Newform orbit 600.3.g.e

• Label: 600.3.g.e
• Level: $600$
• Weight: $3$
• Character orbit: 600.g
• Analytic conductor: $16.349$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3488158616\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} - 6 q^{6} + 72 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
451.1	−1.99893	−	0.0655209i	−1.73205			3.99141	+	0.261943i		0		3.46224	+	0.113485i			6.07322i	−7.96138	−	0.785125i	3.00000				0
451.2	−1.99893	+	0.0655209i	−1.73205			3.99141	−	0.261943i		0		3.46224	−	0.113485i		−	6.07322i	−7.96138	+	0.785125i	3.00000				0
451.3	−1.94804	−	0.452908i	1.73205			3.58975	+	1.76457i		0		−3.37411	−	0.784460i		−	9.26960i	−6.19380	−	5.06329i	3.00000				0
451.4	−1.94804	+	0.452908i	1.73205			3.58975	−	1.76457i		0		−3.37411	+	0.784460i			9.26960i	−6.19380	+	5.06329i	3.00000				0
451.5	−1.46803	−	1.35826i	1.73205			0.310252	+	3.98795i		0		−2.54271	−	2.35258i			5.41487i	4.96122	−	6.27585i	3.00000				0
451.6	−1.46803	+	1.35826i	1.73205			0.310252	−	3.98795i		0		−2.54271	+	2.35258i		−	5.41487i	4.96122	+	6.27585i	3.00000				0
451.7	−1.03181	−	1.71329i	−1.73205			−1.87075	+	3.53557i		0		1.78714	+	2.96751i		−	13.5060i	7.98773	−	0.442871i	3.00000				0
451.8	−1.03181	+	1.71329i	−1.73205			−1.87075	−	3.53557i		0		1.78714	−	2.96751i			13.5060i	7.98773	+	0.442871i	3.00000				0
451.9	−0.798974	−	1.83348i	1.73205			−2.72328	+	2.92980i		0		−1.38386	−	3.17568i			0.116466i	7.54756	+	2.65225i	3.00000				0
451.10	−0.798974	+	1.83348i	1.73205			−2.72328	−	2.92980i		0		−1.38386	+	3.17568i		−	0.116466i	7.54756	−	2.65225i	3.00000				0
451.11	−0.318295	−	1.97451i	−1.73205			−3.79738	+	1.25695i		0		0.551302	+	3.41995i			1.20123i	3.69055	+	7.09788i	3.00000				0
451.12	−0.318295	+	1.97451i	−1.73205			−3.79738	−	1.25695i		0		0.551302	−	3.41995i		−	1.20123i	3.69055	−	7.09788i	3.00000				0
451.13	0.318295	−	1.97451i	1.73205			−3.79738	−	1.25695i		0		0.551302	−	3.41995i			1.20123i	−3.69055	+	7.09788i	3.00000				0
451.14	0.318295	+	1.97451i	1.73205			−3.79738	+	1.25695i		0		0.551302	+	3.41995i		−	1.20123i	−3.69055	−	7.09788i	3.00000				0
451.15	0.798974	−	1.83348i	−1.73205			−2.72328	−	2.92980i		0		−1.38386	+	3.17568i			0.116466i	−7.54756	+	2.65225i	3.00000				0
451.16	0.798974	+	1.83348i	−1.73205			−2.72328	+	2.92980i		0		−1.38386	−	3.17568i		−	0.116466i	−7.54756	−	2.65225i	3.00000				0
451.17	1.03181	−	1.71329i	1.73205			−1.87075	−	3.53557i		0		1.78714	−	2.96751i		−	13.5060i	−7.98773	−	0.442871i	3.00000				0
451.18	1.03181	+	1.71329i	1.73205			−1.87075	+	3.53557i		0		1.78714	+	2.96751i			13.5060i	−7.98773	+	0.442871i	3.00000				0
451.19	1.46803	−	1.35826i	−1.73205			0.310252	−	3.98795i		0		−2.54271	+	2.35258i			5.41487i	−4.96122	−	6.27585i	3.00000				0
451.20	1.46803	+	1.35826i	−1.73205			0.310252	+	3.98795i		0		−2.54271	−	2.35258i		−	5.41487i	−4.96122	+	6.27585i	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.g.e		24
4.b	odd	2	1		2400.3.g.e		24
5.b	even	2	1	inner	600.3.g.e		24
5.c	odd	4	2		120.3.p.a	✓	24
8.b	even	2	1		2400.3.g.e		24
8.d	odd	2	1	inner	600.3.g.e		24
15.e	even	4	2		360.3.p.i		24
20.d	odd	2	1		2400.3.g.e		24
20.e	even	4	2		480.3.p.a		24
40.e	odd	2	1	inner	600.3.g.e		24
40.f	even	2	1		2400.3.g.e		24
40.i	odd	4	2		480.3.p.a		24
40.k	even	4	2		120.3.p.a	✓	24
60.l	odd	4	2		1440.3.p.i		24
120.q	odd	4	2		360.3.p.i		24
120.w	even	4	2		1440.3.p.i		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.3.p.a	✓	24	5.c	odd	4	2
120.3.p.a	✓	24	40.k	even	4	2
360.3.p.i		24	15.e	even	4	2
360.3.p.i		24	120.q	odd	4	2
480.3.p.a		24	20.e	even	4	2
480.3.p.a		24	40.i	odd	4	2
600.3.g.e		24	1.a	even	1	1	trivial
600.3.g.e		24	5.b	even	2	1	inner
600.3.g.e		24	8.d	odd	2	1	inner
600.3.g.e		24	40.e	odd	2	1	inner
1440.3.p.i		24	60.l	odd	4	2
1440.3.p.i		24	120.w	even	4	2
2400.3.g.e		24	4.b	odd	2	1
2400.3.g.e		24	8.b	even	2	1
2400.3.g.e		24	20.d	odd	2	1
2400.3.g.e		24	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{12} + 336T_{7}^{10} + 35008T_{7}^{8} + 1378176T_{7}^{6} + 18885632T_{7}^{4} + 24715264T_{7}^{2} + 331776 \)

T17^12 - 2248*T17^10 + 1828976*T17^8 - 640282496*T17^6 + 89728131328*T17^4 - 4880188563456*T17^2 + 71070565072896