Embedded newform 45.3.i.a.11.3

• Label: 45.3.i.a.11.3
• Level: $45$
• Weight: $3$
• Character: 45.11
• Analytic conductor: $1.226$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.22616118962\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.3
Root			\(-1.39204i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.11
Dual form			45.3.i.a.41.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20554 - 0.696021i) q^{2} +(2.33148 - 1.88791i) q^{3} +(-1.03111 - 1.78593i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(-4.12473 + 0.653207i) q^{6} +(4.41004 - 7.63842i) q^{7} +8.43886i q^{8} +(1.87156 - 8.80325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20554	−	0.696021i	−0.602772	−	0.348011i	0.167359	−	0.985896i	\(-0.446476\pi\)
							−0.770131	+	0.637885i	\(0.779809\pi\)
\(3\)	2.33148	−	1.88791i	0.777159	−	0.629305i
							\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
\(7\)	4.41004	−	7.63842i	0.630006	−	1.09120i								−0.357544	−	0.933896i	\(-0.616386\pi\)
							0.987550	−	0.157306i	\(-0.0502810\pi\)
\(11\)	−0.805373	−	0.464982i	−0.0732157	−	0.0422711i	0.462945	−	0.886387i	\(-0.346793\pi\)
							−0.536161	+	0.844116i	\(0.680126\pi\)
\(13\)	12.2405	+	21.2011i	0.941576	+	1.63086i	0.762467	+	0.647027i	\(0.223988\pi\)
							0.179109	+	0.983829i	\(0.442679\pi\)
\(17\)			18.3007i			1.07651i	0.842781	+	0.538256i	\(0.180917\pi\)
							−0.842781	+	0.538256i	\(0.819083\pi\)
\(19\)	−5.58727			−0.294067			−0.147033	−	0.989132i	\(-0.546972\pi\)
							−0.147033	+	0.989132i	\(0.546972\pi\)
\(23\)	20.6131	−	11.9010i	0.896221	−	0.517433i	0.0202485	−	0.999795i	\(-0.493554\pi\)
							0.875972	+	0.482362i	\(0.160221\pi\)
\(29\)	−23.7620	−	13.7190i	−0.819381	−	0.473070i	0.0308220	−	0.999525i	\(-0.490187\pi\)
							−0.850203	+	0.526455i	\(0.823521\pi\)
\(31\)	4.66760	+	8.08452i	0.150568	+	0.260791i	0.931436	−	0.363904i	\(-0.118556\pi\)
							−0.780869	+	0.624695i	\(0.785223\pi\)
\(37\)	−24.7588			−0.669156			−0.334578	−	0.942368i	\(-0.608594\pi\)
							−0.334578	+	0.942368i	\(0.608594\pi\)
\(41\)	−6.45555	+	3.72712i	−0.157453	+	0.0909053i	−0.576656	−	0.816987i	\(-0.695643\pi\)
							0.419203	+	0.907892i	\(0.362309\pi\)
\(43\)	−17.7288	+	30.7071i	−0.412297	+	0.714119i	−0.995140	−	0.0984653i	\(-0.968607\pi\)
							0.582844	+	0.812584i	\(0.301940\pi\)
\(47\)	−0.298523	−	0.172352i	−0.00635154	−	0.00366707i	0.496821	−	0.867853i	\(-0.334501\pi\)
							−0.503172	+	0.864186i	\(0.667834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.i.a.11.3	✓	16
3.2	odd	2		135.3.i.a.116.6		16
4.3	odd	2		720.3.bs.c.641.3		16
5.2	odd	4		225.3.i.b.74.11		32
5.3	odd	4		225.3.i.b.74.6		32
5.4	even	2		225.3.j.b.101.6		16
9.2	odd	6		405.3.c.a.161.6		16
9.4	even	3		135.3.i.a.71.6		16
9.5	odd	6	inner	45.3.i.a.41.3	yes	16
9.7	even	3		405.3.c.a.161.11		16
12.11	even	2		2160.3.bs.c.1601.5		16
15.2	even	4		675.3.i.c.224.6		32
15.8	even	4		675.3.i.c.224.11		32
15.14	odd	2		675.3.j.b.251.3		16
36.23	even	6		720.3.bs.c.401.3		16
36.31	odd	6		2160.3.bs.c.881.5		16
45.4	even	6		675.3.j.b.476.3		16
45.13	odd	12		675.3.i.c.449.6		32
45.14	odd	6		225.3.j.b.176.6		16
45.22	odd	12		675.3.i.c.449.11		32
45.23	even	12		225.3.i.b.149.11		32
45.32	even	12		225.3.i.b.149.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.i.a.11.3	✓	16	1.1	even	1	trivial
45.3.i.a.41.3	yes	16	9.5	odd	6	inner
135.3.i.a.71.6		16	9.4	even	3
135.3.i.a.116.6		16	3.2	odd	2
225.3.i.b.74.6		32	5.3	odd	4
225.3.i.b.74.11		32	5.2	odd	4
225.3.i.b.149.6		32	45.32	even	12
225.3.i.b.149.11		32	45.23	even	12
225.3.j.b.101.6		16	5.4	even	2
225.3.j.b.176.6		16	45.14	odd	6
405.3.c.a.161.6		16	9.2	odd	6
405.3.c.a.161.11		16	9.7	even	3
675.3.i.c.224.6		32	15.2	even	4
675.3.i.c.224.11		32	15.8	even	4
675.3.i.c.449.6		32	45.13	odd	12
675.3.i.c.449.11		32	45.22	odd	12
675.3.j.b.251.3		16	15.14	odd	2
675.3.j.b.476.3		16	45.4	even	6
720.3.bs.c.401.3		16	36.23	even	6
720.3.bs.c.641.3		16	4.3	odd	2
2160.3.bs.c.881.5		16	36.31	odd	6
2160.3.bs.c.1601.5		16	12.11	even	2