Embedded newform 105.2.u.a.52.7

• Label: 105.2.u.a.52.7
• Level: $105$
• Weight: $2$
• Character: 105.52
• 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			52.7
Character	\(\chi\)	\(=\)	105.52
Dual form			105.2.u.a.103.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.602389 - 2.24814i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-2.95923 - 1.70851i) q^{4} +(-1.28534 + 1.82973i) q^{5} -2.32745i q^{6} +(0.519864 - 2.59417i) q^{7} +(-2.33208 + 2.33208i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.602389	−	2.24814i	0.425953	−	1.58968i	−0.335880	−	0.941905i	\(-0.609034\pi\)
							0.761833	−	0.647774i	\(-0.224300\pi\)
\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
							\(5\)	−1.28534	+	1.82973i	−0.574820	+	0.818280i
\(7\)	0.519864	−	2.59417i	0.196490	−	0.980506i
							\(11\)	−1.76389	+	3.05515i	−0.531834	+	0.921163i	0.467476	+	0.884006i	\(0.345163\pi\)
−0.999309	+	0.0371569i	\(0.988170\pi\)
\(13\)	4.49057	+	4.49057i	1.24546	+	1.24546i	0.957704	+	0.287756i	\(0.0929093\pi\)
							0.287756	+	0.957704i	\(0.407091\pi\)
\(17\)	0.481759	+	1.79795i	0.116844	+	0.436066i	0.999418	−	0.0341060i	\(-0.0108584\pi\)
							−0.882575	+	0.470172i	\(0.844192\pi\)
\(19\)	−0.0699116	−	0.121090i	−0.0160388	−	0.0277800i	0.857895	−	0.513826i	\(-0.171772\pi\)
							−0.873933	+	0.486046i	\(0.838439\pi\)
\(23\)	−3.72112	−	0.997072i	−0.775908	−	0.207904i	−0.150928	−	0.988545i	\(-0.548226\pi\)
							−0.624980	+	0.780641i	\(0.714893\pi\)
\(29\)		−	2.01969i		−	0.375047i	−0.982260	−	0.187524i	\(-0.939954\pi\)
							0.982260	−	0.187524i	\(-0.0600462\pi\)
\(31\)	−4.56612	−	2.63625i	−0.820100	−	0.473485i	0.0303510	−	0.999539i	\(-0.490337\pi\)
							−0.850451	+	0.526054i	\(0.823671\pi\)
\(37\)	−1.50406	+	5.61323i	−0.247266	+	0.922810i	0.724965	+	0.688786i	\(0.241856\pi\)
							−0.972231	+	0.234024i	\(0.924811\pi\)
\(41\)			0.903323i			0.141075i	0.997509	+	0.0705377i	\(0.0224715\pi\)
							−0.997509	+	0.0705377i	\(0.977529\pi\)
\(43\)	2.38469	−	2.38469i	0.363663	−	0.363663i	−0.501497	−	0.865159i	\(-0.667217\pi\)
							0.865159	+	0.501497i	\(0.167217\pi\)
\(47\)	−2.38655	−	0.639474i	−0.348114	−	0.0932768i	0.0805254	−	0.996753i	\(-0.474340\pi\)
							−0.428639	+	0.903476i	\(0.641007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.u.a.52.7	✓	32
3.2	odd	2		315.2.bz.d.262.2		32
5.2	odd	4		525.2.bc.e.493.7		32
5.3	odd	4	inner	105.2.u.a.73.2	yes	32
5.4	even	2		525.2.bc.e.157.2		32
7.2	even	3		735.2.v.b.607.2		32
7.3	odd	6		735.2.m.c.97.14		32
7.4	even	3		735.2.m.c.97.13		32
7.5	odd	6	inner	105.2.u.a.82.2	yes	32
7.6	odd	2		735.2.v.b.472.7		32
15.8	even	4		315.2.bz.d.73.7		32
21.5	even	6		315.2.bz.d.82.7		32
35.3	even	12		735.2.m.c.538.13		32
35.12	even	12		525.2.bc.e.418.2		32
35.13	even	4		735.2.v.b.178.2		32
35.18	odd	12		735.2.m.c.538.14		32
35.19	odd	6		525.2.bc.e.82.7		32
35.23	odd	12		735.2.v.b.313.7		32
35.33	even	12	inner	105.2.u.a.103.7	yes	32
105.68	odd	12		315.2.bz.d.208.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.7	✓	32	1.1	even	1	trivial
105.2.u.a.73.2	yes	32	5.3	odd	4	inner
105.2.u.a.82.2	yes	32	7.5	odd	6	inner
105.2.u.a.103.7	yes	32	35.33	even	12	inner
315.2.bz.d.73.7		32	15.8	even	4
315.2.bz.d.82.7		32	21.5	even	6
315.2.bz.d.208.2		32	105.68	odd	12
315.2.bz.d.262.2		32	3.2	odd	2
525.2.bc.e.82.7		32	35.19	odd	6
525.2.bc.e.157.2		32	5.4	even	2
525.2.bc.e.418.2		32	35.12	even	12
525.2.bc.e.493.7		32	5.2	odd	4
735.2.m.c.97.13		32	7.4	even	3
735.2.m.c.97.14		32	7.3	odd	6
735.2.m.c.538.13		32	35.3	even	12
735.2.m.c.538.14		32	35.18	odd	12
735.2.v.b.178.2		32	35.13	even	4
735.2.v.b.313.7		32	35.23	odd	12
735.2.v.b.472.7		32	7.6	odd	2
735.2.v.b.607.2		32	7.2	even	3