Embedded newform 43.7.h.a.12.11

• Label: 43.7.h.a.12.11
• 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			12.11
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.758988 + 1.57606i) q^{2} +(6.18403 + 9.07030i) q^{3} +(37.9955 - 47.6448i) q^{4} +(-99.6397 + 107.386i) q^{5} +(-9.60169 + 16.6306i) q^{6} +(-494.102 + 285.270i) q^{7} +(213.077 + 48.6333i) q^{8} +(222.305 - 566.425i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.758988	+	1.57606i	0.0948735	+	0.197007i	0.943005	−	0.332779i	\(-0.107986\pi\)
							−0.848131	+	0.529786i	\(0.822272\pi\)
\(3\)	6.18403	+	9.07030i	0.229038	+	0.335937i	0.923504	−	0.383588i	\(-0.125312\pi\)
							−0.694466	+	0.719525i	\(0.744359\pi\)
\(5\)	−99.6397	+	107.386i	−0.797117	+	0.859088i	−0.992576	−	0.121624i	\(-0.961190\pi\)
							0.195459	+	0.980712i	\(0.437380\pi\)
\(7\)	−494.102	+	285.270i	−1.44053	+	0.831691i	−0.997885	−	0.0650062i	\(-0.979293\pi\)
							−0.442645	+	0.896697i	\(0.645960\pi\)
\(11\)	1481.81	+	1858.14i	1.11331	+	1.39604i	0.908827	+	0.417174i	\(0.136979\pi\)
							0.204482	+	0.978870i	\(0.434449\pi\)
\(13\)	−3550.76	+	1095.26i	−1.61619	+	0.498527i	−0.965409	−	0.260741i	\(-0.916033\pi\)
							−0.650777	+	0.759269i	\(0.725557\pi\)
\(17\)	−5513.15	+	5115.45i	−1.12215	+	1.04121i	−0.123348	+	0.992363i	\(0.539363\pi\)
							−0.998806	+	0.0488442i	\(0.984446\pi\)
\(19\)	−789.362	+	309.802i	−0.115084	+	0.0451672i	−0.422184	−	0.906510i	\(-0.638737\pi\)
							0.307100	+	0.951677i	\(0.400641\pi\)
\(23\)	789.621	−	119.016i	0.0648986	−	0.00978189i	−0.116513	−	0.993189i	\(-0.537172\pi\)
							0.181411	+	0.983407i	\(0.441933\pi\)
\(29\)	10248.1	−	15031.2i	0.420195	−	0.616312i	−0.556525	−	0.830831i	\(-0.687866\pi\)
							0.976720	+	0.214518i	\(0.0688181\pi\)
\(31\)	11.1854	−	149.259i	0.000375464	−	0.00501022i	−0.997013	−	0.0772350i	\(-0.975391\pi\)
							0.997388	+	0.0722247i	\(0.0230099\pi\)
\(37\)	45473.4	+	26254.1i	0.897744	+	0.518313i	0.876468	−	0.481461i	\(-0.159894\pi\)
							0.0212764	+	0.999774i	\(0.493227\pi\)
\(41\)	−35500.2	+	17096.0i	−0.515086	+	0.248052i	−0.673322	−	0.739349i	\(-0.735133\pi\)
							0.158237	+	0.987401i	\(0.449419\pi\)
\(43\)	68072.6	+	41079.0i	0.856184	+	0.516671i
							\(47\)	89983.6	−	112836.i	0.866702	−	1.08681i	−0.128762	−	0.991675i	\(-0.541100\pi\)
0.995464	−	0.0951345i	\(-0.0303281\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.11	✓	252
43.18	odd	42	inner	43.7.h.a.18.11	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.11	✓	252	1.1	even	1	trivial
43.7.h.a.18.11	yes	252	43.18	odd	42	inner