Embedded newform 210.4.i.j.121.2

• Label: 210.4.i.j.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:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 5x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.2
Root			\(-1.93649 - 1.11803i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.121
Dual form			210.4.i.j.151.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(2.50000 - 4.33013i) q^{5} +6.00000 q^{6} +(0.809475 + 18.5026i) q^{7} -8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} + 24 q^{6} - 20 q^{7} - 32 q^{8} - 18 q^{9}+O(q^{10}) \)

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
							\(7\)	0.809475	+	18.5026i	0.0437075	+	0.999044i
\(11\)	21.9284	+	37.9811i	0.601061	+	1.04107i								0.992661	+	0.120932i	\(0.0385884\pi\)
							−0.391600	+	0.920136i	\(0.628078\pi\)
\(13\)	66.8569			1.42637			0.713183	−	0.700978i	\(-0.247253\pi\)
							0.713183	+	0.700978i	\(0.247253\pi\)
\(17\)	−8.07157	−	13.9804i	−0.115156	−	0.199455i	0.802686	−	0.596401i	\(-0.203403\pi\)
							−0.917842	+	0.396946i	\(0.870070\pi\)
\(19\)	38.7379	−	67.0960i	0.467741	−	0.810152i	−0.531579	−	0.847009i	\(-0.678401\pi\)
							0.999321	+	0.0368569i	\(0.0117346\pi\)
\(23\)	−51.1663	+	88.6227i	−0.463866	+	0.803439i	−0.999150	−	0.0412330i	\(-0.986871\pi\)
							0.535284	+	0.844672i	\(0.320205\pi\)
\(29\)	192.333			1.23156			0.615781	−	0.787918i	\(-0.288841\pi\)
							0.615781	+	0.787918i	\(0.288841\pi\)
\(31\)	151.927	+	263.146i	0.880225	+	1.52459i	0.851091	+	0.525018i	\(0.175942\pi\)
							0.0291339	+	0.999576i	\(0.490725\pi\)
\(37\)	183.666	−	318.119i	0.816069	−	1.41347i	−0.0924885	−	0.995714i	\(-0.529482\pi\)
							0.908558	−	0.417759i	\(-0.137185\pi\)
\(41\)	−7.85685			−0.0299277			−0.0149638	−	0.999888i	\(-0.504763\pi\)
							−0.0149638	+	0.999888i	\(0.504763\pi\)
\(43\)	−182.095			−0.645795			−0.322898	−	0.946434i	\(-0.604657\pi\)
							−0.322898	+	0.946434i	\(0.604657\pi\)
\(47\)	−134.238	+	232.507i	−0.416609	+	0.721587i	−0.995596	−	0.0937492i	\(-0.970115\pi\)
							0.578987	+	0.815337i	\(0.303448\pi\)

(See \(a_n\) instead)

Twists

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

    

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