Embedded newform 43.7.h.a.18.13

• Label: 43.7.h.a.18.13
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.13
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04638 - 2.17283i) q^{2} +(-3.26146 + 4.78369i) q^{3} +(36.2771 + 45.4900i) q^{4} +(-105.155 - 113.330i) q^{5} +(6.98143 + 12.0922i) q^{6} +(260.538 + 150.422i) q^{7} +(287.278 - 65.5694i) q^{8} +(254.087 + 647.403i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04638	−	2.17283i	0.130798	−	0.271604i	−0.825277	−	0.564729i	\(-0.808981\pi\)
							0.956074	+	0.293125i	\(0.0946951\pi\)
\(3\)	−3.26146	+	4.78369i	−0.120795	+	0.177174i	−0.881874	−	0.471486i	\(-0.843718\pi\)
							0.761079	+	0.648660i	\(0.224670\pi\)
\(5\)	−105.155	−	113.330i	−0.841239	−	0.906640i	0.155459	−	0.987842i	\(-0.450314\pi\)
							−0.996698	+	0.0812024i	\(0.974124\pi\)
\(7\)	260.538	+	150.422i	0.759587	+	0.438548i	0.829147	−	0.559030i	\(-0.188826\pi\)
							−0.0695606	+	0.997578i	\(0.522160\pi\)
\(11\)	−671.040	+	841.457i	−0.504162	+	0.632199i	−0.967163	−	0.254159i	\(-0.918201\pi\)
							0.463000	+	0.886358i	\(0.346773\pi\)
\(13\)	608.041	+	187.556i	0.276760	+	0.0853691i	0.430028	−	0.902816i	\(-0.358504\pi\)
							−0.153268	+	0.988185i	\(0.548980\pi\)
\(17\)	5150.10	+	4778.59i	1.04826	+	0.972643i	0.999657	−	0.0261754i	\(-0.00833283\pi\)
							0.0486023	+	0.998818i	\(0.484523\pi\)
\(19\)	10285.6	+	4036.80i	1.49958	+	0.588541i	0.966305	−	0.257401i	\(-0.0828663\pi\)
							0.533274	+	0.845943i	\(0.320962\pi\)
\(23\)	−7816.97	−	1178.22i	−0.642473	−	0.0968373i	−0.180278	−	0.983616i	\(-0.557700\pi\)
							−0.462195	+	0.886778i	\(0.652938\pi\)
\(29\)	−22314.7	−	32729.7i	−0.914951	−	1.34199i	−0.939594	−	0.342290i	\(-0.888797\pi\)
							0.0246432	−	0.999696i	\(-0.492155\pi\)
\(31\)	1399.25	+	18671.7i	0.0469688	+	0.626756i	0.970168	+	0.242434i	\(0.0779457\pi\)
							−0.923199	+	0.384322i	\(0.874435\pi\)
\(37\)	37950.6	−	21910.8i	0.749228	−	0.432567i	−0.0761871	−	0.997094i	\(-0.524275\pi\)
							0.825415	+	0.564527i	\(0.190941\pi\)
\(41\)	−92667.0	−	44626.1i	−1.34454	−	0.647496i	−0.383406	−	0.923580i	\(-0.625249\pi\)
							−0.961134	+	0.276084i	\(0.910963\pi\)
\(43\)	−61784.8	+	50040.0i	−0.777099	+	0.629379i
							\(47\)	−18307.7	−	22957.1i	−0.176336	−	0.221118i	0.685807	−	0.727783i	\(-0.259449\pi\)
−0.862143	+	0.506665i	\(0.830878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.18.13	yes	252
43.12	odd	42	inner	43.7.h.a.12.13	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.13	✓	252	43.12	odd	42	inner
43.7.h.a.18.13	yes	252	1.1	even	1	trivial