Embedded newform 350.3.s.b.113.3

• Label: 350.3.s.b.113.3
• Level: $350$
• Weight: $3$
• Character: 350.113
• Analytic conductor: $9.537$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.s (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.53680925261\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			113.3
Character	\(\chi\)	\(=\)	350.113
Dual form			350.3.s.b.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(-1.81921 + 3.57041i) q^{3} +(1.90211 - 0.618034i) q^{4} +(-3.95517 + 3.05886i) q^{5} +(1.75119 - 5.38962i) q^{6} +(1.87083 - 1.87083i) q^{7} +(-2.52015 + 1.28408i) q^{8} +(-4.14819 - 5.70950i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 32 q^{2} - 4 q^{3} + 4 q^{5} - 8 q^{6} + 64 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39680	+	0.221232i	−0.698401	+	0.110616i
							\(3\)	−1.81921	+	3.57041i	−0.606404	+	1.19014i	0.359963	+	0.932967i	\(0.382789\pi\)
−0.966367	+	0.257168i	\(0.917211\pi\)
\(5\)	−3.95517	+	3.05886i	−0.791034	+	0.611772i
							\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
\(11\)	−0.649853	−	0.472146i	−0.0590776	−	0.0429224i								0.557855	−	0.829939i	\(-0.311625\pi\)
							−0.616932	+	0.787016i	\(0.711625\pi\)
\(13\)	−24.9372	−	3.94966i	−1.91825	−	0.303820i	−0.921773	−	0.387731i	\(-0.873259\pi\)
							−0.996473	+	0.0839105i	\(0.973259\pi\)
\(17\)	5.30699	+	10.4156i	0.312176	+	0.612680i	0.992777	−	0.119975i	\(-0.0382813\pi\)
							−0.680601	+	0.732654i	\(0.738281\pi\)
\(19\)	−5.74152	−	1.86553i	−0.302185	−	0.0981859i	0.154000	−	0.988071i	\(-0.450784\pi\)
							−0.456185	+	0.889885i	\(0.650784\pi\)
\(23\)	−1.60685	−	10.1453i	−0.0698630	−	0.441098i	−0.997681	−	0.0680644i	\(-0.978318\pi\)
							0.927818	−	0.373033i	\(-0.121682\pi\)
\(29\)	43.1552	−	14.0220i	1.48811	−	0.483516i	0.551586	−	0.834118i	\(-0.314023\pi\)
							0.936525	+	0.350602i	\(0.114023\pi\)
\(31\)	6.79183	−	20.9031i	0.219091	−	0.674294i	−0.779746	−	0.626096i	\(-0.784652\pi\)
							0.998838	−	0.0481986i	\(-0.0153480\pi\)
\(37\)	−6.16997	+	38.9557i	−0.166756	+	1.05286i	0.752327	+	0.658790i	\(0.228932\pi\)
							−0.919083	+	0.394065i	\(0.871068\pi\)
\(41\)	20.2063	−	14.6808i	0.492837	−	0.358067i	−0.313437	−	0.949609i	\(-0.601480\pi\)
							0.806274	+	0.591542i	\(0.201480\pi\)
\(43\)	0.457353	+	0.457353i	0.0106361	+	0.0106361i	0.712405	−	0.701769i	\(-0.247606\pi\)
							−0.701769	+	0.712405i	\(0.747606\pi\)
\(47\)	58.2108	+	29.6599i	1.23853	+	0.631061i	0.945680	−	0.325098i	\(-0.105397\pi\)
							0.292847	+	0.956159i	\(0.405397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.s.b.113.3	✓	128
25.2	odd	20	inner	350.3.s.b.127.3	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.s.b.113.3	✓	128	1.1	even	1	trivial
350.3.s.b.127.3	yes	128	25.2	odd	20	inner