Embedded newform 105.2.u.a.73.6

• Label: 105.2.u.a.73.6
• 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			73.6
Character	\(\chi\)	\(=\)	105.73
Dual form			105.2.u.a.82.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.49657 + 0.401003i) q^{2} +(0.258819 + 0.965926i) q^{3} +(0.346853 + 0.200256i) q^{4} +(1.61490 - 1.54664i) q^{5} +1.54936i q^{6} +(-2.44171 + 1.01885i) q^{7} +(-1.75234 - 1.75234i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.49657	+	0.401003i	1.05823	+	0.283552i	0.745649	−	0.666338i	\(-0.232139\pi\)
							0.312582	+	0.949891i	\(0.398806\pi\)
\(3\)	0.258819	+	0.965926i	0.149429	+	0.557678i
							\(5\)	1.61490	−	1.54664i	0.722204	−	0.691680i
\(7\)	−2.44171	+	1.01885i	−0.922880	+	0.385088i
							\(11\)	−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	\(-0.525208\pi\)
							0.996866	+	0.0791106i	\(0.0252080\pi\)
\(17\)	0.0194028	−	0.00519896i	0.00470587	−	0.00126093i	−0.256465	−	0.966553i	\(-0.582558\pi\)
							0.261171	+	0.965292i	\(0.415891\pi\)
\(19\)	−1.24048	−	2.14858i	−0.284586	−	0.492918i	0.687922	−	0.725784i	\(-0.258523\pi\)
							−0.972509	+	0.232866i	\(0.925190\pi\)
\(23\)	0.601811	−	2.24599i	0.125486	−	0.468321i	−0.874370	−	0.485259i	\(-0.838725\pi\)
							0.999857	+	0.0169383i	\(0.00539189\pi\)
\(29\)			10.2081i			1.89559i	0.318874	+	0.947797i	\(0.396695\pi\)
							−0.318874	+	0.947797i	\(0.603305\pi\)
\(31\)	5.69268	+	3.28667i	1.02244	+	0.590303i	0.914808	−	0.403888i	\(-0.132342\pi\)
							0.107627	+	0.994191i	\(0.465675\pi\)
\(37\)	2.66573	+	0.714279i	0.438243	+	0.117427i	0.471193	−	0.882030i	\(-0.343824\pi\)
							−0.0329501	+	0.999457i	\(0.510490\pi\)
\(41\)			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	0.460884	−	0.887460i	\(-0.347532\pi\)
							−0.887460	+	0.460884i	\(0.847532\pi\)
\(47\)	0.303190	−	1.13152i	0.0442248	−	0.165049i	−0.940282	−	0.340397i	\(-0.889438\pi\)
							0.984506	+	0.175348i	\(0.0561051\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.3	✓	32	5.2	odd	4	inner
105.2.u.a.73.6	yes	32	1.1	even	1	trivial
105.2.u.a.82.6	yes	32	35.12	even	12	inner
105.2.u.a.103.3	yes	32	7.5	odd	6	inner
315.2.bz.d.73.3		32	3.2	odd	2
315.2.bz.d.82.3		32	105.47	odd	12
315.2.bz.d.208.6		32	21.5	even	6
315.2.bz.d.262.6		32	15.2	even	4
525.2.bc.e.82.3		32	35.33	even	12
525.2.bc.e.157.6		32	5.3	odd	4
525.2.bc.e.418.6		32	35.19	odd	6
525.2.bc.e.493.3		32	5.4	even	2
735.2.m.c.97.5		32	35.32	odd	12
735.2.m.c.97.6		32	35.17	even	12
735.2.m.c.538.5		32	7.3	odd	6
735.2.m.c.538.6		32	7.4	even	3
735.2.v.b.178.6		32	7.6	odd	2
735.2.v.b.313.3		32	7.2	even	3
735.2.v.b.472.3		32	35.27	even	4
735.2.v.b.607.6		32	35.2	odd	12