Embedded newform 403.2.ba.a.6.13

• Label: 403.2.ba.a.6.13
• 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			6.13
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.216378 - 0.807533i) q^{2} +1.94866i q^{3} +(1.12676 - 0.650535i) q^{4} +(-0.0513533 - 0.191653i) q^{5} +(1.57361 - 0.421647i) q^{6} +(1.59671 - 0.427837i) q^{7} +(-1.95145 - 1.95145i) q^{8} -0.797286 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.216378	−	0.807533i	−0.153002	−	0.571012i	−0.999268	−	0.0382514i	\(-0.987821\pi\)
							0.846266	−	0.532761i	\(-0.178845\pi\)
\(3\)			1.94866i			1.12506i	0.826777	+	0.562530i	\(0.190172\pi\)
							−0.826777	+	0.562530i	\(0.809828\pi\)
\(5\)	−0.0513533	−	0.191653i	−0.0229659	−	0.0857099i	0.953492	−	0.301419i	\(-0.0974603\pi\)
							−0.976458	+	0.215709i	\(0.930794\pi\)
\(7\)	1.59671	−	0.427837i	0.603500	−	0.161707i	0.0558842	−	0.998437i	\(-0.482202\pi\)
							0.547616	+	0.836730i	\(0.315536\pi\)
\(11\)	0.492757	+	0.132034i	0.148572	+	0.0398097i	0.332338	−	0.943160i	\(-0.392162\pi\)
							−0.183767	+	0.982970i	\(0.558829\pi\)
\(13\)	2.92880	−	2.10288i	0.812304	−	0.583235i
							\(17\)	−0.427765	+	0.740910i	−0.103748	+	0.179697i	−0.913226	−	0.407453i	\(-0.866417\pi\)
0.809478	+	0.587150i	\(0.199750\pi\)
\(19\)	−0.0858299	−	0.320322i	−0.0196907	−	0.0734868i	0.955381	−	0.295375i	\(-0.0954446\pi\)
							−0.975072	+	0.221888i	\(0.928778\pi\)
\(23\)	−3.72205	+	6.44679i	−0.776102	+	1.34425i	0.158071	+	0.987428i	\(0.449473\pi\)
							−0.934173	+	0.356820i	\(0.883861\pi\)
\(29\)	7.83555	+	4.52386i	1.45502	+	0.840059i	0.998760	−	0.0497854i	\(-0.0158537\pi\)
							0.456265	+	0.889844i	\(0.349187\pi\)
\(31\)	−2.64786	−	4.89784i	−0.475570	−	0.879678i
							\(37\)	−3.31662	+	3.31662i	−0.545248	+	0.545248i	−0.925063	−	0.379814i	\(-0.875988\pi\)
0.379814	+	0.925063i	\(0.375988\pi\)
\(41\)	8.28474	+	2.21989i	1.29386	+	0.346688i	0.839125	−	0.543939i	\(-0.183068\pi\)
							0.454734	+	0.890627i	\(0.349734\pi\)
\(43\)	5.75280	−	9.96414i	0.877294	−	1.51952i	0.0229944	−	0.999736i	\(-0.492680\pi\)
							0.854299	−	0.519782i	\(-0.173987\pi\)
\(47\)	−4.29720	−	4.29720i	−0.626812	−	0.626812i	0.320453	−	0.947265i	\(-0.396165\pi\)
							−0.947265	+	0.320453i	\(0.896165\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.13	✓	140
13.11	odd	12		403.2.bf.a.37.13	yes	140
31.26	odd	6		403.2.bf.a.305.13	yes	140
403.336	even	12	inner	403.2.ba.a.336.13	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.13	✓	140	1.1	even	1	trivial
403.2.ba.a.336.13	yes	140	403.336	even	12	inner
403.2.bf.a.37.13	yes	140	13.11	odd	12
403.2.bf.a.305.13	yes	140	31.26	odd	6