Embedded newform 350.3.w.a.23.15

• Label: 350.3.w.a.23.15
• 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			23.15
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(2.65474 + 1.72401i) q^{3} +(0.415823 + 1.95630i) q^{4} +(2.92674 + 4.05391i) q^{5} +(1.38334 + 4.25748i) q^{6} +(1.01529 + 6.92598i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(0.414820 + 0.931702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

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
							\(3\)	2.65474	+	1.72401i	0.884914	+	0.574670i	0.905134	−	0.425125i	\(-0.139770\pi\)
−0.0202200	+	0.999796i	\(0.506437\pi\)
\(5\)	2.92674	+	4.05391i	0.585348	+	0.810782i
							\(7\)	1.01529	+	6.92598i	0.145041	+	0.989426i
\(11\)	−4.97513	−	2.21507i	−0.452284	−	0.201370i								0.167936	−	0.985798i	\(-0.446290\pi\)
							−0.620220	+	0.784428i	\(0.712957\pi\)
\(13\)	−0.0956358	−	0.603820i	−0.00735660	−	0.0464477i	0.983737	−	0.179615i	\(-0.0574852\pi\)
							−0.991094	+	0.133167i	\(0.957485\pi\)
\(17\)	−0.751469	−	14.3389i	−0.0442041	−	0.843464i	−0.927311	−	0.374292i	\(-0.877886\pi\)
							0.883107	−	0.469172i	\(-0.155448\pi\)
\(19\)	2.75544	−	12.9633i	0.145023	−	0.682281i	−0.844219	−	0.535998i	\(-0.819936\pi\)
							0.989243	−	0.146283i	\(-0.0467312\pi\)
\(23\)	9.64763	−	11.9138i	0.419462	−	0.517993i	−0.522988	−	0.852340i	\(-0.675183\pi\)
							0.942450	+	0.334347i	\(0.108516\pi\)
\(29\)	22.1167	+	7.18614i	0.762644	+	0.247798i	0.664413	−	0.747366i	\(-0.268682\pi\)
							0.0982310	+	0.995164i	\(0.468682\pi\)
\(31\)	−1.81843	+	2.01957i	−0.0586591	+	0.0651476i	−0.771760	−	0.635914i	\(-0.780623\pi\)
							0.713101	+	0.701061i	\(0.247290\pi\)
\(37\)	−1.69025	+	4.40325i	−0.0456824	+	0.119007i	−0.954523	−	0.298136i	\(-0.903635\pi\)
							0.908841	+	0.417143i	\(0.136968\pi\)
\(41\)	15.9527	+	11.5903i	0.389089	+	0.282690i	0.765082	−	0.643933i	\(-0.222698\pi\)
							−0.375993	+	0.926622i	\(0.622698\pi\)
\(43\)	35.4592	+	35.4592i	0.824633	+	0.824633i	0.986768	−	0.162136i	\(-0.0518382\pi\)
							−0.162136	+	0.986768i	\(0.551838\pi\)
\(47\)	50.5664	+	2.65007i	1.07588	+	0.0563845i	0.582034	−	0.813165i	\(-0.302257\pi\)
							0.493847	+	0.869549i	\(0.335590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.15	✓	320
7.4	even	3	inner	350.3.w.a.123.15	yes	320
25.12	odd	20	inner	350.3.w.a.37.15	yes	320
175.137	odd	60	inner	350.3.w.a.137.15	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.15	✓	320	1.1	even	1	trivial
350.3.w.a.37.15	yes	320	25.12	odd	20	inner
350.3.w.a.123.15	yes	320	7.4	even	3	inner
350.3.w.a.137.15	yes	320	175.137	odd	60	inner