Embedded newform 105.4.j.a.8.12

• Label: 105.4.j.a.8.12
• Level: $105$
• Weight: $4$
• Character: 105.8
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.12
Character	\(\chi\)	\(=\)	105.8
Dual form			105.4.j.a.92.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59773 - 1.59773i) q^{2} +(3.12266 + 4.15319i) q^{3} -2.89452i q^{4} +(-1.68350 - 11.0529i) q^{5} +(1.64652 - 11.6248i) q^{6} +(4.94975 - 4.94975i) q^{7} +(-17.4065 + 17.4065i) q^{8} +(-7.49804 + 25.9380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 8 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59773	−	1.59773i	−0.564883	−	0.564883i	0.365808	−	0.930690i	\(-0.380793\pi\)
							−0.930690	+	0.365808i	\(0.880793\pi\)
\(3\)	3.12266	+	4.15319i	0.600955	+	0.799283i
							\(5\)	−1.68350	−	11.0529i	−0.150577	−	0.988598i
\(7\)	4.94975	−	4.94975i	0.267261	−	0.267261i
							\(11\)		−	48.1207i		−	1.31899i	−0.751707	−	0.659497i	\(-0.770769\pi\)
0.751707	−	0.659497i	\(-0.229231\pi\)
\(13\)	−18.4701	−	18.4701i	−0.394054	−	0.394054i	0.482076	−	0.876129i	\(-0.339883\pi\)
							−0.876129	+	0.482076i	\(0.839883\pi\)
\(17\)	−4.83084	−	4.83084i	−0.0689207	−	0.0689207i	0.671806	−	0.740727i	\(-0.265519\pi\)
							−0.740727	+	0.671806i	\(0.765519\pi\)
\(19\)		−	146.471i		−	1.76857i	−0.466949	−	0.884284i	\(-0.654647\pi\)
							0.466949	−	0.884284i	\(-0.345353\pi\)
\(23\)	28.8804	−	28.8804i	0.261825	−	0.261825i	−0.563970	−	0.825795i	\(-0.690727\pi\)
							0.825795	+	0.563970i	\(0.190727\pi\)
\(29\)	−57.4414			−0.367813			−0.183907	−	0.982944i	\(-0.558874\pi\)
							−0.183907	+	0.982944i	\(0.558874\pi\)
\(31\)	−2.83297			−0.0164134			−0.00820671	−	0.999966i	\(-0.502612\pi\)
							−0.00820671	+	0.999966i	\(0.502612\pi\)
\(37\)	299.547	−	299.547i	1.33095	−	1.33095i	0.426429	−	0.904521i	\(-0.359771\pi\)
							0.904521	−	0.426429i	\(-0.140229\pi\)
\(41\)			377.756i			1.43891i	0.694537	+	0.719457i	\(0.255609\pi\)
							−0.694537	+	0.719457i	\(0.744391\pi\)
\(43\)	137.812	+	137.812i	0.488749	+	0.488749i	0.907911	−	0.419163i	\(-0.137676\pi\)
							−0.419163	+	0.907911i	\(0.637676\pi\)
\(47\)	433.756	+	433.756i	1.34617	+	1.34617i	0.889785	+	0.456380i	\(0.150854\pi\)
							0.456380	+	0.889785i	\(0.349146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.j.a.8.12	✓	72
3.2	odd	2	inner	105.4.j.a.8.25	yes	72
5.2	odd	4	inner	105.4.j.a.92.25	yes	72
15.2	even	4	inner	105.4.j.a.92.12	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.j.a.8.12	✓	72	1.1	even	1	trivial
105.4.j.a.8.25	yes	72	3.2	odd	2	inner
105.4.j.a.92.12	yes	72	15.2	even	4	inner
105.4.j.a.92.25	yes	72	5.2	odd	4	inner