Embedded newform 45.3.h.a.14.2

• Label: 45.3.h.a.14.2
• Level: $45$
• Weight: $3$
• Character: 45.14
• 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			14.2
Root			\(-0.346576 - 1.69702i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.14
Dual form			45.3.h.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42081 - 2.46092i) q^{2} +(2.93933 - 0.600288i) q^{3} +(-2.03740 + 3.52889i) q^{4} +(3.58357 - 3.48684i) q^{5} +(-5.65349 - 6.38055i) q^{6} +(-8.42581 + 4.86464i) q^{7} +0.212574 q^{8} +(8.27931 - 3.52889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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	−0.964705	−	0.263332i	\(-0.915178\pi\)
							0.254300	−	0.967125i	\(-0.418155\pi\)
\(3\)	2.93933	−	0.600288i	0.979776	−	0.200096i
							\(5\)	3.58357	−	3.48684i	0.716714	−	0.697367i
\(7\)	−8.42581	+	4.86464i	−1.20369	+	0.694949i								−0.961373	−	0.275249i	\(-0.911240\pi\)
							−0.242314	+	0.970198i	\(0.577906\pi\)
\(11\)	0.370966	−	0.214177i	0.0337242	−	0.0194707i	−0.483043	−	0.875597i	\(-0.660469\pi\)
							0.516767	+	0.856126i	\(0.327135\pi\)
\(13\)	16.2396	+	9.37595i	1.24920	+	0.721227i	0.970950	−	0.239281i	\(-0.0769117\pi\)
							0.278252	+	0.960508i	\(0.410245\pi\)
\(17\)	−2.85718			−0.168069			−0.0840347	−	0.996463i	\(-0.526781\pi\)
							−0.0840347	+	0.996463i	\(0.526781\pi\)
\(19\)	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	\(-0.990278\pi\)
							0.526214	+	0.850352i	\(0.323611\pi\)
\(29\)	−21.0633	+	12.1609i	−0.726320	+	0.419341i	−0.817074	−	0.576532i	\(-0.804405\pi\)
							0.0907547	+	0.995873i	\(0.471072\pi\)
\(31\)	6.33163	−	10.9667i	0.204246	−	0.353765i	−0.745646	−	0.666342i	\(-0.767859\pi\)
							0.949892	+	0.312577i	\(0.101192\pi\)
\(37\)			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	0.0383889	−	0.999263i	\(-0.487777\pi\)
							−0.846193	+	0.532877i	\(0.821111\pi\)
\(43\)	−50.0269	+	28.8831i	−1.16342	+	0.671699i	−0.952121	−	0.305723i	\(-0.901102\pi\)
							−0.211297	+	0.977422i	\(0.567769\pi\)
\(47\)	−24.7850	−	42.9289i	−0.527341	−	0.913381i	−0.999492	−	0.0318635i	\(-0.989856\pi\)
							0.472152	−	0.881517i	\(-0.343478\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.2	✓	20	1.1	even	1	trivial
45.3.h.a.14.9	yes	20	5.4	even	2	inner
45.3.h.a.29.2	yes	20	45.29	odd	6	inner
45.3.h.a.29.9	yes	20	9.2	odd	6	inner
135.3.h.a.44.2		20	15.14	odd	2
135.3.h.a.44.9		20	3.2	odd	2
135.3.h.a.89.2		20	9.7	even	3
135.3.h.a.89.9		20	45.34	even	6
225.3.j.e.101.2		20	45.38	even	12
225.3.j.e.101.9		20	45.2	even	12
225.3.j.e.176.2		20	5.3	odd	4
225.3.j.e.176.9		20	5.2	odd	4
405.3.d.a.404.3		20	45.4	even	6
405.3.d.a.404.4		20	9.5	odd	6
405.3.d.a.404.17		20	9.4	even	3
405.3.d.a.404.18		20	45.14	odd	6
675.3.j.e.251.2		20	45.7	odd	12
675.3.j.e.251.9		20	45.43	odd	12
675.3.j.e.476.2		20	15.2	even	4
675.3.j.e.476.9		20	15.8	even	4