Embedded newform 43.4.g.a.13.7

• Label: 43.4.g.a.13.7
• Level: $43$
• Weight: $4$
• Character: 43.13
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			13.7
Character	\(\chi\)	\(=\)	43.13
Dual form			43.4.g.a.10.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.399395 - 1.74986i) q^{2} +(-2.95769 + 2.74433i) q^{3} +(4.30524 + 2.07330i) q^{4} +(12.0373 + 1.81433i) q^{5} +(3.62092 + 6.27163i) q^{6} +(7.86579 - 13.6240i) q^{7} +(14.3001 - 17.9318i) q^{8} +(-0.801160 + 10.6907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.399395	−	1.74986i	0.141207	−	0.618670i	−0.853948	−	0.520358i	\(-0.825798\pi\)
							0.995156	−	0.0983123i	\(-0.0313444\pi\)
\(3\)	−2.95769	+	2.74433i	−0.569207	+	0.528147i	−0.911421	−	0.411475i	\(-0.865014\pi\)
							0.342214	+	0.939622i	\(0.388823\pi\)
\(5\)	12.0373	+	1.81433i	1.07665	+	0.162278i	0.663343	−	0.748315i	\(-0.269137\pi\)
							0.413303	+	0.910594i	\(0.364375\pi\)
\(7\)	7.86579	−	13.6240i	0.424713	−	0.735624i	−0.571681	−	0.820476i	\(-0.693708\pi\)
							0.996394	+	0.0848518i	\(0.0270417\pi\)
\(11\)	−19.4809	+	9.38152i	−0.533974	+	0.257148i	−0.681383	−	0.731927i	\(-0.738621\pi\)
							0.147408	+	0.989076i	\(0.452907\pi\)
\(13\)	1.46689	+	3.73758i	0.0312956	+	0.0797399i	0.945667	−	0.325136i	\(-0.105410\pi\)
							−0.914372	+	0.404876i	\(0.867315\pi\)
\(17\)	−24.7041	+	3.72355i	−0.352449	+	0.0531231i	−0.322881	−	0.946440i	\(-0.604651\pi\)
							−0.0295681	+	0.999563i	\(0.509413\pi\)
\(19\)	0.505800	+	6.74943i	0.00610729	+	0.0814961i	0.999425	−	0.0339196i	\(-0.0107990\pi\)
							−0.993317	+	0.115416i	\(0.963180\pi\)
\(23\)	−147.392	−	100.490i	−1.33623	−	0.911028i	−0.336701	−	0.941612i	\(-0.609311\pi\)
							−0.999532	+	0.0305832i	\(0.990264\pi\)
\(29\)	−124.930	−	115.918i	−0.799962	−	0.742256i	0.170018	−	0.985441i	\(-0.445617\pi\)
							−0.969980	+	0.243185i	\(0.921808\pi\)
\(31\)	−40.5849	+	12.5188i	−0.235138	+	0.0725303i	−0.410085	−	0.912048i	\(-0.634501\pi\)
							0.174947	+	0.984578i	\(0.444025\pi\)
\(37\)	220.843	+	382.512i	0.981254	+	1.69958i	0.657526	+	0.753432i	\(0.271603\pi\)
							0.323728	+	0.946150i	\(0.395064\pi\)
\(41\)	−11.8359	+	51.8565i	−0.0450844	+	0.197528i	−0.992455	−	0.122612i	\(-0.960873\pi\)
							0.947370	+	0.320140i	\(0.103730\pi\)
\(43\)	85.2599	−	268.771i	0.302372	−	0.953190i
							\(47\)	213.150	+	102.648i	0.661514	+	0.318568i	0.734346	−	0.678776i	\(-0.237489\pi\)
−0.0728313	+	0.997344i	\(0.523203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.13.7	yes	120
43.10	even	21	inner	43.4.g.a.10.7	✓	120
43.15	even	21		1849.4.a.l.1.21		60
43.28	odd	42		1849.4.a.k.1.40		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.7	✓	120	43.10	even	21	inner
43.4.g.a.13.7	yes	120	1.1	even	1	trivial
1849.4.a.k.1.40		60	43.28	odd	42
1849.4.a.l.1.21		60	43.15	even	21