Embedded newform 43.5.h.a.12.10

• Label: 43.5.h.a.12.10
• Level: $43$
• Weight: $5$
• Character: 43.12
• 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			12.10
Character	\(\chi\)	\(=\)	43.12
Dual form			43.5.h.a.18.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65115 + 3.42866i) q^{2} +(-2.58658 - 3.79382i) q^{3} +(0.946460 - 1.18682i) q^{4} +(30.9309 - 33.3355i) q^{5} +(8.73686 - 15.1327i) q^{6} +(-43.6643 + 25.2096i) q^{7} +(64.9937 + 14.8344i) q^{8} +(21.8900 - 55.7747i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65115	+	3.42866i	0.412788	+	0.857164i	0.998898	+	0.0469323i	\(0.0149445\pi\)
							−0.586110	+	0.810232i	\(0.699341\pi\)
\(3\)	−2.58658	−	3.79382i	−0.287398	−	0.421536i	0.655205	−	0.755451i	\(-0.272582\pi\)
							−0.942603	+	0.333916i	\(0.891630\pi\)
\(5\)	30.9309	−	33.3355i	1.23723	−	1.33342i	0.313885	−	0.949461i	\(-0.398369\pi\)
							0.923349	−	0.383961i	\(-0.125440\pi\)
\(7\)	−43.6643	+	25.2096i	−0.891108	+	0.514481i	−0.874305	−	0.485378i	\(-0.838682\pi\)
							−0.0168030	+	0.999859i	\(0.505349\pi\)
\(11\)	81.5151	+	102.217i	0.673679	+	0.844766i	0.994755	−	0.102287i	\(-0.0326160\pi\)
							−0.321076	+	0.947053i	\(0.604045\pi\)
\(13\)	−124.580	+	38.4277i	−0.737157	+	0.227383i	−0.640525	−	0.767937i	\(-0.721283\pi\)
							−0.0966323	+	0.995320i	\(0.530807\pi\)
\(17\)	213.946	−	198.513i	0.740297	−	0.686895i	−0.216945	−	0.976184i	\(-0.569609\pi\)
							0.957242	+	0.289288i	\(0.0934187\pi\)
\(19\)	−307.687	+	120.758i	−0.852319	+	0.334511i	−0.750967	−	0.660340i	\(-0.770412\pi\)
							−0.101353	+	0.994851i	\(0.532317\pi\)
\(23\)	179.291	−	27.0238i	0.338925	−	0.0510847i	0.0226252	−	0.999744i	\(-0.492798\pi\)
							0.316300	+	0.948659i	\(0.397559\pi\)
\(29\)	−671.387	+	984.744i	−0.798320	+	1.17092i	0.183855	+	0.982953i	\(0.441142\pi\)
							−0.982175	+	0.187967i	\(0.939810\pi\)
\(31\)	−92.4166	+	1233.21i	−0.0961672	+	1.28326i	0.715281	+	0.698837i	\(0.246299\pi\)
							−0.811448	+	0.584425i	\(0.801320\pi\)
\(37\)	567.733	+	327.781i	0.414706	+	0.239431i	0.692810	−	0.721120i	\(-0.256373\pi\)
							−0.278104	+	0.960551i	\(0.589706\pi\)
\(41\)	−2226.96	+	1072.45i	−1.32478	+	0.637982i	−0.956500	−	0.291732i	\(-0.905769\pi\)
							−0.368283	+	0.929714i	\(0.620054\pi\)
\(43\)	1841.10	−	170.758i	0.995726	−	0.0923516i
							\(47\)	−318.124	+	398.915i	−0.144013	+	0.180586i	−0.848606	−	0.529025i	\(-0.822558\pi\)
0.704594	+	0.709611i	\(0.251129\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.12.10	✓	168
43.18	odd	42	inner	43.5.h.a.18.10	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.12.10	✓	168	1.1	even	1	trivial
43.5.h.a.18.10	yes	168	43.18	odd	42	inner