Embedded newform 105.3.v.a.37.14

• Label: 105.3.v.a.37.14
• 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			37.14
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.88660 - 0.773463i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(4.27013 - 2.46536i) q^{4} +(0.160937 - 4.99741i) q^{5} -5.17611 q^{6} +(5.19830 - 4.68804i) q^{7} +(1.96674 - 1.96674i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.88660	−	0.773463i	1.44330	−	0.386731i	0.549613	−	0.835419i	\(-0.314775\pi\)
							0.893688	+	0.448688i	\(0.148109\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	0.160937	−	4.99741i	0.0321874	−	0.999482i
\(7\)	5.19830	−	4.68804i	0.742614	−	0.669719i
							\(11\)	6.71313	+	11.6275i	0.610285	+	1.05704i	0.991192	+	0.132431i	\(0.0422784\pi\)
−0.380907	+	0.924613i	\(0.624388\pi\)
\(13\)	−3.94411	+	3.94411i	−0.303393	+	0.303393i	−0.842340	−	0.538947i	\(-0.818822\pi\)
							0.538947	+	0.842340i	\(0.318822\pi\)
\(17\)	−2.53643	+	9.46609i	−0.149202	+	0.556829i	0.850330	+	0.526249i	\(0.176402\pi\)
							−0.999532	+	0.0305799i	\(0.990265\pi\)
\(19\)	−4.57992	−	2.64422i	−0.241049	−	0.139169i	0.374610	−	0.927182i	\(-0.377777\pi\)
							−0.615659	+	0.788013i	\(0.711110\pi\)
\(23\)	−1.98091	−	7.39285i	−0.0861264	−	0.321428i	0.909399	−	0.415925i	\(-0.136542\pi\)
							−0.995525	+	0.0944974i	\(0.969876\pi\)
\(29\)			36.7188i			1.26616i	0.774085	+	0.633082i	\(0.218211\pi\)
							−0.774085	+	0.633082i	\(0.781789\pi\)
\(31\)	−9.71723	−	16.8307i	−0.313459	−	0.542927i	0.665650	−	0.746264i	\(-0.268155\pi\)
							−0.979109	+	0.203337i	\(0.934821\pi\)
\(37\)	43.7832	−	11.7317i	1.18333	−	0.317072i	0.387083	−	0.922045i	\(-0.373483\pi\)
							0.796246	+	0.604973i	\(0.206816\pi\)
\(41\)	57.9301			1.41293			0.706464	−	0.707749i	\(-0.250289\pi\)
							0.706464	+	0.707749i	\(0.250289\pi\)
\(43\)	46.3359	−	46.3359i	1.07758	−	1.07758i	0.0808525	−	0.996726i	\(-0.474236\pi\)
							0.996726	−	0.0808525i	\(-0.0257643\pi\)
\(47\)	−61.3721	+	16.4446i	−1.30579	+	0.349885i	−0.843636	−	0.536915i	\(-0.819590\pi\)
							−0.462153	+	0.886800i	\(0.652923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.14	✓	64
3.2	odd	2		315.3.ca.b.37.3		64
5.3	odd	4	inner	105.3.v.a.58.3	yes	64
7.4	even	3	inner	105.3.v.a.67.3	yes	64
15.8	even	4		315.3.ca.b.163.14		64
21.11	odd	6		315.3.ca.b.172.14		64
35.18	odd	12	inner	105.3.v.a.88.14	yes	64
105.53	even	12		315.3.ca.b.298.3		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.14	✓	64	1.1	even	1	trivial
105.3.v.a.58.3	yes	64	5.3	odd	4	inner
105.3.v.a.67.3	yes	64	7.4	even	3	inner
105.3.v.a.88.14	yes	64	35.18	odd	12	inner
315.3.ca.b.37.3		64	3.2	odd	2
315.3.ca.b.163.14		64	15.8	even	4
315.3.ca.b.172.14		64	21.11	odd	6
315.3.ca.b.298.3		64	105.53	even	12