Embedded newform 403.2.bg.a.123.20

• Label: 403.2.bg.a.123.20
• Level: $403$
• Weight: $2$
• Character: 403.123
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.20
Character	\(\chi\)	\(=\)	403.123
Dual form			403.2.bg.a.154.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0151761 + 0.0566379i) q^{2} +(2.67732 + 1.54575i) q^{3} +(1.72907 - 0.998281i) q^{4} +(-2.77132 + 2.77132i) q^{5} +(-0.0469169 + 0.175096i) q^{6} +(0.222097 - 0.828876i) q^{7} +(0.165705 + 0.165705i) q^{8} +(3.27869 + 5.67887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 8 q^{2} - 12 q^{4} - 8 q^{5} - 16 q^{7} - 28 q^{8} + 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0151761	+	0.0566379i	0.0107311	+	0.0400491i	0.971084	−	0.238739i	\(-0.0767340\pi\)
							−0.960353	+	0.278788i	\(0.910067\pi\)
\(3\)	2.67732	+	1.54575i	1.54575	+	0.892440i	0.998459	+	0.0554986i	\(0.0176748\pi\)
							0.547293	+	0.836941i	\(0.315658\pi\)
\(5\)	−2.77132	+	2.77132i	−1.23937	+	1.23937i	−0.279114	+	0.960258i	\(0.590041\pi\)
							−0.960258	+	0.279114i	\(0.909959\pi\)
\(7\)	0.222097	−	0.828876i	0.0839447	−	0.313286i	−0.911168	−	0.412036i	\(-0.864818\pi\)
							0.995112	+	0.0987502i	\(0.0314845\pi\)
\(11\)	−1.23746	−	4.61826i	−0.373108	−	1.39246i	−0.856090	−	0.516827i	\(-0.827113\pi\)
							0.482982	−	0.875630i	\(-0.339554\pi\)
\(13\)	−0.621749	+	3.55154i	−0.172442	+	0.985020i
							\(17\)	0.324591	+	0.562208i	0.0787249	+	0.136355i	0.902700	−	0.430270i	\(-0.141582\pi\)
−0.823975	+	0.566626i	\(0.808248\pi\)
\(19\)	5.73218	+	1.53593i	1.31505	+	0.352367i	0.847122	−	0.531399i	\(-0.178333\pi\)
							0.467930	+	0.883766i	\(0.345000\pi\)
\(23\)	−0.953361	+	1.65127i	−0.198790	+	0.344314i	−0.948136	−	0.317864i	\(-0.897034\pi\)
							0.749347	+	0.662178i	\(0.230368\pi\)
\(29\)	−4.24661	−	2.45178i	−0.788575	−	0.455284i	0.0508856	−	0.998704i	\(-0.483796\pi\)
							−0.839461	+	0.543420i	\(0.817129\pi\)
\(31\)	−5.33895	−	1.57975i	−0.958904	−	0.283731i
							\(37\)	−1.74308	−	6.50526i	−0.286560	−	1.06946i	−0.947692	−	0.319188i	\(-0.896590\pi\)
0.661131	−	0.750270i	\(-0.270077\pi\)
\(41\)	−2.27358	−	8.48510i	−0.355073	−	1.32515i	−0.880393	−	0.474246i	\(-0.842721\pi\)
							0.525320	−	0.850905i	\(-0.323946\pi\)
\(43\)	−3.37020	−	5.83735i	−0.513950	−	0.890187i	−0.999869	−	0.0161836i	\(-0.994848\pi\)
							0.485919	−	0.874004i	\(-0.338485\pi\)
\(47\)	6.39951	+	6.39951i	0.933464	+	0.933464i	0.997921	−	0.0644563i	\(-0.0205313\pi\)
							−0.0644563	+	0.997921i	\(0.520531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bg.a.123.20	yes	144
13.11	odd	12	inner	403.2.bg.a.154.19	yes	144
31.30	odd	2	inner	403.2.bg.a.123.19	✓	144
403.154	even	12	inner	403.2.bg.a.154.20	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bg.a.123.19	✓	144	31.30	odd	2	inner
403.2.bg.a.123.20	yes	144	1.1	even	1	trivial
403.2.bg.a.154.19	yes	144	13.11	odd	12	inner
403.2.bg.a.154.20	yes	144	403.154	even	12	inner