LMFDB - Embedded newform 1308.1.br.a.785.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1308,1,Mod(5,1308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1308, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([0, 27, 38]))

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

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

chi := DirichletCharacter("1308.5");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1308 = 2^{2} \cdot 3 \cdot 109 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1308.br (of order $54$, degree $18$, 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:	$0.652777036524$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\Q(\zeta_{54})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{9} + 1 $ x^18 - x^9 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

Embedding invariants

Embedding label			785.1
Root			$0.0581448 - 0.998308i$ of defining polynomial
Character	$\chi$	$=$	1308.785
Dual form			1308.1.br.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.286803 - 0.957990i) q^{3} +(-1.22650 + 0.615969i) q^{7} +(-0.835488 + 0.549509i) q^{9} +O(q^{10})$	$q+(-0.286803 - 0.957990i) q^{3} +(-1.22650 + 0.615969i) q^{7} +(-0.835488 + 0.549509i) q^{9} +(-1.67948 + 0.843467i) q^{13} +(-1.05138 - 0.882215i) q^{19} +(0.941855 + 0.998308i) q^{21} +(-0.993238 - 0.116093i) q^{25} +(0.766044 + 0.642788i) q^{27} +(-1.67948 - 0.843467i) q^{31} +(0.0971586 + 1.66815i) q^{37} +(1.28971 + 1.36702i) q^{39} +(1.86668 - 0.679415i) q^{43} +(0.527715 - 0.708845i) q^{49} +(-0.543613 + 1.26024i) q^{57} +(1.65968 - 0.193988i) q^{61} +(0.686242 - 1.18861i) q^{63} +(-0.893633 - 0.448799i) q^{67} +(-0.342534 + 1.14414i) q^{73} +(0.173648 + 0.984808i) q^{75} +(-0.997837 + 0.656288i) q^{79} +(0.396080 - 0.918216i) q^{81} +(1.54033 - 2.06902i) q^{91} +(-0.326352 + 1.85083i) q^{93} +(-0.290162 - 0.190842i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q+O(q^{10}) $ 18 * q	$ 18 q + 18 q^{21} - 9 q^{37} - 9 q^{91} - 9 q^{93}+O(q^{100}) $ 18 * q + 18 * q^21 - 9 * q^37 - 9 * q^91 - 9 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1308\mathbb{Z}\right)^\times$.

$n$	$437$	$655$	$769$
$\chi(n)$	$-1$	$1$	$e\left(\frac{8}{27}\right)$

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			0
$2$		0			0
$3$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$3$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$4$		0			0
$5$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$5$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$6$		0			0
$7$	−1.22650	+	0.615969i	−1.22650	+	0.615969i	−0.939693	−	0.342020i	$-0.888889\pi$
$7$	−1.22650	+	0.615969i	−1.22650	+	0.615969i	−0.286803	+	0.957990i	$0.592593\pi$
$8$		0			0
$9$	−0.835488	+	0.549509i	−0.835488	+	0.549509i
$10$		0			0
$11$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$11$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$12$		0			0
$13$	−1.67948	+	0.843467i	−1.67948	+	0.843467i	−0.686242	+	0.727374i	$0.740741\pi$
$13$	−1.67948	+	0.843467i	−1.67948	+	0.843467i	−0.993238	+	0.116093i	$0.962963\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$17$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$18$		0			0
$19$	−1.05138	−	0.882215i	−1.05138	−	0.882215i	−0.0581448	−	0.998308i	$-0.518519\pi$
$19$	−1.05138	−	0.882215i	−1.05138	−	0.882215i	−0.993238	+	0.116093i	$0.962963\pi$
$20$		0			0
$21$	0.941855	+	0.998308i	0.941855	+	0.998308i
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		0			0
$25$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$26$		0			0
$27$	0.766044	+	0.642788i	0.766044	+	0.642788i
$28$		0			0
$29$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$29$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$30$		0			0
$31$	−1.67948	−	0.843467i	−1.67948	−	0.843467i	−0.993238	−	0.116093i	$-0.962963\pi$
$31$	−1.67948	−	0.843467i	−1.67948	−	0.843467i	−0.686242	−	0.727374i	$-0.740741\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.0971586	+	1.66815i	0.0971586	+	1.66815i	0.597159	+	0.802123i	$0.296296\pi$
$37$	0.0971586	+	1.66815i	0.0971586	+	1.66815i	−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	1.28971	+	1.36702i	1.28971	+	1.36702i
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	1.86668	−	0.679415i	1.86668	−	0.679415i	0.893633	−	0.448799i	$-0.148148\pi$
$43$	1.86668	−	0.679415i	1.86668	−	0.679415i	0.973045	−	0.230616i	$-0.0740741\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$47$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$48$		0			0
$49$	0.527715	−	0.708845i	0.527715	−	0.708845i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$53$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.543613	+	1.26024i	−0.543613	+	1.26024i
$58$		0			0
$59$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$59$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$60$		0			0
$61$	1.65968	−	0.193988i	1.65968	−	0.193988i	0.766044	−	0.642788i	$-0.222222\pi$
$61$	1.65968	−	0.193988i	1.65968	−	0.193988i	0.893633	+	0.448799i	$0.148148\pi$
$62$		0			0
$63$	0.686242	−	1.18861i	0.686242	−	1.18861i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.893633	−	0.448799i	−0.893633	−	0.448799i	−0.0581448	−	0.998308i	$-0.518519\pi$
$67$	−0.893633	−	0.448799i	−0.893633	−	0.448799i	−0.835488	+	0.549509i	$0.814815\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$71$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$72$		0			0
$73$	−0.342534	+	1.14414i	−0.342534	+	1.14414i	0.597159	+	0.802123i	$0.296296\pi$
$73$	−0.342534	+	1.14414i	−0.342534	+	1.14414i	−0.939693	+	0.342020i	$0.888889\pi$
$74$		0			0
$75$	0.173648	+	0.984808i	0.173648	+	0.984808i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.997837	+	0.656288i	−0.997837	+	0.656288i	−0.939693	−	0.342020i	$-0.888889\pi$
$79$	−0.997837	+	0.656288i	−0.997837	+	0.656288i	−0.0581448	+	0.998308i	$0.518519\pi$
$80$		0			0
$81$	0.396080	−	0.918216i	0.396080	−	0.918216i
$82$		0			0
$83$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$83$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$89$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$90$		0			0
$91$	1.54033	−	2.06902i	1.54033	−	2.06902i
$92$		0			0
$93$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.290162	−	0.190842i	−0.290162	−	0.190842i	0.396080	−	0.918216i	$-0.370370\pi$
$97$	−0.290162	−	0.190842i	−0.290162	−	0.190842i	−0.686242	+	0.727374i	$0.740741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−1.18624	−	1.59340i	−1.18624	−	1.59340i	−0.686242	−	0.727374i	$-0.740741\pi$
$103$	−1.18624	−	1.59340i	−1.18624	−	1.59340i	−0.500000	−	0.866025i	$-0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$107$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$108$		0			0
$109$	0.766044	−	0.642788i	0.766044	−	0.642788i
$109$	0.766044	−	0.642788i	0.766044	−	0.642788i
$110$		0			0
$111$	1.57020	−	0.571507i	1.57020	−	0.571507i
$112$		0			0
$113$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$113$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.939693	−	1.62760i	0.939693	−	1.62760i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.18624	+	1.59340i	−1.18624	+	1.59340i	−0.500000	+	0.866025i	$0.666667\pi$
$127$	−1.18624	+	1.59340i	−1.18624	+	1.59340i	−0.686242	+	0.727374i	$0.740741\pi$
$128$		0			0
$129$	−1.18624	−	1.59340i	−1.18624	−	1.59340i
$130$		0			0
$131$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$131$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$132$		0			0
$133$	1.83293	+	0.434414i	1.83293	+	0.434414i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$137$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$138$		0			0
$139$	−0.661840	+	0.435299i	−0.661840	+	0.435299i	−0.835488	−	0.549509i	$-0.814815\pi$
$139$	−0.661840	+	0.435299i	−0.661840	+	0.435299i	0.173648	+	0.984808i	$0.444444\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.830416	−	0.302247i	−0.830416	−	0.302247i
$148$		0			0
$149$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$149$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$150$		0			0
$151$	1.06728	+	0.536009i	1.06728	+	0.536009i	0.893633	−	0.448799i	$-0.148148\pi$
$151$	1.06728	+	0.536009i	1.06728	+	0.536009i	0.173648	+	0.984808i	$0.444444\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.786803	+	0.0919641i	−0.786803	+	0.0919641i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−0.786803	+	0.0919641i	−0.786803	+	0.0919641i	−0.286803	+	0.957990i	$0.592593\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.539014	+	0.196185i	0.539014	+	0.196185i	0.597159	−	0.802123i	$-0.296296\pi$
$163$	0.539014	+	0.196185i	0.539014	+	0.196185i	−0.0581448	+	0.998308i	$0.518519\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$167$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$168$		0			0
$169$	1.51206	−	2.03105i	1.51206	−	2.03105i
$170$		0			0
$171$	1.36320	+	0.159336i	1.36320	+	0.159336i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	1.28971	−	0.469417i	1.28971	−	0.469417i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$179$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$180$		0			0
$181$	−0.113155	−	1.94280i	−0.113155	−	1.94280i	−0.286803	−	0.957990i	$-0.592593\pi$
$181$	−0.113155	−	1.94280i	−0.113155	−	1.94280i	0.173648	−	0.984808i	$-0.444444\pi$
$182$		0			0
$183$	−0.661840	−	1.53432i	−0.661840	−	1.53432i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.33549	−	0.316516i	−1.33549	−	0.316516i
$190$		0			0
$191$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$191$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$192$		0			0
$193$	−1.77518	−	0.207489i	−1.77518	−	0.207489i	−0.835488	−	0.549509i	$-0.814815\pi$
$193$	−1.77518	−	0.207489i	−1.77518	−	0.207489i	−0.939693	+	0.342020i	$0.888889\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$197$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$198$		0			0
$199$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$199$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$200$		0			0
$201$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.36912	−	0.687600i	1.36912	−	0.687600i	0.396080	−	0.918216i	$-0.370370\pi$
$211$	1.36912	−	0.687600i	1.36912	−	0.687600i	0.973045	+	0.230616i	$0.0740741\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.57942			2.57942
$218$		0			0
$219$	1.19432			1.19432
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.103920	+	1.78424i	−0.103920	+	1.78424i	0.396080	+	0.918216i	$0.370370\pi$
$223$	−0.103920	+	1.78424i	−0.103920	+	1.78424i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	0.893633	−	0.448799i	0.893633	−	0.448799i
$226$		0			0
$227$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$227$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$228$		0			0
$229$	−0.439408	+	1.46773i	−0.439408	+	1.46773i	0.396080	+	0.918216i	$0.370370\pi$
$229$	−0.439408	+	1.46773i	−0.439408	+	1.46773i	−0.835488	+	0.549509i	$0.814815\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$233$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.914900	+	0.767692i	0.914900	+	0.767692i
$238$		0			0
$239$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$239$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$240$		0			0
$241$	−0.0890830	+	0.0747496i	−0.0890830	+	0.0747496i	−0.686242	−	0.727374i	$-0.740741\pi$
$241$	−0.0890830	+	0.0747496i	−0.0890830	+	0.0747496i	0.597159	+	0.802123i	$0.296296\pi$
$242$		0			0
$243$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.50990	+	0.594856i	2.50990	+	0.594856i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$257$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$258$		0			0
$259$	−1.14669	−	1.98613i	−1.14669	−	1.98613i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$269$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$270$		0			0
$271$	0.00676164	−	0.116093i	0.00676164	−	0.116093i	−0.993238	−	0.116093i	$-0.962963\pi$
$271$	0.00676164	−	0.116093i	0.00676164	−	0.116093i	1.00000			$0$
$272$		0			0
$273$	−2.42387	−	0.882215i	−2.42387	−	0.882215i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.337935	−	0.0800921i	0.337935	−	0.0800921i	−0.0581448	−	0.998308i	$-0.518519\pi$
$277$	0.337935	−	0.0800921i	0.337935	−	0.0800921i	0.396080	+	0.918216i	$0.370370\pi$
$278$		0			0
$279$	1.86668	−	0.218183i	1.86668	−	0.218183i
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	0.707900	+	1.64110i	0.707900	+	1.64110i	0.766044	+	0.642788i	$0.222222\pi$
$283$	0.707900	+	1.64110i	0.707900	+	1.64110i	−0.0581448	+	0.998308i	$0.518519\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$290$		0			0
$291$	−0.0996057	+	0.332706i	−0.0996057	+	0.332706i
$292$		0			0
$293$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$293$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.87097	+	1.98312i	−1.87097	+	1.98312i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.473045	+	0.635410i	0.473045	+	0.635410i	0.973045	−	0.230616i	$-0.0740741\pi$
$307$	0.473045	+	0.635410i	0.473045	+	0.635410i	−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$	−1.18624	+	1.59340i	−1.18624	+	1.59340i
$310$		0			0
$311$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$311$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$312$		0			0
$313$	0.393633	−	0.417226i	0.393633	−	0.417226i	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	0.393633	−	0.417226i	0.393633	−	0.417226i	0.893633	+	0.448799i	$0.148148\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$317$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.76604	−	0.642788i	1.76604	−	0.642788i
$326$		0			0
$327$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.52173	+	1.27688i	−1.52173	+	1.27688i	−0.686242	+	0.727374i	$0.740741\pi$
$331$	−1.52173	+	1.27688i	−1.52173	+	1.27688i	−0.835488	+	0.549509i	$0.814815\pi$
$332$		0			0
$333$	−0.997837	−	1.34033i	−0.997837	−	1.34033i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.479241	+	1.60078i	0.479241	+	1.60078i	0.766044	+	0.642788i	$0.222222\pi$
$337$	0.479241	+	1.60078i	0.479241	+	1.60078i	−0.286803	+	0.957990i	$0.592593\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.0277152	−	0.157181i	0.0277152	−	0.157181i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$347$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$348$		0			0
$349$	0.0971586	+	1.66815i	0.0971586	+	1.66815i	0.597159	+	0.802123i	$0.296296\pi$
$349$	0.0971586	+	1.66815i	0.0971586	+	1.66815i	−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$	−1.82873	−	0.433416i	−1.82873	−	0.433416i
$352$		0			0
$353$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$353$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$359$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$360$		0			0
$361$	0.153455	+	0.870285i	0.153455	+	0.870285i
$362$		0			0
$363$	−0.286803	+	0.957990i	−0.286803	+	0.957990i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.28004	−	0.841897i	−1.28004	−	0.841897i	−0.286803	−	0.957990i	$-0.592593\pi$
$367$	−1.28004	−	0.841897i	−1.28004	−	0.841897i	−0.993238	+	0.116093i	$0.962963\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$373$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0460600	+	0.106779i	−0.0460600	+	0.106779i	−0.939693	−	0.342020i	$-0.888889\pi$
$379$	−0.0460600	+	0.106779i	−0.0460600	+	0.106779i	0.893633	+	0.448799i	$0.148148\pi$
$380$		0			0
$381$	1.86668	+	0.679415i	1.86668	+	0.679415i
$382$		0			0
$383$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$383$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.18624	+	1.59340i	−1.18624	+	1.59340i
$388$		0			0
$389$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$389$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.286803	−	0.957990i	$-0.592593\pi$
$398$		0			0
$399$	−0.109528	−	1.88052i	−0.109528	−	1.88052i
$400$		0			0
$401$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$401$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$402$		0			0
$403$	3.53209			3.53209
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.28004	−	1.07408i	−1.28004	−	1.07408i	−0.993238	−	0.116093i	$-0.962963\pi$
$409$	−1.28004	−	1.07408i	−1.28004	−	1.07408i	−0.286803	−	0.957990i	$-0.592593\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.606829	+	0.509190i	0.606829	+	0.509190i
$418$		0			0
$419$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$419$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$420$		0			0
$421$	−0.113155	+	0.0268182i	−0.113155	+	0.0268182i	−0.286803	−	0.957990i	$-0.592593\pi$
$421$	−0.113155	+	0.0268182i	−0.113155	+	0.0268182i	0.173648	+	0.984808i	$0.444444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.91610	+	1.26024i	−1.91610	+	1.26024i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$431$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$432$		0			0
$433$	0.164512	+	0.549509i	0.164512	+	0.549509i	1.00000			$0$
$433$	0.164512	+	0.549509i	0.164512	+	0.549509i	−0.835488	+	0.549509i	$0.814815\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.539014	+	1.80043i	0.539014	+	1.80043i	0.597159	+	0.802123i	$0.296296\pi$
$439$	0.539014	+	1.80043i	0.539014	+	1.80043i	−0.0581448	+	0.998308i	$0.518519\pi$
$440$		0			0
$441$	−0.0513832	+	0.882215i	−0.0513832	+	0.882215i
$442$		0			0
$443$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$443$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$449$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.207391	−	1.17617i	0.207391	−	1.17617i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.238329	−	0.252614i	−0.238329	−	0.252614i	0.597159	−	0.802123i	$-0.296296\pi$
$457$	−0.238329	−	0.252614i	−0.238329	−	0.252614i	−0.835488	+	0.549509i	$0.814815\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$461$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$462$		0			0
$463$	−0.439408	−	0.368707i	−0.439408	−	0.368707i	0.396080	−	0.918216i	$-0.370370\pi$
$463$	−0.439408	−	0.368707i	−0.439408	−	0.368707i	−0.835488	+	0.549509i	$0.814815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$467$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$468$		0			0
$469$	1.37248			1.37248
$470$		0			0
$471$	0.313758	+	0.727374i	0.313758	+	0.727374i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.941855	+	0.998308i	0.941855	+	0.998308i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$479$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$480$		0			0
$481$	−1.57020	−	2.71967i	−1.57020	−	2.71967i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.569728	−	0.0665916i	0.569728	−	0.0665916i	0.173648	−	0.984808i	$-0.444444\pi$
$487$	0.569728	−	0.0665916i	0.569728	−	0.0665916i	0.396080	+	0.918216i	$0.370370\pi$
$488$		0			0
$489$	0.0333522	−	0.572636i	0.0333522	−	0.572636i
$490$		0			0
$491$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$491$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.597159	+	1.03431i	−0.597159	+	1.03431i	0.396080	+	0.918216i	$0.370370\pi$
$499$	−0.597159	+	1.03431i	−0.597159	+	1.03431i	−0.993238	+	0.116093i	$0.962963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$503$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.37939	−	0.866025i	−2.37939	−	0.866025i
$508$		0			0
$509$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$509$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$510$		0			0
$511$	−0.284641	−	1.61428i	−0.284641	−	1.61428i
$512$		0			0
$513$	−0.238329	−	1.35163i	−0.238329	−	1.35163i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$521$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$522$		0			0
$523$	0.109277	+	1.87621i	0.109277	+	1.87621i	0.396080	+	0.918216i	$0.370370\pi$
$523$	0.109277	+	1.87621i	0.109277	+	1.87621i	−0.286803	+	0.957990i	$0.592593\pi$
$524$		0			0
$525$	−0.819590	−	1.10090i	−0.819590	−	1.10090i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.173648	−	0.984808i	0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.49079	−	1.25092i	1.49079	−	1.25092i	0.597159	−	0.802123i	$-0.296296\pi$
$541$	1.49079	−	1.25092i	1.49079	−	1.25092i	0.893633	−	0.448799i	$-0.148148\pi$
$542$		0			0
$543$	−1.82873	+	0.665602i	−1.82873	+	0.665602i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.82873	+	0.665602i	−1.82873	+	0.665602i	−0.835488	+	0.549509i	$0.814815\pi$
$547$	−1.82873	+	0.665602i	−1.82873	+	0.665602i	−0.993238	+	0.116093i	$0.962963\pi$
$548$		0			0
$549$	−1.28004	+	1.07408i	−1.28004	+	1.07408i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.819590	−	1.41957i	0.819590	−	1.41957i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$557$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$558$		0			0
$559$	−2.56198	+	2.71554i	−2.56198	+	2.71554i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$563$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.0798028	+	1.37016i	0.0798028	+	1.37016i
$568$		0			0
$569$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$569$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$570$		0			0
$571$	0.393633	−	0.417226i	0.393633	−	0.417226i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	0.393633	−	0.417226i	0.393633	−	0.417226i	0.893633	+	0.448799i	$0.148148\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.344948	−	1.95630i	−0.344948	−	1.95630i	−0.286803	−	0.957990i	$-0.592593\pi$
$577$	−0.344948	−	1.95630i	−0.344948	−	1.95630i	−0.0581448	−	0.998308i	$-0.518519\pi$
$578$		0			0
$579$	0.310355	+	1.76011i	0.310355	+	1.76011i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$587$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$588$		0			0
$589$	1.02166	+	2.36847i	1.02166	+	2.36847i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$593$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.396080	+	0.918216i	−0.396080	+	0.918216i
$598$		0			0
$599$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$599$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$600$		0			0
$601$	0.0581448	−	0.998308i	0.0581448	−	0.998308i	−0.835488	−	0.549509i	$-0.814815\pi$
$601$	0.0581448	−	0.998308i	0.0581448	−	0.998308i	0.893633	−	0.448799i	$-0.148148\pi$
$602$		0			0
$603$	0.993238	−	0.116093i	0.993238	−	0.116093i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.344948	−	0.0403186i	−0.344948	−	0.0403186i	−0.0581448	−	0.998308i	$-0.518519\pi$
$607$	−0.344948	−	0.0403186i	−0.344948	−	0.0403186i	−0.286803	+	0.957990i	$0.592593\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.835488	+	1.44711i	0.835488	+	1.44711i	0.893633	+	0.448799i	$0.148148\pi$
$613$	0.835488	+	1.44711i	0.835488	+	1.44711i	−0.0581448	+	0.998308i	$0.518519\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$617$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$618$		0			0
$619$	−0.661840	−	1.53432i	−0.661840	−	1.53432i	−0.835488	−	0.549509i	$-0.814815\pi$
$619$	−0.661840	−	1.53432i	−0.661840	−	1.53432i	0.173648	−	0.984808i	$-0.444444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.973045	+	0.230616i	0.973045	+	0.230616i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.05138	+	0.882215i	−1.05138	+	0.882215i	−0.993238	−	0.116093i	$-0.962963\pi$
$631$	−1.05138	+	0.882215i	−1.05138	+	0.882215i	−0.0581448	+	0.998308i	$0.518519\pi$
$632$		0			0
$633$	−1.05138	−	1.11440i	−1.05138	−	1.11440i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.288400	+	1.63560i	−0.288400	+	1.63560i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$641$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$642$		0			0
$643$	0.164512	−	0.549509i	0.164512	−	0.549509i	−0.835488	−	0.549509i	$-0.814815\pi$
$643$	0.164512	−	0.549509i	0.164512	−	0.549509i	1.00000			$0$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$647$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.739787	−	2.47106i	−0.739787	−	2.47106i
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.342534	−	1.14414i	−0.342534	−	1.14414i
$658$		0			0
$659$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$659$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$660$		0			0
$661$	−0.512593	+	0.257434i	−0.512593	+	0.257434i	−0.686242	−	0.727374i	$-0.740741\pi$
$661$	−0.512593	+	0.257434i	−0.512593	+	0.257434i	0.173648	+	0.984808i	$0.444444\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.73909	−	0.412172i	1.73909	−	0.412172i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.36912	+	1.14883i	1.36912	+	1.14883i	0.973045	+	0.230616i	$0.0740741\pi$
$673$	1.36912	+	1.14883i	1.36912	+	1.14883i	0.396080	+	0.918216i	$0.370370\pi$
$674$		0			0
$675$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$676$		0			0
$677$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$677$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$678$		0			0
$679$	0.473435	+	0.0553367i	0.473435	+	0.0553367i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$683$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.53209			1.53209
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.0201935	−	0.346709i	−0.0201935	−	0.346709i	−0.993238	−	0.116093i	$-0.962963\pi$
$691$	−0.0201935	−	0.346709i	−0.0201935	−	0.346709i	0.973045	−	0.230616i	$-0.0740741\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$701$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$702$		0			0
$703$	1.36952	−	1.83958i	1.36952	−	1.83958i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$709$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	0.473045	−	1.09664i	0.473045	−	1.09664i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$719$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$720$		0			0
$721$	2.43641	+	1.22361i	2.43641	+	1.22361i
$722$		0			0
$723$	0.0971586	+	0.0639022i	0.0971586	+	0.0639022i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.342534	+	1.14414i	−0.342534	+	1.14414i	0.597159	+	0.802123i	$0.296296\pi$
$727$	−0.342534	+	1.14414i	−0.342534	+	1.14414i	−0.939693	+	0.342020i	$0.888889\pi$
$728$		0			0
$729$	0.173648	+	0.984808i	0.173648	+	0.984808i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.835488	−	0.549509i	0.835488	−	0.549509i	−0.0581448	−	0.998308i	$-0.518519\pi$
$733$	0.835488	−	0.549509i	0.835488	−	0.549509i	0.893633	+	0.448799i	$0.148148\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.973045	−	0.230616i	−0.973045	−	0.230616i	−0.286803	−	0.957990i	$-0.592593\pi$
$739$	−0.973045	−	0.230616i	−0.973045	−	0.230616i	−0.686242	+	0.727374i	$0.740741\pi$
$740$		0			0
$741$	−0.149980	−	2.57506i	−0.149980	−	2.57506i
$742$		0			0
$743$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$743$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0971586	+	0.0639022i	0.0971586	+	0.0639022i	0.597159	−	0.802123i	$-0.296296\pi$
$751$	0.0971586	+	0.0639022i	0.0971586	+	0.0639022i	−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.914900	+	1.22892i	0.914900	+	1.22892i	0.973045	+	0.230616i	$0.0740741\pi$
$757$	0.914900	+	1.22892i	0.914900	+	1.22892i	−0.0581448	+	0.998308i	$0.518519\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$761$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$762$		0			0
$763$	−0.543613	+	1.26024i	−0.543613	+	1.26024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.16212	+	1.56100i	1.16212	+	1.56100i	0.766044	+	0.642788i	$0.222222\pi$
$769$	1.16212	+	1.56100i	1.16212	+	1.56100i	0.396080	+	0.918216i	$0.370370\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$773$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$774$		0			0
$775$	1.57020	+	1.03274i	1.57020	+	1.03274i
$776$		0			0
$777$	−1.57382	+	1.66815i	−1.57382	+	1.66815i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.33549	−	0.316516i	−1.33549	−	0.316516i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	−1.33549	−	0.316516i	−1.33549	−	0.316516i	−0.835488	+	0.549509i	$0.814815\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.62377	+	1.72568i	−2.62377	+	1.72568i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$797$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	0.115503	−	0.0135004i	0.115503	−	0.0135004i	−0.0581448	−	0.998308i	$-0.518519\pi$
$811$	0.115503	−	0.0135004i	0.115503	−	0.0135004i	0.173648	+	0.984808i	$0.444444\pi$
$812$		0			0
$813$	−0.113155	+	0.0268182i	−0.113155	+	0.0268182i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.56198	−	0.932486i	−2.56198	−	0.932486i
$818$		0			0
$819$	−0.149980	+	2.57506i	−0.149980	+	2.57506i
$820$		0			0
$821$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$821$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$822$		0			0
$823$	1.16212	−	1.56100i	1.16212	−	1.56100i	0.396080	−	0.918216i	$-0.370370\pi$
$823$	1.16212	−	1.56100i	1.16212	−	1.56100i	0.766044	−	0.642788i	$-0.222222\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	1.86668	−	0.679415i	1.86668	−	0.679415i	0.893633	−	0.448799i	$-0.148148\pi$
$829$	1.86668	−	0.679415i	1.86668	−	0.679415i	0.973045	−	0.230616i	$-0.0740741\pi$
$830$		0			0
$831$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.744386	−	1.72568i	−0.744386	−	1.72568i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	0.893633	+	0.448799i	0.893633	+	0.448799i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.36320	+	0.159336i	1.36320	+	0.159336i
$848$		0			0
$849$	1.36912	−	1.14883i	1.36912	−	1.14883i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.0890830	−	0.0747496i	−0.0890830	−	0.0747496i	0.597159	−	0.802123i	$-0.296296\pi$
$853$	−0.0890830	−	0.0747496i	−0.0890830	−	0.0747496i	−0.686242	+	0.727374i	$0.740741\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$857$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$858$		0			0
$859$	1.36912	−	0.687600i	1.36912	−	0.687600i	0.396080	−	0.918216i	$-0.370370\pi$
$859$	1.36912	−	0.687600i	1.36912	−	0.687600i	0.973045	+	0.230616i	$0.0740741\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$863$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.0581448	+	0.998308i	−0.0581448	+	0.998308i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.87939			1.87939
$872$		0			0
$873$	0.347296			0.347296
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.0333522	−	0.572636i	0.0333522	−	0.572636i	−0.939693	−	0.342020i	$-0.888889\pi$
$877$	0.0333522	−	0.572636i	0.0333522	−	0.572636i	0.973045	−	0.230616i	$-0.0740741\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$881$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$882$		0			0
$883$	0.0333522	−	0.111404i	0.0333522	−	0.111404i	−0.939693	−	0.342020i	$-0.888889\pi$
$883$	0.0333522	−	0.111404i	0.0333522	−	0.111404i	0.973045	+	0.230616i	$0.0740741\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$887$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$888$		0			0
$889$	0.473435	−	2.68499i	0.473435	−	2.68499i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	2.43641	+	1.22361i	2.43641	+	1.22361i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.661840	−	1.53432i	−0.661840	−	1.53432i	−0.835488	−	0.549509i	$-0.814815\pi$
$907$	−0.661840	−	1.53432i	−0.661840	−	1.53432i	0.173648	−	0.984808i	$-0.444444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$911$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.569728	+	0.0665916i	0.569728	+	0.0665916i	0.396080	−	0.918216i	$-0.370370\pi$
$919$	0.569728	+	0.0665916i	0.569728	+	0.0665916i	0.173648	+	0.984808i	$0.444444\pi$
$920$		0			0
$921$	0.473045	−	0.635410i	0.473045	−	0.635410i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0971586	−	1.66815i	0.0971586	−	1.66815i
$926$		0			0
$927$	1.86668	+	0.679415i	1.86668	+	0.679415i
$928$		0			0
$929$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$929$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$930$		0			0
$931$	−1.18018	+	0.279709i	−1.18018	+	0.279709i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.744386	−	1.72568i	−0.744386	−	1.72568i	−0.686242	−	0.727374i	$-0.740741\pi$
$937$	−0.744386	−	1.72568i	−0.744386	−	1.72568i	−0.0581448	−	0.998308i	$-0.518519\pi$
$938$		0			0
$939$	−0.512593	−	0.257434i	−0.512593	−	0.257434i
$940$		0			0
$941$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$941$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$947$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$948$		0			0
$949$	−0.389768	−	2.21048i	−0.389768	−	2.21048i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$953$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.51206	+	2.03105i	1.51206	+	2.03105i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	−0.835488	−	0.549509i	$-0.814815\pi$
$967$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	0.597159	+	0.802123i	$0.296296\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$971$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$972$		0			0
$973$	0.543613	−	0.941565i	0.543613	−	0.941565i
$974$		0			0
$975$	−1.12229	−	1.50750i	−1.12229	−	1.50750i
$976$		0			0
$977$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$977$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.286803	+	0.957990i	−0.286803	+	0.957990i
$982$		0			0
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.479241	+	1.60078i	0.479241	+	1.60078i	0.766044	+	0.642788i	$0.222222\pi$
$991$	0.479241	+	1.60078i	0.479241	+	1.60078i	−0.286803	+	0.957990i	$0.592593\pi$
$992$		0			0
$993$	1.65968	+	1.09159i	1.65968	+	1.09159i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.266044	−	1.50881i	0.266044	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$997$	0.266044	−	1.50881i	0.266044	−	1.50881i	0.766044	−	0.642788i	$-0.222222\pi$
$998$		0			0
$999$	−0.997837	+	1.34033i	−0.997837	+	1.34033i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1308.1.br.a.785.1	yes	18
3.2	odd	2	CM	1308.1.br.a.785.1	yes	18
109.5	even	27	inner	1308.1.br.a.5.1	✓	18
327.5	odd	54	inner	1308.1.br.a.5.1	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1308.1.br.a.5.1	✓	18	109.5	even	27	inner
1308.1.br.a.5.1	✓	18	327.5	odd	54	inner
1308.1.br.a.785.1	yes	18	1.1	even	1	trivial
1308.1.br.a.785.1	yes	18	3.2	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:	\( N \)	\(=\)	\( 1308 = 2^{2} \cdot 3 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1308.br (of order \(54\), degree \(18\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.286803 - 0.957990i) q^{3} +(-1.22650 + 0.615969i) q^{7} +(-0.835488 + 0.549509i) q^{9} +O(q^{10})\)	\(q+(-0.286803 - 0.957990i) q^{3} +(-1.22650 + 0.615969i) q^{7} +(-0.835488 + 0.549509i) q^{9} +(-1.67948 + 0.843467i) q^{13} +(-1.05138 - 0.882215i) q^{19} +(0.941855 + 0.998308i) q^{21} +(-0.993238 - 0.116093i) q^{25} +(0.766044 + 0.642788i) q^{27} +(-1.67948 - 0.843467i) q^{31} +(0.0971586 + 1.66815i) q^{37} +(1.28971 + 1.36702i) q^{39} +(1.86668 - 0.679415i) q^{43} +(0.527715 - 0.708845i) q^{49} +(-0.543613 + 1.26024i) q^{57} +(1.65968 - 0.193988i) q^{61} +(0.686242 - 1.18861i) q^{63} +(-0.893633 - 0.448799i) q^{67} +(-0.342534 + 1.14414i) q^{73} +(0.173648 + 0.984808i) q^{75} +(-0.997837 + 0.656288i) q^{79} +(0.396080 - 0.918216i) q^{81} +(1.54033 - 2.06902i) q^{91} +(-0.326352 + 1.85083i) q^{93} +(-0.290162 - 0.190842i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q+O(q^{10}) \) 18 * q	\( 18 q + 18 q^{21} - 9 q^{37} - 9 q^{91} - 9 q^{93}+O(q^{100}) \) 18 * q + 18 * q^21 - 9 * q^37 - 9 * q^91 - 9 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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