Embedded newform 390.2.bj.a.59.17

• Label: 390.2.bj.a.59.17
• Level: $390$
• Weight: $2$
• Character: 390.59
• Analytic conductor: $3.114$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.bj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.11416567883\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			59.17
Character	\(\chi\)	\(=\)	390.59
Dual form			390.2.bj.a.119.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.965926 - 0.258819i) q^{2} +(-1.52381 - 0.823413i) q^{3} +(0.866025 - 0.500000i) q^{4} +(1.09695 - 1.94851i) q^{5} +(-1.68500 - 0.400965i) q^{6} +(0.391563 - 1.46133i) q^{7} +(0.707107 - 0.707107i) q^{8} +(1.64398 + 2.50945i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\)	\(131\)	\(157\)	\(301\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.965926	−	0.258819i	0.683013	−	0.183013i
							\(3\)	−1.52381	−	0.823413i	−0.879771	−	0.475398i
\(5\)	1.09695	−	1.94851i	0.490571	−	0.871401i
							\(7\)	0.391563	−	1.46133i	0.147997	−	0.552332i	−0.851607	−	0.524181i	\(-0.824371\pi\)
0.999604	−	0.0281509i	\(-0.00896190\pi\)
\(11\)	−1.75011	+	0.468942i	−0.527679	+	0.141391i	−0.512816	−	0.858498i	\(-0.671398\pi\)
							−0.0148629	+	0.999890i	\(0.504731\pi\)
\(13\)	−2.94633	−	2.07826i	−0.817165	−	0.576404i
							\(17\)	0.174178	−	0.100562i	0.0422444	−	0.0243898i	−0.478729	−	0.877963i	\(-0.658902\pi\)
0.520973	+	0.853573i	\(0.325569\pi\)
\(19\)	0.975349	−	3.64005i	0.223760	−	0.835085i	−0.759137	−	0.650931i	\(-0.774379\pi\)
							0.982897	−	0.184154i	\(-0.0589546\pi\)
\(23\)	1.36020	+	0.785312i	0.283621	+	0.163749i	0.635062	−	0.772461i	\(-0.280975\pi\)
							−0.351440	+	0.936210i	\(0.614308\pi\)
\(29\)	6.92417	+	3.99767i	1.28579	+	0.742349i	0.977900	−	0.209073i	\(-0.0670448\pi\)
							0.307887	+	0.951423i	\(0.400378\pi\)
\(31\)	−6.25168	−	6.25168i	−1.12284	−	1.12284i	−0.991313	−	0.131522i	\(-0.958014\pi\)
							−0.131522	−	0.991313i	\(-0.541986\pi\)
\(37\)	4.71004	−	1.26205i	0.774326	−	0.207480i	0.150044	−	0.988679i	\(-0.452058\pi\)
							0.624282	+	0.781199i	\(0.285392\pi\)
\(41\)	2.38343	+	8.89507i	0.372229	+	1.38918i	0.857352	+	0.514730i	\(0.172108\pi\)
							−0.485124	+	0.874446i	\(0.661225\pi\)
\(43\)	1.71457	+	2.96971i	0.261469	+	0.452877i	0.966632	−	0.256167i	\(-0.0824598\pi\)
							−0.705164	+	0.709045i	\(0.749126\pi\)
\(47\)	−6.03201	+	6.03201i	−0.879859	+	0.879859i	−0.993520	−	0.113661i	\(-0.963742\pi\)
							0.113661	+	0.993520i	\(0.463742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.bj.a.59.17	yes	112
3.2	odd	2	inner	390.2.bj.a.59.2	✓	112
5.4	even	2	inner	390.2.bj.a.59.12	yes	112
13.2	odd	12	inner	390.2.bj.a.119.27	yes	112
15.14	odd	2	inner	390.2.bj.a.59.27	yes	112
39.2	even	12	inner	390.2.bj.a.119.12	yes	112
65.54	odd	12	inner	390.2.bj.a.119.2	yes	112
195.119	even	12	inner	390.2.bj.a.119.17	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bj.a.59.2	✓	112	3.2	odd	2	inner
390.2.bj.a.59.12	yes	112	5.4	even	2	inner
390.2.bj.a.59.17	yes	112	1.1	even	1	trivial
390.2.bj.a.59.27	yes	112	15.14	odd	2	inner
390.2.bj.a.119.2	yes	112	65.54	odd	12	inner
390.2.bj.a.119.12	yes	112	39.2	even	12	inner
390.2.bj.a.119.17	yes	112	195.119	even	12	inner
390.2.bj.a.119.27	yes	112	13.2	odd	12	inner