Embedded newform 41.4.f.a.23.6

• Label: 41.4.f.a.23.6
• Level: $41$
• Weight: $4$
• Character: 41.23
• Analytic conductor: $2.419$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	41.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.41907831024\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.6
Character	\(\chi\)	\(=\)	41.23
Dual form			41.4.f.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.239215 + 0.736229i) q^{2} +6.90238i q^{3} +(5.98733 - 4.35005i) q^{4} +(-13.1428 + 9.54883i) q^{5} +(-5.08173 + 1.65116i) q^{6} +(5.83421 + 1.89565i) q^{7} +(9.64508 + 7.00756i) q^{8} -20.6429 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + q^{2} - 43 q^{4} - q^{5} - 15 q^{6} - 5 q^{7} + 112 q^{8} - 370 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.239215	+	0.736229i	0.0845754	+	0.260296i	0.984397	−	0.175962i	\(-0.0563035\pi\)
							−0.899822	+	0.436258i	\(0.856303\pi\)
\(3\)			6.90238i			1.32836i	0.747571	+	0.664182i	\(0.231220\pi\)
							−0.747571	+	0.664182i	\(0.768780\pi\)
\(5\)	−13.1428	+	9.54883i	−1.17553	+	0.854074i	−0.991661	−	0.128877i	\(-0.958863\pi\)
							−0.183871	+	0.982950i	\(0.558863\pi\)
\(7\)	5.83421	+	1.89565i	0.315018	+	0.102356i	0.462259	−	0.886745i	\(-0.347039\pi\)
							−0.147241	+	0.989101i	\(0.547039\pi\)
\(11\)	−14.9430	+	20.5673i	−0.409591	+	0.563753i	−0.963119	−	0.269077i	\(-0.913281\pi\)
							0.553528	+	0.832831i	\(0.313281\pi\)
\(13\)	73.9579	−	24.0304i	1.57786	−	0.512679i	0.616359	−	0.787465i	\(-0.288607\pi\)
							0.961505	+	0.274786i	\(0.0886069\pi\)
\(17\)	44.9458	−	61.8625i	0.641232	−	0.882581i	−0.357448	−	0.933933i	\(-0.616353\pi\)
							0.998681	+	0.0513523i	\(0.0163531\pi\)
\(19\)	−29.5808	−	9.61137i	−0.357173	−	0.116053i	0.124933	−	0.992165i	\(-0.460128\pi\)
							−0.482107	+	0.876112i	\(0.660128\pi\)
\(23\)	−34.1778	−	105.189i	−0.309851	−	0.953623i	−0.977822	−	0.209436i	\(-0.932837\pi\)
							0.667971	−	0.744187i	\(-0.267163\pi\)
\(29\)	47.0508	+	64.7599i	0.301280	+	0.414676i	0.932637	−	0.360816i	\(-0.117502\pi\)
							−0.631357	+	0.775492i	\(0.717502\pi\)
\(31\)	147.970	+	107.506i	0.857295	+	0.622861i	0.927148	−	0.374696i	\(-0.122253\pi\)
							−0.0698526	+	0.997557i	\(0.522253\pi\)
\(37\)	−356.892	+	259.297i	−1.58575	+	1.15211i	−0.676040	+	0.736865i	\(0.736305\pi\)
							−0.909708	+	0.415249i	\(0.863695\pi\)
\(41\)	−57.2776	−	256.204i	−0.218177	−	0.975909i
							\(43\)	−49.7936	−	153.249i	−0.176592	−	0.543494i	0.823111	−	0.567881i	\(-0.192237\pi\)
−0.999703	+	0.0243870i	\(0.992237\pi\)
\(47\)	−54.5050	+	17.7098i	−0.169157	+	0.0549624i	−0.392371	−	0.919807i	\(-0.628345\pi\)
							0.223214	+	0.974769i	\(0.428345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.4.f.a.23.6	✓	40
41.5	even	20		1681.4.a.l.1.17		40
41.25	even	10	inner	41.4.f.a.25.6	yes	40
41.36	even	20		1681.4.a.l.1.18		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.4.f.a.23.6	✓	40	1.1	even	1	trivial
41.4.f.a.25.6	yes	40	41.25	even	10	inner
1681.4.a.l.1.17		40	41.5	even	20
1681.4.a.l.1.18		40	41.36	even	20