Embedded newform 403.2.cb.a.15.5

• Label: 403.2.cb.a.15.5
• Level: $403$
• Weight: $2$
• Character: 403.15
• 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			15.5
Character	\(\chi\)	\(=\)	403.15
Dual form			403.2.cb.a.215.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.18231 + 0.837709i) q^{2} +(-1.82986 - 0.192326i) q^{3} +(2.57441 - 2.31801i) q^{4} +(2.00072 - 2.00072i) q^{5} +(4.15442 - 1.11317i) q^{6} +(0.230296 - 4.39430i) q^{7} +(-1.55387 + 3.04965i) q^{8} +(0.376947 + 0.0801226i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 32 q^{5} - 4 q^{7} - 22 q^{8} - 74 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.18231	+	0.837709i	−1.54312	+	0.592350i	−0.973324	−	0.229434i	\(-0.926312\pi\)
							−0.569799	+	0.821784i	\(0.692979\pi\)
\(3\)	−1.82986	−	0.192326i	−1.05647	−	0.111039i	−0.439669	−	0.898160i	\(-0.644904\pi\)
							−0.616799	+	0.787120i	\(0.711571\pi\)
\(5\)	2.00072	−	2.00072i	0.894750	−	0.894750i	−0.100216	−	0.994966i	\(-0.531953\pi\)
							0.994966	+	0.100216i	\(0.0319534\pi\)
\(7\)	0.230296	−	4.39430i	0.0870435	−	1.66089i	−0.511115	−	0.859512i	\(-0.670767\pi\)
							0.598159	−	0.801378i	\(-0.295899\pi\)
\(11\)	2.86112	−	4.40574i	0.862661	−	1.32838i	−0.0807428	−	0.996735i	\(-0.525729\pi\)
							0.943404	−	0.331646i	\(-0.107604\pi\)
\(13\)	−0.963852	+	3.47433i	−0.267324	+	0.963607i
							\(17\)	−3.66428	−	0.778866i	−0.888718	−	0.188903i	−0.259140	−	0.965840i	\(-0.583439\pi\)
−0.629578	+	0.776937i	\(0.716772\pi\)
\(19\)	−1.13208	−	0.916739i	−0.259716	−	0.210314i	0.490614	−	0.871377i	\(-0.336772\pi\)
							−0.750331	+	0.661063i	\(0.770106\pi\)
\(23\)	0.0695932	−	0.0772911i	0.0145112	−	0.0161163i	−0.735846	−	0.677149i	\(-0.763215\pi\)
							0.750357	+	0.661033i	\(0.229882\pi\)
\(29\)	1.34021	−	3.01017i	0.248872	−	0.558975i	−0.745307	−	0.666722i	\(-0.767697\pi\)
							0.994178	+	0.107747i	\(0.0343637\pi\)
\(31\)	1.08344	+	5.46133i	0.194592	+	0.980884i
							\(37\)	−10.7883	−	2.89073i	−1.77359	−	0.475233i	−0.784202	−	0.620506i	\(-0.786927\pi\)
−0.989392	+	0.145273i	\(0.953594\pi\)
\(41\)	8.55300	−	3.28319i	1.33575	−	0.512748i	0.417478	−	0.908687i	\(-0.362914\pi\)
							0.918277	+	0.395939i	\(0.129581\pi\)
\(43\)	−0.0269085	−	0.256017i	−0.00410350	−	0.0390422i	0.992279	−	0.124029i	\(-0.0395817\pi\)
							−0.996382	+	0.0849872i	\(0.972915\pi\)
\(47\)	−0.211372	−	1.33455i	−0.0308317	−	0.194664i	0.967465	−	0.253005i	\(-0.0814189\pi\)
							−0.998297	+	0.0583410i	\(0.981419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cb.a.15.5	✓	576
13.7	odd	12	inner	403.2.cb.a.46.5	yes	576
31.29	odd	10	inner	403.2.cb.a.184.5	yes	576
403.215	even	60	inner	403.2.cb.a.215.5	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cb.a.15.5	✓	576	1.1	even	1	trivial
403.2.cb.a.46.5	yes	576	13.7	odd	12	inner
403.2.cb.a.184.5	yes	576	31.29	odd	10	inner
403.2.cb.a.215.5	yes	576	403.215	even	60	inner