Embedded newform 403.2.bf.a.37.13

• Label: 403.2.bf.a.37.13
• 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			37.13
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.216378 + 0.807533i) q^{2} +(-1.68759 - 0.974331i) q^{3} +(1.12676 + 0.650535i) q^{4} +(0.191653 - 0.0513533i) q^{5} +(1.15196 - 1.15196i) q^{6} +(-1.16887 + 1.16887i) q^{7} +(-1.95145 + 1.95145i) q^{8} +(0.398643 + 0.690470i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.216378	+	0.807533i	−0.153002	+	0.571012i	0.846266	+	0.532761i	\(0.178845\pi\)
							−0.999268	+	0.0382514i	\(0.987821\pi\)
\(3\)	−1.68759	−	0.974331i	−0.974331	−	0.562530i	−0.0737774	−	0.997275i	\(-0.523505\pi\)
							−0.900554	+	0.434744i	\(0.856839\pi\)
\(5\)	0.191653	−	0.0513533i	0.0857099	−	0.0229659i	−0.215709	−	0.976458i	\(-0.569206\pi\)
							0.301419	+	0.953492i	\(0.402540\pi\)
\(7\)	−1.16887	+	1.16887i	−0.441793	+	0.441793i	−0.892614	−	0.450821i	\(-0.851131\pi\)
							0.450821	+	0.892614i	\(0.351131\pi\)
\(11\)	0.360723	+	0.360723i	0.108762	+	0.108762i	0.759394	−	0.650631i	\(-0.225496\pi\)
							−0.650631	+	0.759394i	\(0.725496\pi\)
\(13\)	−0.356749	+	3.58786i	−0.0989444	+	0.995093i
							\(17\)	−0.855529			−0.207496			−0.103748	−	0.994604i	\(-0.533084\pi\)
−0.103748	+	0.994604i	\(0.533084\pi\)
\(19\)	−0.234492	−	0.234492i	−0.0537961	−	0.0537961i	0.679697	−	0.733493i	\(-0.262111\pi\)
							−0.733493	+	0.679697i	\(0.762111\pi\)
\(23\)	3.72205	+	6.44679i	0.776102	+	1.34425i	0.934173	+	0.356820i	\(0.116139\pi\)
							−0.158071	+	0.987428i	\(0.550527\pi\)
\(29\)	−7.83555	+	4.52386i	−1.45502	+	0.840059i	−0.998760	−	0.0497854i	\(-0.984146\pi\)
							−0.456265	+	0.889844i	\(0.650813\pi\)
\(31\)	4.89784	−	2.64786i	0.879678	−	0.475570i
							\(37\)	−4.53058	+	1.21397i	−0.744823	+	0.199575i	−0.611221	−	0.791460i	\(-0.709321\pi\)
−0.133603	+	0.991035i	\(0.542655\pi\)
\(41\)	−6.06485	−	6.06485i	−0.947170	−	0.947170i	0.0515026	−	0.998673i	\(-0.483599\pi\)
							−0.998673	+	0.0515026i	\(0.983599\pi\)
\(43\)	11.5056			1.75459			0.877294	−	0.479954i	\(-0.159347\pi\)
							0.877294	+	0.479954i	\(0.159347\pi\)
\(47\)	−4.29720	+	4.29720i	−0.626812	+	0.626812i	−0.947265	−	0.320453i	\(-0.896165\pi\)
							0.320453	+	0.947265i	\(0.396165\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.13	✓	140	13.6	odd	12
403.2.ba.a.336.13	yes	140	31.26	odd	6
403.2.bf.a.37.13	yes	140	1.1	even	1	trivial
403.2.bf.a.305.13	yes	140	403.305	even	12	inner