Embedded newform 48.7.i.a.5.17

• Label: 48.7.i.a.5.17
• 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			5.17
Character	\(\chi\)	\(=\)	48.5
Dual form			48.7.i.a.29.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23345 - 7.31743i) q^{2} +(9.91420 + 25.1139i) q^{3} +(-43.0896 + 47.3211i) q^{4} +(41.0809 - 41.0809i) q^{5} +(151.712 - 153.751i) q^{6} -200.806i q^{7} +(485.597 + 162.295i) q^{8} +(-532.417 + 497.969i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.23345	−	7.31743i	−0.404181	−	0.914679i
							\(3\)	9.91420	+	25.1139i	0.367192	+	0.930145i
\(5\)	41.0809	−	41.0809i	0.328647	−	0.328647i								−0.523425	−	0.852072i	\(-0.675346\pi\)
							0.852072	+	0.523425i	\(0.175346\pi\)
\(7\)		−	200.806i		−	0.585441i	−0.956198	−	0.292721i	\(-0.905439\pi\)
							0.956198	−	0.292721i	\(-0.0945606\pi\)
\(11\)	−1513.17	+	1513.17i	−1.13686	+	1.13686i	−0.147854	+	0.989009i	\(0.547237\pi\)
							−0.989009	+	0.147854i	\(0.952763\pi\)
\(13\)	−232.531	+	232.531i	−0.105840	+	0.105840i	−0.758044	−	0.652204i	\(-0.773845\pi\)
							0.652204	+	0.758044i	\(0.273845\pi\)
\(17\)			7607.14i			1.54837i	0.632959	+	0.774185i	\(0.281840\pi\)
							−0.632959	+	0.774185i	\(0.718160\pi\)
\(19\)	−984.204	+	984.204i	−0.143491	+	0.143491i	−0.775203	−	0.631712i	\(-0.782352\pi\)
							0.631712	+	0.775203i	\(0.282352\pi\)
\(23\)	−9776.32			−0.803511			−0.401755	−	0.915747i	\(-0.631600\pi\)
							−0.401755	+	0.915747i	\(0.631600\pi\)
\(29\)	23865.7	+	23865.7i	0.978544	+	0.978544i	0.999775	−	0.0212307i	\(-0.00675846\pi\)
							−0.0212307	+	0.999775i	\(0.506758\pi\)
\(31\)	−4886.36			−0.164021			−0.0820107	−	0.996631i	\(-0.526134\pi\)
							−0.0820107	+	0.996631i	\(0.526134\pi\)
\(37\)	−60342.1	−	60342.1i	−1.19128	−	1.19128i	−0.976707	−	0.214577i	\(-0.931163\pi\)
							−0.214577	−	0.976707i	\(-0.568837\pi\)
\(41\)	−723.442			−0.0104967			−0.00524834	−	0.999986i	\(-0.501671\pi\)
							−0.00524834	+	0.999986i	\(0.501671\pi\)
\(43\)	−23334.3	−	23334.3i	−0.293488	−	0.293488i	0.544969	−	0.838456i	\(-0.316542\pi\)
							−0.838456	+	0.544969i	\(0.816542\pi\)
\(47\)			109520.i			1.05487i	0.849595	+	0.527436i	\(0.176847\pi\)
							−0.849595	+	0.527436i	\(0.823153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.i.a.5.17	✓	92
3.2	odd	2	inner	48.7.i.a.5.30	yes	92
4.3	odd	2		192.7.i.a.113.18		92
12.11	even	2		192.7.i.a.113.41		92
16.3	odd	4		192.7.i.a.17.41		92
16.13	even	4	inner	48.7.i.a.29.30	yes	92
48.29	odd	4	inner	48.7.i.a.29.17	yes	92
48.35	even	4		192.7.i.a.17.18		92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.i.a.5.17	✓	92	1.1	even	1	trivial
48.7.i.a.5.30	yes	92	3.2	odd	2	inner
48.7.i.a.29.17	yes	92	48.29	odd	4	inner
48.7.i.a.29.30	yes	92	16.13	even	4	inner
192.7.i.a.17.18		92	48.35	even	4
192.7.i.a.17.41		92	16.3	odd	4
192.7.i.a.113.18		92	4.3	odd	2
192.7.i.a.113.41		92	12.11	even	2