Embedded newform 43.5.f.a.2.8

• Label: 43.5.f.a.2.8
• Level: $43$
• Weight: $5$
• Character: 43.2
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.8
Character	\(\chi\)	\(=\)	43.2
Dual form			43.5.f.a.22.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.607375 + 1.26123i) q^{2} +(-1.76429 + 3.66358i) q^{3} +(8.75405 - 10.9772i) q^{4} +(19.0995 + 4.35934i) q^{5} -5.69219 q^{6} +6.49631i q^{7} +(40.9979 + 9.35750i) q^{8} +(40.1936 + 50.4011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.607375	+	1.26123i	0.151844	+	0.315307i	0.962990	−	0.269536i	\(-0.0868704\pi\)
							−0.811147	+	0.584843i	\(0.801156\pi\)
\(3\)	−1.76429	+	3.66358i	−0.196032	+	0.407065i	−0.975694	−	0.219135i	\(-0.929676\pi\)
							0.779663	+	0.626200i	\(0.215391\pi\)
\(5\)	19.0995	+	4.35934i	0.763981	+	0.174374i	0.586719	−	0.809790i	\(-0.300419\pi\)
							0.177261	+	0.984164i	\(0.443276\pi\)
\(7\)			6.49631i			0.132578i	0.997800	+	0.0662889i	\(0.0211159\pi\)
							−0.997800	+	0.0662889i	\(0.978884\pi\)
\(11\)	54.2359	+	68.0097i	0.448231	+	0.562064i	0.953692	−	0.300786i	\(-0.0972489\pi\)
							−0.505461	+	0.862850i	\(0.668677\pi\)
\(13\)	2.70293	−	11.8423i	0.0159937	−	0.0700728i	−0.966301	−	0.257414i	\(-0.917130\pi\)
							0.982295	+	0.187341i	\(0.0599869\pi\)
\(17\)	−46.0525	−	201.769i	−0.159351	−	0.698163i	−0.989965	−	0.141314i	\(-0.954867\pi\)
							0.830614	−	0.556849i	\(-0.187990\pi\)
\(19\)	−106.511	−	84.9394i	−0.295043	−	0.235289i	0.464769	−	0.885432i	\(-0.346138\pi\)
							−0.759812	+	0.650143i	\(0.774709\pi\)
\(23\)	5.36077	+	6.72219i	0.0101338	+	0.0127073i	0.786873	−	0.617116i	\(-0.211699\pi\)
							−0.776739	+	0.629823i	\(0.783127\pi\)
\(29\)	−390.962	−	811.842i	−0.464878	−	0.965329i	−0.993215	−	0.116294i	\(-0.962898\pi\)
							0.528337	−	0.849035i	\(-0.322816\pi\)
\(31\)	−716.611	+	345.102i	−0.745693	+	0.359107i	−0.767835	−	0.640648i	\(-0.778666\pi\)
							0.0221417	+	0.999755i	\(0.492951\pi\)
\(37\)		−	0.166172i		−	0.000121382i	−1.00000		6.06910e-5i	\(-0.999981\pi\)
							1.00000		6.06910e-5i	\(-1.93185e-5\pi\)
\(41\)	−231.871	+	111.663i	−0.137936	+	0.0664267i	−0.501579	−	0.865112i	\(-0.667247\pi\)
							0.363642	+	0.931539i	\(0.381533\pi\)
\(43\)	−834.149	+	1650.15i	−0.451135	+	0.892456i
							\(47\)	−622.596	+	780.710i	−0.281845	+	0.353422i	−0.902522	−	0.430644i	\(-0.858286\pi\)
0.620677	+	0.784066i	\(0.286858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.2.8	✓	78
43.22	odd	14	inner	43.5.f.a.22.8	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.8	✓	78	1.1	even	1	trivial
43.5.f.a.22.8	yes	78	43.22	odd	14	inner