Embedded newform 310.2.w.a.53.15

• Label: 310.2.w.a.53.15
• Level: $310$
• Weight: $2$
• Character: 310.53
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47536246266\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			53.15
Character	\(\chi\)	\(=\)	310.53
Dual form			310.2.w.a.117.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.453990 + 0.891007i) q^{2} +(1.39807 - 0.907916i) q^{3} +(-0.587785 + 0.809017i) q^{4} +(1.70977 + 1.44108i) q^{5} +(1.44367 + 0.833503i) q^{6} +(-1.17558 + 3.06249i) q^{7} +(-0.987688 - 0.156434i) q^{8} +(-0.0899259 + 0.201977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{17}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.453990	+	0.891007i	0.321020	+	0.630037i
							\(3\)	1.39807	−	0.907916i	0.807175	−	0.524186i	−0.0738419	−	0.997270i	\(-0.523526\pi\)
0.881017	+	0.473084i	\(0.156859\pi\)
\(5\)	1.70977	+	1.44108i	0.764630	+	0.644469i
							\(7\)	−1.17558	+	3.06249i	−0.444327	+	1.15751i	0.510375	+	0.859952i	\(0.329506\pi\)
−0.954703	+	0.297561i	\(0.903827\pi\)
\(11\)	−0.643025	−	0.0675846i	−0.193879	−	0.0203775i	0.00709109	−	0.999975i	\(-0.497743\pi\)
							−0.200970	+	0.979597i	\(0.564409\pi\)
\(13\)	−0.164320	−	3.13541i	−0.0455741	−	0.869605i	−0.921791	−	0.387688i	\(-0.873274\pi\)
							0.876217	−	0.481917i	\(-0.160059\pi\)
\(17\)	0.505900	−	0.624734i	0.122699	−	0.151520i	−0.712095	−	0.702083i	\(-0.752253\pi\)
							0.834794	+	0.550563i	\(0.185587\pi\)
\(19\)	4.16895	−	3.75374i	0.956423	−	0.861167i	−0.0339733	−	0.999423i	\(-0.510816\pi\)
							0.990397	+	0.138255i	\(0.0441495\pi\)
\(23\)	0.984764	−	6.21756i	0.205338	−	1.29645i	−0.642537	−	0.766254i	\(-0.722118\pi\)
							0.847875	−	0.530196i	\(-0.177882\pi\)
\(29\)	0.356381	−	1.09683i	0.0661782	−	0.203676i	−0.912499	−	0.409078i	\(-0.865850\pi\)
							0.978678	+	0.205403i	\(0.0658504\pi\)
\(31\)	−3.10939	−	4.61863i	−0.558463	−	0.829530i
							\(37\)	3.82683	−	1.02540i	0.629128	−	0.168574i	0.0698538	−	0.997557i	\(-0.477747\pi\)
0.559274	+	0.828983i	\(0.311080\pi\)
\(41\)	0.464812	−	0.0987989i	0.0725915	−	0.0154298i	−0.171473	−	0.985189i	\(-0.554853\pi\)
							0.244064	+	0.969759i	\(0.421519\pi\)
\(43\)	−2.90784	−	0.152393i	−0.443441	−	0.0232397i	−0.170691	−	0.985325i	\(-0.554600\pi\)
							−0.272750	+	0.962085i	\(0.587933\pi\)
\(47\)	−7.56015	−	3.85209i	−1.10276	−	0.561885i	−0.194760	−	0.980851i	\(-0.562393\pi\)
							−0.908002	+	0.418966i	\(0.862393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.2.w.a.53.15	✓	256
5.2	odd	4	inner	310.2.w.a.177.2	yes	256
31.24	odd	30	inner	310.2.w.a.303.2	yes	256
155.117	even	60	inner	310.2.w.a.117.15	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.15	✓	256	1.1	even	1	trivial
310.2.w.a.117.15	yes	256	155.117	even	60	inner
310.2.w.a.177.2	yes	256	5.2	odd	4	inner
310.2.w.a.303.2	yes	256	31.24	odd	30	inner