Newform orbit 225.2.p.c

• Label: 225.2.p.c
• Level: $225$
• Weight: $2$
• Character orbit: 225.p
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−2.50229	−	0.670487i	1.72810	−	0.116950i	4.07987	+	2.35551i		0		−4.40262	−	0.866025i	0.588179	−	2.19511i	−4.96607	−	4.96607i	2.97265	−	0.404201i		0
32.2	−1.82596	−	0.489263i	−1.49488	+	0.874837i	1.36269	+	0.786747i		0		3.15761	−	0.866025i	0.792231	−	2.95665i	0.570108	+	0.570108i	1.46932	−	2.61555i		0
32.3	−1.16638	−	0.312529i	1.70849	+	0.284704i	−0.469294	−	0.270947i		0		−1.90376	−	0.866025i	−1.17061	+	4.36876i	2.17039	+	2.17039i	2.83789	+	0.972829i		0
32.4	−0.490038	−	0.131305i	0.158548	+	1.72478i	−1.50915	−	0.871311i		0		0.148778	−	0.866025i	−0.518267	+	1.93420i	1.34260	+	1.34260i	−2.94972	+	0.546922i		0
32.5	0.490038	+	0.131305i	−0.158548	−	1.72478i	−1.50915	−	0.871311i		0		0.148778	−	0.866025i	0.518267	−	1.93420i	−1.34260	−	1.34260i	−2.94972	+	0.546922i		0
32.6	1.16638	+	0.312529i	−1.70849	−	0.284704i	−0.469294	−	0.270947i		0		−1.90376	−	0.866025i	1.17061	−	4.36876i	−2.17039	−	2.17039i	2.83789	+	0.972829i		0
32.7	1.82596	+	0.489263i	1.49488	−	0.874837i	1.36269	+	0.786747i		0		3.15761	−	0.866025i	−0.792231	+	2.95665i	−0.570108	−	0.570108i	1.46932	−	2.61555i		0
32.8	2.50229	+	0.670487i	−1.72810	+	0.116950i	4.07987	+	2.35551i		0		−4.40262	−	0.866025i	−0.588179	+	2.19511i	4.96607	+	4.96607i	2.97265	−	0.404201i		0
68.1	−0.670487	+	2.50229i	0.116950	+	1.72810i	−4.07987	−	2.35551i		0		−4.40262	−	0.866025i	−2.19511	−	0.588179i	4.96607	−	4.96607i	−2.97265	+	0.404201i		0
68.2	−0.489263	+	1.82596i	−0.874837	−	1.49488i	−1.36269	−	0.786747i		0		3.15761	−	0.866025i	−2.95665	−	0.792231i	−0.570108	+	0.570108i	−1.46932	+	2.61555i		0
68.3	−0.312529	+	1.16638i	−0.284704	+	1.70849i	0.469294	+	0.270947i		0		−1.90376	−	0.866025i	4.36876	+	1.17061i	−2.17039	+	2.17039i	−2.83789	−	0.972829i		0
68.4	−0.131305	+	0.490038i	−1.72478	+	0.158548i	1.50915	+	0.871311i		0		0.148778	−	0.866025i	1.93420	+	0.518267i	−1.34260	+	1.34260i	2.94972	−	0.546922i		0
68.5	0.131305	−	0.490038i	1.72478	−	0.158548i	1.50915	+	0.871311i		0		0.148778	−	0.866025i	−1.93420	−	0.518267i	1.34260	−	1.34260i	2.94972	−	0.546922i		0
68.6	0.312529	−	1.16638i	0.284704	−	1.70849i	0.469294	+	0.270947i		0		−1.90376	−	0.866025i	−4.36876	−	1.17061i	2.17039	−	2.17039i	−2.83789	−	0.972829i		0
68.7	0.489263	−	1.82596i	0.874837	+	1.49488i	−1.36269	−	0.786747i		0		3.15761	−	0.866025i	2.95665	+	0.792231i	0.570108	−	0.570108i	−1.46932	+	2.61555i		0
68.8	0.670487	−	2.50229i	−0.116950	−	1.72810i	−4.07987	−	2.35551i		0		−4.40262	−	0.866025i	2.19511	+	0.588179i	−4.96607	+	4.96607i	−2.97265	+	0.404201i		0
182.1	−0.670487	−	2.50229i	0.116950	−	1.72810i	−4.07987	+	2.35551i		0		−4.40262	+	0.866025i	−2.19511	+	0.588179i	4.96607	+	4.96607i	−2.97265	−	0.404201i		0
182.2	−0.489263	−	1.82596i	−0.874837	+	1.49488i	−1.36269	+	0.786747i		0		3.15761	+	0.866025i	−2.95665	+	0.792231i	−0.570108	−	0.570108i	−1.46932	−	2.61555i		0
182.3	−0.312529	−	1.16638i	−0.284704	−	1.70849i	0.469294	−	0.270947i		0		−1.90376	+	0.866025i	4.36876	−	1.17061i	−2.17039	−	2.17039i	−2.83789	+	0.972829i		0
182.4	−0.131305	−	0.490038i	−1.72478	−	0.158548i	1.50915	−	0.871311i		0		0.148778	+	0.866025i	1.93420	−	0.518267i	−1.34260	−	1.34260i	2.94972	+	0.546922i		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.d	odd	6	1	inner
45.h	odd	6	1	inner
45.l	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.2.p.c	✓	32
3.b	odd	2	1		675.2.q.c		32
5.b	even	2	1	inner	225.2.p.c	✓	32
5.c	odd	4	2	inner	225.2.p.c	✓	32
9.c	even	3	1		675.2.q.c		32
9.d	odd	6	1	inner	225.2.p.c	✓	32
15.d	odd	2	1		675.2.q.c		32
15.e	even	4	2		675.2.q.c		32
45.h	odd	6	1	inner	225.2.p.c	✓	32
45.j	even	6	1		675.2.q.c		32
45.k	odd	12	2		675.2.q.c		32
45.l	even	12	2	inner	225.2.p.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.2.p.c	✓	32	1.a	even	1	1	trivial
225.2.p.c	✓	32	5.b	even	2	1	inner
225.2.p.c	✓	32	5.c	odd	4	2	inner
225.2.p.c	✓	32	9.d	odd	6	1	inner
225.2.p.c	✓	32	45.h	odd	6	1	inner
225.2.p.c	✓	32	45.l	even	12	2	inner
675.2.q.c		32	3.b	odd	2	1
675.2.q.c		32	9.c	even	3	1
675.2.q.c		32	15.d	odd	2	1
675.2.q.c		32	15.e	even	4	2
675.2.q.c		32	45.j	even	6	1
675.2.q.c		32	45.k	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 60*T2^28 + 2898*T2^24 - 39582*T2^20 + 416583*T2^16 - 881118*T2^12 + 1553499*T2^8 - 102789*T2^4 + 6561