Embedded newform 105.4.x.a.2.4

• Label: 105.4.x.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26251 - 4.71175i) q^{2} +(-4.27909 - 2.94778i) q^{3} +(-13.6784 + 7.89724i) q^{4} +(-3.51839 - 10.6123i) q^{5} +(-8.48683 + 23.8836i) q^{6} +(-15.2757 - 10.4716i) q^{7} +(26.8850 + 26.8850i) q^{8} +(9.62114 + 25.2276i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26251	−	4.71175i	−0.446364	−	1.66585i	−0.712309	−	0.701866i	\(-0.752351\pi\)
							0.265945	−	0.963988i	\(-0.414316\pi\)
\(3\)	−4.27909	−	2.94778i	−0.823510	−	0.567301i
							\(5\)	−3.51839	−	10.6123i	−0.314695	−	0.949193i
\(7\)	−15.2757	−	10.4716i	−0.824808	−	0.565413i
							\(11\)	22.0741	−	12.7445i	0.605054	−	0.349328i	−0.165973	−	0.986130i	\(-0.553077\pi\)
0.771027	+	0.636802i	\(0.219743\pi\)
\(13\)	27.8250	−	27.8250i	0.593636	−	0.593636i	−0.344976	−	0.938612i	\(-0.612113\pi\)
							0.938612	+	0.344976i	\(0.112113\pi\)
\(17\)	−25.9954	−	6.96544i	−0.370871	−	0.0993746i	0.0685694	−	0.997646i	\(-0.478157\pi\)
							−0.439440	+	0.898272i	\(0.644823\pi\)
\(19\)	−37.1432	−	21.4447i	−0.448487	−	0.258934i	0.258704	−	0.965957i	\(-0.416705\pi\)
							−0.707191	+	0.707023i	\(0.750038\pi\)
\(23\)	179.331	−	48.0515i	1.62578	−	0.435628i	0.673091	−	0.739560i	\(-0.264966\pi\)
							0.952694	+	0.303932i	\(0.0982995\pi\)
\(29\)	−156.565			−1.00253			−0.501266	−	0.865293i	\(-0.667132\pi\)
							−0.501266	+	0.865293i	\(0.667132\pi\)
\(31\)	78.3505	+	135.707i	0.453941	+	0.786248i	0.998627	−	0.0523918i	\(-0.0166845\pi\)
							−0.544686	+	0.838640i	\(0.683351\pi\)
\(37\)	−421.701	+	112.994i	−1.87371	+	0.502058i	−0.873832	+	0.486229i	\(0.838372\pi\)
							−0.999875	+	0.0158296i	\(0.994961\pi\)
\(41\)			103.957i			0.395985i	0.980204	+	0.197993i	\(0.0634422\pi\)
							−0.980204	+	0.197993i	\(0.936558\pi\)
\(43\)	−217.750	+	217.750i	−0.772244	+	0.772244i	−0.978498	−	0.206254i	\(-0.933873\pi\)
							0.206254	+	0.978498i	\(0.433873\pi\)
\(47\)	−27.9677	−	104.377i	−0.0867982	−	0.323935i	0.908850	−	0.417122i	\(-0.136961\pi\)
							−0.995649	+	0.0931869i	\(0.970295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.4	✓	176
3.2	odd	2	inner	105.4.x.a.2.41	yes	176
5.3	odd	4	inner	105.4.x.a.23.4	yes	176
7.4	even	3	inner	105.4.x.a.32.41	yes	176
15.8	even	4	inner	105.4.x.a.23.41	yes	176
21.11	odd	6	inner	105.4.x.a.32.4	yes	176
35.18	odd	12	inner	105.4.x.a.53.41	yes	176
105.53	even	12	inner	105.4.x.a.53.4	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.4	✓	176	1.1	even	1	trivial
105.4.x.a.2.41	yes	176	3.2	odd	2	inner
105.4.x.a.23.4	yes	176	5.3	odd	4	inner
105.4.x.a.23.41	yes	176	15.8	even	4	inner
105.4.x.a.32.4	yes	176	21.11	odd	6	inner
105.4.x.a.32.41	yes	176	7.4	even	3	inner
105.4.x.a.53.4	yes	176	105.53	even	12	inner
105.4.x.a.53.41	yes	176	35.18	odd	12	inner