Embedded newform 43.7.h.a.12.12

• Label: 43.7.h.a.12.12
• 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			12.12
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.965900 + 2.00571i) q^{2} +(-22.0783 - 32.3829i) q^{3} +(36.8134 - 46.1626i) q^{4} +(89.0872 - 96.0132i) q^{5} +(43.6253 - 75.5612i) q^{6} +(-260.183 + 150.217i) q^{7} +(267.050 + 60.9524i) q^{8} +(-294.867 + 751.308i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.965900	+	2.00571i	0.120737	+	0.250714i	0.952574	−	0.304307i	\(-0.0984247\pi\)
							−0.831837	+	0.555020i	\(0.812710\pi\)
\(3\)	−22.0783	−	32.3829i	−0.817713	−	1.19937i	−0.977233	−	0.212171i	\(-0.931947\pi\)
							0.159519	−	0.987195i	\(-0.449006\pi\)
\(5\)	89.0872	−	96.0132i	0.712698	−	0.768105i	−0.268046	−	0.963406i	\(-0.586378\pi\)
							0.980743	+	0.195301i	\(0.0625683\pi\)
\(7\)	−260.183	+	150.217i	−0.758550	+	0.437949i	−0.828775	−	0.559582i	\(-0.810962\pi\)
							0.0702248	+	0.997531i	\(0.477628\pi\)
\(11\)	−688.672	−	863.568i	−0.517410	−	0.648811i	0.452647	−	0.891690i	\(-0.350480\pi\)
							−0.970057	+	0.242879i	\(0.921908\pi\)
\(13\)	−194.444	+	59.9779i	−0.0885042	+	0.0272999i	−0.338691	−	0.940898i	\(-0.609984\pi\)
							0.250187	+	0.968198i	\(0.419508\pi\)
\(17\)	−2896.09	+	2687.18i	−0.589476	+	0.546954i	−0.917510	−	0.397712i	\(-0.869804\pi\)
							0.328034	+	0.944666i	\(0.393614\pi\)
\(19\)	6018.67	−	2362.15i	0.877485	−	0.344388i	0.116526	−	0.993188i	\(-0.462824\pi\)
							0.760959	+	0.648800i	\(0.224729\pi\)
\(23\)	−9947.15	+	1499.29i	−0.817552	+	0.123226i	−0.544487	−	0.838770i	\(-0.683275\pi\)
							−0.273065	+	0.961996i	\(0.588037\pi\)
\(29\)	−8256.13	+	12109.5i	−0.338518	+	0.496515i	−0.957319	−	0.289033i	\(-0.906666\pi\)
							0.618801	+	0.785548i	\(0.287619\pi\)
\(31\)	1510.66	−	20158.4i	0.0507087	−	0.676661i	−0.912701	−	0.408628i	\(-0.866007\pi\)
							0.963410	−	0.268033i	\(-0.0863736\pi\)
\(37\)	−22051.8	−	12731.6i	−0.435350	−	0.251349i	0.266273	−	0.963898i	\(-0.414208\pi\)
							−0.701623	+	0.712548i	\(0.747541\pi\)
\(41\)	110543.	−	53234.5i	1.60390	−	0.772398i	0.604200	−	0.796833i	\(-0.293493\pi\)
							0.999701	+	0.0244341i	\(0.00777841\pi\)
\(43\)	66717.8	−	43244.7i	0.839143	−	0.543910i
							\(47\)	126501.	−	158627.i	1.21842	−	1.52786i	0.442857	−	0.896592i	\(-0.353965\pi\)
0.775568	−	0.631264i	\(-0.217464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.12	✓	252
43.18	odd	42	inner	43.7.h.a.18.12	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.12	✓	252	1.1	even	1	trivial
43.7.h.a.18.12	yes	252	43.18	odd	42	inner