Embedded newform 41.10.f.a.4.1

• Label: 41.10.f.a.4.1
• Level: $41$
• Weight: $10$
• Character: 41.4
• Analytic conductor: $21.116$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(21.1164692827\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	41.4
Dual form			41.10.f.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-33.1388 + 24.0768i) q^{2} -179.086i q^{3} +(360.274 - 1108.81i) q^{4} +(395.503 - 1217.23i) q^{5} +(4311.80 + 5934.69i) q^{6} +(-5863.08 + 8069.83i) q^{7} +(8276.65 + 25472.9i) q^{8} -12388.7 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} - 7171 q^{4} + 1023 q^{5} + 10155 q^{6} - 5 q^{7} + 15352 q^{8} - 690166 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−33.1388	+	24.0768i	−1.46454	+	1.06405i	−0.482392	+	0.875955i	\(0.660232\pi\)
							−0.982150	+	0.188097i	\(0.939768\pi\)
\(3\)		−	179.086i		−	1.27648i	−0.769836	−	0.638242i	\(-0.779662\pi\)
							0.769836	−	0.638242i	\(-0.220338\pi\)
\(5\)	395.503	−	1217.23i	0.282999	−	0.870981i	−0.703992	−	0.710208i	\(-0.748601\pi\)
							0.986991	−	0.160774i	\(-0.0513989\pi\)
\(7\)	−5863.08	+	8069.83i	−0.922963	+	1.27035i	0.0395787	+	0.999216i	\(0.487398\pi\)
							−0.962542	+	0.271133i	\(0.912602\pi\)
\(11\)	12818.4	−	4164.96i	0.263978	−	0.0857717i	−0.174038	−	0.984739i	\(-0.555681\pi\)
							0.438016	+	0.898967i	\(0.355681\pi\)
\(13\)	72703.5	+	100068.i	0.706009	+	0.971738i	0.999874	+	0.0158913i	\(0.00505857\pi\)
							−0.293864	+	0.955847i	\(0.594941\pi\)
\(17\)	252339.	−	81989.8i	0.732763	−	0.238089i	0.0812152	−	0.996697i	\(-0.474120\pi\)
							0.651548	+	0.758607i	\(0.274120\pi\)
\(19\)	−49864.5	+	68632.7i	−0.0877810	+	0.120820i	−0.850649	−	0.525734i	\(-0.823791\pi\)
							0.762868	+	0.646554i	\(0.223791\pi\)
\(23\)	−1.63665e6	+	1.18910e6i	−1.21950	+	0.886017i	−0.996058	−	0.0887011i	\(-0.971728\pi\)
							−0.223439	+	0.974718i	\(0.571728\pi\)
\(29\)	6.87972e6	+	2.23536e6i	1.80626	+	0.586889i	0.999989	−	0.00466027i	\(-0.00148341\pi\)
							0.806269	+	0.591549i	\(0.201483\pi\)
\(31\)	−2.49045e6	−	7.66481e6i	−0.484340	−	1.49064i	−0.832935	−	0.553371i	\(-0.813341\pi\)
							0.348595	−	0.937273i	\(-0.386659\pi\)
\(37\)	3.54643e6	−	1.09148e7i	0.311088	−	0.957430i	−0.666247	−	0.745731i	\(-0.732100\pi\)
							0.977335	−	0.211699i	\(-0.0678997\pi\)
\(41\)	1.49140e7	+	1.02448e7i	0.824262	+	0.566208i
							\(43\)	−7.61099e6	+	5.52971e6i	−0.339495	+	0.246657i	−0.744449	−	0.667680i	\(-0.767288\pi\)
0.404954	+	0.914337i	\(0.367288\pi\)
\(47\)	9.65250e6	+	1.32855e7i	0.288536	+	0.397135i	0.928538	−	0.371238i	\(-0.121067\pi\)
							−0.640002	+	0.768373i	\(0.721067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.10.f.a.4.1	✓	120
41.31	even	10	inner	41.10.f.a.31.1	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.10.f.a.4.1	✓	120	1.1	even	1	trivial
41.10.f.a.31.1	yes	120	41.31	even	10	inner