Embedded newform 105.4.u.a.52.17

• Label: 105.4.u.a.52.17
• Level: $105$
• Weight: $4$
• Character: 105.52
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			52.17
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.662502 - 2.47249i) q^{2} +(2.89778 - 0.776457i) q^{3} +(1.25390 + 0.723938i) q^{4} +(4.16719 + 10.3747i) q^{5} -7.67914i q^{6} +(6.62391 + 17.2952i) q^{7} +(17.1006 - 17.1006i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.662502	−	2.47249i	0.234230	−	0.874158i	−0.744265	−	0.667885i	\(-0.767200\pi\)
							0.978494	−	0.206273i	\(-0.0661335\pi\)
\(3\)	2.89778	−	0.776457i	0.557678	−	0.149429i
							\(5\)	4.16719	+	10.3747i	0.372724	+	0.927942i
\(7\)	6.62391	+	17.2952i	0.357658	+	0.933853i
							\(11\)	−19.5340	+	33.8338i	−0.535428	+	0.927389i	0.463714	+	0.885985i	\(0.346516\pi\)
−0.999142	+	0.0414039i	\(0.986817\pi\)
\(13\)	20.7293	+	20.7293i	0.442252	+	0.442252i	0.892768	−	0.450516i	\(-0.148760\pi\)
							−0.450516	+	0.892768i	\(0.648760\pi\)
\(17\)	−34.6677	−	129.382i	−0.494598	−	1.84586i	−0.532271	−	0.846574i	\(-0.678661\pi\)
							0.0376729	−	0.999290i	\(-0.488006\pi\)
\(19\)	−49.7699	−	86.2041i	−0.600948	−	1.04087i	−0.992678	−	0.120792i	\(-0.961457\pi\)
							0.391730	−	0.920080i	\(-0.371877\pi\)
\(23\)	−8.68009	−	2.32582i	−0.0786924	−	0.0210856i	0.219258	−	0.975667i	\(-0.429636\pi\)
							−0.297950	+	0.954581i	\(0.596303\pi\)
\(29\)		−	240.426i		−	1.53952i	−0.638334	−	0.769760i	\(-0.720376\pi\)
							0.638334	−	0.769760i	\(-0.279624\pi\)
\(31\)	124.225	+	71.7215i	0.719727	+	0.415534i	0.814652	−	0.579950i	\(-0.196928\pi\)
							−0.0949255	+	0.995484i	\(0.530261\pi\)
\(37\)	−38.8466	+	144.977i	−0.172604	+	0.644166i	0.824344	+	0.566090i	\(0.191544\pi\)
							−0.996947	+	0.0780763i	\(0.975122\pi\)
\(41\)		−	271.656i		−	1.03477i	−0.855753	−	0.517384i	\(-0.826906\pi\)
							0.855753	−	0.517384i	\(-0.173094\pi\)
\(43\)	−142.106	+	142.106i	−0.503977	+	0.503977i	−0.912671	−	0.408694i	\(-0.865984\pi\)
							0.408694	+	0.912671i	\(0.365984\pi\)
\(47\)	−67.3724	−	18.0524i	−0.209091	−	0.0560258i	0.152753	−	0.988264i	\(-0.451186\pi\)
							−0.361844	+	0.932239i	\(0.617853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.17	✓	96
5.3	odd	4	inner	105.4.u.a.73.8	yes	96
7.5	odd	6	inner	105.4.u.a.82.8	yes	96
35.33	even	12	inner	105.4.u.a.103.17	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.17	✓	96	1.1	even	1	trivial
105.4.u.a.73.8	yes	96	5.3	odd	4	inner
105.4.u.a.82.8	yes	96	7.5	odd	6	inner
105.4.u.a.103.17	yes	96	35.33	even	12	inner