Embedded newform 105.4.p.a.59.13

• Label: 105.4.p.a.59.13
• 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			59.13
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29175 - 2.23737i) q^{2} +(-4.90178 - 1.72412i) q^{3} +(0.662782 - 1.14797i) q^{4} +(-9.00517 - 6.62623i) q^{5} +(2.47436 + 13.1942i) q^{6} +(1.81286 - 18.4313i) q^{7} -24.0925 q^{8} +(21.0548 + 16.9025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29175	−	2.23737i	−0.456701	−	0.791030i	0.542083	−	0.840325i	\(-0.317636\pi\)
							−0.998784	+	0.0492950i	\(0.984303\pi\)
\(3\)	−4.90178	−	1.72412i	−0.943347	−	0.331807i
							\(5\)	−9.00517	−	6.62623i	−0.805447	−	0.592668i
\(7\)	1.81286	−	18.4313i	0.0978853	−	0.995198i
							\(11\)	10.5104	+	6.06820i	0.288092	+	0.166330i	0.637081	−	0.770797i	\(-0.280142\pi\)
−0.348989	+	0.937127i	\(0.613475\pi\)
\(13\)	−32.6517			−0.696613			−0.348306	−	0.937381i	\(-0.613243\pi\)
							−0.348306	+	0.937381i	\(0.613243\pi\)
\(17\)	66.2469	+	38.2477i	0.945131	+	0.545672i	0.891565	−	0.452892i	\(-0.149608\pi\)
							0.0535662	+	0.998564i	\(0.482941\pi\)
\(19\)	−41.3069	+	23.8486i	−0.498761	+	0.287960i	−0.728202	−	0.685363i	\(-0.759644\pi\)
							0.229441	+	0.973323i	\(0.426310\pi\)
\(23\)	34.5475	+	59.8381i	0.313203	+	0.542483i	0.979054	−	0.203602i	\(-0.0652648\pi\)
							−0.665851	+	0.746085i	\(0.731931\pi\)
\(29\)		−	135.453i		−	0.867343i	−0.901071	−	0.433671i	\(-0.857218\pi\)
							0.901071	−	0.433671i	\(-0.142782\pi\)
\(31\)	−142.890	−	82.4973i	−0.827862	−	0.477966i	0.0252582	−	0.999681i	\(-0.491959\pi\)
							−0.853120	+	0.521715i	\(0.825293\pi\)
\(37\)	71.4057	−	41.2261i	0.317271	−	0.183176i	−0.332905	−	0.942960i	\(-0.608029\pi\)
							0.650175	+	0.759784i	\(0.274695\pi\)
\(41\)	−411.728			−1.56832			−0.784160	−	0.620559i	\(-0.786906\pi\)
							−0.784160	+	0.620559i	\(0.786906\pi\)
\(43\)		−	432.567i		−	1.53409i	−0.641594	−	0.767045i	\(-0.721726\pi\)
							0.641594	−	0.767045i	\(-0.278274\pi\)
\(47\)	−179.518	+	103.645i	−0.557136	+	0.321662i	−0.751995	−	0.659169i	\(-0.770908\pi\)
							0.194859	+	0.980831i	\(0.437575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.13	✓	88
3.2	odd	2	inner	105.4.p.a.59.31	yes	88
5.4	even	2	inner	105.4.p.a.59.32	yes	88
7.5	odd	6	inner	105.4.p.a.89.14	yes	88
15.14	odd	2	inner	105.4.p.a.59.14	yes	88
21.5	even	6	inner	105.4.p.a.89.32	yes	88
35.19	odd	6	inner	105.4.p.a.89.31	yes	88
105.89	even	6	inner	105.4.p.a.89.13	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.13	✓	88	1.1	even	1	trivial
105.4.p.a.59.14	yes	88	15.14	odd	2	inner
105.4.p.a.59.31	yes	88	3.2	odd	2	inner
105.4.p.a.59.32	yes	88	5.4	even	2	inner
105.4.p.a.89.13	yes	88	105.89	even	6	inner
105.4.p.a.89.14	yes	88	7.5	odd	6	inner
105.4.p.a.89.31	yes	88	35.19	odd	6	inner
105.4.p.a.89.32	yes	88	21.5	even	6	inner