Embedded newform 41.9.h.a.6.18

• Label: 41.9.h.a.6.18
• Level: $41$
• Weight: $9$
• Character: 41.6
• Analytic conductor: $16.703$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	41.h (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.7025230125\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(27\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			6.18
Character	\(\chi\)	\(=\)	41.6
Dual form			41.9.h.a.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.80077 - 7.45943i) q^{2} +(44.5773 + 107.619i) q^{3} +(109.276 + 150.405i) q^{4} +(-108.558 + 685.408i) q^{5} +(972.206 + 76.5142i) q^{6} +(2347.49 - 184.752i) q^{7} +(3654.09 - 578.752i) q^{8} +(-4955.41 + 4955.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 16 q^{2} - 240 q^{3} - 20 q^{4} - 16 q^{5} - 6484 q^{6} - 16 q^{7} - 1040 q^{8} + 21520 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.80077	−	7.45943i	0.237548	−	0.466215i	−0.741199	−	0.671285i	\(-0.765742\pi\)
							0.978747	+	0.205071i	\(0.0657425\pi\)
\(3\)	44.5773	+	107.619i	0.550337	+	1.32863i	0.917226	+	0.398366i	\(0.130423\pi\)
							−0.366889	+	0.930265i	\(0.619577\pi\)
\(5\)	−108.558	+	685.408i	−0.173693	+	1.09665i	0.734656	+	0.678440i	\(0.237344\pi\)
							−0.908349	+	0.418214i	\(0.862656\pi\)
\(7\)	2347.49	−	184.752i	0.977715	−	0.0769478i	0.420479	−	0.907302i	\(-0.361862\pi\)
							0.557236	+	0.830354i	\(0.311862\pi\)
\(11\)	−9713.71	−	2332.05i	−0.663459	−	0.159282i	−0.112289	−	0.993676i	\(-0.535818\pi\)
							−0.551170	+	0.834393i	\(0.685818\pi\)
\(13\)	−6130.15	+	5235.64i	−0.214633	+	0.183314i	−0.750264	−	0.661138i	\(-0.770074\pi\)
							0.535631	+	0.844452i	\(0.320074\pi\)
\(17\)	97465.2	−	59726.8i	1.16695	−	0.715111i	0.202480	−	0.979286i	\(-0.435100\pi\)
							0.964475	+	0.264175i	\(0.0850998\pi\)
\(19\)	−133514.	−	114032.i	−1.02450	−	0.875006i	−0.0322774	−	0.999479i	\(-0.510276\pi\)
							−0.992223	+	0.124473i	\(0.960276\pi\)
\(23\)	−203403.	+	66089.6i	−0.726851	+	0.236168i	−0.648991	−	0.760796i	\(-0.724809\pi\)
							−0.0778601	+	0.996964i	\(0.524809\pi\)
\(29\)	662003.	+	405676.i	0.935983	+	0.573571i	0.904751	−	0.425941i	\(-0.140057\pi\)
							0.0312323	+	0.999512i	\(0.490057\pi\)
\(31\)	103026.	−	141803.i	0.111558	−	0.153546i	−0.749587	−	0.661906i	\(-0.769748\pi\)
							0.861145	+	0.508359i	\(0.169748\pi\)
\(37\)	−361121.	+	262370.i	−0.192684	+	0.139993i	−0.679945	−	0.733263i	\(-0.737996\pi\)
							0.487260	+	0.873257i	\(0.337996\pi\)
\(41\)	−2.64992e6	−	981259.i	−0.937771	−	0.347255i
							\(43\)	2.18530e6	−	4.28889e6i	0.639200	−	1.25450i	−0.313210	−	0.949684i	\(-0.601404\pi\)
0.952411	−	0.304818i	\(-0.0985955\pi\)
\(47\)	−1.11459e6	−	87720.2i	−0.228415	−	0.0179766i	−0.0362648	−	0.999342i	\(-0.511546\pi\)
							−0.192150	+	0.981366i	\(0.561546\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.9.h.a.6.18	✓	432
41.7	odd	40	inner	41.9.h.a.7.18	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.9.h.a.6.18	✓	432	1.1	even	1	trivial
41.9.h.a.7.18	yes	432	41.7	odd	40	inner