Embedded newform 90.3.j.a.29.4

• Label: 90.3.j.a.29.4
• Level: $90$
• Weight: $3$
• Character: 90.29
• Analytic conductor: $2.452$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	90.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45232237924\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.4
Root			\(-0.965926 + 0.258819i\) of defining polynomial
Character	\(\chi\)	\(=\)	90.29
Dual form			90.3.j.a.59.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(2.59808 - 1.50000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(3.38865 + 3.67656i) q^{5} -4.24264i q^{6} +(1.81954 + 1.05051i) q^{7} -2.82843 q^{8} +(4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 12 q^{5} + 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	−	1.22474i	0.353553	−	0.612372i
							\(3\)	2.59808	−	1.50000i	0.866025	−	0.500000i
\(5\)	3.38865	+	3.67656i	0.677729	+	0.735311i
							\(7\)	1.81954	+	1.05051i	0.259934	+	0.150073i	0.624304	−	0.781181i	\(-0.285382\pi\)
−0.364371	+	0.931254i	\(0.618716\pi\)
\(11\)	−11.5732	−	6.68180i	−1.05211	−	0.607436i	−0.128871	−	0.991661i	\(-0.541135\pi\)
							−0.923239	+	0.384225i	\(0.874469\pi\)
\(13\)	−20.0454	+	11.5732i	−1.54195	+	0.890247i	−0.543238	+	0.839579i	\(0.682802\pi\)
							−0.998716	+	0.0506684i	\(0.983865\pi\)
\(17\)	27.5057			1.61798			0.808992	−	0.587819i	\(-0.200013\pi\)
							0.808992	+	0.587819i	\(0.200013\pi\)
\(19\)	−15.5505			−0.818448			−0.409224	−	0.912434i	\(-0.634201\pi\)
							−0.409224	+	0.912434i	\(0.634201\pi\)
\(23\)	7.47639	+	12.9495i	0.325060	+	0.563021i	0.981525	−	0.191336i	\(-0.0612820\pi\)
							−0.656464	+	0.754357i	\(0.727949\pi\)
\(29\)	2.84847	+	1.64456i	0.0982231	+	0.0567091i	0.548307	−	0.836277i	\(-0.315273\pi\)
							−0.450084	+	0.892986i	\(0.648606\pi\)
\(31\)	4.92168	+	8.52461i	0.158764	+	0.274987i	0.934423	−	0.356165i	\(-0.115916\pi\)
							−0.775659	+	0.631152i	\(0.782582\pi\)
\(37\)			33.3485i			0.901310i	0.892698	+	0.450655i	\(0.148810\pi\)
							−0.892698	+	0.450655i	\(0.851190\pi\)
\(41\)	9.39898	−	5.42650i	0.229243	−	0.132354i	−0.380980	−	0.924583i	\(-0.624413\pi\)
							0.610223	+	0.792230i	\(0.291080\pi\)
\(43\)	−37.4052	−	21.5959i	−0.869889	−	0.502231i	−0.00257765	−	0.999997i	\(-0.500820\pi\)
							−0.867311	+	0.497766i	\(0.834154\pi\)
\(47\)	18.0437	−	31.2526i	0.383908	−	0.664948i	−0.607709	−	0.794160i	\(-0.707911\pi\)
							0.991617	+	0.129212i	\(0.0412447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.3.j.a.29.4	yes	8
3.2	odd	2		270.3.j.a.89.1		8
5.2	odd	4		450.3.i.a.101.2		4
5.3	odd	4		450.3.i.c.101.1		4
5.4	even	2	inner	90.3.j.a.29.1	✓	8
9.2	odd	6		810.3.b.a.809.7		8
9.4	even	3		270.3.j.a.179.3		8
9.5	odd	6	inner	90.3.j.a.59.1	yes	8
9.7	even	3		810.3.b.a.809.2		8
15.2	even	4		1350.3.i.a.251.1		4
15.8	even	4		1350.3.i.c.251.2		4
15.14	odd	2		270.3.j.a.89.3		8
45.4	even	6		270.3.j.a.179.1		8
45.13	odd	12		1350.3.i.c.1151.2		4
45.14	odd	6	inner	90.3.j.a.59.4	yes	8
45.22	odd	12		1350.3.i.a.1151.1		4
45.23	even	12		450.3.i.c.401.1		4
45.29	odd	6		810.3.b.a.809.1		8
45.32	even	12		450.3.i.a.401.2		4
45.34	even	6		810.3.b.a.809.8		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.a.29.1	✓	8	5.4	even	2	inner
90.3.j.a.29.4	yes	8	1.1	even	1	trivial
90.3.j.a.59.1	yes	8	9.5	odd	6	inner
90.3.j.a.59.4	yes	8	45.14	odd	6	inner
270.3.j.a.89.1		8	3.2	odd	2
270.3.j.a.89.3		8	15.14	odd	2
270.3.j.a.179.1		8	45.4	even	6
270.3.j.a.179.3		8	9.4	even	3
450.3.i.a.101.2		4	5.2	odd	4
450.3.i.a.401.2		4	45.32	even	12
450.3.i.c.101.1		4	5.3	odd	4
450.3.i.c.401.1		4	45.23	even	12
810.3.b.a.809.1		8	45.29	odd	6
810.3.b.a.809.2		8	9.7	even	3
810.3.b.a.809.7		8	9.2	odd	6
810.3.b.a.809.8		8	45.34	even	6
1350.3.i.a.251.1		4	15.2	even	4
1350.3.i.a.1151.1		4	45.22	odd	12
1350.3.i.c.251.2		4	15.8	even	4
1350.3.i.c.1151.2		4	45.13	odd	12