Embedded newform 403.2.ba.a.6.9

• Label: 403.2.ba.a.6.9
• 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			6.9
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.495867 - 1.85060i) q^{2} -0.831703i q^{3} +(-1.44679 + 0.835304i) q^{4} +(0.740052 + 2.76191i) q^{5} +(-1.53915 + 0.412414i) q^{6} +(1.04383 - 0.279694i) q^{7} +(-0.446239 - 0.446239i) q^{8} +2.30827 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.495867	−	1.85060i	−0.350631	−	1.30857i	−0.885894	−	0.463887i	\(-0.846454\pi\)
							0.535263	−	0.844685i	\(-0.320212\pi\)
\(3\)		−	0.831703i		−	0.480184i	−0.970750	−	0.240092i	\(-0.922822\pi\)
							0.970750	−	0.240092i	\(-0.0771776\pi\)
\(5\)	0.740052	+	2.76191i	0.330961	+	1.23517i	0.908182	+	0.418576i	\(0.137471\pi\)
							−0.577220	+	0.816589i	\(0.695863\pi\)
\(7\)	1.04383	−	0.279694i	0.394531	−	0.105714i	−0.0560987	−	0.998425i	\(-0.517866\pi\)
							0.450630	+	0.892711i	\(0.351199\pi\)
\(11\)	2.40774	+	0.645153i	0.725962	+	0.194521i	0.602830	−	0.797869i	\(-0.294040\pi\)
							0.123132	+	0.992390i	\(0.460706\pi\)
\(13\)	−1.08861	−	3.43729i	−0.301925	−	0.953332i
							\(17\)	2.29000	−	3.96640i	0.555407	−	0.961993i	−0.442465	−	0.896786i	\(-0.645896\pi\)
0.997872	−	0.0652072i	\(-0.0207708\pi\)
\(19\)	−1.17250	−	4.37583i	−0.268990	−	1.00388i	−0.959763	−	0.280813i	\(-0.909396\pi\)
							0.690772	−	0.723072i	\(-0.257271\pi\)
\(23\)	−0.0230993	+	0.0400091i	−0.00481653	+	0.00834247i	−0.868424	−	0.495823i	\(-0.834866\pi\)
							0.863607	+	0.504165i	\(0.168200\pi\)
\(29\)	6.76189	+	3.90398i	1.25565	+	0.724951i	0.972226	−	0.234044i	\(-0.0751959\pi\)
							0.283425	+	0.958994i	\(0.408529\pi\)
\(31\)	3.41527	+	4.39726i	0.613400	+	0.789772i
							\(37\)	−5.55983	+	5.55983i	−0.914030	+	0.914030i	−0.996586	−	0.0825565i	\(-0.973692\pi\)
0.0825565	+	0.996586i	\(0.473692\pi\)
\(41\)	−9.94629	−	2.66510i	−1.55335	−	0.416219i	−0.622799	−	0.782382i	\(-0.714004\pi\)
							−0.930550	+	0.366164i	\(0.880671\pi\)
\(43\)	2.10659	−	3.64872i	0.321252	−	0.556425i	−0.659495	−	0.751709i	\(-0.729230\pi\)
							0.980747	+	0.195285i	\(0.0625630\pi\)
\(47\)	1.52443	+	1.52443i	0.222360	+	0.222360i	0.809492	−	0.587131i	\(-0.199743\pi\)
							−0.587131	+	0.809492i	\(0.699743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.9	✓	140
13.11	odd	12		403.2.bf.a.37.9	yes	140
31.26	odd	6		403.2.bf.a.305.9	yes	140
403.336	even	12	inner	403.2.ba.a.336.9	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.9	✓	140	1.1	even	1	trivial
403.2.ba.a.336.9	yes	140	403.336	even	12	inner
403.2.bf.a.37.9	yes	140	13.11	odd	12
403.2.bf.a.305.9	yes	140	31.26	odd	6