Embedded newform 105.4.p.a.59.6

• Label: 105.4.p.a.59.6
• Level: $105$
• Weight: $4$
• Character: 105.59
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.6
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.26674 - 3.92610i) q^{2} +(-2.11487 - 4.74630i) q^{3} +(-6.27618 + 10.8707i) q^{4} +(5.01525 - 9.99236i) q^{5} +(-13.8406 + 19.0618i) q^{6} +(-18.5202 - 0.0273336i) q^{7} +20.6380 q^{8} +(-18.0547 + 20.0756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.26674	−	3.92610i	−0.801412	−	1.38809i	−0.918687	−	0.394987i	\(-0.870749\pi\)
							0.117275	−	0.993099i	\(-0.462584\pi\)
\(3\)	−2.11487	−	4.74630i	−0.407007	−	0.913425i
							\(5\)	5.01525	−	9.99236i	0.448577	−	0.893744i
\(7\)	−18.5202	−	0.0273336i	−0.999999	−	0.00147588i
							\(11\)	−7.20183	−	4.15798i	−0.197403	−	0.113971i	0.398041	−	0.917368i	\(-0.369690\pi\)
−0.595444	+	0.803397i	\(0.703024\pi\)
\(13\)	18.3690			0.391896			0.195948	−	0.980614i	\(-0.437222\pi\)
							0.195948	+	0.980614i	\(0.437222\pi\)
\(17\)	−28.7352	−	16.5903i	−0.409960	−	0.236691i	0.280813	−	0.959763i	\(-0.409396\pi\)
							−0.690773	+	0.723072i	\(0.742729\pi\)
\(19\)	108.334	−	62.5466i	1.30808	−	0.755220i	0.326304	−	0.945265i	\(-0.394197\pi\)
							0.981775	+	0.190045i	\(0.0608633\pi\)
\(23\)	−10.5542	−	18.2803i	−0.0956823	−	0.165727i	0.814211	−	0.580569i	\(-0.197170\pi\)
							−0.909893	+	0.414843i	\(0.863837\pi\)
\(29\)			236.100i			1.51181i	0.654679	+	0.755907i	\(0.272804\pi\)
							−0.654679	+	0.755907i	\(0.727196\pi\)
\(31\)	−117.849	−	68.0399i	−0.682781	−	0.394204i	0.118121	−	0.992999i	\(-0.462313\pi\)
							−0.800902	+	0.598795i	\(0.795646\pi\)
\(37\)	−335.069	+	193.452i	−1.48879	+	0.859550i	−0.999918	−	0.0128079i	\(-0.995923\pi\)
							−0.488867	+	0.872358i	\(0.662590\pi\)
\(41\)	−359.300			−1.36861			−0.684307	−	0.729194i	\(-0.739895\pi\)
							−0.684307	+	0.729194i	\(0.739895\pi\)
\(43\)		−	148.175i		−	0.525501i	−0.964864	−	0.262750i	\(-0.915370\pi\)
							0.964864	−	0.262750i	\(-0.0846296\pi\)
\(47\)	−257.466	+	148.648i	−0.799048	+	0.461330i	−0.843138	−	0.537697i	\(-0.819294\pi\)
							0.0440904	+	0.999028i	\(0.485961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.6	yes	88
3.2	odd	2	inner	105.4.p.a.59.40	yes	88
5.4	even	2	inner	105.4.p.a.59.39	yes	88
7.5	odd	6	inner	105.4.p.a.89.5	yes	88
15.14	odd	2	inner	105.4.p.a.59.5	✓	88
21.5	even	6	inner	105.4.p.a.89.39	yes	88
35.19	odd	6	inner	105.4.p.a.89.40	yes	88
105.89	even	6	inner	105.4.p.a.89.6	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.5	✓	88	15.14	odd	2	inner
105.4.p.a.59.6	yes	88	1.1	even	1	trivial
105.4.p.a.59.39	yes	88	5.4	even	2	inner
105.4.p.a.59.40	yes	88	3.2	odd	2	inner
105.4.p.a.89.5	yes	88	7.5	odd	6	inner
105.4.p.a.89.6	yes	88	105.89	even	6	inner
105.4.p.a.89.39	yes	88	21.5	even	6	inner
105.4.p.a.89.40	yes	88	35.19	odd	6	inner