Embedded newform 48.7.i.a.5.7

• Label: 48.7.i.a.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	48.5
Dual form			48.7.i.a.29.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.59869 + 2.50199i) q^{2} +(26.4071 + 5.62718i) q^{3} +(51.4801 - 38.0237i) q^{4} +(-81.2421 + 81.2421i) q^{5} +(-214.738 + 23.3111i) q^{6} +51.0073i q^{7} +(-296.046 + 417.733i) q^{8} +(665.670 + 297.195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.59869	+	2.50199i	−0.949836	+	0.312749i
							\(3\)	26.4071	+	5.62718i	0.978041	+	0.208414i
\(5\)	−81.2421	+	81.2421i	−0.649937	+	0.649937i								−0.952978	−	0.303041i	\(-0.901998\pi\)
							0.303041	+	0.952978i	\(0.401998\pi\)
\(7\)			51.0073i			0.148709i	0.997232	+	0.0743547i	\(0.0236897\pi\)
							−0.997232	+	0.0743547i	\(0.976310\pi\)
\(11\)	164.267	−	164.267i	0.123416	−	0.123416i	−0.642701	−	0.766117i	\(-0.722186\pi\)
							0.766117	+	0.642701i	\(0.222186\pi\)
\(13\)	−124.342	+	124.342i	−0.0565964	+	0.0565964i	−0.734839	−	0.678242i	\(-0.762742\pi\)
							0.678242	+	0.734839i	\(0.262742\pi\)
\(17\)			7333.58i			1.49269i	0.665560	+	0.746344i	\(0.268193\pi\)
							−0.665560	+	0.746344i	\(0.731807\pi\)
\(19\)	−6513.12	+	6513.12i	−0.949573	+	0.949573i	−0.998788	−	0.0492151i	\(-0.984328\pi\)
							0.0492151	+	0.998788i	\(0.484328\pi\)
\(23\)	−20753.9			−1.70575			−0.852877	−	0.522111i	\(-0.825145\pi\)
							−0.852877	+	0.522111i	\(0.825145\pi\)
\(29\)	1261.67	+	1261.67i	0.0517312	+	0.0517312i	0.732499	−	0.680768i	\(-0.238354\pi\)
							−0.680768	+	0.732499i	\(0.738354\pi\)
\(31\)	15632.4			0.524737			0.262368	−	0.964968i	\(-0.415496\pi\)
							0.262368	+	0.964968i	\(0.415496\pi\)
\(37\)	−10764.5	−	10764.5i	−0.212516	−	0.212516i	0.592820	−	0.805335i	\(-0.298015\pi\)
							−0.805335	+	0.592820i	\(0.798015\pi\)
\(41\)	−44260.4			−0.642191			−0.321095	−	0.947047i	\(-0.604051\pi\)
							−0.321095	+	0.947047i	\(0.604051\pi\)
\(43\)	86724.4	+	86724.4i	1.09078	+	1.09078i	0.995446	+	0.0953315i	\(0.0303911\pi\)
							0.0953315	+	0.995446i	\(0.469609\pi\)
\(47\)		−	79712.4i		−	0.767772i	−0.923381	−	0.383886i	\(-0.874586\pi\)
							0.923381	−	0.383886i	\(-0.125414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.i.a.5.7	✓	92
3.2	odd	2	inner	48.7.i.a.5.40	yes	92
4.3	odd	2		192.7.i.a.113.4		92
12.11	even	2		192.7.i.a.113.26		92
16.3	odd	4		192.7.i.a.17.26		92
16.13	even	4	inner	48.7.i.a.29.40	yes	92
48.29	odd	4	inner	48.7.i.a.29.7	yes	92
48.35	even	4		192.7.i.a.17.4		92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.i.a.5.7	✓	92	1.1	even	1	trivial
48.7.i.a.5.40	yes	92	3.2	odd	2	inner
48.7.i.a.29.7	yes	92	48.29	odd	4	inner
48.7.i.a.29.40	yes	92	16.13	even	4	inner
192.7.i.a.17.4		92	48.35	even	4
192.7.i.a.17.26		92	16.3	odd	4
192.7.i.a.113.4		92	4.3	odd	2
192.7.i.a.113.26		92	12.11	even	2