Embedded newform 105.3.v.a.88.13

• Label: 105.3.v.a.88.13
• Level: $105$
• Weight: $3$
• Character: 105.88
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			88.13
Character	\(\chi\)	\(=\)	105.88
Dual form			105.3.v.a.37.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.56059 + 0.686107i) q^{2} +(1.67303 - 0.448288i) q^{3} +(2.62176 + 1.51367i) q^{4} +(1.32982 + 4.81992i) q^{5} +4.59152 q^{6} +(0.158915 - 6.99820i) q^{7} +(-1.82323 - 1.82323i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.56059	+	0.686107i	1.28029	+	0.343053i	0.833966	−	0.551816i	\(-0.186065\pi\)
							0.446327	+	0.894870i	\(0.352732\pi\)
\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
							\(5\)	1.32982	+	4.81992i	0.265964	+	0.963983i
\(7\)	0.158915	−	6.99820i	0.0227021	−	0.999742i
							\(11\)	−8.05696	+	13.9551i	−0.732451	+	1.26864i	0.223382	+	0.974731i	\(0.428290\pi\)
−0.955833	+	0.293911i	\(0.905043\pi\)
\(13\)	−0.148341	−	0.148341i	−0.0114109	−	0.0114109i	0.701378	−	0.712789i	\(-0.252568\pi\)
							−0.712789	+	0.701378i	\(0.752568\pi\)
\(17\)	−5.20277	−	19.4170i	−0.306045	−	1.14218i	−0.932042	−	0.362351i	\(-0.881974\pi\)
							0.625996	−	0.779826i	\(-0.284692\pi\)
\(19\)	10.9984	−	6.34991i	0.578861	−	0.334206i	−0.181819	−	0.983332i	\(-0.558199\pi\)
							0.760681	+	0.649126i	\(0.224865\pi\)
\(23\)	−4.18723	+	15.6270i	−0.182054	+	0.679433i	0.813189	+	0.582000i	\(0.197730\pi\)
							−0.995242	+	0.0974328i	\(0.968937\pi\)
\(29\)			21.7589i			0.750308i	0.926962	+	0.375154i	\(0.122410\pi\)
							−0.926962	+	0.375154i	\(0.877590\pi\)
\(31\)	7.17206	−	12.4224i	0.231357	−	0.400721i	−0.726851	−	0.686795i	\(-0.759017\pi\)
							0.958208	+	0.286074i	\(0.0923503\pi\)
\(37\)	−62.8854	−	16.8501i	−1.69960	−	0.455408i	−0.726765	−	0.686886i	\(-0.758977\pi\)
							−0.972840	+	0.231479i	\(0.925644\pi\)
\(41\)	2.26939			0.0553509			0.0276754	−	0.999617i	\(-0.491190\pi\)
							0.0276754	+	0.999617i	\(0.491190\pi\)
\(43\)	35.1154	+	35.1154i	0.816636	+	0.816636i	0.985619	−	0.168983i	\(-0.0540482\pi\)
							−0.168983	+	0.985619i	\(0.554048\pi\)
\(47\)	55.8413	+	14.9626i	1.18811	+	0.318354i	0.798140	−	0.602472i	\(-0.205818\pi\)
							0.389973	+	0.920826i	\(0.372484\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.88.13	yes	64
3.2	odd	2		315.3.ca.b.298.4		64
5.2	odd	4	inner	105.3.v.a.67.4	yes	64
7.2	even	3	inner	105.3.v.a.58.4	yes	64
15.2	even	4		315.3.ca.b.172.13		64
21.2	odd	6		315.3.ca.b.163.13		64
35.2	odd	12	inner	105.3.v.a.37.13	✓	64
105.2	even	12		315.3.ca.b.37.4		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.13	✓	64	35.2	odd	12	inner
105.3.v.a.58.4	yes	64	7.2	even	3	inner
105.3.v.a.67.4	yes	64	5.2	odd	4	inner
105.3.v.a.88.13	yes	64	1.1	even	1	trivial
315.3.ca.b.37.4		64	105.2	even	12
315.3.ca.b.163.13		64	21.2	odd	6
315.3.ca.b.172.13		64	15.2	even	4
315.3.ca.b.298.4		64	3.2	odd	2