Embedded newform 210.3.v.a.67.1

• Label: 210.3.v.a.67.1
• Level: $210$
• Weight: $3$
• Character: 210.67
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.1
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.a.163.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-4.97092 - 0.538514i) q^{5} -2.44949 q^{6} +(3.26756 - 6.19056i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	−	1.36603i	0.183013	−	0.683013i
							\(3\)	−0.448288	−	1.67303i	−0.149429	−	0.557678i
\(5\)	−4.97092	−	0.538514i	−0.994183	−	0.107703i
							\(7\)	3.26756	−	6.19056i	0.466794	−	0.884366i
\(11\)	1.81875	−	3.15016i	0.165340	−	0.286378i								−0.771436	−	0.636307i	\(-0.780461\pi\)
							0.936776	+	0.349929i	\(0.113794\pi\)
\(13\)	−14.2920	+	14.2920i	−1.09938	+	1.09938i	−0.104901	+	0.994483i	\(0.533453\pi\)
							−0.994483	+	0.104901i	\(0.966547\pi\)
\(17\)	−20.6697	+	5.53844i	−1.21587	+	0.325790i	−0.809061	−	0.587724i	\(-0.800024\pi\)
							−0.406805	+	0.913515i	\(0.633357\pi\)
\(19\)	0.949547	−	0.548221i	0.0499761	−	0.0288537i	−0.474804	−	0.880092i	\(-0.657481\pi\)
							0.524780	+	0.851238i	\(0.324148\pi\)
\(23\)	−24.3662	−	6.52891i	−1.05940	−	0.283866i	−0.313270	−	0.949664i	\(-0.601424\pi\)
							−0.746132	+	0.665798i	\(0.768091\pi\)
\(29\)		−	1.99080i		−	0.0686481i	−0.999411	−	0.0343241i	\(-0.989072\pi\)
							0.999411	−	0.0343241i	\(-0.0109278\pi\)
\(31\)	25.3707	−	43.9434i	0.818411	−	1.41753i	−0.0884411	−	0.996081i	\(-0.528189\pi\)
							0.906852	−	0.421448i	\(-0.138478\pi\)
\(37\)	11.8876	−	44.3653i	0.321288	−	1.19906i	−0.596704	−	0.802461i	\(-0.703523\pi\)
							0.917992	−	0.396600i	\(-0.129810\pi\)
\(41\)	−18.6962			−0.456004			−0.228002	−	0.973661i	\(-0.573219\pi\)
							−0.228002	+	0.973661i	\(0.573219\pi\)
\(43\)	−6.49605	+	6.49605i	−0.151071	+	0.151071i	−0.778596	−	0.627525i	\(-0.784068\pi\)
							0.627525	+	0.778596i	\(0.284068\pi\)
\(47\)	−0.305039	+	1.13842i	−0.00649020	+	0.0242218i	−0.969095	−	0.246688i	\(-0.920658\pi\)
							0.962605	+	0.270910i	\(0.0873244\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.67.1	yes	32
5.3	odd	4	inner	210.3.v.a.193.7	yes	32
7.2	even	3	inner	210.3.v.a.37.7	✓	32
35.23	odd	12	inner	210.3.v.a.163.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.7	✓	32	7.2	even	3	inner
210.3.v.a.67.1	yes	32	1.1	even	1	trivial
210.3.v.a.163.1	yes	32	35.23	odd	12	inner
210.3.v.a.193.7	yes	32	5.3	odd	4	inner