Embedded newform 105.4.s.a.101.4

• Label: 105.4.s.a.101.4
• Level: $105$
• Weight: $4$
• Character: 105.101
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.4
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.06378 + 1.76887i) q^{2} +(-0.433556 + 5.17803i) q^{3} +(2.25781 - 3.91065i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-7.83096 - 16.6312i) q^{6} +(5.77785 - 17.5959i) q^{7} -12.3268i q^{8} +(-26.6241 - 4.48993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.06378	+	1.76887i	−1.08321	+	0.625391i	−0.931760	−	0.363075i	\(-0.881727\pi\)
							−0.151448	+	0.988465i	\(0.548394\pi\)
\(3\)	−0.433556	+	5.17803i	−0.0834379	+	0.996513i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	5.77785	−	17.5959i	0.311975	−	0.950090i
							\(11\)	3.05657	+	1.76471i	0.0837811	+	0.0483710i	0.541305	−	0.840826i	\(-0.317930\pi\)
−0.457524	+	0.889197i	\(0.651264\pi\)
\(13\)		−	14.5404i		−	0.310214i	−0.987898	−	0.155107i	\(-0.950428\pi\)
							0.987898	−	0.155107i	\(-0.0495723\pi\)
\(17\)	30.9412	−	53.5917i	0.441432	−	0.764582i	−0.556364	−	0.830939i	\(-0.687804\pi\)
							0.997796	+	0.0663561i	\(0.0211373\pi\)
\(19\)	−39.8972	+	23.0346i	−0.481739	+	0.278132i	−0.721141	−	0.692788i	\(-0.756382\pi\)
							0.239402	+	0.970921i	\(0.423049\pi\)
\(23\)	99.5813	−	57.4933i	0.902788	−	0.521225i	0.0246846	−	0.999695i	\(-0.492142\pi\)
							0.878104	+	0.478470i	\(0.158809\pi\)
\(29\)		−	127.376i		−	0.815628i	−0.913065	−	0.407814i	\(-0.866291\pi\)
							0.913065	−	0.407814i	\(-0.133709\pi\)
\(31\)	183.725	+	106.074i	1.06445	+	0.614561i	0.926660	−	0.375900i	\(-0.122666\pi\)
							0.137791	+	0.990461i	\(0.456000\pi\)
\(37\)	−142.232	−	246.353i	−0.631969	−	1.09460i	−0.987149	−	0.159804i	\(-0.948914\pi\)
							0.355180	−	0.934798i	\(-0.384419\pi\)
\(41\)	−328.199			−1.25015			−0.625075	−	0.780565i	\(-0.714932\pi\)
							−0.625075	+	0.780565i	\(0.714932\pi\)
\(43\)	−108.351			−0.384264			−0.192132	−	0.981369i	\(-0.561540\pi\)
							−0.192132	+	0.981369i	\(0.561540\pi\)
\(47\)	−194.765	−	337.343i	−0.604455	−	1.04695i	−0.992137	−	0.125154i	\(-0.960058\pi\)
							0.387682	−	0.921793i	\(-0.373276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.4	yes	32
3.2	odd	2		105.4.s.b.101.13	yes	32
7.5	odd	6		105.4.s.b.26.13	yes	32
21.5	even	6	inner	105.4.s.a.26.4	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.4	✓	32	21.5	even	6	inner
105.4.s.a.101.4	yes	32	1.1	even	1	trivial
105.4.s.b.26.13	yes	32	7.5	odd	6
105.4.s.b.101.13	yes	32	3.2	odd	2