Embedded newform 45.3.i.a.41.2

• Label: 45.3.i.a.41.2
• Level: $45$
• Weight: $3$
• Character: 45.41
• 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			41.2
Root			\(3.27064i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.41
Dual form			45.3.i.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.83245 + 1.63532i) q^{2} +(-2.26486 + 1.96733i) q^{3} +(3.34853 - 5.79983i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(3.19790 - 9.27615i) q^{6} +(-3.16931 - 5.48940i) q^{7} +8.82112i q^{8} +(1.25919 - 8.91148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.83245	+	1.63532i	−1.41623	+	0.817659i	−0.995965	−	0.0897435i	\(-0.971395\pi\)
							−0.420262	+	0.907403i	\(0.638062\pi\)
\(3\)	−2.26486	+	1.96733i	−0.754954	+	0.655778i
							\(5\)	−1.93649	−	1.11803i	−0.387298	−	0.223607i
\(7\)	−3.16931	−	5.48940i	−0.452758	−	0.784200i								0.545798	−	0.837917i	\(-0.316227\pi\)
							−0.998556	+	0.0537168i	\(0.982893\pi\)
\(11\)	−12.8821	+	7.43751i	−1.17110	+	0.676137i	−0.953940	−	0.299998i	\(-0.903014\pi\)
							−0.217164	+	0.976135i	\(0.569681\pi\)
\(13\)	−7.73683	+	13.4006i	−0.595141	+	1.03081i	0.398386	+	0.917218i	\(0.369570\pi\)
							−0.993527	+	0.113596i	\(0.963763\pi\)
\(17\)		−	20.0994i		−	1.18232i	−0.806555	−	0.591159i	\(-0.798671\pi\)
							0.806555	−	0.591159i	\(-0.201329\pi\)
\(19\)	−25.1235			−1.32229			−0.661144	−	0.750259i	\(-0.729929\pi\)
							−0.661144	+	0.750259i	\(0.729929\pi\)
\(23\)	1.40763	+	0.812693i	0.0612011	+	0.0353345i	0.530288	−	0.847817i	\(-0.322084\pi\)
							−0.469087	+	0.883152i	\(0.655417\pi\)
\(29\)	−1.07318	+	0.619600i	−0.0370062	+	0.0213655i	−0.518389	−	0.855145i	\(-0.673468\pi\)
							0.481383	+	0.876510i	\(0.340135\pi\)
\(31\)	−6.69310	+	11.5928i	−0.215907	+	0.373961i	−0.953553	−	0.301226i	\(-0.902604\pi\)
							0.737646	+	0.675188i	\(0.235937\pi\)
\(37\)	3.89313			0.105220			0.0526099	−	0.998615i	\(-0.483246\pi\)
							0.0526099	+	0.998615i	\(0.483246\pi\)
\(41\)	50.3757	+	29.0844i	1.22868	+	0.709376i	0.966753	−	0.255713i	\(-0.0823101\pi\)
							0.261923	+	0.965089i	\(0.415643\pi\)
\(43\)	13.6159	+	23.5834i	0.316648	+	0.548451i	0.979787	−	0.200046i	\(-0.0641091\pi\)
							−0.663138	+	0.748497i	\(0.730776\pi\)
\(47\)	−54.2876	+	31.3429i	−1.15505	+	0.666871i	−0.950114	−	0.311903i	\(-0.899034\pi\)
							−0.204941	+	0.978774i	\(0.565700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.i.a.41.2	yes	16
3.2	odd	2		135.3.i.a.71.7		16
4.3	odd	2		720.3.bs.c.401.7		16
5.2	odd	4		225.3.i.b.149.2		32
5.3	odd	4		225.3.i.b.149.15		32
5.4	even	2		225.3.j.b.176.7		16
9.2	odd	6	inner	45.3.i.a.11.2	✓	16
9.4	even	3		405.3.c.a.161.2		16
9.5	odd	6		405.3.c.a.161.15		16
9.7	even	3		135.3.i.a.116.7		16
12.11	even	2		2160.3.bs.c.881.7		16
15.2	even	4		675.3.i.c.449.15		32
15.8	even	4		675.3.i.c.449.2		32
15.14	odd	2		675.3.j.b.476.2		16
36.7	odd	6		2160.3.bs.c.1601.7		16
36.11	even	6		720.3.bs.c.641.7		16
45.2	even	12		225.3.i.b.74.15		32
45.7	odd	12		675.3.i.c.224.2		32
45.29	odd	6		225.3.j.b.101.7		16
45.34	even	6		675.3.j.b.251.2		16
45.38	even	12		225.3.i.b.74.2		32
45.43	odd	12		675.3.i.c.224.15		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.i.a.11.2	✓	16	9.2	odd	6	inner
45.3.i.a.41.2	yes	16	1.1	even	1	trivial
135.3.i.a.71.7		16	3.2	odd	2
135.3.i.a.116.7		16	9.7	even	3
225.3.i.b.74.2		32	45.38	even	12
225.3.i.b.74.15		32	45.2	even	12
225.3.i.b.149.2		32	5.2	odd	4
225.3.i.b.149.15		32	5.3	odd	4
225.3.j.b.101.7		16	45.29	odd	6
225.3.j.b.176.7		16	5.4	even	2
405.3.c.a.161.2		16	9.4	even	3
405.3.c.a.161.15		16	9.5	odd	6
675.3.i.c.224.2		32	45.7	odd	12
675.3.i.c.224.15		32	45.43	odd	12
675.3.i.c.449.2		32	15.8	even	4
675.3.i.c.449.15		32	15.2	even	4
675.3.j.b.251.2		16	45.34	even	6
675.3.j.b.476.2		16	15.14	odd	2
720.3.bs.c.401.7		16	4.3	odd	2
720.3.bs.c.641.7		16	36.11	even	6
2160.3.bs.c.881.7		16	12.11	even	2
2160.3.bs.c.1601.7		16	36.7	odd	6