Embedded newform 105.4.s.a.101.5

• Label: 105.4.s.a.101.5
• 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			101.5
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.51429 + 1.45163i) q^{2} +(-2.26232 - 4.67781i) q^{3} +(0.214449 - 0.371437i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(12.4786 + 8.47733i) q^{6} +(17.9959 - 4.37591i) q^{7} -21.9809i q^{8} +(-16.7638 + 21.1654i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.51429	+	1.45163i	−0.888937	+	0.513228i	−0.873595	−	0.486654i	\(-0.838217\pi\)
							−0.0153424	+	0.999882i	\(0.504884\pi\)
\(3\)	−2.26232	−	4.67781i	−0.435384	−	0.900245i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	17.9959	−	4.37591i	0.971686	−	0.236277i
							\(11\)	−28.8369	−	16.6490i	−0.790423	−	0.456351i	0.0496886	−	0.998765i	\(-0.484177\pi\)
−0.840111	+	0.542414i	\(0.817510\pi\)
\(13\)			62.4189i			1.33168i	0.746093	+	0.665842i	\(0.231928\pi\)
							−0.746093	+	0.665842i	\(0.768072\pi\)
\(17\)	−34.7249	+	60.1453i	−0.495413	+	0.858080i	−0.999986	−	0.00528856i	\(-0.998317\pi\)
							0.504573	+	0.863369i	\(0.331650\pi\)
\(19\)	−71.6103	+	41.3442i	−0.864659	+	0.499211i	−0.865570	−	0.500788i	\(-0.833044\pi\)
							0.000910397		1.00000i	\(0.499710\pi\)
\(23\)	−49.8101	+	28.7578i	−0.451570	+	0.260714i	−0.708493	−	0.705718i	\(-0.750625\pi\)
							0.256923	+	0.966432i	\(0.417291\pi\)
\(29\)			114.270i			0.731702i	0.930673	+	0.365851i	\(0.119222\pi\)
							−0.930673	+	0.365851i	\(0.880778\pi\)
\(31\)	−187.647	−	108.338i	−1.08717	−	0.627679i	−0.154350	−	0.988016i	\(-0.549328\pi\)
							−0.932822	+	0.360337i	\(0.882662\pi\)
\(37\)	136.662	+	236.705i	0.607218	+	1.05173i	0.991697	+	0.128598i	\(0.0410477\pi\)
							−0.384479	+	0.923134i	\(0.625619\pi\)
\(41\)	36.2347			0.138022			0.0690111	−	0.997616i	\(-0.478016\pi\)
							0.0690111	+	0.997616i	\(0.478016\pi\)
\(43\)	185.189			0.656769			0.328385	−	0.944544i	\(-0.393496\pi\)
							0.328385	+	0.944544i	\(0.393496\pi\)
\(47\)	−25.2220	−	43.6858i	−0.0782767	−	0.135579i	0.824230	−	0.566255i	\(-0.191608\pi\)
							−0.902507	+	0.430676i	\(0.858275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.5	yes	32
3.2	odd	2		105.4.s.b.101.12	yes	32
7.5	odd	6		105.4.s.b.26.12	yes	32
21.5	even	6	inner	105.4.s.a.26.5	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.5	✓	32	21.5	even	6	inner
105.4.s.a.101.5	yes	32	1.1	even	1	trivial
105.4.s.b.26.12	yes	32	7.5	odd	6
105.4.s.b.101.12	yes	32	3.2	odd	2