Embedded newform 43.7.f.a.2.10

• Label: 43.7.f.a.2.10
• Level: $43$
• Weight: $7$
• Character: 43.2
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	43.2
Dual form			43.7.f.a.22.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.716329 - 1.48747i) q^{2} +(3.40007 - 7.06031i) q^{3} +(38.2039 - 47.9062i) q^{4} +(69.5466 + 15.8736i) q^{5} -12.9376 q^{6} -168.130i q^{7} +(-201.638 - 46.0227i) q^{8} +(416.237 + 521.944i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.716329	−	1.48747i	−0.0895411	−	0.185934i	0.851389	−	0.524535i	\(-0.175761\pi\)
							−0.940930	+	0.338601i	\(0.890046\pi\)
\(3\)	3.40007	−	7.06031i	0.125928	−	0.261493i	−0.828467	−	0.560037i	\(-0.810787\pi\)
							0.954396	+	0.298544i	\(0.0965011\pi\)
\(5\)	69.5466	+	15.8736i	0.556373	+	0.126988i	0.491456	−	0.870902i	\(-0.336465\pi\)
							0.0649165	+	0.997891i	\(0.479322\pi\)
\(7\)		−	168.130i		−	0.490175i	−0.969501	−	0.245087i	\(-0.921183\pi\)
							0.969501	−	0.245087i	\(-0.0788167\pi\)
\(11\)	−494.032	−	619.497i	−0.371174	−	0.465437i	0.560806	−	0.827947i	\(-0.310491\pi\)
							−0.931980	+	0.362510i	\(0.881920\pi\)
\(13\)	419.543	−	1838.14i	0.190962	−	0.836658i	−0.785136	−	0.619324i	\(-0.787407\pi\)
							0.976097	−	0.217334i	\(-0.0697361\pi\)
\(17\)	−1487.47	−	6517.04i	−0.302763	−	1.32649i	−0.865938	−	0.500152i	\(-0.833277\pi\)
							0.563175	−	0.826338i	\(-0.309580\pi\)
\(19\)	7489.56	+	5972.73i	1.09193	+	0.870787i	0.992253	−	0.124233i	\(-0.0396470\pi\)
							0.0996791	+	0.995020i	\(0.468218\pi\)
\(23\)	−10784.3	−	13523.1i	−0.886359	−	1.11146i	−0.993112	−	0.117173i	\(-0.962617\pi\)
							0.106753	−	0.994286i	\(-0.465955\pi\)
\(29\)	16175.3	+	33588.4i	0.663222	+	1.37720i	0.912626	+	0.408796i	\(0.134051\pi\)
							−0.249403	+	0.968400i	\(0.580234\pi\)
\(31\)	22421.8	−	10797.8i	0.752635	−	0.362450i	−0.0179063	−	0.999840i	\(-0.505700\pi\)
							0.770542	+	0.637390i	\(0.219986\pi\)
\(37\)			93504.2i			1.84597i	0.384830	+	0.922987i	\(0.374260\pi\)
							−0.384830	+	0.922987i	\(0.625740\pi\)
\(41\)	−68207.6	+	32847.0i	−0.989648	+	0.476590i	−0.857413	−	0.514629i	\(-0.827930\pi\)
							−0.132235	+	0.991218i	\(0.542215\pi\)
\(43\)	44101.9	+	66154.2i	0.554692	+	0.832055i
							\(47\)	41930.8	−	52579.6i	0.403868	−	0.506435i	−0.537756	−	0.843101i	\(-0.680728\pi\)
0.941624	+	0.336666i	\(0.109299\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.2.10	✓	126
43.22	odd	14	inner	43.7.f.a.22.10	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.10	✓	126	1.1	even	1	trivial
43.7.f.a.22.10	yes	126	43.22	odd	14	inner