Embedded newform 43.5.f.a.8.2

• Label: 43.5.f.a.8.2
• 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			8.2
Character	\(\chi\)	\(=\)	43.8
Dual form			43.5.f.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.68339 - 1.52544i) q^{2} +(15.1977 - 3.46877i) q^{3} +(27.9252 + 13.4481i) q^{4} +(1.95321 + 1.55764i) q^{5} -106.863 q^{6} +79.0264i q^{7} +(-80.3660 - 64.0897i) q^{8} +(145.958 - 70.2899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.68339	−	1.52544i	−1.67085	−	0.381360i	−0.720720	−	0.693226i	\(-0.756189\pi\)
							−0.950126	+	0.311866i	\(0.899046\pi\)
\(3\)	15.1977	−	3.46877i	1.68863	−	0.385419i	0.733056	−	0.680168i	\(-0.238093\pi\)
							0.955575	+	0.294750i	\(0.0952362\pi\)
\(5\)	1.95321	+	1.55764i	0.0781285	+	0.0623054i	0.661775	−	0.749702i	\(-0.269803\pi\)
							−0.583647	+	0.812008i	\(0.698375\pi\)
\(7\)			79.0264i			1.61278i	0.591381	+	0.806392i	\(0.298583\pi\)
							−0.591381	+	0.806392i	\(0.701417\pi\)
\(11\)	150.233	−	72.3486i	1.24160	−	0.597922i	0.306351	−	0.951919i	\(-0.400892\pi\)
							0.935247	+	0.353997i	\(0.115178\pi\)
\(13\)	−64.0354	+	80.2979i	−0.378908	+	0.475135i	−0.934318	−	0.356441i	\(-0.883990\pi\)
							0.555410	+	0.831577i	\(0.312561\pi\)
\(17\)	47.4390	+	59.4866i	0.164149	+	0.205836i	0.857102	−	0.515146i	\(-0.172262\pi\)
							−0.692954	+	0.720982i	\(0.743691\pi\)
\(19\)	108.039	−	224.346i	0.299278	−	0.621458i	−0.696051	−	0.717992i	\(-0.745061\pi\)
							0.995330	+	0.0965343i	\(0.0307757\pi\)
\(23\)	566.081	−	272.610i	1.07010	−	0.515331i	0.185958	−	0.982558i	\(-0.440461\pi\)
							0.884138	+	0.467227i	\(0.154747\pi\)
\(29\)	−1462.12	−	333.719i	−1.73855	−	0.396812i	−0.768522	−	0.639824i	\(-0.779007\pi\)
							−0.970023	+	0.243012i	\(0.921865\pi\)
\(31\)	−308.422	+	1351.28i	−0.320938	+	1.40612i	0.514948	+	0.857221i	\(0.327811\pi\)
							−0.835887	+	0.548902i	\(0.815046\pi\)
\(37\)			69.4893i			0.0507592i	0.999678	+	0.0253796i	\(0.00807945\pi\)
							−0.999678	+	0.0253796i	\(0.991921\pi\)
\(41\)	−27.4113	+	120.097i	−0.0163066	+	0.0714437i	−0.982427	−	0.186650i	\(-0.940237\pi\)
							0.966120	+	0.258093i	\(0.0830942\pi\)
\(43\)	1090.43	+	1493.24i	0.589739	+	0.807594i
							\(47\)	−2024.07	−	974.743i	−0.916285	−	0.441260i	−0.0845418	−	0.996420i	\(-0.526943\pi\)
−0.831743	+	0.555160i	\(0.812657\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.8.2	✓	78
43.27	odd	14	inner	43.5.f.a.27.2	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.8.2	✓	78	1.1	even	1	trivial
43.5.f.a.27.2	yes	78	43.27	odd	14	inner