Embedded newform 105.4.s.a.26.11

• Label: 105.4.s.a.26.11
• Level: $105$
• Weight: $4$
• Character: 105.26
• 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			26.11
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.a.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.48940 + 0.859905i) q^{2} +(-4.11977 + 3.16663i) q^{3} +(-2.52113 - 4.36672i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-8.85898 + 1.17376i) q^{6} +(2.28179 - 18.3792i) q^{7} -22.4302i q^{8} +(6.94494 - 26.0915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.48940	+	0.859905i	0.526582	+	0.304022i	0.739624	−	0.673021i	\(-0.235004\pi\)
							−0.213041	+	0.977043i	\(0.568337\pi\)
\(3\)	−4.11977	+	3.16663i	−0.792849	+	0.609418i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	2.28179	−	18.3792i	0.123205	−	0.992381i
							\(11\)	49.7733	−	28.7366i	1.36429	−	0.787674i	0.374100	−	0.927388i	\(-0.377952\pi\)
0.990192	+	0.139714i	\(0.0446183\pi\)
\(13\)			44.6643i			0.952896i	0.879203	+	0.476448i	\(0.158076\pi\)
							−0.879203	+	0.476448i	\(0.841924\pi\)
\(17\)	−54.8920	−	95.0758i	−0.783134	−	1.35643i	−0.930108	−	0.367287i	\(-0.880287\pi\)
							0.146974	−	0.989140i	\(-0.453047\pi\)
\(19\)	−79.5306	−	45.9170i	−0.960293	−	0.554426i	−0.0640300	−	0.997948i	\(-0.520395\pi\)
							−0.896263	+	0.443522i	\(0.853729\pi\)
\(23\)	−27.1329	−	15.6652i	−0.245983	−	0.142018i	0.371941	−	0.928257i	\(-0.378693\pi\)
							−0.617923	+	0.786238i	\(0.712026\pi\)
\(29\)		−	183.561i		−	1.17539i	−0.809082	−	0.587696i	\(-0.800035\pi\)
							0.809082	−	0.587696i	\(-0.199965\pi\)
\(31\)	123.493	−	71.2990i	0.715486	−	0.413086i	−0.0976027	−	0.995225i	\(-0.531117\pi\)
							0.813089	+	0.582139i	\(0.197784\pi\)
\(37\)	−114.912	+	199.033i	−0.510579	+	0.884349i	0.489346	+	0.872090i	\(0.337236\pi\)
							−0.999925	+	0.0122589i	\(0.996098\pi\)
\(41\)	185.753			0.707555			0.353778	−	0.935330i	\(-0.384897\pi\)
							0.353778	+	0.935330i	\(0.384897\pi\)
\(43\)	−22.3617			−0.0793054			−0.0396527	−	0.999214i	\(-0.512625\pi\)
							−0.0396527	+	0.999214i	\(0.512625\pi\)
\(47\)	−116.516	+	201.811i	−0.361608	+	0.626323i	−0.988226	−	0.153004i	\(-0.951105\pi\)
							0.626618	+	0.779326i	\(0.284439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.26.11	✓	32
3.2	odd	2		105.4.s.b.26.6	yes	32
7.3	odd	6		105.4.s.b.101.6	yes	32
21.17	even	6	inner	105.4.s.a.101.11	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.11	✓	32	1.1	even	1	trivial
105.4.s.a.101.11	yes	32	21.17	even	6	inner
105.4.s.b.26.6	yes	32	3.2	odd	2
105.4.s.b.101.6	yes	32	7.3	odd	6