Embedded newform 105.4.u.a.52.19

• Label: 105.4.u.a.52.19
• 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			52.19
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.815920 - 3.04505i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(-1.67842 - 0.969037i) q^{4} +(10.1281 - 4.73506i) q^{5} +9.45741i q^{6} +(13.1472 + 13.0442i) q^{7} +(13.5128 - 13.5128i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.815920	−	3.04505i	0.288471	−	1.07659i	−0.657794	−	0.753198i	\(-0.728510\pi\)
							0.946265	−	0.323391i	\(-0.104823\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	10.1281	−	4.73506i	0.905888	−	0.423517i
\(7\)	13.1472	+	13.0442i	0.709882	+	0.704321i
							\(11\)	5.25944	−	9.10962i	0.144162	−	0.249696i	−0.784898	−	0.619625i	\(-0.787285\pi\)
0.929060	+	0.369929i	\(0.120618\pi\)
\(13\)	−32.9347	−	32.9347i	−0.702650	−	0.702650i	0.262328	−	0.964979i	\(-0.415510\pi\)
							−0.964979	+	0.262328i	\(0.915510\pi\)
\(17\)	−21.2051	−	79.1384i	−0.302529	−	1.12905i	−0.935052	−	0.354511i	\(-0.884647\pi\)
							0.632523	−	0.774541i	\(-0.282019\pi\)
\(19\)	51.3702	+	88.9757i	0.620270	+	1.07434i	0.989435	+	0.144975i	\(0.0463101\pi\)
							−0.369166	+	0.929364i	\(0.620357\pi\)
\(23\)	111.715	+	29.9339i	1.01279	+	0.271376i	0.726794	−	0.686856i	\(-0.241010\pi\)
							0.285994	+	0.958231i	\(0.407676\pi\)
\(29\)			95.6033i			0.612175i	0.952003	+	0.306088i	\(0.0990200\pi\)
							−0.952003	+	0.306088i	\(0.900980\pi\)
\(31\)	−225.780	−	130.354i	−1.30810	−	0.755234i	−0.326324	−	0.945258i	\(-0.605810\pi\)
							−0.981779	+	0.190024i	\(0.939143\pi\)
\(37\)	−61.5775	+	229.810i	−0.273602	+	1.02110i	0.683171	+	0.730259i	\(0.260601\pi\)
							−0.956772	+	0.290837i	\(0.906066\pi\)
\(41\)			412.534i			1.57139i	0.618614	+	0.785695i	\(0.287695\pi\)
							−0.618614	+	0.785695i	\(0.712305\pi\)
\(43\)	226.541	−	226.541i	0.803423	−	0.803423i	−0.180205	−	0.983629i	\(-0.557676\pi\)
							0.983629	+	0.180205i	\(0.0576763\pi\)
\(47\)	167.520	+	44.8868i	0.519900	+	0.139307i	0.509220	−	0.860636i	\(-0.329934\pi\)
							0.0106796	+	0.999943i	\(0.496601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.19	✓	96
5.3	odd	4	inner	105.4.u.a.73.6	yes	96
7.5	odd	6	inner	105.4.u.a.82.6	yes	96
35.33	even	12	inner	105.4.u.a.103.19	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.19	✓	96	1.1	even	1	trivial
105.4.u.a.73.6	yes	96	5.3	odd	4	inner
105.4.u.a.82.6	yes	96	7.5	odd	6	inner
105.4.u.a.103.19	yes	96	35.33	even	12	inner