Embedded newform 403.2.y.a.64.4

• Label: 403.2.y.a.64.4
• 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			64.4
Character	\(\chi\)	\(=\)	403.64
Dual form			403.2.y.a.233.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37219 + 1.88866i) q^{2} +(1.37165 - 0.996564i) q^{3} +(-1.06609 - 3.28109i) q^{4} +0.910017i q^{5} +3.95806i q^{6} +(1.80401 - 0.586159i) q^{7} +(3.21924 + 1.04600i) q^{8} +(-0.0387606 + 0.119293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37219	+	1.88866i	−0.970286	+	1.33548i	−0.0283836	+	0.999597i	\(0.509036\pi\)
							−0.941902	+	0.335887i	\(0.890964\pi\)
\(3\)	1.37165	−	0.996564i	0.791924	−	0.575366i	−0.116610	−	0.993178i	\(-0.537203\pi\)
							0.908534	+	0.417812i	\(0.137203\pi\)
\(5\)			0.910017i			0.406972i	0.979078	+	0.203486i	\(0.0652271\pi\)
							−0.979078	+	0.203486i	\(0.934773\pi\)
\(7\)	1.80401	−	0.586159i	0.681852	−	0.221547i	0.0524460	−	0.998624i	\(-0.483298\pi\)
							0.629406	+	0.777077i	\(0.283298\pi\)
\(11\)	1.75828	−	0.571300i	0.530142	−	0.172254i	−0.0317011	−	0.999497i	\(-0.510092\pi\)
							0.561843	+	0.827244i	\(0.310092\pi\)
\(13\)	2.27378	−	2.79820i	0.630634	−	0.776081i
							\(17\)	−0.117217	+	0.360758i	−0.0284294	+	0.0874967i	−0.964264	−	0.264942i	\(-0.914647\pi\)
0.935835	+	0.352438i	\(0.114647\pi\)
\(19\)	−0.635016	+	0.874025i	−0.145683	+	0.200515i	−0.875622	−	0.482997i	\(-0.839548\pi\)
							0.729939	+	0.683512i	\(0.239548\pi\)
\(23\)	−0.126679	+	0.389878i	−0.0264144	+	0.0812951i	−0.963395	−	0.268087i	\(-0.913609\pi\)
							0.936980	+	0.349382i	\(0.113609\pi\)
\(29\)	5.40500	+	3.92696i	1.00368	+	0.729219i	0.962875	−	0.269948i	\(-0.0870066\pi\)
							0.0408085	+	0.999167i	\(0.487007\pi\)
\(31\)	3.55106	−	4.28835i	0.637789	−	0.770211i
							\(37\)			2.92373i			0.480659i	0.970691	+	0.240330i	\(0.0772555\pi\)
−0.970691	+	0.240330i	\(0.922745\pi\)
\(41\)	−1.29335	+	1.78014i	−0.201987	+	0.278012i	−0.897979	−	0.440038i	\(-0.854965\pi\)
							0.695992	+	0.718050i	\(0.254965\pi\)
\(43\)	−5.57462	−	4.05019i	−0.850121	−	0.617649i	0.0750585	−	0.997179i	\(-0.476086\pi\)
							−0.925179	+	0.379530i	\(0.876086\pi\)
\(47\)	0.248152	+	0.341552i	0.0361966	+	0.0498204i	0.826732	−	0.562596i	\(-0.190197\pi\)
							−0.790535	+	0.612417i	\(0.790197\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.y.a.64.4	✓	136
13.12	even	2	inner	403.2.y.a.64.31	yes	136
31.16	even	5	inner	403.2.y.a.233.31	yes	136
403.233	even	10	inner	403.2.y.a.233.4	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.y.a.64.4	✓	136	1.1	even	1	trivial
403.2.y.a.64.31	yes	136	13.12	even	2	inner
403.2.y.a.233.4	yes	136	403.233	even	10	inner
403.2.y.a.233.31	yes	136	31.16	even	5	inner