Embedded newform 43.10.e.a.4.18

• Label: 43.10.e.a.4.18
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.18
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.75125 - 3.44996i) q^{2} +(-150.990 - 189.336i) q^{3} +(109.598 + 480.180i) q^{4} +(-2399.77 - 1155.67i) q^{5} -1068.61 q^{6} -8095.40 q^{7} +(3993.68 + 1923.26i) q^{8} +(-8670.08 + 37986.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.75125	−	3.44996i	0.121589	−	0.152468i	−0.717311	−	0.696753i	\(-0.754627\pi\)
							0.838901	+	0.544285i	\(0.183199\pi\)
\(3\)	−150.990	−	189.336i	−1.07622	−	1.34954i	−0.933010	−	0.359851i	\(-0.882828\pi\)
							−0.143215	−	0.989692i	\(-0.545744\pi\)
\(5\)	−2399.77	−	1155.67i	−1.71713	−	0.826929i	−0.990106	−	0.140322i	\(-0.955186\pi\)
							−0.727029	−	0.686607i	\(-0.759099\pi\)
\(7\)	−8095.40			−1.27437			−0.637187	−	0.770709i	\(-0.719902\pi\)
							−0.637187	+	0.770709i	\(0.719902\pi\)
\(11\)	5611.54	−	24585.8i	0.115562	−	0.506310i	−0.883706	−	0.468043i	\(-0.844959\pi\)
							0.999268	−	0.0382669i	\(-0.0121837\pi\)
\(13\)	−85673.9	−	41258.4i	−0.831962	−	0.400652i	−0.0311114	−	0.999516i	\(-0.509905\pi\)
							−0.800851	+	0.598864i	\(0.795619\pi\)
\(17\)	5566.49	−	2680.68i	0.0161645	−	0.00778440i	−0.425784	−	0.904825i	\(-0.640002\pi\)
							0.441949	+	0.897040i	\(0.354287\pi\)
\(19\)	−189202.	−	828946.i	−0.333068	−	1.45927i	−0.813155	−	0.582047i	\(-0.802252\pi\)
							0.480087	−	0.877221i	\(-0.340605\pi\)
\(23\)	317039.	−	1.38904e6i	0.236231	−	1.03500i	−0.708129	−	0.706083i	\(-0.750461\pi\)
							0.944360	−	0.328913i	\(-0.106682\pi\)
\(29\)	1.63029e6	−	2.04432e6i	0.428030	−	0.536733i	−0.520315	−	0.853975i	\(-0.674185\pi\)
							0.948345	+	0.317242i	\(0.102757\pi\)
\(31\)	−2.86815e6	+	3.59654e6i	−0.557794	+	0.699451i	−0.978148	−	0.207909i	\(-0.933334\pi\)
							0.420354	+	0.907360i	\(0.361906\pi\)
\(37\)	1.19228e7			1.04585			0.522926	−	0.852378i	\(-0.324840\pi\)
							0.522926	+	0.852378i	\(0.324840\pi\)
\(41\)	−5.63307e6	+	7.06365e6i	−0.311328	+	0.390393i	−0.912736	−	0.408549i	\(-0.866035\pi\)
							0.601409	+	0.798942i	\(0.294606\pi\)
\(43\)	−2.24022e7	+	855642.i	−0.999271	+	0.0381667i
							\(47\)	−1.04180e6	−	4.56442e6i	−0.0311418	−	0.136441i	0.956967	−	0.290196i	\(-0.0937205\pi\)
−0.988109	+	0.153755i	\(0.950863\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.18	✓	192
43.11	even	7	inner	43.10.e.a.11.18	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.18	✓	192	1.1	even	1	trivial
43.10.e.a.11.18	yes	192	43.11	even	7	inner