Embedded newform 735.2.p.f.374.9

• Label: 735.2.p.f.374.9
• Level: $735$
• Weight: $2$
• Character: 735.374
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			374.9
Character	\(\chi\)	\(=\)	735.374
Dual form			735.2.p.f.509.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.757344 + 1.31176i) q^{2} +(-1.24551 + 1.20362i) q^{3} +(-0.147140 + 0.254854i) q^{4} +(2.20411 - 0.376714i) q^{5} +(-2.52214 - 0.722254i) q^{6} +2.58363 q^{8} +(0.102593 - 2.99825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.757344	+	1.31176i	0.535523	+	0.927553i	0.999138	+	0.0415164i	\(0.0132189\pi\)
							−0.463615	+	0.886037i	\(0.653448\pi\)
\(3\)	−1.24551	+	1.20362i	−0.719096	+	0.694911i
							\(5\)	2.20411	−	0.376714i	0.985706	−	0.168472i
\(7\)		0			0
							\(11\)	−1.86048	−	1.07415i	−0.560957	−	0.323869i	0.192573	−	0.981283i	\(-0.438317\pi\)
−0.753529	+	0.657414i	\(0.771650\pi\)
\(13\)	3.48097			0.965448			0.482724	−	0.875773i	\(-0.339647\pi\)
							0.482724	+	0.875773i	\(0.339647\pi\)
\(17\)	3.09793	+	1.78859i	0.751359	+	0.433797i	0.826185	−	0.563399i	\(-0.190507\pi\)
							−0.0748259	+	0.997197i	\(0.523840\pi\)
\(19\)	1.05858	−	0.611171i	0.242855	−	0.140212i	−0.373633	−	0.927576i	\(-0.621888\pi\)
							0.616488	+	0.787364i	\(0.288555\pi\)
\(23\)	−0.757344	−	1.31176i	−0.157917	−	0.273521i	0.776200	−	0.630486i	\(-0.217145\pi\)
							−0.934117	+	0.356966i	\(0.883811\pi\)
\(29\)			5.95645i			1.10608i	0.833153	+	0.553042i	\(0.186533\pi\)
							−0.833153	+	0.553042i	\(0.813467\pi\)
\(31\)	−2.75098	−	1.58828i	−0.494091	−	0.285263i	0.232179	−	0.972673i	\(-0.425414\pi\)
							−0.726270	+	0.687410i	\(0.758748\pi\)
\(37\)	6.75803	−	3.90175i	1.11101	−	0.641444i	0.171921	−	0.985111i	\(-0.445002\pi\)
							0.939092	+	0.343667i	\(0.111669\pi\)
\(41\)	−11.8685			−1.85355			−0.926773	−	0.375622i	\(-0.877429\pi\)
							−0.926773	+	0.375622i	\(0.877429\pi\)
\(43\)			2.99294i			0.456419i	0.973612	+	0.228209i	\(0.0732871\pi\)
							−0.973612	+	0.228209i	\(0.926713\pi\)
\(47\)	−5.28420	+	3.05084i	−0.770780	+	0.445010i	−0.833153	−	0.553043i	\(-0.813467\pi\)
							0.0623727	+	0.998053i	\(0.480133\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.p.f.374.9		24
3.2	odd	2	inner	735.2.p.f.374.3		24
5.4	even	2	inner	735.2.p.f.374.4		24
7.2	even	3		105.2.p.a.89.9	yes	24
7.3	odd	6		735.2.g.b.734.5		24
7.4	even	3		735.2.g.b.734.8		24
7.5	odd	6	inner	735.2.p.f.509.10		24
7.6	odd	2		105.2.p.a.59.10	yes	24
15.14	odd	2	inner	735.2.p.f.374.10		24
21.2	odd	6		105.2.p.a.89.3	yes	24
21.5	even	6	inner	735.2.p.f.509.4		24
21.11	odd	6		735.2.g.b.734.19		24
21.17	even	6		735.2.g.b.734.18		24
21.20	even	2		105.2.p.a.59.4	yes	24
35.2	odd	12		525.2.t.j.26.9		24
35.4	even	6		735.2.g.b.734.17		24
35.9	even	6		105.2.p.a.89.4	yes	24
35.13	even	4		525.2.t.j.101.10		24
35.19	odd	6	inner	735.2.p.f.509.3		24
35.23	odd	12		525.2.t.j.26.4		24
35.24	odd	6		735.2.g.b.734.20		24
35.27	even	4		525.2.t.j.101.3		24
35.34	odd	2		105.2.p.a.59.3	✓	24
105.2	even	12		525.2.t.j.26.3		24
105.23	even	12		525.2.t.j.26.10		24
105.44	odd	6		105.2.p.a.89.10	yes	24
105.59	even	6		735.2.g.b.734.7		24
105.62	odd	4		525.2.t.j.101.9		24
105.74	odd	6		735.2.g.b.734.6		24
105.83	odd	4		525.2.t.j.101.4		24
105.89	even	6	inner	735.2.p.f.509.9		24
105.104	even	2		105.2.p.a.59.9	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.p.a.59.3	✓	24	35.34	odd	2
105.2.p.a.59.4	yes	24	21.20	even	2
105.2.p.a.59.9	yes	24	105.104	even	2
105.2.p.a.59.10	yes	24	7.6	odd	2
105.2.p.a.89.3	yes	24	21.2	odd	6
105.2.p.a.89.4	yes	24	35.9	even	6
105.2.p.a.89.9	yes	24	7.2	even	3
105.2.p.a.89.10	yes	24	105.44	odd	6
525.2.t.j.26.3		24	105.2	even	12
525.2.t.j.26.4		24	35.23	odd	12
525.2.t.j.26.9		24	35.2	odd	12
525.2.t.j.26.10		24	105.23	even	12
525.2.t.j.101.3		24	35.27	even	4
525.2.t.j.101.4		24	105.83	odd	4
525.2.t.j.101.9		24	105.62	odd	4
525.2.t.j.101.10		24	35.13	even	4
735.2.g.b.734.5		24	7.3	odd	6
735.2.g.b.734.6		24	105.74	odd	6
735.2.g.b.734.7		24	105.59	even	6
735.2.g.b.734.8		24	7.4	even	3
735.2.g.b.734.17		24	35.4	even	6
735.2.g.b.734.18		24	21.17	even	6
735.2.g.b.734.19		24	21.11	odd	6
735.2.g.b.734.20		24	35.24	odd	6
735.2.p.f.374.3		24	3.2	odd	2	inner
735.2.p.f.374.4		24	5.4	even	2	inner
735.2.p.f.374.9		24	1.1	even	1	trivial
735.2.p.f.374.10		24	15.14	odd	2	inner
735.2.p.f.509.3		24	35.19	odd	6	inner
735.2.p.f.509.4		24	21.5	even	6	inner
735.2.p.f.509.9		24	105.89	even	6	inner
735.2.p.f.509.10		24	7.5	odd	6	inner