Embedded newform 350.3.w.a.23.10

• Label: 350.3.w.a.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(-0.372709 - 0.242040i) q^{3} +(0.415823 + 1.95630i) q^{4} +(-4.59473 + 1.97191i) q^{5} +(-0.194212 - 0.597722i) q^{6} +(-3.14459 + 6.25392i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(-3.58030 - 8.04149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.372709	−	0.242040i	−0.124236	−	0.0806799i	0.481025	−	0.876707i	\(-0.340265\pi\)
−0.605261	+	0.796027i	\(0.706931\pi\)
\(5\)	−4.59473	+	1.97191i	−0.918947	+	0.394381i
							\(7\)	−3.14459	+	6.25392i	−0.449227	+	0.893418i
\(11\)	−0.737059	−	0.328160i	−0.0670053	−	0.0298327i								0.372960	−	0.927847i	\(-0.378343\pi\)
							−0.439966	+	0.898015i	\(0.645009\pi\)
\(13\)	−2.75603	−	17.4009i	−0.212002	−	1.33853i	−0.832371	−	0.554219i	\(-0.813017\pi\)
							0.620369	−	0.784310i	\(-0.286983\pi\)
\(17\)	−0.754945	−	14.4052i	−0.0444085	−	0.847365i	−0.926502	−	0.376290i	\(-0.877200\pi\)
							0.882093	−	0.471075i	\(-0.156134\pi\)
\(19\)	2.37491	−	11.1731i	0.124995	−	0.588056i	−0.870412	−	0.492324i	\(-0.836147\pi\)
							0.995407	−	0.0957320i	\(-0.0305192\pi\)
\(23\)	−19.7947	+	24.4445i	−0.860641	+	1.06280i	0.136749	+	0.990606i	\(0.456335\pi\)
							−0.997390	+	0.0721976i	\(0.976999\pi\)
\(29\)	−44.3873	−	14.4223i	−1.53059	−	0.497320i	−0.581832	−	0.813309i	\(-0.697664\pi\)
							−0.948763	+	0.315989i	\(0.897664\pi\)
\(31\)	−9.27089	+	10.2964i	−0.299061	+	0.332141i	−0.873882	−	0.486138i	\(-0.838405\pi\)
							0.574821	+	0.818279i	\(0.305072\pi\)
\(37\)	−23.0349	+	60.0080i	−0.622565	+	1.62184i	0.151300	+	0.988488i	\(0.451654\pi\)
							−0.773866	+	0.633350i	\(0.781679\pi\)
\(41\)	−0.0850908	−	0.0618221i	−0.00207539	−	0.00150786i	0.586747	−	0.809770i	\(-0.300408\pi\)
							−0.588822	+	0.808262i	\(0.700408\pi\)
\(43\)	7.24023	+	7.24023i	0.168378	+	0.168378i	0.786266	−	0.617888i	\(-0.212012\pi\)
							−0.617888	+	0.786266i	\(0.712012\pi\)
\(47\)	−59.8986	−	3.13915i	−1.27444	−	0.0667904i	−0.596922	−	0.802299i	\(-0.703610\pi\)
							−0.677515	+	0.735509i	\(0.736943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.10	✓	320
7.4	even	3	inner	350.3.w.a.123.10	yes	320
25.12	odd	20	inner	350.3.w.a.37.10	yes	320
175.137	odd	60	inner	350.3.w.a.137.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.10	✓	320	1.1	even	1	trivial
350.3.w.a.37.10	yes	320	25.12	odd	20	inner
350.3.w.a.123.10	yes	320	7.4	even	3	inner
350.3.w.a.137.10	yes	320	175.137	odd	60	inner