Embedded newform 45.3.h.a.29.8

• Label: 45.3.h.a.29.8
• Level: $45$
• Weight: $3$
• Character: 45.29
• Analytic conductor: $1.226$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	45.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.22616118962\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 3*x^18 - 19*x^16 + 66*x^14 + 109*x^12 - 813*x^10 + 981*x^8 + 5346*x^6 - 13851*x^4 - 19683*x^2 + 59049
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.8
Root			\(1.72886 - 0.105167i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.29
Dual form			45.3.h.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14670 - 1.98614i) q^{2} +(-0.182154 - 2.99446i) q^{3} +(-0.629835 - 1.09091i) q^{4} +(0.662169 + 4.95596i) q^{5} +(-6.15630 - 3.07197i) q^{6} +(-6.04559 - 3.49042i) q^{7} +6.28466 q^{8} +(-8.93364 + 1.09091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14670	−	1.98614i	0.573349	−	0.993070i	−0.422870	−	0.906191i	\(-0.638977\pi\)
							0.996219	−	0.0868795i	\(-0.0276895\pi\)
\(3\)	−0.182154	−	2.99446i	−0.0607179	−	0.998155i
							\(5\)	0.662169	+	4.95596i	0.132434	+	0.991192i
\(7\)	−6.04559	−	3.49042i	−0.863655	−	0.498632i								0.00157923	−	0.999999i	\(-0.499497\pi\)
							−0.865235	+	0.501367i	\(0.832831\pi\)
\(11\)	15.8064	+	9.12584i	1.43695	+	0.829622i	0.997636	−	0.0687126i	\(-0.0218892\pi\)
							0.439311	+	0.898335i	\(0.355222\pi\)
\(13\)	−3.66955	+	2.11862i	−0.282273	+	0.162970i	−0.634452	−	0.772962i	\(-0.718774\pi\)
							0.352179	+	0.935933i	\(0.385441\pi\)
\(17\)	−17.3066			−1.01804			−0.509019	−	0.860756i	\(-0.669992\pi\)
							−0.509019	+	0.860756i	\(0.669992\pi\)
\(19\)	−3.96601			−0.208737			−0.104369	−	0.994539i	\(-0.533282\pi\)
							−0.104369	+	0.994539i	\(0.533282\pi\)
\(23\)	0.287675	+	0.498269i	0.0125076	+	0.0216639i	0.872211	−	0.489129i	\(-0.162685\pi\)
							−0.859704	+	0.510793i	\(0.829352\pi\)
\(29\)	−18.1762	−	10.4940i	−0.626766	−	0.361863i	0.152733	−	0.988268i	\(-0.451193\pi\)
							−0.779498	+	0.626404i	\(0.784526\pi\)
\(31\)	−16.7326	−	28.9818i	−0.539763	−	0.934897i	−0.998916	−	0.0465398i	\(-0.985181\pi\)
							0.459154	−	0.888357i	\(-0.348153\pi\)
\(37\)			21.4222i			0.578978i	0.957181	+	0.289489i	\(0.0934854\pi\)
							−0.957181	+	0.289489i	\(0.906515\pi\)
\(41\)	44.1003	−	25.4613i	1.07562	−	0.621008i	0.145907	−	0.989298i	\(-0.453390\pi\)
							0.929711	+	0.368290i	\(0.120057\pi\)
\(43\)	−7.15514	−	4.13102i	−0.166399	−	0.0960703i	0.414488	−	0.910055i	\(-0.363961\pi\)
							−0.580887	+	0.813984i	\(0.697294\pi\)
\(47\)	−7.57789	+	13.1253i	−0.161232	+	0.279262i	−0.935311	−	0.353828i	\(-0.884880\pi\)
							0.774079	+	0.633089i	\(0.218213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.h.a.29.8	yes	20
3.2	odd	2		135.3.h.a.89.3		20
5.2	odd	4		225.3.j.e.101.8		20
5.3	odd	4		225.3.j.e.101.3		20
5.4	even	2	inner	45.3.h.a.29.3	yes	20
9.2	odd	6		405.3.d.a.404.16		20
9.4	even	3		135.3.h.a.44.8		20
9.5	odd	6	inner	45.3.h.a.14.3	✓	20
9.7	even	3		405.3.d.a.404.5		20
15.2	even	4		675.3.j.e.251.3		20
15.8	even	4		675.3.j.e.251.8		20
15.14	odd	2		135.3.h.a.89.8		20
45.4	even	6		135.3.h.a.44.3		20
45.13	odd	12		675.3.j.e.476.8		20
45.14	odd	6	inner	45.3.h.a.14.8	yes	20
45.22	odd	12		675.3.j.e.476.3		20
45.23	even	12		225.3.j.e.176.3		20
45.29	odd	6		405.3.d.a.404.6		20
45.32	even	12		225.3.j.e.176.8		20
45.34	even	6		405.3.d.a.404.15		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.3	✓	20	9.5	odd	6	inner
45.3.h.a.14.8	yes	20	45.14	odd	6	inner
45.3.h.a.29.3	yes	20	5.4	even	2	inner
45.3.h.a.29.8	yes	20	1.1	even	1	trivial
135.3.h.a.44.3		20	45.4	even	6
135.3.h.a.44.8		20	9.4	even	3
135.3.h.a.89.3		20	3.2	odd	2
135.3.h.a.89.8		20	15.14	odd	2
225.3.j.e.101.3		20	5.3	odd	4
225.3.j.e.101.8		20	5.2	odd	4
225.3.j.e.176.3		20	45.23	even	12
225.3.j.e.176.8		20	45.32	even	12
405.3.d.a.404.5		20	9.7	even	3
405.3.d.a.404.6		20	45.29	odd	6
405.3.d.a.404.15		20	45.34	even	6
405.3.d.a.404.16		20	9.2	odd	6
675.3.j.e.251.3		20	15.2	even	4
675.3.j.e.251.8		20	15.8	even	4
675.3.j.e.476.3		20	45.22	odd	12
675.3.j.e.476.8		20	45.13	odd	12