Embedded newform 43.9.f.a.2.16

• Label: 43.9.f.a.2.16
• Level: $43$
• Weight: $9$
• Character: 43.2
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.16
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53552 + 3.18853i) q^{2} +(57.5019 - 119.404i) q^{3} +(151.804 - 190.357i) q^{4} +(-620.261 - 141.571i) q^{5} +469.018 q^{6} -2752.04i q^{7} +(1723.33 + 393.338i) q^{8} +(-6860.10 - 8602.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53552	+	3.18853i	0.0959697	+	0.199283i	0.943427	−	0.331581i	\(-0.107582\pi\)
							−0.847457	+	0.530864i	\(0.821868\pi\)
\(3\)	57.5019	−	119.404i	0.709900	−	1.47412i	−0.163207	−	0.986592i	\(-0.552184\pi\)
							0.873106	−	0.487530i	\(-0.162102\pi\)
\(5\)	−620.261	−	141.571i	−0.992418	−	0.226513i	−0.304667	−	0.952459i	\(-0.598545\pi\)
							−0.687752	+	0.725946i	\(0.741402\pi\)
\(7\)		−	2752.04i		−	1.14621i	−0.819483	−	0.573103i	\(-0.805740\pi\)
							0.819483	−	0.573103i	\(-0.194260\pi\)
\(11\)	3916.43	+	4911.05i	0.267498	+	0.335432i	0.897380	−	0.441259i	\(-0.145468\pi\)
							−0.629882	+	0.776691i	\(0.716897\pi\)
\(13\)	−4456.23	+	19524.0i	−0.156025	+	0.683591i	0.835037	+	0.550193i	\(0.185446\pi\)
							−0.991063	+	0.133398i	\(0.957411\pi\)
\(17\)	20747.0	+	90898.7i	0.248405	+	1.08833i	0.933132	+	0.359534i	\(0.117064\pi\)
							−0.684727	+	0.728800i	\(0.740079\pi\)
\(19\)	199437.	+	159046.i	1.53035	+	1.22042i	0.893185	+	0.449690i	\(0.148466\pi\)
							0.637168	+	0.770725i	\(0.280106\pi\)
\(23\)	−293415.	−	367930.i	−1.04850	−	1.31478i	−0.947451	−	0.319902i	\(-0.896350\pi\)
							−0.101054	−	0.994881i	\(-0.532222\pi\)
\(29\)	−255149.	−	529822.i	−0.360746	−	0.749097i	0.639053	−	0.769163i	\(-0.279327\pi\)
							−0.999799	+	0.0200663i	\(0.993612\pi\)
\(31\)	584457.	−	281460.i	0.632857	−	0.304768i	−0.0898155	−	0.995958i	\(-0.528628\pi\)
							0.722673	+	0.691190i	\(0.242913\pi\)
\(37\)		−	2.13515e6i		−	1.13925i	−0.821903	−	0.569627i	\(-0.807088\pi\)
							0.821903	−	0.569627i	\(-0.192912\pi\)
\(41\)	−2.47691e6	+	1.19282e6i	−0.876546	+	0.422122i	−0.817362	−	0.576125i	\(-0.804564\pi\)
							−0.0591842	+	0.998247i	\(0.518850\pi\)
\(43\)	2.17351e6	−	2.63895e6i	0.635752	−	0.771893i
							\(47\)	−3.14123e6	+	3.93898e6i	−0.643738	+	0.807221i	−0.991465	−	0.130374i	\(-0.958382\pi\)
0.347727	+	0.937596i	\(0.386954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.16	✓	174
43.22	odd	14	inner	43.9.f.a.22.16	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.16	✓	174	1.1	even	1	trivial
43.9.f.a.22.16	yes	174	43.22	odd	14	inner