Newform orbit 225.3.i.c

• Label: 225.3.i.c
• Level: $225$
• Weight: $3$
• Character orbit: 225.i
• Analytic conductor: $6.131$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13080594811\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 2 q^{6} + 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
74.1	−1.91334	+	3.31400i	2.84135	+	0.962657i	−5.32172	−	9.21749i		0		−8.62671	+	7.57435i	5.17511	+	2.98785i	25.4223			7.14658	+	5.47050i		0
74.2	−1.68092	+	2.91143i	−2.98432	+	0.306336i	−3.65096	−	6.32364i		0		4.12451	−	9.20356i	11.9069	+	6.87444i	11.1005			8.81232	−	1.82841i		0
74.3	−1.46810	+	2.54282i	1.83839	−	2.37072i	−2.31062	−	4.00210i		0		3.32938	+	8.15514i	−5.65462	−	3.26469i	1.82406			−2.24064	−	8.71662i		0
74.4	−1.22636	+	2.12412i	−0.862751	−	2.87327i	−1.00793	−	1.74578i		0		7.16121	+	1.69108i	−2.19292	−	1.26608i	−4.86656			−7.51132	+	4.95783i		0
74.5	−1.10877	+	1.92045i	−1.99458	+	2.24090i	−0.458755	−	0.794587i		0		−2.09201	−	6.31515i	2.03278	+	1.17362i	−6.83556			−1.04331	−	8.93932i		0
74.6	−0.578744	+	1.00241i	−1.50856	+	2.59312i	1.33011	+	2.30382i		0		−1.72631	−	3.01295i	−9.26150	−	5.34713i	−7.70913			−4.44852	−	7.82373i		0
74.7	−0.538293	+	0.932351i	2.76965	+	1.15284i	1.42048	+	2.46035i		0		−2.56574	+	1.96172i	8.25202	+	4.76431i	−7.36488			6.34191	+	6.38593i		0
74.8	−0.0175770	+	0.0304442i	1.86319	+	2.35128i	1.99938	+	3.46303i		0		−0.104332	+	0.0153951i	−0.638204	−	0.368467i	−0.281188			−2.05703	+	8.76177i		0
74.9	0.0175770	−	0.0304442i	−1.86319	−	2.35128i	1.99938	+	3.46303i		0		−0.104332	+	0.0153951i	0.638204	+	0.368467i	0.281188			−2.05703	+	8.76177i		0
74.10	0.538293	−	0.932351i	−2.76965	−	1.15284i	1.42048	+	2.46035i		0		−2.56574	+	1.96172i	−8.25202	−	4.76431i	7.36488			6.34191	+	6.38593i		0
74.11	0.578744	−	1.00241i	1.50856	−	2.59312i	1.33011	+	2.30382i		0		−1.72631	−	3.01295i	9.26150	+	5.34713i	7.70913			−4.44852	−	7.82373i		0
74.12	1.10877	−	1.92045i	1.99458	−	2.24090i	−0.458755	−	0.794587i		0		−2.09201	−	6.31515i	−2.03278	−	1.17362i	6.83556			−1.04331	−	8.93932i		0
74.13	1.22636	−	2.12412i	0.862751	+	2.87327i	−1.00793	−	1.74578i		0		7.16121	+	1.69108i	2.19292	+	1.26608i	4.86656			−7.51132	+	4.95783i		0
74.14	1.46810	−	2.54282i	−1.83839	+	2.37072i	−2.31062	−	4.00210i		0		3.32938	+	8.15514i	5.65462	+	3.26469i	−1.82406			−2.24064	−	8.71662i		0
74.15	1.68092	−	2.91143i	2.98432	−	0.306336i	−3.65096	−	6.32364i		0		4.12451	−	9.20356i	−11.9069	−	6.87444i	−11.1005			8.81232	−	1.82841i		0
74.16	1.91334	−	3.31400i	−2.84135	−	0.962657i	−5.32172	−	9.21749i		0		−8.62671	+	7.57435i	−5.17511	−	2.98785i	−25.4223			7.14658	+	5.47050i		0
149.1	−1.91334	−	3.31400i	2.84135	−	0.962657i	−5.32172	+	9.21749i		0		−8.62671	−	7.57435i	5.17511	−	2.98785i	25.4223			7.14658	−	5.47050i		0
149.2	−1.68092	−	2.91143i	−2.98432	−	0.306336i	−3.65096	+	6.32364i		0		4.12451	+	9.20356i	11.9069	−	6.87444i	11.1005			8.81232	+	1.82841i		0
149.3	−1.46810	−	2.54282i	1.83839	+	2.37072i	−2.31062	+	4.00210i		0		3.32938	−	8.15514i	−5.65462	+	3.26469i	1.82406			−2.24064	+	8.71662i		0
149.4	−1.22636	−	2.12412i	−0.862751	+	2.87327i	−1.00793	+	1.74578i		0		7.16121	−	1.69108i	−2.19292	+	1.26608i	−4.86656			−7.51132	−	4.95783i		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	225.3.i.c		32
3.b	odd	2	1		675.3.i.b		32
5.b	even	2	1	inner	225.3.i.c		32
5.c	odd	4	1		225.3.j.c	✓	16
5.c	odd	4	1		225.3.j.d	yes	16
9.c	even	3	1		675.3.i.b		32
9.d	odd	6	1	inner	225.3.i.c		32
15.d	odd	2	1		675.3.i.b		32
15.e	even	4	1		675.3.j.c		16
15.e	even	4	1		675.3.j.d		16
45.h	odd	6	1	inner	225.3.i.c		32
45.j	even	6	1		675.3.i.b		32
45.k	odd	12	1		675.3.j.c		16
45.k	odd	12	1		675.3.j.d		16
45.l	even	12	1		225.3.j.c	✓	16
45.l	even	12	1		225.3.j.d	yes	16

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 + 48*T2^30 + 1392*T2^28 + 26242*T2^26 + 365814*T2^24 + 3784986*T2^22 + 30231321*T2^20 + 183013893*T2^18 + 851083677*T2^16 + 2902652860*T2^14 + 7261920075*T2^12 + 11860104795*T2^10 + 13903486240*T2^8 + 9689946336*T2^6 + 4308072885*T2^4 + 5323887*T2^2 + 6561