Embedded newform 45.3.h.a.29.10

• Label: 45.3.h.a.29.10
• 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			29.10
Root			\(-1.44078 + 0.961330i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.29
Dual form			45.3.h.a.14.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.84364 - 3.19328i) q^{2} +(1.66507 + 2.49550i) q^{3} +(-4.79800 - 8.31039i) q^{4} +(-3.30943 + 3.74802i) q^{5} +(11.0386 - 0.716233i) q^{6} +(2.22343 + 1.28370i) q^{7} -20.6340 q^{8} +(-3.45506 + 8.31039i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.84364	−	3.19328i	0.921819	−	1.59664i	0.125221	−	0.992129i	\(-0.460036\pi\)
							0.796599	−	0.604509i	\(-0.206631\pi\)
\(3\)	1.66507	+	2.49550i	0.555024	+	0.831834i
							\(5\)	−3.30943	+	3.74802i	−0.661886	+	0.749605i
\(7\)	2.22343	+	1.28370i	0.317633	+	0.183385i								0.650337	−	0.759646i	\(-0.274628\pi\)
							−0.332704	+	0.943031i	\(0.607961\pi\)
\(11\)	−8.51311	−	4.91505i	−0.773919	−	0.446823i	0.0603516	−	0.998177i	\(-0.480778\pi\)
							−0.834271	+	0.551355i	\(0.814111\pi\)
\(13\)	10.4471	−	6.03166i	0.803626	−	0.463974i	−0.0411114	−	0.999155i	\(-0.513090\pi\)
							0.844738	+	0.535181i	\(0.179757\pi\)
\(17\)	4.28451			0.252030			0.126015	−	0.992028i	\(-0.459781\pi\)
							0.126015	+	0.992028i	\(0.459781\pi\)
\(19\)	7.16698			0.377210			0.188605	−	0.982053i	\(-0.439603\pi\)
							0.188605	+	0.982053i	\(0.439603\pi\)
\(23\)	−0.255270	−	0.442140i	−0.0110987	−	0.0192235i	0.860423	−	0.509581i	\(-0.170200\pi\)
							−0.871521	+	0.490357i	\(0.836866\pi\)
\(29\)	−26.4589	−	15.2761i	−0.912376	−	0.526760i	−0.0311810	−	0.999514i	\(-0.509927\pi\)
							−0.881195	+	0.472753i	\(0.843260\pi\)
\(31\)	9.61361	+	16.6513i	0.310116	+	0.537137i	0.978387	−	0.206781i	\(-0.0662986\pi\)
							−0.668271	+	0.743918i	\(0.732965\pi\)
\(37\)			1.31851i			0.0356355i	0.999841	+	0.0178177i	\(0.00567186\pi\)
							−0.999841	+	0.0178177i	\(0.994328\pi\)
\(41\)	−29.9735	+	17.3052i	−0.731061	+	0.422078i	−0.818810	−	0.574064i	\(-0.805366\pi\)
							0.0877493	+	0.996143i	\(0.472033\pi\)
\(43\)	44.9872	+	25.9734i	1.04621	+	0.604032i	0.921587	−	0.388172i	\(-0.126893\pi\)
							0.124627	+	0.992204i	\(0.460227\pi\)
\(47\)	−25.4656	+	44.1078i	−0.541822	+	0.938463i	0.456977	+	0.889478i	\(0.348932\pi\)
							−0.998800	+	0.0489851i	\(0.984401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.h.a.29.10	yes	20
3.2	odd	2		135.3.h.a.89.1		20
5.2	odd	4		225.3.j.e.101.10		20
5.3	odd	4		225.3.j.e.101.1		20
5.4	even	2	inner	45.3.h.a.29.1	yes	20
9.2	odd	6		405.3.d.a.404.20		20
9.4	even	3		135.3.h.a.44.10		20
9.5	odd	6	inner	45.3.h.a.14.1	✓	20
9.7	even	3		405.3.d.a.404.1		20
15.2	even	4		675.3.j.e.251.1		20
15.8	even	4		675.3.j.e.251.10		20
15.14	odd	2		135.3.h.a.89.10		20
45.4	even	6		135.3.h.a.44.1		20
45.13	odd	12		675.3.j.e.476.10		20
45.14	odd	6	inner	45.3.h.a.14.10	yes	20
45.22	odd	12		675.3.j.e.476.1		20
45.23	even	12		225.3.j.e.176.1		20
45.29	odd	6		405.3.d.a.404.2		20
45.32	even	12		225.3.j.e.176.10		20
45.34	even	6		405.3.d.a.404.19		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.1	✓	20	9.5	odd	6	inner
45.3.h.a.14.10	yes	20	45.14	odd	6	inner
45.3.h.a.29.1	yes	20	5.4	even	2	inner
45.3.h.a.29.10	yes	20	1.1	even	1	trivial
135.3.h.a.44.1		20	45.4	even	6
135.3.h.a.44.10		20	9.4	even	3
135.3.h.a.89.1		20	3.2	odd	2
135.3.h.a.89.10		20	15.14	odd	2
225.3.j.e.101.1		20	5.3	odd	4
225.3.j.e.101.10		20	5.2	odd	4
225.3.j.e.176.1		20	45.23	even	12
225.3.j.e.176.10		20	45.32	even	12
405.3.d.a.404.1		20	9.7	even	3
405.3.d.a.404.2		20	45.29	odd	6
405.3.d.a.404.19		20	45.34	even	6
405.3.d.a.404.20		20	9.2	odd	6
675.3.j.e.251.1		20	15.2	even	4
675.3.j.e.251.10		20	15.8	even	4
675.3.j.e.476.1		20	45.22	odd	12
675.3.j.e.476.10		20	45.13	odd	12