Embedded newform 350.3.w.a.37.16

• Label: 350.3.w.a.37.16
• Level: $350$
• Weight: $3$
• Character: 350.37
• 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			37.16
Character	\(\chi\)	\(=\)	350.37
Dual form			350.3.w.a.123.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32028 - 0.506809i) q^{2} +(0.180324 + 3.44080i) q^{3} +(1.48629 + 1.33826i) q^{4} +(3.20606 - 3.83682i) q^{5} +(1.50575 - 4.63421i) q^{6} +(-3.12573 + 6.26337i) q^{7} +(-1.28408 - 2.52015i) q^{8} +(-2.85586 + 0.300163i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32028	−	0.506809i	−0.660141	−	0.253404i
							\(3\)	0.180324	+	3.44080i	0.0601082	+	1.14693i	0.847610	+	0.530620i	\(0.178041\pi\)
−0.787502	+	0.616312i	\(0.788626\pi\)
\(5\)	3.20606	−	3.83682i	0.641211	−	0.767365i
							\(7\)	−3.12573	+	6.26337i	−0.446533	+	0.894767i
\(11\)	−0.588559	+	5.59977i	−0.0535054	+	0.509070i								0.934645	+	0.355582i	\(0.115717\pi\)
							−0.988151	+	0.153488i	\(0.950949\pi\)
\(13\)	−0.878802	+	5.54854i	−0.0676001	+	0.426810i	0.930558	+	0.366145i	\(0.119322\pi\)
							−0.998158	+	0.0606659i	\(0.980678\pi\)
\(17\)	9.37281	+	6.08677i	0.551342	+	0.358046i	0.790047	−	0.613047i	\(-0.210056\pi\)
							−0.238705	+	0.971092i	\(0.576723\pi\)
\(19\)	1.24094	−	1.11735i	0.0653125	−	0.0588077i	−0.635828	−	0.771831i	\(-0.719341\pi\)
							0.701141	+	0.713023i	\(0.252675\pi\)
\(23\)	−3.91117	+	10.1889i	−0.170051	+	0.442997i	−0.991949	−	0.126637i	\(-0.959582\pi\)
							0.821898	+	0.569634i	\(0.192915\pi\)
\(29\)	16.5871	−	5.38947i	0.571969	−	0.185844i	−0.00873086	−	0.999962i	\(-0.502779\pi\)
							0.580699	+	0.814118i	\(0.302779\pi\)
\(31\)	−11.7992	+	2.50800i	−0.380619	+	0.0809031i	−0.394248	−	0.919004i	\(-0.628995\pi\)
							0.0136290	+	0.999907i	\(0.495662\pi\)
\(37\)	−45.2707	+	55.9047i	−1.22353	+	1.51094i	−0.429493	+	0.903070i	\(0.641308\pi\)
							−0.794039	+	0.607867i	\(0.792026\pi\)
\(41\)	−39.1998	+	28.4803i	−0.956093	+	0.694642i	−0.952240	−	0.305350i	\(-0.901226\pi\)
							−0.00385300	+	0.999993i	\(0.501226\pi\)
\(43\)	−18.3115	+	18.3115i	−0.425849	+	0.425849i	−0.887212	−	0.461363i	\(-0.847361\pi\)
							0.461363	+	0.887212i	\(0.347361\pi\)
\(47\)	29.4415	+	45.3360i	0.626416	+	0.964596i	0.999325	+	0.0367279i	\(0.0116935\pi\)
							−0.372910	+	0.927868i	\(0.621640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.37.16	yes	320
7.4	even	3	inner	350.3.w.a.137.16	yes	320
25.23	odd	20	inner	350.3.w.a.23.16	✓	320
175.123	odd	60	inner	350.3.w.a.123.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.16	✓	320	25.23	odd	20	inner
350.3.w.a.37.16	yes	320	1.1	even	1	trivial
350.3.w.a.123.16	yes	320	175.123	odd	60	inner
350.3.w.a.137.16	yes	320	7.4	even	3	inner