Embedded newform 403.2.bf.a.37.15

• Label: 403.2.bf.a.37.15
• Level: $403$
• Weight: $2$
• Character: 403.37
• 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.bf (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			37.15
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.137083 + 0.511599i) q^{2} +(1.59451 + 0.920592i) q^{3} +(1.48911 + 0.859737i) q^{4} +(-3.24004 + 0.868165i) q^{5} +(-0.689555 + 0.689555i) q^{6} +(-0.996271 + 0.996271i) q^{7} +(-1.39301 + 1.39301i) q^{8} +(0.194981 + 0.337717i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.137083	+	0.511599i	−0.0969321	+	0.361755i	−0.997305	−	0.0733630i	\(-0.976627\pi\)
							0.900373	+	0.435118i	\(0.143293\pi\)
\(3\)	1.59451	+	0.920592i	0.920592	+	0.531504i	0.883824	−	0.467820i	\(-0.154960\pi\)
							0.0367684	+	0.999324i	\(0.488294\pi\)
\(5\)	−3.24004	+	0.868165i	−1.44899	+	0.388255i	−0.895672	−	0.444716i	\(-0.853305\pi\)
							−0.553317	+	0.832971i	\(0.686638\pi\)
\(7\)	−0.996271	+	0.996271i	−0.376555	+	0.376555i	−0.869858	−	0.493303i	\(-0.835789\pi\)
							0.493303	+	0.869858i	\(0.335789\pi\)
\(11\)	−0.247900	−	0.247900i	−0.0747447	−	0.0747447i	0.668746	−	0.743491i	\(-0.266831\pi\)
							−0.743491	+	0.668746i	\(0.766831\pi\)
\(13\)	1.21805	+	3.39357i	0.337827	+	0.941208i
							\(17\)	−2.63974			−0.640231			−0.320116	−	0.947378i	\(-0.603722\pi\)
−0.320116	+	0.947378i	\(0.603722\pi\)
\(19\)	1.47191	+	1.47191i	0.337680	+	0.337680i	0.855493	−	0.517814i	\(-0.173254\pi\)
							−0.517814	+	0.855493i	\(0.673254\pi\)
\(23\)	1.65057	+	2.85887i	0.344167	+	0.596115i	0.985202	−	0.171397i	\(-0.0548280\pi\)
							−0.641035	+	0.767512i	\(0.721495\pi\)
\(29\)	7.43967	−	4.29529i	1.38151	−	0.797616i	0.389173	−	0.921165i	\(-0.372761\pi\)
							0.992339	+	0.123548i	\(0.0394274\pi\)
\(31\)	5.56006	−	0.292881i	0.998616	−	0.0526029i
							\(37\)	1.34052	−	0.359192i	0.220381	−	0.0590508i	−0.146939	−	0.989146i	\(-0.546942\pi\)
0.367320	+	0.930095i	\(0.380275\pi\)
\(41\)	0.868737	+	0.868737i	0.135674	+	0.135674i	0.771682	−	0.636008i	\(-0.219416\pi\)
							−0.636008	+	0.771682i	\(0.719416\pi\)
\(43\)	−4.02713			−0.614132			−0.307066	−	0.951688i	\(-0.599347\pi\)
							−0.307066	+	0.951688i	\(0.599347\pi\)
\(47\)	3.48537	−	3.48537i	0.508393	−	0.508393i	−0.405640	−	0.914033i	\(-0.632951\pi\)
							0.914033	+	0.405640i	\(0.132951\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.15	yes	140
13.6	odd	12		403.2.ba.a.6.15	✓	140
31.26	odd	6		403.2.ba.a.336.15	yes	140
403.305	even	12	inner	403.2.bf.a.305.15	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.15	✓	140	13.6	odd	12
403.2.ba.a.336.15	yes	140	31.26	odd	6
403.2.bf.a.37.15	yes	140	1.1	even	1	trivial
403.2.bf.a.305.15	yes	140	403.305	even	12	inner