Embedded newform 105.3.v.a.37.10

• Label: 105.3.v.a.37.10
• 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			37.10
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.901161 - 0.241465i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(-2.71032 + 1.56480i) q^{4} +(2.44472 + 4.36158i) q^{5} -1.61592 q^{6} +(5.41162 + 4.44009i) q^{7} +(-4.70337 + 4.70337i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.901161	−	0.241465i	0.450581	−	0.120733i	−0.0263905	−	0.999652i	\(-0.508401\pi\)
							0.476971	+	0.878919i	\(0.341735\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	2.44472	+	4.36158i	0.488943	+	0.872316i
\(7\)	5.41162	+	4.44009i	0.773088	+	0.634298i
							\(11\)	3.17246	+	5.49486i	0.288405	+	0.499533i	0.973429	−	0.228988i	\(-0.0735416\pi\)
−0.685024	+	0.728521i	\(0.740208\pi\)
\(13\)	−5.74921	+	5.74921i	−0.442247	+	0.442247i	−0.892767	−	0.450520i	\(-0.851239\pi\)
							0.450520	+	0.892767i	\(0.351239\pi\)
\(17\)	0.226953	−	0.847000i	0.0133502	−	0.0498235i	−0.958929	−	0.283645i	\(-0.908456\pi\)
							0.972280	+	0.233821i	\(0.0751230\pi\)
\(19\)	5.02493	+	2.90114i	0.264470	+	0.152692i	0.626372	−	0.779524i	\(-0.284539\pi\)
							−0.361902	+	0.932216i	\(0.617872\pi\)
\(23\)	−9.99491	−	37.3015i	−0.434561	−	1.62181i	−0.742114	−	0.670274i	\(-0.766177\pi\)
							0.307552	−	0.951531i	\(-0.400490\pi\)
\(29\)		−	28.7187i		−	0.990300i	−0.868808	−	0.495150i	\(-0.835113\pi\)
							0.868808	−	0.495150i	\(-0.164887\pi\)
\(31\)	30.5551	+	52.9230i	0.985648	+	1.70719i	0.639022	+	0.769189i	\(0.279339\pi\)
							0.346626	+	0.938003i	\(0.387327\pi\)
\(37\)	−11.3239	+	3.03422i	−0.306051	+	0.0820061i	−0.408576	−	0.912724i	\(-0.633974\pi\)
							0.102525	+	0.994730i	\(0.467308\pi\)
\(41\)	21.6632			0.528370			0.264185	−	0.964472i	\(-0.414897\pi\)
							0.264185	+	0.964472i	\(0.414897\pi\)
\(43\)	45.3759	−	45.3759i	1.05525	−	1.05525i	0.0568731	−	0.998381i	\(-0.481887\pi\)
							0.998381	−	0.0568731i	\(-0.0181131\pi\)
\(47\)	49.7271	−	13.3243i	1.05802	−	0.283497i	0.312460	−	0.949931i	\(-0.398847\pi\)
							0.745564	+	0.666434i	\(0.232180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.10	✓	64
3.2	odd	2		315.3.ca.b.37.7		64
5.3	odd	4	inner	105.3.v.a.58.7	yes	64
7.4	even	3	inner	105.3.v.a.67.7	yes	64
15.8	even	4		315.3.ca.b.163.10		64
21.11	odd	6		315.3.ca.b.172.10		64
35.18	odd	12	inner	105.3.v.a.88.10	yes	64
105.53	even	12		315.3.ca.b.298.7		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.10	✓	64	1.1	even	1	trivial
105.3.v.a.58.7	yes	64	5.3	odd	4	inner
105.3.v.a.67.7	yes	64	7.4	even	3	inner
105.3.v.a.88.10	yes	64	35.18	odd	12	inner
315.3.ca.b.37.7		64	3.2	odd	2
315.3.ca.b.163.10		64	15.8	even	4
315.3.ca.b.172.10		64	21.11	odd	6
315.3.ca.b.298.7		64	105.53	even	12