Embedded newform 43.7.h.a.18.4

• Label: 43.7.h.a.18.4
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.4
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.4

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.4	✓	252	43.12	odd	42	inner
43.7.h.a.18.4	yes	252	1.1	even	1	trivial