Embedded newform 43.9.h.a.18.24

• Label: 43.9.h.a.18.24
• 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			18.24
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.24

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.24	✓	336	43.12	odd	42	inner
43.9.h.a.18.24	yes	336	1.1	even	1	trivial