Embedded newform 43.4.e.a.11.8

• Label: 43.4.e.a.11.8
• Level: $43$
• Weight: $4$
• Character: 43.11
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	43.11
Dual form			43.4.e.a.4.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.80704 + 2.26595i) q^{2} +(-3.86902 + 4.85159i) q^{3} +(-0.0889949 + 0.389912i) q^{4} +(-19.3746 + 9.33033i) q^{5} -17.9850 q^{6} +25.5918 q^{7} +(19.8456 - 9.55715i) q^{8} +(-2.56060 - 11.2187i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.80704	+	2.26595i	0.638885	+	0.801136i	0.990863	−	0.134872i	\(-0.0430624\pi\)
							−0.351978	+	0.936008i	\(0.614491\pi\)
\(3\)	−3.86902	+	4.85159i	−0.744593	+	0.933690i	−0.999446	−	0.0332901i	\(-0.989401\pi\)
							0.254853	+	0.966980i	\(0.417973\pi\)
\(5\)	−19.3746	+	9.33033i	−1.73292	+	0.834530i	−0.747526	+	0.664232i	\(0.768759\pi\)
							−0.985393	+	0.170297i	\(0.945527\pi\)
\(7\)	25.5918			1.38183			0.690913	−	0.722938i	\(-0.257209\pi\)
							0.690913	+	0.722938i	\(0.257209\pi\)
\(11\)	9.85812	+	43.1912i	0.270212	+	1.18388i	0.909763	+	0.415128i	\(0.136263\pi\)
							−0.639551	+	0.768749i	\(0.720880\pi\)
\(13\)	−20.3427	+	9.79652i	−0.434004	+	0.209005i	−0.638113	−	0.769943i	\(-0.720285\pi\)
							0.204109	+	0.978948i	\(0.434570\pi\)
\(17\)	28.8475	+	13.8922i	0.411562	+	0.198198i	0.628196	−	0.778055i	\(-0.283794\pi\)
							−0.216634	+	0.976253i	\(0.569508\pi\)
\(19\)	17.0592	−	74.7412i	0.205982	−	0.902464i	−0.761228	−	0.648484i	\(-0.775403\pi\)
							0.967210	−	0.253980i	\(-0.0817396\pi\)
\(23\)	−11.2158	−	49.1397i	−0.101681	−	0.445493i	−0.999981	−	0.00611113i	\(-0.998055\pi\)
							0.898301	−	0.439382i	\(-0.144802\pi\)
\(29\)	−32.4464	−	40.6865i	−0.207764	−	0.260527i	0.667021	−	0.745039i	\(-0.267569\pi\)
							−0.874785	+	0.484511i	\(0.838998\pi\)
\(31\)	167.082	+	209.514i	0.968025	+	1.21386i	0.976855	+	0.213904i	\(0.0686179\pi\)
							−0.00882972	+	0.999961i	\(0.502811\pi\)
\(37\)	73.1460			0.325004			0.162502	−	0.986708i	\(-0.448044\pi\)
							0.162502	+	0.986708i	\(0.448044\pi\)
\(41\)	80.0687	+	100.403i	0.304991	+	0.382446i	0.910581	−	0.413330i	\(-0.135634\pi\)
							−0.605590	+	0.795776i	\(0.707063\pi\)
\(43\)	−241.708	−	145.204i	−0.857212	−	0.514963i
							\(47\)	−35.1914	+	154.183i	−0.109217	+	0.478510i	0.890506	+	0.454971i	\(0.150350\pi\)
−0.999723	+	0.0235388i	\(0.992507\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.11.8	yes	60
43.2	odd	14		1849.4.a.g.1.9		30
43.4	even	7	inner	43.4.e.a.4.8	✓	60
43.41	even	7		1849.4.a.h.1.22		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.8	✓	60	43.4	even	7	inner
43.4.e.a.11.8	yes	60	1.1	even	1	trivial
1849.4.a.g.1.9		30	43.2	odd	14
1849.4.a.h.1.22		30	43.41	even	7