Embedded newform 105.3.v.a.37.4

• Label: 105.3.v.a.37.4
• 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			37.4
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38023 + 0.637781i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.79464 - 1.03614i) q^{4} +(2.91424 - 4.06291i) q^{5} +4.26812 q^{6} +(-4.25379 + 5.55925i) q^{7} +(3.35897 - 3.35897i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38023	+	0.637781i	−1.19012	+	0.318891i	−0.798932	−	0.601422i	\(-0.794601\pi\)
							−0.391184	+	0.920312i	\(0.627935\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	2.91424	−	4.06291i	0.582848	−	0.812581i
\(7\)	−4.25379	+	5.55925i	−0.607684	+	0.794179i
							\(11\)	4.95365	+	8.57997i	0.450331	+	0.779997i	0.998406	−	0.0564323i	\(-0.0179725\pi\)
−0.548075	+	0.836429i	\(0.684639\pi\)
\(13\)	−14.1612	+	14.1612i	−1.08932	+	1.08932i	−0.0937246	+	0.995598i	\(0.529877\pi\)
							−0.995598	+	0.0937246i	\(0.970123\pi\)
\(17\)	−4.22139	+	15.7544i	−0.248317	+	0.926731i	0.723370	+	0.690460i	\(0.242592\pi\)
							−0.971687	+	0.236271i	\(0.924075\pi\)
\(19\)	23.3583	+	13.4859i	1.22939	+	0.709787i	0.966902	−	0.255150i	\(-0.0821247\pi\)
							0.262485	+	0.964936i	\(0.415458\pi\)
\(23\)	1.79608	+	6.70305i	0.0780903	+	0.291437i	0.993916	−	0.110138i	\(-0.0351294\pi\)
							−0.915826	+	0.401575i	\(0.868463\pi\)
\(29\)		−	18.5702i		−	0.640350i	−0.947358	−	0.320175i	\(-0.896258\pi\)
							0.947358	−	0.320175i	\(-0.103742\pi\)
\(31\)	19.6183	+	33.9798i	0.632847	+	1.09612i	0.986967	+	0.160923i	\(0.0514470\pi\)
							−0.354120	+	0.935200i	\(0.615220\pi\)
\(37\)	42.8463	−	11.4806i	1.15801	−	0.310288i	0.371839	−	0.928297i	\(-0.378727\pi\)
							0.786170	+	0.618010i	\(0.212061\pi\)
\(41\)	−55.1266			−1.34455			−0.672275	−	0.740301i	\(-0.734683\pi\)
							−0.672275	+	0.740301i	\(0.734683\pi\)
\(43\)	−36.0471	+	36.0471i	−0.838304	+	0.838304i	−0.988636	−	0.150331i	\(-0.951966\pi\)
							0.150331	+	0.988636i	\(0.451966\pi\)
\(47\)	24.5013	−	6.56509i	0.521303	−	0.139683i	0.0114339	−	0.999935i	\(-0.496360\pi\)
							0.509869	+	0.860252i	\(0.329694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.4	✓	64
3.2	odd	2		315.3.ca.b.37.13		64
5.3	odd	4	inner	105.3.v.a.58.13	yes	64
7.4	even	3	inner	105.3.v.a.67.13	yes	64
15.8	even	4		315.3.ca.b.163.4		64
21.11	odd	6		315.3.ca.b.172.4		64
35.18	odd	12	inner	105.3.v.a.88.4	yes	64
105.53	even	12		315.3.ca.b.298.13		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.4	✓	64	1.1	even	1	trivial
105.3.v.a.58.13	yes	64	5.3	odd	4	inner
105.3.v.a.67.13	yes	64	7.4	even	3	inner
105.3.v.a.88.4	yes	64	35.18	odd	12	inner
315.3.ca.b.37.13		64	3.2	odd	2
315.3.ca.b.163.4		64	15.8	even	4
315.3.ca.b.172.4		64	21.11	odd	6
315.3.ca.b.298.13		64	105.53	even	12