Embedded newform 310.2.w.a.117.15

• Label: 310.2.w.a.117.15
• Level: $310$
• Weight: $2$
• Character: 310.117
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• 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			117.15
Character	\(\chi\)	\(=\)	310.117
Dual form			310.2.w.a.53.15

$q$-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 \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{30}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.15	✓	256	155.53	even	60	inner
310.2.w.a.117.15	yes	256	1.1	even	1	trivial
310.2.w.a.177.2	yes	256	31.22	odd	30	inner
310.2.w.a.303.2	yes	256	5.3	odd	4	inner