Embedded newform 403.2.e.a.191.12

• Label: 403.2.e.a.191.12
• Level: $403$
• Weight: $2$
• Character: 403.191
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			191.12
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.447963 + 0.775894i) q^{2} +(0.0864423 + 0.149722i) q^{3} +(0.598659 + 1.03691i) q^{4} +(-0.364951 - 0.632113i) q^{5} -0.154892 q^{6} +3.22911 q^{7} -2.86456 q^{8} +(1.48506 - 2.57219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.447963	+	0.775894i	−0.316758	+	0.548640i	−0.979810	−	0.199933i	\(-0.935927\pi\)
							0.663052	+	0.748573i	\(0.269261\pi\)
\(3\)	0.0864423	+	0.149722i	0.0499075	+	0.0864423i	0.889900	−	0.456156i	\(-0.150774\pi\)
							−0.839992	+	0.542598i	\(0.817441\pi\)
\(5\)	−0.364951	−	0.632113i	−0.163211	−	0.282690i	0.772808	−	0.634640i	\(-0.218852\pi\)
							−0.936018	+	0.351951i	\(0.885518\pi\)
\(7\)	3.22911			1.22049			0.610244	−	0.792213i	\(-0.291071\pi\)
							0.610244	+	0.792213i	\(0.291071\pi\)
\(11\)	1.81678			0.547779			0.273889	−	0.961761i	\(-0.411690\pi\)
							0.273889	+	0.961761i	\(0.411690\pi\)
\(13\)	2.89061	+	2.15508i	0.801712	+	0.597711i
							\(17\)	0.0925515			0.0224470			0.0112235	−	0.999937i	\(-0.496427\pi\)
0.0112235	+	0.999937i	\(0.496427\pi\)
\(19\)	−1.12258			−0.257537			−0.128769	−	0.991675i	\(-0.541102\pi\)
							−0.128769	+	0.991675i	\(0.541102\pi\)
\(23\)	−1.73386	+	3.00313i	−0.361534	+	0.626196i	−0.988214	−	0.153082i	\(-0.951080\pi\)
							0.626679	+	0.779277i	\(0.284414\pi\)
\(29\)	−4.24841	+	7.35845i	−0.788909	+	1.36643i	0.137727	+	0.990470i	\(0.456020\pi\)
							−0.926636	+	0.375960i	\(0.877313\pi\)
\(31\)	−1.19086	+	5.43892i	−0.213884	+	0.976859i
							\(37\)	−2.93416	−	5.08212i	−0.482373	−	0.835495i	0.517422	−	0.855730i	\(-0.326892\pi\)
−0.999795	+	0.0202353i	\(0.993558\pi\)
\(41\)	2.44280			0.381501			0.190750	−	0.981639i	\(-0.438908\pi\)
							0.190750	+	0.981639i	\(0.438908\pi\)
\(43\)	11.2082			1.70924			0.854618	−	0.519257i	\(-0.173791\pi\)
							0.854618	+	0.519257i	\(0.173791\pi\)
\(47\)	−5.88890			−0.858985			−0.429493	−	0.903070i	\(-0.641308\pi\)
							−0.429493	+	0.903070i	\(0.641308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.12	✓	70
13.3	even	3		403.2.g.a.315.12	yes	70
31.25	even	3		403.2.g.a.87.12	yes	70
403.211	even	3	inner	403.2.e.a.211.12	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.12	✓	70	1.1	even	1	trivial
403.2.e.a.211.12	yes	70	403.211	even	3	inner
403.2.g.a.87.12	yes	70	31.25	even	3
403.2.g.a.315.12	yes	70	13.3	even	3