Embedded newform 105.3.w.a.17.12

• Label: 105.3.w.a.17.12
• Level: $105$
• Weight: $3$
• Character: 105.17
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.12
Character	\(\chi\)	\(=\)	105.17
Dual form			105.3.w.a.68.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.667244 - 0.178787i) q^{2} +(-2.94454 - 0.574167i) q^{3} +(-3.05085 - 1.76141i) q^{4} +(1.04868 + 4.88879i) q^{5} +(1.86207 + 0.909557i) q^{6} +(6.73061 - 1.92324i) q^{7} +(3.67457 + 3.67457i) q^{8} +(8.34066 + 3.38132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.667244	−	0.178787i	−0.333622	−	0.0893937i	0.0881194	−	0.996110i	\(-0.471914\pi\)
							−0.421741	+	0.906716i	\(0.638581\pi\)
\(3\)	−2.94454	−	0.574167i	−0.981514	−	0.191389i
							\(5\)	1.04868	+	4.88879i	0.209737	+	0.977758i
\(7\)	6.73061	−	1.92324i	0.961516	−	0.274748i
							\(11\)	13.5279	+	7.81032i	1.22981	+	0.710029i	0.966989	−	0.254816i	\(-0.0820150\pi\)
0.262817	+	0.964846i	\(0.415348\pi\)
\(13\)	−6.69274	−	6.69274i	−0.514826	−	0.514826i	0.401175	−	0.916001i	\(-0.368602\pi\)
							−0.916001	+	0.401175i	\(0.868602\pi\)
\(17\)	−0.667233	+	0.178784i	−0.0392490	+	0.0105167i	−0.278390	−	0.960468i	\(-0.589801\pi\)
							0.239141	+	0.970985i	\(0.423134\pi\)
\(19\)	7.94736	+	13.7652i	0.418282	+	0.724486i	0.995767	−	0.0919159i	\(-0.0292991\pi\)
							−0.577485	+	0.816401i	\(0.695966\pi\)
\(23\)	−5.76473	+	21.5143i	−0.250640	+	0.935403i	0.719824	+	0.694157i	\(0.244223\pi\)
							−0.970464	+	0.241246i	\(0.922444\pi\)
\(29\)	50.8410			1.75314			0.876569	−	0.481276i	\(-0.159826\pi\)
							0.876569	+	0.481276i	\(0.159826\pi\)
\(31\)	23.6917	+	13.6784i	0.764247	+	0.441238i	0.830819	−	0.556543i	\(-0.187873\pi\)
							−0.0665712	+	0.997782i	\(0.521206\pi\)
\(37\)	9.69523	−	36.1831i	0.262033	−	0.977922i	−0.702008	−	0.712169i	\(-0.747713\pi\)
							0.964041	−	0.265753i	\(-0.0856204\pi\)
\(41\)	−48.7887			−1.18997			−0.594984	−	0.803738i	\(-0.702842\pi\)
							−0.594984	+	0.803738i	\(0.702842\pi\)
\(43\)	11.8345	−	11.8345i	0.275221	−	0.275221i	−0.555977	−	0.831198i	\(-0.687656\pi\)
							0.831198	+	0.555977i	\(0.187656\pi\)
\(47\)	−13.8676	+	51.7545i	−0.295055	+	1.10116i	0.646119	+	0.763237i	\(0.276391\pi\)
							−0.941174	+	0.337923i	\(0.890276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.17.12	✓	112
3.2	odd	2	inner	105.3.w.a.17.17	yes	112
5.3	odd	4	inner	105.3.w.a.38.12	yes	112
7.5	odd	6	inner	105.3.w.a.47.17	yes	112
15.8	even	4	inner	105.3.w.a.38.17	yes	112
21.5	even	6	inner	105.3.w.a.47.12	yes	112
35.33	even	12	inner	105.3.w.a.68.17	yes	112
105.68	odd	12	inner	105.3.w.a.68.12	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.12	✓	112	1.1	even	1	trivial
105.3.w.a.17.17	yes	112	3.2	odd	2	inner
105.3.w.a.38.12	yes	112	5.3	odd	4	inner
105.3.w.a.38.17	yes	112	15.8	even	4	inner
105.3.w.a.47.12	yes	112	21.5	even	6	inner
105.3.w.a.47.17	yes	112	7.5	odd	6	inner
105.3.w.a.68.12	yes	112	105.68	odd	12	inner
105.3.w.a.68.17	yes	112	35.33	even	12	inner