Embedded newform 48.7.i.a.5.15

• Label: 48.7.i.a.5.15
• Level: $48$
• Weight: $7$
• Character: 48.5
• Analytic conductor: $11.043$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0425960138\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	48.5
Dual form			48.7.i.a.29.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.22435 - 6.79374i) q^{2} +(-18.2268 - 19.9194i) q^{3} +(-28.3097 + 57.3983i) q^{4} +(-14.1816 + 14.1816i) q^{5} +(-58.3307 + 207.975i) q^{6} +191.419i q^{7} +(509.539 - 50.1415i) q^{8} +(-64.5655 + 726.135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.22435	−	6.79374i	−0.528044	−	0.849217i
							\(3\)	−18.2268	−	19.9194i	−0.675068	−	0.737756i
\(5\)	−14.1816	+	14.1816i	−0.113453	+	0.113453i								−0.761554	−	0.648101i	\(-0.775563\pi\)
							0.648101	+	0.761554i	\(0.275563\pi\)
\(7\)			191.419i			0.558072i	0.960281	+	0.279036i	\(0.0900149\pi\)
							−0.960281	+	0.279036i	\(0.909985\pi\)
\(11\)	44.8622	−	44.8622i	0.0337056	−	0.0337056i	−0.690053	−	0.723759i	\(-0.742413\pi\)
							0.723759	+	0.690053i	\(0.242413\pi\)
\(13\)	−64.9762	+	64.9762i	−0.0295749	+	0.0295749i	−0.721740	−	0.692165i	\(-0.756657\pi\)
							0.692165	+	0.721740i	\(0.256657\pi\)
\(17\)		−	1390.76i		−	0.283078i	−0.989933	−	0.141539i	\(-0.954795\pi\)
							0.989933	−	0.141539i	\(-0.0452050\pi\)
\(19\)	5707.42	−	5707.42i	0.832106	−	0.832106i	−0.155699	−	0.987805i	\(-0.549763\pi\)
							0.987805	+	0.155699i	\(0.0497628\pi\)
\(23\)	6203.68			0.509878			0.254939	−	0.966957i	\(-0.417945\pi\)
							0.254939	+	0.966957i	\(0.417945\pi\)
\(29\)	10405.2	+	10405.2i	0.426636	+	0.426636i	0.887481	−	0.460845i	\(-0.152454\pi\)
							−0.460845	+	0.887481i	\(0.652454\pi\)
\(31\)	36582.1			1.22796			0.613980	−	0.789322i	\(-0.289568\pi\)
							0.613980	+	0.789322i	\(0.289568\pi\)
\(37\)	39456.5	+	39456.5i	0.778957	+	0.778957i	0.979654	−	0.200696i	\(-0.0643204\pi\)
							−0.200696	+	0.979654i	\(0.564320\pi\)
\(41\)	−111617.			−1.61950			−0.809749	−	0.586776i	\(-0.800397\pi\)
							−0.809749	+	0.586776i	\(0.800397\pi\)
\(43\)	57937.2	+	57937.2i	0.728706	+	0.728706i	0.970362	−	0.241656i	\(-0.0776905\pi\)
							−0.241656	+	0.970362i	\(0.577691\pi\)
\(47\)			126602.i			1.21940i	0.792631	+	0.609701i	\(0.208710\pi\)
							−0.792631	+	0.609701i	\(0.791290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.i.a.5.15	✓	92
3.2	odd	2	inner	48.7.i.a.5.32	yes	92
4.3	odd	2		192.7.i.a.113.34		92
12.11	even	2		192.7.i.a.113.11		92
16.3	odd	4		192.7.i.a.17.11		92
16.13	even	4	inner	48.7.i.a.29.32	yes	92
48.29	odd	4	inner	48.7.i.a.29.15	yes	92
48.35	even	4		192.7.i.a.17.34		92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.i.a.5.15	✓	92	1.1	even	1	trivial
48.7.i.a.5.32	yes	92	3.2	odd	2	inner
48.7.i.a.29.15	yes	92	48.29	odd	4	inner
48.7.i.a.29.32	yes	92	16.13	even	4	inner
192.7.i.a.17.11		92	16.3	odd	4
192.7.i.a.17.34		92	48.35	even	4
192.7.i.a.113.11		92	12.11	even	2
192.7.i.a.113.34		92	4.3	odd	2