Embedded newform 105.4.s.b.26.7

• Label: 105.4.s.b.26.7
• 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			26.7
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.b.101.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.815639 - 0.470909i) q^{2} +(-0.600797 + 5.16130i) q^{3} +(-3.55649 - 6.16002i) q^{4} +(2.50000 - 4.33013i) q^{5} +(2.92054 - 3.92684i) q^{6} +(13.9941 + 12.1312i) q^{7} +14.2337i q^{8} +(-26.2781 - 6.20179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{3} + 64 q^{4} + 80 q^{5} + 28 q^{6} + 46 q^{7} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.815639	−	0.470909i	−0.288372	−	0.166492i	0.348835	−	0.937184i	\(-0.386577\pi\)
							−0.637207	+	0.770692i	\(0.719911\pi\)
\(3\)	−0.600797	+	5.16130i	−0.115623	+	0.993293i
							\(5\)	2.50000	−	4.33013i	0.223607	−	0.387298i
\(7\)	13.9941	+	12.1312i	0.755608	+	0.655024i
							\(11\)	55.6111	−	32.1071i	1.52431	−	0.880059i	0.524720	−	0.851275i	\(-0.324170\pi\)
0.999586	−	0.0287839i	\(-0.00916347\pi\)
\(13\)			67.4233i			1.43845i	0.694777	+	0.719225i	\(0.255503\pi\)
							−0.694777	+	0.719225i	\(0.744497\pi\)
\(17\)	25.5674	+	44.2840i	0.364764	+	0.631791i	0.988738	−	0.149655i	\(-0.0478162\pi\)
							−0.623974	+	0.781445i	\(0.714483\pi\)
\(19\)	99.0631	+	57.1941i	1.19614	+	0.690591i	0.959692	−	0.281053i	\(-0.0906838\pi\)
							0.236447	+	0.971644i	\(0.424017\pi\)
\(23\)	74.0258	+	42.7388i	0.671106	+	0.387463i	0.796496	−	0.604644i	\(-0.206685\pi\)
							−0.125389	+	0.992108i	\(0.540018\pi\)
\(29\)		−	138.431i		−	0.886411i	−0.896420	−	0.443205i	\(-0.853841\pi\)
							0.896420	−	0.443205i	\(-0.146159\pi\)
\(31\)	−23.9598	+	13.8332i	−0.138816	+	0.0801455i	−0.567800	−	0.823167i	\(-0.692205\pi\)
							0.428984	+	0.903312i	\(0.358872\pi\)
\(37\)	63.3925	−	109.799i	0.281667	−	0.487861i	−0.690129	−	0.723687i	\(-0.742446\pi\)
							0.971795	+	0.235826i	\(0.0757794\pi\)
\(41\)	−72.8096			−0.277340			−0.138670	−	0.990339i	\(-0.544283\pi\)
							−0.138670	+	0.990339i	\(0.544283\pi\)
\(43\)	−550.446			−1.95215			−0.976073	−	0.217444i	\(-0.930228\pi\)
							−0.976073	+	0.217444i	\(0.930228\pi\)
\(47\)	86.4414	−	149.721i	0.268272	−	0.464660i	−0.700144	−	0.714002i	\(-0.746881\pi\)
							0.968416	+	0.249341i	\(0.0802142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.b.26.7	yes	32
3.2	odd	2		105.4.s.a.26.10	✓	32
7.3	odd	6		105.4.s.a.101.10	yes	32
21.17	even	6	inner	105.4.s.b.101.7	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.10	✓	32	3.2	odd	2
105.4.s.a.101.10	yes	32	7.3	odd	6
105.4.s.b.26.7	yes	32	1.1	even	1	trivial
105.4.s.b.101.7	yes	32	21.17	even	6	inner