Embedded newform 105.3.v.a.58.1

• Label: 105.3.v.a.58.1
• 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			58.1
Character	\(\chi\)	\(=\)	105.58
Dual form			105.3.v.a.67.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.945463 - 3.52852i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-8.09242 + 4.67216i) q^{4} +(-4.69698 - 1.71417i) q^{5} -6.32716 q^{6} +(1.65300 + 6.80203i) q^{7} +(13.8047 + 13.8047i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.945463	−	3.52852i	−0.472731	−	1.76426i	−0.629890	−	0.776684i	\(-0.716900\pi\)
							0.157159	−	0.987573i	\(-0.449767\pi\)
\(3\)	0.448288	−	1.67303i	0.149429	−	0.557678i
							\(5\)	−4.69698	−	1.71417i	−0.939396	−	0.342835i
\(7\)	1.65300	+	6.80203i	0.236143	+	0.971718i
							\(11\)	−7.38858	−	12.7974i	−0.671689	−	1.16340i	−0.977425	−	0.211283i	\(-0.932236\pi\)
0.305736	−	0.952116i	\(-0.401098\pi\)
\(13\)	−7.90282	−	7.90282i	−0.607909	−	0.607909i	0.334490	−	0.942399i	\(-0.391436\pi\)
							−0.942399	+	0.334490i	\(0.891436\pi\)
\(17\)	8.26159	+	2.21369i	0.485976	+	0.130217i	0.493484	−	0.869755i	\(-0.335723\pi\)
							−0.00750800	+	0.999972i	\(0.502390\pi\)
\(19\)	−7.14233	−	4.12363i	−0.375912	−	0.217033i	0.300126	−	0.953900i	\(-0.402971\pi\)
							−0.676038	+	0.736867i	\(0.736305\pi\)
\(23\)	−18.1839	+	4.87235i	−0.790603	+	0.211842i	−0.631454	−	0.775413i	\(-0.717542\pi\)
							−0.159149	+	0.987255i	\(0.550875\pi\)
\(29\)		−	5.44111i		−	0.187624i	−0.995590	−	0.0938122i	\(-0.970095\pi\)
							0.995590	−	0.0938122i	\(-0.0299053\pi\)
\(31\)	−26.5427	−	45.9734i	−0.856217	−	1.48301i	−0.875511	−	0.483198i	\(-0.839475\pi\)
							0.0192943	−	0.999814i	\(-0.493858\pi\)
\(37\)	−9.25110	−	34.5256i	−0.250030	−	0.933124i	−0.970788	−	0.239938i	\(-0.922873\pi\)
							0.720758	−	0.693186i	\(-0.243794\pi\)
\(41\)	−40.8709			−0.996850			−0.498425	−	0.866933i	\(-0.666088\pi\)
							−0.498425	+	0.866933i	\(0.666088\pi\)
\(43\)	53.7395	+	53.7395i	1.24976	+	1.24976i	0.955828	+	0.293927i	\(0.0949624\pi\)
							0.293927	+	0.955828i	\(0.405038\pi\)
\(47\)	−10.7959	−	40.2908i	−0.229700	−	0.857251i	−0.980467	−	0.196684i	\(-0.936983\pi\)
							0.750767	−	0.660567i	\(-0.229684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.58.1	yes	64
3.2	odd	2		315.3.ca.b.163.16		64
5.2	odd	4	inner	105.3.v.a.37.16	✓	64
7.4	even	3	inner	105.3.v.a.88.16	yes	64
15.2	even	4		315.3.ca.b.37.1		64
21.11	odd	6		315.3.ca.b.298.1		64
35.32	odd	12	inner	105.3.v.a.67.1	yes	64
105.32	even	12		315.3.ca.b.172.16		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.16	✓	64	5.2	odd	4	inner
105.3.v.a.58.1	yes	64	1.1	even	1	trivial
105.3.v.a.67.1	yes	64	35.32	odd	12	inner
105.3.v.a.88.16	yes	64	7.4	even	3	inner
315.3.ca.b.37.1		64	15.2	even	4
315.3.ca.b.163.16		64	3.2	odd	2
315.3.ca.b.172.16		64	105.32	even	12
315.3.ca.b.298.1		64	21.11	odd	6