Embedded newform 43.9.h.a.12.1

• Label: 43.9.h.a.12.1
• 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			12.1
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.7525 - 28.5573i) q^{2} +(12.3641 + 18.1347i) q^{3} +(-466.775 + 585.317i) q^{4} +(484.880 - 522.576i) q^{5} +(347.843 - 602.481i) q^{6} +(709.697 - 409.744i) q^{7} +(15223.6 + 3474.68i) q^{8} +(2221.00 - 5659.03i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.7525	−	28.5573i	−0.859529	−	1.78483i	−0.553620	−	0.832769i	\(-0.686754\pi\)
							−0.305909	−	0.952061i	\(-0.598960\pi\)
\(3\)	12.3641	+	18.1347i	0.152643	+	0.223886i	0.894942	−	0.446183i	\(-0.147217\pi\)
							−0.742299	+	0.670069i	\(0.766265\pi\)
\(5\)	484.880	−	522.576i	0.775808	−	0.836122i	−0.214250	−	0.976779i	\(-0.568731\pi\)
							0.990058	+	0.140657i	\(0.0449213\pi\)
\(7\)	709.697	−	409.744i	0.295584	−	0.170655i	−0.344873	−	0.938649i	\(-0.612078\pi\)
							0.640457	+	0.767994i	\(0.278745\pi\)
\(11\)	−3407.24	−	4272.54i	−0.232719	−	0.291820i	0.651736	−	0.758446i	\(-0.274041\pi\)
							−0.884455	+	0.466626i	\(0.845470\pi\)
\(13\)	−10849.5	+	3346.62i	−0.379870	+	0.117174i	−0.478809	−	0.877919i	\(-0.658931\pi\)
							0.0989389	+	0.995094i	\(0.468455\pi\)
\(17\)	72910.0	−	67650.6i	0.872955	−	0.809984i	−0.109981	−	0.993934i	\(-0.535079\pi\)
							0.982936	+	0.183950i	\(0.0588885\pi\)
\(19\)	659.646	−	258.892i	0.00506170	−	0.00198657i	−0.362809	−	0.931864i	\(-0.618182\pi\)
							0.367871	+	0.929877i	\(0.380087\pi\)
\(23\)	−29313.5	+	4418.30i	−0.104751	+	0.0157886i	−0.201208	−	0.979548i	\(-0.564487\pi\)
							0.0964578	+	0.995337i	\(0.469249\pi\)
\(29\)	−592569.	+	869139.i	−0.837813	+	1.22885i	0.133502	+	0.991049i	\(0.457378\pi\)
							−0.971315	+	0.237797i	\(0.923575\pi\)
\(31\)	119973.	−	1.60093e6i	0.129908	−	1.73351i	−0.428151	−	0.903707i	\(-0.640835\pi\)
							0.558060	−	0.829801i	\(-0.311546\pi\)
\(37\)	−975423.	−	563161.i	−0.520458	−	0.300487i	0.216664	−	0.976246i	\(-0.430482\pi\)
							−0.737122	+	0.675760i	\(0.763816\pi\)
\(41\)	2.69722e6	−	1.29891e6i	0.954512	−	0.459669i	0.109246	−	0.994015i	\(-0.465156\pi\)
							0.845266	+	0.534346i	\(0.179442\pi\)
\(43\)	−3.24029e6	−	1.09030e6i	−0.947784	−	0.318912i
							\(47\)	−2.94725e6	+	3.69573e6i	−0.603984	+	0.757372i	−0.985993	−	0.166787i	\(-0.946661\pi\)
0.382009	+	0.924159i	\(0.375232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.1	✓	336
43.18	odd	42	inner	43.9.h.a.18.1	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.1	✓	336	1.1	even	1	trivial
43.9.h.a.18.1	yes	336	43.18	odd	42	inner