Embedded newform 43.9.h.a.18.26

• Label: 43.9.h.a.18.26
• Level: $43$
• Weight: $9$
• Character: 43.18
• 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			18.26
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.7003 - 24.2960i) q^{2} +(87.7083 - 128.644i) q^{3} +(-293.785 - 368.395i) q^{4} +(94.6988 + 102.061i) q^{5} +(-2099.33 - 3636.15i) q^{6} +(2428.86 + 1402.31i) q^{7} +(-5657.57 + 1291.30i) q^{8} +(-6459.65 - 16458.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.7003	−	24.2960i	0.731272	−	1.51850i	−0.119424	−	0.992843i	\(-0.538105\pi\)
							0.850696	−	0.525658i	\(-0.176181\pi\)
\(3\)	87.7083	−	128.644i	1.08282	−	1.58820i	0.313214	−	0.949683i	\(-0.398594\pi\)
							0.769605	−	0.638521i	\(-0.220453\pi\)
\(5\)	94.6988	+	102.061i	0.151518	+	0.163298i	0.804250	−	0.594292i	\(-0.202567\pi\)
							−0.652732	+	0.757589i	\(0.726377\pi\)
\(7\)	2428.86	+	1402.31i	1.01161	+	0.584051i	0.911661	−	0.410942i	\(-0.134800\pi\)
							0.0999441	+	0.994993i	\(0.468134\pi\)
\(11\)	−8571.12	+	10747.8i	−0.585419	+	0.734092i	−0.983027	−	0.183463i	\(-0.941269\pi\)
							0.397608	+	0.917556i	\(0.369841\pi\)
\(13\)	−22133.7	−	6827.35i	−0.774963	−	0.239044i	−0.118047	−	0.993008i	\(-0.537663\pi\)
							−0.656916	+	0.753964i	\(0.728139\pi\)
\(17\)	36035.2	+	33435.8i	0.431451	+	0.400328i	0.865779	−	0.500427i	\(-0.166824\pi\)
							−0.434328	+	0.900755i	\(0.643014\pi\)
\(19\)	139940.	+	54922.3i	1.07381	+	0.421439i	0.835353	−	0.549714i	\(-0.185263\pi\)
							0.238456	+	0.971153i	\(0.423359\pi\)
\(23\)	307857.	+	46402.0i	1.10011	+	0.165816i	0.673912	−	0.738812i	\(-0.264613\pi\)
							0.426203	+	0.904628i	\(0.359851\pi\)
\(29\)	768627.	+	1.12737e6i	1.08673	+	1.59395i	0.761321	+	0.648375i	\(0.224551\pi\)
							0.325414	+	0.945572i	\(0.394496\pi\)
\(31\)	−31706.7	−	423096.i	−0.0343324	−	0.458134i	−0.987704	−	0.156335i	\(-0.950032\pi\)
							0.953372	−	0.301799i	\(-0.0975870\pi\)
\(37\)	−655772.	+	378610.i	−0.349902	+	0.202016i	−0.664642	−	0.747162i	\(-0.731416\pi\)
							0.314740	+	0.949178i	\(0.398083\pi\)
\(41\)	534182.	+	257248.i	0.189040	+	0.0910369i	0.526009	−	0.850479i	\(-0.323688\pi\)
							−0.336969	+	0.941516i	\(0.609402\pi\)
\(43\)	−2.81427e6	−	1.94116e6i	−0.823175	−	0.567788i
							\(47\)	−3.20941e6	−	4.02447e6i	−0.657709	−	0.824741i	0.335383	−	0.942082i	\(-0.391134\pi\)
−0.993092	+	0.117341i	\(0.962563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.26	yes	336
43.12	odd	42	inner	43.9.h.a.12.26	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.26	✓	336	43.12	odd	42	inner
43.9.h.a.18.26	yes	336	1.1	even	1	trivial