Embedded newform 403.2.y.a.64.16

• Label: 403.2.y.a.64.16
• Level: $403$
• Weight: $2$
• Character: 403.64
• 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.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			64.16
Character	\(\chi\)	\(=\)	403.64
Dual form			403.2.y.a.233.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.160682 + 0.221160i) q^{2} +(0.324832 - 0.236004i) q^{3} +(0.594941 + 1.83104i) q^{4} +0.569297i q^{5} +0.109762i q^{6} +(-3.20061 + 1.03994i) q^{7} +(-1.02053 - 0.331590i) q^{8} +(-0.877233 + 2.69985i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.160682	+	0.221160i	−0.113620	+	0.156384i	−0.862039	−	0.506841i	\(-0.830813\pi\)
							0.748420	+	0.663225i	\(0.230813\pi\)
\(3\)	0.324832	−	0.236004i	0.187542	−	0.136257i	−0.490053	−	0.871693i	\(-0.663023\pi\)
							0.677594	+	0.735436i	\(0.263023\pi\)
\(5\)			0.569297i			0.254597i	0.991864	+	0.127299i	\(0.0406306\pi\)
							−0.991864	+	0.127299i	\(0.959369\pi\)
\(7\)	−3.20061	+	1.03994i	−1.20972	+	0.393060i	−0.843327	−	0.537401i	\(-0.819406\pi\)
							−0.366389	+	0.930462i	\(0.619406\pi\)
\(11\)	0.412804	−	0.134128i	0.124465	−	0.0404412i	−0.246122	−	0.969239i	\(-0.579156\pi\)
							0.370587	+	0.928798i	\(0.379156\pi\)
\(13\)	−3.60356	+	0.119848i	−0.999447	+	0.0332399i
							\(17\)	1.74829	−	5.38069i	0.424023	−	1.30501i	−0.479903	−	0.877321i	\(-0.659328\pi\)
0.903927	−	0.427688i	\(-0.140672\pi\)
\(19\)	−2.09792	+	2.88754i	−0.481297	+	0.662448i	−0.978753	−	0.205041i	\(-0.934267\pi\)
							0.497457	+	0.867489i	\(0.334267\pi\)
\(23\)	−0.802619	+	2.47021i	−0.167358	+	0.515074i	−0.999202	−	0.0399347i	\(-0.987285\pi\)
							0.831845	+	0.555008i	\(0.187285\pi\)
\(29\)	2.99915	+	2.17901i	0.556927	+	0.404631i	0.830333	−	0.557267i	\(-0.188150\pi\)
							−0.273406	+	0.961899i	\(0.588150\pi\)
\(31\)	5.46295	+	1.07523i	0.981176	+	0.193117i
							\(37\)		−	5.33199i		−	0.876574i	−0.898835	−	0.438287i	\(-0.855585\pi\)
0.898835	−	0.438287i	\(-0.144415\pi\)
\(41\)	−6.39050	+	8.79577i	−0.998029	+	1.37367i	−0.0715019	+	0.997440i	\(0.522779\pi\)
							−0.926527	+	0.376228i	\(0.877221\pi\)
\(43\)	1.53784	+	1.11731i	0.234519	+	0.170388i	0.698838	−	0.715280i	\(-0.253701\pi\)
							−0.464319	+	0.885668i	\(0.653701\pi\)
\(47\)	6.38465	+	8.78772i	0.931298	+	1.28182i	0.959351	+	0.282215i	\(0.0910691\pi\)
							−0.0280536	+	0.999606i	\(0.508931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.y.a.64.16	✓	136
13.12	even	2	inner	403.2.y.a.64.19	yes	136
31.16	even	5	inner	403.2.y.a.233.19	yes	136
403.233	even	10	inner	403.2.y.a.233.16	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.y.a.64.16	✓	136	1.1	even	1	trivial
403.2.y.a.64.19	yes	136	13.12	even	2	inner
403.2.y.a.233.16	yes	136	403.233	even	10	inner
403.2.y.a.233.19	yes	136	31.16	even	5	inner