Embedded newform 403.2.v.a.56.18

• Label: 403.2.v.a.56.18
• Level: $403$
• Weight: $2$
• Character: 403.56
• 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.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			56.18
Character	\(\chi\)	\(=\)	403.56
Dual form			403.2.v.a.36.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0898111 - 0.0518525i) q^{2} -0.467545 q^{3} +(-0.994623 - 1.72274i) q^{4} +(1.88817 + 1.09014i) q^{5} +(0.0419908 + 0.0242434i) q^{6} +(-2.42419 - 1.39960i) q^{7} +0.413704i q^{8} -2.78140 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0898111	−	0.0518525i	−0.0635060	−	0.0366652i	0.467911	−	0.883776i	\(-0.345007\pi\)
							−0.531417	+	0.847110i	\(0.678340\pi\)
\(3\)	−0.467545			−0.269937			−0.134969	−	0.990850i	\(-0.543093\pi\)
							−0.134969	+	0.990850i	\(0.543093\pi\)
\(5\)	1.88817	+	1.09014i	0.844417	+	0.487525i	0.858763	−	0.512372i	\(-0.171233\pi\)
							−0.0143459	+	0.999897i	\(0.504567\pi\)
\(7\)	−2.42419	−	1.39960i	−0.916256	−	0.529001i	−0.0338176	−	0.999428i	\(-0.510767\pi\)
							−0.882439	+	0.470427i	\(0.844100\pi\)
\(11\)	2.24155	−	1.29416i	0.675854	−	0.390205i	−0.122437	−	0.992476i	\(-0.539071\pi\)
							0.798291	+	0.602272i	\(0.205738\pi\)
\(13\)	−0.248332	−	3.59699i	−0.0688749	−	0.997625i
							\(17\)	1.32604	−	2.29677i	0.321612	−	0.557048i	−0.659209	−	0.751960i	\(-0.729109\pi\)
0.980821	+	0.194912i	\(0.0624420\pi\)
\(19\)	−6.58957	−	3.80449i	−1.51175	−	0.872809i	−0.999906	−	0.0137313i	\(-0.995629\pi\)
							−0.511845	−	0.859078i	\(-0.671038\pi\)
\(23\)	−3.60890	+	6.25080i	−0.752508	+	1.30338i	0.194095	+	0.980983i	\(0.437823\pi\)
							−0.946603	+	0.322400i	\(0.895510\pi\)
\(29\)	4.67329	−	8.09438i	0.867808	−	1.50309i	0.00357667	−	0.999994i	\(-0.498862\pi\)
							0.864232	−	0.503094i	\(-0.167805\pi\)
\(31\)	−5.23399	+	1.89878i	−0.940052	+	0.341031i
							\(37\)		−	1.96464i		−	0.322984i	−0.986874	−	0.161492i	\(-0.948369\pi\)
0.986874	−	0.161492i	\(-0.0516306\pi\)
\(41\)	−0.383246	+	0.221267i	−0.0598530	+	0.0345561i	−0.529628	−	0.848230i	\(-0.677668\pi\)
							0.469775	+	0.882786i	\(0.344335\pi\)
\(43\)	3.17167	−	5.49349i	0.483675	−	0.837749i	−0.516150	−	0.856498i	\(-0.672635\pi\)
							0.999824	+	0.0187494i	\(0.00596847\pi\)
\(47\)		−	1.47978i		−	0.215848i	−0.994159	−	0.107924i	\(-0.965580\pi\)
							0.994159	−	0.107924i	\(-0.0344203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.56.18	yes	70
13.10	even	6		403.2.s.a.335.18	yes	70
31.5	even	3		403.2.s.a.160.18	✓	70
403.36	even	6	inner	403.2.v.a.36.18	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.18	✓	70	31.5	even	3
403.2.s.a.335.18	yes	70	13.10	even	6
403.2.v.a.36.18	yes	70	403.36	even	6	inner
403.2.v.a.56.18	yes	70	1.1	even	1	trivial