LMFDB - Embedded newform 668.2.b.a.667.20

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			667.20
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.28818 + 0.583606i) q^{2} -1.66053i q^{3} +(1.31881 + 1.50358i) q^{4} +(0.969093 - 2.13906i) q^{6} +2.84773i q^{7} +(0.821364 + 2.70654i) q^{8} +0.242651 q^{9} +O(q^{10})$	\(q+(1.28818 + 0.583606i) q^{2} -1.66053i q^{3} +(1.31881 + 1.50358i) q^{4} +(0.969093 - 2.13906i) q^{6} +2.84773i q^{7} +(0.821364 + 2.70654i) q^{8} +0.242651 q^{9} +5.38837i q^{11} +(2.49673 - 2.18992i) q^{12} +(-1.66195 + 3.66839i) q^{14} +(-0.521489 + 3.96586i) q^{16} +(0.312577 + 0.141612i) q^{18} -5.77787i q^{19} +4.72874 q^{21} +(-3.14469 + 6.94119i) q^{22} +(4.49428 - 1.36390i) q^{24} +5.00000 q^{25} -5.38451i q^{27} +(-4.28179 + 3.75561i) q^{28} -1.43452 q^{29} -5.69724i q^{31} +(-2.98627 + 4.80439i) q^{32} +8.94754 q^{33} +(0.320010 + 0.364844i) q^{36} +(3.37200 - 7.44293i) q^{38} +(6.09146 + 2.75972i) q^{42} +(-8.10183 + 7.10623i) q^{44} -4.31870i q^{47} +(6.58542 + 0.865947i) q^{48} -1.10959 q^{49} +(6.44089 + 2.91803i) q^{50} +(3.14243 - 6.93621i) q^{54} +(-7.70751 + 2.33903i) q^{56} -9.59431 q^{57} +(-1.84791 - 0.837193i) q^{58} -11.2065 q^{61} +(3.32494 - 7.33906i) q^{62} +0.691004i q^{63} +(-6.65072 + 4.44611i) q^{64} +(11.5260 + 5.22184i) q^{66} +(0.199305 + 0.656744i) q^{72} -8.30263i q^{75} +(8.68748 - 7.61991i) q^{76} -15.3446 q^{77} -8.21317 q^{81} +(6.23630 + 7.11002i) q^{84} +2.38205i q^{87} +(-14.5838 + 4.42582i) q^{88} +17.2868 q^{89} -9.46042 q^{93} +(2.52042 - 5.56326i) q^{94} +(7.97782 + 4.95878i) q^{96} -16.8808 q^{97} +(-1.42934 - 0.647561i) q^{98} +1.30749i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.28818	+	0.583606i	0.910880	+	0.412672i
$2$	1.28818	+	0.583606i	0.910880	+	0.412672i
$3$		−	1.66053i		−	0.958706i	−0.877622	−	0.479353i	$-0.840871\pi$
$3$		−	1.66053i		−	0.958706i	0.877622	−	0.479353i	$-0.159129\pi$
$4$	1.31881	+	1.50358i	0.659404	+	0.751789i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	0.969093	−	2.13906i	0.395631	−	0.873266i
$7$			2.84773i			1.07634i	0.842836	+	0.538171i	$0.180885\pi$
$7$			2.84773i			1.07634i	−0.842836	+	0.538171i	$0.819115\pi$
$8$	0.821364	+	2.70654i	0.290396	+	0.956907i
$9$	0.242651			0.0808836
$10$		0			0
$11$			5.38837i			1.62466i	0.583201	+	0.812328i	$0.301800\pi$
$11$			5.38837i			1.62466i	−0.583201	+	0.812328i	$0.698200\pi$
$12$	2.49673	−	2.18992i	0.720744	−	0.632174i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−1.66195	+	3.66839i	−0.444176	+	0.980418i
$15$		0			0
$16$	−0.521489	+	3.96586i	−0.130372	+	0.991465i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.312577	+	0.141612i	0.0736752	+	0.0333784i
$19$		−	5.77787i		−	1.32554i	−0.748825	−	0.662768i	$-0.769382\pi$
$19$		−	5.77787i		−	1.32554i	0.748825	−	0.662768i	$-0.230618\pi$
$20$		0			0
$21$	4.72874			1.03190
$22$	−3.14469	+	6.94119i	−0.670449	+	1.47987i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	4.49428	−	1.36390i	0.917392	−	0.278404i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	5.38451i		−	1.03625i
$28$	−4.28179	+	3.75561i	−0.809182	+	0.709744i
$29$	−1.43452			−0.266383			−0.133192	−	0.991090i	$-0.542523\pi$
$29$	−1.43452			−0.266383			−0.133192	+	0.991090i	$0.542523\pi$
$30$		0			0
$31$		−	5.69724i		−	1.02325i	−0.859207	−	0.511627i	$-0.829043\pi$
$31$		−	5.69724i		−	1.02325i	0.859207	−	0.511627i	$-0.170957\pi$
$32$	−2.98627	+	4.80439i	−0.527903	+	0.849305i
$33$	8.94754			1.55757
$34$		0			0
$35$		0			0
$36$	0.320010	+	0.364844i	0.0533350	+	0.0608073i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	3.37200	−	7.44293i	0.547011	−	1.20740i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	6.09146	+	2.75972i	0.939932	+	0.425834i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−8.10183	+	7.10623i	−1.22140	+	1.07130i
$45$		0			0
$46$		0			0
$47$		−	4.31870i		−	0.629948i	−0.949100	−	0.314974i	$-0.898004\pi$
$47$		−	4.31870i		−	0.629948i	0.949100	−	0.314974i	$-0.101996\pi$
$48$	6.58542	+	0.865947i	0.950523	+	0.124989i
$49$	−1.10959			−0.158512
$50$	6.44089	+	2.91803i	0.910880	+	0.412672i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	3.14243	−	6.93621i	0.427631	−	0.943898i
$55$		0			0
$56$	−7.70751	+	2.33903i	−1.02996	+	0.312565i
$57$	−9.59431			−1.27080
$58$	−1.84791	−	0.837193i	−0.242643	−	0.109929i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−11.2065			−1.43484			−0.717421	−	0.696640i	$-0.754678\pi$
$61$	−11.2065			−1.43484			−0.717421	+	0.696640i	$0.754678\pi$
$62$	3.32494	−	7.33906i	0.422268	−	0.932062i
$63$			0.691004i			0.0870584i
$64$	−6.65072	+	4.44611i	−0.831340	+	0.555764i
$65$		0			0
$66$	11.5260	+	5.22184i	1.41876	+	0.642764i
$67$		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	$0.5\pi$
$72$	0.199305	+	0.656744i	0.0234883	+	0.0773980i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	8.30263i		−	0.958706i
$76$	8.68748	−	7.61991i	0.996522	−	0.874063i
$77$	−15.3446			−1.74869
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−8.21317			−0.912574
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	6.23630	+	7.11002i	0.680436	+	0.775767i
$85$		0			0
$86$		0			0
$87$			2.38205i			0.255383i
$88$	−14.5838	+	4.42582i	−1.55464	+	0.471794i
$89$	17.2868			1.83240			0.916200	−	0.400722i	$-0.131241\pi$
$89$	17.2868			1.83240			0.916200	+	0.400722i	$0.131241\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−9.46042			−0.981000
$94$	2.52042	−	5.56326i	0.259962	−	0.573807i
$95$		0			0
$96$	7.97782	+	4.95878i	0.814233	+	0.506104i
$97$	−16.8808			−1.71398			−0.856992	−	0.515329i	$-0.827670\pi$
$97$	−16.8808			−1.71398			−0.856992	+	0.515329i	$0.827670\pi$
$98$	−1.42934	−	0.647561i	−0.144386	−	0.0654135i
$99$			1.30749i			0.131408i
$100$	6.59404	+	7.51789i	0.659404	+	0.751789i
$101$		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
$107$		−	12.8055i		−	1.23796i	−0.785407	−	0.618979i	$-0.787547\pi$
$107$		−	12.8055i		−	1.23796i	0.785407	−	0.618979i	$-0.212453\pi$
$108$	8.09602	−	7.10113i	0.779040	−	0.683307i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−11.2937	−	1.48506i	−1.06716	−	0.140325i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−12.3592	−	5.59930i	−1.15754	−	0.524422i
$115$		0			0
$116$	−1.89185	−	2.15691i	−0.175654	−	0.200264i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−18.0346			−1.63951
$122$	−14.4359	−	6.54017i	−1.30697	−	0.592119i
$123$		0			0
$124$	8.56624	−	7.51357i	0.769271	−	0.674738i
$125$		0			0
$126$	−0.403274	+	0.890137i	−0.0359265	+	0.0792997i
$127$		−	4.13650i		−	0.367055i	−0.983015	−	0.183528i	$-0.941248\pi$
$127$		−	4.13650i		−	0.367055i	0.983015	−	0.183528i	$-0.0587516\pi$
$128$	−11.1621	+	1.84598i	−0.986599	+	0.163163i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	11.8001	+	13.4533i	1.02707	+	1.17096i
$133$	16.4538			1.42673
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.6942			1.68259			0.841295	−	0.540576i	$-0.181794\pi$
$137$	19.6942			1.68259			0.841295	+	0.540576i	$0.181794\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−7.17132			−0.603934
$142$		0			0
$143$		0			0
$144$	−0.126540	+	0.962319i	−0.0105450	+	0.0801932i
$145$		0			0
$146$		0			0
$147$			1.84250i			0.151967i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	4.84547	−	10.6953i	0.395631	−	0.873266i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	15.6380	−	4.74574i	1.26841	−	0.384930i
$153$		0			0
$154$	−19.7666	−	8.95523i	−1.59284	−	0.721633i
$155$		0			0
$156$		0			0
$157$	−25.0259			−1.99728			−0.998641	−	0.0521162i	$-0.983403\pi$
$157$	−25.0259			−1.99728			−0.998641	+	0.0521162i	$0.983403\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−10.5800	−	4.79325i	−0.831246	−	0.376594i
$163$		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
$168$	3.88402	+	12.7985i	0.299658	+	0.987427i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	1.40200i		−	0.107214i
$172$		0			0
$173$	−11.9572			−0.909088			−0.454544	−	0.890724i	$-0.650198\pi$
$173$	−11.9572			−0.909088			−0.454544	+	0.890724i	$0.650198\pi$
$174$	−1.39018	+	3.06851i	−0.105389	+	0.232623i
$175$			14.2387i			1.07634i
$176$	−21.3695	−	2.80998i	−1.61079	−	0.211810i
$177$		0			0
$178$	22.2685	+	10.0887i	1.66910	+	0.756179i
$179$		−	25.9710i		−	1.94117i	−0.240767	−	0.970583i	$-0.577399\pi$
$179$		−	25.9710i		−	1.94117i	0.240767	−	0.970583i	$-0.422601\pi$
$180$		0			0
$181$	−3.32826			−0.247388			−0.123694	−	0.992320i	$-0.539474\pi$
$181$	−3.32826			−0.247388			−0.123694	+	0.992320i	$0.539474\pi$
$182$		0			0
$183$			18.6087i			1.37559i
$184$		0			0
$185$		0			0
$186$	−12.1867	−	5.52116i	−0.893573	−	0.404831i
$187$		0			0
$188$	6.49350	−	5.69554i	0.473588	−	0.415390i
$189$	15.3336			1.11536
$190$		0			0
$191$			26.4294i			1.91237i	0.292770	+	0.956183i	$0.405423\pi$
$191$			26.4294i			1.91237i	−0.292770	+	0.956183i	$0.594577\pi$
$192$	7.38289	+	11.0437i	0.532814	+	0.797011i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−21.7455	−	9.85173i	−1.56123	−	0.707313i
$195$		0			0
$196$	−1.46333	−	1.66835i	−0.104524	−	0.119168i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.763060	+	1.68428i	−0.0542283	+	0.119697i
$199$		−	28.1937i		−	1.99860i	−0.0373967	−	0.999300i	$-0.511907\pi$
$199$		−	28.1937i		−	1.99860i	0.0373967	−	0.999300i	$-0.488093\pi$
$200$	4.10682	+	13.5327i	0.290396	+	0.956907i
$201$		0			0
$202$		0			0
$203$		−	4.08512i		−	0.286719i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	31.1333			2.15354
$210$		0			0
$211$		−	19.7038i		−	1.35646i	−0.734848	−	0.678232i	$-0.762746\pi$
$211$		−	19.7038i		−	1.35646i	0.734848	−	0.678232i	$-0.237254\pi$
$212$		0			0
$213$		0			0
$214$	7.47339	−	16.4958i	0.510870	−	1.12763i
$215$		0			0
$216$	14.5734	−	4.42264i	0.991594	−	0.300923i
$217$	16.2242			1.10137
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	26.7453i		−	1.79100i	−0.445064	−	0.895499i	$-0.646819\pi$
$223$		−	26.7453i		−	1.79100i	0.445064	−	0.895499i	$-0.353181\pi$
$224$	−13.6816	−	8.50410i	−0.914142	−	0.568204i
$225$	1.21325			0.0808836
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−12.6531	−	14.4258i	−0.837969	−	0.955371i
$229$	22.1430			1.46325			0.731626	−	0.681706i	$-0.238762\pi$
$229$	22.1430			1.46325			0.731626	+	0.681706i	$0.238762\pi$
$230$		0			0
$231$			25.4802i			1.67647i
$232$	−1.17826	−	3.88258i	−0.0773566	−	0.254904i
$233$	23.5505			1.54284			0.771421	−	0.636325i	$-0.219546\pi$
$233$	23.5505			1.54284			0.771421	+	0.636325i	$0.219546\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			30.0664i			1.94483i	0.233256	+	0.972415i	$0.425062\pi$
$239$			30.0664i			1.94483i	−0.233256	+	0.972415i	$0.574938\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−23.2317	−	10.5251i	−1.49339	−	0.676578i
$243$		−	2.51534i		−	0.161359i
$244$	−14.7792	−	16.8498i	−0.946141	−	1.07870i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	15.4198	−	4.67951i	0.979159	−	0.297149i
$249$		0			0
$250$		0			0
$251$			29.8919i			1.88676i	0.331711	+	0.943381i	$0.392374\pi$
$251$			29.8919i			1.88676i	−0.331711	+	0.943381i	$0.607626\pi$
$252$	−1.03898	+	0.911302i	−0.0654495	+	0.0574067i
$253$		0			0
$254$	2.41409	−	5.32855i	0.151473	−	0.334343i
$255$		0			0
$256$	−15.4561	−	4.13631i	−0.966006	−	0.258519i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.348087			−0.0215460
$262$		0			0
$263$		−	20.1032i		−	1.23962i	−0.784753	−	0.619808i	$-0.787210\pi$
$263$		−	20.1032i		−	1.23962i	0.784753	−	0.619808i	$-0.212790\pi$
$264$	7.34919	+	24.2169i	0.452311	+	1.49045i
$265$		0			0
$266$	21.1955	+	9.60256i	1.29958	+	0.588771i
$267$		−	28.7052i		−	1.75673i
$268$		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
$274$	25.3697	+	11.4937i	1.53264	+	0.694358i
$275$			26.9419i			1.62466i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		−	1.38244i		−	0.0827645i
$280$		0			0
$281$	−19.1953			−1.14509			−0.572546	−	0.819872i	$-0.694044\pi$
$281$	−19.1953			−1.14509			−0.572546	+	0.819872i	$0.694044\pi$
$282$	−9.23795	−	4.18523i	−0.550112	−	0.249227i
$283$			7.89212i			0.469138i	0.972100	+	0.234569i	$0.0753678\pi$
$283$			7.89212i			0.469138i	−0.972100	+	0.234569i	$0.924632\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.724621	+	1.16579i	−0.0426987	+	0.0686948i
$289$	17.0000			1.00000
$290$		0			0
$291$			28.0310i			1.64321i
$292$		0			0
$293$	−7.36581			−0.430315			−0.215158	−	0.976579i	$-0.569027\pi$
$293$	−7.36581			−0.430315			−0.215158	+	0.976579i	$0.569027\pi$
$294$	−1.07529	+	2.37346i	−0.0627123	+	0.138423i
$295$		0			0
$296$		0			0
$297$	29.0137			1.68355
$298$		0			0
$299$		0			0
$300$	12.4837	−	10.9496i	0.720744	−	0.632174i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	22.9142	+	3.01310i	1.31422	+	0.172813i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−20.2367	−	23.0719i	−1.15309	−	1.31464i
$309$		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			$\pi$
$314$	−32.2378	−	14.6052i	−1.81928	−	0.824222i
$315$		0			0
$316$		0			0
$317$	−35.5279			−1.99544			−0.997722	−	0.0674648i	$-0.978509\pi$
$317$	−35.5279			−1.99544			−0.997722	+	0.0674648i	$0.978509\pi$
$318$		0			0
$319$		−	7.72972i		−	0.432781i
$320$		0			0
$321$	−21.2639			−1.18684
$322$		0			0
$323$		0			0
$324$	−10.8316	−	12.3491i	−0.601755	−	0.686063i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.2985			0.678039
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	−7.54185	+	16.6469i	−0.412672	+	0.910880i
$335$		0			0
$336$	−2.46599	+	18.7535i	−0.134531	+	1.02309i
$337$	−35.8404			−1.95235			−0.976175	−	0.216985i	$-0.930378\pi$
$337$	−35.8404			−1.95235			−0.976175	+	0.216985i	$0.930378\pi$
$338$	16.7463	+	7.58688i	0.910880	+	0.412672i
$339$		0			0
$340$		0			0
$341$	30.6989			1.66244
$342$	0.818218	−	1.80603i	0.0442442	−	0.0976591i
$343$			16.7743i			0.905729i
$344$		0			0
$345$		0			0
$346$	−15.4030	−	6.97828i	−0.828070	−	0.375155i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−3.58160	+	3.14147i	−0.191994	+	0.168401i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−8.30977	+	18.3419i	−0.444176	+	0.980418i
$351$		0			0
$352$	−25.8879	−	16.0911i	−1.37983	−	0.857661i
$353$	−25.3247			−1.34790			−0.673948	−	0.738778i	$-0.735403\pi$
$353$	−25.3247			−1.34790			−0.673948	+	0.738778i	$0.735403\pi$
$354$		0			0
$355$		0			0
$356$	22.7980	+	25.9921i	1.20829	+	1.37758i
$357$		0			0
$358$	15.1568	−	33.4553i	0.801064	−	1.76817i
$359$			35.7671i			1.88772i	0.330351	+	0.943858i	$0.392833\pi$
$359$			35.7671i			1.88772i	−0.330351	+	0.943858i	$0.607167\pi$
$360$		0			0
$361$	−14.3838			−0.757043
$362$	−4.28740	−	1.94239i	−0.225341	−	0.102090i
$363$			29.9469i			1.57180i
$364$		0			0
$365$		0			0
$366$	−10.8601	+	23.9713i	−0.567668	+	1.25300i
$367$			36.0676i			1.88271i	0.337413	+	0.941357i	$0.390448\pi$
$367$			36.0676i			1.88271i	−0.337413	+	0.941357i	$0.609552\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−12.4765	−	14.2245i	−0.646876	−	0.737505i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	11.6887	−	3.54723i	0.602801	−	0.182934i
$377$		0			0
$378$	19.7525	+	8.94881i	1.01596	+	0.460277i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−6.86877			−0.351898
$382$	−15.4244	+	34.0458i	−0.789179	+	1.74194i
$383$			34.1543i			1.74520i	0.488434	+	0.872601i	$0.337568\pi$
$383$			34.1543i			1.74520i	−0.488434	+	0.872601i	$0.662432\pi$
$384$	3.06531	+	18.5350i	0.156426	+	0.945858i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−22.2625	−	25.3816i	−1.13021	−	1.28855i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−0.911374	−	3.00314i	−0.0460313	−	0.151681i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.96592	+	1.72433i	−0.0987910	+	0.0866509i
$397$	25.2756			1.26855			0.634273	−	0.773109i	$-0.281300\pi$
$397$	25.2756			1.26855			0.634273	+	0.773109i	$0.281300\pi$
$398$	16.4540	−	36.3186i	0.824766	−	1.82049i
$399$		−	27.3220i		−	1.36781i
$400$	−2.60745	+	19.8293i	−0.130372	+	0.991465i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	2.38410	−	5.26237i	0.118321	−	0.261167i
$407$		0			0
$408$		0			0
$409$	−37.9817			−1.87808			−0.939038	−	0.343814i	$-0.888281\pi$
$409$	−37.9817			−1.87808			−0.939038	+	0.343814i	$0.888281\pi$
$410$		0			0
$411$		−	32.7028i		−	1.61311i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	40.1053	+	18.1696i	1.96161	+	0.888704i
$419$			12.4229i			0.606897i	0.952848	+	0.303449i	$0.0981381\pi$
$419$			12.4229i			0.606897i	−0.952848	+	0.303449i	$0.901862\pi$
$420$		0			0
$421$	39.9825			1.94863			0.974314	−	0.225195i	$-0.0723020\pi$
$421$	39.9825			1.94863			0.974314	+	0.225195i	$0.0723020\pi$
$422$	11.4992	−	25.3820i	0.559774	−	1.23558i
$423$		−	1.04794i		−	0.0509524i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	31.9131i		−	1.54438i
$428$	19.2541	−	16.8881i	0.930683	−	0.816315i
$429$		0			0
$430$		0			0
$431$		−	40.0295i		−	1.92815i	−0.265623	−	0.964077i	$-0.585578\pi$
$431$		−	40.0295i		−	1.92815i	0.265623	−	0.964077i	$-0.414422\pi$
$432$	21.3542	+	2.80796i	1.02740	+	0.135098i
$433$	16.3845			0.787388			0.393694	−	0.919242i	$-0.371197\pi$
$433$	16.3845			0.787388			0.393694	+	0.919242i	$0.371197\pi$
$434$	20.8997	+	9.46855i	1.00322	+	0.454505i
$435$		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
$441$	−0.269242			−0.0128210
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	15.6087	−	34.4527i	0.739094	−	1.63138i
$447$		0			0
$448$	−12.6613	−	18.9395i	−0.598192	−	0.894806i
$449$	42.3257			1.99747			0.998736	−	0.0502593i	$-0.0160048\pi$
$449$	42.3257			1.99747			0.998736	+	0.0502593i	$0.0160048\pi$
$450$	1.56289	+	0.708062i	0.0736752	+	0.0333784i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−7.88042	−	25.9674i	−0.369035	−	1.21603i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	28.5242	+	12.9228i	1.33285	+	0.603843i
$459$		0			0
$460$		0			0
$461$	35.7548			1.66527			0.832633	−	0.553825i	$-0.186833\pi$
$461$	35.7548			1.66527			0.832633	+	0.553825i	$0.186833\pi$
$462$	−14.8704	+	32.8231i	−0.691833	+	1.52707i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	0.748086	−	5.68910i	0.0347290	−	0.264110i
$465$		0			0
$466$	30.3372	+	13.7442i	1.40534	+	0.636688i
$467$		−	4.28086i		−	0.198095i	−0.995083	−	0.0990473i	$-0.968420\pi$
$467$		−	4.28086i		−	0.198095i	0.995083	−	0.0990473i	$-0.0315795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			41.5561i			1.91481i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	28.8894i		−	1.32554i
$476$		0			0
$477$		0			0
$478$	−17.5469	+	38.7308i	−0.802577	+	1.77151i
$479$		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
$484$	−23.7841	−	27.1164i	−1.08110	−	1.23256i
$485$		0			0
$486$	1.46797	−	3.24021i	0.0665883	−	0.146979i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−9.20460	−	30.3308i	−0.416673	−	1.37301i
$489$		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
$496$	22.5945	+	2.97105i	1.01452	+	0.133404i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	21.4587			0.958706
$502$	−17.4451	+	38.5062i	−0.778613	+	1.71861i
$503$			10.1400i			0.452122i	0.974113	+	0.226061i	$0.0725848\pi$
$503$			10.1400i			0.452122i	−0.974113	+	0.226061i	$0.927415\pi$
$504$	−1.87023	+	0.567566i	−0.0833067	+	0.0252814i
$505$		0			0
$506$		0			0
$507$		−	21.5868i		−	0.958706i
$508$	6.21955	−	5.45525i	0.275948	−	0.242038i
$509$	3.51690			0.155884			0.0779418	−	0.996958i	$-0.475165\pi$
$509$	3.51690			0.155884			0.0779418	+	0.996958i	$0.475165\pi$
$510$		0			0
$511$		0			0
$512$	−17.4962	−	14.3486i	−0.773232	−	0.634123i
$513$	−31.1110			−1.37358
$514$		0			0
$515$		0			0
$516$		0			0
$517$	23.2708			1.02345
$518$		0			0
$519$			19.8552i			0.871547i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−0.448398	−	0.203145i	−0.0196258	−	0.00889143i
$523$		−	42.4928i		−	1.85808i	−0.369982	−	0.929039i	$-0.620636\pi$
$523$		−	42.4928i		−	1.85808i	0.369982	−	0.929039i	$-0.379364\pi$
$524$		0			0
$525$	23.6437			1.03190
$526$	11.7323	−	25.8965i	0.511554	−	1.12914i
$527$		0			0
$528$	−4.66605	+	35.4847i	−0.203064	+	1.54427i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	21.6995	+	24.7396i	0.940791	+	1.07260i
$533$		0			0
$534$	16.7525	−	36.9775i	0.724953	−	1.60017i
$535$		0			0
$536$		0			0
$537$	−43.1256			−1.86101
$538$		0			0
$539$		−	5.97886i		−	0.257528i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$			5.52667i			0.237172i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	25.9729	+	29.6118i	1.10951	+	1.26495i
$549$	−2.71926			−0.116055
$550$	−15.7234	+	34.7059i	−0.670449	+	1.47987i
$551$			8.28846i			0.353100i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−42.9053			−1.81796			−0.908979	−	0.416843i	$-0.863137\pi$
$557$	−42.9053			−1.81796			−0.908979	+	0.416843i	$0.863137\pi$
$558$	0.806800	−	1.78083i	0.0341546	−	0.0753885i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−24.7269	−	11.2025i	−1.04304	−	0.472547i
$563$			34.6084i			1.45857i	0.684209	+	0.729286i	$0.260148\pi$
$563$			34.6084i			1.45857i	−0.684209	+	0.729286i	$0.739852\pi$
$564$	−9.45760	−	10.7826i	−0.398237	−	0.454031i
$565$		0			0
$566$	−4.60589	+	10.1665i	−0.193600	+	0.427328i
$567$		−	23.3889i		−	0.982242i
$568$		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
$573$	43.8868			1.83340
$574$		0			0
$575$		0			0
$576$	−1.61380	+	1.07885i	−0.0672418	+	0.0449522i
$577$	−32.3805			−1.34802			−0.674009	−	0.738723i	$-0.735429\pi$
$577$	−32.3805			−1.34802			−0.674009	+	0.738723i	$0.735429\pi$
$578$	21.8990	+	9.92130i	0.910880	+	0.412672i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−16.3591	+	36.1089i	−0.678105	+	1.49676i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−9.48848	−	4.29873i	−0.391966	−	0.177579i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−2.77034	+	2.42990i	−0.114247	+	0.100207i
$589$	−32.9179			−1.35636
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	37.3749	+	16.9326i	1.53351	+	0.694753i
$595$		0			0
$596$		0			0
$597$	−46.8165			−1.91607
$598$		0			0
$599$		−	17.0679i		−	0.697375i	−0.937239	−	0.348687i	$-0.886628\pi$
$599$		−	17.0679i		−	0.697375i	0.937239	−	0.348687i	$-0.113372\pi$
$600$	22.4714	−	6.81949i	0.917392	−	0.278404i
$601$	−48.9022			−1.99476			−0.997382	−	0.0723176i	$-0.976960\pi$
$601$	−48.9022			−1.99476			−0.997382	+	0.0723176i	$0.976960\pi$
$602$		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	$0.5\pi$
$608$	27.7592	+	17.2543i	1.12578	+	0.699754i
$609$	−6.78346			−0.274880
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.46421			−0.0591391			−0.0295695	−	0.999563i	$-0.509414\pi$
$613$	−1.46421			−0.0591391			−0.0295695	+	0.999563i	$0.509414\pi$
$614$		0			0
$615$		0			0
$616$	−12.6035	−	41.5309i	−0.507811	−	1.67333i
$617$	33.9318			1.36604			0.683021	−	0.730399i	$-0.260666\pi$
$617$	33.9318			1.36604			0.683021	+	0.730399i	$0.260666\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−15.0837	+	33.2939i	−0.604801	+	1.33496i
$623$			49.2283i			1.97229i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	51.6977i		−	2.06461i
$628$	−33.0043	−	37.6283i	−1.31702	−	1.50153i
$629$		0			0
$630$		0			0
$631$		−	6.81898i		−	0.271459i	−0.990746	−	0.135730i	$-0.956662\pi$
$631$		−	6.81898i		−	0.271459i	0.990746	−	0.135730i	$-0.0433379\pi$
$632$		0			0
$633$	−32.7187			−1.30045
$634$	−45.7662	−	20.7343i	−1.81761	−	0.823463i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	4.51111	−	9.95725i	0.178596	−	0.394211i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−27.3918	−	12.4098i	−1.08107	−	0.489774i
$643$		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	$0.5\pi$
$648$	−6.74600	−	22.2293i	−0.265008	−	0.873248i
$649$		0			0
$650$		0			0
$651$		−	26.9408i		−	1.05589i
$652$		0			0
$653$	−27.9767			−1.09481			−0.547406	−	0.836867i	$-0.684385\pi$
$653$	−27.9767			−1.09481			−0.547406	+	0.836867i	$0.684385\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	15.8427	+	7.17749i	0.617612	+	0.279808i
$659$		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
$668$	−19.4305	+	17.0428i	−0.751789	+	0.659404i
$669$	−44.4113			−1.71704
$670$		0			0
$671$		−	60.3847i		−	2.33112i
$672$	−14.1213	+	22.7187i	−0.544741	+	0.876393i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−46.1688	−	20.9167i	−1.77836	−	0.805680i
$675$		−	26.9225i		−	1.03625i
$676$	17.1445	+	19.5465i	0.659404	+	0.751789i
$677$	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
$679$		−	48.0720i		−	1.84483i
$680$		0			0
$681$		0			0
$682$	39.5456	+	17.9160i	1.51428	+	0.686041i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	2.10802	−	1.84898i	0.0806023	−	0.0706974i
$685$		0			0
$686$	−9.78960	+	21.6083i	−0.373769	+	0.825010i
$687$		−	36.7691i		−	1.40283i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−15.7692	−	17.9785i	−0.599456	−	0.683442i
$693$	−3.72339			−0.141440
$694$		0			0
$695$		0			0
$696$	−6.44713	+	1.95653i	−0.244378	+	0.0741622i
$697$		0			0
$698$		0			0
$699$		−	39.1062i		−	1.47913i
$700$	−21.4089	+	18.7781i	−0.809182	+	0.709744i
$701$	37.4363			1.41395			0.706975	−	0.707239i	$-0.250060\pi$
$701$	37.4363			1.41395			0.706975	+	0.707239i	$0.250060\pi$
$702$		0			0
$703$		0			0
$704$	−23.9573	−	35.8366i	−0.902925	−	1.35064i
$705$		0			0
$706$	−32.6227	−	14.7796i	−1.22777	−	0.556239i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	14.1988	+	46.7875i	0.532122	+	1.75343i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	39.0494	−	34.2508i	1.45935	−	1.28001i
$717$	49.9260			1.86452
$718$	−20.8739	+	46.0744i	−0.779007	+	1.71948i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−18.5289	−	8.39448i	−0.689575	−	0.312410i
$723$		0			0
$724$	−4.38934	−	5.00430i	−0.163129	−	0.185983i
$725$	−7.17259			−0.266383
$726$	−17.4772	+	38.5769i	−0.648639	+	1.43172i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−28.8163			−1.06727
$730$		0			0
$731$		0			0
$732$	−27.9795	+	24.5412i	−1.03415	+	0.907071i
$733$	−42.2401			−1.56017			−0.780086	−	0.625672i	$-0.784825\pi$
$733$	−42.2401			−1.56017			−0.780086	+	0.625672i	$0.784825\pi$
$734$	−21.0493	+	46.4615i	−0.776943	+	1.71493i
$735$		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	$0.842776\pi$
$744$	−7.77045	−	25.6050i	−0.284879	−	0.938725i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.4668			1.33247
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	17.1274	+	2.25216i	0.624571	+	0.0821277i
$753$	49.6364			1.80885
$754$		0			0
$755$		0			0
$756$	20.2221	+	23.0553i	0.735472	+	0.838514i
$757$	−48.5322			−1.76393			−0.881966	−	0.471313i	$-0.843780\pi$
$757$	−48.5322			−1.76393			−0.881966	+	0.471313i	$0.843780\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	53.7689			1.94912			0.974560	−	0.224125i	$-0.0719523\pi$
$761$	53.7689			1.94912			0.974560	+	0.224125i	$0.0719523\pi$
$762$	−8.84820	−	4.00865i	−0.320537	−	0.145218i
$763$		0			0
$764$	−39.7387	+	34.8553i	−1.43769	+	1.26102i
$765$		0			0
$766$	−19.9326	+	43.9968i	−0.720195	+	1.58967i
$767$		0			0
$768$	−6.86845	+	25.6653i	−0.247844	+	0.926115i
$769$		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
$775$		−	28.4862i		−	1.02325i
$776$	−13.8653	−	45.6885i	−0.497734	−	1.64012i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			7.72417i			0.276039i
$784$	0.578637	−	4.40046i	0.0206656	−	0.157159i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	−33.3819			−1.18843
$790$		0			0
$791$		0			0
$792$	−3.53878	+	1.07393i	−0.125745	+	0.0381603i
$793$		0			0
$794$	32.5595	+	14.7510i	1.15549	+	0.523493i
$795$		0			0
$796$	42.3915	−	37.1821i	1.50253	−	1.31789i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	15.9453	−	35.1957i	0.564458	−	1.24591i
$799$		0			0
$800$	−14.9314	+	24.0220i	−0.527903	+	0.849305i
$801$	4.19466			0.148211
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.2539			−0.395667			−0.197834	−	0.980236i	$-0.563391\pi$
$809$	−11.2539			−0.395667			−0.197834	+	0.980236i	$0.563391\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	6.14230	−	5.38750i	0.215552	−	0.189064i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−48.9272	−	22.1664i	−1.71070	−	0.775029i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	19.0855	−	42.1270i	0.665684	−	1.46935i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	44.7377			1.55757
$826$		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
$836$	41.0589	+	46.8114i	1.42005	+	1.61901i
$837$	−30.6768			−1.06035
$838$	−7.25006	+	16.0029i	−0.250449	+	0.552811i
$839$			49.9927i			1.72594i	0.505256	+	0.862969i	$0.331398\pi$
$839$			49.9927i			1.72594i	−0.505256	+	0.862969i	$0.668602\pi$
$840$		0			0
$841$	−26.9422			−0.929040
$842$	51.5046	+	23.3340i	1.77497	+	0.804143i
$843$			31.8742i			1.09781i
$844$	29.6262	−	25.9855i	1.01977	−	0.894458i
$845$		0			0
$846$	0.611582	−	1.34993i	0.0210266	−	0.0464115i
$847$		−	51.3576i		−	1.76467i
$848$		0			0
$849$	13.1051			0.449765
$850$		0			0
$851$		0			0
$852$		0			0
$853$	40.0919			1.37272			0.686360	−	0.727262i	$-0.259207\pi$
$853$	40.0919			1.37272			0.686360	+	0.727262i	$0.259207\pi$
$854$	18.6246	−	41.1097i	0.637322	−	1.40675i
$855$		0			0
$856$	34.6587	−	10.5180i	1.18461	−	0.359498i
$857$	3.41387			0.116616			0.0583078	−	0.998299i	$-0.481430\pi$
$857$	3.41387			0.116616			0.0583078	+	0.998299i	$0.481430\pi$
$858$		0			0
$859$			49.8075i			1.69941i	0.527257	+	0.849706i	$0.323220\pi$
$859$			49.8075i			1.69941i	−0.527257	+	0.849706i	$0.676780\pi$
$860$		0			0
$861$		0			0
$862$	23.3615	−	51.5652i	0.795695	−	1.75632i
$863$		−	1.63275i		−	0.0555796i	−0.999614	−	0.0277898i	$-0.991153\pi$
$863$		−	1.63275i		−	0.0555796i	0.999614	−	0.0277898i	$-0.00884691\pi$
$864$	25.8693	+	16.0796i	0.880091	+	0.547039i
$865$		0			0
$866$	21.1061	+	9.56208i	0.717216	+	0.324933i
$867$		−	28.2290i		−	0.958706i
$868$	21.3966	+	24.3944i	0.726249	+	0.827999i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−4.09614			−0.138633
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−16.1276			−0.544592			−0.272296	−	0.962214i	$-0.587783\pi$
$877$	−16.1276			−0.544592			−0.272296	+	0.962214i	$0.587783\pi$
$878$		0			0
$879$			12.2311i			0.412546i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.346831	−	0.157131i	−0.0116784	−	0.00529088i
$883$		−	44.1121i		−	1.48449i	−0.670129	−	0.742245i	$-0.733761\pi$
$883$		−	44.1121i		−	1.48449i	0.670129	−	0.742245i	$-0.266239\pi$
$884$		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
$889$	11.7796			0.395077
$890$		0			0
$891$		−	44.2556i		−	1.48262i
$892$	40.2136	−	35.2719i	1.34645	−	1.18099i
$893$	−24.9529			−0.835018
$894$		0			0
$895$		0			0
$896$	−5.25687	−	31.7867i	−0.175620	−	1.06192i
$897$		0			0
$898$	54.5230	+	24.7015i	1.81946	+	0.824300i
$899$			8.17279i			0.272578i
$900$	1.60005	+	1.82422i	0.0533350	+	0.0608073i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			48.5090i			1.61072i	0.592789	+	0.805358i	$0.298027\pi$
$907$			48.5090i			1.61072i	−0.592789	+	0.805358i	$0.701973\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	30.3485i		−	1.00549i	−0.864434	−	0.502746i	$-0.832323\pi$
$911$		−	30.3485i		−	1.00549i	0.864434	−	0.502746i	$-0.167677\pi$
$912$	5.00333	−	38.0497i	0.165677	−	1.25995i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	29.2024	+	33.2937i	0.964874	+	1.10006i
$917$		0			0
$918$		0			0
$919$			59.1791i			1.95214i	0.217460	+	0.976069i	$0.430223\pi$
$919$			59.1791i			1.95214i	−0.217460	+	0.976069i	$0.569777\pi$
$920$		0			0
$921$		0			0
$922$	46.0585	+	20.8667i	1.51686	+	0.687208i
$923$		0			0
$924$	−38.3115	+	33.6035i	−1.26035	+	1.10547i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	4.28386	−	6.89199i	0.140625	−	0.226241i
$929$	−38.2821			−1.25600			−0.627998	−	0.778215i	$-0.716125\pi$
$929$	−38.2821			−1.25600			−0.627998	+	0.778215i	$0.716125\pi$
$930$		0			0
$931$			6.41104i			0.210113i
$932$	31.0586	+	35.4100i	1.01736	+	1.15989i
$933$	42.9175			1.40505
$934$	2.49834	−	5.51452i	0.0817481	−	0.180440i
$935$		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			$\pi$
$942$	−24.2524	+	53.5317i	−0.790186	+	1.74416i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	45.1880i		−	1.46841i	−0.678927	−	0.734206i	$-0.737555\pi$
$947$		−	45.1880i		−	1.46841i	0.678927	−	0.734206i	$-0.262445\pi$
$948$		0			0
$949$		0			0
$950$	16.8600	−	37.2147i	0.547011	−	1.20740i
$951$			58.9950i			1.91304i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−45.2071	+	39.6518i	−1.46210	+	1.28243i
$957$	−12.8354			−0.414910
$958$		0			0
$959$			56.0839i			1.81104i
$960$		0			0
$961$	−1.45856			−0.0470504
$962$		0			0
$963$		−	3.10727i		−	0.100130i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	33.7103i		−	1.08405i	−0.840362	−	0.542025i	$-0.817658\pi$
$967$		−	33.7103i		−	1.08405i	0.840362	−	0.542025i	$-0.182342\pi$
$968$	−14.8129	−	48.8113i	−0.476106	−	1.56885i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	3.78201	−	3.31725i	0.121308	−	0.106401i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	5.84406	−	44.4433i	0.187064	−	1.42260i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$			93.1478i			2.97702i
$980$		0			0
$981$		0			0
$982$	15.0837	−	33.2939i	0.481340	−	1.06245i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	20.4220i		−	0.650040i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	27.3718	+	17.0135i	0.869055	+	0.540179i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−31.3754			−0.993668			−0.496834	−	0.867845i	$-0.665504\pi$
$997$	−31.3754			−0.993668			−0.496834	+	0.867845i	$0.665504\pi$
$998$		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.667.20	yes	22
4.3	odd	2	inner	668.2.b.a.667.19	✓	22
167.166	odd	2	CM	668.2.b.a.667.20	yes	22
668.667	even	2	inner	668.2.b.a.667.19	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.19	✓	22	4.3	odd	2	inner
668.2.b.a.667.19	✓	22	668.667	even	2	inner
668.2.b.a.667.20	yes	22	1.1	even	1	trivial
668.2.b.a.667.20	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)

\(f(q)\)	\(=\)	\(q+(1.28818 + 0.583606i) q^{2} -1.66053i q^{3} +(1.31881 + 1.50358i) q^{4} +(0.969093 - 2.13906i) q^{6} +2.84773i q^{7} +(0.821364 + 2.70654i) q^{8} +0.242651 q^{9} +O(q^{10})\)	\(q+(1.28818 + 0.583606i) q^{2} -1.66053i q^{3} +(1.31881 + 1.50358i) q^{4} +(0.969093 - 2.13906i) q^{6} +2.84773i q^{7} +(0.821364 + 2.70654i) q^{8} +0.242651 q^{9} +5.38837i q^{11} +(2.49673 - 2.18992i) q^{12} +(-1.66195 + 3.66839i) q^{14} +(-0.521489 + 3.96586i) q^{16} +(0.312577 + 0.141612i) q^{18} -5.77787i q^{19} +4.72874 q^{21} +(-3.14469 + 6.94119i) q^{22} +(4.49428 - 1.36390i) q^{24} +5.00000 q^{25} -5.38451i q^{27} +(-4.28179 + 3.75561i) q^{28} -1.43452 q^{29} -5.69724i q^{31} +(-2.98627 + 4.80439i) q^{32} +8.94754 q^{33} +(0.320010 + 0.364844i) q^{36} +(3.37200 - 7.44293i) q^{38} +(6.09146 + 2.75972i) q^{42} +(-8.10183 + 7.10623i) q^{44} -4.31870i q^{47} +(6.58542 + 0.865947i) q^{48} -1.10959 q^{49} +(6.44089 + 2.91803i) q^{50} +(3.14243 - 6.93621i) q^{54} +(-7.70751 + 2.33903i) q^{56} -9.59431 q^{57} +(-1.84791 - 0.837193i) q^{58} -11.2065 q^{61} +(3.32494 - 7.33906i) q^{62} +0.691004i q^{63} +(-6.65072 + 4.44611i) q^{64} +(11.5260 + 5.22184i) q^{66} +(0.199305 + 0.656744i) q^{72} -8.30263i q^{75} +(8.68748 - 7.61991i) q^{76} -15.3446 q^{77} -8.21317 q^{81} +(6.23630 + 7.11002i) q^{84} +2.38205i q^{87} +(-14.5838 + 4.42582i) q^{88} +17.2868 q^{89} -9.46042 q^{93} +(2.52042 - 5.56326i) q^{94} +(7.97782 + 4.95878i) q^{96} -16.8808 q^{97} +(-1.42934 - 0.647561i) q^{98} +1.30749i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 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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