Embedded newform 775.2.cx.a.223.16

• Label: 775.2.cx.a.223.16
• Level: $775$
• Weight: $2$
• Character: 775.223
• 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			223.16
Character	\(\chi\)	\(=\)	775.223
Dual form			775.2.cx.a.212.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.307697 + 1.94272i) q^{2} +(-1.75267 + 2.69888i) q^{3} +(-1.77739 - 0.577509i) q^{4} +(-0.245104 - 2.22259i) q^{5} +(-4.70389 - 4.23540i) q^{6} +(-1.92290 - 0.515241i) q^{7} +(-0.117106 + 0.229834i) q^{8} +(-2.99188 - 6.71988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1248 q - 32 q^{2} - 24 q^{3} - 40 q^{4} - 8 q^{5} - 18 q^{6} - 4 q^{7} - 60 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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	0.600970	+	0.799272i	\(0.294781\pi\)
							−0.818545	+	0.574442i	\(0.805219\pi\)
\(3\)	−1.75267	+	2.69888i	−1.01191	+	1.55820i	−0.195515	+	0.980701i	\(0.562638\pi\)
							−0.816392	+	0.577499i	\(0.804029\pi\)
\(5\)	−0.245104	−	2.22259i	−0.109614	−	0.993974i
							\(7\)	−1.92290	−	0.515241i	−0.726789	−	0.194743i	−0.123590	−	0.992333i	\(-0.539441\pi\)
−0.603199	+	0.797591i	\(0.706108\pi\)
\(11\)	4.49457	−	0.472398i	1.35516	−	0.142433i	0.601068	−	0.799198i	\(-0.294742\pi\)
							0.754096	+	0.656764i	\(0.228075\pi\)
\(13\)	1.58233	−	4.12212i	0.438860	−	1.14327i	−0.518680	−	0.854969i	\(-0.673576\pi\)
							0.957540	−	0.288301i	\(-0.0930903\pi\)
\(17\)	−4.77025	+	0.249998i	−1.15696	+	0.0606335i	−0.621139	−	0.783701i	\(-0.713330\pi\)
							−0.535817	+	0.844334i	\(0.679996\pi\)
\(19\)	4.15693	+	3.74292i	0.953666	+	0.858684i	0.990059	−	0.140653i	\(-0.0449202\pi\)
							−0.0363933	+	0.999338i	\(0.511587\pi\)
\(23\)	−0.771102	+	4.86855i	−0.160786	+	1.01516i	0.766891	+	0.641778i	\(0.221803\pi\)
							−0.927677	+	0.373385i	\(0.878197\pi\)
\(29\)	2.18875	−	6.73627i	0.406440	−	1.25089i	−0.513247	−	0.858241i	\(-0.671558\pi\)
							0.919687	−	0.392652i	\(-0.128442\pi\)
\(31\)	−2.14223	−	5.13915i	−0.384756	−	0.923019i
							\(37\)	5.26472	−	4.26328i	0.865514	−	0.700880i	−0.0899628	−	0.995945i	\(-0.528675\pi\)
0.955477	+	0.295066i	\(0.0953415\pi\)
\(41\)	−1.00653	+	0.448134i	−0.157193	+	0.0699868i	−0.483825	−	0.875165i	\(-0.660753\pi\)
							0.326632	+	0.945152i	\(0.394086\pi\)
\(43\)	2.33032	+	0.624408i	0.355371	+	0.0952214i	0.432088	−	0.901832i	\(-0.357777\pi\)
							−0.0767168	+	0.997053i	\(0.524444\pi\)
\(47\)	−5.32325	−	10.4475i	−0.776476	−	1.52392i	−0.850093	−	0.526633i	\(-0.823454\pi\)
							0.0736173	−	0.997287i	\(-0.476546\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.cx.a.37.16	✓	1248	25.12	odd	20	inner
775.2.cx.a.212.16	yes	1248	775.212	even	60	inner
775.2.cx.a.223.16	yes	1248	1.1	even	1	trivial
775.2.cx.a.398.16	yes	1248	31.26	odd	6	inner