Embedded newform 216.3.x.a.5.12

• Label: 216.3.x.a.5.12
• Level: $216$
• Weight: $3$
• Character: 216.5
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.12
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68433 + 1.07844i) q^{2} +(-2.97464 + 0.389273i) q^{3} +(1.67392 - 3.63290i) q^{4} +(-3.75258 + 1.36583i) q^{5} +(4.59046 - 3.86364i) q^{6} +(2.21734 + 12.5752i) q^{7} +(1.09844 + 7.92423i) q^{8} +(8.69693 - 2.31589i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68433	+	1.07844i	−0.842164	+	0.539221i
							\(3\)	−2.97464	+	0.389273i	−0.991546	+	0.129758i
\(5\)	−3.75258	+	1.36583i	−0.750516	+	0.273166i								−0.688823	−	0.724929i	\(-0.741872\pi\)
							−0.0616934	+	0.998095i	\(0.519650\pi\)
\(7\)	2.21734	+	12.5752i	0.316763	+	1.79645i	0.562160	+	0.827028i	\(0.309970\pi\)
							−0.245398	+	0.969422i	\(0.578919\pi\)
\(11\)	−2.44466	−	0.889783i	−0.222242	−	0.0808893i	0.228500	−	0.973544i	\(-0.426618\pi\)
							−0.450741	+	0.892655i	\(0.648840\pi\)
\(13\)	−5.59866	+	6.67223i	−0.430666	+	0.513248i	−0.937114	−	0.349022i	\(-0.886514\pi\)
							0.506448	+	0.862270i	\(0.330958\pi\)
\(17\)	−26.5782	−	15.3449i	−1.56342	−	0.902641i	−0.996907	−	0.0785890i	\(-0.974959\pi\)
							−0.566514	−	0.824052i	\(-0.691708\pi\)
\(19\)	24.4175	−	14.0975i	1.28513	−	0.741972i	0.307351	−	0.951596i	\(-0.400557\pi\)
							0.977782	+	0.209624i	\(0.0672240\pi\)
\(23\)	−10.6035	−	1.86968i	−0.461020	−	0.0812903i	−0.0616858	−	0.998096i	\(-0.519648\pi\)
							−0.399335	+	0.916805i	\(0.630759\pi\)
\(29\)	9.26262	−	7.77226i	0.319401	−	0.268009i	−0.468964	−	0.883217i	\(-0.655373\pi\)
							0.788365	+	0.615208i	\(0.210928\pi\)
\(31\)	5.13142	−	29.1018i	0.165530	−	0.938766i	−0.782987	−	0.622039i	\(-0.786305\pi\)
							0.948516	−	0.316728i	\(-0.102584\pi\)
\(37\)	−32.2600	−	18.6253i	−0.871891	−	0.503387i	−0.00391506	−	0.999992i	\(-0.501246\pi\)
							−0.867976	+	0.496606i	\(0.834580\pi\)
\(41\)	13.0252	−	15.5228i	0.317688	−	0.378606i	−0.583442	−	0.812155i	\(-0.698294\pi\)
							0.901130	+	0.433549i	\(0.142739\pi\)
\(43\)	5.88159	−	16.1595i	0.136781	−	0.375803i	−0.852324	−	0.523014i	\(-0.824807\pi\)
							0.989105	+	0.147211i	\(0.0470296\pi\)
\(47\)	14.9615	−	2.63812i	0.318330	−	0.0561301i	−0.0122003	−	0.999926i	\(-0.503884\pi\)
							0.330530	+	0.943795i	\(0.392772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.12	✓	420
8.5	even	2	inner	216.3.x.a.5.52	yes	420
27.11	odd	18	inner	216.3.x.a.173.52	yes	420
216.173	odd	18	inner	216.3.x.a.173.12	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.12	✓	420	1.1	even	1	trivial
216.3.x.a.5.52	yes	420	8.5	even	2	inner
216.3.x.a.173.12	yes	420	216.173	odd	18	inner
216.3.x.a.173.52	yes	420	27.11	odd	18	inner