Embedded newform 403.2.bf.a.37.4

• Label: 403.2.bf.a.37.4
• 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			37.4
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.616245 + 2.29986i) q^{2} +(2.49411 + 1.43998i) q^{3} +(-3.17753 - 1.83455i) q^{4} +(0.879300 - 0.235608i) q^{5} +(-4.84872 + 4.84872i) q^{6} +(1.37504 - 1.37504i) q^{7} +(2.81012 - 2.81012i) q^{8} +(2.64706 + 4.58484i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.616245	+	2.29986i	−0.435751	+	1.62624i	0.303512	+	0.952828i	\(0.401841\pi\)
							−0.739263	+	0.673417i	\(0.764826\pi\)
\(3\)	2.49411	+	1.43998i	1.43998	+	0.831370i	0.997847	−	0.0655818i	\(-0.0208903\pi\)
							0.442128	+	0.896952i	\(0.354224\pi\)
\(5\)	0.879300	−	0.235608i	0.393235	−	0.105367i	−0.0567829	−	0.998387i	\(-0.518084\pi\)
							0.450018	+	0.893020i	\(0.351418\pi\)
\(7\)	1.37504	−	1.37504i	0.519716	−	0.519716i	−0.397769	−	0.917486i	\(-0.630216\pi\)
							0.917486	+	0.397769i	\(0.130216\pi\)
\(11\)	1.87754	+	1.87754i	0.566100	+	0.566100i	0.931034	−	0.364933i	\(-0.118908\pi\)
							−0.364933	+	0.931034i	\(0.618908\pi\)
\(13\)	−1.87803	+	3.07783i	−0.520871	+	0.853635i
							\(17\)	−3.47765			−0.843455			−0.421727	−	0.906723i	\(-0.638576\pi\)
−0.421727	+	0.906723i	\(0.638576\pi\)
\(19\)	−2.41412	−	2.41412i	−0.553838	−	0.553838i	0.373708	−	0.927546i	\(-0.378086\pi\)
							−0.927546	+	0.373708i	\(0.878086\pi\)
\(23\)	0.466472	+	0.807953i	0.0972661	+	0.168470i	0.910552	−	0.413394i	\(-0.135657\pi\)
							−0.813286	+	0.581864i	\(0.802324\pi\)
\(29\)	5.68854	−	3.28428i	1.05634	−	0.609876i	0.131920	−	0.991260i	\(-0.457886\pi\)
							0.924417	+	0.381384i	\(0.124553\pi\)
\(31\)	0.527588	−	5.54271i	0.0947577	−	0.995500i
							\(37\)	6.63793	−	1.77863i	1.09127	−	0.292404i	0.332064	−	0.943257i	\(-0.392255\pi\)
0.759204	+	0.650852i	\(0.225588\pi\)
\(41\)	2.00103	+	2.00103i	0.312509	+	0.312509i	0.845881	−	0.533372i	\(-0.179075\pi\)
							−0.533372	+	0.845881i	\(0.679075\pi\)
\(43\)	2.00492			0.305747			0.152874	−	0.988246i	\(-0.451147\pi\)
							0.152874	+	0.988246i	\(0.451147\pi\)
\(47\)	7.48134	−	7.48134i	1.09127	−	1.09127i	0.0958713	−	0.995394i	\(-0.469436\pi\)
							0.995394	−	0.0958713i	\(-0.0305637\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.4	yes	140
13.6	odd	12		403.2.ba.a.6.4	✓	140
31.26	odd	6		403.2.ba.a.336.4	yes	140
403.305	even	12	inner	403.2.bf.a.305.4	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.4	✓	140	13.6	odd	12
403.2.ba.a.336.4	yes	140	31.26	odd	6
403.2.bf.a.37.4	yes	140	1.1	even	1	trivial
403.2.bf.a.305.4	yes	140	403.305	even	12	inner