Newform orbit 180.3.u.a

• Label: 180.3.u.a
• Level: $180$
• Weight: $3$
• Character orbit: 180.u
• Analytic conductor: $4.905$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.90464475849\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1		0		−2.93271	+	0.631853i		0		−4.98610	+	0.372540i		0		5.98052	+	1.60248i		0		8.20152	−	3.70608i		0
13.2		0		−2.69602	−	1.31586i		0		4.02568	+	2.96545i		0		−4.41867	−	1.18398i		0		5.53700	+	7.09518i		0
13.3		0		−2.08827	−	2.15386i		0		−2.95142	−	4.03598i		0		−4.02302	−	1.07796i		0		−0.278258	+	8.99570i		0
13.4		0		−1.80382	+	2.39713i		0		1.85957	+	4.64134i		0		3.33405	+	0.893356i		0		−2.49245	−	8.64799i		0
13.5		0		−1.70695	+	2.46705i		0		2.79014	−	4.14911i		0		−10.5029	−	2.81425i		0		−3.17264	−	8.42226i		0
13.6		0		−0.412115	−	2.97156i		0		−3.05431	+	3.95868i		0		−2.46810	−	0.661325i		0		−8.66032	+	2.44925i		0
13.7		0		0.165444	−	2.99543i		0		4.61203	−	1.93111i		0		9.89538	+	2.65146i		0		−8.94526	−	0.991152i		0
13.8		0		0.594985	+	2.94041i		0		−1.82257	−	4.65599i		0		12.3966	+	3.32166i		0		−8.29199	+	3.49900i		0
13.9		0		1.55644	+	2.56466i		0		−4.88163	+	1.08150i		0		−10.5922	−	2.83818i		0		−4.15500	+	7.98348i		0
13.10		0		2.49269	−	1.66927i		0		−0.806137	−	4.93459i		0		−4.69726	−	1.25863i		0		3.42704	−	8.32198i		0
13.11		0		2.62769	+	1.44749i		0		4.94178	+	0.760767i		0		0.709316	+	0.190061i		0		4.80952	+	7.60713i		0
13.12		0		2.83660	−	0.976571i		0		−2.32510	+	4.42650i		0		4.38636	+	1.17532i		0		7.09262	−	5.54029i		0
97.1		0		−2.93271	−	0.631853i		0		−4.98610	−	0.372540i		0		5.98052	−	1.60248i		0		8.20152	+	3.70608i		0
97.2		0		−2.69602	+	1.31586i		0		4.02568	−	2.96545i		0		−4.41867	+	1.18398i		0		5.53700	−	7.09518i		0
97.3		0		−2.08827	+	2.15386i		0		−2.95142	+	4.03598i		0		−4.02302	+	1.07796i		0		−0.278258	−	8.99570i		0
97.4		0		−1.80382	−	2.39713i		0		1.85957	−	4.64134i		0		3.33405	−	0.893356i		0		−2.49245	+	8.64799i		0
97.5		0		−1.70695	−	2.46705i		0		2.79014	+	4.14911i		0		−10.5029	+	2.81425i		0		−3.17264	+	8.42226i		0
97.6		0		−0.412115	+	2.97156i		0		−3.05431	−	3.95868i		0		−2.46810	+	0.661325i		0		−8.66032	−	2.44925i		0
97.7		0		0.165444	+	2.99543i		0		4.61203	+	1.93111i		0		9.89538	−	2.65146i		0		−8.94526	+	0.991152i		0
97.8		0		0.594985	−	2.94041i		0		−1.82257	+	4.65599i		0		12.3966	−	3.32166i		0		−8.29199	−	3.49900i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
9.c	even	3	1	inner
45.k	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	180.3.u.a	✓	48
3.b	odd	2	1		540.3.v.a		48
5.c	odd	4	1	inner	180.3.u.a	✓	48
9.c	even	3	1	inner	180.3.u.a	✓	48
9.c	even	3	1		1620.3.l.f		24
9.d	odd	6	1		540.3.v.a		48
9.d	odd	6	1		1620.3.l.g		24
15.e	even	4	1		540.3.v.a		48
45.k	odd	12	1	inner	180.3.u.a	✓	48
45.k	odd	12	1		1620.3.l.f		24
45.l	even	12	1		540.3.v.a		48
45.l	even	12	1		1620.3.l.g		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.3.u.a	✓	48	1.a	even	1	1	trivial
180.3.u.a	✓	48	5.c	odd	4	1	inner
180.3.u.a	✓	48	9.c	even	3	1	inner
180.3.u.a	✓	48	45.k	odd	12	1	inner
540.3.v.a		48	3.b	odd	2	1
540.3.v.a		48	9.d	odd	6	1
540.3.v.a		48	15.e	even	4	1
540.3.v.a		48	45.l	even	12	1
1620.3.l.f		24	9.c	even	3	1
1620.3.l.f		24	45.k	odd	12	1
1620.3.l.g		24	9.d	odd	6	1
1620.3.l.g		24	45.l	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(180, [\chi])\).