Embedded newform 210.8.i.b.121.2

• Label: 210.8.i.b.121.2
• 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			121.2
Root			\(37.6407 + 65.1956i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.121
Dual form			210.8.i.b.151.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.00000 + 6.92820i) q^{2} +(-13.5000 - 23.3827i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(62.5000 - 108.253i) q^{5} +216.000 q^{6} +(765.954 - 486.680i) q^{7} +512.000 q^{8} +(-364.500 + 631.333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 250 q^{5} + 864 q^{6} - 77 q^{7} + 2048 q^{8} - 1458 q^{9}+O(q^{10}) \)

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
							\(7\)	765.954	−	486.680i	0.844033	−	0.536291i
\(11\)	3911.21	+	6774.41i	0.886005	+	1.53461i								0.844557	+	0.535466i	\(0.179864\pi\)
							0.0414484	+	0.999141i	\(0.486803\pi\)
\(13\)	−13749.8			−1.73578			−0.867892	−	0.496754i	\(-0.834525\pi\)
							−0.867892	+	0.496754i	\(0.834525\pi\)
\(17\)	−12849.9	−	22256.6i	−0.634347	−	1.09872i	−0.986653	−	0.162837i	\(-0.947936\pi\)
							0.352306	−	0.935885i	\(-0.385398\pi\)
\(19\)	−628.830	+	1089.17i	−0.0210327	+	0.0364298i	−0.876350	−	0.481675i	\(-0.840029\pi\)
							0.855317	+	0.518104i	\(0.173362\pi\)
\(23\)	5366.29	−	9294.68i	0.0919659	−	0.159290i	−0.816372	−	0.577526i	\(-0.804018\pi\)
							0.908338	+	0.418236i	\(0.137352\pi\)
\(29\)	75009.5			0.571115			0.285557	−	0.958362i	\(-0.407821\pi\)
							0.285557	+	0.958362i	\(0.407821\pi\)
\(31\)	74601.1	+	129213.i	0.449758	+	0.779004i	0.998370	−	0.0570731i	\(-0.0181768\pi\)
							−0.548612	+	0.836077i	\(0.684843\pi\)
\(37\)	−82885.9	+	143563.i	−0.269014	+	0.465945i	−0.968607	−	0.248595i	\(-0.920031\pi\)
							0.699594	+	0.714541i	\(0.253364\pi\)
\(41\)	−2775.64			−0.00628954			−0.00314477	−	0.999995i	\(-0.501001\pi\)
							−0.00314477	+	0.999995i	\(0.501001\pi\)
\(43\)	−773907.			−1.48439			−0.742197	−	0.670182i	\(-0.766216\pi\)
							−0.742197	+	0.670182i	\(0.766216\pi\)
\(47\)	−364206.	+	630824.i	−0.511688	+	0.886269i	0.488221	+	0.872720i	\(0.337646\pi\)
							−0.999908	+	0.0135487i	\(0.995687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.8.i.b.121.2	✓	4
7.4	even	3	inner	210.8.i.b.151.2	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.8.i.b.121.2	✓	4	1.1	even	1	trivial
210.8.i.b.151.2	yes	4	7.4	even	3	inner