Embedded newform 403.2.ba.a.6.7

• Label: 403.2.ba.a.6.7
• 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			6.7
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.505205 - 1.88545i) q^{2} -2.78647i q^{3} +(-1.56764 + 0.905077i) q^{4} +(-0.0340828 - 0.127199i) q^{5} +(-5.25374 + 1.40774i) q^{6} +(4.21322 - 1.12893i) q^{7} +(-0.262034 - 0.262034i) q^{8} -4.76440 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.505205	−	1.88545i	−0.357234	−	1.33321i	−0.877650	−	0.479302i	\(-0.840890\pi\)
							0.520416	−	0.853913i	\(-0.325777\pi\)
\(3\)		−	2.78647i		−	1.60877i	−0.594110	−	0.804384i	\(-0.702496\pi\)
							0.594110	−	0.804384i	\(-0.297504\pi\)
\(5\)	−0.0340828	−	0.127199i	−0.0152423	−	0.0568850i	0.957886	−	0.287149i	\(-0.0927076\pi\)
							−0.973128	+	0.230264i	\(0.926041\pi\)
\(7\)	4.21322	−	1.12893i	1.59245	−	0.426695i	0.649698	−	0.760192i	\(-0.274895\pi\)
							0.942751	+	0.333497i	\(0.108229\pi\)
\(11\)	−4.49982	−	1.20572i	−1.35675	−	0.363539i	−0.494125	−	0.869391i	\(-0.664512\pi\)
							−0.862620	+	0.505852i	\(0.831178\pi\)
\(13\)	−0.472153	+	3.57450i	−0.130952	+	0.991389i
							\(17\)	3.31930	−	5.74920i	0.805050	−	1.39439i	−0.111208	−	0.993797i	\(-0.535472\pi\)
0.916258	−	0.400590i	\(-0.131195\pi\)
\(19\)	0.380642	+	1.42058i	0.0873253	+	0.325903i	0.995744	−	0.0921583i	\(-0.0293766\pi\)
							−0.908419	+	0.418061i	\(0.862710\pi\)
\(23\)	−3.20452	+	5.55039i	−0.668189	+	1.15734i	0.310222	+	0.950664i	\(0.399597\pi\)
							−0.978410	+	0.206672i	\(0.933737\pi\)
\(29\)	−0.0238858	−	0.0137905i	−0.00443548	−	0.00256083i	0.497781	−	0.867303i	\(-0.334148\pi\)
							−0.502216	+	0.864742i	\(0.667482\pi\)
\(31\)	4.45431	−	3.34053i	0.800017	−	0.599978i
							\(37\)	2.52584	−	2.52584i	0.415246	−	0.415246i	−0.468316	−	0.883561i	\(-0.655139\pi\)
0.883561	+	0.468316i	\(0.155139\pi\)
\(41\)	2.54365	+	0.681570i	0.397252	+	0.106443i	0.451914	−	0.892062i	\(-0.350741\pi\)
							−0.0546620	+	0.998505i	\(0.517408\pi\)
\(43\)	−1.17416	+	2.03370i	−0.179057	+	0.310136i	−0.941558	−	0.336851i	\(-0.890638\pi\)
							0.762501	+	0.646987i	\(0.223971\pi\)
\(47\)	3.34217	+	3.34217i	0.487506	+	0.487506i	0.907518	−	0.420013i	\(-0.137974\pi\)
							−0.420013	+	0.907518i	\(0.637974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.7	✓	140
13.11	odd	12		403.2.bf.a.37.7	yes	140
31.26	odd	6		403.2.bf.a.305.7	yes	140
403.336	even	12	inner	403.2.ba.a.336.7	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.7	✓	140	1.1	even	1	trivial
403.2.ba.a.336.7	yes	140	403.336	even	12	inner
403.2.bf.a.37.7	yes	140	13.11	odd	12
403.2.bf.a.305.7	yes	140	31.26	odd	6