Embedded newform 105.2.u.a.73.5

• Label: 105.2.u.a.73.5
• Level: $105$
• Weight: $2$
• Character: 105.73
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.5
Character	\(\chi\)	\(=\)	105.73
Dual form			105.2.u.a.82.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.648264 + 0.173702i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-1.34198 - 0.774791i) q^{4} +(2.13259 - 0.672361i) q^{5} -0.671132i q^{6} +(2.57939 + 0.588837i) q^{7} +(-1.68450 - 1.68450i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.648264	+	0.173702i	0.458392	+	0.122826i	0.480623	−	0.876927i	\(-0.340410\pi\)
							−0.0222315	+	0.999753i	\(0.507077\pi\)
\(3\)	−0.258819	−	0.965926i	−0.149429	−	0.557678i
							\(5\)	2.13259	−	0.672361i	0.953722	−	0.300689i
\(7\)	2.57939	+	0.588837i	0.974919	+	0.222559i
							\(11\)	0.0701413	−	0.121488i	0.0211484	−	0.0366301i	−0.855258	−	0.518203i	\(-0.826601\pi\)
0.876406	+	0.481573i	\(0.159934\pi\)
\(13\)	−2.35788	+	2.35788i	−0.653958	+	0.653958i	−0.953944	−	0.299986i	\(-0.903018\pi\)
							0.299986	+	0.953944i	\(0.403018\pi\)
\(17\)	−7.37059	+	1.97494i	−1.78763	+	0.478994i	−0.991940	−	0.126710i	\(-0.959558\pi\)
							−0.795690	+	0.605704i	\(0.792892\pi\)
\(19\)	3.89475	+	6.74590i	0.893517	+	1.54762i	0.835630	+	0.549293i	\(0.185103\pi\)
							0.0578866	+	0.998323i	\(0.481564\pi\)
\(23\)	0.671527	−	2.50617i	0.140023	−	0.522573i	−0.859904	−	0.510456i	\(-0.829476\pi\)
							0.999927	−	0.0121164i	\(-0.00385688\pi\)
\(29\)		−	5.09095i		−	0.945365i	−0.881233	−	0.472683i	\(-0.843286\pi\)
							0.881233	−	0.472683i	\(-0.156714\pi\)
\(31\)	2.54499	+	1.46935i	0.457093	+	0.263903i	0.710821	−	0.703373i	\(-0.248324\pi\)
							−0.253728	+	0.967276i	\(0.581657\pi\)
\(37\)	−5.73455	−	1.53657i	−0.942754	−	0.252610i	−0.245469	−	0.969404i	\(-0.578942\pi\)
							−0.697285	+	0.716794i	\(0.745609\pi\)
\(41\)		−	0.261637i		−	0.0408609i	−0.999791	−	0.0204304i	\(-0.993496\pi\)
							0.999791	−	0.0204304i	\(-0.00650367\pi\)
\(43\)	−2.11921	−	2.11921i	−0.323177	−	0.323177i	0.526808	−	0.849984i	\(-0.323389\pi\)
							−0.849984	+	0.526808i	\(0.823389\pi\)
\(47\)	0.402594	−	1.50250i	0.0587244	−	0.219162i	−0.930328	−	0.366729i	\(-0.880478\pi\)
							0.989052	+	0.147567i	\(0.0471442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.u.a.73.5	yes	32
3.2	odd	2		315.2.bz.d.73.4		32
5.2	odd	4	inner	105.2.u.a.52.4	✓	32
5.3	odd	4		525.2.bc.e.157.5		32
5.4	even	2		525.2.bc.e.493.4		32
7.2	even	3		735.2.v.b.313.4		32
7.3	odd	6		735.2.m.c.538.8		32
7.4	even	3		735.2.m.c.538.7		32
7.5	odd	6	inner	105.2.u.a.103.4	yes	32
7.6	odd	2		735.2.v.b.178.5		32
15.2	even	4		315.2.bz.d.262.5		32
21.5	even	6		315.2.bz.d.208.5		32
35.2	odd	12		735.2.v.b.607.5		32
35.12	even	12	inner	105.2.u.a.82.5	yes	32
35.17	even	12		735.2.m.c.97.7		32
35.19	odd	6		525.2.bc.e.418.5		32
35.27	even	4		735.2.v.b.472.4		32
35.32	odd	12		735.2.m.c.97.8		32
35.33	even	12		525.2.bc.e.82.4		32
105.47	odd	12		315.2.bz.d.82.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.4	✓	32	5.2	odd	4	inner
105.2.u.a.73.5	yes	32	1.1	even	1	trivial
105.2.u.a.82.5	yes	32	35.12	even	12	inner
105.2.u.a.103.4	yes	32	7.5	odd	6	inner
315.2.bz.d.73.4		32	3.2	odd	2
315.2.bz.d.82.4		32	105.47	odd	12
315.2.bz.d.208.5		32	21.5	even	6
315.2.bz.d.262.5		32	15.2	even	4
525.2.bc.e.82.4		32	35.33	even	12
525.2.bc.e.157.5		32	5.3	odd	4
525.2.bc.e.418.5		32	35.19	odd	6
525.2.bc.e.493.4		32	5.4	even	2
735.2.m.c.97.7		32	35.17	even	12
735.2.m.c.97.8		32	35.32	odd	12
735.2.m.c.538.7		32	7.4	even	3
735.2.m.c.538.8		32	7.3	odd	6
735.2.v.b.178.5		32	7.6	odd	2
735.2.v.b.313.4		32	7.2	even	3
735.2.v.b.472.4		32	35.27	even	4
735.2.v.b.607.5		32	35.2	odd	12