Embedded newform 43.9.f.a.2.9

• Label: 43.9.f.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.86394 - 12.1766i) q^{2} +(-67.7996 + 140.787i) q^{3} +(45.7298 - 57.3434i) q^{4} +(-958.314 - 218.729i) q^{5} +2111.88 q^{6} -747.870i q^{7} +(-4339.50 - 990.462i) q^{8} +(-11133.6 - 13961.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.86394	−	12.1766i	−0.366496	−	0.761037i	0.633422	−	0.773806i	\(-0.281650\pi\)
							−0.999918	+	0.0127694i	\(0.995935\pi\)
\(3\)	−67.7996	+	140.787i	−0.837032	+	1.73811i	−0.180852	+	0.983510i	\(0.557886\pi\)
							−0.656180	+	0.754604i	\(0.727829\pi\)
\(5\)	−958.314	−	218.729i	−1.53330	−	0.349966i	−0.629186	−	0.777254i	\(-0.716612\pi\)
							−0.904115	+	0.427288i	\(0.859469\pi\)
\(7\)		−	747.870i		−	0.311483i	−0.987798	−	0.155741i	\(-0.950223\pi\)
							0.987798	−	0.155741i	\(-0.0497767\pi\)
\(11\)	10.2101	+	12.8031i	0.000697363	+	0.000874466i	0.782180	−	0.623052i	\(-0.214108\pi\)
							−0.781483	+	0.623927i	\(0.785536\pi\)
\(13\)	−1945.35	+	8523.15i	−0.0681122	+	0.298419i	−0.997498	−	0.0706898i	\(-0.977480\pi\)
							0.929386	+	0.369109i	\(0.120337\pi\)
\(17\)	29569.0	+	129550.i	0.354031	+	1.55111i	0.767776	+	0.640719i	\(0.221364\pi\)
							−0.413745	+	0.910393i	\(0.635779\pi\)
\(19\)	38943.1	+	31056.1i	0.298824	+	0.238304i	0.761411	−	0.648270i	\(-0.224507\pi\)
							−0.462586	+	0.886574i	\(0.653079\pi\)
\(23\)	−111007.	−	139198.i	−0.396677	−	0.497418i	0.542880	−	0.839811i	\(-0.317334\pi\)
							−0.939557	+	0.342393i	\(0.888763\pi\)
\(29\)	−376338.	−	781473.i	−0.532091	−	1.10490i	−0.977765	−	0.209703i	\(-0.932750\pi\)
							0.445675	−	0.895195i	\(-0.352964\pi\)
\(31\)	864972.	−	416549.i	0.936603	−	0.451044i	0.0976332	−	0.995222i	\(-0.468873\pi\)
							0.838970	+	0.544178i	\(0.183159\pi\)
\(37\)			2.71467e6i			1.44847i	0.689553	+	0.724235i	\(0.257807\pi\)
							−0.689553	+	0.724235i	\(0.742193\pi\)
\(41\)	2.80502e6	−	1.35083e6i	0.992661	−	0.478040i	0.134220	−	0.990952i	\(-0.457147\pi\)
							0.858442	+	0.512911i	\(0.171433\pi\)
\(43\)	−3.03494e6	+	1.57396e6i	−0.887720	+	0.460385i
							\(47\)	600496.	−	752998.i	0.123060	−	0.154313i	−0.716485	−	0.697603i	\(-0.754250\pi\)
0.839545	+	0.543290i	\(0.182822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.9	✓	174
43.22	odd	14	inner	43.9.f.a.22.9	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.9	✓	174	1.1	even	1	trivial
43.9.f.a.22.9	yes	174	43.22	odd	14	inner