Embedded newform 43.9.h.a.18.9

• Label: 43.9.h.a.18.9
• Level: $43$
• Weight: $9$
• Character: 43.18
• 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			18.9
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.18476 + 10.7663i) q^{2} +(68.5364 - 100.524i) q^{3} +(70.5826 + 88.5078i) q^{4} +(713.104 + 768.544i) q^{5} +(726.928 + 1259.08i) q^{6} +(2048.19 + 1182.52i) q^{7} +(-4301.27 + 981.736i) q^{8} +(-3010.92 - 7671.70i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.18476	+	10.7663i	−0.324048	+	0.672892i	−0.997816	−	0.0660506i	\(-0.978960\pi\)
							0.673769	+	0.738942i	\(0.264674\pi\)
\(3\)	68.5364	−	100.524i	0.846128	−	1.24104i	−0.122485	−	0.992470i	\(-0.539086\pi\)
							0.968613	−	0.248572i	\(-0.0799612\pi\)
\(5\)	713.104	+	768.544i	1.14097	+	1.22967i	0.969661	+	0.244452i	\(0.0786081\pi\)
							0.171306	+	0.985218i	\(0.445201\pi\)
\(7\)	2048.19	+	1182.52i	0.853058	+	0.492513i	0.861681	−	0.507450i	\(-0.169412\pi\)
							−0.00862380	+	0.999963i	\(0.502745\pi\)
\(11\)	−5936.51	+	7444.15i	−0.405472	+	0.508445i	−0.942081	−	0.335385i	\(-0.891133\pi\)
							0.536609	+	0.843831i	\(0.319705\pi\)
\(13\)	−4149.30	−	1279.89i	−0.145279	−	0.0448125i	0.221263	−	0.975214i	\(-0.428982\pi\)
							−0.366542	+	0.930402i	\(0.619458\pi\)
\(17\)	33692.3	+	31261.9i	0.403399	+	0.374299i	0.855618	−	0.517607i	\(-0.173177\pi\)
							−0.452220	+	0.891907i	\(0.649368\pi\)
\(19\)	−159821.	−	62725.3i	−1.22637	−	0.481314i	−0.338305	−	0.941037i	\(-0.609853\pi\)
							−0.888062	+	0.459723i	\(0.847949\pi\)
\(23\)	144894.	+	21839.3i	0.517773	+	0.0780418i	0.402730	−	0.915319i	\(-0.368061\pi\)
							0.115043	+	0.993361i	\(0.463299\pi\)
\(29\)	−300368.	−	440559.i	−0.424680	−	0.622892i	0.552952	−	0.833213i	\(-0.313501\pi\)
							−0.977632	+	0.210322i	\(0.932549\pi\)
\(31\)	61700.0	+	823329.i	0.0668095	+	0.891511i	0.925435	+	0.378907i	\(0.123700\pi\)
							−0.858625	+	0.512604i	\(0.828681\pi\)
\(37\)	2.80149e6	−	1.61744e6i	1.49480	−	0.863020i	0.494813	−	0.868999i	\(-0.335237\pi\)
							0.999982	+	0.00597906i	\(0.00190321\pi\)
\(41\)	1.94093e6	+	934704.i	0.686871	+	0.330779i	0.744560	−	0.667555i	\(-0.232659\pi\)
							−0.0576898	+	0.998335i	\(0.518373\pi\)
\(43\)	1.49928e6	−	3.07252e6i	0.438539	−	0.898712i
							\(47\)	1.13124e6	+	1.41853e6i	0.231827	+	0.290702i	0.884115	−	0.467269i	\(-0.154762\pi\)
−0.652288	+	0.757971i	\(0.726191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.9	yes	336
43.12	odd	42	inner	43.9.h.a.12.9	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.9	✓	336	43.12	odd	42	inner
43.9.h.a.18.9	yes	336	1.1	even	1	trivial