Embedded newform 105.3.v.a.37.11

• Label: 105.3.v.a.37.11
• 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			37.11
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.984292 - 0.263740i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(-2.56483 + 1.48081i) q^{4} +(-4.04958 - 2.93273i) q^{5} -1.76498 q^{6} +(-5.81621 + 3.89508i) q^{7} +(-5.01620 + 5.01620i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.984292	−	0.263740i	0.492146	−	0.131870i	−0.00420656	−	0.999991i	\(-0.501339\pi\)
							0.496353	+	0.868121i	\(0.334672\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	−4.04958	−	2.93273i	−0.809916	−	0.586546i
\(7\)	−5.81621	+	3.89508i	−0.830888	+	0.556440i
							\(11\)	−2.58885	−	4.48403i	−0.235350	−	0.407639i	0.724024	−	0.689775i	\(-0.242290\pi\)
−0.959374	+	0.282136i	\(0.908957\pi\)
\(13\)	−3.12416	+	3.12416i	−0.240320	+	0.240320i	−0.816982	−	0.576663i	\(-0.804355\pi\)
							0.576663	+	0.816982i	\(0.304355\pi\)
\(17\)	4.02869	−	15.0353i	0.236982	−	0.884429i	−0.740263	−	0.672317i	\(-0.765299\pi\)
							0.977245	−	0.212112i	\(-0.0680341\pi\)
\(19\)	−17.1897	−	9.92450i	−0.904723	−	0.522342i	−0.0259936	−	0.999662i	\(-0.508275\pi\)
							−0.878730	+	0.477320i	\(0.841608\pi\)
\(23\)	9.71406	+	36.2534i	0.422350	+	1.57623i	0.769642	+	0.638476i	\(0.220435\pi\)
							−0.347292	+	0.937757i	\(0.612899\pi\)
\(29\)		−	11.8306i		−	0.407951i	−0.978976	−	0.203975i	\(-0.934614\pi\)
							0.978976	−	0.203975i	\(-0.0653863\pi\)
\(31\)	−9.02952	−	15.6396i	−0.291275	−	0.504503i	0.682837	−	0.730571i	\(-0.260746\pi\)
							−0.974111	+	0.226068i	\(0.927413\pi\)
\(37\)	−70.9224	+	19.0036i	−1.91682	+	0.513611i	−0.926189	+	0.377059i	\(0.876935\pi\)
							−0.990633	+	0.136552i	\(0.956398\pi\)
\(41\)	60.6872			1.48018			0.740088	−	0.672510i	\(-0.234784\pi\)
							0.740088	+	0.672510i	\(0.234784\pi\)
\(43\)	−33.1804	+	33.1804i	−0.771637	+	0.771637i	−0.978393	−	0.206756i	\(-0.933709\pi\)
							0.206756	+	0.978393i	\(0.433709\pi\)
\(47\)	−19.9094	+	5.33470i	−0.423604	+	0.113504i	−0.464322	−	0.885666i	\(-0.653702\pi\)
							0.0407183	+	0.999171i	\(0.487035\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.11	✓	64
3.2	odd	2		315.3.ca.b.37.6		64
5.3	odd	4	inner	105.3.v.a.58.6	yes	64
7.4	even	3	inner	105.3.v.a.67.6	yes	64
15.8	even	4		315.3.ca.b.163.11		64
21.11	odd	6		315.3.ca.b.172.11		64
35.18	odd	12	inner	105.3.v.a.88.11	yes	64
105.53	even	12		315.3.ca.b.298.6		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.11	✓	64	1.1	even	1	trivial
105.3.v.a.58.6	yes	64	5.3	odd	4	inner
105.3.v.a.67.6	yes	64	7.4	even	3	inner
105.3.v.a.88.11	yes	64	35.18	odd	12	inner
315.3.ca.b.37.6		64	3.2	odd	2
315.3.ca.b.163.11		64	15.8	even	4
315.3.ca.b.172.11		64	21.11	odd	6
315.3.ca.b.298.6		64	105.53	even	12