Embedded newform 350.3.w.a.23.16

• Label: 350.3.w.a.23.16
• 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			23.16
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(2.88965 + 1.87656i) q^{3} +(0.415823 + 1.95630i) q^{4} +(1.71976 - 4.69494i) q^{5} +(1.50575 + 4.63421i) q^{6} +(6.26337 + 3.12573i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(1.16798 + 2.62333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

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
							\(3\)	2.88965	+	1.87656i	0.963218	+	0.625521i	0.927493	−	0.373840i	\(-0.121959\pi\)
0.0357251	+	0.999362i	\(0.488626\pi\)
\(5\)	1.71976	−	4.69494i	0.343952	−	0.938987i
							\(7\)	6.26337	+	3.12573i	0.894767	+	0.446533i
\(11\)	5.14382	+	2.29018i	0.467620	+	0.208198i								0.627000	−	0.779019i	\(-0.284283\pi\)
							−0.159380	+	0.987217i	\(0.550949\pi\)
\(13\)	−0.878802	−	5.54854i	−0.0676001	−	0.426810i	−0.998158	−	0.0606659i	\(-0.980678\pi\)
							0.930558	−	0.366145i	\(-0.119322\pi\)
\(17\)	0.584896	+	11.1605i	0.0344056	+	0.656499i	0.960343	+	0.278822i	\(0.0899438\pi\)
							−0.925937	+	0.377677i	\(0.876723\pi\)
\(19\)	0.347181	−	1.63336i	0.0182727	−	0.0859662i	−0.968066	−	0.250694i	\(-0.919341\pi\)
							0.986339	+	0.164728i	\(0.0526746\pi\)
\(23\)	−6.86829	+	8.48164i	−0.298621	+	0.368767i	−0.904267	−	0.426968i	\(-0.859582\pi\)
							0.605645	+	0.795735i	\(0.292915\pi\)
\(29\)	16.5871	+	5.38947i	0.571969	+	0.185844i	0.580699	−	0.814118i	\(-0.302779\pi\)
							−0.00873086	+	0.999962i	\(0.502779\pi\)
\(31\)	8.07158	−	8.96440i	0.260374	−	0.289174i	−0.598757	−	0.800931i	\(-0.704338\pi\)
							0.859130	+	0.511757i	\(0.171005\pi\)
\(37\)	−25.7795	+	67.1579i	−0.696743	+	1.81508i	−0.129408	+	0.991591i	\(0.541308\pi\)
							−0.567335	+	0.823487i	\(0.692026\pi\)
\(41\)	−39.1998	−	28.4803i	−0.956093	−	0.694642i	−0.00385300	−	0.999993i	\(-0.501226\pi\)
							−0.952240	+	0.305350i	\(0.901226\pi\)
\(43\)	−18.3115	−	18.3115i	−0.425849	−	0.425849i	0.461363	−	0.887212i	\(-0.347361\pi\)
							−0.887212	+	0.461363i	\(0.847361\pi\)
\(47\)	−53.9829	−	2.82912i	−1.14857	−	0.0601941i	−0.531470	−	0.847077i	\(-0.678360\pi\)
							−0.617102	+	0.786883i	\(0.711693\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.16	✓	320	1.1	even	1	trivial
350.3.w.a.37.16	yes	320	25.12	odd	20	inner
350.3.w.a.123.16	yes	320	7.4	even	3	inner
350.3.w.a.137.16	yes	320	175.137	odd	60	inner