Embedded newform 43.7.h.a.12.18

• Label: 43.7.h.a.12.18
• Level: $43$
• Weight: $7$
• Character: 43.12
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.18
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.45532 + 11.3281i) q^{2} +(-9.73743 - 14.2822i) q^{3} +(-58.6616 + 73.5593i) q^{4} +(-58.7717 + 63.3408i) q^{5} +(108.669 - 188.220i) q^{6} +(-290.866 + 167.932i) q^{7} +(-368.793 - 84.1745i) q^{8} +(157.170 - 400.464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.45532	+	11.3281i	0.681915	+	1.41601i	0.898159	+	0.439671i	\(0.144905\pi\)
							−0.216244	+	0.976339i	\(0.569381\pi\)
\(3\)	−9.73743	−	14.2822i	−0.360645	−	0.528970i	0.602441	−	0.798164i	\(-0.294195\pi\)
							−0.963086	+	0.269194i	\(0.913243\pi\)
\(5\)	−58.7717	+	63.3408i	−0.470174	+	0.506727i	−0.922962	−	0.384892i	\(-0.874239\pi\)
							0.452788	+	0.891618i	\(0.350429\pi\)
\(7\)	−290.866	+	167.932i	−0.848006	+	0.489597i	−0.859978	−	0.510332i	\(-0.829523\pi\)
							0.0119713	+	0.999928i	\(0.496189\pi\)
\(11\)	−1328.18	−	1665.49i	−0.997882	−	1.25130i	−0.967791	−	0.251757i	\(-0.918992\pi\)
							−0.0300911	−	0.999547i	\(-0.509580\pi\)
\(13\)	−1680.67	+	518.418i	−0.764984	+	0.235966i	−0.652604	−	0.757699i	\(-0.726323\pi\)
							−0.112380	+	0.993665i	\(0.535847\pi\)
\(17\)	−1047.07	+	971.543i	−0.213123	+	0.197750i	−0.779509	−	0.626390i	\(-0.784532\pi\)
							0.566386	+	0.824140i	\(0.308341\pi\)
\(19\)	−4169.06	+	1636.23i	−0.607823	+	0.238553i	−0.649233	−	0.760589i	\(-0.724910\pi\)
							0.0414105	+	0.999142i	\(0.486815\pi\)
\(23\)	13769.4	−	2075.40i	1.13170	−	0.170576i	0.443622	−	0.896214i	\(-0.353693\pi\)
							0.688077	+	0.725638i	\(0.258455\pi\)
\(29\)	−13235.5	+	19412.9i	−0.542682	+	0.795969i	−0.995412	−	0.0956789i	\(-0.969498\pi\)
							0.452730	+	0.891648i	\(0.350450\pi\)
\(31\)	−2782.80	+	37133.9i	−0.0934107	+	1.24648i	0.731747	+	0.681576i	\(0.238705\pi\)
							−0.825158	+	0.564902i	\(0.808914\pi\)
\(37\)	−22650.2	−	13077.1i	−0.447165	−	0.258171i	0.259467	−	0.965752i	\(-0.416453\pi\)
							−0.706632	+	0.707581i	\(0.749786\pi\)
\(41\)	3409.35	−	1641.86i	0.0494675	−	0.0238223i	−0.408986	−	0.912540i	\(-0.634118\pi\)
							0.458454	+	0.888718i	\(0.348403\pi\)
\(43\)	28881.1	−	74075.9i	0.363252	−	0.931691i
							\(47\)	−26846.6	+	33664.5i	−0.258580	+	0.324249i	−0.894127	−	0.447813i	\(-0.852203\pi\)
0.635547	+	0.772062i	\(0.280775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.18	✓	252
43.18	odd	42	inner	43.7.h.a.18.18	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.18	✓	252	1.1	even	1	trivial
43.7.h.a.18.18	yes	252	43.18	odd	42	inner