0.00904220	+	0.00253704i
$337$	36.6331			1.99553			0.997765	−	0.0668248i	$-0.0212868\pi$
$337$	36.6331			1.99553			0.997765	+	0.0668248i	$0.0212868\pi$
$338$	13.8579	−	12.0813i	0.753773	−	0.657135i
$339$		0			0
$340$		0			0
$341$	−6.69727			−0.362678
$342$	−16.1436	−	18.5176i	−0.872946	−	1.00132i
$343$		−	0.217282i		−	0.0117321i
$344$		0			0
$345$		0			0
$346$	−10.5298	+	9.17983i	−0.566086	+	0.493511i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−25.0759	−	3.45123i	−1.34421	−	0.185005i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.0721179	−	0.0827234i	−0.00385486	−	0.00442175i
$351$		0			0
$352$	1.46847	+	3.09136i	0.0782696	+	0.164770i
$353$	−36.3131			−1.93275			−0.966375	−	0.257136i	$-0.917221\pi$
$353$	−36.3131			−1.93275			−0.966375	+	0.257136i	$0.917221\pi$
$354$		0			0
$355$		0			0
$356$	1.36992	−	9.95353i	0.0726055	−	0.527536i
$357$		0			0
$358$	24.8448	+	28.4984i	1.31309	+	1.50619i
$359$			3.47094i			0.183189i	0.995796	+	0.0915946i	$0.0291964\pi$
$359$			3.47094i			0.183189i	−0.995796	+	0.0915946i	$0.970804\pi$
$360$		0			0
$361$	5.27558			0.277662
$362$	11.4231	−	9.95858i	0.600383	−	0.523411i
$363$		−	29.4871i		−	1.54767i
$364$		0			0
$365$		0			0
$366$	23.6467	+	27.1242i	1.23603	+	1.41780i
$367$		−	38.2488i		−	1.99657i	−0.0585343	−	0.998285i	$-0.518643\pi$
$367$		−	38.2488i		−	1.99657i	0.0585343	−	0.998285i	$-0.481357\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−8.37046	+	60.8180i	−0.433988	+	3.15327i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	18.4747	−	12.1104i	0.952761	−	0.624546i
$377$		0			0
$378$	0.0774876	−	0.0675534i	0.00398553	−	0.00347457i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−38.9194			−1.99390
$382$	17.0441	+	19.5506i	0.872053	+	1.00030i
$383$			39.0682i			1.99629i	0.0608672	+	0.998146i	$0.480613\pi$
$383$			39.0682i			1.99629i	−0.0608672	+	0.998146i	$0.519387\pi$
$384$	29.9080	−	9.47150i	1.52623	−	0.483341i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	5.10644	−	37.1023i	0.259240	−	1.88358i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−10.8539	−	16.5579i	−0.548206	−	0.836302i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	5.62077	+	0.773594i	0.282454	+	0.0388745i
$397$	−34.0917			−1.71101			−0.855507	−	0.517792i	$-0.826754\pi$
$397$	−34.0917			−1.71101			−0.855507	+	0.517792i	$0.826754\pi$
$398$	16.4172	+	18.8314i	0.822918	+	0.943934i
$399$		−	0.159436i		−	0.00798177i
$400$	−19.2564	−	5.40291i	−0.962820	−	0.270146i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.0658322	−	0.0755133i	−0.00326720	−	0.00374766i
$407$		0			0
$408$		0			0
$409$	14.3630			0.710203			0.355101	−	0.934828i	$-0.384446\pi$
$409$	14.3630			0.710203			0.355101	+	0.934828i	$0.384446\pi$
$410$		0			0
$411$			9.24293i			0.455920i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	2.38924	−	2.08293i	0.116861	−	0.101879i
$419$			40.6441i			1.98560i	0.119801	+	0.992798i	$0.461774\pi$
$419$			40.6441i			1.98560i	−0.119801	+	0.992798i	$0.538226\pi$
$420$		0			0
$421$	28.6392			1.39579			0.697895	−	0.716200i	$-0.254120\pi$
$421$	28.6392			1.39579			0.697895	+	0.716200i	$0.254120\pi$
$422$	10.4402	+	11.9755i	0.508219	+	0.582957i
$423$		−	36.6217i		−	1.78061i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			0.142419i			0.00689215i
$428$	−38.7495	−	5.33314i	−1.87303	−	0.257787i
$429$		0			0
$430$		0			0
$431$		−	5.21990i		−	0.251434i	−0.992066	−	0.125717i	$-0.959877\pi$
$431$		−	5.21990i		−	0.251434i	0.992066	−	0.125717i	$-0.0401231\pi$
$432$	5.06095	−	18.0376i	0.243495	−	0.867835i
$433$	−26.4989			−1.27345			−0.636727	−	0.771090i	$-0.719712\pi$
$433$	−26.4989			−1.27345			−0.636727	+	0.771090i	$0.719712\pi$
$434$	−0.183147	+	0.159667i	−0.00879133	+	0.00766424i
$435$		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
$441$	−32.8221			−1.56296
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	15.7646	+	18.0829i	0.746475	+	0.856250i
$447$		0			0
$448$	0.113857	+	0.0495286i	0.00537924	+	0.00234001i
$449$	−40.0111			−1.88824			−0.944121	−	0.329600i	$-0.893086\pi$
$449$	−40.0111			−1.88824			−0.944121	+	0.329600i	$0.893086\pi$
$450$	−24.9924	+	21.7883i	−1.17815	+	1.02711i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−15.9289	−	24.3000i	−0.745940	−	1.13795i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	7.96650	−	6.94516i	0.372250	−	0.324526i
$459$		0			0
$460$		0			0
$461$	−27.6062			−1.28575			−0.642875	−	0.765971i	$-0.722258\pi$
$461$	−27.6062			−1.28575			−0.642875	+	0.765971i	$0.722258\pi$
$462$	0.0241972	+	0.0277556i	0.00112576	+	0.00129131i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−17.5780	−	4.93200i	−0.816040	−	0.228962i
$465$		0			0
$466$	−8.40796	+	7.33002i	−0.389491	+	0.339557i
$467$			19.6505i			0.909317i	0.890666	+	0.454658i	$0.150239\pi$
$467$			19.6505i			0.909317i	−0.890666	+	0.454658i	$0.849761\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			32.1218i			1.48009i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			18.5232i			0.849905i
$476$		0			0
$477$		0			0
$478$	−10.6107	−	12.1711i	−0.485321	−	0.556692i
$479$		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
$484$	2.89980	−	21.0693i	0.131809	−	0.957697i
$485$		0			0
$486$	15.8409	+	18.1705i	0.718559	+	0.824229i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	14.2288	+	21.7064i	0.644108	+	0.982604i
$489$		0			0
$490$		0			0
$491$		−	25.8457i		−	1.16640i	−0.812329	−	0.583200i	$-0.801800\pi$
$491$		−	25.8457i		−	1.16640i	0.812329	−	0.583200i	$-0.198200\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−11.9619	+	42.6330i	−0.537103	+	1.91428i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	35.8339			1.60094
$502$	18.0887	+	20.7488i	0.807339	+	0.926065i
$503$		−	44.6925i		−	1.99274i	−0.0851292	−	0.996370i	$-0.527130\pi$
$503$		−	44.6925i		−	1.99274i	0.0851292	−	0.996370i	$-0.472870\pi$
$504$	0.172151	−	0.112847i	0.00766822	−	0.00502660i
$505$		0			0
$506$		0			0
$507$		−	36.0478i		−	1.60094i
$508$	−27.8090	−	3.82739i	−1.23382	−	0.169813i
$509$	−36.3003			−1.60898			−0.804492	−	0.593964i	$-0.797562\pi$
$509$	−36.3003			−1.60898			−0.804492	+	0.593964i	$0.797562\pi$
$510$		0			0
$511$		0			0
$512$	22.3015	−	3.82646i	0.985598	−	0.169107i
$513$	−17.3509			−0.766059
$514$		0			0
$515$		0			0
$516$		0			0
$517$	4.72511			0.207810
$518$		0			0
$519$			27.3906i			1.20231i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−22.8141	+	19.8893i	−0.998547	+	0.870529i
$523$		−	33.0452i		−	1.44497i	−0.691388	−	0.722484i	$-0.743000\pi$
$523$		−	33.0452i		−	1.44497i	0.691388	−	0.722484i	$-0.257000\pi$
$524$		0			0
$525$	−0.215183			−0.00939138
$526$	26.0724	+	29.9065i	1.13681	+	1.30399i
$527$		0			0
$528$	6.46097	+	1.81280i	0.281178	+	0.0788921i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	0.0156792	−	0.113921i	0.000679778	−	0.00493912i
$533$		0			0
$534$	−12.9458	−	14.8495i	−0.560218	−	0.642602i
$535$		0			0
$536$		0			0
$537$	74.1312			3.19900
$538$		0			0
$539$		−	4.23487i		−	0.182409i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	29.7142i		−	1.27516i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−0.908964	+	6.60433i	−0.0388290	+	0.282123i
$549$	43.0278			1.83638
$550$	−2.81123	−	3.22465i	−0.119871	−	0.137499i
$551$			16.9088i			0.720338i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	42.9669			1.82057			0.910283	−	0.413986i	$-0.135864\pi$
$557$	42.9669			1.82057			0.910283	+	0.413986i	$0.135864\pi$
$558$	48.2385	+	55.3324i	2.04210	+	2.34241i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	31.9150	−	27.8233i	1.34625	−	1.17366i
$563$		−	42.3542i		−	1.78502i	−0.451030	−	0.892509i	$-0.648943\pi$
$563$		−	42.3542i		−	1.78502i	0.451030	−	0.892509i	$-0.351057\pi$
$564$	5.90559	−	42.9087i	0.248670	−	1.80678i
$565$		0			0
$566$	−27.7741	−	31.8585i	−1.16743	−	1.33911i
$567$			0.0167631i			0.000703983i
$568$		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
$573$	50.8558			2.12453
$574$		0			0
$575$		0			0
$576$	14.9636	−	34.3986i	0.623484	−	1.43327i
$577$	18.8310			0.783946			0.391973	−	0.919977i	$-0.371793\pi$
$577$	18.8310			0.783946			0.391973	+	0.919977i	$0.371793\pi$
$578$	18.1219	−	15.7986i	0.753773	−	0.657135i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−48.2560	−	55.3525i	−2.00028	−	2.29443i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−35.6803	+	31.1059i	−1.47394	+	1.28497i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−38.4569	−	5.29287i	−1.58594	−	0.218274i
$589$	41.0098			1.68978
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	3.02055	−	2.63330i	0.123935	−	0.108046i
$595$		0			0
$596$		0			0
$597$	48.9851			2.00483
$598$		0			0
$599$		−	39.1613i		−	1.60009i	−0.599941	−	0.800044i	$-0.704809\pi$
$599$		−	39.1613i		−	1.60009i	0.599941	−	0.800044i	$-0.295191\pi$
$600$	−32.7966	+	21.4986i	−1.33892	+	0.877675i
$601$	−39.2222			−1.59991			−0.799953	−	0.600063i	$-0.795142\pi$
$601$	−39.2222			−1.59991			−0.799953	+	0.600063i	$0.795142\pi$
$602$		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	$0.5\pi$
$608$	−8.99197	−	18.9295i	−0.364672	−	0.767693i
$609$	−0.196428			−0.00795968
$610$		0			0
$611$		0			0
$612$		0			0
$613$	44.4149			1.79390			0.896951	−	0.442131i	$-0.145777\pi$
$613$	44.4149			1.79390			0.896951	+	0.442131i	$0.145777\pi$
$614$		0			0
$615$		0			0
$616$	0.0145601	+	0.0222118i	0.000586642	+	0.000894938i
$617$	8.92785			0.359422			0.179711	−	0.983719i	$-0.442484\pi$
$617$	8.92785			0.359422			0.179711	+	0.983719i	$0.442484\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	24.0192	+	27.5514i	0.963081	+	1.10471i
$623$		−	0.0779695i		−	0.00312378i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	6.21498i		−	0.248202i
$628$	−3.15890	+	22.9519i	−0.126054	+	0.915880i
$629$		0			0
$630$		0			0
$631$			50.2383i			1.99996i	0.00664970	+	0.999978i	$0.497883\pi$
$631$			50.2383i			1.99996i	−0.00664970	+	0.999978i	$0.502117\pi$
$632$		0			0
$633$	31.1511			1.23815
$634$	7.92465	−	6.90868i	0.314728	−	0.274379i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−2.56621	−	2.94359i	−0.101597	−	0.116538i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−57.8099	+	50.3984i	−2.28157	+	1.98906i
$643$		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	$0.5\pi$
$648$	1.67477	+	2.55490i	0.0657910	+	0.100366i
$649$		0			0
$650$		0			0
$651$			0.476409i			0.0186719i
$652$		0			0
$653$	−50.5272			−1.97728			−0.988642	−	0.150291i	$-0.951979\pi$
$653$	−50.5272			−1.97728			−0.988642	+	0.150291i	$0.951979\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.129215	−	0.112649i	0.00503733	−	0.00439152i
$659$		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
$668$	25.6043	+	3.52396i	0.990661	+	0.136346i
$669$	47.0380			1.81859
$670$		0			0
$671$			5.55165i			0.214319i
$672$	0.219903	−	0.104459i	0.00848294	−	0.00402960i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	39.0507	−	34.0442i	1.50418	−	1.31133i
$675$			23.4177i			0.901347i
$676$	3.54500	−	25.7572i	0.136346	−	0.990661i
$677$	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
$679$		−	0.290636i		−	0.0111536i
$680$		0			0
$681$		0			0
$682$	−7.13926	+	6.22398i	−0.273376	+	0.238328i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−34.4180	−	4.73699i	−1.31600	−	0.181124i
$685$		0			0
$686$	−0.201927	−	0.231622i	−0.00770959	−	0.00884335i
$687$		−	20.7228i		−	0.790624i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−2.69363	+	19.5713i	−0.102396	+	0.743990i
$693$	0.0440295			0.00167254
$694$		0			0
$695$		0			0
$696$	−29.9381	+	19.6248i	−1.13480	+	0.743875i
$697$		0			0
$698$		0			0
$699$			21.8711i			0.827242i
$700$	−0.153755	−	0.0211615i	−0.00581138	−	0.000799828i
$701$	−42.3968			−1.60131			−0.800653	−	0.599128i	$-0.795514\pi$
$701$	−42.3968			−1.60131			−0.800653	+	0.599128i	$0.795514\pi$
$702$		0			0
$703$		0			0
$704$	4.43827	+	1.93068i	0.167274	+	0.0727652i
$705$		0			0
$706$	−38.7096	+	33.7468i	−1.45685	+	1.27008i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	−7.78978	−	11.8835i	−0.291934	−	0.445354i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	52.9688	+	7.29017i	1.97954	+	0.272447i
$717$	−31.6599			−1.18236
$718$	3.22565	+	3.70001i	0.120380	+	0.138083i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	5.62374	−	4.90275i	0.209294	−	0.182462i
$723$		0			0
$724$	2.92213	−	21.2316i	0.108600	−	0.789066i
$725$	22.8210			0.847552
$726$	−27.4032	−	31.4331i	−1.01703	−	1.16659i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	44.0256			1.63058
$730$		0			0
$731$		0			0
$732$	50.4146	+	6.93863i	1.86338	+	0.256459i
$733$	30.9843			1.14443			0.572215	−	0.820104i	$-0.306084\pi$
$733$	30.9843			1.14443			0.572215	+	0.820104i	$0.306084\pi$
$734$	−35.5457	−	40.7730i	−1.31202	−	1.50496i
$735$		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	$0.157224\pi$
$743$			25.8457i			0.948187i	−0.880475	+	0.474093i	$0.842776\pi$
$744$	47.5971	+	72.6106i	1.74499	+	2.66203i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.303538			−0.0110911
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	8.43942	−	30.0787i	0.307754	−	1.09686i
$753$	53.9726			1.96687
$754$		0			0
$755$		0			0
$756$	0.0198221	−	0.144023i	0.000720923	−	0.00523807i
$757$	12.1814			0.442741			0.221371	−	0.975190i	$-0.428947\pi$
$757$	12.1814			0.442741			0.221371	+	0.975190i	$0.428947\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.8918			−0.721076			−0.360538	−	0.932745i	$-0.617407\pi$
$761$	−19.8918			−0.721076			−0.360538	+	0.932745i	$0.617407\pi$
$762$	−41.4879	+	36.1690i	−1.50295	+	1.31026i
$763$		0			0
$764$	36.3379	+	5.00123i	1.31466	+	0.180938i
$765$		0			0
$766$	36.3072	+	41.6465i	1.31183	+	1.50475i
$767$		0			0
$768$	23.0796	−	37.8909i	0.832814	−	1.36727i
$769$		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
$775$		−	55.3492i		−	1.98820i
$776$	−29.0368	−	44.2965i	−1.04236	−	1.59015i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			21.3766i			0.763939i
$784$	−26.9580	−	7.56381i	−0.962786	−	0.270136i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	77.7941			2.76954
$790$		0			0
$791$		0			0
$792$	6.71063	−	4.39890i	0.238452	−	0.156308i
$793$		0			0
$794$	−36.3416	+	31.6824i	−1.28971	+	1.12437i
$795$		0			0
$796$	35.0012	+	4.81727i	1.24059	+	0.170743i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	−0.148168	−	0.169958i	−0.00524511	−	0.00601644i
$799$		0			0
$800$	−25.5483	+	12.1361i	−0.903269	+	0.429075i
$801$	−23.5562			−0.832317
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.5115			1.74073			0.870366	−	0.492405i	$-0.163882\pi$
$809$	49.5115			1.74073			0.870366	+	0.492405i	$0.163882\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−0.140354	−	0.0193171i	−0.00492545	−	0.000677896i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	15.3109	−	13.3479i	0.535331	−	0.466699i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	8.58973	+	9.85292i	0.299601	+	0.343660i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−8.38809			−0.292036
$826$		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
$836$	0.611190	−	4.44078i	0.0211385	−	0.153588i
$837$	51.8460			1.79206
$838$	37.7718	+	43.3265i	1.30481	+	1.49669i
$839$			21.8574i			0.754600i	0.926091	+	0.377300i	$0.123147\pi$
$839$			21.8574i			0.754600i	−0.926091	+	0.377300i	$0.876853\pi$
$840$		0			0
$841$	−8.16802			−0.281656
$842$	30.5293	−	26.6153i	1.05211	−	0.917223i
$843$		−	83.0185i		−	2.85931i
$844$	22.2583	+	3.06345i	0.766164	+	0.105448i
$845$		0			0
$846$	−34.0336	−	39.0385i	−1.17010	−	1.34217i
$847$		−	0.165044i		−	0.00567096i
$848$		0			0
$849$	−82.8716			−2.84415
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−58.3596			−1.99820			−0.999098	−	0.0424614i	$-0.986480\pi$
$853$	−58.3596			−1.99820			−0.999098	+	0.0424614i	$0.986480\pi$
$854$	0.132354	+	0.151818i	0.00452907	+	0.00519511i
$855$		0			0
$856$	−46.2630	+	30.3259i	−1.58124	+	1.03652i
$857$	−19.7427			−0.674398			−0.337199	−	0.941433i	$-0.609480\pi$
$857$	−19.7427			−0.674398			−0.337199	+	0.941433i	$0.609480\pi$
$858$		0			0
$859$			58.6101i			1.99975i	0.0158288	+	0.999875i	$0.494961\pi$
$859$			58.6101i			1.99975i	−0.0158288	+	0.999875i	$0.505039\pi$
$860$		0			0
$861$		0			0
$862$	−4.85101	−	5.56439i	−0.165226	−	0.189524i
$863$		−	43.3167i		−	1.47452i	−0.675610	−	0.737259i	$-0.736120\pi$
$863$		−	43.3167i		−	1.47452i	0.675610	−	0.737259i	$-0.263880\pi$
$864$	−11.3679	−	23.9313i	−0.386745	−	0.814159i
$865$		0			0
$866$	−28.2477	+	24.6262i	−0.959894	+	0.836831i
$867$		−	47.1395i		−	1.60094i
$868$	−0.0468508	+	0.340408i	−0.00159022	+	0.0115542i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−87.8070			−2.97182
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.581676			−0.0196418			−0.00982090	−	0.999952i	$-0.503126\pi$
$877$	−0.581676			−0.0196418			−0.00982090	+	0.999952i	$0.503126\pi$
$878$		0			0
$879$			92.8130i			3.13050i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−34.9882	+	30.5026i	−1.17811	+	1.02707i
$883$		−	58.6411i		−	1.97343i	−0.162460	−	0.986715i	$-0.551943\pi$
$883$		−	58.6411i		−	1.97343i	0.162460	−	0.986715i	$-0.448057\pi$
$884$		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
$889$	−0.217838			−0.00730604
$890$		0			0
$891$			0.653443i			0.0218912i
$892$	33.6099	+	4.62578i	1.12534	+	0.154883i
$893$	−28.9336			−0.968224
$894$		0			0
$895$		0			0
$896$	0.167400	−	0.0530134i	0.00559243	−	0.00177105i
$897$		0			0
$898$	−42.6517	+	37.1835i	−1.42330	+	1.24083i
$899$		−	50.5250i		−	1.68510i
$900$	−6.39331	+	46.4524i	−0.213110	+	1.54841i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	42.2455i		−	1.40274i	−0.712798	−	0.701370i	$-0.752572\pi$
$907$		−	42.2455i		−	1.40274i	0.712798	−	0.701370i	$-0.247428\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			14.4178i			0.477684i	0.971058	+	0.238842i	$0.0767678\pi$
$911$			14.4178i			0.477684i	−0.971058	+	0.238842i	$0.923232\pi$
$912$	−39.5629	−	11.1005i	−1.31006	−	0.367573i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	2.03791	−	14.8070i	0.0673344	−	0.489238i
$917$		0			0
$918$		0			0
$919$		−	53.0674i		−	1.75053i	−0.483642	−	0.875266i	$-0.660686\pi$
$919$		−	53.0674i		−	1.75053i	0.483642	−	0.875266i	$-0.339314\pi$
$920$		0			0
$921$		0			0
$922$	−29.4281	+	25.6553i	−0.969163	+	0.844911i
$923$		0			0
$924$	0.0515883	+	0.00710017i	0.00169713	+	0.000233578i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−23.3216	+	11.0783i	−0.765567	+	0.363663i
$929$	−59.0552			−1.93754			−0.968769	−	0.247964i	$-0.920238\pi$
$929$	−59.0552			−1.93754			−0.968769	+	0.247964i	$0.920238\pi$
$930$		0			0
$931$			25.9316i			0.849875i
$932$	−2.15084	+	15.6275i	−0.0704531	+	0.511897i
$933$	71.6678			2.34630
$934$	18.2618	+	20.9473i	0.597544	+	0.685418i
$935$		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			$\pi$
$942$	29.8517	+	34.2416i	0.972621	+	1.11565i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			47.7913i			1.55301i	0.630111	+	0.776505i	$0.283009\pi$
$947$			47.7913i			1.55301i	−0.630111	+	0.776505i	$0.716991\pi$
$948$		0			0
$949$		0			0
$950$	17.2142	+	19.7457i	0.558502	+	0.640635i
$951$		−	20.6139i		−	0.668453i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−22.6219	−	3.11348i	−0.731644	−	0.100697i
$957$	−7.65699			−0.247515
$958$		0			0
$959$			0.0517341i			0.00167058i
$960$		0			0
$961$	−91.5412			−2.95294
$962$		0			0
$963$			91.7052i			2.95516i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			33.5380i			1.07851i	0.842143	+	0.539255i	$0.181294\pi$
$967$			33.5380i			1.07851i	−0.842143	+	0.539255i	$0.818706\pi$
$968$	−16.4892	−	25.1547i	−0.529982	−	0.808502i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	33.7727	+	4.64819i	1.08326	+	0.149091i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	35.3403	+	9.91569i	1.13122	+	0.317394i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		−	3.03934i		−	0.0971376i
$980$		0			0
$981$		0			0
$982$	−24.0192	−	27.5514i	−0.766483	−	0.879200i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	0.336119i		−	0.0106988i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	26.8688	+	56.5631i	0.853086	+	1.79588i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−56.0245			−1.77431			−0.887156	−	0.461469i	$-0.847322\pi$
$997$	−56.0245			−1.77431			−0.887156	+	0.461469i	$0.847322\pi$
$998$		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.667.17	✓	22
4.3	odd	2	inner	668.2.b.a.667.18	yes	22
167.166	odd	2	CM	668.2.b.a.667.17	✓	22
668.667	even	2	inner	668.2.b.a.667.18	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.17	✓	22	1.1	even	1	trivial
668.2.b.a.667.17	✓	22	167.166	odd	2	CM
668.2.b.a.667.18	yes	22	4.3	odd	2	inner
668.2.b.a.667.18	yes	22	668.667	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.06600 - 0.929330i) q^{2} -2.77291i q^{3} +(0.272692 - 1.98132i) q^{4} +(-2.57695 - 2.95591i) q^{6} -0.0155204i q^{7} +(-1.55061 - 2.36550i) q^{8} -4.68903 q^{9} +O(q^{10})\)	\(q+(1.06600 - 0.929330i) q^{2} -2.77291i q^{3} +(0.272692 - 1.98132i) q^{4} +(-2.57695 - 2.95591i) q^{6} -0.0155204i q^{7} +(-1.55061 - 2.36550i) q^{8} -4.68903 q^{9} -0.605002i q^{11} +(-5.49403 - 0.756151i) q^{12} +(-0.0144236 - 0.0165447i) q^{14} +(-3.85128 - 1.08058i) q^{16} +(-4.99849 + 4.35766i) q^{18} +3.70465i q^{19} -0.0430367 q^{21} +(-0.562247 - 0.644930i) q^{22} +(-6.55932 + 4.29971i) q^{24} +5.00000 q^{25} +4.68354i q^{27} +(-0.0307509 - 0.00423229i) q^{28} +4.56421 q^{29} -11.0698i q^{31} +(-5.10966 + 2.42721i) q^{32} -1.67762 q^{33} +(-1.27866 + 9.29049i) q^{36} +(3.44284 + 3.94914i) q^{38} +(-0.0458769 + 0.0399953i) q^{42} +(-1.19870 - 0.164979i) q^{44} +7.81007i q^{47} +(-2.99636 + 10.6792i) q^{48} +6.99976 q^{49} +(5.32998 - 4.64665i) q^{50} +(4.35255 + 4.99263i) q^{54} +(-0.0367135 + 0.0240662i) q^{56} +10.2727 q^{57} +(4.86542 - 4.24165i) q^{58} -9.17625 q^{61} +(-10.2875 - 11.8004i) q^{62} +0.0727757i q^{63} +(-3.19119 + 7.33596i) q^{64} +(-1.78833 + 1.55906i) q^{66} +(7.27088 + 11.0919i) q^{72} -13.8646i q^{75} +(7.34010 + 1.01023i) q^{76} -0.00938988 q^{77} -1.08007 q^{81} +(-0.0117358 + 0.0852696i) q^{84} -12.6561i q^{87} +(-1.43113 + 0.938125i) q^{88} +5.02368 q^{89} -30.6957 q^{93} +(7.25813 + 8.32549i) q^{94} +(6.73044 + 14.1686i) q^{96} +18.7260 q^{97} +(7.46171 - 6.50508i) q^{98} +2.83688i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 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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