Embedded newform 43.9.h.a.12.25

• Label: 43.9.h.a.12.25
• 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			12.25
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.1133 + 21.0004i) q^{2} +(9.28830 + 13.6234i) q^{3} +(-179.125 + 224.616i) q^{4} +(619.797 - 667.983i) q^{5} +(-192.163 + 332.835i) q^{6} +(1111.86 - 641.931i) q^{7} +(-711.136 - 162.312i) q^{8} +(2297.68 - 5854.39i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.1133	+	21.0004i	0.632078	+	1.31252i	0.933344	+	0.358984i	\(0.116877\pi\)
							−0.301265	+	0.953540i	\(0.597409\pi\)
\(3\)	9.28830	+	13.6234i	0.114670	+	0.168191i	0.879283	−	0.476299i	\(-0.158022\pi\)
							−0.764613	+	0.644490i	\(0.777070\pi\)
\(5\)	619.797	−	667.983i	0.991676	−	1.06877i	−0.00595519	−	0.999982i	\(-0.501896\pi\)
							0.997631	−	0.0687902i	\(-0.0219139\pi\)
\(7\)	1111.86	−	641.931i	0.463081	−	0.267360i	−0.250258	−	0.968179i	\(-0.580515\pi\)
							0.713339	+	0.700819i	\(0.247182\pi\)
\(11\)	−9040.83	−	11336.8i	−0.617501	−	0.774321i	0.370490	−	0.928837i	\(-0.379190\pi\)
							−0.987990	+	0.154515i	\(0.950618\pi\)
\(13\)	36778.8	−	11344.8i	1.28773	−	0.397211i	0.426065	−	0.904693i	\(-0.359899\pi\)
							0.861663	+	0.507481i	\(0.169423\pi\)
\(17\)	−12673.6	+	11759.4i	−0.151742	+	0.140796i	−0.752391	−	0.658717i	\(-0.771099\pi\)
							0.600649	+	0.799513i	\(0.294909\pi\)
\(19\)	4964.91	−	1948.58i	0.0380975	−	0.0149522i	−0.346216	−	0.938155i	\(-0.612533\pi\)
							0.384313	+	0.923203i	\(0.374438\pi\)
\(23\)	−503457.	+	75884.0i	−1.79908	+	0.271168i	−0.961433	−	0.275038i	\(-0.911310\pi\)
							−0.837651	+	0.546206i	\(0.816071\pi\)
\(29\)	−149245.	+	218902.i	−0.211012	+	0.309498i	−0.917124	−	0.398603i	\(-0.869495\pi\)
							0.706112	+	0.708101i	\(0.250448\pi\)
\(31\)	−99617.2	+	1.32930e6i	−0.107867	+	1.43938i	0.636328	+	0.771418i	\(0.280452\pi\)
							−0.744195	+	0.667962i	\(0.767167\pi\)
\(37\)	986197.	+	569381.i	0.526207	+	0.303806i	0.739471	−	0.673189i	\(-0.235076\pi\)
							−0.213263	+	0.976995i	\(0.568409\pi\)
\(41\)	4.18617e6	−	2.01595e6i	1.48143	−	0.713419i	0.493707	−	0.869629i	\(-0.335642\pi\)
							0.987724	+	0.156209i	\(0.0499274\pi\)
\(43\)	1.96497e6	−	2.79770e6i	0.574753	−	0.818327i
							\(47\)	−2.11935e6	+	2.65759e6i	−0.434322	+	0.544623i	−0.950037	−	0.312137i	\(-0.898955\pi\)
0.515715	+	0.856760i	\(0.327527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.25	✓	336
43.18	odd	42	inner	43.9.h.a.18.25	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.25	✓	336	1.1	even	1	trivial
43.9.h.a.18.25	yes	336	43.18	odd	42	inner