Embedded newform 105.4.u.a.52.12

• Label: 105.4.u.a.52.12
• 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			52.12
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0422777 - 0.157783i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(6.90510 + 3.98666i) q^{4} +(9.25781 + 6.26841i) q^{5} +0.490046i q^{6} +(-18.5183 - 0.270750i) q^{7} +(1.84500 - 1.84500i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0422777	−	0.157783i	0.0149474	−	0.0557846i	−0.958049	−	0.286603i	\(-0.907474\pi\)
							0.972997	+	0.230819i	\(0.0741405\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	9.25781	+	6.26841i	0.828044	+	0.560664i
\(7\)	−18.5183	−	0.270750i	−0.999893	−	0.0146191i
							\(11\)	−17.7903	+	30.8136i	−0.487633	+	0.844605i	−0.999899	−	0.0142216i	\(-0.995473\pi\)
0.512266	+	0.858827i	\(0.328806\pi\)
\(13\)	24.9069	+	24.9069i	0.531380	+	0.531380i	0.920983	−	0.389603i	\(-0.127388\pi\)
							−0.389603	+	0.920983i	\(0.627388\pi\)
\(17\)	−4.80586	−	17.9357i	−0.0685642	−	0.255885i	0.923133	−	0.384481i	\(-0.125620\pi\)
							−0.991697	+	0.128596i	\(0.958953\pi\)
\(19\)	77.8896	+	134.909i	0.940479	+	1.62896i	0.764559	+	0.644554i	\(0.222957\pi\)
							0.175921	+	0.984404i	\(0.443710\pi\)
\(23\)	−69.1942	−	18.5405i	−0.627303	−	0.168085i	−0.0688577	−	0.997626i	\(-0.521935\pi\)
							−0.558446	+	0.829541i	\(0.688602\pi\)
\(29\)		−	234.129i		−	1.49919i	−0.661895	−	0.749596i	\(-0.730248\pi\)
							0.661895	−	0.749596i	\(-0.269752\pi\)
\(31\)	−19.7310	−	11.3917i	−0.114316	−	0.0660002i	0.441752	−	0.897137i	\(-0.354357\pi\)
							−0.556068	+	0.831137i	\(0.687690\pi\)
\(37\)	81.4145	−	303.843i	0.361742	−	1.35004i	−0.510042	−	0.860149i	\(-0.670370\pi\)
							0.871784	−	0.489890i	\(-0.162963\pi\)
\(41\)			26.4665i			0.100814i	0.998729	+	0.0504070i	\(0.0160519\pi\)
							−0.998729	+	0.0504070i	\(0.983948\pi\)
\(43\)	−195.603	+	195.603i	−0.693701	+	0.693701i	−0.963044	−	0.269343i	\(-0.913193\pi\)
							0.269343	+	0.963044i	\(0.413193\pi\)
\(47\)	387.113	+	103.727i	1.20141	+	0.321917i	0.803386	−	0.595458i	\(-0.203030\pi\)
							0.398024	+	0.917375i	\(0.369696\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.12	✓	96
5.3	odd	4	inner	105.4.u.a.73.13	yes	96
7.5	odd	6	inner	105.4.u.a.82.13	yes	96
35.33	even	12	inner	105.4.u.a.103.12	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.12	✓	96	1.1	even	1	trivial
105.4.u.a.73.13	yes	96	5.3	odd	4	inner
105.4.u.a.82.13	yes	96	7.5	odd	6	inner
105.4.u.a.103.12	yes	96	35.33	even	12	inner