Embedded newform 105.3.v.a.58.2

• Label: 105.3.v.a.58.2
• Level: $105$
• Weight: $3$
• Character: 105.58
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			58.2
Character	\(\chi\)	\(=\)	105.58
Dual form			105.3.v.a.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.914152 - 3.41166i) q^{2} +(-0.448288 + 1.67303i) q^{3} +(-7.33966 + 4.23756i) q^{4} +(2.17799 + 4.50071i) q^{5} +6.11763 q^{6} +(-2.92953 + 6.35750i) q^{7} +(11.1766 + 11.1766i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.914152	−	3.41166i	−0.457076	−	1.70583i	−0.681912	−	0.731435i	\(-0.738851\pi\)
							0.224836	−	0.974397i	\(-0.427815\pi\)
\(3\)	−0.448288	+	1.67303i	−0.149429	+	0.557678i
							\(5\)	2.17799	+	4.50071i	0.435597	+	0.900142i
\(7\)	−2.92953	+	6.35750i	−0.418505	+	0.908215i
							\(11\)	8.72135	+	15.1058i	0.792850	+	1.37326i	0.924195	+	0.381920i	\(0.124737\pi\)
−0.131345	+	0.991337i	\(0.541930\pi\)
\(13\)	−8.72144	−	8.72144i	−0.670880	−	0.670880i	0.287039	−	0.957919i	\(-0.407329\pi\)
							−0.957919	+	0.287039i	\(0.907329\pi\)
\(17\)	−11.2788	−	3.02215i	−0.663461	−	0.177774i	−0.0886535	−	0.996063i	\(-0.528256\pi\)
							−0.574807	+	0.818289i	\(0.694923\pi\)
\(19\)	3.12903	+	1.80655i	0.164686	+	0.0950814i	0.580078	−	0.814561i	\(-0.303022\pi\)
							−0.415392	+	0.909643i	\(0.636356\pi\)
\(23\)	23.1890	−	6.21349i	1.00822	−	0.270152i	0.283334	−	0.959021i	\(-0.408559\pi\)
							0.724885	+	0.688870i	\(0.241893\pi\)
\(29\)			46.5831i			1.60631i	0.595767	+	0.803157i	\(0.296848\pi\)
							−0.595767	+	0.803157i	\(0.703152\pi\)
\(31\)	−1.10507	−	1.91403i	−0.0356474	−	0.0617430i	0.847651	−	0.530554i	\(-0.178016\pi\)
							−0.883299	+	0.468811i	\(0.844683\pi\)
\(37\)	7.87211	+	29.3791i	0.212760	+	0.794030i	0.986943	+	0.161069i	\(0.0514942\pi\)
							−0.774183	+	0.632961i	\(0.781839\pi\)
\(41\)	29.9435			0.730329			0.365165	−	0.930943i	\(-0.381013\pi\)
							0.365165	+	0.930943i	\(0.381013\pi\)
\(43\)	−19.6771	−	19.6771i	−0.457608	−	0.457608i	0.440262	−	0.897869i	\(-0.354886\pi\)
							−0.897869	+	0.440262i	\(0.854886\pi\)
\(47\)	−21.2083	−	79.1503i	−0.451240	−	1.68405i	−0.698913	−	0.715206i	\(-0.746333\pi\)
							0.247674	−	0.968844i	\(-0.420334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.58.2	yes	64
3.2	odd	2		315.3.ca.b.163.15		64
5.2	odd	4	inner	105.3.v.a.37.15	✓	64
7.4	even	3	inner	105.3.v.a.88.15	yes	64
15.2	even	4		315.3.ca.b.37.2		64
21.11	odd	6		315.3.ca.b.298.2		64
35.32	odd	12	inner	105.3.v.a.67.2	yes	64
105.32	even	12		315.3.ca.b.172.15		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.15	✓	64	5.2	odd	4	inner
105.3.v.a.58.2	yes	64	1.1	even	1	trivial
105.3.v.a.67.2	yes	64	35.32	odd	12	inner
105.3.v.a.88.15	yes	64	7.4	even	3	inner
315.3.ca.b.37.2		64	15.2	even	4
315.3.ca.b.163.15		64	3.2	odd	2
315.3.ca.b.172.15		64	105.32	even	12
315.3.ca.b.298.2		64	21.11	odd	6