Embedded newform 43.4.e.a.4.10

• Label: 43.4.e.a.4.10
• Level: $43$
• Weight: $4$
• Character: 43.4
• 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			4.10
Character	\(\chi\)	\(=\)	43.4
Dual form			43.4.e.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.96047 - 3.71232i) q^{2} +(4.10366 + 5.14583i) q^{3} +(-3.23672 - 14.1810i) q^{4} +(-6.52318 - 3.14140i) q^{5} +31.2517 q^{6} +6.49418 q^{7} +(-28.0025 - 13.4853i) q^{8} +(-3.63147 + 15.9105i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.96047	−	3.71232i	1.04669	−	1.31250i	0.0983775	−	0.995149i	\(-0.468635\pi\)
							0.948308	−	0.317353i	\(-0.102794\pi\)
\(3\)	4.10366	+	5.14583i	0.789751	+	0.990316i	0.999920	+	0.0126510i	\(0.00402704\pi\)
							−0.210169	+	0.977665i	\(0.567402\pi\)
\(5\)	−6.52318	−	3.14140i	−0.583451	−	0.280975i	0.118785	−	0.992920i	\(-0.462100\pi\)
							−0.702235	+	0.711945i	\(0.747814\pi\)
\(7\)	6.49418			0.350653			0.175326	−	0.984510i	\(-0.443902\pi\)
							0.175326	+	0.984510i	\(0.443902\pi\)
\(11\)	−14.2157	+	62.2832i	−0.389655	+	1.70719i	0.276194	+	0.961102i	\(0.410927\pi\)
							−0.665848	+	0.746087i	\(0.731930\pi\)
\(13\)	−60.9667	−	29.3600i	−1.30070	−	0.626385i	−0.350076	−	0.936721i	\(-0.613844\pi\)
							−0.950627	+	0.310336i	\(0.899558\pi\)
\(17\)	3.84960	−	1.85387i	0.0549214	−	0.0264488i	−0.406221	−	0.913775i	\(-0.633154\pi\)
							0.461143	+	0.887326i	\(0.347440\pi\)
\(19\)	25.0832	+	109.897i	0.302867	+	1.32695i	0.865777	+	0.500429i	\(0.166825\pi\)
							−0.562910	+	0.826518i	\(0.690318\pi\)
\(23\)	21.8573	−	95.7630i	0.198155	−	0.868172i	−0.773879	−	0.633333i	\(-0.781686\pi\)
							0.972034	−	0.234839i	\(-0.0754564\pi\)
\(29\)	179.507	−	225.095i	1.14944	−	1.44135i	0.271584	−	0.962415i	\(-0.412452\pi\)
							0.877852	−	0.478933i	\(-0.158976\pi\)
\(31\)	−56.8270	+	71.2588i	−0.329239	+	0.412853i	−0.918708	−	0.394938i	\(-0.870766\pi\)
							0.589468	+	0.807792i	\(0.299337\pi\)
\(37\)	−207.629			−0.922542			−0.461271	−	0.887259i	\(-0.652606\pi\)
							−0.461271	+	0.887259i	\(0.652606\pi\)
\(41\)	151.751	−	190.290i	0.578039	−	0.724837i	−0.403738	−	0.914874i	\(-0.632289\pi\)
							0.981777	+	0.190037i	\(0.0608608\pi\)
\(43\)	101.435	+	263.093i	0.359737	+	0.933054i
							\(47\)	16.8156	+	73.6740i	0.0521874	+	0.228648i	0.994295	−	0.106662i	\(-0.0340162\pi\)
−0.942108	+	0.335310i	\(0.891159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.4.10	✓	60
43.11	even	7	inner	43.4.e.a.11.10	yes	60
43.21	even	7		1849.4.a.h.1.27		30
43.22	odd	14		1849.4.a.g.1.4		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.10	✓	60	1.1	even	1	trivial
43.4.e.a.11.10	yes	60	43.11	even	7	inner
1849.4.a.g.1.4		30	43.22	odd	14
1849.4.a.h.1.27		30	43.21	even	7