Embedded newform 210.4.i.h.121.2

• Label: 210.4.i.h.121.2
• Level: $210$
• Weight: $4$
• Character: 210.121
• Analytic conductor: $12.390$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.3904011012\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{46})\)
Defining polynomial:	\( x^{4} + 46x^{2} + 2116 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.2
Root			\(3.39116 + 5.87367i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.121
Dual form			210.4.i.h.151.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} -6.00000 q^{6} +(15.4558 - 10.2038i) q^{7} +8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} - 10 q^{5} - 24 q^{6} - 6 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	1.50000	+	2.59808i	0.288675	+	0.500000i
\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
							\(7\)	15.4558	−	10.2038i	0.834536	−	0.550953i
\(11\)	21.9558	+	38.0286i	0.601812	+	1.04237i								0.992547	+	0.121865i	\(0.0388876\pi\)
							−0.390735	+	0.920503i	\(0.627779\pi\)
\(13\)	14.2177			0.303329			0.151664	−	0.988432i	\(-0.451537\pi\)
							0.151664	+	0.988432i	\(0.451537\pi\)
\(17\)	18.3028	+	31.7014i	0.261123	+	0.452278i	0.966541	−	0.256514i	\(-0.0825740\pi\)
							−0.705418	+	0.708792i	\(0.749241\pi\)
\(19\)	−29.2823	+	50.7185i	−0.353570	+	0.612401i	−0.986872	−	0.161504i	\(-0.948366\pi\)
							0.633302	+	0.773905i	\(0.281699\pi\)
\(23\)	−48.3028	+	83.6629i	−0.437906	+	0.758475i	−0.997528	−	0.0702729i	\(-0.977613\pi\)
							0.559622	+	0.828748i	\(0.310946\pi\)
\(29\)	−46.0884			−0.295117			−0.147558	−	0.989053i	\(-0.547141\pi\)
							−0.147558	+	0.989053i	\(0.547141\pi\)
\(31\)	11.6293	+	20.1426i	0.0673770	+	0.116700i	0.897746	−	0.440514i	\(-0.145204\pi\)
							−0.830369	+	0.557214i	\(0.811870\pi\)
\(37\)	−15.6261	+	27.0652i	−0.0694302	+	0.120257i	−0.898651	−	0.438665i	\(-0.855451\pi\)
							0.829220	+	0.558922i	\(0.188785\pi\)
\(41\)	201.123			0.766101			0.383050	−	0.923728i	\(-0.374874\pi\)
							0.383050	+	0.923728i	\(0.374874\pi\)
\(43\)	−250.123			−0.887055			−0.443528	−	0.896261i	\(-0.646273\pi\)
							−0.443528	+	0.896261i	\(0.646273\pi\)
\(47\)	−293.082	+	507.633i	−0.909583	+	1.57544i	−0.0949383	+	0.995483i	\(0.530265\pi\)
							−0.814645	+	0.579961i	\(0.803068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.4.i.h.121.2	✓	4
3.2	odd	2		630.4.k.n.541.2		4
7.2	even	3		1470.4.a.bo.1.1		2
7.4	even	3	inner	210.4.i.h.151.2	yes	4
7.5	odd	6		1470.4.a.bp.1.1		2
21.11	odd	6		630.4.k.n.361.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.i.h.121.2	✓	4	1.1	even	1	trivial
210.4.i.h.151.2	yes	4	7.4	even	3	inner
630.4.k.n.361.2		4	21.11	odd	6
630.4.k.n.541.2		4	3.2	odd	2
1470.4.a.bo.1.1		2	7.2	even	3
1470.4.a.bp.1.1		2	7.5	odd	6