Embedded newform 210.8.i.b.121.1

• Label: 210.8.i.b.121.1
• Level: $210$
• Weight: $8$
• Character: 210.121
• Analytic conductor: $65.601$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	210.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(65.6008553517\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial:	\( x^{4} - x^{3} + 5593x^{2} + 5592x + 31270464 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.1
Root			\(-37.1407 - 64.3295i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.121
Dual form			210.8.i.b.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 + 6.92820i) q^{2} +(-13.5000 - 23.3827i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(62.5000 - 108.253i) q^{5} +216.000 q^{6} +(-804.454 + 419.996i) q^{7} +512.000 q^{8} +(-364.500 + 631.333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 250 q^{5} + 864 q^{6} - 77 q^{7} + 2048 q^{8} - 1458 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.00000	+	6.92820i	−0.353553	+	0.612372i
							\(3\)	−13.5000	−	23.3827i	−0.288675	−	0.500000i
\(5\)	62.5000	−	108.253i	0.223607	−	0.387298i
							\(7\)	−804.454	+	419.996i	−0.886458	+	0.462809i
\(11\)	−1248.71	−	2162.82i	−0.282869	−	0.489944i								0.689221	−	0.724551i	\(-0.257953\pi\)
							−0.972090	+	0.234607i	\(0.924620\pi\)
\(13\)	6889.83			0.869774			0.434887	−	0.900485i	\(-0.356788\pi\)
							0.434887	+	0.900485i	\(0.356788\pi\)
\(17\)	1956.85	+	3389.37i	0.0966022	+	0.167320i	0.910276	−	0.414002i	\(-0.135869\pi\)
							−0.813674	+	0.581322i	\(0.802536\pi\)
\(19\)	−21717.2	+	37615.2i	−0.726383	+	1.25813i	0.232019	+	0.972711i	\(0.425467\pi\)
							−0.958402	+	0.285421i	\(0.907867\pi\)
\(23\)	17705.2	−	30666.3i	0.303427	−	0.525550i	−0.673483	−	0.739203i	\(-0.735203\pi\)
							0.976910	+	0.213652i	\(0.0685359\pi\)
\(29\)	165644.			1.26120			0.630600	−	0.776108i	\(-0.282809\pi\)
							0.630600	+	0.776108i	\(0.282809\pi\)
\(31\)	96362.4	+	166905.i	0.580954	+	1.00624i	0.995367	+	0.0961529i	\(0.0306538\pi\)
							−0.414412	+	0.910089i	\(0.636013\pi\)
\(37\)	−46542.1	+	80613.3i	−0.151057	+	0.261638i	−0.931616	−	0.363444i	\(-0.881601\pi\)
							0.780559	+	0.625082i	\(0.214934\pi\)
\(41\)	−318203.			−0.721043			−0.360522	−	0.932751i	\(-0.617401\pi\)
							−0.360522	+	0.932751i	\(0.617401\pi\)
\(43\)	−216188.			−0.414660			−0.207330	−	0.978271i	\(-0.566477\pi\)
							−0.207330	+	0.978271i	\(0.566477\pi\)
\(47\)	−497242.	+	861249.i	−0.698595	+	1.21000i	0.270359	+	0.962760i	\(0.412858\pi\)
							−0.968954	+	0.247242i	\(0.920476\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.8.i.b.121.1	✓	4
7.4	even	3	inner	210.8.i.b.151.1	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.8.i.b.121.1	✓	4	1.1	even	1	trivial
210.8.i.b.151.1	yes	4	7.4	even	3	inner