Embedded newform 43.9.h.a.12.24

• Label: 43.9.h.a.12.24
• 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			12.24
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.94599 + 20.6531i) q^{2} +(-21.7887 - 31.9581i) q^{3} +(-168.013 + 210.682i) q^{4} +(-41.6658 + 44.9050i) q^{5} +(443.323 - 767.858i) q^{6} +(2732.76 - 1577.76i) q^{7} +(-301.080 - 68.7196i) q^{8} +(1850.43 - 4714.81i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.94599	+	20.6531i	0.621625	+	1.29082i	0.939456	+	0.342670i	\(0.111331\pi\)
							−0.317831	+	0.948147i	\(0.602955\pi\)
\(3\)	−21.7887	−	31.9581i	−0.268996	−	0.394545i	0.667845	−	0.744300i	\(-0.267217\pi\)
							−0.936841	+	0.349756i	\(0.886265\pi\)
\(5\)	−41.6658	+	44.9050i	−0.0666652	+	0.0718480i	−0.765504	−	0.643431i	\(-0.777510\pi\)
							0.698838	+	0.715280i	\(0.253701\pi\)
\(7\)	2732.76	−	1577.76i	1.13818	−	0.657126i	0.192198	−	0.981356i	\(-0.438439\pi\)
							0.945978	+	0.324230i	\(0.105105\pi\)
\(11\)	10969.1	+	13754.9i	0.749207	+	0.939476i	0.999589	−	0.0286712i	\(-0.00912756\pi\)
							−0.250382	+	0.968147i	\(0.580556\pi\)
\(13\)	−25059.8	+	7729.92i	−0.877413	+	0.270646i	−0.700562	−	0.713592i	\(-0.747067\pi\)
							−0.176851	+	0.984238i	\(0.556591\pi\)
\(17\)	2104.53	−	1952.72i	0.0251976	−	0.0233800i	−0.667473	−	0.744634i	\(-0.732624\pi\)
							0.692671	+	0.721254i	\(0.256434\pi\)
\(19\)	95892.6	−	37635.1i	0.735819	−	0.288788i	0.0323136	−	0.999478i	\(-0.489712\pi\)
							0.703505	+	0.710690i	\(0.251617\pi\)
\(23\)	521247.	−	78565.4i	1.86265	−	0.280750i	0.881243	−	0.472663i	\(-0.156707\pi\)
							0.981412	+	0.191913i	\(0.0614693\pi\)
\(29\)	−595255.	+	873078.i	−0.841610	+	1.23442i	0.128490	+	0.991711i	\(0.458987\pi\)
							−0.970100	+	0.242704i	\(0.921965\pi\)
\(31\)	79992.5	−	1.06743e6i	0.0866168	−	1.15582i	−0.769363	−	0.638812i	\(-0.779426\pi\)
							0.855980	−	0.517009i	\(-0.172955\pi\)
\(37\)	930217.	+	537061.i	0.496338	+	0.286561i	0.727200	−	0.686426i	\(-0.240821\pi\)
							−0.230862	+	0.972986i	\(0.574155\pi\)
\(41\)	1.94501e6	−	936668.i	0.688314	−	0.331475i	−0.0568244	−	0.998384i	\(-0.518098\pi\)
							0.745139	+	0.666909i	\(0.232383\pi\)
\(43\)	534895.	−	3.37670e6i	0.156457	−	0.987685i
							\(47\)	5.15778e6	−	6.46766e6i	1.05699	−	1.32543i	0.113681	−	0.993517i	\(-0.463736\pi\)
0.943311	−	0.331909i	\(-0.107693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.24	✓	336
43.18	odd	42	inner	43.9.h.a.18.24	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.24	✓	336	1.1	even	1	trivial
43.9.h.a.18.24	yes	336	43.18	odd	42	inner