Embedded newform 403.2.g.a.87.4

• Label: 403.2.g.a.87.4
• Level: $403$
• Weight: $2$
• Character: 403.87
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			87.4
Character	\(\chi\)	\(=\)	403.87
Dual form			403.2.g.a.315.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18452 + 2.05166i) q^{2} +0.601104 q^{3} +(-1.80620 - 3.12843i) q^{4} +(2.14607 - 3.71711i) q^{5} +(-0.712022 + 1.23326i) q^{6} +(-0.998859 + 1.73007i) q^{7} +3.81984 q^{8} -2.63867 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 8 q^{3} - 34 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18452	+	2.05166i	−0.837585	+	1.45074i	0.0543224	+	0.998523i	\(0.482700\pi\)
							−0.891908	+	0.452217i	\(0.850633\pi\)
\(3\)	0.601104			0.347047			0.173524	−	0.984830i	\(-0.444485\pi\)
							0.173524	+	0.984830i	\(0.444485\pi\)
\(5\)	2.14607	−	3.71711i	0.959753	−	1.66234i	0.236657	−	0.971593i	\(-0.423948\pi\)
							0.723096	−	0.690748i	\(-0.242719\pi\)
\(7\)	−0.998859	+	1.73007i	−0.377533	+	0.653907i	−0.990703	−	0.136045i	\(-0.956561\pi\)
							0.613170	+	0.789951i	\(0.289894\pi\)
\(11\)	−2.48681	−	4.30728i	−0.749802	−	1.29870i	−0.947917	−	0.318516i	\(-0.896815\pi\)
							0.198115	−	0.980179i	\(-0.436518\pi\)
\(13\)	−3.54574	−	0.654007i	−0.983411	−	0.181389i
							\(17\)	−0.00256545	+	0.00444349i	−0.000622213	+	0.00107770i	−0.866336	−	0.499461i	\(-0.833531\pi\)
0.865714	+	0.500539i	\(0.166865\pi\)
\(19\)	0.164804	−	0.285449i	0.0378086	−	0.0654864i	−0.846502	−	0.532386i	\(-0.821296\pi\)
							0.884310	+	0.466899i	\(0.154629\pi\)
\(23\)	3.12651	−	5.41527i	0.651922	−	1.12916i	−0.330734	−	0.943724i	\(-0.607296\pi\)
							0.982656	−	0.185438i	\(-0.0593705\pi\)
\(29\)	−3.13159	+	5.42407i	−0.581521	+	1.00722i	0.413778	+	0.910378i	\(0.364209\pi\)
							−0.995299	+	0.0968466i	\(0.969124\pi\)
\(31\)	−4.25706	−	3.58852i	−0.764590	−	0.644517i
							\(37\)	5.60298			0.921124			0.460562	−	0.887628i	\(-0.347648\pi\)
0.460562	+	0.887628i	\(0.347648\pi\)
\(41\)	3.12052	+	5.40490i	0.487343	+	0.844103i	0.999894	−	0.0145536i	\(-0.00463272\pi\)
							−0.512551	+	0.858657i	\(0.671299\pi\)
\(43\)	0.389825	−	0.675196i	0.0594477	−	0.102966i	−0.834770	−	0.550599i	\(-0.814399\pi\)
							0.894218	+	0.447632i	\(0.147733\pi\)
\(47\)	10.6914			1.55950			0.779752	−	0.626089i	\(-0.215345\pi\)
							0.779752	+	0.626089i	\(0.215345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.g.a.87.4	yes	70
13.3	even	3		403.2.e.a.211.4	yes	70
31.5	even	3		403.2.e.a.191.4	✓	70
403.315	even	3	inner	403.2.g.a.315.4	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.4	✓	70	31.5	even	3
403.2.e.a.211.4	yes	70	13.3	even	3
403.2.g.a.87.4	yes	70	1.1	even	1	trivial
403.2.g.a.315.4	yes	70	403.315	even	3	inner