Embedded newform 403.2.bf.a.37.6

• Label: 403.2.bf.a.37.6
• 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			37.6
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.529121 + 1.97471i) q^{2} +(-0.0550086 - 0.0317592i) q^{3} +(-1.88744 - 1.08972i) q^{4} +(-1.35641 + 0.363448i) q^{5} +(0.0918213 - 0.0918213i) q^{6} +(2.81815 - 2.81815i) q^{7} +(0.259387 - 0.259387i) q^{8} +(-1.49798 - 2.59458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.529121	+	1.97471i	−0.374145	+	1.39633i	0.480445	+	0.877025i	\(0.340475\pi\)
							−0.854590	+	0.519303i	\(0.826191\pi\)
\(3\)	−0.0550086	−	0.0317592i	−0.0317592	−	0.0183362i	0.484036	−	0.875048i	\(-0.339170\pi\)
							−0.515796	+	0.856712i	\(0.672504\pi\)
\(5\)	−1.35641	+	0.363448i	−0.606603	+	0.162539i	−0.549030	−	0.835802i	\(-0.685003\pi\)
							−0.0575729	+	0.998341i	\(0.518336\pi\)
\(7\)	2.81815	−	2.81815i	1.06516	−	1.06516i	0.0674355	−	0.997724i	\(-0.478518\pi\)
							0.997724	−	0.0674355i	\(-0.0214817\pi\)
\(11\)	−3.38347	−	3.38347i	−1.02016	−	1.02016i	−0.999793	−	0.0203633i	\(-0.993518\pi\)
							−0.0203633	−	0.999793i	\(-0.506482\pi\)
\(13\)	−2.25079	−	2.81672i	−0.624257	−	0.781219i
							\(17\)	−5.27890			−1.28032			−0.640160	−	0.768241i	\(-0.721132\pi\)
−0.640160	+	0.768241i	\(0.721132\pi\)
\(19\)	0.00735902	+	0.00735902i	0.00168827	+	0.00168827i	0.707950	−	0.706262i	\(-0.249620\pi\)
							−0.706262	+	0.707950i	\(0.749620\pi\)
\(23\)	2.73351	+	4.73458i	0.569977	+	0.987229i	0.996567	+	0.0827842i	\(0.0263812\pi\)
							−0.426591	+	0.904445i	\(0.640285\pi\)
\(29\)	4.66755	−	2.69481i	0.866742	−	0.500414i	0.000477773	−	1.00000i	\(-0.499848\pi\)
							0.866264	+	0.499586i	\(0.166515\pi\)
\(31\)	5.54034	+	0.551933i	0.995074	+	0.0991300i
							\(37\)	0.476213	−	0.127601i	0.0782890	−	0.0209775i	−0.219462	−	0.975621i	\(-0.570430\pi\)
0.297751	+	0.954644i	\(0.403764\pi\)
\(41\)	−5.05486	−	5.05486i	−0.789437	−	0.789437i	0.191965	−	0.981402i	\(-0.438514\pi\)
							−0.981402	+	0.191965i	\(0.938514\pi\)
\(43\)	−2.30084			−0.350875			−0.175438	−	0.984491i	\(-0.556134\pi\)
							−0.175438	+	0.984491i	\(0.556134\pi\)
\(47\)	−8.27293	+	8.27293i	−1.20673	+	1.20673i	−0.234651	+	0.972080i	\(0.575395\pi\)
							−0.972080	+	0.234651i	\(0.924605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.6	yes	140
13.6	odd	12		403.2.ba.a.6.6	✓	140
31.26	odd	6		403.2.ba.a.336.6	yes	140
403.305	even	12	inner	403.2.bf.a.305.6	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.6	✓	140	13.6	odd	12
403.2.ba.a.336.6	yes	140	31.26	odd	6
403.2.bf.a.37.6	yes	140	1.1	even	1	trivial
403.2.bf.a.305.6	yes	140	403.305	even	12	inner