Embedded newform 105.3.v.a.37.15

• Label: 105.3.v.a.37.15
• Level: $105$
• Weight: $3$
• Character: 105.37
• 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			37.15
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.41166 - 0.914152i) q^{2} +(1.67303 + 0.448288i) q^{3} +(7.33966 - 4.23756i) q^{4} +(-4.98672 + 0.364163i) q^{5} +6.11763 q^{6} +(-6.35750 - 2.92953i) q^{7} +(11.1766 - 11.1766i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.41166	−	0.914152i	1.70583	−	0.457076i	0.731435	−	0.681912i	\(-0.238851\pi\)
							0.974397	+	0.224836i	\(0.0721845\pi\)
\(3\)	1.67303	+	0.448288i	0.557678	+	0.149429i
							\(5\)	−4.98672	+	0.364163i	−0.997344	+	0.0728326i
\(7\)	−6.35750	−	2.92953i	−0.908215	−	0.418505i
							\(11\)	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	−0.957919	−	0.287039i	\(-0.907329\pi\)
							0.287039	+	0.957919i	\(0.407329\pi\)
\(17\)	3.02215	−	11.2788i	0.177774	−	0.663461i	−0.818289	−	0.574807i	\(-0.805077\pi\)
							0.996063	−	0.0886535i	\(-0.0282564\pi\)
\(19\)	−3.12903	−	1.80655i	−0.164686	−	0.0950814i	0.415392	−	0.909643i	\(-0.363644\pi\)
							−0.580078	+	0.814561i	\(0.696978\pi\)
\(23\)	−6.21349	−	23.1890i	−0.270152	−	1.00822i	−0.959021	−	0.283334i	\(-0.908559\pi\)
							0.688870	−	0.724885i	\(-0.258107\pi\)
\(29\)		−	46.5831i		−	1.60631i	−0.595767	−	0.803157i	\(-0.703152\pi\)
							0.595767	−	0.803157i	\(-0.296848\pi\)
\(31\)	−1.10507	−	1.91403i	−0.0356474	−	0.0617430i	0.847651	−	0.530554i	\(-0.178016\pi\)
							−0.883299	+	0.468811i	\(0.844683\pi\)
\(37\)	−29.3791	+	7.87211i	−0.794030	+	0.212760i	−0.632961	−	0.774183i	\(-0.718161\pi\)
							−0.161069	+	0.986943i	\(0.551494\pi\)
\(41\)	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	−0.897869	−	0.440262i	\(-0.854886\pi\)
							0.440262	+	0.897869i	\(0.354886\pi\)
\(47\)	79.1503	−	21.2083i	1.68405	−	0.451240i	0.715206	−	0.698913i	\(-0.246333\pi\)
							0.968844	+	0.247674i	\(0.0796661\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.15	✓	64	1.1	even	1	trivial
105.3.v.a.58.2	yes	64	5.3	odd	4	inner
105.3.v.a.67.2	yes	64	7.4	even	3	inner
105.3.v.a.88.15	yes	64	35.18	odd	12	inner
315.3.ca.b.37.2		64	3.2	odd	2
315.3.ca.b.163.15		64	15.8	even	4
315.3.ca.b.172.15		64	21.11	odd	6
315.3.ca.b.298.2		64	105.53	even	12