Embedded newform 403.2.cb.a.215.14

• Label: 403.2.cb.a.215.14
• Level: $403$
• Weight: $2$
• Character: 403.215
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.cb (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(36\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			215.14
Character	\(\chi\)	\(=\)	403.215
Dual form			403.2.cb.a.15.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.857856 - 0.329300i) q^{2} +(-2.24412 + 0.235866i) q^{3} +(-0.858811 - 0.773277i) q^{4} +(1.02734 + 1.02734i) q^{5} +(2.00280 + 0.536649i) q^{6} +(0.00452335 + 0.0863108i) q^{7} +(1.31643 + 2.58364i) q^{8} +(2.04600 - 0.434890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 32 q^{5} - 4 q^{7} - 22 q^{8} - 74 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.857856	−	0.329300i	−0.606596	−	0.232850i	0.0356310	−	0.999365i	\(-0.488656\pi\)
							−0.642227	+	0.766515i	\(0.721989\pi\)
\(3\)	−2.24412	+	0.235866i	−1.29564	+	0.136178i	−0.727132	−	0.686498i	\(-0.759147\pi\)
							−0.568511	+	0.822675i	\(0.692480\pi\)
\(5\)	1.02734	+	1.02734i	0.459440	+	0.459440i	0.898472	−	0.439032i	\(-0.144678\pi\)
							−0.439032	+	0.898472i	\(0.644678\pi\)
\(7\)	0.00452335	+	0.0863108i	0.00170967	+	0.0326224i	0.999351	−	0.0360178i	\(-0.0114673\pi\)
							−0.997641	+	0.0686402i	\(0.978134\pi\)
\(11\)	0.549157	+	0.845628i	0.165577	+	0.254966i	0.911713	−	0.410827i	\(-0.134760\pi\)
							−0.746136	+	0.665793i	\(0.768093\pi\)
\(13\)	−3.52397	−	0.762656i	−0.977373	−	0.211523i
							\(17\)	6.67387	−	1.41857i	1.61865	−	0.344055i	0.692561	−	0.721359i	\(-0.256482\pi\)
0.926089	+	0.377304i	\(0.123149\pi\)
\(19\)	−0.975790	+	0.790179i	−0.223862	+	0.181280i	−0.734690	−	0.678403i	\(-0.762672\pi\)
							0.510828	+	0.859683i	\(0.329339\pi\)
\(23\)	−4.08257	−	4.53415i	−0.851274	−	0.945435i	0.147776	−	0.989021i	\(-0.452789\pi\)
							−0.999050	+	0.0435855i	\(0.986122\pi\)
\(29\)	−3.28689	−	7.38247i	−0.610359	−	1.37089i	−0.909109	−	0.416558i	\(-0.863236\pi\)
							0.298750	−	0.954332i	\(-0.403430\pi\)
\(31\)	−4.94565	−	2.55745i	−0.888265	−	0.459332i
							\(37\)	5.17495	−	1.38662i	0.850756	−	0.227959i	0.193008	−	0.981197i	\(-0.438176\pi\)
0.657748	+	0.753238i	\(0.271509\pi\)
\(41\)	−9.39441	−	3.60618i	−1.46716	−	0.563190i	−0.511923	−	0.859031i	\(-0.671067\pi\)
							−0.955237	+	0.295841i	\(0.904400\pi\)
\(43\)	−0.128195	+	1.21970i	−0.0195496	+	0.186002i	−0.999939	−	0.0110694i	\(-0.996476\pi\)
							0.980389	+	0.197071i	\(0.0631431\pi\)
\(47\)	0.810664	−	5.11833i	0.118247	−	0.746585i	−0.855306	−	0.518124i	\(-0.826631\pi\)
							0.973553	−	0.228461i	\(-0.0733694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cb.a.215.14	yes	576
13.2	odd	12	inner	403.2.cb.a.184.14	yes	576
31.15	odd	10	inner	403.2.cb.a.46.14	yes	576
403.15	even	60	inner	403.2.cb.a.15.14	✓	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cb.a.15.14	✓	576	403.15	even	60	inner
403.2.cb.a.46.14	yes	576	31.15	odd	10	inner
403.2.cb.a.184.14	yes	576	13.2	odd	12	inner
403.2.cb.a.215.14	yes	576	1.1	even	1	trivial