Embedded newform 43.9.h.a.12.15

• Label: 43.9.h.a.12.15
• Level: $43$
• Weight: $9$
• Character: 43.12
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.15
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.645627 + 1.34066i) q^{2} +(42.1660 + 61.8461i) q^{3} +(158.233 - 198.418i) q^{4} +(-171.137 + 184.442i) q^{5} +(-55.6910 + 96.4597i) q^{6} +(2100.77 - 1212.88i) q^{7} +(739.552 + 168.798i) q^{8} +(350.028 - 891.857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.645627	+	1.34066i	0.0403517	+	0.0837911i	0.920170	−	0.391519i	\(-0.128050\pi\)
							−0.879818	+	0.475310i	\(0.842336\pi\)
\(3\)	42.1660	+	61.8461i	0.520568	+	0.763533i	0.993036	−	0.117815i	\(-0.0375891\pi\)
							−0.472468	+	0.881348i	\(0.656637\pi\)
\(5\)	−171.137	+	184.442i	−0.273820	+	0.295108i	−0.854958	−	0.518697i	\(-0.826417\pi\)
							0.581138	+	0.813805i	\(0.302608\pi\)
\(7\)	2100.77	−	1212.88i	0.874957	−	0.505156i	0.00596440	−	0.999982i	\(-0.498101\pi\)
							0.868992	+	0.494826i	\(0.164768\pi\)
\(11\)	−7362.46	−	9232.24i	−0.502866	−	0.630574i	0.464007	−	0.885832i	\(-0.346411\pi\)
							−0.966873	+	0.255257i	\(0.917840\pi\)
\(13\)	28444.9	−	8774.10i	0.995937	−	0.307206i	0.246400	−	0.969168i	\(-0.420752\pi\)
							0.749537	+	0.661962i	\(0.230276\pi\)
\(17\)	−114067.	+	105839.i	−1.36573	+	1.26722i	−0.435528	+	0.900175i	\(0.643438\pi\)
							−0.930205	+	0.367040i	\(0.880371\pi\)
\(19\)	179309.	−	70373.5i	1.37590	−	0.540001i	0.441833	−	0.897097i	\(-0.354328\pi\)
							0.934068	+	0.357096i	\(0.116233\pi\)
\(23\)	332656.	−	50139.9i	1.18873	−	0.179173i	0.475254	−	0.879849i	\(-0.342356\pi\)
							0.713479	+	0.700676i	\(0.247118\pi\)
\(29\)	229549.	−	336687.i	0.324552	−	0.476030i	−0.628938	−	0.777456i	\(-0.716510\pi\)
							0.953490	+	0.301425i	\(0.0974624\pi\)
\(31\)	23206.3	−	309667.i	0.0251281	−	0.335311i	−0.970445	−	0.241322i	\(-0.922419\pi\)
							0.995573	−	0.0939892i	\(-0.0299619\pi\)
\(37\)	1.03669e6	+	598533.i	0.553148	+	0.319360i	0.750391	−	0.660994i	\(-0.229865\pi\)
							−0.197242	+	0.980355i	\(0.563199\pi\)
\(41\)	−1.22849e6	+	591610.i	−0.434747	+	0.209363i	−0.638440	−	0.769672i	\(-0.720420\pi\)
							0.203693	+	0.979035i	\(0.434706\pi\)
\(43\)	3.41878e6	−	11644.1i	0.999994	−	0.00340590i
							\(47\)	−5.47918e6	+	6.87067e6i	−1.12286	+	1.40802i	−0.221380	+	0.975188i	\(0.571056\pi\)
−0.901476	+	0.432829i	\(0.857515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.15	✓	336
43.18	odd	42	inner	43.9.h.a.18.15	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.15	✓	336	1.1	even	1	trivial
43.9.h.a.18.15	yes	336	43.18	odd	42	inner