Embedded newform 43.9.h.a.3.4

• Label: 43.9.h.a.3.4
• Level: $43$
• Weight: $9$
• Character: 43.3
• 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			3.4
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.2950 + 23.4544i) q^{2} +(75.5829 + 5.66415i) q^{3} +(-262.917 - 329.688i) q^{4} +(125.810 - 407.865i) q^{5} +(-986.561 + 1708.77i) q^{6} +(-2233.56 + 1289.54i) q^{7} +(4205.09 - 959.784i) q^{8} +(-807.029 - 121.640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.2950	+	23.4544i	−0.705940	+	1.46590i	0.170985	+	0.985274i	\(0.445305\pi\)
							−0.876925	+	0.480626i	\(0.840409\pi\)
\(3\)	75.5829	+	5.66415i	0.933122	+	0.0699278i	0.532607	−	0.846362i	\(-0.321212\pi\)
							0.400515	+	0.916290i	\(0.368831\pi\)
\(5\)	125.810	−	407.865i	0.201296	−	0.652584i	−0.797491	−	0.603331i	\(-0.793840\pi\)
							0.998786	−	0.0492532i	\(-0.0156841\pi\)
\(7\)	−2233.56	+	1289.54i	−0.930261	+	0.537087i	−0.886894	−	0.461973i	\(-0.847142\pi\)
							−0.0433671	+	0.999059i	\(0.513809\pi\)
\(11\)	16470.3	−	20653.0i	1.12494	−	1.41063i	0.225142	−	0.974326i	\(-0.427715\pi\)
							0.899799	−	0.436305i	\(-0.143713\pi\)
\(13\)	−1152.21	+	1069.10i	−0.0403422	+	0.0374321i	−0.700086	−	0.714058i	\(-0.746855\pi\)
							0.659744	+	0.751490i	\(0.270665\pi\)
\(17\)	103895.	−	32047.4i	1.24394	−	0.383704i	0.398198	−	0.917299i	\(-0.369636\pi\)
							0.845740	+	0.533595i	\(0.179159\pi\)
\(19\)	−4186.87	−	27778.0i	−0.0321273	−	0.213151i	0.967007	−	0.254750i	\(-0.0819930\pi\)
							−0.999134	+	0.0415986i	\(0.986755\pi\)
\(23\)	−73366.8	−	186935.i	−0.262173	−	0.668006i	0.737802	−	0.675017i	\(-0.235864\pi\)
							−0.999975	+	0.00701077i	\(0.997768\pi\)
\(29\)	777650.	−	58276.8i	1.09949	−	0.0823956i	0.487398	−	0.873180i	\(-0.337946\pi\)
							0.612095	+	0.790784i	\(0.290327\pi\)
\(31\)	420005.	−	286355.i	0.454787	−	0.310069i	−0.314173	−	0.949366i	\(-0.601727\pi\)
							0.768960	+	0.639297i	\(0.220775\pi\)
\(37\)	−395772.	−	228499.i	−0.211173	−	0.121921i	0.390683	−	0.920525i	\(-0.372239\pi\)
							−0.601856	+	0.798604i	\(0.705572\pi\)
\(41\)	−3.93623e6	−	1.89559e6i	−1.39298	−	0.670825i	−0.421257	−	0.906941i	\(-0.638411\pi\)
							−0.971725	+	0.236117i	\(0.924125\pi\)
\(43\)	−1.68891e6	−	2.97251e6i	−0.494006	−	0.869458i
							\(47\)	−292691.	−	367023.i	−0.0599816	−	0.0752145i	0.750936	−	0.660375i	\(-0.229603\pi\)
−0.810917	+	0.585161i	\(0.801031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.4	✓	336
43.29	odd	42	inner	43.9.h.a.29.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.4	✓	336	1.1	even	1	trivial
43.9.h.a.29.4	yes	336	43.29	odd	42	inner