Embedded newform 310.2.w.a.303.2

• Label: 310.2.w.a.303.2
• Level: $310$
• Weight: $2$
• Character: 310.303
• 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			303.2
Character	\(\chi\)	\(=\)	310.303
Dual form			310.2.w.a.177.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.891007 - 0.453990i) q^{2} +(-0.907916 + 1.39807i) q^{3} +(0.587785 + 0.809017i) q^{4} +(0.393126 - 2.20124i) q^{5} +(1.44367 - 0.833503i) q^{6} +(-3.06249 + 1.17558i) q^{7} +(-0.156434 - 0.987688i) q^{8} +(0.0899259 + 0.201977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.891007	−	0.453990i	−0.630037	−	0.321020i
							\(3\)	−0.907916	+	1.39807i	−0.524186	+	0.807175i	−0.997270	−	0.0738419i	\(-0.976474\pi\)
0.473084	+	0.881017i	\(0.343141\pi\)
\(5\)	0.393126	−	2.20124i	0.175811	−	0.984424i
							\(7\)	−3.06249	+	1.17558i	−1.15751	+	0.444327i	−0.859952	−	0.510375i	\(-0.829506\pi\)
−0.297561	+	0.954703i	\(0.596173\pi\)
\(11\)	−0.643025	+	0.0675846i	−0.193879	+	0.0203775i	−0.200970	−	0.979597i	\(-0.564409\pi\)
							0.00709109	+	0.999975i	\(0.497743\pi\)
\(13\)	−3.13541	−	0.164320i	−0.869605	−	0.0455741i	−0.387688	−	0.921791i	\(-0.626726\pi\)
							−0.481917	+	0.876217i	\(0.660059\pi\)
\(17\)	0.624734	−	0.505900i	0.151520	−	0.122699i	−0.550563	−	0.834794i	\(-0.685587\pi\)
							0.702083	+	0.712095i	\(0.252253\pi\)
\(19\)	−4.16895	−	3.75374i	−0.956423	−	0.861167i	0.0339733	−	0.999423i	\(-0.489184\pi\)
							−0.990397	+	0.138255i	\(0.955851\pi\)
\(23\)	−6.21756	+	0.984764i	−1.29645	+	0.205338i	−0.766254	−	0.642537i	\(-0.777882\pi\)
							−0.530196	+	0.847875i	\(0.677882\pi\)
\(29\)	−0.356381	−	1.09683i	−0.0661782	−	0.203676i	0.912499	−	0.409078i	\(-0.134150\pi\)
							−0.978678	+	0.205403i	\(0.934150\pi\)
\(31\)	−3.10939	+	4.61863i	−0.558463	+	0.829530i
							\(37\)	1.02540	−	3.82683i	0.168574	−	0.629128i	−0.828983	−	0.559274i	\(-0.811080\pi\)
0.997557	−	0.0698538i	\(-0.0222533\pi\)
\(41\)	0.464812	+	0.0987989i	0.0725915	+	0.0154298i	0.244064	−	0.969759i	\(-0.421519\pi\)
							−0.171473	+	0.985189i	\(0.554853\pi\)
\(43\)	−0.152393	−	2.90784i	−0.0232397	−	0.443441i	−0.985325	−	0.170691i	\(-0.945400\pi\)
							0.962085	−	0.272750i	\(-0.0879332\pi\)
\(47\)	3.85209	+	7.56015i	0.561885	+	1.10276i	0.980851	+	0.194760i	\(0.0623927\pi\)
							−0.418966	+	0.908002i	\(0.637607\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.15	✓	256	31.22	odd	30	inner
310.2.w.a.117.15	yes	256	5.2	odd	4	inner
310.2.w.a.177.2	yes	256	155.22	even	60	inner
310.2.w.a.303.2	yes	256	1.1	even	1	trivial