Embedded newform 105.4.x.a.2.6

• Label: 105.4.x.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19670 - 4.46614i) q^{2} +(3.39774 - 3.93133i) q^{3} +(-11.5861 + 6.68922i) q^{4} +(2.95094 - 10.7839i) q^{5} +(-21.6239 - 10.4701i) q^{6} +(16.8977 - 7.58076i) q^{7} +(17.5845 + 17.5845i) q^{8} +(-3.91074 - 26.7153i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19670	−	4.46614i	−0.423096	−	1.57902i	−0.768045	−	0.640396i	\(-0.778770\pi\)
							0.344949	−	0.938622i	\(-0.387896\pi\)
\(3\)	3.39774	−	3.93133i	0.653895	−	0.756585i
							\(5\)	2.95094	−	10.7839i	0.263940	−	0.964539i
\(7\)	16.8977	−	7.58076i	0.912390	−	0.409323i
							\(11\)	−46.0859	+	26.6077i	−1.26322	+	0.729320i	−0.973696	−	0.227850i	\(-0.926830\pi\)
−0.289524	+	0.957171i	\(0.593497\pi\)
\(13\)	16.1509	−	16.1509i	0.344572	−	0.344572i	−0.513511	−	0.858083i	\(-0.671655\pi\)
							0.858083	+	0.513511i	\(0.171655\pi\)
\(17\)	98.9831	+	26.5225i	1.41217	+	0.378391i	0.882700	−	0.469936i	\(-0.155723\pi\)
							0.529473	+	0.848327i	\(0.322390\pi\)
\(19\)	100.194	+	57.8473i	1.20980	+	0.698478i	0.962715	−	0.270517i	\(-0.0871945\pi\)
							0.247083	+	0.968994i	\(0.420528\pi\)
\(23\)	−105.150	+	28.1747i	−0.953269	+	0.255428i	−0.701749	−	0.712424i	\(-0.747597\pi\)
							−0.251520	+	0.967852i	\(0.580931\pi\)
\(29\)	90.0714			0.576753			0.288376	−	0.957517i	\(-0.406885\pi\)
							0.288376	+	0.957517i	\(0.406885\pi\)
\(31\)	23.8963	+	41.3896i	0.138448	+	0.239800i	0.926909	−	0.375285i	\(-0.122455\pi\)
							−0.788461	+	0.615085i	\(0.789122\pi\)
\(37\)	−112.526	+	30.1513i	−0.499977	+	0.133969i	−0.499989	−	0.866031i	\(-0.666663\pi\)
							1.21375e−5		1.00000i	\(0.499996\pi\)
\(41\)		−	200.101i		−	0.762207i	−0.924532	−	0.381103i	\(-0.875544\pi\)
							0.924532	−	0.381103i	\(-0.124456\pi\)
\(43\)	85.6207	−	85.6207i	0.303652	−	0.303652i	−0.538789	−	0.842441i	\(-0.681118\pi\)
							0.842441	+	0.538789i	\(0.181118\pi\)
\(47\)	−107.202	−	400.083i	−0.332702	−	1.24166i	−0.906339	−	0.422551i	\(-0.861135\pi\)
							0.573637	−	0.819110i	\(-0.305532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.6	✓	176
3.2	odd	2	inner	105.4.x.a.2.39	yes	176
5.3	odd	4	inner	105.4.x.a.23.6	yes	176
7.4	even	3	inner	105.4.x.a.32.39	yes	176
15.8	even	4	inner	105.4.x.a.23.39	yes	176
21.11	odd	6	inner	105.4.x.a.32.6	yes	176
35.18	odd	12	inner	105.4.x.a.53.39	yes	176
105.53	even	12	inner	105.4.x.a.53.6	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.6	✓	176	1.1	even	1	trivial
105.4.x.a.2.39	yes	176	3.2	odd	2	inner
105.4.x.a.23.6	yes	176	5.3	odd	4	inner
105.4.x.a.23.39	yes	176	15.8	even	4	inner
105.4.x.a.32.6	yes	176	21.11	odd	6	inner
105.4.x.a.32.39	yes	176	7.4	even	3	inner
105.4.x.a.53.6	yes	176	105.53	even	12	inner
105.4.x.a.53.39	yes	176	35.18	odd	12	inner