LMFDB - Embedded newform 1813.1.cb.a.165.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1813,1,Mod(79,1813)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1813, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([12, 23]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1813.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1813 = 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1813.cb (of order $36$, degree $12$, not 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:	$0.904804867904$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{36}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{36} - \cdots)$

Embedding invariants

Embedding label			165.1
Root			$0.984808 + 0.173648i$ of defining polynomial
Character	$\chi$	$=$	1813.165
Dual form			1813.1.cb.a.912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.173648 - 1.98481i) q^{2} +(-2.92450 - 0.515668i) q^{4} +(-1.01567 + 3.79053i) q^{8} +(0.173648 + 0.984808i) q^{9} +O(q^{10})$	$q+(0.173648 - 1.98481i) q^{2} +(-2.92450 - 0.515668i) q^{4} +(-1.01567 + 3.79053i) q^{8} +(0.173648 + 0.984808i) q^{9} +1.53209i q^{11} +(4.55657 + 1.65846i) q^{16} +(1.98481 - 0.173648i) q^{18} +(3.04090 + 0.266044i) q^{22} +(0.597672 + 0.597672i) q^{23} +(-0.342020 + 0.939693i) q^{25} +(-1.10806 - 0.296905i) q^{29} +(2.42450 - 5.19936i) q^{32} -2.96962i q^{36} +(-0.866025 + 0.500000i) q^{37} +(-0.366025 - 0.366025i) q^{43} +(0.790050 - 4.48059i) q^{44} +(1.29005 - 1.08248i) q^{46} +(1.80572 + 0.842020i) q^{50} +(0.118782 + 0.673648i) q^{53} +(-0.781713 + 2.14774i) q^{58} +(-5.69936 - 3.29053i) q^{64} +(0.592396 + 1.62760i) q^{67} +(0.173648 - 0.984808i) q^{71} +(-3.90931 - 0.342020i) q^{72} +(0.842020 + 1.80572i) q^{74} +(1.48481 - 1.03967i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(-0.790050 + 0.662930i) q^{86} +(-5.80742 - 1.55609i) q^{88} +(-1.43969 - 2.05609i) q^{92} +(-1.50881 + 0.266044i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} - 6 q^{8}+O(q^{10}) $ 12 * q - 12 * q^4 - 6 * q^8	$ 12 q - 12 q^{4} - 6 q^{8} + 12 q^{16} + 12 q^{18} + 6 q^{32} + 6 q^{43} - 6 q^{44} + 6 q^{58} - 18 q^{64} - 12 q^{72} + 6 q^{74} + 6 q^{79} + 6 q^{86} - 6 q^{88} - 6 q^{92}+O(q^{100}) $ 12 * q - 12 * q^4 - 6 * q^8 + 12 * q^16 + 12 * q^18 + 6 * q^32 + 6 * q^43 - 6 * q^44 + 6 * q^58 - 18 * q^64 - 12 * q^72 + 6 * q^74 + 6 * q^79 + 6 * q^86 - 6 * q^88 - 6 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1813\mathbb{Z}\right)^\times$.

$n$	$297$	$1667$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{36}\right)$

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 $

$2$	0.173648	−	1.98481i	0.173648	−	1.98481i		−	1.00000i	$-0.5\pi$
$2$	0.173648	−	1.98481i	0.173648	−	1.98481i	0.173648	−	0.984808i	$-0.444444\pi$
$3$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$3$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$4$	−2.92450	−	0.515668i	−2.92450	−	0.515668i
$5$		0			0		−0.573576	−	0.819152i	$-0.694444\pi$
$5$		0			0		0.573576	+	0.819152i	$0.305556\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.01567	+	3.79053i	−1.01567	+	3.79053i
$9$	0.173648	+	0.984808i	0.173648	+	0.984808i
$10$		0			0
$11$			1.53209i			1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$11$			1.53209i			1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$12$		0			0
$13$		0			0		0.422618	−	0.906308i	$-0.361111\pi$
$13$		0			0		−0.422618	+	0.906308i	$0.638889\pi$
$14$		0			0
$15$		0			0
$16$	4.55657	+	1.65846i	4.55657	+	1.65846i
$17$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$17$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$18$	1.98481	−	0.173648i	1.98481	−	0.173648i
$19$		0			0		0.422618	−	0.906308i	$-0.361111\pi$
$19$		0			0		−0.422618	+	0.906308i	$0.638889\pi$
$20$		0			0
$21$		0			0
$22$	3.04090	+	0.266044i	3.04090	+	0.266044i
$23$	0.597672	+	0.597672i	0.597672	+	0.597672i	0.939693	−	0.342020i	$-0.111111\pi$
$23$	0.597672	+	0.597672i	0.597672	+	0.597672i	−0.342020	+	0.939693i	$0.611111\pi$
$24$		0			0
$25$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.10806	−	0.296905i	−1.10806	−	0.296905i	−0.342020	−	0.939693i	$-0.611111\pi$
$29$	−1.10806	−	0.296905i	−1.10806	−	0.296905i	−0.766044	+	0.642788i	$0.777778\pi$
$30$		0			0
$31$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$31$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$32$	2.42450	−	5.19936i	2.42450	−	5.19936i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	2.96962i		−	2.96962i
$37$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$37$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$41$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$43$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$44$	0.790050	−	4.48059i	0.790050	−	4.48059i
$45$		0			0
$46$	1.29005	−	1.08248i	1.29005	−	1.08248i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$		0			0
$50$	1.80572	+	0.842020i	1.80572	+	0.842020i
$51$		0			0
$52$		0			0
$53$	0.118782	+	0.673648i	0.118782	+	0.673648i	0.984808	+	0.173648i	$0.0555556\pi$
$53$	0.118782	+	0.673648i	0.118782	+	0.673648i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.781713	+	2.14774i	−0.781713	+	2.14774i
$59$		0			0		−0.819152	−	0.573576i	$-0.805556\pi$
$59$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$60$		0			0
$61$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$61$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$62$		0			0
$63$		0			0
$64$	−5.69936	−	3.29053i	−5.69936	−	3.29053i
$65$		0			0
$66$		0			0
$67$	0.592396	+	1.62760i	0.592396	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$67$	0.592396	+	1.62760i	0.592396	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.173648	−	0.984808i	0.173648	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$71$	0.173648	−	0.984808i	0.173648	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$72$	−3.90931	−	0.342020i	−3.90931	−	0.342020i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	0.842020	+	1.80572i	0.842020	+	1.80572i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.48481	−	1.03967i	1.48481	−	1.03967i	0.500000	−	0.866025i	$-0.333333\pi$
$79$	1.48481	−	1.03967i	1.48481	−	1.03967i	0.984808	−	0.173648i	$-0.0555556\pi$
$80$		0			0
$81$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$82$		0			0
$83$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$83$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$84$		0			0
$85$		0			0
$86$	−0.790050	+	0.662930i	−0.790050	+	0.662930i
$87$		0			0
$88$	−5.80742	−	1.55609i	−5.80742	−	1.55609i
$89$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$89$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$90$		0			0
$91$		0			0
$92$	−1.43969	−	2.05609i	−1.43969	−	2.05609i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$97$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$98$		0			0
$99$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$100$	1.48481	−	2.57176i	1.48481	−	2.57176i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$103$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$104$		0			0
$105$		0			0
$106$	1.35769	−	0.118782i	1.35769	−	0.118782i
$107$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$107$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−0.692377	−	1.48481i	−0.692377	−	1.48481i	−0.866025	−	0.500000i	$-0.833333\pi$
$109$	−0.692377	−	1.48481i	−0.692377	−	1.48481i	0.173648	−	0.984808i	$-0.444444\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.15846	+	0.811160i	1.15846	+	0.811160i	0.984808	−	0.173648i	$-0.0555556\pi$
$113$	1.15846	+	0.811160i	1.15846	+	0.811160i	0.173648	+	0.984808i	$0.444444\pi$
$114$		0			0
$115$		0			0
$116$	3.08743	+	1.43969i	3.08743	+	1.43969i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.34730			−1.34730
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.984808	−	0.826352i	0.984808	−	0.826352i		−	1.00000i	$-0.5\pi$
$127$	0.984808	−	0.826352i	0.984808	−	0.826352i	0.984808	+	0.173648i	$0.0555556\pi$
$128$	−4.23022	+	6.04138i	−4.23022	+	6.04138i
$129$		0			0
$130$		0			0
$131$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$131$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$132$		0			0
$133$		0			0
$134$	3.33333	−	0.893164i	3.33333	−	0.893164i
$135$		0			0
$136$		0			0
$137$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$139$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$140$		0			0
$141$		0			0
$142$	−1.92450	−	0.515668i	−1.92450	−	0.515668i
$143$		0			0
$144$	−0.842020	+	4.77533i	−0.842020	+	4.77533i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	2.79053	−	1.01567i	2.79053	−	1.01567i
$149$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$149$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$150$		0			0
$151$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.342020	−	0.939693i	$-0.388889\pi$
$151$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.866025	−	0.500000i	$-0.166667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$157$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$158$	−1.80572	−	3.12760i	−1.80572	−	3.12760i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.515668	+	1.92450i	0.515668	+	1.92450i
$163$	1.28171	−	0.597672i	1.28171	−	0.597672i	0.342020	−	0.939693i	$-0.388889\pi$
$163$	1.28171	−	0.597672i	1.28171	−	0.597672i	0.939693	+	0.342020i	$0.111111\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.573576	−	0.819152i	$-0.694444\pi$
$167$		0			0		0.573576	+	0.819152i	$0.305556\pi$
$168$		0			0
$169$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$170$		0			0
$171$		0			0
$172$	0.881694	+	1.25919i	0.881694	+	1.25919i
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$		0			0
$176$	−2.54090	+	6.98107i	−2.54090	+	6.98107i
$177$		0			0
$178$		0			0
$179$	−0.0451151	−	0.168372i	−0.0451151	−	0.168372i	0.939693	−	0.342020i	$-0.111111\pi$
$179$	−0.0451151	−	0.168372i	−0.0451151	−	0.168372i	−0.984808	+	0.173648i	$0.944444\pi$
$180$		0			0
$181$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$181$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$182$		0			0
$183$		0			0
$184$	−2.87253	+	1.65846i	−2.87253	+	1.65846i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.424024	+	1.58248i	−0.424024	+	1.58248i	0.342020	+	0.939693i	$0.388889\pi$
$191$	−0.424024	+	1.58248i	−0.424024	+	1.58248i	−0.766044	+	0.642788i	$0.777778\pi$
$192$		0			0
$193$	−0.123257	−	0.123257i	−0.123257	−	0.123257i	0.642788	−	0.766044i	$-0.277778\pi$
$193$	−0.123257	−	0.123257i	−0.123257	−	0.123257i	−0.766044	+	0.642788i	$0.777778\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$197$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$198$	0.266044	+	3.04090i	0.266044	+	3.04090i
$199$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$199$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$200$	−3.21455	−	2.25085i	−3.21455	−	2.25085i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.484808	+	0.692377i	−0.484808	+	0.692377i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.342020	−	0.592396i	−0.342020	−	0.592396i	0.642788	−	0.766044i	$-0.277778\pi$
$211$	−0.342020	−	0.592396i	−0.342020	−	0.592396i	−0.984808	+	0.173648i	$0.944444\pi$
$212$		−	2.03134i		−	2.03134i
$213$		0			0
$214$	0.489282	−	0.489282i	0.489282	−	0.489282i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−3.06729	+	1.11640i	−3.06729	+	1.11640i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$226$	1.81116	−	2.15846i	1.81116	−	2.15846i
$227$		0			0		0.422618	−	0.906308i	$-0.361111\pi$
$227$		0			0		−0.422618	+	0.906308i	$0.638889\pi$
$228$		0			0
$229$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$229$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$230$		0			0
$231$		0			0
$232$	2.25085	−	3.89859i	2.25085	−	3.89859i
$233$	1.62760	+	0.939693i	1.62760	+	0.939693i	0.984808	+	0.173648i	$0.0555556\pi$
$233$	1.62760	+	0.939693i	1.62760	+	0.939693i	0.642788	+	0.766044i	$0.277778\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.597672	−	1.28171i	−0.597672	−	1.28171i	−0.939693	−	0.342020i	$-0.888889\pi$
$239$	−0.597672	−	1.28171i	−0.597672	−	1.28171i	0.342020	−	0.939693i	$-0.388889\pi$
$240$		0			0
$241$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$241$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$242$	−0.233956	+	2.67412i	−0.233956	+	2.67412i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$251$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$252$		0			0
$253$	−0.915687	+	0.915687i	−0.915687	+	0.915687i
$254$	−1.46914	−	2.09815i	−1.46914	−	2.09815i
$255$		0			0
$256$	6.21503	+	5.21503i	6.21503	+	5.21503i
$257$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$257$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.0999810	−	1.14279i	0.0999810	−	1.14279i
$262$		0			0
$263$	0.673648	+	0.118782i	0.673648	+	0.118782i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	0.673648	+	0.118782i	0.673648	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.893164	−	5.06538i	−0.893164	−	5.06538i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$271$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$272$		0			0
$273$		0			0
$274$	−0.173648	+	1.98481i	−0.173648	+	1.98481i
$275$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$276$		0			0
$277$	−0.173648	+	0.0151922i	−0.173648	+	0.0151922i	−0.173648	−	0.984808i	$-0.555556\pi$
$277$	−0.173648	+	0.0151922i	−0.173648	+	0.0151922i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.515668	−	0.0451151i	−0.515668	−	0.0451151i	−0.173648	−	0.984808i	$-0.555556\pi$
$281$	−0.515668	−	0.0451151i	−0.515668	−	0.0451151i	−0.342020	+	0.939693i	$0.611111\pi$
$282$		0			0
$283$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$283$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$284$	−1.01567	+	2.79053i	−1.01567	+	2.79053i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	5.54138	+	1.48481i	5.54138	+	1.48481i
$289$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$293$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$294$		0			0
$295$		0			0
$296$	−1.01567	−	3.79053i	−1.01567	−	3.79053i
$297$		0			0
$298$	3.12760	−	1.45842i	3.12760	−	1.45842i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2.64774	−	2.64774i	−2.64774	−	2.64774i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$311$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$312$		0			0
$313$		0			0		−0.819152	−	0.573576i	$-0.805556\pi$
$313$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$314$		0			0
$315$		0			0
$316$	−4.87845	+	2.27486i	−4.87845	+	2.27486i
$317$	0.118782	−	0.326352i	0.118782	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$317$	0.118782	−	0.326352i	0.118782	−	0.326352i	0.984808	+	0.173648i	$0.0555556\pi$
$318$		0			0
$319$	0.454885	−	1.69765i	0.454885	−	1.69765i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	2.92450	−	0.515668i	2.92450	−	0.515668i
$325$		0			0
$326$	−0.963698	−	2.64774i	−0.963698	−	2.64774i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.14279	−	0.0999810i	−1.14279	−	0.0999810i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.14279	−	0.0999810i	−1.14279	−	0.0999810i	−0.642788	+	0.766044i	$0.722222\pi$
$332$		0			0
$333$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.592396	+	1.62760i	0.592396	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$337$	0.592396	+	1.62760i	0.592396	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$338$	−1.63207	+	1.14279i	−1.63207	+	1.14279i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.75919	−	1.01567i	1.75919	−	1.01567i
$345$		0			0
$346$		0			0
$347$	1.86603	+	0.500000i	1.86603	+	0.500000i	1.00000			$0$
$347$	1.86603	+	0.500000i	1.86603	+	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$349$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$350$		0			0
$351$		0			0
$352$	7.96588	+	3.71455i	7.96588	+	3.71455i
$353$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$353$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.342020	+	0.0603074i	−0.342020	+	0.0603074i
$359$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$359$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$360$		0			0
$361$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$367$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$368$	1.73212	+	3.71455i	1.73212	+	3.71455i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.342020	−	0.939693i	−0.342020	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$373$	−0.342020	−	0.939693i	−0.342020	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.20805	+	0.439693i	1.20805	+	0.439693i	0.866025	−	0.500000i	$-0.166667\pi$
$379$	1.20805	+	0.439693i	1.20805	+	0.439693i	0.342020	+	0.939693i	$0.388889\pi$
$380$		0			0
$381$		0			0
$382$	3.06729	+	1.11640i	3.06729	+	1.11640i
$383$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$383$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$384$		0			0
$385$		0			0
$386$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$387$	0.296905	−	0.424024i	0.296905	−	0.424024i
$388$		0			0
$389$	−0.123257	−	1.40883i	−0.123257	−	1.40883i	−0.766044	−	0.642788i	$-0.777778\pi$
$389$	−0.123257	−	1.40883i	−0.123257	−	1.40883i	0.642788	−	0.766044i	$-0.277778\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	3.73022	−	0.326352i	3.73022	−	0.326352i
$395$		0			0
$396$	4.54971			4.54971
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−3.11688	+	3.71455i	−3.11688	+	3.71455i
$401$	1.86603	+	0.500000i	1.86603	+	0.500000i	1.00000			$0$
$401$	1.86603	+	0.500000i	1.86603	+	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$408$		0			0
$409$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$409$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.29005	+	1.08248i	1.29005	+	1.08248i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$419$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$420$		0			0
$421$	−0.218763	−	0.816436i	−0.218763	−	0.816436i	−0.984808	−	0.173648i	$-0.944444\pi$
$421$	−0.218763	−	0.816436i	−0.218763	−	0.816436i	0.766044	−	0.642788i	$-0.222222\pi$
$422$	−1.23518	+	0.575976i	−1.23518	+	0.575976i
$423$		0			0
$424$	−2.67412	−	0.233956i	−2.67412	−	0.233956i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.662930	−	0.790050i	−0.662930	−	0.790050i
$429$		0			0
$430$		0			0
$431$	−0.657980	−	0.939693i	−0.657980	−	0.939693i	0.342020	−	0.939693i	$-0.388889\pi$
$431$	−0.657980	−	0.939693i	−0.657980	−	0.939693i	−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$	1.25919	+	4.69936i	1.25919	+	4.69936i
$437$		0			0
$438$		0			0
$439$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$439$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$443$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.984808	+	0.173648i	$0.944444\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.142788	+	1.63207i	0.142788	+	1.63207i	0.642788	+	0.766044i	$0.277778\pi$
$449$	0.142788	+	1.63207i	0.142788	+	1.63207i	−0.500000	+	0.866025i	$0.666667\pi$
$450$	−0.515668	+	1.92450i	−0.515668	+	1.92450i
$451$		0			0
$452$	−2.96962	−	2.96962i	−2.96962	−	2.96962i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.168372	−	1.92450i	−0.168372	−	1.92450i	−0.342020	−	0.939693i	$-0.611111\pi$
$457$	−0.168372	−	1.92450i	−0.168372	−	1.92450i	0.173648	−	0.984808i	$-0.444444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$461$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$462$		0			0
$463$	0.939693	−	0.657980i	0.939693	−	0.657980i		−	1.00000i	$-0.5\pi$
$463$	0.939693	−	0.657980i	0.939693	−	0.657980i	0.939693	+	0.342020i	$0.111111\pi$
$464$	−4.55657	−	3.19054i	−4.55657	−	3.19054i
$465$		0			0
$466$	2.14774	−	3.06729i	2.14774	−	3.06729i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.560783	−	0.560783i	0.560783	−	0.560783i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.642788	+	0.233956i	−0.642788	+	0.233956i
$478$	−2.64774	+	0.963698i	−2.64774	+	0.963698i
$479$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$479$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	3.94017	+	0.694758i	3.94017	+	0.694758i
$485$		0			0
$486$		0			0
$487$	−0.168372	+	0.0451151i	−0.168372	+	0.0451151i	−0.342020	−	0.939693i	$-0.611111\pi$
$487$	−0.168372	+	0.0451151i	−0.168372	+	0.0451151i	0.173648	+	0.984808i	$0.444444\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.642788	+	1.11334i	−0.642788	+	1.11334i	0.342020	+	0.939693i	$0.388889\pi$
$491$	−0.642788	+	1.11334i	−0.642788	+	1.11334i	−0.984808	+	0.173648i	$0.944444\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.766044	−	1.64279i	0.766044	−	1.64279i		−	1.00000i	$-0.5\pi$
$499$	0.766044	−	1.64279i	0.766044	−	1.64279i	0.766044	−	0.642788i	$-0.222222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$503$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$504$		0			0
$505$		0			0
$506$	1.65846	+	1.97647i	1.65846	+	1.97647i
$507$		0			0
$508$	−3.30619	+	1.90883i	−3.30619	+	1.90883i
$509$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$509$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$510$		0			0
$511$		0			0
$512$	6.21503	−	6.21503i	6.21503	−	6.21503i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$521$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$522$	−2.25085	−	0.396886i	−2.25085	−	0.396886i
$523$		0			0		−0.573576	−	0.819152i	$-0.694444\pi$
$523$		0			0		0.573576	+	0.819152i	$0.305556\pi$
$524$		0			0
$525$		0			0
$526$	0.352738	−	1.31644i	0.352738	−	1.31644i
$527$		0			0
$528$		0			0
$529$		−	0.285575i		−	0.285575i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−6.77112	+	0.592396i	−6.77112	+	0.592396i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$541$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.86603	−	0.500000i	−1.86603	−	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$	−1.86603	−	0.500000i	−1.86603	−	0.500000i	−1.00000			$\pi$
$548$	2.92450	+	0.515668i	2.92450	+	0.515668i
$549$		0			0
$550$	−1.29005	+	2.76652i	−1.29005	+	2.76652i
$551$		0			0
$552$		0			0
$553$		0			0
$554$			0.347296i			0.347296i
$555$		0			0
$556$		0			0
$557$	−0.157980	+	0.0736672i	−0.157980	+	0.0736672i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−0.157980	+	0.0736672i	−0.157980	+	0.0736672i	0.342020	+	0.939693i	$0.388889\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.179090	+	1.01567i	−0.179090	+	1.01567i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	3.55657	+	1.65846i	3.55657	+	1.65846i
$569$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$571$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.766044	+	0.357212i	−0.766044	+	0.357212i
$576$	2.25085	−	6.18417i	2.25085	−	6.18417i
$577$		0			0		−0.819152	−	0.573576i	$-0.805556\pi$
$577$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$578$	−0.515668	+	1.92450i	−0.515668	+	1.92450i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.03209	+	0.181985i	−1.03209	+	0.181985i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$587$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−4.77533	+	0.842020i	−4.77533	+	0.842020i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$	−1.75919	−	4.83333i	−1.75919	−	4.83333i
$597$		0			0
$598$		0			0
$599$	1.85083	−	0.673648i	1.85083	−	0.673648i	0.866025	−	0.500000i	$-0.166667\pi$
$599$	1.85083	−	0.673648i	1.85083	−	0.673648i	0.984808	−	0.173648i	$-0.0555556\pi$
$600$		0			0
$601$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$601$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$602$		0			0
$603$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$604$	−4.27533	+	3.58743i	−4.27533	+	3.58743i
$605$		0			0
$606$		0			0
$607$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$607$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.93969	+	0.342020i	−1.93969	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$613$	−1.93969	+	0.342020i	−1.93969	+	0.342020i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.93969	+	0.342020i	−1.93969	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$617$	−1.93969	+	0.342020i	−1.93969	+	0.342020i	−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.58248	+	1.10806i	1.58248	+	1.10806i	0.939693	+	0.342020i	$0.111111\pi$
$631$	1.58248	+	1.10806i	1.58248	+	1.10806i	0.642788	+	0.766044i	$0.277778\pi$
$632$	2.43284	+	6.68417i	2.43284	+	6.68417i
$633$		0			0
$634$	−0.627119	−	0.292431i	−0.627119	−	0.292431i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−3.29053	−	1.19765i	−3.29053	−	1.19765i
$639$	1.00000			1.00000
$640$		0			0
$641$	−1.20805	−	0.439693i	−1.20805	−	0.439693i	−0.342020	−	0.939693i	$-0.611111\pi$
$641$	−1.20805	−	0.439693i	−1.20805	−	0.439693i	−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$643$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$647$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$648$	−0.342020	−	3.90931i	−0.342020	−	3.90931i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−4.05657	+	1.08695i	−4.05657	+	1.08695i
$653$	−0.515668	+	0.0451151i	−0.515668	+	0.0451151i	−0.342020	−	0.939693i	$-0.611111\pi$
$653$	−0.515668	+	0.0451151i	−0.515668	+	0.0451151i	−0.173648	+	0.984808i	$0.555556\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.26604	−	1.50881i	1.26604	−	1.50881i	0.500000	−	0.866025i	$-0.333333\pi$
$659$	1.26604	−	1.50881i	1.26604	−	1.50881i	0.766044	−	0.642788i	$-0.222222\pi$
$660$		0			0
$661$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$661$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$662$	−0.396886	+	2.25085i	−0.396886	+	2.25085i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.63207	+	1.14279i	−1.63207	+	1.14279i
$667$	−0.484808	−	0.839712i	−0.484808	−	0.839712i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.939693	+	0.342020i	$0.111111\pi$
$673$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.500000	+	0.866025i	$0.333333\pi$
$674$	3.33333	−	0.893164i	3.33333	−	0.893164i
$675$		0			0
$676$	1.48481	+	2.57176i	1.48481	+	2.57176i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.80572	+	0.157980i	1.80572	+	0.157980i	0.939693	−	0.342020i	$-0.111111\pi$
$683$	1.80572	+	0.157980i	1.80572	+	0.157980i	0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.06078	−	2.27486i	−1.06078	−	2.27486i
$689$		0			0
$690$		0			0
$691$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$691$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$692$		0			0
$693$		0			0
$694$	1.31644	−	3.61688i	1.31644	−	3.61688i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.03967	−	1.48481i	1.03967	−	1.48481i	0.173648	−	0.984808i	$-0.444444\pi$
$701$	1.03967	−	1.48481i	1.03967	−	1.48481i	0.866025	−	0.500000i	$-0.166667\pi$
$702$		0			0
$703$		0			0
$704$	5.04138	−	8.73192i	5.04138	−	8.73192i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.218763	+	0.816436i	−0.218763	+	0.816436i	0.766044	+	0.642788i	$0.222222\pi$
$709$	−0.218763	+	0.816436i	−0.218763	+	0.816436i	−0.984808	+	0.173648i	$0.944444\pi$
$710$		0			0
$711$	1.28171	+	1.28171i	1.28171	+	1.28171i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.0451151	+	0.515668i	0.0451151	+	0.515668i
$717$		0			0
$718$	0.566812	+	0.396886i	0.566812	+	0.396886i
$719$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$719$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$720$		0			0
$721$		0			0
$722$	−1.63207	+	1.14279i	−1.63207	+	1.14279i
$723$		0			0
$724$		0			0
$725$	0.657980	−	0.939693i	0.657980	−	0.939693i
$726$		0			0
$727$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$727$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$728$		0			0
$729$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$733$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$734$		0			0
$735$		0			0
$736$	4.55657	−	1.65846i	4.55657	−	1.65846i
$737$	−2.49362	+	0.907604i	−2.49362	+	0.907604i
$738$		0			0
$739$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$739$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$743$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$744$		0			0
$745$		0			0
$746$	−1.92450	+	0.515668i	−1.92450	+	0.515668i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.300767	+	0.173648i	0.300767	+	0.173648i	0.642788	−	0.766044i	$-0.277778\pi$
$751$	0.300767	+	0.173648i	0.300767	+	0.173648i	−0.342020	+	0.939693i	$0.611111\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.816436	+	1.75085i	0.816436	+	1.75085i	0.642788	+	0.766044i	$0.277778\pi$
$757$	0.816436	+	1.75085i	0.816436	+	1.75085i	0.173648	+	0.984808i	$0.444444\pi$
$758$	1.08248	−	2.32139i	1.08248	−	2.32139i
$759$		0			0
$760$		0			0
$761$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$761$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$762$		0			0
$763$		0			0
$764$	2.05609	−	4.40931i	2.05609	−	4.40931i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$769$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$770$		0			0
$771$		0			0
$772$	0.296905	+	0.424024i	0.296905	+	0.424024i
$773$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$773$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$774$	−0.790050	−	0.662930i	−0.790050	−	0.662930i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−2.81766			−2.81766
$779$		0			0
$780$		0			0
$781$	1.50881	+	0.266044i	1.50881	+	0.266044i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		−	5.58105i		−	5.58105i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.524005	−	5.98941i	0.524005	−	5.98941i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$797$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$798$		0			0
$799$		0			0
$800$	4.05657	+	4.05657i	4.05657	+	4.05657i
$801$		0			0
$802$	1.31644	−	3.61688i	1.31644	−	3.61688i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.0736672	−	0.157980i	0.0736672	−	0.157980i	−0.866025	−	0.500000i	$-0.833333\pi$
$809$	0.0736672	−	0.157980i	0.0736672	−	0.157980i	0.939693	+	0.342020i	$0.111111\pi$
$810$		0			0
$811$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$811$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$812$		0			0
$813$		0			0
$814$	−2.76652	+	1.29005i	−2.76652	+	1.29005i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$821$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$822$		0			0
$823$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$823$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.03967	−	0.484808i	−1.03967	−	0.484808i	−0.173648	−	0.984808i	$-0.555556\pi$
$827$	−1.03967	−	0.484808i	−1.03967	−	0.484808i	−0.866025	+	0.500000i	$0.833333\pi$
$828$	1.77486	−	1.77486i	1.77486	−	1.77486i
$829$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$829$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$839$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$840$		0			0
$841$	0.273629	+	0.157980i	0.273629	+	0.157980i
$842$	−1.65846	+	0.292431i	−1.65846	+	0.292431i
$843$		0			0
$844$	0.694758	+	1.90883i	0.694758	+	1.90883i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−0.575976	+	3.26652i	−0.575976	+	3.26652i
$849$		0			0
$850$		0			0
$851$	−0.816436	−	0.218763i	−0.816436	−	0.218763i
$852$		0			0
$853$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$853$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$854$		0			0
$855$		0			0
$856$	−1.11640	+	0.781713i	−1.11640	+	0.781713i
$857$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$857$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$858$		0			0
$859$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$859$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$860$		0			0
$861$		0			0
$862$	−1.97937	+	1.14279i	−1.97937	+	1.14279i
$863$	1.50881	−	1.26604i	1.50881	−	1.26604i	0.642788	−	0.766044i	$-0.277778\pi$
$863$	1.50881	−	1.26604i	1.50881	−	1.26604i	0.866025	−	0.500000i	$-0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.59287	+	2.27486i	1.59287	+	2.27486i
$870$		0			0
$871$		0			0
$872$	6.33143	−	1.11640i	6.33143	−	1.11640i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$877$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$881$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$882$		0			0
$883$	1.63207	−	0.142788i	1.63207	−	0.142788i	0.766044	−	0.642788i	$-0.222222\pi$
$883$	1.63207	−	0.142788i	1.63207	−	0.142788i	0.866025	+	0.500000i	$0.166667\pi$
$884$		0			0
$885$		0			0
$886$	1.58248	+	3.39364i	1.58248	+	3.39364i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.524005	−	1.43969i	−0.524005	−	1.43969i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	3.26414			3.26414
$899$		0			0
$900$	2.79053	+	1.01567i	2.79053	+	1.01567i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−4.25133	+	3.56729i	−4.25133	+	3.56729i
$905$		0			0
$906$		0			0
$907$	0.0151922	+	0.173648i	0.0151922	+	0.173648i	1.00000			$0$
$907$	0.0151922	+	0.173648i	0.0151922	+	0.173648i	−0.984808	+	0.173648i	$0.944444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.500000	+	0.133975i	−0.500000	+	0.133975i	−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−0.500000	+	0.133975i	−0.500000	+	0.133975i			1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	−3.84900			−3.84900
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.58248	+	0.424024i	1.58248	+	0.424024i	0.939693	−	0.342020i	$-0.111111\pi$
$919$	1.58248	+	0.424024i	1.58248	+	0.424024i	0.642788	+	0.766044i	$0.277778\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$926$	−1.14279	−	1.97937i	−1.14279	−	1.97937i
$927$		0			0
$928$	−4.23022	+	5.04138i	−4.23022	+	5.04138i
$929$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$929$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$930$		0			0
$931$		0			0
$932$	−4.27533	−	3.58743i	−4.27533	−	3.58743i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$937$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$941$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−1.01567	−	1.21043i	−1.01567	−	1.21043i
$947$	−0.766044	−	1.64279i	−0.766044	−	1.64279i	−0.766044	−	0.642788i	$-0.777778\pi$
$947$	−0.766044	−	1.64279i	−0.766044	−	1.64279i		−	1.00000i	$-0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.524005	−	1.43969i	0.524005	−	1.43969i	−0.342020	−	0.939693i	$-0.611111\pi$
$953$	0.524005	−	1.43969i	0.524005	−	1.43969i	0.866025	−	0.500000i	$-0.166667\pi$
$954$	0.352738	+	1.31644i	0.352738	+	1.31644i
$955$		0			0
$956$	1.08695	+	4.05657i	1.08695	+	4.05657i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.866025	−	0.500000i	0.866025	−	0.500000i
$962$		0			0
$963$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.0451151	+	0.515668i	0.0451151	+	0.515668i	0.984808	+	0.173648i	$0.0555556\pi$
$967$	0.0451151	+	0.515668i	0.0451151	+	0.515668i	−0.939693	+	0.342020i	$0.888889\pi$
$968$	1.36841	−	5.10696i	1.36841	−	5.10696i
$969$		0			0
$970$		0			0
$971$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$971$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$972$		0			0
$973$		0			0
$974$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$975$		0			0
$976$		0			0
$977$	1.48481	+	1.03967i	1.48481	+	1.03967i	0.984808	+	0.173648i	$0.0555556\pi$
$977$	1.48481	+	1.03967i	1.48481	+	1.03967i	0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.34202	−	0.939693i	1.34202	−	0.939693i
$982$	2.09815	+	1.46914i	2.09815	+	1.46914i
$983$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$983$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	0.437527i		−	0.437527i
$990$		0			0
$991$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$991$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$997$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$998$	−3.12760	−	1.80572i	−3.12760	−	1.80572i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1813.1.cb.a.165.1		12
7.2	even	3		1813.1.cf.a.128.1		12
7.3	odd	6		1813.1.cc.a.1275.1	yes	12
7.4	even	3		1813.1.cc.a.1275.1	yes	12
7.5	odd	6		1813.1.cf.a.128.1		12
7.6	odd	2	CM	1813.1.cb.a.165.1		12
37.24	odd	36		1813.1.cf.a.949.1		12
259.24	even	36		1813.1.cc.a.246.1	✓	12
259.61	even	36	inner	1813.1.cb.a.912.1		12
259.135	odd	36	inner	1813.1.cb.a.912.1		12
259.172	odd	36		1813.1.cc.a.246.1	✓	12
259.209	even	36		1813.1.cf.a.949.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1813.1.cb.a.165.1		12	1.1	even	1	trivial
1813.1.cb.a.165.1		12	7.6	odd	2	CM
1813.1.cb.a.912.1		12	259.61	even	36	inner
1813.1.cb.a.912.1		12	259.135	odd	36	inner
1813.1.cc.a.246.1	✓	12	259.24	even	36
1813.1.cc.a.246.1	✓	12	259.172	odd	36
1813.1.cc.a.1275.1	yes	12	7.3	odd	6
1813.1.cc.a.1275.1	yes	12	7.4	even	3
1813.1.cf.a.128.1		12	7.2	even	3
1813.1.cf.a.128.1		12	7.5	odd	6
1813.1.cf.a.949.1		12	37.24	odd	36
1813.1.cf.a.949.1		12	259.209	even	36

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:	\( N \)	\(=\)	\( 1813 = 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1813.cb (of order \(36\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.173648 - 1.98481i) q^{2} +(-2.92450 - 0.515668i) q^{4} +(-1.01567 + 3.79053i) q^{8} +(0.173648 + 0.984808i) q^{9} +O(q^{10})\)	\(q+(0.173648 - 1.98481i) q^{2} +(-2.92450 - 0.515668i) q^{4} +(-1.01567 + 3.79053i) q^{8} +(0.173648 + 0.984808i) q^{9} +1.53209i q^{11} +(4.55657 + 1.65846i) q^{16} +(1.98481 - 0.173648i) q^{18} +(3.04090 + 0.266044i) q^{22} +(0.597672 + 0.597672i) q^{23} +(-0.342020 + 0.939693i) q^{25} +(-1.10806 - 0.296905i) q^{29} +(2.42450 - 5.19936i) q^{32} -2.96962i q^{36} +(-0.866025 + 0.500000i) q^{37} +(-0.366025 - 0.366025i) q^{43} +(0.790050 - 4.48059i) q^{44} +(1.29005 - 1.08248i) q^{46} +(1.80572 + 0.842020i) q^{50} +(0.118782 + 0.673648i) q^{53} +(-0.781713 + 2.14774i) q^{58} +(-5.69936 - 3.29053i) q^{64} +(0.592396 + 1.62760i) q^{67} +(0.173648 - 0.984808i) q^{71} +(-3.90931 - 0.342020i) q^{72} +(0.842020 + 1.80572i) q^{74} +(1.48481 - 1.03967i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(-0.790050 + 0.662930i) q^{86} +(-5.80742 - 1.55609i) q^{88} +(-1.43969 - 2.05609i) q^{92} +(-1.50881 + 0.266044i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} - 6 q^{8}+O(q^{10}) \) 12 * q - 12 * q^4 - 6 * q^8	\( 12 q - 12 q^{4} - 6 q^{8} + 12 q^{16} + 12 q^{18} + 6 q^{32} + 6 q^{43} - 6 q^{44} + 6 q^{58} - 18 q^{64} - 12 q^{72} + 6 q^{74} + 6 q^{79} + 6 q^{86} - 6 q^{88} - 6 q^{92}+O(q^{100}) \) 12 * q - 12 * q^4 - 6 * q^8 + 12 * q^16 + 12 * q^18 + 6 * q^32 + 6 * q^43 - 6 * q^44 + 6 * q^58 - 18 * q^64 - 12 * q^72 + 6 * q^74 + 6 * q^79 + 6 * q^86 - 6 * q^88 - 6 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more