Newform orbit 600.3.p.b

• Label: 600.3.p.b
• Level: $600$
• Weight: $3$
• Character orbit: 600.p
• Analytic conductor: $16.349$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3488158616\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 32 * q - 28 * q^4 + 12 * q^6 - 96 * q^9 + 128 * q^11 + 40 * q^14 - 28 * q^16 + 64 * q^19 + 108 * q^24 + 72 * q^26 + 144 * q^34 + 84 * q^36 + 200 * q^44 + 424 * q^46 + 160 * q^49 + 192 * q^51 - 36 * q^54 - 232 * q^56 - 256 * q^59 - 28 * q^64 - 360 * q^66 - 600 * q^74 + 840 * q^76 + 288 * q^81 - 360 * q^84 + 624 * q^86 - 384 * q^91 - 920 * q^94 - 372 * q^96 - 384 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
499.1	−1.94989	−	0.444906i			1.73205i	3.60412	+	1.73503i		0		0.770599	−	3.37730i	5.48395			−6.25570	−	4.98661i	−3.00000				0
499.2	−1.94989	+	0.444906i		−	1.73205i	3.60412	−	1.73503i		0		0.770599	+	3.37730i	5.48395			−6.25570	+	4.98661i	−3.00000				0
499.3	−1.68225	−	1.08168i			1.73205i	1.65995	+	3.63931i		0		1.87352	−	2.91375i	−3.06445			1.14409	−	7.91777i	−3.00000				0
499.4	−1.68225	+	1.08168i		−	1.73205i	1.65995	−	3.63931i		0		1.87352	+	2.91375i	−3.06445			1.14409	+	7.91777i	−3.00000				0
499.5	−1.64881	−	1.13200i		−	1.73205i	1.43715	+	3.73291i		0		−1.96068	+	2.85582i	−11.6935			1.85607	−	7.78171i	−3.00000				0
499.6	−1.64881	+	1.13200i			1.73205i	1.43715	−	3.73291i		0		−1.96068	−	2.85582i	−11.6935			1.85607	+	7.78171i	−3.00000				0
499.7	−1.24999	−	1.56126i			1.73205i	−0.875058	+	3.90311i		0		2.70418	−	2.16504i	7.58970			7.18758	−	3.51265i	−3.00000				0
499.8	−1.24999	+	1.56126i		−	1.73205i	−0.875058	−	3.90311i		0		2.70418	+	2.16504i	7.58970			7.18758	+	3.51265i	−3.00000				0
499.9	−0.840753	−	1.81470i			1.73205i	−2.58627	+	3.05143i		0		3.14315	−	1.45623i	−6.88771			7.71184	+	2.12781i	−3.00000				0
499.10	−0.840753	+	1.81470i		−	1.73205i	−2.58627	−	3.05143i		0		3.14315	+	1.45623i	−6.88771			7.71184	−	2.12781i	−3.00000				0
499.11	−0.714481	−	1.86802i			1.73205i	−2.97903	+	2.66934i		0		3.23551	−	1.23752i	0.274624			7.11485	+	3.65772i	−3.00000				0
499.12	−0.714481	+	1.86802i		−	1.73205i	−2.97903	−	2.66934i		0		3.23551	+	1.23752i	0.274624			7.11485	−	3.65772i	−3.00000				0
499.13	−0.436996	−	1.95167i		−	1.73205i	−3.61807	+	1.70575i		0		−3.38040	+	0.756900i	8.35441			4.91015	+	6.31589i	−3.00000				0
499.14	−0.436996	+	1.95167i			1.73205i	−3.61807	−	1.70575i		0		−3.38040	−	0.756900i	8.35441			4.91015	−	6.31589i	−3.00000				0
499.15	−0.422617	−	1.95484i		−	1.73205i	−3.64279	+	1.65230i		0		−3.38588	+	0.731994i	−8.99346			4.76948	+	6.42278i	−3.00000				0
499.16	−0.422617	+	1.95484i			1.73205i	−3.64279	−	1.65230i		0		−3.38588	−	0.731994i	−8.99346			4.76948	−	6.42278i	−3.00000				0
499.17	0.422617	−	1.95484i		−	1.73205i	−3.64279	−	1.65230i		0		−3.38588	−	0.731994i	8.99346			−4.76948	+	6.42278i	−3.00000				0
499.18	0.422617	+	1.95484i			1.73205i	−3.64279	+	1.65230i		0		−3.38588	+	0.731994i	8.99346			−4.76948	−	6.42278i	−3.00000				0
499.19	0.436996	−	1.95167i		−	1.73205i	−3.61807	−	1.70575i		0		−3.38040	−	0.756900i	−8.35441			−4.91015	+	6.31589i	−3.00000				0
499.20	0.436996	+	1.95167i			1.73205i	−3.61807	+	1.70575i		0		−3.38040	+	0.756900i	−8.35441			−4.91015	−	6.31589i	−3.00000				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.d	odd	2	1	inner
40.e	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.3.p.b		32
4.b	odd	2	1		2400.3.p.b		32
5.b	even	2	1	inner	600.3.p.b		32
5.c	odd	4	1		120.3.g.a	✓	16
5.c	odd	4	1		600.3.g.d		16
8.b	even	2	1		2400.3.p.b		32
8.d	odd	2	1	inner	600.3.p.b		32
15.e	even	4	1		360.3.g.c		16
20.d	odd	2	1		2400.3.p.b		32
20.e	even	4	1		480.3.g.a		16
20.e	even	4	1		2400.3.g.b		16
40.e	odd	2	1	inner	600.3.p.b		32
40.f	even	2	1		2400.3.p.b		32
40.i	odd	4	1		480.3.g.a		16
40.i	odd	4	1		2400.3.g.b		16
40.k	even	4	1		120.3.g.a	✓	16
40.k	even	4	1		600.3.g.d		16
60.l	odd	4	1		1440.3.g.c		16
120.q	odd	4	1		360.3.g.c		16
120.w	even	4	1		1440.3.g.c		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.3.g.a	✓	16	5.c	odd	4	1
120.3.g.a	✓	16	40.k	even	4	1
360.3.g.c		16	15.e	even	4	1
360.3.g.c		16	120.q	odd	4	1
480.3.g.a		16	20.e	even	4	1
480.3.g.a		16	40.i	odd	4	1
600.3.g.d		16	5.c	odd	4	1
600.3.g.d		16	40.k	even	4	1
600.3.p.b		32	1.a	even	1	1	trivial
600.3.p.b		32	5.b	even	2	1	inner
600.3.p.b		32	8.d	odd	2	1	inner
600.3.p.b		32	40.e	odd	2	1	inner
1440.3.g.c		16	60.l	odd	4	1
1440.3.g.c		16	120.w	even	4	1
2400.3.g.b		16	20.e	even	4	1
2400.3.g.b		16	40.i	odd	4	1
2400.3.p.b		32	4.b	odd	2	1
2400.3.p.b		32	8.b	even	2	1
2400.3.p.b		32	20.d	odd	2	1
2400.3.p.b		32	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^16 - 432*T7^14 + 74976*T7^12 - 6766336*T7^10 + 340312320*T7^8 - 9384443904*T7^6 + 127112740864*T7^4 - 605282107392*T7^2 + 44930433024