Newform orbit 225.3.o.a

• Label: 225.3.o.a
• Level: $225$
• Weight: $3$
• Character orbit: 225.o
• Analytic conductor: $6.131$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13080594811\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−3.52187	+	0.943683i	−1.08886	+	2.79542i	8.04894	−	4.64706i		0		1.19684	−	10.8727i	9.18317	−	2.46062i	−13.6492	+	13.6492i	−6.62876	−	6.08766i		0
7.2	−2.57494	+	0.689954i	−1.60805	−	2.53262i	2.69019	−	1.55318i		0		5.88803	+	5.41188i	−7.87998	+	2.11143i	1.68449	−	1.68449i	−3.82835	+	8.14517i		0
7.3	−1.82579	+	0.489219i	2.88086	+	0.837030i	−0.369922	+	0.213575i		0		−5.66935	−	0.118868i	−1.32263	+	0.354398i	5.91720	−	5.91720i	7.59876	+	4.82274i		0
7.4	−0.157726	+	0.0422626i	−1.26453	+	2.72047i	−3.44101	+	1.98667i		0		0.0844760	−	0.482532i	−7.85836	+	2.10564i	0.920630	−	0.920630i	−5.80191	−	6.88025i		0
7.5	0.157726	−	0.0422626i	1.26453	−	2.72047i	−3.44101	+	1.98667i		0		0.0844760	−	0.482532i	7.85836	−	2.10564i	−0.920630	+	0.920630i	−5.80191	−	6.88025i		0
7.6	1.82579	−	0.489219i	−2.88086	−	0.837030i	−0.369922	+	0.213575i		0		−5.66935	−	0.118868i	1.32263	−	0.354398i	−5.91720	+	5.91720i	7.59876	+	4.82274i		0
7.7	2.57494	−	0.689954i	1.60805	+	2.53262i	2.69019	−	1.55318i		0		5.88803	+	5.41188i	7.87998	−	2.11143i	−1.68449	+	1.68449i	−3.82835	+	8.14517i		0
7.8	3.52187	−	0.943683i	1.08886	−	2.79542i	8.04894	−	4.64706i		0		1.19684	−	10.8727i	−9.18317	+	2.46062i	13.6492	−	13.6492i	−6.62876	−	6.08766i		0
43.1	−0.943683	−	3.52187i	2.79542	+	1.08886i	−8.04894	+	4.64706i		0		1.19684	−	10.8727i	2.46062	+	9.18317i	13.6492	+	13.6492i	6.62876	+	6.08766i		0
43.2	−0.689954	−	2.57494i	−2.53262	+	1.60805i	−2.69019	+	1.55318i		0		5.88803	+	5.41188i	−2.11143	−	7.87998i	−1.68449	−	1.68449i	3.82835	−	8.14517i		0
43.3	−0.489219	−	1.82579i	0.837030	−	2.88086i	0.369922	−	0.213575i		0		−5.66935	−	0.118868i	−0.354398	−	1.32263i	−5.91720	−	5.91720i	−7.59876	−	4.82274i		0
43.4	−0.0422626	−	0.157726i	2.72047	+	1.26453i	3.44101	−	1.98667i		0		0.0844760	−	0.482532i	−2.10564	−	7.85836i	−0.920630	−	0.920630i	5.80191	+	6.88025i		0
43.5	0.0422626	+	0.157726i	−2.72047	−	1.26453i	3.44101	−	1.98667i		0		0.0844760	−	0.482532i	2.10564	+	7.85836i	0.920630	+	0.920630i	5.80191	+	6.88025i		0
43.6	0.489219	+	1.82579i	−0.837030	+	2.88086i	0.369922	−	0.213575i		0		−5.66935	−	0.118868i	0.354398	+	1.32263i	5.91720	+	5.91720i	−7.59876	−	4.82274i		0
43.7	0.689954	+	2.57494i	2.53262	−	1.60805i	−2.69019	+	1.55318i		0		5.88803	+	5.41188i	2.11143	+	7.87998i	1.68449	+	1.68449i	3.82835	−	8.14517i		0
43.8	0.943683	+	3.52187i	−2.79542	−	1.08886i	−8.04894	+	4.64706i		0		1.19684	−	10.8727i	−2.46062	−	9.18317i	−13.6492	−	13.6492i	6.62876	+	6.08766i		0
157.1	−0.943683	+	3.52187i	2.79542	−	1.08886i	−8.04894	−	4.64706i		0		1.19684	+	10.8727i	2.46062	−	9.18317i	13.6492	−	13.6492i	6.62876	−	6.08766i		0
157.2	−0.689954	+	2.57494i	−2.53262	−	1.60805i	−2.69019	−	1.55318i		0		5.88803	−	5.41188i	−2.11143	+	7.87998i	−1.68449	+	1.68449i	3.82835	+	8.14517i		0
157.3	−0.489219	+	1.82579i	0.837030	+	2.88086i	0.369922	+	0.213575i		0		−5.66935	+	0.118868i	−0.354398	+	1.32263i	−5.91720	+	5.91720i	−7.59876	+	4.82274i		0
157.4	−0.0422626	+	0.157726i	2.72047	−	1.26453i	3.44101	+	1.98667i		0		0.0844760	+	0.482532i	−2.10564	+	7.85836i	−0.920630	+	0.920630i	5.80191	−	6.88025i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
9.c	even	3	1	inner
45.j	even	6	1	inner
45.k	odd	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.3.o.a	✓	32
5.b	even	2	1	inner	225.3.o.a	✓	32
5.c	odd	4	2	inner	225.3.o.a	✓	32
9.c	even	3	1	inner	225.3.o.a	✓	32
45.j	even	6	1	inner	225.3.o.a	✓	32
45.k	odd	12	2	inner	225.3.o.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.3.o.a	✓	32	1.a	even	1	1	trivial
225.3.o.a	✓	32	5.b	even	2	1	inner
225.3.o.a	✓	32	5.c	odd	4	2	inner
225.3.o.a	✓	32	9.c	even	3	1	inner
225.3.o.a	✓	32	45.j	even	6	1	inner
225.3.o.a	✓	32	45.k	odd	12	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 240*T2^28 + 45774*T2^24 - 2610360*T2^20 + 112508595*T2^16 - 1347415560*T2^12 + 12981365694*T2^8 - 9229140*T2^4 + 6561