Embedded newform 105.3.v.a.37.13

• Label: 105.3.v.a.37.13
• Level: $105$
• Weight: $3$
• Character: 105.37
• 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			37.13
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.13

$q$-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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

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

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	0.446327	−	0.894870i	\(-0.352732\pi\)
							0.833966	+	0.551816i	\(0.186065\pi\)
\(3\)	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	−0.955833	−	0.293911i	\(-0.905043\pi\)
0.223382	−	0.974731i	\(-0.428290\pi\)
\(13\)	−0.148341	+	0.148341i	−0.0114109	+	0.0114109i	−0.712789	−	0.701378i	\(-0.752568\pi\)
							0.701378	+	0.712789i	\(0.252568\pi\)
\(17\)	−5.20277	+	19.4170i	−0.306045	+	1.14218i	0.625996	+	0.779826i	\(0.284692\pi\)
							−0.932042	+	0.362351i	\(0.881974\pi\)
\(19\)	10.9984	+	6.34991i	0.578861	+	0.334206i	0.760681	−	0.649126i	\(-0.224865\pi\)
							−0.181819	+	0.983332i	\(0.558199\pi\)
\(23\)	−4.18723	−	15.6270i	−0.182054	−	0.679433i	−0.995242	−	0.0974328i	\(-0.968937\pi\)
							0.813189	−	0.582000i	\(-0.197730\pi\)
\(29\)		−	21.7589i		−	0.750308i	−0.926962	−	0.375154i	\(-0.877590\pi\)
							0.926962	−	0.375154i	\(-0.122410\pi\)
\(31\)	7.17206	+	12.4224i	0.231357	+	0.400721i	0.958208	−	0.286074i	\(-0.0923503\pi\)
							−0.726851	+	0.686795i	\(0.759017\pi\)
\(37\)	−62.8854	+	16.8501i	−1.69960	+	0.455408i	−0.972840	−	0.231479i	\(-0.925644\pi\)
							−0.726765	+	0.686886i	\(0.758977\pi\)
\(41\)	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	−0.168983	−	0.985619i	\(-0.554048\pi\)
							0.985619	+	0.168983i	\(0.0540482\pi\)
\(47\)	55.8413	−	14.9626i	1.18811	−	0.318354i	0.389973	−	0.920826i	\(-0.372484\pi\)
							0.798140	+	0.602472i	\(0.205818\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.13	✓	64
3.2	odd	2		315.3.ca.b.37.4		64
5.3	odd	4	inner	105.3.v.a.58.4	yes	64
7.4	even	3	inner	105.3.v.a.67.4	yes	64
15.8	even	4		315.3.ca.b.163.13		64
21.11	odd	6		315.3.ca.b.172.13		64
35.18	odd	12	inner	105.3.v.a.88.13	yes	64
105.53	even	12		315.3.ca.b.298.4		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.13	✓	64	1.1	even	1	trivial
105.3.v.a.58.4	yes	64	5.3	odd	4	inner
105.3.v.a.67.4	yes	64	7.4	even	3	inner
105.3.v.a.88.13	yes	64	35.18	odd	12	inner
315.3.ca.b.37.4		64	3.2	odd	2
315.3.ca.b.163.13		64	15.8	even	4
315.3.ca.b.172.13		64	21.11	odd	6
315.3.ca.b.298.4		64	105.53	even	12