Embedded newform 403.2.be.c.57.19

• Label: 403.2.be.c.57.19
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.19
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0798971 - 0.0798971i) q^{2} +(-1.31416 - 0.758731i) q^{3} +1.98723i q^{4} +(1.61887 + 0.433775i) q^{5} +(-0.165618 + 0.0443772i) q^{6} +(-4.54713 + 1.21840i) q^{7} +(0.318568 + 0.318568i) q^{8} +(-0.348656 - 0.603889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0798971	−	0.0798971i	0.0564958	−	0.0564958i	−0.678294	−	0.734790i	\(-0.737281\pi\)
							0.734790	+	0.678294i	\(0.237281\pi\)
\(3\)	−1.31416	−	0.758731i	−0.758731	−	0.438053i	0.0701091	−	0.997539i	\(-0.477665\pi\)
							−0.828840	+	0.559486i	\(0.810999\pi\)
\(5\)	1.61887	+	0.433775i	0.723981	+	0.193990i	0.601947	−	0.798536i	\(-0.294392\pi\)
							0.122034	+	0.992526i	\(0.461058\pi\)
\(7\)	−4.54713	+	1.21840i	−1.71865	+	0.460512i	−0.977520	−	0.210844i	\(-0.932379\pi\)
							−0.741135	+	0.671356i	\(0.765712\pi\)
\(11\)	−4.23639	−	1.13514i	−1.27732	−	0.342257i	−0.444489	−	0.895784i	\(-0.646615\pi\)
							−0.832831	+	0.553527i	\(0.813282\pi\)
\(13\)	2.60457	−	2.49323i	0.722379	−	0.691497i
							\(17\)	0.334298	−	0.579020i	0.0810791	−	0.140433i	−0.822635	−	0.568570i	\(-0.807497\pi\)
0.903714	+	0.428137i	\(0.140830\pi\)
\(19\)	−1.36971	−	5.11183i	−0.314233	−	1.17273i	−0.924701	−	0.380693i	\(-0.875685\pi\)
							0.610468	−	0.792041i	\(-0.290981\pi\)
\(23\)	−5.31244			−1.10772			−0.553860	−	0.832610i	\(-0.686846\pi\)
							−0.553860	+	0.832610i	\(0.686846\pi\)
\(29\)			4.72098i			0.876664i	0.898813	+	0.438332i	\(0.144431\pi\)
							−0.898813	+	0.438332i	\(0.855569\pi\)
\(31\)	4.44287	+	3.35572i	0.797963	+	0.602706i
							\(37\)	0.501471	+	1.87151i	0.0824413	+	0.307675i	0.994817	−	0.101678i	\(-0.0324211\pi\)
−0.912376	+	0.409353i	\(0.865754\pi\)
\(41\)	−5.43026	−	1.45503i	−0.848064	−	0.227238i	−0.191485	−	0.981495i	\(-0.561330\pi\)
							−0.656579	+	0.754257i	\(0.727997\pi\)
\(43\)	−3.14908	+	5.45436i	−0.480230	+	0.831782i	−0.999743	−	0.0226804i	\(-0.992780\pi\)
							0.519513	+	0.854462i	\(0.326113\pi\)
\(47\)	−2.82430	−	2.82430i	−0.411967	−	0.411967i	0.470456	−	0.882423i	\(-0.344089\pi\)
							−0.882423	+	0.470456i	\(0.844089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.19	✓	136
13.8	odd	4	inner	403.2.be.c.398.19	yes	136
31.6	odd	6	inner	403.2.be.c.161.19	yes	136
403.99	even	12	inner	403.2.be.c.99.19	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.19	✓	136	1.1	even	1	trivial
403.2.be.c.99.19	yes	136	403.99	even	12	inner
403.2.be.c.161.19	yes	136	31.6	odd	6	inner
403.2.be.c.398.19	yes	136	13.8	odd	4	inner