Embedded newform 105.3.v.a.37.2

• Label: 105.3.v.a.37.2
• 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			37.2
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.18498 + 0.853412i) q^{2} +(1.67303 + 0.448288i) q^{3} +(5.95167 - 3.43620i) q^{4} +(-4.91224 + 0.932701i) q^{5} -5.71115 q^{6} +(5.00478 + 4.89410i) q^{7} +(-6.69718 + 6.69718i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.18498	+	0.853412i	−1.59249	+	0.426706i	−0.942763	−	0.333463i	\(-0.891783\pi\)
							−0.649726	+	0.760169i	\(0.725116\pi\)
\(3\)	1.67303	+	0.448288i	0.557678	+	0.149429i
							\(5\)	−4.91224	+	0.932701i	−0.982447	+	0.186540i
\(7\)	5.00478	+	4.89410i	0.714968	+	0.699157i
							\(11\)	−0.581984	−	1.00803i	−0.0529076	−	0.0916387i	0.838359	−	0.545119i	\(-0.183516\pi\)
−0.891266	+	0.453480i	\(0.850182\pi\)
\(13\)	−14.6930	+	14.6930i	−1.13023	+	1.13023i	−0.140089	+	0.990139i	\(0.544739\pi\)
							−0.990139	+	0.140089i	\(0.955261\pi\)
\(17\)	−6.97229	+	26.0209i	−0.410135	+	1.53064i	0.384251	+	0.923229i	\(0.374460\pi\)
							−0.794385	+	0.607414i	\(0.792207\pi\)
\(19\)	−24.9464	−	14.4028i	−1.31297	−	0.758042i	−0.330381	−	0.943848i	\(-0.607177\pi\)
							−0.982586	+	0.185806i	\(0.940510\pi\)
\(23\)	3.19577	+	11.9268i	0.138946	+	0.518555i	0.999950	+	0.00995696i	\(0.00316945\pi\)
							−0.861004	+	0.508598i	\(0.830164\pi\)
\(29\)			12.5542i			0.432905i	0.976293	+	0.216452i	\(0.0694486\pi\)
							−0.976293	+	0.216452i	\(0.930551\pi\)
\(31\)	8.24013	+	14.2723i	0.265811	+	0.460398i	0.967776	−	0.251814i	\(-0.0810271\pi\)
							−0.701965	+	0.712211i	\(0.747694\pi\)
\(37\)	13.6641	−	3.66129i	0.369300	−	0.0989538i	−0.0693957	−	0.997589i	\(-0.522107\pi\)
							0.438696	+	0.898635i	\(0.355440\pi\)
\(41\)	48.3009			1.17807			0.589036	−	0.808107i	\(-0.299508\pi\)
							0.589036	+	0.808107i	\(0.299508\pi\)
\(43\)	−0.357576	+	0.357576i	−0.00831573	+	0.00831573i	−0.711252	−	0.702937i	\(-0.751872\pi\)
							0.702937	+	0.711252i	\(0.251872\pi\)
\(47\)	27.3656	−	7.33258i	0.582246	−	0.156012i	0.0443391	−	0.999017i	\(-0.485882\pi\)
							0.537907	+	0.843004i	\(0.319215\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.2	✓	64
3.2	odd	2		315.3.ca.b.37.15		64
5.3	odd	4	inner	105.3.v.a.58.15	yes	64
7.4	even	3	inner	105.3.v.a.67.15	yes	64
15.8	even	4		315.3.ca.b.163.2		64
21.11	odd	6		315.3.ca.b.172.2		64
35.18	odd	12	inner	105.3.v.a.88.2	yes	64
105.53	even	12		315.3.ca.b.298.15		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.2	✓	64	1.1	even	1	trivial
105.3.v.a.58.15	yes	64	5.3	odd	4	inner
105.3.v.a.67.15	yes	64	7.4	even	3	inner
105.3.v.a.88.2	yes	64	35.18	odd	12	inner
315.3.ca.b.37.15		64	3.2	odd	2
315.3.ca.b.163.2		64	15.8	even	4
315.3.ca.b.172.2		64	21.11	odd	6
315.3.ca.b.298.15		64	105.53	even	12