Embedded newform 105.3.k.d.62.2

• Label: 105.3.k.d.62.2
• Level: $105$
• Weight: $3$
• Character: 105.62
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			62.2
Character	\(\chi\)	\(=\)	105.62
Dual form			105.3.k.d.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63482 + 2.63482i) q^{2} +(2.54097 + 1.59482i) q^{3} -9.88460i q^{4} +(4.15332 + 2.78387i) q^{5} +(-10.8971 + 2.49295i) q^{6} +(6.57626 + 2.39850i) q^{7} +(15.5049 + 15.5049i) q^{8} +(3.91310 + 8.10479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63482	+	2.63482i	−1.31741	+	1.31741i	−0.401595	+	0.915817i	\(0.631544\pi\)
							−0.915817	+	0.401595i	\(0.868456\pi\)
\(3\)	2.54097	+	1.59482i	0.846991	+	0.531606i
							\(5\)	4.15332	+	2.78387i	0.830663	+	0.556775i
\(7\)	6.57626	+	2.39850i	0.939466	+	0.342643i
							\(11\)		−	5.66373i		−	0.514885i	−0.966294	−	0.257442i	\(-0.917120\pi\)
0.966294	−	0.257442i	\(-0.0828797\pi\)
\(13\)	1.68481	+	1.68481i	0.129601	+	0.129601i	0.768932	−	0.639331i	\(-0.220789\pi\)
							−0.639331	+	0.768932i	\(0.720789\pi\)
\(17\)	−14.0118	−	14.0118i	−0.824224	−	0.824224i	0.162486	−	0.986711i	\(-0.448049\pi\)
							−0.986711	+	0.162486i	\(0.948049\pi\)
\(19\)	−24.0066			−1.26351			−0.631753	−	0.775170i	\(-0.717664\pi\)
							−0.631753	+	0.775170i	\(0.717664\pi\)
\(23\)	3.17641	+	3.17641i	0.138105	+	0.138105i	0.772780	−	0.634675i	\(-0.218866\pi\)
							−0.634675	+	0.772780i	\(0.718866\pi\)
\(29\)	24.1064			0.831254			0.415627	−	0.909535i	\(-0.363562\pi\)
							0.415627	+	0.909535i	\(0.363562\pi\)
\(31\)		−	23.8436i		−	0.769148i	−0.923094	−	0.384574i	\(-0.874348\pi\)
							0.923094	−	0.384574i	\(-0.125652\pi\)
\(37\)	−13.8075	−	13.8075i	−0.373177	−	0.373177i	0.495456	−	0.868633i	\(-0.335001\pi\)
							−0.868633	+	0.495456i	\(0.835001\pi\)
\(41\)	−53.4368			−1.30334			−0.651668	−	0.758505i	\(-0.725930\pi\)
							−0.651668	+	0.758505i	\(0.725930\pi\)
\(43\)	25.9017	−	25.9017i	0.602366	−	0.602366i	−0.338574	−	0.940940i	\(-0.609945\pi\)
							0.940940	+	0.338574i	\(0.109945\pi\)
\(47\)	27.3968	+	27.3968i	0.582911	+	0.582911i	0.935702	−	0.352791i	\(-0.114767\pi\)
							−0.352791	+	0.935702i	\(0.614767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.k.d.62.2	yes	32
3.2	odd	2	inner	105.3.k.d.62.16	yes	32
5.3	odd	4	inner	105.3.k.d.83.15	yes	32
7.6	odd	2	inner	105.3.k.d.62.1	✓	32
15.8	even	4	inner	105.3.k.d.83.1	yes	32
21.20	even	2	inner	105.3.k.d.62.15	yes	32
35.13	even	4	inner	105.3.k.d.83.16	yes	32
105.83	odd	4	inner	105.3.k.d.83.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.k.d.62.1	✓	32	7.6	odd	2	inner
105.3.k.d.62.2	yes	32	1.1	even	1	trivial
105.3.k.d.62.15	yes	32	21.20	even	2	inner
105.3.k.d.62.16	yes	32	3.2	odd	2	inner
105.3.k.d.83.1	yes	32	15.8	even	4	inner
105.3.k.d.83.2	yes	32	105.83	odd	4	inner
105.3.k.d.83.15	yes	32	5.3	odd	4	inner
105.3.k.d.83.16	yes	32	35.13	even	4	inner