Embedded newform 41.3.e.b.38.1

• Label: 41.3.e.b.38.1
• Level: $41$
• Weight: $3$
• Character: 41.38
• Analytic conductor: $1.117$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	41.e (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11716908388\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 66*x^18 + 1853*x^16 + 28868*x^14 + 272678*x^12 + 1600296*x^10 + 5739482*x^8 + 11863964*x^6 + 12547553*x^4 + 5476470*x^2 + 776161
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			38.1
Root			\(3.37131i\) of defining polynomial
Character	\(\chi\)	\(=\)	41.38
Dual form			41.3.e.b.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38388 + 2.38388i) q^{2} +(4.75273 + 1.96865i) q^{3} -7.36576i q^{4} +(2.78016 + 2.78016i) q^{5} +(-16.0230 + 6.63692i) q^{6} +(-9.79337 - 4.05655i) q^{7} +(8.02356 + 8.02356i) q^{8} +(12.3489 + 12.3489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 4 q^{3} - 12 q^{5} - 16 q^{6} - 4 q^{7} + 36 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38388	+	2.38388i	−1.19194	+	1.19194i	−0.215418	+	0.976522i	\(0.569111\pi\)
							−0.976522	+	0.215418i	\(0.930889\pi\)
\(3\)	4.75273	+	1.96865i	1.58424	+	0.656215i	0.989079	−	0.147388i	\(-0.0470866\pi\)
							0.595165	+	0.803603i	\(0.297087\pi\)
\(5\)	2.78016	+	2.78016i	0.556032	+	0.556032i	0.928175	−	0.372144i	\(-0.121377\pi\)
							−0.372144	+	0.928175i	\(0.621377\pi\)
\(7\)	−9.79337	−	4.05655i	−1.39905	−	0.579506i	−0.449547	−	0.893257i	\(-0.648415\pi\)
							−0.949506	+	0.313750i	\(0.898415\pi\)
\(11\)	4.61148	−	11.1331i	0.419225	−	1.01210i	−0.563347	−	0.826220i	\(-0.690487\pi\)
							0.982573	−	0.185879i	\(-0.0595133\pi\)
\(13\)	−2.11730	−	0.877014i	−0.162869	−	0.0674626i	0.299759	−	0.954015i	\(-0.403094\pi\)
							−0.462628	+	0.886552i	\(0.653094\pi\)
\(17\)	−2.83394	+	1.17386i	−0.166702	+	0.0690504i	−0.464474	−	0.885587i	\(-0.653757\pi\)
							0.297772	+	0.954637i	\(0.403757\pi\)
\(19\)	8.73721	−	3.61907i	0.459853	−	0.190477i	−0.140716	−	0.990050i	\(-0.544941\pi\)
							0.600569	+	0.799573i	\(0.294941\pi\)
\(23\)			20.5237i			0.892336i	0.894949	+	0.446168i	\(0.147212\pi\)
							−0.894949	+	0.446168i	\(0.852788\pi\)
\(29\)	8.67290	+	3.59243i	0.299066	+	0.123877i	0.527171	−	0.849760i	\(-0.323253\pi\)
							−0.228105	+	0.973637i	\(0.573253\pi\)
\(31\)		−	33.7758i		−	1.08954i	−0.838585	−	0.544771i	\(-0.816617\pi\)
							0.838585	−	0.544771i	\(-0.183383\pi\)
\(37\)	−60.1877			−1.62669			−0.813347	−	0.581778i	\(-0.802357\pi\)
							−0.813347	+	0.581778i	\(0.802357\pi\)
\(41\)	−6.99603	+	40.3987i	−0.170635	+	0.985334i
							\(43\)	6.83594	−	6.83594i	0.158975	−	0.158975i	−0.623137	−	0.782112i	\(-0.714142\pi\)
0.782112	+	0.623137i	\(0.214142\pi\)
\(47\)	0.199100	−	0.0824700i	0.00423618	−	0.00175468i	−0.380564	−	0.924754i	\(-0.624270\pi\)
							0.384801	+	0.923000i	\(0.374270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.3.e.b.38.1	yes	20
3.2	odd	2		369.3.l.b.325.5		20
41.27	odd	8	inner	41.3.e.b.27.1	✓	20
123.68	even	8		369.3.l.b.109.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.3.e.b.27.1	✓	20	41.27	odd	8	inner
41.3.e.b.38.1	yes	20	1.1	even	1	trivial
369.3.l.b.109.5		20	123.68	even	8
369.3.l.b.325.5		20	3.2	odd	2