Embedded newform 43.5.h.a.3.14

• Label: 43.5.h.a.3.14
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.43387 - 7.13051i) q^{2} +(-5.05054 - 0.378485i) q^{3} +(-29.0769 - 36.4612i) q^{4} +(-5.90976 + 19.1590i) q^{5} +(-20.0417 + 34.7132i) q^{6} +(59.2015 - 34.1800i) q^{7} +(-236.380 + 53.9522i) q^{8} +(-54.7306 - 8.24932i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.43387	−	7.13051i	0.858468	−	1.78263i	0.297688	−	0.954663i	\(-0.403785\pi\)
							0.560781	−	0.827964i	\(-0.310501\pi\)
\(3\)	−5.05054	−	0.378485i	−0.561171	−	0.0420539i	−0.208874	−	0.977942i	\(-0.566980\pi\)
							−0.352296	+	0.935889i	\(0.614599\pi\)
\(5\)	−5.90976	+	19.1590i	−0.236391	+	0.766359i	0.757186	+	0.653199i	\(0.226574\pi\)
							−0.993577	+	0.113160i	\(0.963903\pi\)
\(7\)	59.2015	−	34.1800i	1.20819	−	0.697551i	0.245830	−	0.969313i	\(-0.420940\pi\)
							0.962365	+	0.271762i	\(0.0876063\pi\)
\(11\)	78.0594	−	97.8834i	0.645119	−	0.808954i	−0.346514	−	0.938045i	\(-0.612635\pi\)
							0.991633	+	0.129091i	\(0.0412060\pi\)
\(13\)	112.719	−	104.588i	0.666976	−	0.618863i	−0.272224	−	0.962234i	\(-0.587759\pi\)
							0.939200	+	0.343371i	\(0.111569\pi\)
\(17\)	−98.9161	+	30.5116i	−0.342270	+	0.105576i	−0.461125	−	0.887335i	\(-0.652554\pi\)
							0.118855	+	0.992912i	\(0.462078\pi\)
\(19\)	−55.9817	−	371.415i	−0.155074	−	1.02885i	−0.922021	−	0.387139i	\(-0.873463\pi\)
							0.766947	−	0.641710i	\(-0.221775\pi\)
\(23\)	37.7513	+	96.1889i	0.0713636	+	0.181832i	0.962105	−	0.272679i	\(-0.0879096\pi\)
							−0.890742	+	0.454510i	\(0.849814\pi\)
\(29\)	428.042	−	32.0773i	0.508967	−	0.0381418i	0.182228	−	0.983256i	\(-0.441669\pi\)
							0.326739	+	0.945114i	\(0.394050\pi\)
\(31\)	772.247	−	526.509i	0.803587	−	0.547876i	−0.0904334	−	0.995903i	\(-0.528825\pi\)
							0.894020	+	0.448026i	\(0.147873\pi\)
\(37\)	320.928	+	185.288i	0.234425	+	0.135346i	0.612612	−	0.790384i	\(-0.290119\pi\)
							−0.378186	+	0.925729i	\(0.623452\pi\)
\(41\)	−647.171	−	311.661i	−0.384992	−	0.185402i	0.231369	−	0.972866i	\(-0.425679\pi\)
							−0.616361	+	0.787464i	\(0.711394\pi\)
\(43\)	89.9360	+	1846.81i	0.0486404	+	0.998816i
							\(47\)	544.183	+	682.384i	0.246348	+	0.308911i	0.889597	−	0.456746i	\(-0.150985\pi\)
−0.643249	+	0.765657i	\(0.722414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.14	✓	168
43.29	odd	42	inner	43.5.h.a.29.14	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.14	✓	168	1.1	even	1	trivial
43.5.h.a.29.14	yes	168	43.29	odd	42	inner