Embedded newform 403.2.be.c.57.20

• Label: 403.2.be.c.57.20
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.20
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.185549 - 0.185549i) q^{2} +(-2.28373 - 1.31851i) q^{3} +1.93114i q^{4} +(-3.12819 - 0.838196i) q^{5} +(-0.668395 + 0.179096i) q^{6} +(-1.29814 + 0.347834i) q^{7} +(0.729421 + 0.729421i) q^{8} +(1.97696 + 3.42420i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.185549	−	0.185549i	0.131203	−	0.131203i	−0.638455	−	0.769659i	\(-0.720426\pi\)
							0.769659	+	0.638455i	\(0.220426\pi\)
\(3\)	−2.28373	−	1.31851i	−1.31851	−	0.761245i	−0.335025	−	0.942209i	\(-0.608745\pi\)
							−0.983490	+	0.180964i	\(0.942078\pi\)
\(5\)	−3.12819	−	0.838196i	−1.39897	−	0.374853i	−0.520995	−	0.853560i	\(-0.674439\pi\)
							−0.877975	+	0.478707i	\(0.841106\pi\)
\(7\)	−1.29814	+	0.347834i	−0.490649	+	0.131469i	−0.495657	−	0.868518i	\(-0.665073\pi\)
							0.00500785	+	0.999987i	\(0.498406\pi\)
\(11\)	4.72248	+	1.26538i	1.42388	+	0.381528i	0.886859	−	0.462040i	\(-0.152882\pi\)
							0.537022	+	0.843568i	\(0.319549\pi\)
\(13\)	2.62285	−	2.47400i	0.727448	−	0.686163i
							\(17\)	2.10983	−	3.65434i	0.511710	−	0.886308i	−0.488198	−	0.872733i	\(-0.662346\pi\)
0.999908	−	0.0135747i	\(-0.00432109\pi\)
\(19\)	0.423308	+	1.57981i	0.0971135	+	0.362432i	0.997331	−	0.0730072i	\(-0.0232596\pi\)
							−0.900218	+	0.435440i	\(0.856593\pi\)
\(23\)	5.54227			1.15564			0.577822	−	0.816163i	\(-0.303903\pi\)
							0.577822	+	0.816163i	\(0.303903\pi\)
\(29\)			5.99260i			1.11280i	0.830915	+	0.556399i	\(0.187818\pi\)
							−0.830915	+	0.556399i	\(0.812182\pi\)
\(31\)	−3.16867	−	4.57816i	−0.569111	−	0.822261i
							\(37\)	2.34662	+	8.75770i	0.385782	+	1.43976i	0.836931	+	0.547308i	\(0.184348\pi\)
−0.451149	+	0.892448i	\(0.648986\pi\)
\(41\)	1.70669	+	0.457307i	0.266540	+	0.0714193i	0.389614	−	0.920978i	\(-0.372608\pi\)
							−0.123074	+	0.992398i	\(0.539275\pi\)
\(43\)	4.88043	−	8.45316i	0.744259	−	1.28909i	−0.206281	−	0.978493i	\(-0.566136\pi\)
							0.950540	−	0.310602i	\(-0.100531\pi\)
\(47\)	5.44321	+	5.44321i	0.793974	+	0.793974i	0.982138	−	0.188163i	\(-0.0602534\pi\)
							−0.188163	+	0.982138i	\(0.560253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.20	✓	136
13.8	odd	4	inner	403.2.be.c.398.20	yes	136
31.6	odd	6	inner	403.2.be.c.161.20	yes	136
403.99	even	12	inner	403.2.be.c.99.20	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.20	✓	136	1.1	even	1	trivial
403.2.be.c.99.20	yes	136	403.99	even	12	inner
403.2.be.c.161.20	yes	136	31.6	odd	6	inner
403.2.be.c.398.20	yes	136	13.8	odd	4	inner