Embedded newform 105.3.v.a.88.4

• Label: 105.3.v.a.88.4
• Level: $105$
• Weight: $3$
• Character: 105.88
• 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			88.4
Character	\(\chi\)	\(=\)	105.88
Dual form			105.3.v.a.37.4

$q$-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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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	−0.391184	−	0.920312i	\(-0.627935\pi\)
							−0.798932	+	0.601422i	\(0.794601\pi\)
\(3\)	−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	−0.548075	−	0.836429i	\(-0.684639\pi\)
0.998406	+	0.0564323i	\(0.0179725\pi\)
\(13\)	−14.1612	−	14.1612i	−1.08932	−	1.08932i	−0.995598	−	0.0937246i	\(-0.970123\pi\)
							−0.0937246	−	0.995598i	\(-0.529877\pi\)
\(17\)	−4.22139	−	15.7544i	−0.248317	−	0.926731i	−0.971687	−	0.236271i	\(-0.924075\pi\)
							0.723370	−	0.690460i	\(-0.242592\pi\)
\(19\)	23.3583	−	13.4859i	1.22939	−	0.709787i	0.262485	−	0.964936i	\(-0.415458\pi\)
							0.966902	+	0.255150i	\(0.0821247\pi\)
\(23\)	1.79608	−	6.70305i	0.0780903	−	0.291437i	−0.915826	−	0.401575i	\(-0.868463\pi\)
							0.993916	+	0.110138i	\(0.0351294\pi\)
\(29\)			18.5702i			0.640350i	0.947358	+	0.320175i	\(0.103742\pi\)
							−0.947358	+	0.320175i	\(0.896258\pi\)
\(31\)	19.6183	−	33.9798i	0.632847	−	1.09612i	−0.354120	−	0.935200i	\(-0.615220\pi\)
							0.986967	−	0.160923i	\(-0.0514470\pi\)
\(37\)	42.8463	+	11.4806i	1.15801	+	0.310288i	0.786170	−	0.618010i	\(-0.212061\pi\)
							0.371839	+	0.928297i	\(0.378727\pi\)
\(41\)	−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	0.150331	−	0.988636i	\(-0.451966\pi\)
							−0.988636	+	0.150331i	\(0.951966\pi\)
\(47\)	24.5013	+	6.56509i	0.521303	+	0.139683i	0.509869	−	0.860252i	\(-0.329694\pi\)
							0.0114339	+	0.999935i	\(0.496360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.88.4	yes	64
3.2	odd	2		315.3.ca.b.298.13		64
5.2	odd	4	inner	105.3.v.a.67.13	yes	64
7.2	even	3	inner	105.3.v.a.58.13	yes	64
15.2	even	4		315.3.ca.b.172.4		64
21.2	odd	6		315.3.ca.b.163.4		64
35.2	odd	12	inner	105.3.v.a.37.4	✓	64
105.2	even	12		315.3.ca.b.37.13		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.4	✓	64	35.2	odd	12	inner
105.3.v.a.58.13	yes	64	7.2	even	3	inner
105.3.v.a.67.13	yes	64	5.2	odd	4	inner
105.3.v.a.88.4	yes	64	1.1	even	1	trivial
315.3.ca.b.37.13		64	105.2	even	12
315.3.ca.b.163.4		64	21.2	odd	6
315.3.ca.b.172.4		64	15.2	even	4
315.3.ca.b.298.13		64	3.2	odd	2