Embedded newform 43.6.g.a.10.8

• Label: 43.6.g.a.10.8
• Level: $43$
• Weight: $6$
• Character: 43.10
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.8
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.257084 - 1.12636i) q^{2} +(-18.5027 - 17.1680i) q^{3} +(27.6284 - 13.3051i) q^{4} +(37.0106 - 5.57846i) q^{5} +(-14.5806 + 25.2543i) q^{6} +(-53.8830 - 93.3282i) q^{7} +(-45.1399 - 56.6037i) q^{8} +(29.4502 + 392.986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.257084	−	1.12636i	−0.0454465	−	0.199114i	0.947108	−	0.320915i	\(-0.103990\pi\)
							−0.992555	+	0.121801i	\(0.961133\pi\)
\(3\)	−18.5027	−	17.1680i	−1.18695	−	1.10133i	−0.992729	−	0.120367i	\(-0.961593\pi\)
							−0.194217	−	0.980959i	\(-0.562217\pi\)
\(5\)	37.0106	−	5.57846i	0.662066	−	0.0997905i	0.190591	−	0.981670i	\(-0.438960\pi\)
							0.471476	+	0.881879i	\(0.343722\pi\)
\(7\)	−53.8830	−	93.3282i	−0.415630	−	0.719892i	0.579864	−	0.814713i	\(-0.303106\pi\)
							−0.995494	+	0.0948206i	\(0.969772\pi\)
\(11\)	−544.932	−	262.425i	−1.35788	−	0.653919i	−0.393715	−	0.919233i	\(-0.628810\pi\)
							−0.964162	+	0.265313i	\(0.914525\pi\)
\(13\)	−393.225	+	1001.92i	−0.645331	+	1.64428i	0.113929	+	0.993489i	\(0.463656\pi\)
							−0.759260	+	0.650787i	\(0.774439\pi\)
\(17\)	905.750	+	136.520i	0.760126	+	0.114571i	0.517657	−	0.855588i	\(-0.326804\pi\)
							0.242470	+	0.970159i	\(0.422043\pi\)
\(19\)	139.418	−	1860.41i	0.0886003	−	1.18229i	−0.758796	−	0.651328i	\(-0.774212\pi\)
							0.847397	−	0.530961i	\(-0.178169\pi\)
\(23\)	−275.382	+	187.752i	−0.108546	+	0.0740057i	−0.616377	−	0.787451i	\(-0.711400\pi\)
							0.507830	+	0.861457i	\(0.330448\pi\)
\(29\)	3405.37	−	3159.72i	0.751916	−	0.697676i	−0.207948	−	0.978140i	\(-0.566679\pi\)
							0.959865	+	0.280463i	\(0.0904881\pi\)
\(31\)	130.337	+	40.2036i	0.0243592	+	0.00751383i	0.306911	−	0.951738i	\(-0.400705\pi\)
							−0.282552	+	0.959252i	\(0.591181\pi\)
\(37\)	3671.80	−	6359.74i	0.440935	−	0.763721i	−0.556824	−	0.830630i	\(-0.687980\pi\)
							0.997759	+	0.0669089i	\(0.0213137\pi\)
\(41\)	−3982.70	−	17449.3i	−0.370013	−	1.62113i	−0.726731	−	0.686922i	\(-0.758961\pi\)
							0.356717	−	0.934212i	\(-0.383896\pi\)
\(43\)	2316.52	+	11901.4i	0.191058	+	0.981579i
							\(47\)	1746.65	−	841.143i	0.115335	−	0.0555425i	−0.375326	−	0.926893i	\(-0.622469\pi\)
0.490662	+	0.871350i	\(0.336755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.8	✓	204
43.13	even	21	inner	43.6.g.a.13.8	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.8	✓	204	1.1	even	1	trivial
43.6.g.a.13.8	yes	204	43.13	even	21	inner