Embedded newform 210.4.i.i.151.1

• Label: 210.4.i.i.151.1
• 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			151.1
Root			\(-8.58778 - 14.8745i\) of defining polynomial
Character	\(\chi\)	\(=\)	210.151
Dual form			210.4.i.i.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} -6.00000 q^{6} +(-2.58778 - 18.3386i) q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} - 24 q^{6} + 24 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \)

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
							\(7\)	−2.58778	−	18.3386i	−0.139727	−	0.990190i
\(11\)	−3.91222	+	6.77616i	−0.107234	+	0.185735i								−0.914649	−	0.404249i	\(-0.867533\pi\)
							0.807415	+	0.589985i	\(0.200866\pi\)
\(13\)	−32.8244			−0.700297			−0.350148	−	0.936694i	\(-0.613869\pi\)
							−0.350148	+	0.936694i	\(0.613869\pi\)
\(17\)	34.4389	−	59.6499i	0.491333	−	0.851014i	−0.508617	−	0.860993i	\(-0.669843\pi\)
							0.999950	+	0.00997894i	\(0.00317645\pi\)
\(19\)	−22.8511	−	39.5793i	−0.275916	−	0.477901i	0.694450	−	0.719541i	\(-0.255648\pi\)
							−0.970366	+	0.241640i	\(0.922315\pi\)
\(23\)	−108.790	−	188.430i	−0.986274	−	1.70828i	−0.636135	−	0.771578i	\(-0.719468\pi\)
							−0.350139	−	0.936698i	\(-0.613866\pi\)
\(29\)	−126.527			−0.810187			−0.405093	−	0.914275i	\(-0.632761\pi\)
							−0.405093	+	0.914275i	\(0.632761\pi\)
\(31\)	−84.5534	+	146.451i	−0.489879	+	0.848495i	−0.999932	−	0.0116480i	\(-0.996292\pi\)
							0.510054	+	0.860143i	\(0.329626\pi\)
\(37\)	121.290	+	210.081i	0.538918	+	0.933433i	0.998963	+	0.0455374i	\(0.0145000\pi\)
							−0.460045	+	0.887896i	\(0.652167\pi\)
\(41\)	−286.985			−1.09316			−0.546579	−	0.837408i	\(-0.684070\pi\)
							−0.546579	+	0.837408i	\(0.684070\pi\)
\(43\)	−54.2289			−0.192322			−0.0961609	−	0.995366i	\(-0.530656\pi\)
							−0.0961609	+	0.995366i	\(0.530656\pi\)
\(47\)	−113.351	−	196.330i	−0.351786	−	0.609312i	0.634776	−	0.772696i	\(-0.281092\pi\)
							−0.986563	+	0.163384i	\(0.947759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.4.i.i.151.1	yes	4
3.2	odd	2		630.4.k.m.361.1		4
7.2	even	3	inner	210.4.i.i.121.1	✓	4
7.3	odd	6		1470.4.a.bs.1.1		2
7.4	even	3		1470.4.a.bn.1.1		2
21.2	odd	6		630.4.k.m.541.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.i.i.121.1	✓	4	7.2	even	3	inner
210.4.i.i.151.1	yes	4	1.1	even	1	trivial
630.4.k.m.361.1		4	3.2	odd	2
630.4.k.m.541.1		4	21.2	odd	6
1470.4.a.bn.1.1		2	7.4	even	3
1470.4.a.bs.1.1		2	7.3	odd	6