Embedded newform 105.3.v.a.37.7

• Label: 105.3.v.a.37.7
• 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			37.7
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.445275 + 0.119311i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(-3.28007 + 1.89375i) q^{4} +(4.59550 - 1.97013i) q^{5} +0.798445 q^{6} +(-0.171680 - 6.99789i) q^{7} +(2.53844 - 2.53844i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.445275	+	0.119311i	−0.222637	+	0.0596555i	−0.368413	−	0.929662i	\(-0.620099\pi\)
							0.145776	+	0.989318i	\(0.453432\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	4.59550	−	1.97013i	0.919099	−	0.394026i
\(7\)	−0.171680	−	6.99789i	−0.0245257	−	0.999699i
							\(11\)	−7.70109	−	13.3387i	−0.700099	−	1.21261i	−0.968431	−	0.249280i	\(-0.919806\pi\)
0.268333	−	0.963326i	\(-0.413527\pi\)
\(13\)	17.1139	−	17.1139i	1.31646	−	1.31646i	0.399894	−	0.916561i	\(-0.369047\pi\)
							0.916561	−	0.399894i	\(-0.130953\pi\)
\(17\)	−5.59776	+	20.8911i	−0.329280	+	1.22889i	0.580658	+	0.814147i	\(0.302795\pi\)
							−0.909938	+	0.414743i	\(0.863871\pi\)
\(19\)	−15.6219	−	9.01929i	−0.822203	−	0.474699i	0.0289723	−	0.999580i	\(-0.490777\pi\)
							−0.851176	+	0.524881i	\(0.824110\pi\)
\(23\)	2.98781	+	11.1507i	0.129905	+	0.484812i	0.999967	−	0.00813035i	\(-0.00258800\pi\)
							−0.870062	+	0.492942i	\(0.835921\pi\)
\(29\)		−	1.87676i		−	0.0647160i	−0.999476	−	0.0323580i	\(-0.989698\pi\)
							0.999476	−	0.0323580i	\(-0.0103017\pi\)
\(31\)	9.52708	+	16.5014i	0.307325	+	0.532303i	0.977776	−	0.209651i	\(-0.0672327\pi\)
							−0.670451	+	0.741954i	\(0.733899\pi\)
\(37\)	−4.95991	+	1.32900i	−0.134052	+	0.0359190i	−0.325221	−	0.945638i	\(-0.605439\pi\)
							0.191169	+	0.981557i	\(0.438772\pi\)
\(41\)	−51.4301			−1.25439			−0.627196	−	0.778861i	\(-0.715797\pi\)
							−0.627196	+	0.778861i	\(0.715797\pi\)
\(43\)	18.4492	−	18.4492i	0.429051	−	0.429051i	−0.459254	−	0.888305i	\(-0.651883\pi\)
							0.888305	+	0.459254i	\(0.151883\pi\)
\(47\)	22.4470	−	6.01467i	0.477597	−	0.127972i	−0.0119867	−	0.999928i	\(-0.503816\pi\)
							0.489583	+	0.871957i	\(0.337149\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.7	✓	64
3.2	odd	2		315.3.ca.b.37.10		64
5.3	odd	4	inner	105.3.v.a.58.10	yes	64
7.4	even	3	inner	105.3.v.a.67.10	yes	64
15.8	even	4		315.3.ca.b.163.7		64
21.11	odd	6		315.3.ca.b.172.7		64
35.18	odd	12	inner	105.3.v.a.88.7	yes	64
105.53	even	12		315.3.ca.b.298.10		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.7	✓	64	1.1	even	1	trivial
105.3.v.a.58.10	yes	64	5.3	odd	4	inner
105.3.v.a.67.10	yes	64	7.4	even	3	inner
105.3.v.a.88.7	yes	64	35.18	odd	12	inner
315.3.ca.b.37.10		64	3.2	odd	2
315.3.ca.b.163.7		64	15.8	even	4
315.3.ca.b.172.7		64	21.11	odd	6
315.3.ca.b.298.10		64	105.53	even	12