Embedded newform 43.9.h.a.3.14

• Label: 43.9.h.a.3.14
• Level: $43$
• Weight: $9$
• Character: 43.3
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.385973 + 0.801481i) q^{2} +(-82.9635 - 6.21726i) q^{3} +(159.120 + 199.530i) q^{4} +(153.970 - 499.159i) q^{5} +(37.0047 - 64.0940i) q^{6} +(-701.273 + 404.880i) q^{7} +(-443.358 + 101.194i) q^{8} +(356.569 + 53.7442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.385973	+	0.801481i	−0.0241233	+	0.0500926i	−0.912682	−	0.408670i	\(-0.865993\pi\)
							0.888559	+	0.458762i	\(0.151707\pi\)
\(3\)	−82.9635	−	6.21726i	−1.02424	−	0.0767562i	−0.447998	−	0.894035i	\(-0.647863\pi\)
							−0.576243	+	0.817278i	\(0.695482\pi\)
\(5\)	153.970	−	499.159i	0.246352	−	0.798654i	−0.744958	−	0.667111i	\(-0.767531\pi\)
							0.991310	−	0.131543i	\(-0.0419931\pi\)
\(7\)	−701.273	+	404.880i	−0.292075	+	0.168630i	−0.638877	−	0.769309i	\(-0.720601\pi\)
							0.346802	+	0.937938i	\(0.387268\pi\)
\(11\)	12222.9	−	15327.1i	0.834842	−	1.04686i	−0.163339	−	0.986570i	\(-0.552226\pi\)
							0.998181	−	0.0602888i	\(-0.0192022\pi\)
\(13\)	−12477.4	+	11577.3i	−0.436867	+	0.405353i	−0.867708	−	0.497075i	\(-0.834408\pi\)
							0.430841	+	0.902428i	\(0.358217\pi\)
\(17\)	67834.1	−	20924.0i	0.812180	−	0.250524i	0.139275	−	0.990254i	\(-0.455523\pi\)
							0.672904	+	0.739729i	\(0.265047\pi\)
\(19\)	33460.3	+	221995.i	0.256753	+	1.70344i	0.630727	+	0.776005i	\(0.282757\pi\)
							−0.373974	+	0.927439i	\(0.622005\pi\)
\(23\)	127658.	+	325268.i	0.456181	+	1.16233i	0.955258	+	0.295773i	\(0.0955774\pi\)
							−0.499077	+	0.866558i	\(0.666327\pi\)
\(29\)	−100527.	+	7533.45i	−0.142131	+	0.0106513i	−0.145606	−	0.989343i	\(-0.546513\pi\)
							0.00347434	+	0.999994i	\(0.498894\pi\)
\(31\)	−230818.	+	157369.i	−0.249932	+	0.170401i	−0.681806	−	0.731533i	\(-0.738805\pi\)
							0.431873	+	0.901934i	\(0.357853\pi\)
\(37\)	1.73305e6	+	1.00058e6i	0.924709	+	0.533881i	0.885134	−	0.465335i	\(-0.154066\pi\)
							0.0395749	+	0.999217i	\(0.487400\pi\)
\(41\)	2.83557e6	+	1.36554e6i	1.00347	+	0.483246i	0.862115	−	0.506713i	\(-0.169140\pi\)
							0.141355	+	0.989959i	\(0.454854\pi\)
\(43\)	−3.27572e6	−	978711.i	−0.958148	−	0.286273i
							\(47\)	−695338.	−	871926.i	−0.142497	−	0.178685i	0.705462	−	0.708748i	\(-0.250740\pi\)
−0.847958	+	0.530063i	\(0.822168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.14	✓	336
43.29	odd	42	inner	43.9.h.a.29.14	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.14	✓	336	1.1	even	1	trivial
43.9.h.a.29.14	yes	336	43.29	odd	42	inner