Newform orbit 225.4.e.g

• Label: 225.4.e.g
• Level: $225$
• Weight: $4$
• Character orbit: 225.e
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 54 q^{4} - 12 q^{6} + 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
76.1	−2.52718	−	4.37720i	0.559681	−	5.16592i	−8.77324	+	15.1957i		0		−24.0267	+	10.6054i	10.5059	+	18.1967i	48.2513			−26.3735	−	5.78253i		0
76.2	−2.51186	−	4.35066i	−4.79730	+	1.99648i	−8.61884	+	14.9283i		0		20.7361	+	15.8565i	−2.69099	−	4.66092i	46.4074			19.0281	−	19.1554i		0
76.3	−2.03813	−	3.53014i	4.38850	+	2.78227i	−4.30793	+	7.46156i		0		0.877481	−	21.1627i	6.67148	+	11.5553i	2.51043			11.5179	+	24.4200i		0
76.4	−1.54370	−	2.67377i	4.77188	−	2.05649i	−0.766022	+	1.32679i		0		−12.8649	−	9.58430i	−15.6602	−	27.1243i	−19.9692			18.5417	−	19.6266i		0
76.5	−1.33753	−	2.31667i	−0.694136	+	5.14958i	0.422017	−	0.730954i		0		12.8583	−	5.27964i	7.70766	+	13.3501i	−23.6584			−26.0363	−	7.14902i		0
76.6	−1.23192	−	2.13376i	−4.97325	−	1.50557i	0.964722	−	1.67095i		0		2.91415	+	12.4665i	−9.62007	−	16.6624i	−24.4647			22.4665	+	14.9752i		0
76.7	−0.392666	−	0.680118i	−3.69592	−	3.65242i	3.69163	−	6.39408i		0		−1.03281	+	3.94785i	10.4568	+	18.1117i	−12.0810			0.319654	+	26.9981i		0
76.8	−0.236995	−	0.410487i	2.45316	−	4.58061i	3.88767	−	6.73364i		0		−2.46167	+	0.0785933i	4.10329	+	7.10710i	−7.47735			−14.9641	−	22.4739i		0
76.9	0.236995	+	0.410487i	−2.45316	+	4.58061i	3.88767	−	6.73364i		0		−2.46167	+	0.0785933i	−4.10329	−	7.10710i	7.47735			−14.9641	−	22.4739i		0
76.10	0.392666	+	0.680118i	3.69592	+	3.65242i	3.69163	−	6.39408i		0		−1.03281	+	3.94785i	−10.4568	−	18.1117i	12.0810			0.319654	+	26.9981i		0
76.11	1.23192	+	2.13376i	4.97325	+	1.50557i	0.964722	−	1.67095i		0		2.91415	+	12.4665i	9.62007	+	16.6624i	24.4647			22.4665	+	14.9752i		0
76.12	1.33753	+	2.31667i	0.694136	−	5.14958i	0.422017	−	0.730954i		0		12.8583	−	5.27964i	−7.70766	−	13.3501i	23.6584			−26.0363	−	7.14902i		0
76.13	1.54370	+	2.67377i	−4.77188	+	2.05649i	−0.766022	+	1.32679i		0		−12.8649	−	9.58430i	15.6602	+	27.1243i	19.9692			18.5417	−	19.6266i		0
76.14	2.03813	+	3.53014i	−4.38850	−	2.78227i	−4.30793	+	7.46156i		0		0.877481	−	21.1627i	−6.67148	−	11.5553i	−2.51043			11.5179	+	24.4200i		0
76.15	2.51186	+	4.35066i	4.79730	−	1.99648i	−8.61884	+	14.9283i		0		20.7361	+	15.8565i	2.69099	+	4.66092i	−46.4074			19.0281	−	19.1554i		0
76.16	2.52718	+	4.37720i	−0.559681	+	5.16592i	−8.77324	+	15.1957i		0		−24.0267	+	10.6054i	−10.5059	−	18.1967i	−48.2513			−26.3735	−	5.78253i		0
151.1	−2.52718	+	4.37720i	0.559681	+	5.16592i	−8.77324	−	15.1957i		0		−24.0267	−	10.6054i	10.5059	−	18.1967i	48.2513			−26.3735	+	5.78253i		0
151.2	−2.51186	+	4.35066i	−4.79730	−	1.99648i	−8.61884	−	14.9283i		0		20.7361	−	15.8565i	−2.69099	+	4.66092i	46.4074			19.0281	+	19.1554i		0
151.3	−2.03813	+	3.53014i	4.38850	−	2.78227i	−4.30793	−	7.46156i		0		0.877481	+	21.1627i	6.67148	−	11.5553i	2.51043			11.5179	−	24.4200i		0
151.4	−1.54370	+	2.67377i	4.77188	+	2.05649i	−0.766022	−	1.32679i		0		−12.8649	+	9.58430i	−15.6602	+	27.1243i	−19.9692			18.5417	+	19.6266i		0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.4.e.g		32
5.b	even	2	1	inner	225.4.e.g		32
5.c	odd	4	2		45.4.j.a	✓	32
9.c	even	3	1	inner	225.4.e.g		32
9.c	even	3	1		2025.4.a.bk		16
9.d	odd	6	1		2025.4.a.bl		16
15.e	even	4	2		135.4.j.a		32
45.h	odd	6	1		2025.4.a.bl		16
45.j	even	6	1	inner	225.4.e.g		32
45.j	even	6	1		2025.4.a.bk		16
45.k	odd	12	2		45.4.j.a	✓	32
45.k	odd	12	2		405.4.b.e		16
45.l	even	12	2		135.4.j.a		32
45.l	even	12	2		405.4.b.f		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.4.j.a	✓	32	5.c	odd	4	2
45.4.j.a	✓	32	45.k	odd	12	2
135.4.j.a		32	15.e	even	4	2
135.4.j.a		32	45.l	even	12	2
225.4.e.g		32	1.a	even	1	1	trivial
225.4.e.g		32	5.b	even	2	1	inner
225.4.e.g		32	9.c	even	3	1	inner
225.4.e.g		32	45.j	even	6	1	inner
405.4.b.e		16	45.k	odd	12	2
405.4.b.f		16	45.l	even	12	2
2025.4.a.bk		16	9.c	even	3	1
2025.4.a.bk		16	45.j	even	6	1
2025.4.a.bl		16	9.d	odd	6	1
2025.4.a.bl		16	45.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 + 91*T2^30 + 5013*T2^28 + 179132*T2^26 + 4727201*T2^24 + 92011995*T2^22 + 1377832645*T2^20 + 15589192564*T2^18 + 136116950121*T2^16 + 886104438067*T2^14 + 4314960988885*T2^12 + 14267478237804*T2^10 + 30894211908752*T2^8 + 23134575137792*T2^6 + 12531672253440*T2^4 + 2501345050624*T2^2 + 377801998336