Embedded newform 105.3.v.a.37.5

• Label: 105.3.v.a.37.5
• Level: $105$
• Weight: $3$
• Character: 105.37
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.5
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91023 + 0.511845i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(-0.0771041 + 0.0445161i) q^{4} +(-4.99333 - 0.258093i) q^{5} +3.42533 q^{6} +(6.99587 + 0.240410i) q^{7} +(5.71805 - 5.71805i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91023	+	0.511845i	−0.955116	+	0.255922i	−0.702532	−	0.711653i	\(-0.747947\pi\)
							−0.252584	+	0.967575i	\(0.581280\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	−4.99333	−	0.258093i	−0.998667	−	0.0516186i
\(7\)	6.99587	+	0.240410i	0.999410	+	0.0343443i
							\(11\)	1.58449	+	2.74441i	0.144044	+	0.249492i	0.929016	−	0.370040i	\(-0.120656\pi\)
−0.784972	+	0.619532i	\(0.787323\pi\)
\(13\)	11.0592	−	11.0592i	0.850706	−	0.850706i	−0.139514	−	0.990220i	\(-0.544554\pi\)
							0.990220	+	0.139514i	\(0.0445541\pi\)
\(17\)	4.27411	−	15.9512i	0.251418	−	0.938305i	−0.718630	−	0.695392i	\(-0.755231\pi\)
							0.970048	−	0.242912i	\(-0.0781027\pi\)
\(19\)	24.9472	+	14.4033i	1.31301	+	0.758066i	0.982593	−	0.185769i	\(-0.0594778\pi\)
							0.330416	+	0.943836i	\(0.392811\pi\)
\(23\)	−2.86298	−	10.6848i	−0.124477	−	0.464556i	0.875343	−	0.483502i	\(-0.160635\pi\)
							−0.999820	+	0.0189465i	\(0.993969\pi\)
\(29\)			20.9038i			0.720822i	0.932794	+	0.360411i	\(0.117364\pi\)
							−0.932794	+	0.360411i	\(0.882636\pi\)
\(31\)	−30.4426	−	52.7281i	−0.982019	−	1.70091i	−0.654501	−	0.756061i	\(-0.727121\pi\)
							−0.327518	−	0.944845i	\(-0.606212\pi\)
\(37\)	−4.96117	+	1.32934i	−0.134086	+	0.0359281i	−0.325238	−	0.945632i	\(-0.605444\pi\)
							0.191152	+	0.981560i	\(0.438778\pi\)
\(41\)	0.605481			0.0147678			0.00738392	−	0.999973i	\(-0.497650\pi\)
							0.00738392	+	0.999973i	\(0.497650\pi\)
\(43\)	16.0752	−	16.0752i	0.373842	−	0.373842i	−0.495032	−	0.868874i	\(-0.664844\pi\)
							0.868874	+	0.495032i	\(0.164844\pi\)
\(47\)	34.1202	−	9.14247i	0.725961	−	0.194521i	0.123131	−	0.992390i	\(-0.460706\pi\)
							0.602830	+	0.797870i	\(0.294040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.5	✓	64
3.2	odd	2		315.3.ca.b.37.12		64
5.3	odd	4	inner	105.3.v.a.58.12	yes	64
7.4	even	3	inner	105.3.v.a.67.12	yes	64
15.8	even	4		315.3.ca.b.163.5		64
21.11	odd	6		315.3.ca.b.172.5		64
35.18	odd	12	inner	105.3.v.a.88.5	yes	64
105.53	even	12		315.3.ca.b.298.12		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.5	✓	64	1.1	even	1	trivial
105.3.v.a.58.12	yes	64	5.3	odd	4	inner
105.3.v.a.67.12	yes	64	7.4	even	3	inner
105.3.v.a.88.5	yes	64	35.18	odd	12	inner
315.3.ca.b.37.12		64	3.2	odd	2
315.3.ca.b.163.5		64	15.8	even	4
315.3.ca.b.172.5		64	21.11	odd	6
315.3.ca.b.298.12		64	105.53	even	12