Embedded newform 43.4.e.a.41.6

• Label: 43.4.e.a.41.6
• Level: $43$
• Weight: $4$
• Character: 43.41
• 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			41.6
Character	\(\chi\)	\(=\)	43.41
Dual form			43.4.e.a.21.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.170046 - 0.0818897i) q^{2} +(0.966806 + 0.465589i) q^{3} +(-4.96571 + 6.22680i) q^{4} +(-3.57419 + 15.6596i) q^{5} +0.202528 q^{6} +29.7402 q^{7} +(-0.670469 + 2.93752i) q^{8} +(-16.1163 - 20.2092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.170046	−	0.0818897i	0.0601203	−	0.0289524i	−0.403582	−	0.914943i	\(-0.632235\pi\)
							0.463702	+	0.885991i	\(0.346521\pi\)
\(3\)	0.966806	+	0.465589i	0.186062	+	0.0896027i	0.524595	−	0.851352i	\(-0.324217\pi\)
							−0.338534	+	0.940954i	\(0.609931\pi\)
\(5\)	−3.57419	+	15.6596i	−0.319685	+	1.40063i	0.518420	+	0.855126i	\(0.326520\pi\)
							−0.838106	+	0.545508i	\(0.816337\pi\)
\(7\)	29.7402			1.60582			0.802911	−	0.596099i	\(-0.203283\pi\)
							0.802911	+	0.596099i	\(0.203283\pi\)
\(11\)	7.64529	+	9.58689i	0.209558	+	0.262778i	0.875491	−	0.483234i	\(-0.160538\pi\)
							−0.665933	+	0.746011i	\(0.731966\pi\)
\(13\)	−8.30105	+	36.3693i	−0.177100	+	0.775925i	0.805860	+	0.592106i	\(0.201703\pi\)
							−0.982960	+	0.183819i	\(0.941154\pi\)
\(17\)	−7.89957	−	34.6103i	−0.112702	−	0.493778i	−0.999500	−	0.0316214i	\(-0.989933\pi\)
							0.886798	−	0.462157i	\(-0.152924\pi\)
\(19\)	59.7314	−	74.9009i	0.721228	−	0.904391i	−0.277179	−	0.960818i	\(-0.589399\pi\)
							0.998407	+	0.0564270i	\(0.0179708\pi\)
\(23\)	76.9282	+	96.4649i	0.697419	+	0.874536i	0.996828	−	0.0795892i	\(-0.0253609\pi\)
							−0.299409	+	0.954125i	\(0.596789\pi\)
\(29\)	175.440	−	84.4876i	1.12340	−	0.540999i	0.222455	−	0.974943i	\(-0.428593\pi\)
							0.900940	+	0.433944i	\(0.142879\pi\)
\(31\)	57.7065	−	27.7900i	0.334335	−	0.161007i	−0.259182	−	0.965828i	\(-0.583453\pi\)
							0.593518	+	0.804821i	\(0.297739\pi\)
\(37\)	286.712			1.27392			0.636962	−	0.770895i	\(-0.280191\pi\)
							0.636962	+	0.770895i	\(0.280191\pi\)
\(41\)	−398.287	+	191.805i	−1.51712	+	0.730607i	−0.992673	−	0.120835i	\(-0.961443\pi\)
							−0.524448	+	0.851442i	\(0.675728\pi\)
\(43\)	−112.354	−	258.619i	−0.398460	−	0.917186i
							\(47\)	−289.060	+	362.470i	−0.897100	+	1.12493i	0.0944920	+	0.995526i	\(0.469877\pi\)
−0.991592	+	0.129402i	\(0.958694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.41.6	yes	60
43.8	odd	14		1849.4.a.g.1.16		30
43.21	even	7	inner	43.4.e.a.21.6	✓	60
43.35	even	7		1849.4.a.h.1.15		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.21.6	✓	60	43.21	even	7	inner
43.4.e.a.41.6	yes	60	1.1	even	1	trivial
1849.4.a.g.1.16		30	43.8	odd	14
1849.4.a.h.1.15		30	43.35	even	7