Embedded newform 210.3.q.a.149.7

• Label: 210.3.q.a.149.7
• 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			149.7
Character	\(\chi\)	\(=\)	210.149
Dual form			210.3.q.a.179.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.10765 + 2.78803i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-4.37658 - 2.41777i) q^{5} +(-2.63140 - 3.32803i) q^{6} +(5.67547 - 4.09745i) q^{7} +2.82843 q^{8} +(-6.54621 - 6.17633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.10765	+	2.78803i	−0.369217	+	0.929343i
\(5\)	−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								−0.409678	−	0.912230i	\(-0.634359\pi\)
							0.585176	+	0.810906i	\(0.301025\pi\)
\(13\)		−	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	0.993706	−	0.112022i	\(-0.0357327\pi\)
							−0.593867	+	0.804563i	\(0.702399\pi\)
\(19\)	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	−0.969634	−	0.244561i	\(-0.921356\pi\)
							0.696613	+	0.717447i	\(0.254690\pi\)
\(29\)		−	4.91307i		−	0.169416i	−0.996406	−	0.0847081i	\(-0.973004\pi\)
							0.996406	−	0.0847081i	\(-0.0269958\pi\)
\(31\)	−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	\(-0.691281\pi\)
							0.565409	−	0.824811i	\(-0.308719\pi\)
\(43\)		−	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	\(-0.664929\pi\)
							0.999985	−	0.00546003i	\(-0.00173799\pi\)

(See \(a_n\) instead)

Twists

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

    

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