Embedded newform 105.3.v.a.58.4

• Label: 105.3.v.a.58.4
• Level: $105$
• Weight: $3$
• Character: 105.58
• 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			58.4
Character	\(\chi\)	\(=\)	105.58
Dual form			105.3.v.a.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.686107 - 2.56059i) q^{2} +(-0.448288 + 1.67303i) q^{3} +(-2.62176 + 1.51367i) q^{4} +(3.50926 - 3.56161i) q^{5} +4.59152 q^{6} +(6.99820 - 0.158915i) q^{7} +(-1.82323 - 1.82323i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.686107	−	2.56059i	−0.343053	−	1.28029i	−0.894870	−	0.446327i	\(-0.852732\pi\)
							0.551816	−	0.833966i	\(-0.313935\pi\)
\(3\)	−0.448288	+	1.67303i	−0.149429	+	0.557678i
							\(5\)	3.50926	−	3.56161i	0.701852	−	0.712323i
\(7\)	6.99820	−	0.158915i	0.999742	−	0.0227021i
							\(11\)	−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	0.701378	−	0.712789i	\(-0.252568\pi\)
							−0.712789	+	0.701378i	\(0.752568\pi\)
\(17\)	19.4170	+	5.20277i	1.14218	+	0.306045i	0.779826	−	0.625996i	\(-0.215308\pi\)
							0.362351	+	0.932042i	\(0.381974\pi\)
\(19\)	−10.9984	−	6.34991i	−0.578861	−	0.334206i	0.181819	−	0.983332i	\(-0.441801\pi\)
							−0.760681	+	0.649126i	\(0.775135\pi\)
\(23\)	15.6270	−	4.18723i	0.679433	−	0.182054i	0.0974328	−	0.995242i	\(-0.468937\pi\)
							0.582000	+	0.813189i	\(0.302270\pi\)
\(29\)			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	0.958208	−	0.286074i	\(-0.0923503\pi\)
							−0.726851	+	0.686795i	\(0.759017\pi\)
\(37\)	16.8501	+	62.8854i	0.455408	+	1.69960i	0.686886	+	0.726765i	\(0.258977\pi\)
							−0.231479	+	0.972840i	\(0.574356\pi\)
\(41\)	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	\(0.554048\pi\)
\(47\)	−14.9626	−	55.8413i	−0.318354	−	1.18811i	−0.920826	−	0.389973i	\(-0.872484\pi\)
							0.602472	−	0.798140i	\(-0.294182\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.13	✓	64	5.2	odd	4	inner
105.3.v.a.58.4	yes	64	1.1	even	1	trivial
105.3.v.a.67.4	yes	64	35.32	odd	12	inner
105.3.v.a.88.13	yes	64	7.4	even	3	inner
315.3.ca.b.37.4		64	15.2	even	4
315.3.ca.b.163.13		64	3.2	odd	2
315.3.ca.b.172.13		64	105.32	even	12
315.3.ca.b.298.4		64	21.11	odd	6