Embedded newform 45.3.h.a.29.9

• Label: 45.3.h.a.29.9
• 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			29.9
Root			\(0.346576 - 1.69702i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.29
Dual form			45.3.h.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.42081 - 2.46092i) q^{2} +(-2.93933 - 0.600288i) q^{3} +(-2.03740 - 3.52889i) q^{4} +(-1.22790 - 4.84688i) q^{5} +(-5.65349 + 6.38055i) q^{6} +(8.42581 + 4.86464i) q^{7} -0.212574 q^{8} +(8.27931 + 3.52889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.42081	−	2.46092i	0.710405	−	1.23046i	−0.254300	−	0.967125i	\(-0.581845\pi\)
							0.964705	−	0.263332i	\(-0.0848216\pi\)
\(3\)	−2.93933	−	0.600288i	−0.979776	−	0.200096i
							\(5\)	−1.22790	−	4.84688i	−0.245580	−	0.969376i
\(7\)	8.42581	+	4.86464i	1.20369	+	0.694949i								0.961373	−	0.275249i	\(-0.0887603\pi\)
							0.242314	+	0.970198i	\(0.422094\pi\)
\(11\)	0.370966	+	0.214177i	0.0337242	+	0.0194707i	0.516767	−	0.856126i	\(-0.327135\pi\)
							−0.483043	+	0.875597i	\(0.660469\pi\)
\(13\)	−16.2396	+	9.37595i	−1.24920	+	0.721227i	−0.970950	−	0.239281i	\(-0.923088\pi\)
							−0.278252	+	0.960508i	\(0.589755\pi\)
\(17\)	2.85718			0.168069			0.0840347	−	0.996463i	\(-0.473219\pi\)
							0.0840347	+	0.996463i	\(0.473219\pi\)
\(19\)	0.530568			0.0279246			0.0139623	−	0.999903i	\(-0.495556\pi\)
							0.0139623	+	0.999903i	\(0.495556\pi\)
\(23\)	10.8863	+	18.8557i	0.473319	+	0.819813i	0.999534	−	0.0305388i	\(-0.00972230\pi\)
							−0.526214	+	0.850352i	\(0.676389\pi\)
\(29\)	−21.0633	−	12.1609i	−0.726320	−	0.419341i	0.0907547	−	0.995873i	\(-0.471072\pi\)
							−0.817074	+	0.576532i	\(0.804405\pi\)
\(31\)	6.33163	+	10.9667i	0.204246	+	0.353765i	0.949892	−	0.312577i	\(-0.101192\pi\)
							−0.745646	+	0.666342i	\(0.767859\pi\)
\(37\)			14.5875i			0.394257i	0.980378	+	0.197129i	\(0.0631617\pi\)
							−0.980378	+	0.197129i	\(0.936838\pi\)
\(41\)	−33.1200	+	19.1218i	−0.807804	+	0.466386i	−0.846193	−	0.532877i	\(-0.821111\pi\)
							0.0383889	+	0.999263i	\(0.487777\pi\)
\(43\)	50.0269	+	28.8831i	1.16342	+	0.671699i	0.952121	−	0.305723i	\(-0.0988980\pi\)
							0.211297	+	0.977422i	\(0.432231\pi\)
\(47\)	24.7850	−	42.9289i	0.527341	−	0.913381i	−0.472152	−	0.881517i	\(-0.656522\pi\)
							0.999492	−	0.0318635i	\(-0.0101442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.h.a.29.9	yes	20
3.2	odd	2		135.3.h.a.89.2		20
5.2	odd	4		225.3.j.e.101.9		20
5.3	odd	4		225.3.j.e.101.2		20
5.4	even	2	inner	45.3.h.a.29.2	yes	20
9.2	odd	6		405.3.d.a.404.17		20
9.4	even	3		135.3.h.a.44.9		20
9.5	odd	6	inner	45.3.h.a.14.2	✓	20
9.7	even	3		405.3.d.a.404.4		20
15.2	even	4		675.3.j.e.251.2		20
15.8	even	4		675.3.j.e.251.9		20
15.14	odd	2		135.3.h.a.89.9		20
45.4	even	6		135.3.h.a.44.2		20
45.13	odd	12		675.3.j.e.476.9		20
45.14	odd	6	inner	45.3.h.a.14.9	yes	20
45.22	odd	12		675.3.j.e.476.2		20
45.23	even	12		225.3.j.e.176.2		20
45.29	odd	6		405.3.d.a.404.3		20
45.32	even	12		225.3.j.e.176.9		20
45.34	even	6		405.3.d.a.404.18		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.2	✓	20	9.5	odd	6	inner
45.3.h.a.14.9	yes	20	45.14	odd	6	inner
45.3.h.a.29.2	yes	20	5.4	even	2	inner
45.3.h.a.29.9	yes	20	1.1	even	1	trivial
135.3.h.a.44.2		20	45.4	even	6
135.3.h.a.44.9		20	9.4	even	3
135.3.h.a.89.2		20	3.2	odd	2
135.3.h.a.89.9		20	15.14	odd	2
225.3.j.e.101.2		20	5.3	odd	4
225.3.j.e.101.9		20	5.2	odd	4
225.3.j.e.176.2		20	45.23	even	12
225.3.j.e.176.9		20	45.32	even	12
405.3.d.a.404.3		20	45.29	odd	6
405.3.d.a.404.4		20	9.7	even	3
405.3.d.a.404.17		20	9.2	odd	6
405.3.d.a.404.18		20	45.34	even	6
675.3.j.e.251.2		20	15.2	even	4
675.3.j.e.251.9		20	15.8	even	4
675.3.j.e.476.2		20	45.22	odd	12
675.3.j.e.476.9		20	45.13	odd	12