Embedded newform 450.3.i.e.101.3

• Label: 450.3.i.e.101.3
• Level: $450$
• Weight: $3$
• Character: 450.101
• Analytic conductor: $12.262$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2616118962\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 2x^{14} - 230x^{12} + 96x^{10} + 25551x^{8} + 7776x^{6} - 1509030x^{4} - 1062882x^{2} + 43046721 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.3
Root			\(2.93235 + 0.633522i\) of defining polynomial
Character	\(\chi\)	\(=\)	450.101
Dual form			450.3.i.e.401.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(2.01482 - 2.22272i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-4.03934 + 1.29758i) q^{6} +(-4.38322 + 7.59195i) q^{7} -2.82843i q^{8} +(-0.881011 - 8.95678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4} - 8 q^{6} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	−	0.707107i	−0.612372	−	0.353553i
							\(3\)	2.01482	−	2.22272i	0.671606	−	0.740908i
\(5\)		0			0
							\(7\)	−4.38322	+	7.59195i	−0.626174	+	1.08456i	0.362139	+	0.932124i	\(0.382047\pi\)
−0.988313	+	0.152441i	\(0.951287\pi\)
\(11\)	−11.6424	−	6.72177i	−1.05840	−	0.611070i	−0.133414	−	0.991060i	\(-0.542594\pi\)
							−0.924990	+	0.379990i	\(0.875927\pi\)
\(13\)	8.62140	+	14.9327i	0.663185	+	1.14867i	0.979774	+	0.200107i	\(0.0641289\pi\)
							−0.316590	+	0.948563i	\(0.602538\pi\)
\(17\)			0.183062i			0.0107684i	0.999986	+	0.00538419i	\(0.00171385\pi\)
							−0.999986	+	0.00538419i	\(0.998286\pi\)
\(19\)	−34.7310			−1.82795			−0.913974	−	0.405773i	\(-0.867002\pi\)
							−0.913974	+	0.405773i	\(0.867002\pi\)
\(23\)	−29.8095	+	17.2105i	−1.29607	+	0.748284i	−0.979722	−	0.200360i	\(-0.935789\pi\)
							−0.316344	+	0.948645i	\(0.602455\pi\)
\(29\)	8.98484	+	5.18740i	0.309822	+	0.178876i	0.646847	−	0.762620i	\(-0.276087\pi\)
							−0.337025	+	0.941496i	\(0.609421\pi\)
\(31\)	−10.3258	−	17.8848i	−0.333090	−	0.576928i	0.650026	−	0.759912i	\(-0.274758\pi\)
							−0.983116	+	0.182983i	\(0.941425\pi\)
\(37\)	−8.75453			−0.236609			−0.118304	−	0.992977i	\(-0.537746\pi\)
							−0.118304	+	0.992977i	\(0.537746\pi\)
\(41\)	1.36622	−	0.788788i	0.0333224	−	0.0192387i	−0.483246	−	0.875484i	\(-0.660542\pi\)
							0.516569	+	0.856246i	\(0.327209\pi\)
\(43\)	−20.2854	+	35.1354i	−0.471755	+	0.817103i	−0.999478	−	0.0323135i	\(-0.989712\pi\)
							0.527723	+	0.849416i	\(0.323046\pi\)
\(47\)	26.8388	+	15.4954i	0.571037	+	0.329689i	0.757564	−	0.652761i	\(-0.226390\pi\)
							−0.186526	+	0.982450i	\(0.559723\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.3.i.e.101.3		16
3.2	odd	2		1350.3.i.e.251.5		16
5.2	odd	4		90.3.j.b.29.5	yes	16
5.3	odd	4		90.3.j.b.29.4	✓	16
5.4	even	2	inner	450.3.i.e.101.6		16
9.4	even	3		1350.3.i.e.1151.5		16
9.5	odd	6	inner	450.3.i.e.401.3		16
15.2	even	4		270.3.j.b.89.1		16
15.8	even	4		270.3.j.b.89.6		16
15.14	odd	2		1350.3.i.e.251.4		16
45.2	even	12		810.3.b.b.809.13		16
45.4	even	6		1350.3.i.e.1151.4		16
45.7	odd	12		810.3.b.b.809.4		16
45.13	odd	12		270.3.j.b.179.1		16
45.14	odd	6	inner	450.3.i.e.401.6		16
45.22	odd	12		270.3.j.b.179.6		16
45.23	even	12		90.3.j.b.59.5	yes	16
45.32	even	12		90.3.j.b.59.4	yes	16
45.38	even	12		810.3.b.b.809.3		16
45.43	odd	12		810.3.b.b.809.14		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.b.29.4	✓	16	5.3	odd	4
90.3.j.b.29.5	yes	16	5.2	odd	4
90.3.j.b.59.4	yes	16	45.32	even	12
90.3.j.b.59.5	yes	16	45.23	even	12
270.3.j.b.89.1		16	15.2	even	4
270.3.j.b.89.6		16	15.8	even	4
270.3.j.b.179.1		16	45.13	odd	12
270.3.j.b.179.6		16	45.22	odd	12
450.3.i.e.101.3		16	1.1	even	1	trivial
450.3.i.e.101.6		16	5.4	even	2	inner
450.3.i.e.401.3		16	9.5	odd	6	inner
450.3.i.e.401.6		16	45.14	odd	6	inner
810.3.b.b.809.3		16	45.38	even	12
810.3.b.b.809.4		16	45.7	odd	12
810.3.b.b.809.13		16	45.2	even	12
810.3.b.b.809.14		16	45.43	odd	12
1350.3.i.e.251.4		16	15.14	odd	2
1350.3.i.e.251.5		16	3.2	odd	2
1350.3.i.e.1151.4		16	45.4	even	6
1350.3.i.e.1151.5		16	9.4	even	3