Embedded newform 403.2.ba.a.6.11

• Label: 403.2.ba.a.6.11
• Level: $403$
• Weight: $2$
• Character: 403.6
• 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.ba (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			6.11
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.361997 - 1.35099i) q^{2} +2.34712i q^{3} +(0.0379121 - 0.0218885i) q^{4} +(-0.843165 - 3.14674i) q^{5} +(3.17094 - 0.849651i) q^{6} +(-1.42675 + 0.382295i) q^{7} +(-2.02129 - 2.02129i) q^{8} -2.50897 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

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

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

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	−0.967548	−	0.252688i	\(-0.918685\pi\)
							0.711577	−	0.702608i	\(-0.247981\pi\)
\(3\)			2.34712i			1.35511i	0.735472	+	0.677555i	\(0.236960\pi\)
							−0.735472	+	0.677555i	\(0.763040\pi\)
\(5\)	−0.843165	−	3.14674i	−0.377075	−	1.40726i	−0.850289	−	0.526316i	\(-0.823573\pi\)
							0.473214	−	0.880947i	\(-0.343094\pi\)
\(7\)	−1.42675	+	0.382295i	−0.539259	+	0.144494i	−0.518161	−	0.855283i	\(-0.673383\pi\)
							−0.0210985	+	0.999777i	\(0.506716\pi\)
\(11\)	−4.54581	−	1.21805i	−1.37061	−	0.367254i	−0.502910	−	0.864339i	\(-0.667737\pi\)
							−0.867702	+	0.497084i	\(0.834404\pi\)
\(13\)	−3.02105	+	1.96806i	−0.837888	+	0.545842i
							\(17\)	3.19364	−	5.53155i	0.774572	−	1.34160i	−0.160463	−	0.987042i	\(-0.551299\pi\)
0.935035	−	0.354556i	\(-0.115368\pi\)
\(19\)	−0.212000	−	0.791196i	−0.0486362	−	0.181513i	0.937335	−	0.348430i	\(-0.113285\pi\)
							−0.985971	+	0.166918i	\(0.946619\pi\)
\(23\)	0.908696	−	1.57391i	0.189476	−	0.328183i	−0.755599	−	0.655034i	\(-0.772654\pi\)
							0.945076	+	0.326851i	\(0.105988\pi\)
\(29\)	1.19333	+	0.688968i	0.221595	+	0.127938i	0.606689	−	0.794939i	\(-0.292497\pi\)
							−0.385093	+	0.922878i	\(0.625831\pi\)
\(31\)	−4.98347	−	2.48295i	−0.895058	−	0.445951i
							\(37\)	2.86047	−	2.86047i	0.470259	−	0.470259i	−0.431739	−	0.901998i	\(-0.642100\pi\)
0.901998	+	0.431739i	\(0.142100\pi\)
\(41\)	−8.10384	−	2.17142i	−1.26561	−	0.339119i	−0.437261	−	0.899335i	\(-0.644051\pi\)
							−0.828346	+	0.560216i	\(0.810718\pi\)
\(43\)	5.34928	−	9.26523i	0.815758	−	1.41293i	−0.0930242	−	0.995664i	\(-0.529653\pi\)
							0.908782	−	0.417271i	\(-0.137013\pi\)
\(47\)	7.36081	+	7.36081i	1.07368	+	1.07368i	0.997060	+	0.0766243i	\(0.0244142\pi\)
							0.0766243	+	0.997060i	\(0.475586\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.11	✓	140	1.1	even	1	trivial
403.2.ba.a.336.11	yes	140	403.336	even	12	inner
403.2.bf.a.37.11	yes	140	13.11	odd	12
403.2.bf.a.305.11	yes	140	31.26	odd	6