Newform orbit 600.2.w.k

• Label: 600.2.w.k
• Level: $600$
• Weight: $2$
• Character orbit: 600.w
• Analytic conductor: $4.791$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.79102412128\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
293.1	−1.37896	+	0.313810i	−1.69016	−	0.378611i	1.80305	−	0.865461i		0		2.44948	−	0.00830235i	0.699464	−	0.699464i	−2.21473	+	1.75925i	2.71331	+	1.27983i		0
293.2	−1.37896	+	0.313810i	−0.378611	−	1.69016i	1.80305	−	0.865461i		0		1.05248	+	2.21185i	−0.699464	+	0.699464i	−2.21473	+	1.75925i	−2.71331	+	1.27983i		0
293.3	−1.30189	−	0.552337i	−1.65680	+	0.504984i	1.38985	+	1.43817i		0		2.43590	+	0.257678i	−2.83719	+	2.83719i	−1.01508	−	2.64000i	2.48998	−	1.67332i		0
293.4	−1.30189	−	0.552337i	0.504984	−	1.65680i	1.38985	+	1.43817i		0		−1.57255	+	1.87806i	2.83719	−	2.83719i	−1.01508	−	2.64000i	−2.48998	−	1.67332i		0
293.5	−1.21574	−	0.722477i	0.889875	+	1.48597i	0.956054	+	1.75669i		0		−0.00827553	−	2.44948i	2.09015	−	2.09015i	0.106854	−	2.82641i	−1.41624	+	2.64467i		0
293.6	−1.21574	−	0.722477i	1.48597	+	0.889875i	0.956054	+	1.75669i		0		−1.16365	−	2.15544i	−2.09015	+	2.09015i	0.106854	−	2.82641i	1.41624	+	2.64467i		0
293.7	−1.18496	+	0.771921i	−0.713099	+	1.57845i	0.808275	−	1.82940i		0		−0.373439	−	2.42086i	1.44651	−	1.44651i	0.454373	+	2.79169i	−1.98298	−	2.25118i		0
293.8	−1.18496	+	0.771921i	1.57845	−	0.713099i	0.808275	−	1.82940i		0		−1.31994	+	2.06343i	−1.44651	+	1.44651i	0.454373	+	2.79169i	1.98298	−	2.25118i		0
293.9	−0.771921	+	1.18496i	−0.713099	+	1.57845i	−0.808275	−	1.82940i		0		−1.31994	−	2.06343i	−1.44651	+	1.44651i	2.79169	+	0.454373i	−1.98298	−	2.25118i		0
293.10	−0.771921	+	1.18496i	1.57845	−	0.713099i	−0.808275	−	1.82940i		0		−0.373439	+	2.42086i	1.44651	−	1.44651i	2.79169	+	0.454373i	1.98298	−	2.25118i		0
293.11	−0.722477	−	1.21574i	−1.48597	−	0.889875i	−0.956054	+	1.75669i		0		−0.00827553	+	2.44948i	−2.09015	+	2.09015i	2.82641	−	0.106854i	1.41624	+	2.64467i		0
293.12	−0.722477	−	1.21574i	−0.889875	−	1.48597i	−0.956054	+	1.75669i		0		−1.16365	+	2.15544i	2.09015	−	2.09015i	2.82641	−	0.106854i	−1.41624	+	2.64467i		0
293.13	−0.552337	−	1.30189i	−0.504984	+	1.65680i	−1.38985	+	1.43817i		0		2.43590	−	0.257678i	2.83719	−	2.83719i	2.64000	+	1.01508i	−2.48998	−	1.67332i		0
293.14	−0.552337	−	1.30189i	1.65680	−	0.504984i	−1.38985	+	1.43817i		0		−1.57255	−	1.87806i	−2.83719	+	2.83719i	2.64000	+	1.01508i	2.48998	−	1.67332i		0
293.15	−0.313810	+	1.37896i	−1.69016	−	0.378611i	−1.80305	−	0.865461i		0		1.05248	−	2.21185i	−0.699464	+	0.699464i	1.75925	−	2.21473i	2.71331	+	1.27983i		0
293.16	−0.313810	+	1.37896i	−0.378611	−	1.69016i	−1.80305	−	0.865461i		0		2.44948	+	0.00830235i	0.699464	−	0.699464i	1.75925	−	2.21473i	−2.71331	+	1.27983i		0
293.17	0.313810	−	1.37896i	0.378611	+	1.69016i	−1.80305	−	0.865461i		0		2.44948	+	0.00830235i	−0.699464	+	0.699464i	−1.75925	+	2.21473i	−2.71331	+	1.27983i		0
293.18	0.313810	−	1.37896i	1.69016	+	0.378611i	−1.80305	−	0.865461i		0		1.05248	−	2.21185i	0.699464	−	0.699464i	−1.75925	+	2.21473i	2.71331	+	1.27983i		0
293.19	0.552337	+	1.30189i	−1.65680	+	0.504984i	−1.38985	+	1.43817i		0		−1.57255	−	1.87806i	2.83719	−	2.83719i	−2.64000	−	1.01508i	2.48998	−	1.67332i		0
293.20	0.552337	+	1.30189i	0.504984	−	1.65680i	−1.38985	+	1.43817i		0		2.43590	−	0.257678i	−2.83719	+	2.83719i	−2.64000	−	1.01508i	−2.48998	−	1.67332i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
8.b	even	2	1	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner
24.h	odd	2	1	inner
40.f	even	2	1	inner
40.i	odd	4	2	inner
120.i	odd	2	1	inner
120.w	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.2.w.k	✓	64
3.b	odd	2	1	inner	600.2.w.k	✓	64
5.b	even	2	1	inner	600.2.w.k	✓	64
5.c	odd	4	2	inner	600.2.w.k	✓	64
8.b	even	2	1	inner	600.2.w.k	✓	64
15.d	odd	2	1	inner	600.2.w.k	✓	64
15.e	even	4	2	inner	600.2.w.k	✓	64
24.h	odd	2	1	inner	600.2.w.k	✓	64
40.f	even	2	1	inner	600.2.w.k	✓	64
40.i	odd	4	2	inner	600.2.w.k	✓	64
120.i	odd	2	1	inner	600.2.w.k	✓	64
120.w	even	4	2	inner	600.2.w.k	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.2.w.k	✓	64	1.a	even	1	1	trivial
600.2.w.k	✓	64	3.b	odd	2	1	inner
600.2.w.k	✓	64	5.b	even	2	1	inner
600.2.w.k	✓	64	5.c	odd	4	2	inner
600.2.w.k	✓	64	8.b	even	2	1	inner
600.2.w.k	✓	64	15.d	odd	2	1	inner
600.2.w.k	✓	64	15.e	even	4	2	inner
600.2.w.k	✓	64	24.h	odd	2	1	inner
600.2.w.k	✓	64	40.f	even	2	1	inner
600.2.w.k	✓	64	40.i	odd	4	2	inner
600.2.w.k	✓	64	120.i	odd	2	1	inner
600.2.w.k	✓	64	120.w	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{16} + 354T_{7}^{12} + 26001T_{7}^{8} + 371088T_{7}^{4} + 331776 \)

\( T_{11}^{8} - 48T_{11}^{6} + 711T_{11}^{4} - 3144T_{11}^{2} + 1176 \)

\( T_{17}^{16} + 2386T_{17}^{12} + 1377153T_{17}^{8} + 46443376T_{17}^{4} + 1600 \)