Embedded newform 105.3.v.a.37.1

• Label: 105.3.v.a.37.1
• 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			37.1
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.56483 + 0.955194i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(8.33154 - 4.81021i) q^{4} +(-1.05941 + 4.88648i) q^{5} +6.39228 q^{6} +(-6.28878 + 3.07429i) q^{7} +(-14.6673 + 14.6673i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.56483	+	0.955194i	−1.78242	+	0.477597i	−0.991021	−	0.133707i	\(-0.957312\pi\)
							−0.791396	+	0.611304i	\(0.790645\pi\)
\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
							\(5\)	−1.05941	+	4.88648i	−0.211883	+	0.977295i
\(7\)	−6.28878	+	3.07429i	−0.898397	+	0.439185i
							\(11\)	−4.09367	−	7.09045i	−0.372152	−	0.644586i	0.617744	−	0.786379i	\(-0.288047\pi\)
−0.989896	+	0.141793i	\(0.954713\pi\)
\(13\)	14.0569	−	14.0569i	1.08130	−	1.08130i	0.0849086	−	0.996389i	\(-0.472940\pi\)
							0.996389	−	0.0849086i	\(-0.0270598\pi\)
\(17\)	1.81174	−	6.76150i	0.106573	−	0.397735i	−0.891946	−	0.452142i	\(-0.850660\pi\)
							0.998519	+	0.0544067i	\(0.0173267\pi\)
\(19\)	−18.2350	−	10.5280i	−0.959739	−	0.554105i	−0.0636460	−	0.997973i	\(-0.520273\pi\)
							−0.896093	+	0.443867i	\(0.853606\pi\)
\(23\)	−8.43024	−	31.4621i	−0.366532	−	1.36792i	−0.865332	−	0.501199i	\(-0.832892\pi\)
							0.498800	−	0.866717i	\(-0.333774\pi\)
\(29\)		−	22.1129i		−	0.762515i	−0.924469	−	0.381258i	\(-0.875491\pi\)
							0.924469	−	0.381258i	\(-0.124509\pi\)
\(31\)	13.7286	+	23.7786i	0.442857	+	0.767052i	0.997900	−	0.0647702i	\(-0.0206314\pi\)
							−0.555043	+	0.831822i	\(0.687298\pi\)
\(37\)	14.9872	−	4.01580i	0.405058	−	0.108535i	−0.0505367	−	0.998722i	\(-0.516093\pi\)
							0.455595	+	0.890187i	\(0.349427\pi\)
\(41\)	0.496183			0.0121020			0.00605101	−	0.999982i	\(-0.498074\pi\)
							0.00605101	+	0.999982i	\(0.498074\pi\)
\(43\)	−33.4554	+	33.4554i	−0.778032	+	0.778032i	−0.979496	−	0.201464i	\(-0.935430\pi\)
							0.201464	+	0.979496i	\(0.435430\pi\)
\(47\)	24.1272	−	6.46488i	0.513346	−	0.137551i	0.00715874	−	0.999974i	\(-0.497721\pi\)
							0.506187	+	0.862424i	\(0.331055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.1	✓	64
3.2	odd	2		315.3.ca.b.37.16		64
5.3	odd	4	inner	105.3.v.a.58.16	yes	64
7.4	even	3	inner	105.3.v.a.67.16	yes	64
15.8	even	4		315.3.ca.b.163.1		64
21.11	odd	6		315.3.ca.b.172.1		64
35.18	odd	12	inner	105.3.v.a.88.1	yes	64
105.53	even	12		315.3.ca.b.298.16		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.1	✓	64	1.1	even	1	trivial
105.3.v.a.58.16	yes	64	5.3	odd	4	inner
105.3.v.a.67.16	yes	64	7.4	even	3	inner
105.3.v.a.88.1	yes	64	35.18	odd	12	inner
315.3.ca.b.37.16		64	3.2	odd	2
315.3.ca.b.163.1		64	15.8	even	4
315.3.ca.b.172.1		64	21.11	odd	6
315.3.ca.b.298.16		64	105.53	even	12