Embedded newform 45.4.j.a.4.12

• Label: 45.4.j.a.4.12
• Level: $45$
• Weight: $4$
• Character: 45.4
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.65508595026\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.31667 - 1.33753i) q^{2} +(5.14958 + 0.694136i) q^{3} +(-0.422017 + 0.730954i) q^{4} +(3.65904 - 10.5646i) q^{5} +(12.8583 - 5.27964i) q^{6} +(-13.3501 + 7.70766i) q^{7} +23.6584i q^{8} +(26.0363 + 7.14902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.31667	−	1.33753i	0.819068	−	0.472889i	−0.0310272	−	0.999519i	\(-0.509878\pi\)
							0.850095	+	0.526630i	\(0.176545\pi\)
\(3\)	5.14958	+	0.694136i	0.991037	+	0.133587i
							\(5\)	3.65904	−	10.5646i	0.327274	−	0.944929i
\(7\)	−13.3501	+	7.70766i	−0.720835	+	0.416175i								−0.815060	−	0.579376i	\(-0.803296\pi\)
							0.0942246	+	0.995551i	\(0.469963\pi\)
\(11\)	−22.2799	−	38.5899i	−0.610694	−	1.05775i	−0.991124	−	0.132944i	\(-0.957557\pi\)
							0.380429	−	0.924810i	\(-0.375776\pi\)
\(13\)	−24.2689	−	14.0117i	−0.517768	−	0.298933i	0.218253	−	0.975892i	\(-0.429964\pi\)
							−0.736021	+	0.676959i	\(0.763298\pi\)
\(17\)			92.6615i			1.32198i	0.750394	+	0.660991i	\(0.229864\pi\)
							−0.750394	+	0.660991i	\(0.770136\pi\)
\(19\)	−49.5811			−0.598667			−0.299334	−	0.954149i	\(-0.596764\pi\)
							−0.299334	+	0.954149i	\(0.596764\pi\)
\(23\)	−0.799176	−	0.461404i	−0.00724520	−	0.00418302i	0.496373	−	0.868109i	\(-0.334665\pi\)
							−0.503618	+	0.863926i	\(0.667998\pi\)
\(29\)	94.9247	+	164.414i	0.607830	+	1.05279i	0.991597	+	0.129363i	\(0.0412933\pi\)
							−0.383767	+	0.923430i	\(0.625373\pi\)
\(31\)	149.590	−	259.098i	0.866683	−	1.50114i	0.00131648	−	0.999999i	\(-0.499581\pi\)
							0.865366	−	0.501140i	\(-0.167086\pi\)
\(37\)		−	57.8330i		−	0.256965i	−0.991712	−	0.128482i	\(-0.958989\pi\)
							0.991712	−	0.128482i	\(-0.0410105\pi\)
\(41\)	143.940	−	249.311i	0.548284	−	0.949656i	−0.450108	−	0.892974i	\(-0.648615\pi\)
							0.998392	−	0.0566817i	\(-0.0180520\pi\)
\(43\)	−0.512647	+	0.295977i	−0.00181809	+	0.00104967i	−0.500909	−	0.865500i	\(-0.667001\pi\)
							0.499091	+	0.866550i	\(0.333667\pi\)
\(47\)	518.145	−	299.151i	1.60807	−	0.928418i	0.618266	−	0.785969i	\(-0.287836\pi\)
							0.989802	−	0.142449i	\(-0.0454978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.12	yes	32
3.2	odd	2		135.4.j.a.64.5		32
5.2	odd	4		225.4.e.g.76.12		32
5.3	odd	4		225.4.e.g.76.5		32
5.4	even	2	inner	45.4.j.a.4.5	✓	32
9.2	odd	6		135.4.j.a.19.12		32
9.4	even	3		405.4.b.e.244.12		16
9.5	odd	6		405.4.b.f.244.5		16
9.7	even	3	inner	45.4.j.a.34.5	yes	32
15.14	odd	2		135.4.j.a.64.12		32
45.4	even	6		405.4.b.e.244.5		16
45.7	odd	12		225.4.e.g.151.12		32
45.13	odd	12		2025.4.a.bk.1.12		16
45.14	odd	6		405.4.b.f.244.12		16
45.22	odd	12		2025.4.a.bk.1.5		16
45.23	even	12		2025.4.a.bl.1.5		16
45.29	odd	6		135.4.j.a.19.5		32
45.32	even	12		2025.4.a.bl.1.12		16
45.34	even	6	inner	45.4.j.a.34.12	yes	32
45.43	odd	12		225.4.e.g.151.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.5	✓	32	5.4	even	2	inner
45.4.j.a.4.12	yes	32	1.1	even	1	trivial
45.4.j.a.34.5	yes	32	9.7	even	3	inner
45.4.j.a.34.12	yes	32	45.34	even	6	inner
135.4.j.a.19.5		32	45.29	odd	6
135.4.j.a.19.12		32	9.2	odd	6
135.4.j.a.64.5		32	3.2	odd	2
135.4.j.a.64.12		32	15.14	odd	2
225.4.e.g.76.5		32	5.3	odd	4
225.4.e.g.76.12		32	5.2	odd	4
225.4.e.g.151.5		32	45.43	odd	12
225.4.e.g.151.12		32	45.7	odd	12
405.4.b.e.244.5		16	45.4	even	6
405.4.b.e.244.12		16	9.4	even	3
405.4.b.f.244.5		16	9.5	odd	6
405.4.b.f.244.12		16	45.14	odd	6
2025.4.a.bk.1.5		16	45.22	odd	12
2025.4.a.bk.1.12		16	45.13	odd	12
2025.4.a.bl.1.5		16	45.23	even	12
2025.4.a.bl.1.12		16	45.32	even	12