Embedded newform 43.6.c.a.6.9

• Label: 43.6.c.a.6.9
• Level: $43$
• Weight: $6$
• Character: 43.6
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Relative dimension:	\(17\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			6.9
Character	\(\chi\)	\(=\)	43.6
Dual form			43.6.c.a.36.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.24721 q^{2} +(-0.494517 + 0.856528i) q^{3} -30.4445 q^{4} +(24.3642 - 42.2000i) q^{5} +(0.616768 - 1.06827i) q^{6} +(83.1429 + 144.008i) q^{7} +77.8815 q^{8} +(121.011 + 209.597i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 14 q^{2} - 14 q^{3} + 454 q^{4} + 71 q^{5} + 15 q^{6} + 225 q^{7} - 936 q^{8} - 1011 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24721			−0.220478			−0.110239	−	0.993905i	\(-0.535162\pi\)
							−0.110239	+	0.993905i	\(0.535162\pi\)
\(3\)	−0.494517	+	0.856528i	−0.0317233	+	0.0549463i	−0.881451	−	0.472275i	\(-0.843433\pi\)
							0.849728	+	0.527222i	\(0.176766\pi\)
\(5\)	24.3642	−	42.2000i	0.435840	−	0.754896i	−0.561524	−	0.827460i	\(-0.689785\pi\)
							0.997364	+	0.0725640i	\(0.0231182\pi\)
\(7\)	83.1429	+	144.008i	0.641328	+	1.11081i	0.985137	+	0.171773i	\(0.0549495\pi\)
							−0.343809	+	0.939040i	\(0.611717\pi\)
\(11\)	−436.503			−1.08769			−0.543846	−	0.839185i	\(-0.683032\pi\)
							−0.543846	+	0.839185i	\(0.683032\pi\)
\(13\)	486.675	+	842.946i	0.798695	+	1.38338i	0.920466	+	0.390821i	\(0.127809\pi\)
							−0.121772	+	0.992558i	\(0.538858\pi\)
\(17\)	375.545	+	650.464i	0.315167	+	0.545885i	0.979473	−	0.201576i	\(-0.0646063\pi\)
							−0.664306	+	0.747460i	\(0.731273\pi\)
\(19\)	283.197	−	490.512i	0.179972	−	0.311721i	−0.761899	−	0.647696i	\(-0.775733\pi\)
							0.941871	+	0.335975i	\(0.109066\pi\)
\(23\)	−498.615	+	863.626i	−0.196538	+	0.340413i	−0.947404	−	0.320042i	\(-0.896303\pi\)
							0.750866	+	0.660455i	\(0.229636\pi\)
\(29\)	−1998.43	−	3461.38i	−0.441259	−	0.764283i	0.556524	−	0.830832i	\(-0.312135\pi\)
							−0.997783	+	0.0665481i	\(0.978801\pi\)
\(31\)	3892.30	−	6741.67i	0.727449	−	1.25998i	−0.230509	−	0.973070i	\(-0.574039\pi\)
							0.957958	−	0.286909i	\(-0.0926276\pi\)
\(37\)	−6022.45	+	10431.2i	−0.723217	+	1.25265i	0.236486	+	0.971635i	\(0.424004\pi\)
							−0.959704	+	0.281014i	\(0.909329\pi\)
\(41\)	−15533.0			−1.44310			−0.721548	−	0.692364i	\(-0.756569\pi\)
							−0.721548	+	0.692364i	\(0.756569\pi\)
\(43\)	3632.24	+	11567.9i	0.299573	+	0.954073i
							\(47\)	8224.60			0.543088			0.271544	−	0.962426i	\(-0.412466\pi\)
0.271544	+	0.962426i	\(0.412466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.c.a.6.9	✓	34
43.36	even	3	inner	43.6.c.a.36.9	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.c.a.6.9	✓	34	1.1	even	1	trivial
43.6.c.a.36.9	yes	34	43.36	even	3	inner