Embedded newform 43.8.g.a.24.12

• Label: 43.8.g.a.24.12
• Level: $43$
• Weight: $8$
• Character: 43.24
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			24.12
Character	\(\chi\)	\(=\)	43.24
Dual form			43.8.g.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23932 + 1.55406i) q^{2} +(45.7182 - 6.89092i) q^{3} +(27.6035 + 120.939i) q^{4} +(82.5181 - 56.2599i) q^{5} +(-45.9507 + 79.5890i) q^{6} +(577.502 + 1000.26i) q^{7} +(-451.388 - 217.377i) q^{8} +(-47.1653 + 14.5486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23932	+	1.55406i	−0.109542	+	0.137361i	−0.833580	−	0.552399i	\(-0.813712\pi\)
							0.724038	+	0.689760i	\(0.242284\pi\)
\(3\)	45.7182	−	6.89092i	0.977609	−	0.147351i	0.359245	−	0.933243i	\(-0.383034\pi\)
							0.618363	+	0.785892i	\(0.287796\pi\)
\(5\)	82.5181	−	56.2599i	0.295226	−	0.201282i	−0.406644	−	0.913587i	\(-0.633301\pi\)
							0.701870	+	0.712305i	\(0.252349\pi\)
\(7\)	577.502	+	1000.26i	0.636371	+	1.10223i	0.986223	+	0.165422i	\(0.0528987\pi\)
							−0.349851	+	0.936805i	\(0.613768\pi\)
\(11\)	−55.8335	+	244.623i	−0.0126480	+	0.0554143i	−0.980858	−	0.194724i	\(-0.937619\pi\)
							0.968210	+	0.250138i	\(0.0804761\pi\)
\(13\)	−226.493	−	3022.34i	−0.0285926	−	0.381541i	−0.993082	−	0.117421i	\(-0.962537\pi\)
							0.964490	−	0.264121i	\(-0.0850817\pi\)
\(17\)	7974.55	+	5436.95i	0.393672	+	0.268401i	0.743937	−	0.668249i	\(-0.232956\pi\)
							−0.350265	+	0.936651i	\(0.613908\pi\)
\(19\)	7826.69	+	2414.21i	0.261782	+	0.0807492i	0.422868	−	0.906191i	\(-0.361023\pi\)
							−0.161086	+	0.986940i	\(0.551500\pi\)
\(23\)	58069.7	+	53880.8i	0.995181	+	0.923393i	0.997061	−	0.0766056i	\(-0.0244082\pi\)
							−0.00188093	+	0.999998i	\(0.500599\pi\)
\(29\)	96328.3	+	14519.1i	0.733433	+	0.110547i	0.505131	−	0.863043i	\(-0.331444\pi\)
							0.228303	+	0.973590i	\(0.426682\pi\)
\(31\)	2954.01	+	7526.70i	0.0178093	+	0.0453773i	0.939502	−	0.342544i	\(-0.111289\pi\)
							−0.921693	+	0.387921i	\(0.873193\pi\)
\(37\)	204571.	−	354327.i	0.663953	−	1.15000i	−0.315615	−	0.948887i	\(-0.602211\pi\)
							0.979568	−	0.201113i	\(-0.0644558\pi\)
\(41\)	421073.	−	528009.i	0.954144	−	1.19646i	−0.0262987	−	0.999654i	\(-0.508372\pi\)
							0.980443	−	0.196805i	\(-0.0630565\pi\)
\(43\)	−494963.	−	163799.i	−0.949365	−	0.314175i
							\(47\)	−174477.	−	764433.i	−0.245129	−	1.07398i	−0.936275	−	0.351267i	\(-0.885751\pi\)
0.691146	−	0.722715i	\(-0.257106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.24.12	yes	300
43.9	even	21	inner	43.8.g.a.9.12	✓	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.12	✓	300	43.9	even	21	inner
43.8.g.a.24.12	yes	300	1.1	even	1	trivial