Embedded newform 43.8.e.a.4.2

• Label: 43.8.e.a.4.2
• Level: $43$
• Weight: $8$
• Character: 43.4
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	43.4
Dual form			43.8.e.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.8404 + 14.8473i) q^{2} +(-33.9228 - 42.5378i) q^{3} +(-51.7667 - 226.805i) q^{4} +(73.7187 + 35.5010i) q^{5} +1033.23 q^{6} +537.033 q^{7} +(1790.33 + 862.177i) q^{8} +(-172.058 + 753.837i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.8404	+	14.8473i	−1.04655	+	1.31233i	−0.0981787	+	0.995169i	\(0.531302\pi\)
							−0.948371	+	0.317163i	\(0.897270\pi\)
\(3\)	−33.9228	−	42.5378i	−0.725383	−	0.909601i	0.273246	−	0.961944i	\(-0.411903\pi\)
							−0.998629	+	0.0523429i	\(0.983331\pi\)
\(5\)	73.7187	+	35.5010i	0.263744	+	0.127012i	0.561083	−	0.827760i	\(-0.310385\pi\)
							−0.297339	+	0.954772i	\(0.596099\pi\)
\(7\)	537.033			0.591777			0.295888	−	0.955223i	\(-0.404384\pi\)
							0.295888	+	0.955223i	\(0.404384\pi\)
\(11\)	−427.128	+	1871.37i	−0.0967572	+	0.423921i	−0.999986	−	0.00523780i	\(-0.998333\pi\)
							0.903229	+	0.429159i	\(0.141190\pi\)
\(13\)	83.6416	+	40.2797i	0.0105589	+	0.00508492i	0.439156	−	0.898411i	\(-0.355278\pi\)
							−0.428597	+	0.903496i	\(0.640992\pi\)
\(17\)	−550.067	+	264.898i	−0.0271546	+	0.0130770i	−0.447412	−	0.894328i	\(-0.647654\pi\)
							0.420257	+	0.907405i	\(0.361940\pi\)
\(19\)	−4246.15	−	18603.6i	−0.142023	−	0.622242i	−0.994964	−	0.100235i	\(-0.968041\pi\)
							0.852941	−	0.522007i	\(-0.174816\pi\)
\(23\)	820.637	−	3595.44i	0.0140638	−	0.0616177i	−0.967409	−	0.253219i	\(-0.918511\pi\)
							0.981473	+	0.191601i	\(0.0613679\pi\)
\(29\)	−83424.0	+	104610.i	−0.635182	+	0.796493i	−0.990391	−	0.138294i	\(-0.955838\pi\)
							0.355209	+	0.934787i	\(0.384410\pi\)
\(31\)	−93091.4	+	116733.i	−0.561233	+	0.703765i	−0.978785	−	0.204889i	\(-0.934317\pi\)
							0.417552	+	0.908653i	\(0.362888\pi\)
\(37\)	−109391.			−0.355038			−0.177519	−	0.984117i	\(-0.556807\pi\)
							−0.177519	+	0.984117i	\(0.556807\pi\)
\(41\)	−323038.	+	405077.i	−0.731999	+	0.917898i	−0.998950	−	0.0458119i	\(-0.985413\pi\)
							0.266951	+	0.963710i	\(0.413984\pi\)
\(43\)	−508430.	+	115403.i	−0.975194	+	0.221350i
							\(47\)	62056.9	+	271889.i	0.0871862	+	0.381988i	0.999630	−	0.0272153i	\(-0.00866398\pi\)
−0.912443	+	0.409203i	\(0.865807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.e.a.4.2	✓	144
43.11	even	7	inner	43.8.e.a.11.2	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.e.a.4.2	✓	144	1.1	even	1	trivial
43.8.e.a.11.2	yes	144	43.11	even	7	inner