Embedded newform 43.5.h.a.29.5

• Label: 43.5.h.a.29.5
• Level: $43$
• Weight: $5$
• Character: 43.29
• 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			29.5
Character	\(\chi\)	\(=\)	43.29
Dual form			43.5.h.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59415 - 3.31028i) q^{2} +(12.0593 - 0.903718i) q^{3} +(1.55919 - 1.95516i) q^{4} +(-2.23527 - 7.24655i) q^{5} +(-22.2158 - 38.4789i) q^{6} +(79.2542 + 45.7575i) q^{7} +(-66.2699 - 15.1257i) q^{8} +(64.5140 - 9.72393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59415	−	3.31028i	−0.398537	−	0.827570i	−0.999598	−	0.0283667i	\(-0.990969\pi\)
							0.601061	−	0.799203i	\(-0.294745\pi\)
\(3\)	12.0593	−	0.903718i	1.33992	−	0.100413i	0.614607	−	0.788833i	\(-0.289315\pi\)
							0.725312	+	0.688420i	\(0.241696\pi\)
\(5\)	−2.23527	−	7.24655i	−0.0894106	−	0.289862i	0.899813	−	0.436276i	\(-0.143703\pi\)
							−0.989223	+	0.146414i	\(0.953227\pi\)
\(7\)	79.2542	+	45.7575i	1.61743	+	0.933826i	0.987581	+	0.157111i	\(0.0502180\pi\)
							0.629853	+	0.776715i	\(0.283115\pi\)
\(11\)	−75.2301	−	94.3355i	−0.621736	−	0.779633i	0.366851	−	0.930280i	\(-0.380436\pi\)
							−0.988588	+	0.150647i	\(0.951864\pi\)
\(13\)	−95.4437	−	88.5588i	−0.564755	−	0.524016i	0.345306	−	0.938490i	\(-0.387775\pi\)
							−0.910061	+	0.414474i	\(0.863966\pi\)
\(17\)	88.0327	+	27.1545i	0.304611	+	0.0939601i	0.443293	−	0.896377i	\(-0.353810\pi\)
							−0.138681	+	0.990337i	\(0.544286\pi\)
\(19\)	−36.1727	+	239.990i	−0.100201	+	0.664793i	0.881188	+	0.472766i	\(0.156744\pi\)
							−0.981390	+	0.192027i	\(0.938494\pi\)
\(23\)	−332.295	+	846.673i	−0.628156	+	1.60052i	0.161435	+	0.986883i	\(0.448388\pi\)
							−0.789591	+	0.613633i	\(0.789707\pi\)
\(29\)	360.394	+	27.0078i	0.428531	+	0.0321139i	0.287250	−	0.957856i	\(-0.407259\pi\)
							0.141281	+	0.989970i	\(0.454878\pi\)
\(31\)	927.689	+	632.488i	0.965337	+	0.658156i	0.939875	−	0.341520i	\(-0.110942\pi\)
							0.0254629	+	0.999676i	\(0.491894\pi\)
\(37\)	−630.876	+	364.237i	−0.460830	+	0.266060i	−0.712393	−	0.701781i	\(-0.752389\pi\)
							0.251563	+	0.967841i	\(0.419055\pi\)
\(41\)	438.360	−	211.103i	0.260773	−	0.125582i	−0.298929	−	0.954275i	\(-0.596630\pi\)
							0.559703	+	0.828693i	\(0.310915\pi\)
\(43\)	−1839.49	−	187.300i	−0.994856	−	0.101298i
							\(47\)	−1005.47	+	1260.82i	−0.455168	+	0.570763i	−0.955470	−	0.295088i	\(-0.904651\pi\)
0.500302	+	0.865851i	\(0.333222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.29.5	yes	168
43.3	odd	42	inner	43.5.h.a.3.5	✓	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.5	✓	168	43.3	odd	42	inner
43.5.h.a.29.5	yes	168	1.1	even	1	trivial