Embedded newform 105.3.k.d.62.8

• Label: 105.3.k.d.62.8
• 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			62.8
Character	\(\chi\)	\(=\)	105.62
Dual form			105.3.k.d.83.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67168 + 1.67168i) q^{2} +(2.55943 - 1.56503i) q^{3} -1.58906i q^{4} +(-0.529219 - 4.97191i) q^{5} +(-1.66231 + 6.89480i) q^{6} +(6.78135 - 1.73590i) q^{7} +(-4.03033 - 4.03033i) q^{8} +(4.10134 - 8.01118i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67168	+	1.67168i	−0.835842	+	0.835842i	−0.988309	−	0.152466i	\(-0.951278\pi\)
							0.152466	+	0.988309i	\(0.451278\pi\)
\(3\)	2.55943	−	1.56503i	0.853143	−	0.521678i
							\(5\)	−0.529219	−	4.97191i	−0.105844	−	0.994383i
\(7\)	6.78135	−	1.73590i	0.968764	−	0.247985i
							\(11\)		−	4.41779i		−	0.401617i	−0.979630	−	0.200809i	\(-0.935643\pi\)
0.979630	−	0.200809i	\(-0.0643570\pi\)
\(13\)	1.62244	+	1.62244i	0.124803	+	0.124803i	0.766750	−	0.641946i	\(-0.221873\pi\)
							−0.641946	+	0.766750i	\(0.721873\pi\)
\(17\)	13.9255	+	13.9255i	0.819145	+	0.819145i	0.985984	−	0.166839i	\(-0.0533560\pi\)
							−0.166839	+	0.985984i	\(0.553356\pi\)
\(19\)	−0.694013			−0.0365270			−0.0182635	−	0.999833i	\(-0.505814\pi\)
							−0.0182635	+	0.999833i	\(0.505814\pi\)
\(23\)	23.1818	+	23.1818i	1.00790	+	1.00790i	0.999969	+	0.00793606i	\(0.00252615\pi\)
							0.00793606	+	0.999969i	\(0.497474\pi\)
\(29\)	−49.1234			−1.69391			−0.846956	−	0.531663i	\(-0.821567\pi\)
							−0.846956	+	0.531663i	\(0.821567\pi\)
\(31\)		−	33.8768i		−	1.09280i	−0.837524	−	0.546401i	\(-0.815998\pi\)
							0.837524	−	0.546401i	\(-0.184002\pi\)
\(37\)	2.02579	+	2.02579i	0.0547510	+	0.0547510i	0.733952	−	0.679201i	\(-0.237674\pi\)
							−0.679201	+	0.733952i	\(0.737674\pi\)
\(41\)	−32.5085			−0.792891			−0.396446	−	0.918058i	\(-0.629756\pi\)
							−0.396446	+	0.918058i	\(0.629756\pi\)
\(43\)	−30.4591	+	30.4591i	−0.708351	+	0.708351i	−0.966188	−	0.257838i	\(-0.916990\pi\)
							0.257838	+	0.966188i	\(0.416990\pi\)
\(47\)	18.7790	+	18.7790i	0.399554	+	0.399554i	0.878076	−	0.478522i	\(-0.158827\pi\)
							−0.478522	+	0.878076i	\(0.658827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.k.d.62.8	yes	32
3.2	odd	2	inner	105.3.k.d.62.9	yes	32
5.3	odd	4	inner	105.3.k.d.83.10	yes	32
7.6	odd	2	inner	105.3.k.d.62.7	✓	32
15.8	even	4	inner	105.3.k.d.83.7	yes	32
21.20	even	2	inner	105.3.k.d.62.10	yes	32
35.13	even	4	inner	105.3.k.d.83.9	yes	32
105.83	odd	4	inner	105.3.k.d.83.8	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.k.d.62.7	✓	32	7.6	odd	2	inner
105.3.k.d.62.8	yes	32	1.1	even	1	trivial
105.3.k.d.62.9	yes	32	3.2	odd	2	inner
105.3.k.d.62.10	yes	32	21.20	even	2	inner
105.3.k.d.83.7	yes	32	15.8	even	4	inner
105.3.k.d.83.8	yes	32	105.83	odd	4	inner
105.3.k.d.83.9	yes	32	35.13	even	4	inner
105.3.k.d.83.10	yes	32	5.3	odd	4	inner