Embedded newform 403.2.bf.a.37.11

• Label: 403.2.bf.a.37.11
• Level: $403$
• Weight: $2$
• Character: 403.37
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.11
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.361997 + 1.35099i) q^{2} +(-2.03266 - 1.17356i) q^{3} +(0.0379121 + 0.0218885i) q^{4} +(3.14674 - 0.843165i) q^{5} +(2.32129 - 2.32129i) q^{6} +(1.04445 - 1.04445i) q^{7} +(-2.02129 + 2.02129i) q^{8} +(1.25448 + 2.17283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.361997	+	1.35099i	−0.255971	+	0.955296i	0.711577	+	0.702608i	\(0.247981\pi\)
							−0.967548	+	0.252688i	\(0.918685\pi\)
\(3\)	−2.03266	−	1.17356i	−1.17356	−	0.677555i	−0.219044	−	0.975715i	\(-0.570294\pi\)
							−0.954516	+	0.298160i	\(0.903627\pi\)
\(5\)	3.14674	−	0.843165i	1.40726	−	0.377075i	0.526316	−	0.850289i	\(-0.323573\pi\)
							0.880947	+	0.473214i	\(0.156906\pi\)
\(7\)	1.04445	−	1.04445i	0.394765	−	0.394765i	−0.481617	−	0.876382i	\(-0.659950\pi\)
							0.876382	+	0.481617i	\(0.159950\pi\)
\(11\)	−3.32776	−	3.32776i	−1.00336	−	1.00336i	−0.999994	−	0.00336358i	\(-0.998929\pi\)
							−0.00336358	−	0.999994i	\(-0.501071\pi\)
\(13\)	0.193866	−	3.60034i	0.0537688	−	0.998553i
							\(17\)	6.38728			1.54914			0.774572	−	0.632486i	\(-0.217965\pi\)
0.774572	+	0.632486i	\(0.217965\pi\)
\(19\)	−0.579196	−	0.579196i	−0.132877	−	0.132877i	0.637540	−	0.770417i	\(-0.279952\pi\)
							−0.770417	+	0.637540i	\(0.779952\pi\)
\(23\)	−0.908696	−	1.57391i	−0.189476	−	0.328183i	0.755599	−	0.655034i	\(-0.227346\pi\)
							−0.945076	+	0.326851i	\(0.894012\pi\)
\(29\)	−1.19333	+	0.688968i	−0.221595	+	0.127938i	−0.606689	−	0.794939i	\(-0.707503\pi\)
							0.385093	+	0.922878i	\(0.374169\pi\)
\(31\)	2.48295	−	4.98347i	0.445951	−	0.895058i
							\(37\)	3.90748	−	1.04701i	0.642386	−	0.172127i	0.0771018	−	0.997023i	\(-0.475433\pi\)
0.565284	+	0.824897i	\(0.308767\pi\)
\(41\)	5.93242	+	5.93242i	0.926489	+	0.926489i	0.997477	−	0.0709881i	\(-0.0226152\pi\)
							−0.0709881	+	0.997477i	\(0.522615\pi\)
\(43\)	10.6986			1.63152			0.815758	−	0.578393i	\(-0.196320\pi\)
							0.815758	+	0.578393i	\(0.196320\pi\)
\(47\)	7.36081	−	7.36081i	1.07368	−	1.07368i	0.0766243	−	0.997060i	\(-0.475586\pi\)
							0.997060	−	0.0766243i	\(-0.0244142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.11	yes	140
13.6	odd	12		403.2.ba.a.6.11	✓	140
31.26	odd	6		403.2.ba.a.336.11	yes	140
403.305	even	12	inner	403.2.bf.a.305.11	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.11	✓	140	13.6	odd	12
403.2.ba.a.336.11	yes	140	31.26	odd	6
403.2.bf.a.37.11	yes	140	1.1	even	1	trivial
403.2.bf.a.305.11	yes	140	403.305	even	12	inner