Embedded newform 210.3.v.a.193.7

• Label: 210.3.v.a.193.7
• Level: $210$
• Weight: $3$
• Character: 210.193
• 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			193.7
Character	\(\chi\)	\(=\)	210.193
Dual form			210.3.v.a.37.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.67303 - 0.448288i) q^{3} +(1.73205 + 1.00000i) q^{4} +(2.01909 - 4.57420i) q^{5} -2.44949 q^{6} +(-6.19056 - 3.26756i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	−	0.366025i	−0.683013	−	0.183013i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	2.01909	−	4.57420i	0.403818	−	0.914839i
							\(7\)	−6.19056	−	3.26756i	−0.884366	−	0.466794i
\(11\)	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	−0.994483	−	0.104901i	\(-0.966547\pi\)
							−0.104901	−	0.994483i	\(-0.533453\pi\)
\(17\)	5.53844	+	20.6697i	0.325790	+	1.21587i	0.913515	+	0.406805i	\(0.133357\pi\)
							−0.587724	+	0.809061i	\(0.699976\pi\)
\(19\)	−0.949547	+	0.548221i	−0.0499761	+	0.0288537i	−0.524780	−	0.851238i	\(-0.675852\pi\)
							0.474804	+	0.880092i	\(0.342519\pi\)
\(23\)	6.52891	−	24.3662i	0.283866	−	1.05940i	−0.665798	−	0.746132i	\(-0.731909\pi\)
							0.949664	−	0.313270i	\(-0.101424\pi\)
\(29\)			1.99080i			0.0686481i	0.999411	+	0.0343241i	\(0.0109278\pi\)
							−0.999411	+	0.0343241i	\(0.989072\pi\)
\(31\)	25.3707	−	43.9434i	0.818411	−	1.41753i	−0.0884411	−	0.996081i	\(-0.528189\pi\)
							0.906852	−	0.421448i	\(-0.138478\pi\)
\(37\)	−44.3653	−	11.8876i	−1.19906	−	0.321288i	−0.396600	−	0.917992i	\(-0.629810\pi\)
							−0.802461	+	0.596704i	\(0.796477\pi\)
\(41\)	−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	0.627525	−	0.778596i	\(-0.284068\pi\)
							−0.778596	+	0.627525i	\(0.784068\pi\)
\(47\)	1.13842	+	0.305039i	0.0242218	+	0.00649020i	0.270910	−	0.962605i	\(-0.412676\pi\)
							−0.246688	+	0.969095i	\(0.579342\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.7	✓	32	35.2	odd	12	inner
210.3.v.a.67.1	yes	32	5.2	odd	4	inner
210.3.v.a.163.1	yes	32	7.2	even	3	inner
210.3.v.a.193.7	yes	32	1.1	even	1	trivial