Embedded newform 43.7.f.a.22.5

• Label: 43.7.f.a.22.5
• Level: $43$
• Weight: $7$
• Character: 43.22
• 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			22.5
Character	\(\chi\)	\(=\)	43.22
Dual form			43.7.f.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.79242 + 9.95156i) q^{2} +(-6.97408 - 14.4818i) q^{3} +(-36.1629 - 45.3469i) q^{4} +(117.045 - 26.7147i) q^{5} +177.540 q^{6} +466.480i q^{7} +(-64.6015 + 14.7449i) q^{8} +(293.438 - 367.960i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.79242	+	9.95156i	−0.599052	+	1.24395i	0.352313	+	0.935882i	\(0.385395\pi\)
							−0.951366	+	0.308063i	\(0.900319\pi\)
\(3\)	−6.97408	−	14.4818i	−0.258299	−	0.536364i	0.730980	−	0.682399i	\(-0.239063\pi\)
							−0.989279	+	0.146035i	\(0.953349\pi\)
\(5\)	117.045	−	26.7147i	0.936360	−	0.213718i	0.272996	−	0.962015i	\(-0.411985\pi\)
							0.663364	+	0.748297i	\(0.269128\pi\)
\(7\)			466.480i			1.36000i	0.733212	+	0.680000i	\(0.238020\pi\)
							−0.733212	+	0.680000i	\(0.761980\pi\)
\(11\)	−1084.84	+	1360.35i	−0.815059	+	1.02205i	0.184174	+	0.982894i	\(0.441039\pi\)
							−0.999233	+	0.0391576i	\(0.987533\pi\)
\(13\)	211.677	+	927.416i	0.0963481	+	0.422129i	0.999981	−	0.00617944i	\(-0.00196699\pi\)
							−0.903633	+	0.428308i	\(0.859110\pi\)
\(17\)	1085.44	−	4755.64i	0.220933	−	0.967970i	−0.735845	−	0.677150i	\(-0.763215\pi\)
							0.956778	−	0.290820i	\(-0.0939281\pi\)
\(19\)	−9694.59	+	7731.18i	−1.41341	+	1.12716i	−0.440026	+	0.897985i	\(0.645031\pi\)
							−0.973386	+	0.229173i	\(0.926398\pi\)
\(23\)	−6838.48	+	8575.18i	−0.562052	+	0.704790i	−0.978936	−	0.204169i	\(-0.934551\pi\)
							0.416884	+	0.908960i	\(0.363122\pi\)
\(29\)	−17403.5	+	36138.8i	−0.713582	+	1.48177i	0.155885	+	0.987775i	\(0.450177\pi\)
							−0.869466	+	0.493992i	\(0.835537\pi\)
\(31\)	14896.9	+	7173.99i	0.500048	+	0.240811i	0.666867	−	0.745177i	\(-0.267635\pi\)
							−0.166818	+	0.985988i	\(0.553349\pi\)
\(37\)		−	30453.6i		−	0.601219i	−0.953747	−	0.300610i	\(-0.902810\pi\)
							0.953747	−	0.300610i	\(-0.0971901\pi\)
\(41\)	108345.	+	52176.4i	1.57202	+	0.757047i	0.998085	−	0.0618511i	\(-0.0197004\pi\)
							0.573939	+	0.818898i	\(0.305415\pi\)
\(43\)	−66487.8	−	43597.4i	−0.836251	−	0.548347i
							\(47\)	46855.2	+	58754.5i	0.451299	+	0.565911i	0.954482	−	0.298269i	\(-0.0964093\pi\)
−0.503183	+	0.864180i	\(0.667838\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.22.5	yes	126
43.2	odd	14	inner	43.7.f.a.2.5	✓	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.5	✓	126	43.2	odd	14	inner
43.7.f.a.22.5	yes	126	1.1	even	1	trivial