Embedded newform 105.4.x.a.2.16

• Label: 105.4.x.a.2.16
• Level: $105$
• Weight: $4$
• Character: 105.2
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.16
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.557385 - 2.08019i) q^{2} +(4.92343 + 1.66126i) q^{3} +(2.91169 - 1.68107i) q^{4} +(-2.90737 - 10.7957i) q^{5} +(0.711494 - 11.1676i) q^{6} +(-17.2324 - 6.78566i) q^{7} +(-17.3023 - 17.3023i) q^{8} +(21.4804 + 16.3582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.557385	−	2.08019i	−0.197065	−	0.735458i	−0.991723	−	0.128400i	\(-0.959016\pi\)
							0.794657	−	0.607059i	\(-0.207651\pi\)
\(3\)	4.92343	+	1.66126i	0.947515	+	0.319710i
							\(5\)	−2.90737	−	10.7957i	−0.260043	−	0.965597i
\(7\)	−17.2324	−	6.78566i	−0.930461	−	0.366391i
							\(11\)	7.23712	−	4.17835i	0.198370	−	0.114529i	−0.397525	−	0.917591i	\(-0.630131\pi\)
0.595895	+	0.803062i	\(0.296797\pi\)
\(13\)	28.7390	−	28.7390i	0.613137	−	0.613137i	−0.330625	−	0.943762i	\(-0.607260\pi\)
							0.943762	+	0.330625i	\(0.107260\pi\)
\(17\)	37.7114	+	10.1047i	0.538021	+	0.144162i	0.517589	−	0.855629i	\(-0.326829\pi\)
							0.0204311	+	0.999791i	\(0.493496\pi\)
\(19\)	2.75001	+	1.58772i	0.0332051	+	0.0191710i	0.516511	−	0.856281i	\(-0.327231\pi\)
							−0.483306	+	0.875452i	\(0.660564\pi\)
\(23\)	−71.7083	+	19.2142i	−0.650096	+	0.174193i	−0.568772	−	0.822495i	\(-0.692581\pi\)
							−0.0813239	+	0.996688i	\(0.525915\pi\)
\(29\)	290.651			1.86112			0.930562	−	0.366134i	\(-0.119319\pi\)
							0.930562	+	0.366134i	\(0.119319\pi\)
\(31\)	−52.4273	−	90.8068i	−0.303749	−	0.526109i	0.673233	−	0.739431i	\(-0.264905\pi\)
							−0.976982	+	0.213321i	\(0.931572\pi\)
\(37\)	145.773	−	39.0598i	0.647702	−	0.173551i	0.0800125	−	0.996794i	\(-0.474504\pi\)
							0.567690	+	0.823243i	\(0.307837\pi\)
\(41\)			432.019i			1.64561i	0.568323	+	0.822806i	\(0.307593\pi\)
							−0.568323	+	0.822806i	\(0.692407\pi\)
\(43\)	272.633	−	272.633i	0.966885	−	0.966885i	−0.0325837	−	0.999469i	\(-0.510374\pi\)
							0.999469	+	0.0325837i	\(0.0103736\pi\)
\(47\)	20.5841	+	76.8208i	0.0638829	+	0.238414i	0.990483	−	0.137633i	\(-0.0439496\pi\)
							−0.926600	+	0.376048i	\(0.877283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.16	✓	176
3.2	odd	2	inner	105.4.x.a.2.29	yes	176
5.3	odd	4	inner	105.4.x.a.23.16	yes	176
7.4	even	3	inner	105.4.x.a.32.29	yes	176
15.8	even	4	inner	105.4.x.a.23.29	yes	176
21.11	odd	6	inner	105.4.x.a.32.16	yes	176
35.18	odd	12	inner	105.4.x.a.53.29	yes	176
105.53	even	12	inner	105.4.x.a.53.16	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.16	✓	176	1.1	even	1	trivial
105.4.x.a.2.29	yes	176	3.2	odd	2	inner
105.4.x.a.23.16	yes	176	5.3	odd	4	inner
105.4.x.a.23.29	yes	176	15.8	even	4	inner
105.4.x.a.32.16	yes	176	21.11	odd	6	inner
105.4.x.a.32.29	yes	176	7.4	even	3	inner
105.4.x.a.53.16	yes	176	105.53	even	12	inner
105.4.x.a.53.29	yes	176	35.18	odd	12	inner