Embedded newform 43.6.e.a.4.5

• Label: 43.6.e.a.4.5
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.75577 + 5.96355i) q^{2} +(-3.77448 - 4.73304i) q^{3} +(-5.82590 - 25.5249i) q^{4} +(-49.8748 - 24.0184i) q^{5} +46.1763 q^{6} +187.876 q^{7} +(-39.9875 - 19.2570i) q^{8} +(45.9176 - 201.178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.75577	+	5.96355i	−0.840710	+	1.05422i	0.157068	+	0.987588i	\(0.449796\pi\)
							−0.997778	+	0.0666293i	\(0.978776\pi\)
\(3\)	−3.77448	−	4.73304i	−0.242133	−	0.303625i	0.645884	−	0.763435i	\(-0.276489\pi\)
							−0.888017	+	0.459810i	\(0.847917\pi\)
\(5\)	−49.8748	−	24.0184i	−0.892187	−	0.429655i	−0.0691264	−	0.997608i	\(-0.522021\pi\)
							−0.823061	+	0.567953i	\(0.807735\pi\)
\(7\)	187.876			1.44920			0.724598	−	0.689172i	\(-0.242025\pi\)
							0.724598	+	0.689172i	\(0.242025\pi\)
\(11\)	−98.0027	+	429.378i	−0.244206	+	1.06994i	0.692939	+	0.720996i	\(0.256315\pi\)
							−0.937145	+	0.348940i	\(0.886542\pi\)
\(13\)	551.081	+	265.387i	0.904392	+	0.435532i	0.827473	−	0.561505i	\(-0.189777\pi\)
							0.0769190	+	0.997037i	\(0.475492\pi\)
\(17\)	1027.38	−	494.758i	0.862198	−	0.415213i	0.0501072	−	0.998744i	\(-0.484044\pi\)
							0.812091	+	0.583531i	\(0.198329\pi\)
\(19\)	459.522	+	2013.30i	0.292027	+	1.27945i	0.881700	+	0.471810i	\(0.156399\pi\)
							−0.589673	+	0.807642i	\(0.700744\pi\)
\(23\)	846.113	−	3707.06i	0.333510	−	1.46120i	−0.478773	−	0.877939i	\(-0.658918\pi\)
							0.812283	−	0.583263i	\(-0.198224\pi\)
\(29\)	5099.66	−	6394.78i	1.12602	−	1.41199i	0.227102	−	0.973871i	\(-0.427075\pi\)
							0.898919	−	0.438115i	\(-0.144354\pi\)
\(31\)	3394.40	−	4256.45i	0.634394	−	0.795505i	−0.355895	−	0.934526i	\(-0.615824\pi\)
							0.990289	+	0.139021i	\(0.0443954\pi\)
\(37\)	11002.2			1.32122			0.660608	−	0.750731i	\(-0.270299\pi\)
							0.660608	+	0.750731i	\(0.270299\pi\)
\(41\)	−974.921	+	1222.51i	−0.0905753	+	0.113578i	−0.825054	−	0.565054i	\(-0.808855\pi\)
							0.734479	+	0.678632i	\(0.237427\pi\)
\(43\)	−11269.3	+	4473.31i	−0.929452	+	0.368942i
							\(47\)	3175.91	+	13914.6i	0.209712	+	0.918808i	0.964759	+	0.263137i	\(0.0847570\pi\)
−0.755047	+	0.655671i	\(0.772386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.5	✓	108
43.11	even	7	inner	43.6.e.a.11.5	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.5	✓	108	1.1	even	1	trivial
43.6.e.a.11.5	yes	108	43.11	even	7	inner