Newform orbit 450.3.k.b

• Label: 450.3.k.b
• Level: $450$
• Weight: $3$
• Character orbit: 450.k
• Analytic conductor: $12.262$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2616118962\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
149.1	−0.707107	−	1.22474i	−2.22737	+	2.00968i	−1.00000	+	1.73205i		0		4.03634	+	1.30690i	10.0559	−	5.80576i	2.82843			0.922346	−	8.95261i		0
149.2	−0.707107	−	1.22474i	−0.151259	−	2.99618i	−1.00000	+	1.73205i		0		−3.56261	+	2.30388i	3.00916	−	1.73734i	2.82843			−8.95424	+	0.906401i		0
149.3	−0.707107	−	1.22474i	2.93751	−	0.609109i	−1.00000	+	1.73205i		0		−2.82314	−	3.16700i	−3.03056	+	1.74970i	2.82843			8.25797	−	3.57853i		0
149.4	−0.707107	−	1.22474i	1.96935	+	2.26311i	−1.00000	+	1.73205i		0		1.37919	−	4.01221i	−3.10604	+	1.79327i	2.82843			−1.24333	+	8.91370i		0
149.5	−0.707107	−	1.22474i	−0.337018	+	2.98101i	−1.00000	+	1.73205i		0		3.88928	−	1.69513i	1.91558	−	1.10596i	2.82843			−8.77284	−	2.00931i		0
149.6	−0.707107	−	1.22474i	−2.86392	−	0.893290i	−1.00000	+	1.73205i		0		0.931045	+	4.13922i	−4.62713	+	2.67148i	2.82843			7.40407	+	5.11662i		0
149.7	−0.707107	−	1.22474i	0.841701	−	2.87950i	−1.00000	+	1.73205i		0		−4.12183	+	1.00525i	−7.82064	+	4.51525i	2.82843			−7.58308	−	4.84736i		0
149.8	−0.707107	−	1.22474i	−2.99742	+	0.124284i	−1.00000	+	1.73205i		0		2.27172	+	3.58320i	−9.12416	+	5.26784i	2.82843			8.96911	−	0.745063i		0
149.9	0.707107	+	1.22474i	2.99742	−	0.124284i	−1.00000	+	1.73205i		0		2.27172	+	3.58320i	9.12416	−	5.26784i	−2.82843			8.96911	−	0.745063i		0
149.10	0.707107	+	1.22474i	−0.841701	+	2.87950i	−1.00000	+	1.73205i		0		−4.12183	+	1.00525i	7.82064	−	4.51525i	−2.82843			−7.58308	−	4.84736i		0
149.11	0.707107	+	1.22474i	2.86392	+	0.893290i	−1.00000	+	1.73205i		0		0.931045	+	4.13922i	4.62713	−	2.67148i	−2.82843			7.40407	+	5.11662i		0
149.12	0.707107	+	1.22474i	0.337018	−	2.98101i	−1.00000	+	1.73205i		0		3.88928	−	1.69513i	−1.91558	+	1.10596i	−2.82843			−8.77284	−	2.00931i		0
149.13	0.707107	+	1.22474i	−1.96935	−	2.26311i	−1.00000	+	1.73205i		0		1.37919	−	4.01221i	3.10604	−	1.79327i	−2.82843			−1.24333	+	8.91370i		0
149.14	0.707107	+	1.22474i	−2.93751	+	0.609109i	−1.00000	+	1.73205i		0		−2.82314	−	3.16700i	3.03056	−	1.74970i	−2.82843			8.25797	−	3.57853i		0
149.15	0.707107	+	1.22474i	0.151259	+	2.99618i	−1.00000	+	1.73205i		0		−3.56261	+	2.30388i	−3.00916	+	1.73734i	−2.82843			−8.95424	+	0.906401i		0
149.16	0.707107	+	1.22474i	2.22737	−	2.00968i	−1.00000	+	1.73205i		0		4.03634	+	1.30690i	−10.0559	+	5.80576i	−2.82843			0.922346	−	8.95261i		0
299.1	−0.707107	+	1.22474i	−2.22737	−	2.00968i	−1.00000	−	1.73205i		0		4.03634	−	1.30690i	10.0559	+	5.80576i	2.82843			0.922346	+	8.95261i		0
299.2	−0.707107	+	1.22474i	−0.151259	+	2.99618i	−1.00000	−	1.73205i		0		−3.56261	−	2.30388i	3.00916	+	1.73734i	2.82843			−8.95424	−	0.906401i		0
299.3	−0.707107	+	1.22474i	2.93751	+	0.609109i	−1.00000	−	1.73205i		0		−2.82314	+	3.16700i	−3.03056	−	1.74970i	2.82843			8.25797	+	3.57853i		0
299.4	−0.707107	+	1.22474i	1.96935	−	2.26311i	−1.00000	−	1.73205i		0		1.37919	+	4.01221i	−3.10604	−	1.79327i	2.82843			−1.24333	−	8.91370i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.d	odd	6	1	inner
45.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.3.k.b		32
3.b	odd	2	1		1350.3.k.c		32
5.b	even	2	1	inner	450.3.k.b		32
5.c	odd	4	1		450.3.i.d	✓	16
5.c	odd	4	1		450.3.i.f	yes	16
9.c	even	3	1		1350.3.k.c		32
9.d	odd	6	1	inner	450.3.k.b		32
15.d	odd	2	1		1350.3.k.c		32
15.e	even	4	1		1350.3.i.d		16
15.e	even	4	1		1350.3.i.f		16
45.h	odd	6	1	inner	450.3.k.b		32
45.j	even	6	1		1350.3.k.c		32
45.k	odd	12	1		1350.3.i.d		16
45.k	odd	12	1		1350.3.i.f		16
45.l	even	12	1		450.3.i.d	✓	16
45.l	even	12	1		450.3.i.f	yes	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.3.i.d	✓	16	5.c	odd	4	1
450.3.i.d	✓	16	45.l	even	12	1
450.3.i.f	yes	16	5.c	odd	4	1
450.3.i.f	yes	16	45.l	even	12	1
450.3.k.b		32	1.a	even	1	1	trivial
450.3.k.b		32	5.b	even	2	1	inner
450.3.k.b		32	9.d	odd	6	1	inner
450.3.k.b		32	45.h	odd	6	1	inner
1350.3.i.d		16	15.e	even	4	1
1350.3.i.d		16	45.k	odd	12	1
1350.3.i.f		16	15.e	even	4	1
1350.3.i.f		16	45.k	odd	12	1
1350.3.k.c		32	3.b	odd	2	1
1350.3.k.c		32	9.c	even	3	1
1350.3.k.c		32	15.d	odd	2	1
1350.3.k.c		32	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^32 - 398*T7^30 + 98427*T7^28 - 15232318*T7^26 + 1719933239*T7^24 - 136177416438*T7^22 + 8003428612594*T7^20 - 326919112670492*T7^18 + 9946556857507059*T7^16 - 220788357455091122*T7^14 + 3743578549008890422*T7^12 - 47687049745423851552*T7^10 + 464824951004217698966*T7^8 - 3301583963521607620900*T7^6 + 16989266965374731910000*T7^4 - 53635189250623671875000*T7^2 + 105097788427886962890625