Embedded newform 43.9.h.a.3.19

• Label: 43.9.h.a.3.19
• 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			3.19
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.51335 - 13.5251i) q^{2} +(-137.776 - 10.3249i) q^{3} +(19.1085 + 23.9613i) q^{4} +(-3.15223 + 10.2193i) q^{5} +(-1037.03 + 1796.19i) q^{6} +(-2870.44 + 1657.25i) q^{7} +(4195.19 - 957.525i) q^{8} +(12387.9 + 1867.18i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.51335	−	13.5251i	0.407084	−	0.845319i	−0.592136	−	0.805838i	\(-0.701715\pi\)
							0.999220	−	0.0394813i	\(-0.0125706\pi\)
\(3\)	−137.776	−	10.3249i	−1.70094	−	0.127468i	−0.811450	−	0.584422i	\(-0.801321\pi\)
							−0.889490	+	0.456954i	\(0.848940\pi\)
\(5\)	−3.15223	+	10.2193i	−0.00504356	+	0.0163508i	−0.958060	−	0.286569i	\(-0.907485\pi\)
							0.953016	+	0.302920i	\(0.0979614\pi\)
\(7\)	−2870.44	+	1657.25i	−1.19552	+	0.690234i	−0.959553	−	0.281527i	\(-0.909159\pi\)
							−0.235967	+	0.971761i	\(0.575826\pi\)
\(11\)	−1775.45	+	2226.34i	−0.121266	+	0.152062i	−0.838759	−	0.544503i	\(-0.816718\pi\)
							0.717493	+	0.696566i	\(0.245290\pi\)
\(13\)	−506.728	+	470.175i	−0.0177420	+	0.0164621i	−0.688994	−	0.724767i	\(-0.741947\pi\)
							0.671252	+	0.741229i	\(0.265757\pi\)
\(17\)	51102.9	−	15763.1i	0.611856	−	0.188733i	0.0266871	−	0.999644i	\(-0.491504\pi\)
							0.585169	+	0.810911i	\(0.301028\pi\)
\(19\)	−32538.1	−	215876.i	−0.249676	−	1.65649i	−0.665017	−	0.746829i	\(-0.731576\pi\)
							0.415340	−	0.909666i	\(-0.363663\pi\)
\(23\)	−36252.8	−	92370.5i	−0.129548	−	0.330082i	0.851352	−	0.524594i	\(-0.175783\pi\)
							−0.980900	+	0.194512i	\(0.937688\pi\)
\(29\)	−663391.	+	49714.3i	−0.937945	+	0.0702893i	−0.534923	−	0.844901i	\(-0.679659\pi\)
							−0.403022	+	0.915190i	\(0.632040\pi\)
\(31\)	1.06414e6	−	725518.i	1.15226	−	0.785600i	0.172352	−	0.985035i	\(-0.444863\pi\)
							0.979911	+	0.199435i	\(0.0639108\pi\)
\(37\)	2.47546e6	+	1.42921e6i	1.32084	+	0.762586i	0.983862	−	0.178928i	\(-0.0572629\pi\)
							0.336975	+	0.941514i	\(0.390596\pi\)
\(41\)	2.11501e6	+	1.01853e6i	0.748474	+	0.360446i	0.768920	−	0.639345i	\(-0.220794\pi\)
							−0.0204464	+	0.999791i	\(0.506509\pi\)
\(43\)	2.67836e6	−	2.12476e6i	0.783420	−	0.621493i
							\(47\)	2.56433e6	+	3.21557e6i	0.525512	+	0.658971i	0.971769	−	0.235934i	\(-0.0758148\pi\)
−0.446257	+	0.894905i	\(0.647243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.19	✓	336
43.29	odd	42	inner	43.9.h.a.29.19	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.19	✓	336	1.1	even	1	trivial
43.9.h.a.29.19	yes	336	43.29	odd	42	inner