Embedded newform 403.2.ba.a.6.20

• Label: 403.2.ba.a.6.20
• 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			6.20
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.107992 + 0.403030i) q^{2} +2.83596i q^{3} +(1.58128 - 0.912952i) q^{4} +(-0.961540 - 3.58852i) q^{5} +(-1.14298 + 0.306260i) q^{6} +(1.51235 - 0.405234i) q^{7} +(1.12879 + 1.12879i) q^{8} -5.04265 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.107992	+	0.403030i	0.0763616	+	0.284986i	0.993539	−	0.113495i	\(-0.0362045\pi\)
							−0.917177	+	0.398480i	\(0.869538\pi\)
\(3\)			2.83596i			1.63734i	0.574264	+	0.818670i	\(0.305288\pi\)
							−0.574264	+	0.818670i	\(0.694712\pi\)
\(5\)	−0.961540	−	3.58852i	−0.430014	−	1.60483i	−0.752724	−	0.658336i	\(-0.771261\pi\)
							0.322710	−	0.946498i	\(-0.395406\pi\)
\(7\)	1.51235	−	0.405234i	0.571616	−	0.153164i	0.0385782	−	0.999256i	\(-0.487717\pi\)
							0.533038	+	0.846092i	\(0.321050\pi\)
\(11\)	2.88352	+	0.772637i	0.869414	+	0.232959i	0.665834	−	0.746100i	\(-0.268076\pi\)
							0.203580	+	0.979058i	\(0.434742\pi\)
\(13\)	3.14126	+	1.76989i	0.871228	+	0.490879i
							\(17\)	−2.08763	+	3.61588i	−0.506324	+	0.876979i	0.493649	+	0.869661i	\(0.335663\pi\)
−0.999973	+	0.00731793i	\(0.997671\pi\)
\(19\)	0.875190	+	3.26626i	0.200782	+	0.749330i	0.990694	+	0.136109i	\(0.0434598\pi\)
							−0.789911	+	0.613221i	\(0.789873\pi\)
\(23\)	4.34763	−	7.53032i	0.906543	−	1.57018i	0.0877116	−	0.996146i	\(-0.472045\pi\)
							0.818832	−	0.574033i	\(-0.194622\pi\)
\(29\)	−8.59939	−	4.96486i	−1.59687	−	0.921951i	−0.992086	−	0.125559i	\(-0.959928\pi\)
							−0.604780	−	0.796392i	\(-0.706739\pi\)
\(31\)	−2.39200	+	5.02775i	−0.429617	+	0.903011i
							\(37\)	−0.113397	+	0.113397i	−0.0186423	+	0.0186423i	−0.716367	−	0.697724i	\(-0.754196\pi\)
0.697724	+	0.716367i	\(0.254196\pi\)
\(41\)	6.61228	+	1.77176i	1.03266	+	0.276702i	0.735070	−	0.677991i	\(-0.237149\pi\)
							0.297594	+	0.954692i	\(0.403816\pi\)
\(43\)	0.368983	−	0.639097i	0.0562693	−	0.0974613i	−0.836519	−	0.547938i	\(-0.815413\pi\)
							0.892788	+	0.450477i	\(0.148746\pi\)
\(47\)	−5.14383	−	5.14383i	−0.750304	−	0.750304i	0.224232	−	0.974536i	\(-0.428013\pi\)
							−0.974536	+	0.224232i	\(0.928013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.20	✓	140
13.11	odd	12		403.2.bf.a.37.20	yes	140
31.26	odd	6		403.2.bf.a.305.20	yes	140
403.336	even	12	inner	403.2.ba.a.336.20	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.20	✓	140	1.1	even	1	trivial
403.2.ba.a.336.20	yes	140	403.336	even	12	inner
403.2.bf.a.37.20	yes	140	13.11	odd	12
403.2.bf.a.305.20	yes	140	31.26	odd	6