Embedded newform 105.2.u.a.52.3

• Label: 105.2.u.a.52.3
• 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			52.3
Character	\(\chi\)	\(=\)	105.52
Dual form			105.2.u.a.103.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.401003 + 1.49657i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-0.346853 - 0.200256i) q^{4} +(2.14688 - 0.625221i) q^{5} +1.54936i q^{6} +(-1.01885 - 2.44171i) q^{7} +(-1.75234 + 1.75234i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.401003	+	1.49657i	−0.283552	+	1.05823i	0.666338	+	0.745649i	\(0.267861\pi\)
							−0.949891	+	0.312582i	\(0.898806\pi\)
\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
							\(5\)	2.14688	−	0.625221i	0.960115	−	0.279607i
\(7\)	−1.01885	−	2.44171i	−0.385088	−	0.922880i
							\(11\)	−2.59461	+	4.49400i	−0.782306	+	1.35499i	0.148290	+	0.988944i	\(0.452623\pi\)
−0.930595	+	0.366049i	\(0.880710\pi\)
\(13\)	−3.30901	−	3.30901i	−0.917755	−	0.917755i	0.0791106	−	0.996866i	\(-0.474792\pi\)
							−0.996866	+	0.0791106i	\(0.974792\pi\)
\(17\)	0.00519896	+	0.0194028i	0.00126093	+	0.00470587i	0.966553	−	0.256465i	\(-0.0825580\pi\)
							−0.965292	+	0.261171i	\(0.915891\pi\)
\(19\)	1.24048	+	2.14858i	0.284586	+	0.492918i	0.972509	−	0.232866i	\(-0.0748104\pi\)
							−0.687922	+	0.725784i	\(0.741477\pi\)
\(23\)	−2.24599	−	0.601811i	−0.468321	−	0.125486i	0.0169383	−	0.999857i	\(-0.494608\pi\)
							−0.485259	+	0.874370i	\(0.661275\pi\)
\(29\)		−	10.2081i		−	1.89559i	−0.318874	−	0.947797i	\(-0.603305\pi\)
							0.318874	−	0.947797i	\(-0.396695\pi\)
\(31\)	5.69268	+	3.28667i	1.02244	+	0.590303i	0.914808	−	0.403888i	\(-0.132342\pi\)
							0.107627	+	0.994191i	\(0.465675\pi\)
\(37\)	−0.714279	+	2.66573i	−0.117427	+	0.438243i	−0.999457	−	0.0329501i	\(-0.989510\pi\)
							0.882030	+	0.471193i	\(0.156176\pi\)
\(41\)			3.68910i			0.576141i	0.957609	+	0.288070i	\(0.0930137\pi\)
							−0.957609	+	0.288070i	\(0.906986\pi\)
\(43\)	−2.79725	+	2.79725i	−0.426576	+	0.426576i	−0.887460	−	0.460884i	\(-0.847532\pi\)
							0.460884	+	0.887460i	\(0.347532\pi\)
\(47\)	1.13152	+	0.303190i	0.165049	+	0.0442248i	0.340397	−	0.940282i	\(-0.389438\pi\)
							−0.175348	+	0.984506i	\(0.556105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.u.a.52.3	✓	32
3.2	odd	2		315.2.bz.d.262.6		32
5.2	odd	4		525.2.bc.e.493.3		32
5.3	odd	4	inner	105.2.u.a.73.6	yes	32
5.4	even	2		525.2.bc.e.157.6		32
7.2	even	3		735.2.v.b.607.6		32
7.3	odd	6		735.2.m.c.97.6		32
7.4	even	3		735.2.m.c.97.5		32
7.5	odd	6	inner	105.2.u.a.82.6	yes	32
7.6	odd	2		735.2.v.b.472.3		32
15.8	even	4		315.2.bz.d.73.3		32
21.5	even	6		315.2.bz.d.82.3		32
35.3	even	12		735.2.m.c.538.5		32
35.12	even	12		525.2.bc.e.418.6		32
35.13	even	4		735.2.v.b.178.6		32
35.18	odd	12		735.2.m.c.538.6		32
35.19	odd	6		525.2.bc.e.82.3		32
35.23	odd	12		735.2.v.b.313.3		32
35.33	even	12	inner	105.2.u.a.103.3	yes	32
105.68	odd	12		315.2.bz.d.208.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.3	✓	32	1.1	even	1	trivial
105.2.u.a.73.6	yes	32	5.3	odd	4	inner
105.2.u.a.82.6	yes	32	7.5	odd	6	inner
105.2.u.a.103.3	yes	32	35.33	even	12	inner
315.2.bz.d.73.3		32	15.8	even	4
315.2.bz.d.82.3		32	21.5	even	6
315.2.bz.d.208.6		32	105.68	odd	12
315.2.bz.d.262.6		32	3.2	odd	2
525.2.bc.e.82.3		32	35.19	odd	6
525.2.bc.e.157.6		32	5.4	even	2
525.2.bc.e.418.6		32	35.12	even	12
525.2.bc.e.493.3		32	5.2	odd	4
735.2.m.c.97.5		32	7.4	even	3
735.2.m.c.97.6		32	7.3	odd	6
735.2.m.c.538.5		32	35.3	even	12
735.2.m.c.538.6		32	35.18	odd	12
735.2.v.b.178.6		32	35.13	even	4
735.2.v.b.313.3		32	35.23	odd	12
735.2.v.b.472.3		32	7.6	odd	2
735.2.v.b.607.6		32	7.2	even	3