Embedded newform 45.4.j.a.4.10

• Label: 45.4.j.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	45.4
Dual form			45.4.j.a.34.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.680118 - 0.392666i) q^{2} +(-3.65242 + 3.69592i) q^{3} +(-3.69163 + 6.39408i) q^{4} +(-7.35516 - 8.42031i) q^{5} +(-1.03281 + 3.94785i) q^{6} +(-18.1117 + 10.4568i) q^{7} +12.0810i q^{8} +(-0.319654 - 26.9981i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 54 q^{4} + 3 q^{5} - 12 q^{6} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.680118	−	0.392666i	0.240458	−	0.138829i	−0.374929	−	0.927053i	\(-0.622333\pi\)
							0.615387	+	0.788225i	\(0.289000\pi\)
\(3\)	−3.65242	+	3.69592i	−0.702909	+	0.711280i
							\(5\)	−7.35516	−	8.42031i	−0.657865	−	0.753136i
\(7\)	−18.1117	+	10.4568i	−0.977940	+	0.564614i								−0.901648	−	0.432472i	\(-0.857642\pi\)
							−0.0762923	+	0.997085i	\(0.524308\pi\)
\(11\)	29.6919	+	51.4278i	0.813858	+	1.40964i	0.910145	+	0.414290i	\(0.135970\pi\)
							−0.0962871	+	0.995354i	\(0.530697\pi\)
\(13\)	39.7783	+	22.9660i	0.848655	+	0.489971i	0.860197	−	0.509962i	\(-0.170341\pi\)
							−0.0115421	+	0.999933i	\(0.503674\pi\)
\(17\)		−	43.0872i		−	0.614717i	−0.951594	−	0.307359i	\(-0.900555\pi\)
							0.951594	−	0.307359i	\(-0.0994451\pi\)
\(19\)	−140.178			−1.69258			−0.846292	−	0.532719i	\(-0.821170\pi\)
							−0.846292	+	0.532719i	\(0.821170\pi\)
\(23\)	88.1230	+	50.8778i	0.798909	+	0.461251i	0.843090	−	0.537773i	\(-0.180734\pi\)
							−0.0441803	+	0.999024i	\(0.514068\pi\)
\(29\)	6.19592	+	10.7317i	0.0396743	+	0.0687179i	0.885181	−	0.465247i	\(-0.154035\pi\)
							−0.845506	+	0.533965i	\(0.820701\pi\)
\(31\)	−35.9895	+	62.3357i	−0.208513	+	0.361155i	−0.951246	−	0.308432i	\(-0.900196\pi\)
							0.742733	+	0.669587i	\(0.233529\pi\)
\(37\)			150.734i			0.669742i	0.942264	+	0.334871i	\(0.108693\pi\)
							−0.942264	+	0.334871i	\(0.891307\pi\)
\(41\)	−25.5410	+	44.2384i	−0.0972887	+	0.168509i	−0.910562	−	0.413374i	\(-0.864350\pi\)
							0.813273	+	0.581883i	\(0.197684\pi\)
\(43\)	−29.8533	+	17.2358i	−0.105874	+	0.0611264i	−0.552002	−	0.833843i	\(-0.686136\pi\)
							0.446128	+	0.894969i	\(0.352803\pi\)
\(47\)	−102.170	+	58.9878i	−0.317085	+	0.183069i	−0.650093	−	0.759855i	\(-0.725270\pi\)
							0.333007	+	0.942924i	\(0.391937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.j.a.4.10	yes	32
3.2	odd	2		135.4.j.a.64.7		32
5.2	odd	4		225.4.e.g.76.10		32
5.3	odd	4		225.4.e.g.76.7		32
5.4	even	2	inner	45.4.j.a.4.7	✓	32
9.2	odd	6		135.4.j.a.19.10		32
9.4	even	3		405.4.b.e.244.10		16
9.5	odd	6		405.4.b.f.244.7		16
9.7	even	3	inner	45.4.j.a.34.7	yes	32
15.14	odd	2		135.4.j.a.64.10		32
45.4	even	6		405.4.b.e.244.7		16
45.7	odd	12		225.4.e.g.151.10		32
45.13	odd	12		2025.4.a.bk.1.10		16
45.14	odd	6		405.4.b.f.244.10		16
45.22	odd	12		2025.4.a.bk.1.7		16
45.23	even	12		2025.4.a.bl.1.7		16
45.29	odd	6		135.4.j.a.19.7		32
45.32	even	12		2025.4.a.bl.1.10		16
45.34	even	6	inner	45.4.j.a.34.10	yes	32
45.43	odd	12		225.4.e.g.151.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.7	✓	32	5.4	even	2	inner
45.4.j.a.4.10	yes	32	1.1	even	1	trivial
45.4.j.a.34.7	yes	32	9.7	even	3	inner
45.4.j.a.34.10	yes	32	45.34	even	6	inner
135.4.j.a.19.7		32	45.29	odd	6
135.4.j.a.19.10		32	9.2	odd	6
135.4.j.a.64.7		32	3.2	odd	2
135.4.j.a.64.10		32	15.14	odd	2
225.4.e.g.76.7		32	5.3	odd	4
225.4.e.g.76.10		32	5.2	odd	4
225.4.e.g.151.7		32	45.43	odd	12
225.4.e.g.151.10		32	45.7	odd	12
405.4.b.e.244.7		16	45.4	even	6
405.4.b.e.244.10		16	9.4	even	3
405.4.b.f.244.7		16	9.5	odd	6
405.4.b.f.244.10		16	45.14	odd	6
2025.4.a.bk.1.7		16	45.22	odd	12
2025.4.a.bk.1.10		16	45.13	odd	12
2025.4.a.bl.1.7		16	45.23	even	12
2025.4.a.bl.1.10		16	45.32	even	12