Embedded newform 210.3.q.a.149.17

• Label: 210.3.q.a.149.17
• Level: $210$
• Weight: $3$
• Character: 210.149
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			149.17
Character	\(\chi\)	\(=\)	210.149
Dual form			210.3.q.a.179.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(-2.96833 - 0.434759i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(4.37658 + 2.41777i) q^{5} +(-2.63140 + 3.32803i) q^{6} +(5.67547 - 4.09745i) q^{7} -2.82843 q^{8} +(8.62197 + 2.58102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	−	1.22474i	0.353553	−	0.612372i
							\(3\)	−2.96833	−	0.434759i	−0.989443	−	0.144920i
\(5\)	4.37658	+	2.41777i	0.875315	+	0.483553i
							\(7\)	5.67547	−	4.09745i	0.810781	−	0.585350i
\(11\)	−1.93048	+	1.11456i	−0.175498	+	0.101324i								−0.585176	−	0.810906i	\(-0.698975\pi\)
							0.409678	+	0.912230i	\(0.365641\pi\)
\(13\)		−	9.58505i		−	0.737311i	−0.929566	−	0.368656i	\(-0.879818\pi\)
							0.929566	−	0.368656i	\(-0.120182\pi\)
\(17\)	−6.79726	−	11.7732i	−0.399839	−	0.692541i	0.593867	−	0.804563i	\(-0.297601\pi\)
							−0.993706	+	0.112022i	\(0.964267\pi\)
\(19\)	16.5311	−	28.6327i	0.870058	−	1.50698i	0.00812200	−	0.999967i	\(-0.497415\pi\)
							0.861936	−	0.507017i	\(-0.169252\pi\)
\(23\)	6.27949	−	10.8764i	0.273021	−	0.472887i	−0.696613	−	0.717447i	\(-0.745310\pi\)
							0.969634	+	0.244561i	\(0.0786437\pi\)
\(29\)			4.91307i			0.169416i	0.996406	+	0.0847081i	\(0.0269958\pi\)
							−0.996406	+	0.0847081i	\(0.973004\pi\)
\(31\)	−13.9370	−	24.1396i	−0.449580	−	0.778695i	0.548779	−	0.835968i	\(-0.315093\pi\)
							−0.998359	+	0.0572724i	\(0.981760\pi\)
\(37\)	37.4917	+	21.6459i	1.01329	+	0.585023i	0.912153	−	0.409850i	\(-0.134419\pi\)
							0.101136	+	0.994873i	\(0.467752\pi\)
\(41\)			67.6345i			1.64962i	0.565409	+	0.824811i	\(0.308719\pi\)
							−0.565409	+	0.824811i	\(0.691281\pi\)
\(43\)		−	10.5759i		−	0.245951i	−0.992410	−	0.122976i	\(-0.960756\pi\)
							0.992410	−	0.122976i	\(-0.0392437\pi\)
\(47\)	−23.7219	+	41.0875i	−0.504721	+	0.874203i	0.495264	+	0.868743i	\(0.335071\pi\)
							−0.999985	+	0.00546003i	\(0.998262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.q.a.149.17	yes	64
3.2	odd	2	inner	210.3.q.a.149.7	✓	64
5.4	even	2	inner	210.3.q.a.149.16	yes	64
7.4	even	3	inner	210.3.q.a.179.26	yes	64
15.14	odd	2	inner	210.3.q.a.149.26	yes	64
21.11	odd	6	inner	210.3.q.a.179.16	yes	64
35.4	even	6	inner	210.3.q.a.179.7	yes	64
105.74	odd	6	inner	210.3.q.a.179.17	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.q.a.149.7	✓	64	3.2	odd	2	inner
210.3.q.a.149.16	yes	64	5.4	even	2	inner
210.3.q.a.149.17	yes	64	1.1	even	1	trivial
210.3.q.a.149.26	yes	64	15.14	odd	2	inner
210.3.q.a.179.7	yes	64	35.4	even	6	inner
210.3.q.a.179.16	yes	64	21.11	odd	6	inner
210.3.q.a.179.17	yes	64	105.74	odd	6	inner
210.3.q.a.179.26	yes	64	7.4	even	3	inner