Embedded newform 43.7.h.a.12.4

• Label: 43.7.h.a.12.4
• Level: $43$
• Weight: $7$
• Character: 43.12
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.4
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.10062 - 10.5915i) q^{2} +(-0.905002 - 1.32739i) q^{3} +(-46.2612 + 58.0097i) q^{4} +(140.603 - 151.534i) q^{5} +(-9.44309 + 16.3559i) q^{6} +(243.269 - 140.452i) q^{7} +(116.869 + 26.6747i) q^{8} +(265.391 - 676.204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.10062	−	10.5915i	−0.637577	−	1.32394i	−0.929967	−	0.367644i	\(-0.880165\pi\)
							0.292389	−	0.956299i	\(-0.405550\pi\)
\(3\)	−0.905002	−	1.32739i	−0.0335186	−	0.0491628i	0.809114	−	0.587652i	\(-0.199948\pi\)
							−0.842632	+	0.538489i	\(0.818995\pi\)
\(5\)	140.603	−	151.534i	1.12482	−	1.21227i	0.150144	−	0.988664i	\(-0.452026\pi\)
							0.974679	−	0.223607i	\(-0.0717833\pi\)
\(7\)	243.269	−	140.452i	0.709240	−	0.409480i	−0.101539	−	0.994832i	\(-0.532377\pi\)
							0.810780	+	0.585351i	\(0.199043\pi\)
\(11\)	−66.0129	−	82.7776i	−0.0495965	−	0.0621920i	0.756415	−	0.654092i	\(-0.226949\pi\)
							−0.806011	+	0.591900i	\(0.798378\pi\)
\(13\)	1518.99	−	468.545i	0.691391	−	0.213266i	0.0709047	−	0.997483i	\(-0.477411\pi\)
							0.620486	+	0.784217i	\(0.286935\pi\)
\(17\)	−6423.68	+	5960.31i	−1.30749	+	1.21317i	−0.346028	+	0.938224i	\(0.612470\pi\)
							−0.961460	+	0.274947i	\(0.911340\pi\)
\(19\)	3770.01	−	1479.62i	0.549644	−	0.215719i	−0.0742343	−	0.997241i	\(-0.523651\pi\)
							0.623878	+	0.781521i	\(0.285556\pi\)
\(23\)	−9634.65	+	1452.19i	−0.791867	+	0.119355i	−0.532507	−	0.846426i	\(-0.678750\pi\)
							−0.259360	+	0.965781i	\(0.583512\pi\)
\(29\)	−8907.88	+	13065.5i	−0.365242	+	0.535711i	−0.964237	−	0.265041i	\(-0.914614\pi\)
							0.598995	+	0.800753i	\(0.295567\pi\)
\(31\)	2824.03	−	37684.1i	0.0947949	−	1.26495i	−0.723570	−	0.690251i	\(-0.757500\pi\)
							0.818365	−	0.574699i	\(-0.194881\pi\)
\(37\)	36099.6	+	20842.1i	0.712685	+	0.411469i	0.812054	−	0.583582i	\(-0.198349\pi\)
							−0.0993693	+	0.995051i	\(0.531683\pi\)
\(41\)	−96623.5	+	46531.4i	−1.40195	+	0.675142i	−0.973555	−	0.228452i	\(-0.926634\pi\)
							−0.428391	+	0.903594i	\(0.640919\pi\)
\(43\)	78966.1	−	9258.29i	0.993197	−	0.116446i
							\(47\)	65997.0	−	82757.6i	0.635669	−	0.797103i	−0.354785	−	0.934948i	\(-0.615446\pi\)
0.990454	+	0.137845i	\(0.0440175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.4	✓	252
43.18	odd	42	inner	43.7.h.a.18.4	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.4	✓	252	1.1	even	1	trivial
43.7.h.a.18.4	yes	252	43.18	odd	42	inner