Embedded newform 105.4.g.b.104.1

• Label: 105.4.g.b.104.1
• Level: $105$
• Weight: $4$
• Character: 105.104
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			104.1
Character	\(\chi\)	\(=\)	105.104
Dual form			105.4.g.b.104.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-5.11138 q^{2} +(-1.51783 - 4.96953i) q^{3} +18.1262 q^{4} +(-10.9509 + 2.25323i) q^{5} +(7.75821 + 25.4011i) q^{6} +(-11.4695 - 14.5413i) q^{7} -51.7589 q^{8} +(-22.3924 + 15.0858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 184 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.11138			−1.80715			−0.903573	−	0.428435i	\(-0.859065\pi\)
							−0.903573	+	0.428435i	\(0.859065\pi\)
\(3\)	−1.51783	−	4.96953i	−0.292107	−	0.956386i
							\(5\)	−10.9509	+	2.25323i	−0.979481	+	0.201535i
\(7\)	−11.4695	−	14.5413i	−0.619295	−	0.785158i
							\(11\)		−	55.4308i		−	1.51936i	−0.650294	−	0.759682i	\(-0.725354\pi\)
0.650294	−	0.759682i	\(-0.274646\pi\)
\(13\)	−20.9550			−0.447066			−0.223533	−	0.974696i	\(-0.571759\pi\)
							−0.223533	+	0.974696i	\(0.571759\pi\)
\(17\)			96.8237i			1.38136i	0.723158	+	0.690682i	\(0.242690\pi\)
							−0.723158	+	0.690682i	\(0.757310\pi\)
\(19\)			33.1970i			0.400838i	0.979710	+	0.200419i	\(0.0642303\pi\)
							−0.979710	+	0.200419i	\(0.935770\pi\)
\(23\)	111.288			1.00892			0.504460	−	0.863435i	\(-0.331692\pi\)
							0.504460	+	0.863435i	\(0.331692\pi\)
\(29\)			156.784i			1.00393i	0.864888	+	0.501965i	\(0.167389\pi\)
							−0.864888	+	0.501965i	\(0.832611\pi\)
\(31\)		−	80.4962i		−	0.466372i	−0.972432	−	0.233186i	\(-0.925085\pi\)
							0.972432	−	0.233186i	\(-0.0749152\pi\)
\(37\)			180.365i			0.801402i	0.916209	+	0.400701i	\(0.131233\pi\)
							−0.916209	+	0.400701i	\(0.868767\pi\)
\(41\)	36.5837			0.139351			0.0696757	−	0.997570i	\(-0.477804\pi\)
							0.0696757	+	0.997570i	\(0.477804\pi\)
\(43\)		−	52.5826i		−	0.186483i	−0.995644	−	0.0932416i	\(-0.970277\pi\)
							0.995644	−	0.0932416i	\(-0.0297229\pi\)
\(47\)		−	259.660i		−	0.805858i	−0.915231	−	0.402929i	\(-0.867992\pi\)
							0.915231	−	0.402929i	\(-0.132008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.g.b.104.1	✓	40
3.2	odd	2	inner	105.4.g.b.104.38	yes	40
5.4	even	2	inner	105.4.g.b.104.40	yes	40
7.6	odd	2	inner	105.4.g.b.104.4	yes	40
15.14	odd	2	inner	105.4.g.b.104.3	yes	40
21.20	even	2	inner	105.4.g.b.104.39	yes	40
35.34	odd	2	inner	105.4.g.b.104.37	yes	40
105.104	even	2	inner	105.4.g.b.104.2	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.g.b.104.1	✓	40	1.1	even	1	trivial
105.4.g.b.104.2	yes	40	105.104	even	2	inner
105.4.g.b.104.3	yes	40	15.14	odd	2	inner
105.4.g.b.104.4	yes	40	7.6	odd	2	inner
105.4.g.b.104.37	yes	40	35.34	odd	2	inner
105.4.g.b.104.38	yes	40	3.2	odd	2	inner
105.4.g.b.104.39	yes	40	21.20	even	2	inner
105.4.g.b.104.40	yes	40	5.4	even	2	inner