Embedded newform 775.2.cx.a.37.16

• Label: 775.2.cx.a.37.16
• Level: $775$
• Weight: $2$
• Character: 775.37
• Analytic conductor: $6.188$
• Analytic rank: $0$
• Dimension: $1248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.cx (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(0\)
Dimension:	\(1248\)
Relative dimension:	\(78\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			37.16
Character	\(\chi\)	\(=\)	775.37
Dual form			775.2.cx.a.398.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.307697 - 1.94272i) q^{2} +(-3.21364 + 0.168420i) q^{3} +(-1.77739 + 0.577509i) q^{4} +(2.04738 - 0.899030i) q^{5} +(1.31602 + 6.19139i) q^{6} +(0.515241 - 1.92290i) q^{7} +(-0.117106 - 0.229834i) q^{8} +(7.31552 - 0.768892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1248 q - 32 q^{2} - 24 q^{3} - 40 q^{4} - 8 q^{5} - 18 q^{6} - 4 q^{7} - 60 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{20}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.307697	−	1.94272i	−0.217575	−	1.37371i	−0.818545	−	0.574442i	\(-0.805219\pi\)
							0.600970	−	0.799272i	\(-0.294781\pi\)
\(3\)	−3.21364	+	0.168420i	−1.85539	+	0.0972370i	−0.947069	−	0.321030i	\(-0.895971\pi\)
							−0.908324	+	0.418267i	\(0.862638\pi\)
\(5\)	2.04738	−	0.899030i	0.915614	−	0.402059i
							\(7\)	0.515241	−	1.92290i	0.194743	−	0.726789i	−0.797591	−	0.603199i	\(-0.793892\pi\)
0.992333	−	0.123590i	\(-0.0394409\pi\)
\(11\)	1.83818	+	4.12861i	0.554231	+	1.24482i	0.945823	+	0.324684i	\(0.105258\pi\)
							−0.391592	+	0.920139i	\(0.628075\pi\)
\(13\)	−2.77869	+	3.43140i	−0.770670	+	0.951699i	−0.999764	−	0.0217057i	\(-0.993090\pi\)
							0.229094	+	0.973404i	\(0.426424\pi\)
\(17\)	−2.60163	+	4.00616i	−0.630988	+	0.971636i	0.368136	+	0.929772i	\(0.379996\pi\)
							−0.999123	+	0.0418638i	\(0.986670\pi\)
\(19\)	1.16300	+	5.47147i	0.266810	+	1.25524i	0.883648	+	0.468151i	\(0.155080\pi\)
							−0.616839	+	0.787090i	\(0.711587\pi\)
\(23\)	0.771102	+	4.86855i	0.160786	+	1.01516i	0.927677	+	0.373385i	\(0.121803\pi\)
							−0.766891	+	0.641778i	\(0.778197\pi\)
\(29\)	−2.18875	−	6.73627i	−0.406440	−	1.25089i	−0.919687	−	0.392652i	\(-0.871558\pi\)
							0.513247	−	0.858241i	\(-0.328442\pi\)
\(31\)	−1.28762	+	5.41683i	−0.231263	+	0.972891i
							\(37\)	6.32447	−	2.42774i	1.03974	−	0.399118i	0.222204	−	0.975000i	\(-0.428675\pi\)
0.817532	+	0.575883i	\(0.195341\pi\)
\(41\)	0.115167	+	1.09574i	0.0179861	+	0.171126i	0.999827	−	0.0185777i	\(-0.00591379\pi\)
							−0.981841	+	0.189704i	\(0.939247\pi\)
\(43\)	0.624408	−	2.33032i	0.0952214	−	0.355371i	−0.901832	−	0.432088i	\(-0.857777\pi\)
							0.997053	+	0.0767168i	\(0.0244437\pi\)
\(47\)	−5.32325	+	10.4475i	−0.776476	+	1.52392i	0.0736173	+	0.997287i	\(0.476546\pi\)
							−0.850093	+	0.526633i	\(0.823454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.cx.a.37.16	✓	1248
25.23	odd	20	inner	775.2.cx.a.223.16	yes	1248
31.26	odd	6	inner	775.2.cx.a.212.16	yes	1248
775.398	even	60	inner	775.2.cx.a.398.16	yes	1248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.cx.a.37.16	✓	1248	1.1	even	1	trivial
775.2.cx.a.212.16	yes	1248	31.26	odd	6	inner
775.2.cx.a.223.16	yes	1248	25.23	odd	20	inner
775.2.cx.a.398.16	yes	1248	775.398	even	60	inner