Embedded newform 350.3.w.a.23.7

• Label: 350.3.w.a.23.7
• Level: $350$
• Weight: $3$
• Character: 350.23
• Analytic conductor: $9.537$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.53680925261\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(20\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			23.7
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(-2.04025 - 1.32495i) q^{3} +(0.415823 + 1.95630i) q^{4} +(0.343406 + 4.98819i) q^{5} +(-1.06314 - 3.27200i) q^{6} +(-4.29280 - 5.52918i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(-1.25351 - 2.81543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09905	+	0.889993i	0.549525	+	0.444997i
							\(3\)	−2.04025	−	1.32495i	−0.680084	−	0.441651i	0.157821	−	0.987468i	\(-0.449553\pi\)
−0.837905	+	0.545816i	\(0.816220\pi\)
\(5\)	0.343406	+	4.98819i	0.0686812	+	0.997639i
							\(7\)	−4.29280	−	5.52918i	−0.613257	−	0.789883i
\(11\)	−5.19403	−	2.31253i	−0.472184	−	0.210230i								0.156827	−	0.987626i	\(-0.449873\pi\)
							−0.629011	+	0.777396i	\(0.716540\pi\)
\(13\)	−0.874493	−	5.52133i	−0.0672687	−	0.424718i	−0.998223	−	0.0595830i	\(-0.981023\pi\)
							0.930955	−	0.365135i	\(-0.118977\pi\)
\(17\)	−0.378997	−	7.23169i	−0.0222939	−	0.425394i	−0.986863	−	0.161562i	\(-0.948347\pi\)
							0.964569	−	0.263832i	\(-0.0849864\pi\)
\(19\)	1.18111	−	5.55667i	0.0621635	−	0.292456i	−0.936074	−	0.351803i	\(-0.885569\pi\)
							0.998237	+	0.0593471i	\(0.0189019\pi\)
\(23\)	24.7781	−	30.5984i	1.07731	−	1.33037i	0.136800	−	0.990599i	\(-0.456318\pi\)
							0.940510	−	0.339767i	\(-0.110348\pi\)
\(29\)	−51.5045	−	16.7348i	−1.77602	−	0.577062i	−0.777368	−	0.629047i	\(-0.783445\pi\)
							−0.998648	+	0.0519841i	\(0.983445\pi\)
\(31\)	11.0554	−	12.2783i	0.356626	−	0.396074i	−0.537959	−	0.842971i	\(-0.680804\pi\)
							0.894585	+	0.446897i	\(0.147471\pi\)
\(37\)	4.40738	−	11.4816i	0.119118	−	0.310314i	−0.861038	−	0.508541i	\(-0.830185\pi\)
							0.980156	+	0.198227i	\(0.0635183\pi\)
\(41\)	−39.6196	−	28.7854i	−0.966333	−	0.702082i	−0.0117200	−	0.999931i	\(-0.503731\pi\)
							−0.954613	+	0.297849i	\(0.903731\pi\)
\(43\)	−19.3965	−	19.3965i	−0.451080	−	0.451080i	0.444633	−	0.895713i	\(-0.353334\pi\)
							−0.895713	+	0.444633i	\(0.853334\pi\)
\(47\)	51.3100	+	2.68905i	1.09170	+	0.0572137i	0.589677	−	0.807639i	\(-0.299255\pi\)
							0.502026	+	0.864853i	\(0.332588\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.7	✓	320
7.4	even	3	inner	350.3.w.a.123.7	yes	320
25.12	odd	20	inner	350.3.w.a.37.7	yes	320
175.137	odd	60	inner	350.3.w.a.137.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.7	✓	320	1.1	even	1	trivial
350.3.w.a.37.7	yes	320	25.12	odd	20	inner
350.3.w.a.123.7	yes	320	7.4	even	3	inner
350.3.w.a.137.7	yes	320	175.137	odd	60	inner