Embedded newform 210.4.i.i.151.2

• Label: 210.4.i.i.151.2
• Level: $210$
• Weight: $4$
• Character: 210.151
• 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:	\(\Q(\sqrt{-3}, \sqrt{295})\)
Defining polynomial:	\( x^{4} + 295x^{2} + 87025 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			151.2
Root			\(8.58778 + 14.8745i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.151
Dual form			210.4.i.i.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} -6.00000 q^{6} +(14.5878 + 11.4104i) q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} - 24 q^{6} + 24 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)

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
							\(7\)	14.5878	+	11.4104i	0.787666	+	0.616102i
\(11\)	−21.0878	+	36.5251i	−0.578019	+	1.00116i								0.417688	+	0.908591i	\(0.362841\pi\)
							−0.995706	+	0.0925671i	\(0.970493\pi\)
\(13\)	−67.1756			−1.43317			−0.716583	−	0.697502i	\(-0.754295\pi\)
							−0.716583	+	0.697502i	\(0.754295\pi\)
\(17\)	−51.4389	+	89.0948i	−0.733869	+	1.27110i	0.221349	+	0.975195i	\(0.428954\pi\)
							−0.955218	+	0.295903i	\(0.904379\pi\)
\(19\)	45.8511	+	79.4165i	0.553630	+	0.958915i	0.998009	+	0.0630763i	\(0.0200911\pi\)
							−0.444379	+	0.895839i	\(0.646576\pi\)
\(23\)	45.7900	+	79.3107i	0.415125	+	0.719018i	0.995442	−	0.0953730i	\(-0.0304044\pi\)
							−0.580316	+	0.814391i	\(0.697071\pi\)
\(29\)	−23.4733			−0.150306			−0.0751532	−	0.997172i	\(-0.523945\pi\)
							−0.0751532	+	0.997172i	\(0.523945\pi\)
\(31\)	121.553	−	210.537i	0.704246	−	1.21979i	−0.262717	−	0.964873i	\(-0.584618\pi\)
							0.966963	−	0.254917i	\(-0.0820482\pi\)
\(37\)	−33.2900	−	57.6600i	−0.147915	−	0.256196i	0.782542	−	0.622598i	\(-0.213923\pi\)
							−0.930457	+	0.366402i	\(0.880589\pi\)
\(41\)	296.985			1.13125			0.565624	−	0.824663i	\(-0.308635\pi\)
							0.565624	+	0.824663i	\(0.308635\pi\)
\(43\)	186.229			0.660457			0.330228	−	0.943901i	\(-0.392874\pi\)
							0.330228	+	0.943901i	\(0.392874\pi\)
\(47\)	−44.6489	−	77.3341i	−0.138568	−	0.240007i	0.788387	−	0.615180i	\(-0.210917\pi\)
							−0.926955	+	0.375173i	\(0.877583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.4.i.i.151.2	yes	4
3.2	odd	2		630.4.k.m.361.2		4
7.2	even	3	inner	210.4.i.i.121.2	✓	4
7.3	odd	6		1470.4.a.bs.1.2		2
7.4	even	3		1470.4.a.bn.1.2		2
21.2	odd	6		630.4.k.m.541.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.i.i.121.2	✓	4	7.2	even	3	inner
210.4.i.i.151.2	yes	4	1.1	even	1	trivial
630.4.k.m.361.2		4	3.2	odd	2
630.4.k.m.541.2		4	21.2	odd	6
1470.4.a.bn.1.2		2	7.4	even	3
1470.4.a.bs.1.2		2	7.3	odd	6