LMFDB - Embedded newform 164.2.i.a.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [164,2,Mod(3,164)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(164, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("164.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	164.i (of order $8$, degree $4$, 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:	$1.30954659315$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label			55.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	164.55
Dual form			164.2.i.a.3.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.82843 - 2.82843i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.82843 - 2.82843i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 + 2.12132i) q^{9} +5.65685 q^{10} +(-1.53553 - 3.70711i) q^{13} -4.00000 q^{16} +(-2.53553 + 6.12132i) q^{17} -4.24264 q^{18} +(5.65685 + 5.65685i) q^{20} -11.0000i q^{25} +(2.17157 - 5.24264i) q^{26} +(-2.87868 - 6.94975i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(-8.65685 + 3.58579i) q^{34} +(-4.24264 - 4.24264i) q^{36} +7.07107 q^{37} +11.3137i q^{40} +(5.00000 - 4.00000i) q^{41} +12.0000i q^{45} +(-4.94975 + 4.94975i) q^{49} +(11.0000 - 11.0000i) q^{50} +(7.41421 - 3.07107i) q^{52} +(-10.5355 + 4.36396i) q^{53} +(4.07107 - 9.82843i) q^{58} +(11.0000 + 11.0000i) q^{61} -8.00000i q^{64} +(-14.8284 - 6.14214i) q^{65} +(-12.2426 - 5.07107i) q^{68} -8.48528i q^{72} +(11.3137 + 11.3137i) q^{73} +(7.07107 + 7.07107i) q^{74} +(-11.3137 + 11.3137i) q^{80} -9.00000i q^{81} +(9.00000 + 1.00000i) q^{82} +(10.1421 + 24.4853i) q^{85} +(1.19239 + 2.87868i) q^{89} +(-12.0000 + 12.0000i) q^{90} +(-18.1924 - 7.53553i) q^{97} -9.89949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 8 * q^8	$ 4 q + 4 q^{2} - 8 q^{8} + 8 q^{13} - 16 q^{16} + 4 q^{17} + 20 q^{26} - 20 q^{29} - 16 q^{32} - 12 q^{34} + 20 q^{41} + 44 q^{50} + 24 q^{52} - 28 q^{53} - 12 q^{58} + 44 q^{61} - 48 q^{65} - 32 q^{68} + 36 q^{82} - 16 q^{85} - 32 q^{89} - 48 q^{90} - 36 q^{97}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^8 + 8 * q^13 - 16 * q^16 + 4 * q^17 + 20 * q^26 - 20 * q^29 - 16 * q^32 - 12 * q^34 + 20 * q^41 + 44 * q^50 + 24 * q^52 - 28 * q^53 - 12 * q^58 + 44 * q^61 - 48 * q^65 - 32 * q^68 + 36 * q^82 - 16 * q^85 - 32 * q^89 - 48 * q^90 - 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$-1$	$e\left(\frac{5}{8}\right)$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$3$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$4$			2.00000i			1.00000i
$5$	2.82843	−	2.82843i	1.26491	−	1.26491i	0.316228	−	0.948683i	$-0.397584\pi$
$5$	2.82843	−	2.82843i	1.26491	−	1.26491i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$7$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	−2.12132	+	2.12132i	−0.707107	+	0.707107i
$10$	5.65685			1.78885
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$		0			0
$13$	−1.53553	−	3.70711i	−0.425880	−	1.02817i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	−1.53553	−	3.70711i	−0.425880	−	1.02817i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−2.53553	+	6.12132i	−0.614957	+	1.48464i	0.242536	+	0.970143i	$0.422021\pi$
$17$	−2.53553	+	6.12132i	−0.614957	+	1.48464i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	−4.24264			−1.00000
$19$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$19$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$20$	5.65685	+	5.65685i	1.26491	+	1.26491i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		−	11.0000i		−	2.20000i
$26$	2.17157	−	5.24264i	0.425880	−	1.02817i
$27$		0			0
$28$		0			0
$29$	−2.87868	−	6.94975i	−0.534557	−	1.29054i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−2.87868	−	6.94975i	−0.534557	−	1.29054i	0.393919	−	0.919145i	$-0.371119\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	−8.65685	+	3.58579i	−1.48464	+	0.614957i
$35$		0			0
$36$	−4.24264	−	4.24264i	−0.707107	−	0.707107i
$37$	7.07107			1.16248			0.581238	−	0.813733i	$-0.302568\pi$
$37$	7.07107			1.16248			0.581238	+	0.813733i	$0.302568\pi$
$38$		0			0
$39$		0			0
$40$			11.3137i			1.78885i
$41$	5.00000	−	4.00000i	0.780869	−	0.624695i
$41$	5.00000	−	4.00000i	0.780869	−	0.624695i
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$			12.0000i			1.78885i
$46$		0			0
$47$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$47$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$48$		0			0
$49$	−4.94975	+	4.94975i	−0.707107	+	0.707107i
$50$	11.0000	−	11.0000i	1.55563	−	1.55563i
$51$		0			0
$52$	7.41421	−	3.07107i	1.02817	−	0.425880i
$53$	−10.5355	+	4.36396i	−1.44717	+	0.599436i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−10.5355	+	4.36396i	−1.44717	+	0.599436i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.07107	−	9.82843i	0.534557	−	1.29054i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	11.0000	+	11.0000i	1.40841	+	1.40841i	0.768221	+	0.640184i	$0.221142\pi$
$61$	11.0000	+	11.0000i	1.40841	+	1.40841i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	−14.8284	−	6.14214i	−1.83924	−	0.761838i
$66$		0			0
$67$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$67$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$68$	−12.2426	−	5.07107i	−1.48464	−	0.614957i
$69$		0			0
$70$		0			0
$71$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$71$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$72$		−	8.48528i		−	1.00000i
$73$	11.3137	+	11.3137i	1.32417	+	1.32417i	0.910366	+	0.413803i	$0.135800\pi$
$73$	11.3137	+	11.3137i	1.32417	+	1.32417i	0.413803	+	0.910366i	$0.364200\pi$
$74$	7.07107	+	7.07107i	0.821995	+	0.821995i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$79$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$80$	−11.3137	+	11.3137i	−1.26491	+	1.26491i
$81$		−	9.00000i		−	1.00000i
$82$	9.00000	+	1.00000i	0.993884	+	0.110432i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	10.1421	+	24.4853i	1.10007	+	2.65580i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.19239	+	2.87868i	0.126393	+	0.305139i	0.974391	−	0.224860i	$-0.0721923\pi$
$89$	1.19239	+	2.87868i	0.126393	+	0.305139i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	−12.0000	+	12.0000i	−1.26491	+	1.26491i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.1924	−	7.53553i	−1.84716	−	0.765118i	−0.933346	−	0.358979i	$-0.883125\pi$
$97$	−18.1924	−	7.53553i	−1.84716	−	0.765118i	−0.913812	−	0.406138i	$-0.866875\pi$
$98$	−9.89949			−1.00000
$99$		0			0
$100$	22.0000			2.20000
$101$	3.63604	−	8.77817i	0.361799	−	0.873461i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	3.63604	−	8.77817i	0.361799	−	0.873461i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$	10.4853	+	4.34315i	1.02817	+	0.425880i
$105$		0			0
$106$	−14.8995	−	6.17157i	−1.44717	−	0.599436i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	19.1924	+	7.94975i	1.83830	+	0.761448i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	19.1924	+	7.94975i	1.83830	+	0.761448i	0.880471	+	0.474100i	$0.157226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.41421i			0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$			1.41421i			0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$		0			0
$116$	13.8995	−	5.75736i	1.29054	−	0.534557i
$117$	11.1213	+	4.60660i	1.02817	+	0.425880i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.77817	−	7.77817i	0.707107	−	0.707107i
$122$			22.0000i			1.99179i
$123$		0			0
$124$		0			0
$125$	−16.9706	−	16.9706i	−1.51789	−	1.51789i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		0			0
$130$	−8.68629	−	20.9706i	−0.761838	−	1.83924i
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−7.17157	−	17.3137i	−0.614957	−	1.48464i
$137$	6.05025	−	14.6066i	0.516908	−	1.24793i	−0.422885	−	0.906183i	$-0.638983\pi$
$137$	6.05025	−	14.6066i	0.516908	−	1.24793i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	8.48528	−	8.48528i	0.707107	−	0.707107i
$145$	−27.7990	−	11.5147i	−2.30858	−	0.956245i
$146$			22.6274i			1.87266i
$147$		0			0
$148$			14.1421i			1.16248i
$149$	−9.12132	+	22.0208i	−0.747248	+	1.80402i	−0.173785	+	0.984784i	$0.555600\pi$
$149$	−9.12132	+	22.0208i	−0.747248	+	1.80402i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$	−7.60660	−	18.3640i	−0.614957	−	1.48464i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.46447	+	1.02082i	−0.196686	+	0.0814699i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−2.46447	+	1.02082i	−0.196686	+	0.0814699i	0.282166	+	0.959366i	$0.408947\pi$
$158$		0			0
$159$		0			0
$160$	−22.6274			−1.78885
$161$		0			0
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	8.00000	+	10.0000i	0.624695	+	0.780869i
$165$		0			0
$166$		0			0
$167$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$167$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$168$		0			0
$169$	−2.19239	+	2.19239i	−0.168645	+	0.168645i
$170$	−14.3431	+	34.6274i	−1.10007	+	2.65580i
$171$		0			0
$172$		0			0
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.68629	+	4.07107i	−0.126393	+	0.305139i
$179$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$179$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$180$	−24.0000			−1.78885
$181$	9.70711	−	23.4350i	0.721524	−	1.74191i	0.0525588	−	0.998618i	$-0.483262\pi$
$181$	9.70711	−	23.4350i	0.721524	−	1.74191i	0.668965	−	0.743294i	$-0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.0000	−	20.0000i	1.47043	−	1.47043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$191$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$192$		0			0
$193$	1.43503	+	3.46447i	0.103296	+	0.249378i	0.967075	−	0.254493i	$-0.0819087\pi$
$193$	1.43503	+	3.46447i	0.103296	+	0.249378i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−10.6569	−	25.7279i	−0.765118	−	1.84716i
$195$		0			0
$196$	−9.89949	−	9.89949i	−0.707107	−	0.707107i
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$		0			0
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$	22.0000	+	22.0000i	1.55563	+	1.55563i
$201$		0			0
$202$	12.4142	−	5.14214i	0.873461	−	0.361799i
$203$		0			0
$204$		0			0
$205$	2.82843	−	25.4558i	0.197546	−	1.77791i
$206$		0			0
$207$		0			0
$208$	6.14214	+	14.8284i	0.425880	+	1.02817i
$209$		0			0
$210$		0			0
$211$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$211$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$212$	−8.72792	−	21.0711i	−0.599436	−	1.44717i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	11.2426	+	27.1421i	0.761448	+	1.83830i
$219$		0			0
$220$		0			0
$221$	26.5858			1.78835
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	23.3345	+	23.3345i	1.55563	+	1.55563i
$226$	−1.41421	+	1.41421i	−0.0940721	+	0.0940721i
$227$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$227$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$228$		0			0
$229$	7.19239	+	2.97918i	0.475286	+	0.196870i	0.607450	−	0.794358i	$-0.292192\pi$
$229$	7.19239	+	2.97918i	0.475286	+	0.196870i	−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$		0			0
$232$	19.6569	+	8.14214i	1.29054	+	0.534557i
$233$	−27.8492	+	11.5355i	−1.82446	+	0.755718i	−0.851658	+	0.524097i	$0.824403\pi$
$233$	−27.8492	+	11.5355i	−1.82446	+	0.755718i	−0.972806	+	0.231621i	$0.925597\pi$
$234$	6.51472	+	15.7279i	0.425880	+	1.02817i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$239$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$240$		0			0
$241$	19.0000	+	19.0000i	1.22390	+	1.22390i	0.966235	+	0.257663i	$0.0829523\pi$
$241$	19.0000	+	19.0000i	1.22390	+	1.22390i	0.257663	+	0.966235i	$0.417048\pi$
$242$	15.5563			1.00000
$243$		0			0
$244$	−22.0000	+	22.0000i	−1.40841	+	1.40841i
$245$			28.0000i			1.78885i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		−	33.9411i		−	2.14663i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−26.6066	+	11.0208i	−1.65967	+	0.687460i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	−26.6066	+	11.0208i	−1.65967	+	0.687460i	−0.661622	+	0.749838i	$0.730131\pi$
$258$		0			0
$259$		0			0
$260$	12.2843	−	29.6569i	0.761838	−	1.83924i
$261$	20.8492	+	8.63604i	1.29054	+	0.534557i
$262$		0			0
$263$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$263$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$264$		0			0
$265$	−17.4558	+	42.1421i	−1.07230	+	2.58877i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	26.0000i		−	1.58525i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$		−	26.0000i		−	1.58525i	0.609711	−	0.792624i	$-0.291286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	10.1421	−	24.4853i	0.614957	−	1.48464i
$273$		0			0
$274$	20.6569	−	8.55635i	1.24793	−	0.516908i
$275$		0			0
$276$		0			0
$277$			7.07107i			0.424859i	0.977176	+	0.212430i	$0.0681376\pi$
$277$			7.07107i			0.424859i	−0.977176	+	0.212430i	$0.931862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.77817	+	1.15076i	0.165732	+	0.0686484i	0.464007	−	0.885832i	$-0.346411\pi$
$281$	2.77817	+	1.15076i	0.165732	+	0.0686484i	−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.9706			1.00000
$289$	−19.0208	−	19.0208i	−1.11887	−	1.11887i
$290$	−16.2843	−	39.3137i	−0.956245	−	2.30858i
$291$		0			0
$292$	−22.6274	+	22.6274i	−1.32417	+	1.32417i
$293$	−6.39340	+	15.4350i	−0.373506	+	0.901724i	0.619644	+	0.784883i	$0.287277\pi$
$293$	−6.39340	+	15.4350i	−0.373506	+	0.901724i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$	−14.1421	+	14.1421i	−0.821995	+	0.821995i
$297$		0			0
$298$	−31.1421	+	12.8995i	−1.80402	+	0.747248i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	62.2254			3.56302
$306$	10.7574	−	25.9706i	0.614957	−	1.48464i
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$311$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$312$		0			0
$313$	12.7071	−	30.6777i	0.718248	−	1.73400i	0.0399680	−	0.999201i	$-0.487274\pi$
$313$	12.7071	−	30.6777i	0.718248	−	1.73400i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−3.48528	−	1.44365i	−0.196686	−	0.0814699i
$315$		0			0
$316$		0			0
$317$	−11.8787	−	28.6777i	−0.667173	−	1.61070i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−11.8787	−	28.6777i	−0.667173	−	1.61070i	0.119145	−	0.992877i	$-0.461985\pi$
$318$		0			0
$319$		0			0
$320$	−22.6274	−	22.6274i	−1.26491	−	1.26491i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	−40.7782	+	16.8909i	−2.26197	+	0.936937i
$326$		0			0
$327$		0			0
$328$	−2.00000	+	18.0000i	−0.110432	+	0.993884i
$329$		0			0
$330$		0			0
$331$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$331$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.7279	+	12.7279i	−0.693334	+	0.693334i	−0.962964	−	0.269630i	$-0.913099\pi$
$337$	−12.7279	+	12.7279i	−0.693334	+	0.693334i	0.269630	+	0.962964i	$0.413099\pi$
$338$	−4.38478			−0.238500
$339$		0			0
$340$	−48.9706	+	20.2843i	−2.65580	+	1.10007i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−22.0000			−1.18273
$347$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$347$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$348$		0			0
$349$	−7.07107	+	7.07107i	−0.378506	+	0.378506i	−0.870563	−	0.492057i	$-0.836245\pi$
$349$	−7.07107	+	7.07107i	−0.378506	+	0.378506i	0.492057	+	0.870563i	$0.336245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.7279i			0.677439i	0.940887	+	0.338719i	$0.109994\pi$
$353$			12.7279i			0.677439i	−0.940887	+	0.338719i	$0.890006\pi$
$354$		0			0
$355$		0			0
$356$	−5.75736	+	2.38478i	−0.305139	+	0.126393i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−24.0000	−	24.0000i	−1.26491	−	1.26491i
$361$	−13.4350	−	13.4350i	−0.707107	−	0.707107i
$362$	33.1421	−	13.7279i	1.74191	−	0.721524i
$363$		0			0
$364$		0			0
$365$	64.0000			3.34991
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$	−2.12132	+	19.0919i	−0.110432	+	0.993884i
$370$	40.0000			2.07950
$371$		0			0
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.3431	+	21.3431i	−1.09923	+	1.09923i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$383$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$384$		0			0
$385$		0			0
$386$	−2.02944	+	4.89949i	−0.103296	+	0.249378i
$387$		0			0
$388$	15.0711	−	36.3848i	0.765118	−	1.84716i
$389$	−27.0000	−	27.0000i	−1.36895	−	1.36895i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−27.0000	−	27.0000i	−1.36895	−	1.36895i	−0.507020	−	0.861934i	$-0.669253\pi$
$390$		0			0
$391$		0			0
$392$		−	19.7990i		−	1.00000i
$393$		0			0
$394$		−	30.0000i		−	1.51138i
$395$		0			0
$396$		0			0
$397$	−23.6777	+	9.80761i	−1.18835	+	0.492230i	−0.887217	−	0.461353i	$-0.847364\pi$
$397$	−23.6777	+	9.80761i	−1.18835	+	0.492230i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$			44.0000i			2.20000i
$401$	−1.41421	−	1.41421i	−0.0706225	−	0.0706225i	0.670913	−	0.741536i	$-0.265902\pi$
$401$	−1.41421	−	1.41421i	−0.0706225	−	0.0706225i	−0.741536	+	0.670913i	$0.765902\pi$
$402$		0			0
$403$		0			0
$404$	17.5563	+	7.27208i	0.873461	+	0.361799i
$405$	−25.4558	−	25.4558i	−1.26491	−	1.26491i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	32.5269i		−	1.60835i	−0.594391	−	0.804176i	$-0.702607\pi$
$409$		−	32.5269i		−	1.60835i	0.594391	−	0.804176i	$-0.297393\pi$
$410$	28.2843	−	22.6274i	1.39686	−	1.11749i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−8.68629	+	20.9706i	−0.425880	+	1.02817i
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	15.7071	−	6.50610i	0.765518	−	0.317088i	0.0344623	−	0.999406i	$-0.489028\pi$
$421$	15.7071	−	6.50610i	0.765518	−	0.317088i	0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$	12.3431	−	29.7990i	0.599436	−	1.44717i
$425$	67.3345	+	27.8909i	3.26620	+	1.35291i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$	−15.8995	+	38.3848i	−0.761448	+	1.83830i
$437$		0			0
$438$		0			0
$439$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$439$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$	26.5858	+	26.5858i	1.26456	+	1.26456i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	11.5147	+	4.76955i	0.545850	+	0.226098i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	0.943858	+	0.330350i	$0.107167\pi$
$450$			46.6690i			2.20000i
$451$		0			0
$452$	−2.82843			−0.133038
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.32233	−	8.02082i	0.155412	−	0.375198i	−0.826927	−	0.562310i	$-0.809913\pi$
$457$	3.32233	−	8.02082i	0.155412	−	0.375198i	0.982339	+	0.187112i	$0.0599128\pi$
$458$	4.21320	+	10.1716i	0.196870	+	0.475286i
$459$		0			0
$460$		0			0
$461$	12.7279			0.592798			0.296399	−	0.955064i	$-0.404214\pi$
$461$	12.7279			0.592798			0.296399	+	0.955064i	$0.404214\pi$
$462$		0			0
$463$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$463$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$464$	11.5147	+	27.7990i	0.534557	+	1.29054i
$465$		0			0
$466$	−39.3848	−	16.3137i	−1.82446	−	0.755718i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−9.21320	+	22.2426i	−0.425880	+	1.02817i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	13.0919	−	31.6066i	0.599436	−	1.44717i
$478$		0			0
$479$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$479$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$480$		0			0
$481$	−10.8579	−	26.2132i	−0.495076	−	1.19522i
$482$			38.0000i			1.73085i
$483$		0			0
$484$	15.5563	+	15.5563i	0.707107	+	0.707107i
$485$	−72.7696	+	30.1421i	−3.30430	+	1.36868i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−44.0000			−1.99179
$489$		0			0
$490$	−28.0000	+	28.0000i	−1.26491	+	1.26491i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	49.8406			2.24471
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$499$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$500$	33.9411	−	33.9411i	1.51789	−	1.51789i
$501$		0			0
$502$		0			0
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$		0			0
$505$	−14.5442	−	35.1127i	−0.647206	−	1.56249i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.97918	−	24.0919i	0.442320	−	1.06785i	−0.532813	−	0.846233i	$-0.678865\pi$
$509$	9.97918	−	24.0919i	0.442320	−	1.06785i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	−37.6274	−	15.5858i	−1.65967	−	0.687460i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	41.9411	−	17.3726i	1.83924	−	0.761838i
$521$	13.6360	+	32.9203i	0.597406	+	1.44226i	0.876216	+	0.481919i	$0.160060\pi$
$521$	13.6360	+	32.9203i	0.597406	+	1.44226i	−0.278810	+	0.960346i	$0.589940\pi$
$522$	12.2132	+	29.4853i	0.534557	+	1.29054i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$	−59.5980	+	24.6863i	−2.58877	+	1.07230i
$531$		0			0
$532$		0			0
$533$	−22.5061	−	12.3934i	−0.974847	−	0.536818i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	26.0000	−	26.0000i	1.12094	−	1.12094i
$539$		0			0
$540$		0			0
$541$	−29.6985	+	29.6985i	−1.27684	+	1.27684i	−0.334410	+	0.942428i	$0.608537\pi$
$541$	−29.6985	+	29.6985i	−1.27684	+	1.27684i	−0.942428	+	0.334410i	$0.891463\pi$
$542$		0			0
$543$		0			0
$544$	34.6274	−	14.3431i	1.48464	−	0.614957i
$545$	76.7696	−	31.7990i	3.28845	−	1.36212i
$546$		0			0
$547$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$547$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$548$	29.2132	+	12.1005i	1.24793	+	0.516908i
$549$	−46.6690			−1.99179
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−7.07107	+	7.07107i	−0.300421	+	0.300421i
$555$		0			0
$556$		0			0
$557$	22.5355	+	9.33452i	0.954861	+	0.395516i	0.805056	−	0.593199i	$-0.202135\pi$
$557$	22.5355	+	9.33452i	0.954861	+	0.395516i	0.149805	+	0.988716i	$0.452135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.62742	+	3.92893i	0.0686484	+	0.165732i
$563$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$563$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$564$		0			0
$565$	4.00000	+	4.00000i	0.168281	+	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.2843	−	28.2843i	−1.18574	−	1.18574i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	−28.2843	−	28.2843i	−1.18574	−	1.18574i	−0.207504	−	0.978234i	$-0.566534\pi$
$570$		0			0
$571$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$571$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	16.9706	+	16.9706i	0.707107	+	0.707107i
$577$	16.6777	+	40.2635i	0.694300	+	1.67619i	0.735931	+	0.677057i	$0.236745\pi$
$577$	16.6777	+	40.2635i	0.694300	+	1.67619i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$		−	38.0416i		−	1.58232i
$579$		0			0
$580$	23.0294	−	55.5980i	0.956245	−	2.30858i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−45.2548			−1.87266
$585$	44.4853	−	18.4264i	1.83924	−	0.761838i
$586$	−21.8284	+	9.04163i	−0.901724	+	0.373506i
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−28.2843			−1.16248
$593$	−13.9203	+	33.6066i	−0.571639	+	1.38006i	0.328521	+	0.944497i	$0.393450\pi$
$593$	−13.9203	+	33.6066i	−0.571639	+	1.38006i	−0.900159	+	0.435561i	$0.856550\pi$
$594$		0			0
$595$		0			0
$596$	−44.0416	−	18.2426i	−1.80402	−	0.747248i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	25.5061	+	10.5650i	1.04042	+	0.430954i	0.836461	−	0.548026i	$-0.184621\pi$
$601$	25.5061	+	10.5650i	1.04042	+	0.430954i	0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	44.0000i		−	1.78885i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	62.2254	+	62.2254i	2.51943	+	2.51943i
$611$		0			0
$612$	36.7279	−	15.2132i	1.48464	−	0.614957i
$613$	−25.4558	+	25.4558i	−1.02815	+	1.02815i	−0.0285598	+	0.999592i	$0.509092\pi$
$613$	−25.4558	+	25.4558i	−1.02815	+	1.02815i	−0.999592	+	0.0285598i	$0.990908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.0000	−	35.0000i	−1.40905	−	1.40905i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−35.0000	−	35.0000i	−1.40905	−	1.40905i	−0.644136	−	0.764911i	$-0.722783\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−41.0000			−1.64000
$626$	43.3848	−	17.9706i	1.73400	−	0.718248i
$627$		0			0
$628$	−2.04163	−	4.92893i	−0.0814699	−	0.196686i
$629$	−17.9289	+	43.2843i	−0.714873	+	1.72586i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	16.7990	−	40.5563i	0.667173	−	1.61070i
$635$		0			0
$636$		0			0
$637$	25.9497	+	10.7487i	1.02817	+	0.425880i
$638$		0			0
$639$		0			0
$640$		−	45.2548i		−	1.78885i
$641$	−18.8492	+	45.5061i	−0.744500	+	1.79738i	−0.157991	+	0.987441i	$0.550502\pi$
$641$	−18.8492	+	45.5061i	−0.744500	+	1.79738i	−0.586510	+	0.809942i	$0.699498\pi$
$642$		0			0
$643$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$643$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		0			0
$650$	−57.6690	−	23.8873i	−2.26197	−	0.936937i
$651$		0			0
$652$		0			0
$653$	28.3640	−	11.7487i	1.10997	−	0.459764i	0.249041	−	0.968493i	$-0.419885\pi$
$653$	28.3640	−	11.7487i	1.10997	−	0.459764i	0.860927	+	0.508729i	$0.169885\pi$
$654$		0			0
$655$		0			0
$656$	−20.0000	+	16.0000i	−0.780869	+	0.624695i
$657$	−48.0000			−1.87266
$658$		0			0
$659$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$659$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$660$		0			0
$661$	35.3553	−	35.3553i	1.37516	−	1.37516i	0.522562	−	0.852601i	$-0.324976\pi$
$661$	35.3553	−	35.3553i	1.37516	−	1.37516i	0.852601	−	0.522562i	$-0.175024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−30.0000			−1.16248
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	19.7782	−	47.7487i	0.762393	−	1.84058i	0.299827	−	0.953994i	$-0.403071\pi$
$673$	19.7782	−	47.7487i	0.762393	−	1.84058i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	−25.4558			−0.980522
$675$		0			0
$676$	−4.38478	−	4.38478i	−0.168645	−	0.168645i
$677$	−1.41421	+	1.41421i	−0.0543526	+	0.0543526i	−0.733761	−	0.679408i	$-0.762237\pi$
$677$	−1.41421	+	1.41421i	−0.0543526	+	0.0543526i	0.679408	+	0.733761i	$0.262237\pi$
$678$		0			0
$679$		0			0
$680$	−69.2548	−	28.6863i	−2.65580	−	1.10007i
$681$		0			0
$682$		0			0
$683$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$683$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$684$		0			0
$685$	−24.2010	−	58.4264i	−0.924673	−	2.23236i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	32.3553	+	32.3553i	1.23264	+	1.23264i
$690$		0			0
$691$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$691$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$692$	−22.0000	−	22.0000i	−0.836315	−	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	11.8076	+	40.7487i	0.447245	+	1.54347i
$698$	−14.1421			−0.535288
$699$		0			0
$700$		0			0
$701$			29.6985i			1.12170i	0.827919	+	0.560848i	$0.189525\pi$
$701$			29.6985i			1.12170i	−0.827919	+	0.560848i	$0.810475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−12.7279	+	12.7279i	−0.479022	+	0.479022i
$707$		0			0
$708$		0			0
$709$	48.1630	−	19.9497i	1.80880	−	0.749228i	0.826227	−	0.563337i	$-0.190483\pi$
$709$	48.1630	−	19.9497i	1.80880	−	0.749228i	0.982570	−	0.185892i	$-0.0595174\pi$
$710$		0			0
$711$		0			0
$712$	−8.14214	−	3.37258i	−0.305139	−	0.126393i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$719$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$720$		−	48.0000i		−	1.78885i
$721$		0			0
$722$		−	26.8701i		−	1.00000i
$723$		0			0
$724$	46.8701	+	19.4142i	1.74191	+	0.721524i
$725$	−76.4472	+	31.6655i	−2.83918	+	1.17603i
$726$		0			0
$727$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$727$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$728$		0			0
$729$	19.0919	+	19.0919i	0.707107	+	0.707107i
$730$	64.0000	+	64.0000i	2.36875	+	2.36875i
$731$		0			0
$732$		0			0
$733$	−29.0000	−	29.0000i	−1.07114	−	1.07114i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	−29.0000	−	29.0000i	−1.07114	−	1.07114i	−0.0738717	−	0.997268i	$-0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−21.2132	+	16.9706i	−0.780869	+	0.624695i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	40.0000	+	40.0000i	1.47043	+	1.47043i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	36.4853	+	88.0833i	1.33672	+	3.22712i
$746$	14.0000	−	14.0000i	0.512576	−	0.512576i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$751$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$752$		0			0
$753$		0			0
$754$	−42.6863			−1.55454
$755$		0			0
$756$		0			0
$757$	−1.25126	+	3.02082i	−0.0454779	+	0.109793i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−1.25126	+	3.02082i	−0.0454779	+	0.109793i	0.899508	+	0.436904i	$0.143925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			55.1543i			1.99934i	0.0256326	+	0.999671i	$0.491840\pi$
$761$			55.1543i			1.99934i	−0.0256326	+	0.999671i	$0.508160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−73.4558	−	30.4264i	−2.65580	−	1.10007i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	−6.92893	+	2.87006i	−0.249378	+	0.103296i
$773$	44.5772	+	18.4645i	1.60333	+	0.664121i	0.991882	−	0.127164i	$-0.0405876\pi$
$773$	44.5772	+	18.4645i	1.60333	+	0.664121i	0.611448	+	0.791285i	$0.290588\pi$
$774$		0			0
$775$		0			0
$776$	51.4558	−	21.3137i	1.84716	−	0.765118i
$777$		0			0
$778$		−	54.0000i		−	1.93599i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	19.7990	−	19.7990i	0.707107	−	0.707107i
$785$	−4.08326	+	9.85786i	−0.145738	+	0.351842i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	30.0000	−	30.0000i	1.06871	−	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	23.8873	−	57.6690i	0.848263	−	2.04789i
$794$	−33.4853	−	13.8701i	−1.18835	−	0.492230i
$795$		0			0
$796$		0			0
$797$	−22.0000			−0.779280			−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000			−0.779280			−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	−44.0000	+	44.0000i	−1.55563	+	1.55563i
$801$	−8.63604	−	3.57716i	−0.305139	−	0.126393i
$802$		−	2.82843i		−	0.0998752i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	10.2843	+	24.8284i	0.361799	+	0.873461i
$809$	18.3345	+	44.2635i	0.644608	+	1.55622i	0.820398	+	0.571793i	$0.193752\pi$
$809$	18.3345	+	44.2635i	0.644608	+	1.55622i	−0.175791	+	0.984428i	$0.556248\pi$
$810$		−	50.9117i		−	1.78885i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	32.5269	−	32.5269i	1.13728	−	1.13728i
$819$		0			0
$820$	50.9117	+	5.65685i	1.77791	+	0.197546i
$821$	55.1543			1.92490			0.962450	−	0.271460i	$-0.0875065\pi$
$821$	55.1543			1.92490			0.962450	+	0.271460i	$0.0875065\pi$
$822$		0			0
$823$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$823$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$827$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$828$		0			0
$829$	−14.1421	+	14.1421i	−0.491177	+	0.491177i	−0.908677	−	0.417500i	$-0.862906\pi$
$829$	−14.1421	+	14.1421i	−0.491177	+	0.491177i	0.417500	+	0.908677i	$0.362906\pi$
$830$		0			0
$831$		0			0
$832$	−29.6569	+	12.2843i	−1.02817	+	0.425880i
$833$	−17.7487	−	42.8492i	−0.614957	−	1.48464i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$839$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$840$		0			0
$841$	−19.5061	+	19.5061i	−0.672624	+	0.672624i
$842$	22.2132	+	9.20101i	0.765518	+	0.317088i
$843$		0			0
$844$		0			0
$845$			12.4020i			0.426642i
$846$		0			0
$847$		0			0
$848$	42.1421	−	17.4558i	1.44717	−	0.599436i
$849$		0			0
$850$	39.4437	+	95.2254i	1.35291	+	3.26620i
$851$		0			0
$852$		0			0
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	35.3553			1.20772			0.603858	−	0.797092i	$-0.293630\pi$
$857$	35.3553			1.20772			0.603858	+	0.797092i	$0.293630\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$			62.2254i			2.11573i
$866$	−34.0000	+	34.0000i	−1.15537	+	1.15537i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−54.2843	+	22.4853i	−1.83830	+	0.761448i
$873$	54.5772	−	22.6066i	1.84716	−	0.765118i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	58.0000			1.95852			0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000			1.95852			0.979260	+	0.202606i	$0.0649409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.00000	−	9.00000i	−0.303218	−	0.303218i	0.539054	−	0.842271i	$-0.318782\pi$
$881$	−9.00000	−	9.00000i	−0.303218	−	0.303218i	−0.842271	+	0.539054i	$0.818782\pi$
$882$	21.0000	−	21.0000i	0.707107	−	0.707107i
$883$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$883$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$884$			53.1716i			1.78835i
$885$		0			0
$886$		0			0
$887$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$887$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$888$		0			0
$889$		0			0
$890$	6.74517	+	16.2843i	0.226098	+	0.545850i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	26.0000			0.867631
$899$		0			0
$900$	−46.6690	+	46.6690i	−1.55563	+	1.55563i
$901$		−	75.5563i		−	2.51715i
$902$		0			0
$903$		0			0
$904$	−2.82843	−	2.82843i	−0.0940721	−	0.0940721i
$905$	−38.8284	−	93.7401i	−1.29070	−	3.11603i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	10.9081	+	26.3345i	0.361799	+	0.873461i
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$	11.3431	−	4.69848i	0.375198	−	0.155412i
$915$		0			0
$916$	−5.95837	+	14.3848i	−0.196870	+	0.475286i
$917$		0			0
$918$		0			0
$919$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$919$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$920$		0			0
$921$		0			0
$922$	12.7279	+	12.7279i	0.419172	+	0.419172i
$923$		0			0
$924$		0			0
$925$		−	77.7817i		−	2.55745i
$926$		0			0
$927$		0			0
$928$	−16.2843	+	39.3137i	−0.534557	+	1.29054i
$929$	17.8787	+	7.40559i	0.586580	+	0.242970i	0.656179	−	0.754606i	$-0.272172\pi$
$929$	17.8787	+	7.40559i	0.586580	+	0.242970i	−0.0695983	+	0.997575i	$0.522172\pi$
$930$		0			0
$931$		0			0
$932$	−23.0711	−	55.6985i	−0.755718	−	1.82446i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−31.4558	+	13.0294i	−1.02817	+	0.425880i
$937$	−54.4056	−	22.5355i	−1.77735	−	0.736204i	−0.993307	−	0.115501i	$-0.963153\pi$
$937$	−54.4056	−	22.5355i	−1.77735	−	0.736204i	−0.784046	−	0.620703i	$-0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	−	19.0000i	0.619382	−	0.619382i	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	19.0000	−	19.0000i	0.619382	−	0.619382i	0.945373	+	0.325991i	$0.105698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	24.5685	−	59.3137i	0.797529	−	1.92540i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.2132			0.687163			0.343582	−	0.939123i	$-0.388360\pi$
$953$	21.2132			0.687163			0.343582	+	0.939123i	$0.388360\pi$
$954$	44.6985	−	18.5147i	1.44717	−	0.599436i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	15.3553	−	37.0711i	0.495076	−	1.19522i
$963$		0			0
$964$	−38.0000	+	38.0000i	−1.22390	+	1.22390i
$965$	13.8579	+	5.74012i	0.446100	+	0.184781i
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$			31.1127i			1.00000i
$969$		0			0
$970$	−102.912	−	42.6274i	−3.30430	−	1.36868i
$971$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$971$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−44.0000	−	44.0000i	−1.40841	−	1.40841i
$977$	−15.0919	+	6.25126i	−0.482832	+	0.199996i	−0.610803	−	0.791782i	$-0.709153\pi$
$977$	−15.0919	+	6.25126i	−0.482832	+	0.199996i	0.127971	+	0.991778i	$0.459153\pi$
$978$		0			0
$979$		0			0
$980$	−56.0000			−1.78885
$981$	−57.5772	+	23.8492i	−1.83830	+	0.761448i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−84.8528			−2.70364
$986$	49.8406	+	49.8406i	1.58725	+	1.58725i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$991$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	4.83705	+	11.6777i	0.153191	+	0.369836i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	4.83705	+	11.6777i	0.153191	+	0.369836i	−0.828589	+	0.559857i	$0.810856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.i.a.55.1	yes	4
4.3	odd	2	CM	164.2.i.a.55.1	yes	4
41.3	odd	8	inner	164.2.i.a.3.1	✓	4
164.3	even	8	inner	164.2.i.a.3.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.i.a.3.1	✓	4	41.3	odd	8	inner
164.2.i.a.3.1	✓	4	164.3	even	8	inner
164.2.i.a.55.1	yes	4	1.1	even	1	trivial
164.2.i.a.55.1	yes	4	4.3	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:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.i (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.82843 - 2.82843i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 + 2.12132i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.82843 - 2.82843i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 + 2.12132i) q^{9} +5.65685 q^{10} +(-1.53553 - 3.70711i) q^{13} -4.00000 q^{16} +(-2.53553 + 6.12132i) q^{17} -4.24264 q^{18} +(5.65685 + 5.65685i) q^{20} -11.0000i q^{25} +(2.17157 - 5.24264i) q^{26} +(-2.87868 - 6.94975i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(-8.65685 + 3.58579i) q^{34} +(-4.24264 - 4.24264i) q^{36} +7.07107 q^{37} +11.3137i q^{40} +(5.00000 - 4.00000i) q^{41} +12.0000i q^{45} +(-4.94975 + 4.94975i) q^{49} +(11.0000 - 11.0000i) q^{50} +(7.41421 - 3.07107i) q^{52} +(-10.5355 + 4.36396i) q^{53} +(4.07107 - 9.82843i) q^{58} +(11.0000 + 11.0000i) q^{61} -8.00000i q^{64} +(-14.8284 - 6.14214i) q^{65} +(-12.2426 - 5.07107i) q^{68} -8.48528i q^{72} +(11.3137 + 11.3137i) q^{73} +(7.07107 + 7.07107i) q^{74} +(-11.3137 + 11.3137i) q^{80} -9.00000i q^{81} +(9.00000 + 1.00000i) q^{82} +(10.1421 + 24.4853i) q^{85} +(1.19239 + 2.87868i) q^{89} +(-12.0000 + 12.0000i) q^{90} +(-18.1924 - 7.53553i) q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 8 * q^8	\( 4 q + 4 q^{2} - 8 q^{8} + 8 q^{13} - 16 q^{16} + 4 q^{17} + 20 q^{26} - 20 q^{29} - 16 q^{32} - 12 q^{34} + 20 q^{41} + 44 q^{50} + 24 q^{52} - 28 q^{53} - 12 q^{58} + 44 q^{61} - 48 q^{65} - 32 q^{68} + 36 q^{82} - 16 q^{85} - 32 q^{89} - 48 q^{90} - 36 q^{97}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^8 + 8 * q^13 - 16 * q^16 + 4 * q^17 + 20 * q^26 - 20 * q^29 - 16 * q^32 - 12 * q^34 + 20 * q^41 + 44 * q^50 + 24 * q^52 - 28 * q^53 - 12 * q^58 + 44 * q^61 - 48 * q^65 - 32 * q^68 + 36 * q^82 - 16 * q^85 - 32 * q^89 - 48 * q^90 - 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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