Embedded newform 105.4.u.a.52.7

• Label: 105.4.u.a.52.7
• 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			52.7
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.624851 + 2.33198i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(1.88053 + 1.08572i) q^{4} +(-9.58456 + 5.75641i) q^{5} -7.24272i q^{6} +(-5.69491 - 17.6229i) q^{7} +(-17.3639 + 17.3639i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.624851	+	2.33198i	−0.220918	+	0.824478i	0.763081	+	0.646303i	\(0.223686\pi\)
							−0.983999	+	0.178175i	\(0.942981\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	−9.58456	+	5.75641i	−0.857269	+	0.514869i
\(7\)	−5.69491	−	17.6229i	−0.307496	−	0.951549i
							\(11\)	−2.20086	+	3.81201i	−0.0603260	+	0.104488i	−0.894611	−	0.446846i	\(-0.852547\pi\)
0.834285	+	0.551333i	\(0.185881\pi\)
\(13\)	−19.3324	−	19.3324i	−0.412448	−	0.412448i	0.470142	−	0.882591i	\(-0.344203\pi\)
							−0.882591	+	0.470142i	\(0.844203\pi\)
\(17\)	−20.5221	−	76.5896i	−0.292785	−	1.09269i	−0.942960	−	0.332905i	\(-0.891971\pi\)
							0.650175	−	0.759784i	\(-0.274695\pi\)
\(19\)	−12.1913	−	21.1160i	−0.147204	−	0.254965i	0.782989	−	0.622036i	\(-0.213694\pi\)
							−0.930193	+	0.367071i	\(0.880361\pi\)
\(23\)	−60.5320	−	16.2195i	−0.548774	−	0.147044i	−0.0262301	−	0.999656i	\(-0.508350\pi\)
							−0.522544	+	0.852612i	\(0.675017\pi\)
\(29\)			172.744i			1.10613i	0.833139	+	0.553064i	\(0.186542\pi\)
							−0.833139	+	0.553064i	\(0.813458\pi\)
\(31\)	−115.978	−	66.9596i	−0.671941	−	0.387945i	0.124871	−	0.992173i	\(-0.460148\pi\)
							−0.796812	+	0.604228i	\(0.793482\pi\)
\(37\)	−31.1349	+	116.197i	−0.138339	+	0.516288i	0.861623	+	0.507549i	\(0.169448\pi\)
							−0.999962	+	0.00873920i	\(0.997218\pi\)
\(41\)			381.206i			1.45206i	0.687665	+	0.726029i	\(0.258636\pi\)
							−0.687665	+	0.726029i	\(0.741364\pi\)
\(43\)	−283.810	+	283.810i	−1.00652	+	1.00652i	−0.00654637	+	0.999979i	\(0.502084\pi\)
							−0.999979	+	0.00654637i	\(0.997916\pi\)
\(47\)	−508.556	−	136.267i	−1.57831	−	0.422907i	−0.639907	−	0.768452i	\(-0.721027\pi\)
							−0.938402	+	0.345546i	\(0.887694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.7	✓	96
5.3	odd	4	inner	105.4.u.a.73.18	yes	96
7.5	odd	6	inner	105.4.u.a.82.18	yes	96
35.33	even	12	inner	105.4.u.a.103.7	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.7	✓	96	1.1	even	1	trivial
105.4.u.a.73.18	yes	96	5.3	odd	4	inner
105.4.u.a.82.18	yes	96	7.5	odd	6	inner
105.4.u.a.103.7	yes	96	35.33	even	12	inner