Embedded newform 43.7.h.a.12.2

• Label: 43.7.h.a.12.2
• 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			12.2
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.15640 - 12.7839i) q^{2} +(4.85408 + 7.11962i) q^{3} +(-85.6233 + 107.368i) q^{4} +(-85.2874 + 91.9179i) q^{5} +(61.1329 - 105.885i) q^{6} +(94.7175 - 54.6852i) q^{7} +(1014.38 + 231.527i) q^{8} +(239.207 - 609.489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.15640	−	12.7839i	−0.769550	−	1.59799i	−0.801134	−	0.598486i	\(-0.795769\pi\)
							0.0315838	−	0.999501i	\(-0.489945\pi\)
\(3\)	4.85408	+	7.11962i	0.179781	+	0.263690i	0.905542	−	0.424257i	\(-0.139465\pi\)
							−0.725761	+	0.687947i	\(0.758512\pi\)
\(5\)	−85.2874	+	91.9179i	−0.682299	+	0.735343i	−0.975328	−	0.220760i	\(-0.929146\pi\)
							0.293029	+	0.956104i	\(0.405337\pi\)
\(7\)	94.7175	−	54.6852i	0.276144	−	0.159432i	−0.355532	−	0.934664i	\(-0.615700\pi\)
							0.631677	+	0.775232i	\(0.282367\pi\)
\(11\)	596.633	+	748.154i	0.448259	+	0.562099i	0.953699	−	0.300763i	\(-0.0972413\pi\)
							−0.505440	+	0.862862i	\(0.668670\pi\)
\(13\)	1146.31	−	353.589i	0.521760	−	0.160942i	−0.0226776	−	0.999743i	\(-0.507219\pi\)
							0.544438	+	0.838801i	\(0.316743\pi\)
\(17\)	1928.07	−	1788.99i	0.392442	−	0.364133i	−0.459131	−	0.888368i	\(-0.651839\pi\)
							0.851573	+	0.524235i	\(0.175649\pi\)
\(19\)	320.931	−	125.956i	0.0467897	−	0.0183636i	−0.341831	−	0.939762i	\(-0.611047\pi\)
							0.388620	+	0.921398i	\(0.372952\pi\)
\(23\)	11750.6	−	1771.12i	0.965775	−	0.145567i	0.352826	−	0.935689i	\(-0.385221\pi\)
							0.612950	+	0.790122i	\(0.289983\pi\)
\(29\)	5365.62	−	7869.92i	0.220002	−	0.322683i	−0.700332	−	0.713817i	\(-0.746965\pi\)
							0.920334	+	0.391134i	\(0.127917\pi\)
\(31\)	−3279.17	+	43757.5i	−0.110073	+	1.46882i	0.619545	+	0.784961i	\(0.287317\pi\)
							−0.729617	+	0.683856i	\(0.760302\pi\)
\(37\)	61056.0	+	35250.7i	1.20538	+	0.695926i	0.961746	−	0.273942i	\(-0.0883276\pi\)
							0.243632	+	0.969868i	\(0.421661\pi\)
\(41\)	113053.	−	54443.7i	1.64033	−	0.789943i	0.640577	−	0.767894i	\(-0.278695\pi\)
							0.999757	−	0.0220495i	\(-0.00701913\pi\)
\(43\)	−41138.7	−	68036.5i	−0.517423	−	0.855730i
							\(47\)	69659.2	−	87349.9i	0.670942	−	0.841335i	−0.323543	−	0.946214i	\(-0.604874\pi\)
0.994485	+	0.104879i	\(0.0334454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.2	✓	252
43.18	odd	42	inner	43.7.h.a.18.2	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.2	✓	252	1.1	even	1	trivial
43.7.h.a.18.2	yes	252	43.18	odd	42	inner