Embedded newform 43.9.h.a.12.11

• Label: 43.9.h.a.12.11
• 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			12.11
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75198 - 7.79107i) q^{2} +(-78.0156 - 114.428i) q^{3} +(112.990 - 141.685i) q^{4} +(-270.057 + 291.052i) q^{5} +(-598.802 + 1037.16i) q^{6} +(-498.464 + 287.788i) q^{7} +(-3686.05 - 841.318i) q^{8} +(-4610.30 + 11746.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75198	−	7.79107i	−0.234499	−	0.486942i	0.750199	−	0.661212i	\(-0.229958\pi\)
							−0.984698	+	0.174270i	\(0.944243\pi\)
\(3\)	−78.0156	−	114.428i	−0.963155	−	1.41269i	−0.910236	−	0.414091i	\(-0.864100\pi\)
							−0.0529195	−	0.998599i	\(-0.516853\pi\)
\(5\)	−270.057	+	291.052i	−0.432091	+	0.465683i	−0.911135	−	0.412108i	\(-0.864793\pi\)
							0.479045	+	0.877791i	\(0.340983\pi\)
\(7\)	−498.464	+	287.788i	−0.207607	+	0.119862i	−0.600199	−	0.799851i	\(-0.704912\pi\)
							0.392592	+	0.919713i	\(0.371579\pi\)
\(11\)	−103.919	−	130.311i	−0.00709783	−	0.00890039i	0.778270	−	0.627930i	\(-0.216098\pi\)
							−0.785368	+	0.619030i	\(0.787526\pi\)
\(13\)	−21995.1	+	6784.59i	−0.770109	+	0.237547i	−0.654820	−	0.755785i	\(-0.727256\pi\)
							−0.115289	+	0.993332i	\(0.536779\pi\)
\(17\)	27500.2	−	25516.4i	0.329261	−	0.305509i	−0.498167	−	0.867081i	\(-0.665994\pi\)
							0.827428	+	0.561571i	\(0.189803\pi\)
\(19\)	111387.	−	43716.1i	0.854712	−	0.335450i	0.102791	−	0.994703i	\(-0.467223\pi\)
							0.751921	+	0.659253i	\(0.229128\pi\)
\(23\)	−72922.5	+	10991.3i	−0.260585	+	0.0392769i	−0.278035	−	0.960571i	\(-0.589683\pi\)
							0.0174501	+	0.999848i	\(0.494445\pi\)
\(29\)	−12276.7	+	18006.7i	−0.0173576	+	0.0254590i	−0.834820	−	0.550524i	\(-0.814428\pi\)
							0.817462	+	0.575983i	\(0.195380\pi\)
\(31\)	57051.8	−	761304.i	0.0617764	−	0.824349i	−0.877147	−	0.480222i	\(-0.840556\pi\)
							0.938923	−	0.344127i	\(-0.111825\pi\)
\(37\)	1.39268e6	+	804064.i	0.743095	+	0.429026i	0.823194	−	0.567761i	\(-0.192190\pi\)
							−0.0800986	+	0.996787i	\(0.525524\pi\)
\(41\)	−1.80032e6	+	866987.i	−0.637109	+	0.306816i	−0.724413	−	0.689366i	\(-0.757889\pi\)
							0.0873040	+	0.996182i	\(0.472175\pi\)
\(43\)	2.53937e6	−	2.28907e6i	0.742765	−	0.669552i
							\(47\)	18384.1	−	23053.0i	0.00376749	−	0.00472428i	−0.779944	−	0.625849i	\(-0.784753\pi\)
0.783712	+	0.621125i	\(0.213324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.11	✓	336
43.18	odd	42	inner	43.9.h.a.18.11	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.11	✓	336	1.1	even	1	trivial
43.9.h.a.18.11	yes	336	43.18	odd	42	inner