Embedded newform 43.9.h.a.12.14

• Label: 43.9.h.a.12.14
• Level: $43$
• Weight: $9$
• Character: 43.12
• 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			12.14
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08287 - 2.24859i) q^{2} +(-17.4187 - 25.5485i) q^{3} +(155.730 - 195.279i) q^{4} +(618.476 - 666.559i) q^{5} +(-38.5861 + 66.8331i) q^{6} +(963.598 - 556.334i) q^{7} +(-1230.63 - 280.883i) q^{8} +(2047.69 - 5217.42i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08287	−	2.24859i	−0.0676791	−	0.140537i	0.864385	−	0.502831i	\(-0.167708\pi\)
							−0.932064	+	0.362294i	\(0.881994\pi\)
\(3\)	−17.4187	−	25.5485i	−0.215045	−	0.315413i	0.703526	−	0.710670i	\(-0.251608\pi\)
							−0.918571	+	0.395257i	\(0.870656\pi\)
\(5\)	618.476	−	666.559i	0.989562	−	1.06649i	−0.00822265	−	0.999966i	\(-0.502617\pi\)
							0.997785	−	0.0665279i	\(-0.0211921\pi\)
\(7\)	963.598	−	556.334i	0.401332	−	0.231709i	−0.285727	−	0.958311i	\(-0.592235\pi\)
							0.687059	+	0.726602i	\(0.258902\pi\)
\(11\)	16584.6	+	20796.4i	1.13275	+	1.42042i	0.893269	+	0.449522i	\(0.148406\pi\)
							0.239481	+	0.970901i	\(0.423023\pi\)
\(13\)	−14888.2	+	4592.40i	−0.521277	+	0.160793i	−0.544218	−	0.838944i	\(-0.683173\pi\)
							0.0229405	+	0.999737i	\(0.492697\pi\)
\(17\)	−11432.5	+	10607.8i	−0.136882	+	0.127008i	−0.745643	−	0.666346i	\(-0.767857\pi\)
							0.608761	+	0.793354i	\(0.291667\pi\)
\(19\)	181308.	−	71158.1i	1.39124	−	0.546022i	0.452846	−	0.891589i	\(-0.350409\pi\)
							0.938394	+	0.345567i	\(0.112313\pi\)
\(23\)	−383240.	+	57764.2i	−1.36949	+	0.206418i	−0.792271	−	0.610169i	\(-0.791101\pi\)
							−0.577222	+	0.816587i	\(0.695863\pi\)
\(29\)	232415.	−	340890.i	0.328603	−	0.481973i	−0.626012	−	0.779813i	\(-0.715314\pi\)
							0.954616	+	0.297841i	\(0.0962664\pi\)
\(31\)	−10274.0	+	137097.i	−0.0111248	+	0.148451i	−1.00000		0.000297520i	\(-0.999905\pi\)
							0.988875	+	0.148748i	\(0.0475243\pi\)
\(37\)	−777857.	−	449096.i	−0.415043	−	0.239625i	0.277911	−	0.960607i	\(-0.410358\pi\)
							−0.692954	+	0.720982i	\(0.743691\pi\)
\(41\)	1.02514e6	−	493682.i	0.362784	−	0.174708i	−0.243607	−	0.969874i	\(-0.578331\pi\)
							0.606391	+	0.795166i	\(0.292617\pi\)
\(43\)	−670054.	+	3.35250e6i	−0.195991	+	0.980606i
							\(47\)	−2.36859e6	+	2.97012e6i	−0.485398	+	0.608670i	−0.962866	−	0.269979i	\(-0.912983\pi\)
0.477468	+	0.878649i	\(0.341555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.14	✓	336
43.18	odd	42	inner	43.9.h.a.18.14	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.14	✓	336	1.1	even	1	trivial
43.9.h.a.18.14	yes	336	43.18	odd	42	inner