LMFDB - Embedded newform 2548.1.de.b.779.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,1,Mod(51,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 26, 21]))

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

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

chi := DirichletCharacter("2548.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2548.de (of order $42$, degree $12$, 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:	$1.27161765219$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			779.1
Root			$-0.733052 + 0.680173i$ of defining polynomial
Character	$\chi$	$=$	2548.779
Dual form			2548.1.de.b.2391.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.988831 + 0.149042i) q^{2} +(0.955573 - 0.294755i) q^{4} +(0.955573 + 0.294755i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.365341 + 0.930874i) q^{9} +(0.603718 - 1.53825i) q^{11} +(0.623490 + 0.781831i) q^{13} +(-0.988831 - 0.149042i) q^{14} +(0.826239 - 0.563320i) q^{16} +(1.44973 - 1.34515i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-0.365341 + 0.632789i) q^{19} +(-0.367711 + 1.61105i) q^{22} +(-0.988831 - 0.149042i) q^{25} +(-0.733052 - 0.680173i) q^{26} +1.00000 q^{28} +(-0.0332580 - 0.145713i) q^{29} +(-0.623490 - 1.07992i) q^{31} +(-0.733052 + 0.680173i) q^{32} +(-1.23305 + 1.54620i) q^{34} +(0.623490 + 0.781831i) q^{36} +(0.266948 - 0.680173i) q^{38} +(0.123490 - 1.64786i) q^{44} +(-0.147791 + 0.0222759i) q^{47} +(0.826239 + 0.563320i) q^{49} +1.00000 q^{50} +(0.826239 + 0.563320i) q^{52} +(-0.955573 + 0.294755i) q^{53} +(-0.988831 + 0.149042i) q^{56} +(0.0546039 + 0.139129i) q^{58} +(0.142820 + 1.90580i) q^{59} +(-0.955573 - 0.294755i) q^{61} +(0.777479 + 0.974928i) q^{62} +(0.0747301 + 0.997204i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.733052 + 1.26968i) q^{67} +(0.988831 - 1.71271i) q^{68} +(0.440071 - 1.92808i) q^{71} +(-0.733052 - 0.680173i) q^{72} +(-0.162592 + 0.712362i) q^{76} +(1.03030 - 1.29196i) q^{77} +(-0.733052 + 0.680173i) q^{81} +(-0.277479 + 0.347948i) q^{83} +(0.123490 + 1.64786i) q^{88} +(0.365341 + 0.930874i) q^{91} +(0.142820 - 0.0440542i) q^{94} +(-0.900969 - 0.433884i) q^{98} +1.65248 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + q^{4} + q^{7} - 2 q^{8} + q^{9} - q^{11} - 2 q^{13} + q^{14} + q^{16} - q^{17} - 6 q^{18} - q^{19} + 2 q^{22} + q^{25} + q^{26} + 12 q^{28} + 2 q^{29} + 2 q^{31} + q^{32} - 5 q^{34}+ \cdots + 2 q^{99}+O(q^{100}) $ 12 * q + q^2 + q^4 + q^7 - 2 * q^8 + q^9 - q^11 - 2 * q^13 + q^14 + q^16 - q^17 - 6 * q^18 - q^19 + 2 * q^22 + q^25 + q^26 + 12 * q^28 + 2 * q^29 + 2 * q^31 + q^32 - 5 * q^34 - 2 * q^36 + 13 * q^38 - 8 * q^44 - q^47 + q^49 + 12 * q^50 + q^52 - q^53 + q^56 - q^58 - q^59 - q^61 + 10 * q^62 + q^63 - 2 * q^64 - q^67 - q^68 + 2 * q^71 + q^72 + 2 * q^76 + 2 * q^77 + q^81 - 4 * q^83 - 8 * q^88 + q^91 - q^94 - 2 * q^98 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$.

$n$	$197$	$885$	$1275$
$\chi(n)$	$-1$	$e\left(\frac{4}{21}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$2$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$3$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$3$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$4$	0.955573	−	0.294755i	0.955573	−	0.294755i
$5$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$5$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$6$		0			0
$7$	0.955573	+	0.294755i	0.955573	+	0.294755i
$7$	0.955573	+	0.294755i	0.955573	+	0.294755i
$8$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$9$	0.365341	+	0.930874i	0.365341	+	0.930874i
$10$		0			0
$11$	0.603718	−	1.53825i	0.603718	−	1.53825i	−0.222521	−	0.974928i	$-0.571429\pi$
$11$	0.603718	−	1.53825i	0.603718	−	1.53825i	0.826239	−	0.563320i	$-0.190476\pi$
$12$		0			0
$13$	0.623490	+	0.781831i	0.623490	+	0.781831i
$13$	0.623490	+	0.781831i	0.623490	+	0.781831i
$14$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$15$		0			0
$16$	0.826239	−	0.563320i	0.826239	−	0.563320i
$17$	1.44973	−	1.34515i	1.44973	−	1.34515i	0.623490	−	0.781831i	$-0.285714\pi$
$17$	1.44973	−	1.34515i	1.44973	−	1.34515i	0.826239	−	0.563320i	$-0.190476\pi$
$18$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$19$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$20$		0			0
$21$		0			0
$22$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$23$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$23$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$24$		0			0
$25$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$26$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$27$		0			0
$28$	1.00000			1.00000
$29$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$29$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$30$		0			0
$31$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$31$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$32$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$33$		0			0
$34$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$35$		0			0
$36$	0.623490	+	0.781831i	0.623490	+	0.781831i
$37$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$37$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$38$	0.266948	−	0.680173i	0.266948	−	0.680173i
$39$		0			0
$40$		0			0
$41$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$41$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$42$		0			0
$43$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$43$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$44$	0.123490	−	1.64786i	0.123490	−	1.64786i
$45$		0			0
$46$		0			0
$47$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i	−0.222521	−	0.974928i	$-0.571429\pi$
$47$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i	0.0747301	+	0.997204i	$0.476190\pi$
$48$		0			0
$49$	0.826239	+	0.563320i	0.826239	+	0.563320i
$50$	1.00000			1.00000
$51$		0			0
$52$	0.826239	+	0.563320i	0.826239	+	0.563320i
$53$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$53$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$54$		0			0
$55$		0			0
$56$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$57$		0			0
$58$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$59$	0.142820	+	1.90580i	0.142820	+	1.90580i	0.365341	+	0.930874i	$0.380952\pi$
$59$	0.142820	+	1.90580i	0.142820	+	1.90580i	−0.222521	+	0.974928i	$0.571429\pi$
$60$		0			0
$61$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.222521	−	0.974928i	$-0.571429\pi$
$61$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.733052	+	0.680173i	$0.761905\pi$
$62$	0.777479	+	0.974928i	0.777479	+	0.974928i
$63$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$64$	0.623490	−	0.781831i	0.623490	−	0.781831i
$65$		0			0
$66$		0			0
$67$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$67$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$68$	0.988831	−	1.71271i	0.988831	−	1.71271i
$69$		0			0
$70$		0			0
$71$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.0747301	−	0.997204i	$-0.476190\pi$
$71$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.365341	−	0.930874i	$-0.380952\pi$
$72$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$73$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$73$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$74$		0			0
$75$		0			0
$76$	−0.162592	+	0.712362i	−0.162592	+	0.712362i
$77$	1.03030	−	1.29196i	1.03030	−	1.29196i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$82$		0			0
$83$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$83$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.123490	+	1.64786i	0.123490	+	1.64786i
$89$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$89$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$90$		0			0
$91$	0.365341	+	0.930874i	0.365341	+	0.930874i
$92$		0			0
$93$		0			0
$94$	0.142820	−	0.0440542i	0.142820	−	0.0440542i
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$99$	1.65248			1.65248
$100$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$101$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.988831	−	0.149042i	$-0.952381\pi$
$101$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.500000	−	0.866025i	$-0.666667\pi$
$102$		0			0
$103$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$103$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$104$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$105$		0			0
$106$	0.900969	−	0.433884i	0.900969	−	0.433884i
$107$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$107$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$108$		0			0
$109$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$109$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$110$		0			0
$111$		0			0
$112$	0.955573	−	0.294755i	0.955573	−	0.294755i
$113$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.365341	−	0.930874i	$-0.380952\pi$
$113$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.826239	−	0.563320i	$-0.190476\pi$
$114$		0			0
$115$		0			0
$116$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$117$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$118$	−0.425270	−	1.86323i	−0.425270	−	1.86323i
$119$	1.78181	−	0.858075i	1.78181	−	0.858075i
$120$		0			0
$121$	−1.26868	−	1.17716i	−1.26868	−	1.17716i
$122$	0.988831	+	0.149042i	0.988831	+	0.149042i
$123$		0			0
$124$	−0.914101	−	0.848162i	−0.914101	−	0.848162i
$125$		0			0
$126$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$127$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$127$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$131$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$132$		0			0
$133$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$134$	−0.914101	−	1.14625i	−0.914101	−	1.14625i
$135$		0			0
$136$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$137$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$137$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$138$		0			0
$139$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$139$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$140$		0			0
$141$		0			0
$142$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$143$	1.57906	−	0.487076i	1.57906	−	0.487076i
$144$	0.826239	+	0.563320i	0.826239	+	0.563320i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$149$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$150$		0			0
$151$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.0747301	−	0.997204i	$-0.476190\pi$
$151$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.623490	+	0.781831i	$0.285714\pi$
$152$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$153$	1.78181	+	0.858075i	1.78181	+	0.858075i
$154$	−0.826239	+	1.43109i	−0.826239	+	1.43109i
$155$		0			0
$156$		0			0
$157$	0.123490	+	1.64786i	0.123490	+	1.64786i	0.623490	+	0.781831i	$0.285714\pi$
$157$	0.123490	+	1.64786i	0.123490	+	1.64786i	−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.623490	−	0.781831i	0.623490	−	0.781831i
$163$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.623490	−	0.781831i	$-0.285714\pi$
$163$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.955573	−	0.294755i	$-0.0952381\pi$
$164$		0			0
$165$		0			0
$166$	0.222521	−	0.385418i	0.222521	−	0.385418i
$167$	0.326239	+	1.42935i	0.326239	+	1.42935i	0.826239	+	0.563320i	$0.190476\pi$
$167$	0.326239	+	1.42935i	0.326239	+	1.42935i	−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$170$		0			0
$171$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$172$		0			0
$173$	1.07473	+	0.997204i	1.07473	+	0.997204i	1.00000			$0$
$173$	1.07473	+	0.997204i	1.07473	+	0.997204i	0.0747301	+	0.997204i	$0.476190\pi$
$174$		0			0
$175$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$176$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$177$		0			0
$178$		0			0
$179$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$179$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$180$		0			0
$181$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	0.0747301	+	0.997204i	$0.476190\pi$
$181$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	−0.988831	+	0.149042i	$0.952381\pi$
$182$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.19395	−	3.04213i	−1.19395	−	3.04213i
$188$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i
$189$		0			0
$190$		0			0
$191$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$191$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$192$		0			0
$193$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$193$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$194$		0			0
$195$		0			0
$196$	0.955573	+	0.294755i	0.955573	+	0.294755i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$199$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$199$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$200$	0.955573	−	0.294755i	0.955573	−	0.294755i
$201$		0			0
$202$	1.62349	+	0.781831i	1.62349	+	0.781831i
$203$	0.0111692	−	0.149042i	0.0111692	−	0.149042i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.955573	+	0.294755i	0.955573	+	0.294755i
$209$	0.752824	+	0.944011i	0.752824	+	0.944011i
$210$		0			0
$211$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$211$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$212$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.95557	+	0.294755i	1.95557	+	0.294755i
$222$		0			0
$223$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	0.365341	+	0.930874i	$0.380952\pi$
$223$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	−0.733052	+	0.680173i	$0.761905\pi$
$224$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$225$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$226$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$227$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$227$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$228$		0			0
$229$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$229$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$230$		0			0
$231$		0			0
$232$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$233$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.900969	+	0.433884i	$0.857143\pi$
$234$	0.365341	−	0.930874i	0.365341	−	0.930874i
$235$		0			0
$236$	0.698220	+	1.77904i	0.698220	+	1.77904i
$237$		0			0
$238$	−1.63402	+	1.11406i	−1.63402	+	1.11406i
$239$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.988831	−	0.149042i	$-0.952381\pi$
$239$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.733052	−	0.680173i	$-0.761905\pi$
$240$		0			0
$241$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$241$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$242$	1.42996	+	0.974928i	1.42996	+	0.974928i
$243$		0			0
$244$	−1.00000			−1.00000
$245$		0			0
$246$		0			0
$247$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$248$	1.03030	+	0.702449i	1.03030	+	0.702449i
$249$		0			0
$250$		0			0
$251$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$251$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$252$	0.365341	+	0.930874i	0.365341	+	0.930874i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.365341	−	0.930874i	0.365341	−	0.930874i
$257$	−0.425270	−	0.131178i	−0.425270	−	0.131178i	0.0747301	−	0.997204i	$-0.476190\pi$
$257$	−0.425270	−	0.131178i	−0.425270	−	0.131178i	−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$262$		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
$266$	0.455573	−	0.571270i	0.455573	−	0.571270i
$267$		0			0
$268$	1.07473	+	0.997204i	1.07473	+	0.997204i
$269$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.900969	−	0.433884i	$-0.857143\pi$
$269$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.988831	+	0.149042i	$0.952381\pi$
$270$		0			0
$271$	0.733052	+	0.680173i	0.733052	+	0.680173i	0.955573	−	0.294755i	$-0.0952381\pi$
$271$	0.733052	+	0.680173i	0.733052	+	0.680173i	−0.222521	+	0.974928i	$0.571429\pi$
$272$	0.440071	−	1.92808i	0.440071	−	1.92808i
$273$		0			0
$274$		0			0
$275$	−0.826239	+	1.43109i	−0.826239	+	1.43109i
$276$		0			0
$277$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	0.365341	+	0.930874i	$0.380952\pi$
$278$		0			0
$279$	0.777479	−	0.974928i	0.777479	−	0.974928i
$280$		0			0
$281$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$281$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$282$		0			0
$283$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$283$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$284$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$285$		0			0
$286$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$287$		0			0
$288$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$289$	0.217550	−	2.90301i	0.217550	−	2.90301i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$303$		0			0
$304$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$305$		0			0
$306$	−1.88980	−	0.582926i	−1.88980	−	0.582926i
$307$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$307$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$308$	0.603718	−	1.53825i	0.603718	−	1.53825i
$309$		0			0
$310$		0			0
$311$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$311$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$312$		0			0
$313$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$313$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$314$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$315$		0			0
$316$		0			0
$317$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$317$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$318$		0			0
$319$	−0.244221	−	0.0368104i	−0.244221	−	0.0368104i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.321552	+	1.40881i	0.321552	+	1.40881i
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$326$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$327$		0			0
$328$		0			0
$329$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$330$		0			0
$331$	−1.72188	−	0.531130i	−1.72188	−	0.531130i	−0.733052	−	0.680173i	$-0.761905\pi$
$331$	−1.72188	−	0.531130i	−1.72188	−	0.531130i	−0.988831	+	0.149042i	$0.952381\pi$
$332$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$333$		0			0
$334$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$335$		0			0
$336$		0			0
$337$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$337$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$338$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$339$		0			0
$340$		0			0
$341$	−2.03759	+	0.307117i	−2.03759	+	0.307117i
$342$	0.730682			0.730682
$343$	0.623490	+	0.781831i	0.623490	+	0.781831i
$344$		0			0
$345$		0			0
$346$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$347$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$347$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$348$		0			0
$349$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$349$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$350$	0.955573	+	0.294755i	0.955573	+	0.294755i
$351$		0			0
$352$	0.603718	+	1.53825i	0.603718	+	1.53825i
$353$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$353$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.500000	+	0.866025i	$0.666667\pi$
$359$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.988831	+	0.149042i	$0.952381\pi$
$360$		0			0
$361$	0.233052	+	0.403658i	0.233052	+	0.403658i
$362$	0.733052	−	1.26968i	0.733052	−	1.26968i
$363$		0			0
$364$	0.623490	+	0.781831i	0.623490	+	0.781831i
$365$		0			0
$366$		0			0
$367$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$367$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.00000			−1.00000
$372$		0			0
$373$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$373$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$374$	1.63402	+	2.83021i	1.63402	+	2.83021i
$375$		0			0
$376$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$377$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$378$		0			0
$379$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$379$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.162592	−	0.414278i	−0.162592	−	0.414278i	0.826239	−	0.563320i	$-0.190476\pi$
$383$	−0.162592	−	0.414278i	−0.162592	−	0.414278i	−0.988831	+	0.149042i	$0.952381\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$389$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$390$		0			0
$391$		0			0
$392$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.57906	−	0.487076i	1.57906	−	0.487076i
$397$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$397$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$398$		0			0
$399$		0			0
$400$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$401$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$401$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$402$		0			0
$403$	0.455573	−	1.16078i	0.455573	−	1.16078i
$404$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$405$		0			0
$406$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$407$		0			0
$408$		0			0
$409$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$409$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$414$		0			0
$415$		0			0
$416$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$417$		0			0
$418$	−0.885113	−	0.821265i	−0.885113	−	0.821265i
$419$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$419$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$420$		0			0
$421$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$421$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$422$		0			0
$423$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$424$	0.733052	−	0.680173i	0.733052	−	0.680173i
$425$	−1.63402	+	1.11406i	−1.63402	+	1.11406i
$426$		0			0
$427$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.0931869	+	1.24349i	0.0931869	+	1.24349i	0.826239	+	0.563320i	$0.190476\pi$
$431$	0.0931869	+	1.24349i	0.0931869	+	1.24349i	−0.733052	+	0.680173i	$0.761905\pi$
$432$		0			0
$433$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.733052	+	0.680173i	$0.761905\pi$
$433$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.988831	+	0.149042i	$0.952381\pi$
$434$	0.455573	+	1.16078i	0.455573	+	1.16078i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$439$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$440$		0			0
$441$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$442$	−1.97766			−1.97766
$443$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$443$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$444$		0			0
$445$		0			0
$446$	0.123490	−	1.64786i	0.123490	−	1.64786i
$447$		0			0
$448$	0.826239	−	0.563320i	0.826239	−	0.563320i
$449$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$449$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$450$	0.365341	+	0.930874i	0.365341	+	0.930874i
$451$		0			0
$452$	0.698220	−	1.77904i	0.698220	−	1.77904i
$453$		0			0
$454$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$455$		0			0
$456$		0			0
$457$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$457$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$461$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$462$		0			0
$463$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$463$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$464$	−0.109562	−	0.101659i	−0.109562	−	0.101659i
$465$		0			0
$466$	1.44973	+	0.218511i	1.44973	+	0.218511i
$467$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$467$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$468$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$469$	0.326239	+	1.42935i	0.326239	+	1.42935i
$470$		0			0
$471$		0			0
$472$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$473$		0			0
$474$		0			0
$475$	0.455573	−	0.571270i	0.455573	−	0.571270i
$476$	1.44973	−	1.34515i	1.44973	−	1.34515i
$477$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$478$	1.82624	+	0.563320i	1.82624	+	0.563320i
$479$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	0.623490	+	0.781831i	$0.285714\pi$
$479$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	−0.988831	+	0.149042i	$0.952381\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.55929	−	0.750915i	−1.55929	−	0.750915i
$485$		0			0
$486$		0			0
$487$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$487$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$488$	0.988831	−	0.149042i	0.988831	−	0.149042i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−0.244221	−	0.166507i	−0.244221	−	0.166507i
$494$	0.698220	−	0.215372i	0.698220	−	0.215372i
$495$		0			0
$496$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$497$	0.988831	−	1.71271i	0.988831	−	1.71271i
$498$		0			0
$499$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	−0.733052	−	0.680173i	$-0.761905\pi$
$499$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	0.0747301	−	0.997204i	$-0.476190\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$503$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$504$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$513$		0			0
$514$	0.440071	+	0.0663300i	0.440071	+	0.0663300i
$515$		0			0
$516$		0			0
$517$	−0.0549581	+	0.240787i	−0.0549581	+	0.240787i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$521$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$522$	−0.109562	+	0.101659i	−0.109562	+	0.101659i
$523$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$523$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.35654	−	0.726897i	−2.35654	−	0.726897i
$528$		0			0
$529$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$530$		0			0
$531$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$532$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$537$		0			0
$538$	1.91115			1.91115
$539$	1.36534	−	0.930874i	1.36534	−	0.930874i
$540$		0			0
$541$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$541$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$542$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$543$		0			0
$544$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$545$		0			0
$546$		0			0
$547$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$547$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$548$		0			0
$549$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$550$	0.603718	−	1.53825i	0.603718	−	1.53825i
$551$	0.104356	+	0.0321896i	0.104356	+	0.0321896i
$552$		0			0
$553$		0			0
$554$	0.455573	−	0.571270i	0.455573	−	0.571270i
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$563$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$568$	0.440071	+	1.92808i	0.440071	+	1.92808i
$569$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$569$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$570$		0			0
$571$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$571$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$572$	1.36534	−	0.930874i	1.36534	−	0.930874i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.955573	+	0.294755i	0.955573	+	0.294755i
$577$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$577$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$578$	0.217550	+	2.90301i	0.217550	+	2.90301i
$579$		0			0
$580$		0			0
$581$	−0.367711	+	0.250701i	−0.367711	+	0.250701i
$582$		0			0
$583$	−0.123490	+	1.64786i	−0.123490	+	1.64786i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	0.911146			0.911146
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$593$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$599$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$600$		0			0
$601$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$601$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$602$		0			0
$603$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$604$	0.603718	−	0.411608i	0.603718	−	0.411608i
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$609$		0			0
$610$		0			0
$611$	−0.109562	−	0.101659i	−0.109562	−	0.101659i
$612$	1.95557	+	0.294755i	1.95557	+	0.294755i
$613$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$613$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$614$	0.733052	+	0.680173i	0.733052	+	0.680173i
$615$		0			0
$616$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$617$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$617$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$618$		0			0
$619$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$619$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.955573	+	0.294755i	0.955573	+	0.294755i
$626$	−0.658322	+	1.67738i	−0.658322	+	1.67738i
$627$		0			0
$628$	0.603718	+	1.53825i	0.603718	+	1.53825i
$629$		0			0
$630$		0			0
$631$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$631$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$638$	0.246980			0.246980
$639$	1.95557	−	0.294755i	1.95557	−	0.294755i
$640$		0			0
$641$	−1.72188	+	0.531130i	−1.72188	+	0.531130i	−0.988831	−	0.149042i	$-0.952381\pi$
$641$	−1.72188	+	0.531130i	−1.72188	+	0.531130i	−0.733052	+	0.680173i	$0.761905\pi$
$642$		0			0
$643$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.988831	−	0.149042i	$-0.952381\pi$
$643$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.733052	−	0.680173i	$-0.761905\pi$
$644$		0			0
$645$		0			0
$646$	−0.527933	−	1.34515i	−0.527933	−	1.34515i
$647$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$647$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$648$	0.365341	−	0.930874i	0.365341	−	0.930874i
$649$	3.01782	+	0.930874i	3.01782	+	0.930874i
$650$	0.623490	+	0.781831i	0.623490	+	0.781831i
$651$		0			0
$652$	1.19158	−	1.49419i	1.19158	−	1.49419i
$653$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	−0.733052	−	0.680173i	$-0.761905\pi$
$653$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	0.365341	+	0.930874i	$0.380952\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.149460			0.149460
$659$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$659$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$660$		0			0
$661$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$661$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$662$	1.78181	+	0.268565i	1.78181	+	0.268565i
$663$		0			0
$664$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.733052	+	1.26968i	0.733052	+	1.26968i
$669$		0			0
$670$		0			0
$671$	−1.03030	+	1.29196i	−1.03030	+	1.29196i
$672$		0			0
$673$	0.777479	+	0.974928i	0.777479	+	0.974928i	1.00000			$0$
$673$	0.777479	+	0.974928i	0.777479	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$674$	−1.88980	−	0.582926i	−1.88980	−	0.582926i
$675$		0			0
$676$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$677$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$677$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	1.96906	−	0.607374i	1.96906	−	0.607374i
$683$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.955573	−	0.294755i	$-0.0952381\pi$
$683$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.0747301	+	0.997204i	$0.476190\pi$
$684$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$685$		0			0
$686$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$687$		0			0
$688$		0			0
$689$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$690$		0			0
$691$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$692$	1.32091	+	0.636119i	1.32091	+	0.636119i
$693$	1.57906	+	0.487076i	1.57906	+	0.487076i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$701$	0.777479	−	0.974928i	0.777479	−	0.974928i	−0.222521	−	0.974928i	$-0.571429\pi$
$701$	0.777479	−	0.974928i	0.777479	−	0.974928i	1.00000			$0$
$702$		0			0
$703$		0			0
$704$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$705$		0			0
$706$		0			0
$707$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$708$		0			0
$709$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$709$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.32091	−	1.22563i	1.32091	−	1.22563i
$719$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$719$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$720$		0			0
$721$		0			0
$722$	−0.290611	−	0.364415i	−0.290611	−	0.364415i
$723$		0			0
$724$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$725$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$726$		0			0
$727$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$727$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$728$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$729$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$733$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.39564	−	0.361085i	2.39564	−	0.361085i
$738$		0			0
$739$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.365341	−	0.930874i	$-0.380952\pi$
$739$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.826239	+	0.563320i	$0.190476\pi$
$740$		0			0
$741$		0			0
$742$	0.988831	−	0.149042i	0.988831	−	0.149042i
$743$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	0.365341	+	0.930874i	$0.380952\pi$
$744$		0			0
$745$		0			0
$746$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$747$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$748$	−2.03759	−	2.55506i	−2.03759	−	2.55506i
$749$		0			0
$750$		0			0
$751$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$751$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$752$	−0.109562	+	0.101659i	−0.109562	+	0.101659i
$753$		0			0
$754$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i
$755$		0			0
$756$		0			0
$757$	0.326239	−	1.42935i	0.326239	−	1.42935i	−0.500000	−	0.866025i	$-0.666667\pi$
$757$	0.326239	−	1.42935i	0.326239	−	1.42935i	0.826239	−	0.563320i	$-0.190476\pi$
$758$	0.326239	+	0.302705i	0.326239	+	0.302705i
$759$		0			0
$760$		0			0
$761$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$761$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.222521	+	0.385418i	0.222521	+	0.385418i
$767$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$768$		0			0
$769$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$769$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$773$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$774$		0			0
$775$	0.455573	+	1.16078i	0.455573	+	1.16078i
$776$		0			0
$777$		0			0
$778$	1.78181	+	0.858075i	1.78181	+	0.858075i
$779$		0			0
$780$		0			0
$781$	−2.70018	−	1.84095i	−2.70018	−	1.84095i
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	0.623490	−	0.781831i	$-0.285714\pi$
$787$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.57906	−	1.07659i	1.57906	−	1.07659i
$792$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$793$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$797$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$798$		0			0
$799$	−0.184292	+	0.231095i	−0.184292	+	0.231095i
$800$	0.826239	−	0.563320i	0.826239	−	0.563320i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$807$		0			0
$808$	1.78181	+	0.268565i	1.78181	+	0.268565i
$809$	0.988831	+	0.149042i	0.988831	+	0.149042i	0.623490	−	0.781831i	$-0.285714\pi$
$809$	0.988831	+	0.149042i	0.988831	+	0.149042i	0.365341	+	0.930874i	$0.380952\pi$
$810$		0			0
$811$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$811$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$812$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$820$		0			0
$821$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$821$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$822$		0			0
$823$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$823$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$824$		0			0
$825$		0			0
$826$	0.142820	−	1.90580i	0.142820	−	1.90580i
$827$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	0.0747301	−	0.997204i	$-0.476190\pi$
$827$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	−0.733052	+	0.680173i	$0.761905\pi$
$828$		0			0
$829$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.0747301	−	0.997204i	$-0.476190\pi$
$829$	0.698220	−	0.215372i	0.698220	−	0.215372i	0.623490	+	0.781831i	$0.285714\pi$
$830$		0			0
$831$		0			0
$832$	1.00000			1.00000
$833$	1.95557	−	0.294755i	1.95557	−	0.294755i
$834$		0			0
$835$		0			0
$836$	0.997630	+	0.680173i	0.997630	+	0.680173i
$837$		0			0
$838$		0			0
$839$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$839$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$840$		0			0
$841$	0.880843	−	0.424191i	0.880843	−	0.424191i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$847$	−0.865341	−	1.49881i	−0.865341	−	1.49881i
$848$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$849$		0			0
$850$	1.44973	−	1.34515i	1.44973	−	1.34515i
$851$		0			0
$852$		0			0
$853$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$853$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$854$	0.900969	+	0.433884i	0.900969	+	0.433884i
$855$		0			0
$856$		0			0
$857$	1.44973	+	0.218511i	1.44973	+	0.218511i	0.826239	−	0.563320i	$-0.190476\pi$
$857$	1.44973	+	0.218511i	1.44973	+	0.218511i	0.623490	+	0.781831i	$0.285714\pi$
$858$		0			0
$859$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$859$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$860$		0			0
$861$		0			0
$862$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$863$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$863$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$864$		0			0
$865$		0			0
$866$	1.57906	−	1.07659i	1.57906	−	1.07659i
$867$		0			0
$868$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$869$		0			0
$870$		0			0
$871$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$877$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$881$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$882$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	1.95557	−	0.294755i	1.95557	−	0.294755i
$885$		0			0
$886$		0			0
$887$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$887$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.603718	+	1.53825i	0.603718	+	1.53825i
$892$	0.123490	+	1.64786i	0.123490	+	1.64786i
$893$	0.0398981	−	0.101659i	0.0398981	−	0.101659i
$894$		0			0
$895$		0			0
$896$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$897$		0			0
$898$		0			0
$899$	−0.136622	+	0.126766i	−0.136622	+	0.126766i
$900$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$901$	−0.988831	+	1.71271i	−0.988831	+	1.71271i
$902$		0			0
$903$		0			0
$904$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$905$		0			0
$906$		0			0
$907$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$907$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$908$	1.32091	+	1.22563i	1.32091	+	1.22563i
$909$	0.400969	−	1.75676i	0.400969	−	1.75676i
$910$		0			0
$911$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$911$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$912$		0			0
$913$	0.367711	+	0.636894i	0.367711	+	0.636894i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$919$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.78181	−	0.858075i	1.78181	−	0.858075i
$924$		0			0
$925$		0			0
$926$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i
$927$		0			0
$928$	0.123490	+	0.0841939i	0.123490	+	0.0841939i
$929$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$929$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$930$		0			0
$931$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$932$	−1.46610			−1.46610
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$937$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$937$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$938$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$939$		0			0
$940$		0			0
$941$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$941$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$942$		0			0
$943$		0			0
$944$	1.19158	+	1.49419i	1.19158	+	1.49419i
$945$		0			0
$946$		0			0
$947$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	−0.900969	−	0.433884i	$-0.857143\pi$
$947$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	0.0747301	+	0.997204i	$0.476190\pi$
$948$		0			0
$949$		0			0
$950$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$951$		0			0
$952$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$953$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$953$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$954$	0.733052	+	0.680173i	0.733052	+	0.680173i
$955$		0			0
$956$	−1.88980	−	0.284841i	−1.88980	−	0.284841i
$957$		0			0
$958$	0.222521	−	0.974928i	0.222521	−	0.974928i
$959$		0			0
$960$		0			0
$961$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$967$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$968$	1.65379	+	0.510127i	1.65379	+	0.510127i
$969$		0			0
$970$		0			0
$971$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$971$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$972$		0			0
$973$		0			0
$974$	1.78181	+	0.858075i	1.78181	+	0.858075i
$975$		0			0
$976$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$977$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$977$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$983$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$984$		0			0
$985$		0			0
$986$	0.266310	+	0.128248i	0.266310	+	0.128248i
$987$		0			0
$988$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$989$		0			0
$990$		0			0
$991$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$991$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$992$	1.19158	+	0.367554i	1.19158	+	0.367554i
$993$		0			0
$994$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$995$		0			0
$996$		0			0
$997$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.222521	+	0.974928i	$0.571429\pi$
$997$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.988831	+	0.149042i	$0.952381\pi$
$998$	0.900969	+	1.56052i	0.900969	+	1.56052i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2548.1.de.b.779.1	yes	12
4.3	odd	2		2548.1.de.a.779.1	✓	12
13.12	even	2		2548.1.de.a.779.1	✓	12
49.39	even	21	inner	2548.1.de.b.2391.1	yes	12
52.51	odd	2	CM	2548.1.de.b.779.1	yes	12
196.39	odd	42		2548.1.de.a.2391.1	yes	12
637.480	even	42		2548.1.de.a.2391.1	yes	12
2548.2391	odd	42	inner	2548.1.de.b.2391.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2548.1.de.a.779.1	✓	12	4.3	odd	2
2548.1.de.a.779.1	✓	12	13.12	even	2
2548.1.de.a.2391.1	yes	12	196.39	odd	42
2548.1.de.a.2391.1	yes	12	637.480	even	42
2548.1.de.b.779.1	yes	12	1.1	even	1	trivial
2548.1.de.b.779.1	yes	12	52.51	odd	2	CM
2548.1.de.b.2391.1	yes	12	49.39	even	21	inner
2548.1.de.b.2391.1	yes	12	2548.2391	odd	42	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2548.de (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.988831 + 0.149042i) q^{2} +(0.955573 - 0.294755i) q^{4} +(0.955573 + 0.294755i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.365341 + 0.930874i) q^{9} +(0.603718 - 1.53825i) q^{11} +(0.623490 + 0.781831i) q^{13} +(-0.988831 - 0.149042i) q^{14} +(0.826239 - 0.563320i) q^{16} +(1.44973 - 1.34515i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-0.365341 + 0.632789i) q^{19} +(-0.367711 + 1.61105i) q^{22} +(-0.988831 - 0.149042i) q^{25} +(-0.733052 - 0.680173i) q^{26} +1.00000 q^{28} +(-0.0332580 - 0.145713i) q^{29} +(-0.623490 - 1.07992i) q^{31} +(-0.733052 + 0.680173i) q^{32} +(-1.23305 + 1.54620i) q^{34} +(0.623490 + 0.781831i) q^{36} +(0.266948 - 0.680173i) q^{38} +(0.123490 - 1.64786i) q^{44} +(-0.147791 + 0.0222759i) q^{47} +(0.826239 + 0.563320i) q^{49} +1.00000 q^{50} +(0.826239 + 0.563320i) q^{52} +(-0.955573 + 0.294755i) q^{53} +(-0.988831 + 0.149042i) q^{56} +(0.0546039 + 0.139129i) q^{58} +(0.142820 + 1.90580i) q^{59} +(-0.955573 - 0.294755i) q^{61} +(0.777479 + 0.974928i) q^{62} +(0.0747301 + 0.997204i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.733052 + 1.26968i) q^{67} +(0.988831 - 1.71271i) q^{68} +(0.440071 - 1.92808i) q^{71} +(-0.733052 - 0.680173i) q^{72} +(-0.162592 + 0.712362i) q^{76} +(1.03030 - 1.29196i) q^{77} +(-0.733052 + 0.680173i) q^{81} +(-0.277479 + 0.347948i) q^{83} +(0.123490 + 1.64786i) q^{88} +(0.365341 + 0.930874i) q^{91} +(0.142820 - 0.0440542i) q^{94} +(-0.900969 - 0.433884i) q^{98} +1.65248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + q^{4} + q^{7} - 2 q^{8} + q^{9} - q^{11} - 2 q^{13} + q^{14} + q^{16} - q^{17} - 6 q^{18} - q^{19} + 2 q^{22} + q^{25} + q^{26} + 12 q^{28} + 2 q^{29} + 2 q^{31} + q^{32} - 5 q^{34}+ \cdots + 2 q^{99}+O(q^{100}) \) 12 * q + q^2 + q^4 + q^7 - 2 * q^8 + q^9 - q^11 - 2 * q^13 + q^14 + q^16 - q^17 - 6 * q^18 - q^19 + 2 * q^22 + q^25 + q^26 + 12 * q^28 + 2 * q^29 + 2 * q^31 + q^32 - 5 * q^34 - 2 * q^36 + 13 * q^38 - 8 * q^44 - q^47 + q^49 + 12 * q^50 + q^52 - q^53 + q^56 - q^58 - q^59 - q^61 + 10 * q^62 + q^63 - 2 * q^64 - q^67 - q^68 + 2 * q^71 + q^72 + 2 * q^76 + 2 * q^77 + q^81 - 4 * q^83 - 8 * q^88 + q^91 - q^94 - 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more