Embedded newform 43.5.f.a.8.6

• Label: 43.5.f.a.8.6
• Level: $43$
• Weight: $5$
• Character: 43.8
• 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			8.6
Character	\(\chi\)	\(=\)	43.8
Dual form			43.5.f.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.02384 - 0.461929i) q^{2} +(-16.6744 + 3.80583i) q^{3} +(-10.5329 - 5.07240i) q^{4} +(21.9239 + 17.4837i) q^{5} +35.5044 q^{6} -46.4861i q^{7} +(44.9418 + 35.8399i) q^{8} +(190.574 - 91.7754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.02384	−	0.461929i	−0.505960	−	0.115482i	−0.0380813	−	0.999275i	\(-0.512125\pi\)
							−0.467879	+	0.883792i	\(0.654982\pi\)
\(3\)	−16.6744	+	3.80583i	−1.85271	+	0.422870i	−0.995717	−	0.0924523i	\(-0.970529\pi\)
							−0.856997	+	0.515322i	\(0.827672\pi\)
\(5\)	21.9239	+	17.4837i	0.876957	+	0.699350i	0.954683	−	0.297625i	\(-0.0961944\pi\)
							−0.0777258	+	0.996975i	\(0.524766\pi\)
\(7\)		−	46.4861i		−	0.948696i	−0.880337	−	0.474348i	\(-0.842684\pi\)
							0.880337	−	0.474348i	\(-0.157316\pi\)
\(11\)	−90.6610	+	43.6600i	−0.749264	+	0.360827i	−0.769228	−	0.638974i	\(-0.779359\pi\)
							0.0199640	+	0.999801i	\(0.493645\pi\)
\(13\)	12.8965	−	16.1717i	0.0763109	−	0.0956908i	−0.742211	−	0.670167i	\(-0.766223\pi\)
							0.818522	+	0.574476i	\(0.194794\pi\)
\(17\)	205.099	+	257.186i	0.709684	+	0.889916i	0.997705	−	0.0677080i	\(-0.0215686\pi\)
							−0.288021	+	0.957624i	\(0.592997\pi\)
\(19\)	236.957	−	492.047i	0.656391	−	1.36301i	−0.261123	−	0.965306i	\(-0.584093\pi\)
							0.917514	−	0.397704i	\(-0.130193\pi\)
\(23\)	569.023	−	274.027i	1.07566	−	0.518010i	0.189732	−	0.981836i	\(-0.439238\pi\)
							0.885926	+	0.463826i	\(0.153524\pi\)
\(29\)	299.153	+	68.2798i	0.355711	+	0.0811888i	0.396642	−	0.917973i	\(-0.370175\pi\)
							−0.0409311	+	0.999162i	\(0.513032\pi\)
\(31\)	62.5725	−	274.148i	0.0651119	−	0.285274i	−0.931881	−	0.362763i	\(-0.881833\pi\)
							0.996993	+	0.0774896i	\(0.0246905\pi\)
\(37\)			845.241i			0.617415i	0.951157	+	0.308708i	\(0.0998964\pi\)
							−0.951157	+	0.308708i	\(0.900104\pi\)
\(41\)	671.635	−	2942.62i	0.399545	−	1.75052i	−0.229650	−	0.973273i	\(-0.573758\pi\)
							0.629195	−	0.777247i	\(-0.283385\pi\)
\(43\)	944.556	+	1589.53i	0.510847	+	0.859672i
							\(47\)	3108.49	+	1496.97i	1.40719	+	0.677668i	0.974606	−	0.223925i	\(-0.0718872\pi\)
0.432585	+	0.901593i	\(0.357601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.8.6	✓	78
43.27	odd	14	inner	43.5.f.a.27.6	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.8.6	✓	78	1.1	even	1	trivial
43.5.f.a.27.6	yes	78	43.27	odd	14	inner