Newform orbit 588.2.n.e

• Label: 588.2.n.e
• Level: $588$
• Weight: $2$
• Character orbit: 588.n
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 84)
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
263.1	−1.32175	+	0.502962i	1.72058	+	0.199011i	1.49406	−	1.32958i	−1.79791	+	1.03802i	−2.37428	+	0.602344i		0		−1.30604	+	2.50883i	2.92079	+	0.684828i	1.85431	−	2.27629i
263.2	−1.29776	−	0.561968i	−0.373317	+	1.69134i	1.36838	+	1.45860i	−0.432549	+	0.249732i	1.43496	−	1.98517i		0		−0.956154	−	2.66191i	−2.72127	−	1.26281i	0.701688	−	0.0810151i
263.3	−1.09645	+	0.893190i	−1.72058	−	0.199011i	0.404424	−	1.95868i	−1.79791	+	1.03802i	2.06429	−	1.31860i		0		1.30604	+	2.50883i	2.92079	+	0.684828i	1.04417	−	2.74402i
263.4	−0.951145	−	1.04658i	1.66539	−	0.475894i	−0.190646	+	1.99089i	2.36229	−	1.36387i	−2.08209	−	1.29031i		0		2.26495	−	1.69410i	2.54705	−	1.58510i	−3.67427	−	1.17508i
263.5	−0.430790	−	1.34700i	−0.420559	−	1.68022i	−1.62884	+	1.16055i	−2.36229	+	1.36387i	−2.08209	+	1.29031i		0		2.26495	+	1.69410i	−2.64626	+	1.41326i	2.85479	+	2.59447i
263.6	−0.162204	+	1.40488i	0.373317	−	1.69134i	−1.94738	−	0.455755i	−0.432549	+	0.249732i	2.31558	+	0.798808i		0		0.956154	−	2.66191i	−2.72127	−	1.26281i	−0.280683	−	0.648187i
263.7	0.162204	−	1.40488i	−1.27809	+	1.16897i	−1.94738	−	0.455755i	0.432549	−	0.249732i	1.43496	+	1.98517i		0		−0.956154	+	2.66191i	0.267006	−	2.98809i	−0.280683	−	0.648187i
263.8	0.430790	+	1.34700i	−1.66539	+	0.475894i	−1.62884	+	1.16055i	2.36229	−	1.36387i	−1.35846	−	2.03828i		0		−2.26495	−	1.69410i	2.54705	−	1.58510i	2.85479	+	2.59447i
263.9	0.951145	+	1.04658i	0.420559	+	1.68022i	−0.190646	+	1.99089i	−2.36229	+	1.36387i	−1.35846	+	2.03828i		0		−2.26495	+	1.69410i	−2.64626	+	1.41326i	−3.67427	−	1.17508i
263.10	1.09645	−	0.893190i	−1.03264	−	1.39056i	0.404424	−	1.95868i	1.79791	−	1.03802i	−2.37428	−	0.602344i		0		−1.30604	−	2.50883i	−0.867316	+	2.87189i	1.04417	−	2.74402i
263.11	1.29776	+	0.561968i	1.27809	−	1.16897i	1.36838	+	1.45860i	0.432549	−	0.249732i	2.31558	−	0.798808i		0		0.956154	+	2.66191i	0.267006	−	2.98809i	0.701688	−	0.0810151i
263.12	1.32175	−	0.502962i	1.03264	+	1.39056i	1.49406	−	1.32958i	1.79791	−	1.03802i	2.06429	+	1.31860i		0		1.30604	−	2.50883i	−0.867316	+	2.87189i	1.85431	−	2.27629i
275.1	−1.32175	−	0.502962i	1.72058	−	0.199011i	1.49406	+	1.32958i	−1.79791	−	1.03802i	−2.37428	−	0.602344i		0		−1.30604	−	2.50883i	2.92079	−	0.684828i	1.85431	+	2.27629i
275.2	−1.29776	+	0.561968i	−0.373317	−	1.69134i	1.36838	−	1.45860i	−0.432549	−	0.249732i	1.43496	+	1.98517i		0		−0.956154	+	2.66191i	−2.72127	+	1.26281i	0.701688	+	0.0810151i
275.3	−1.09645	−	0.893190i	−1.72058	+	0.199011i	0.404424	+	1.95868i	−1.79791	−	1.03802i	2.06429	+	1.31860i		0		1.30604	−	2.50883i	2.92079	−	0.684828i	1.04417	+	2.74402i
275.4	−0.951145	+	1.04658i	1.66539	+	0.475894i	−0.190646	−	1.99089i	2.36229	+	1.36387i	−2.08209	+	1.29031i		0		2.26495	+	1.69410i	2.54705	+	1.58510i	−3.67427	+	1.17508i
275.5	−0.430790	+	1.34700i	−0.420559	+	1.68022i	−1.62884	−	1.16055i	−2.36229	−	1.36387i	−2.08209	−	1.29031i		0		2.26495	−	1.69410i	−2.64626	−	1.41326i	2.85479	−	2.59447i
275.6	−0.162204	−	1.40488i	0.373317	+	1.69134i	−1.94738	+	0.455755i	−0.432549	−	0.249732i	2.31558	−	0.798808i		0		0.956154	+	2.66191i	−2.72127	+	1.26281i	−0.280683	+	0.648187i
275.7	0.162204	+	1.40488i	−1.27809	−	1.16897i	−1.94738	+	0.455755i	0.432549	+	0.249732i	1.43496	−	1.98517i		0		−0.956154	−	2.66191i	0.267006	+	2.98809i	−0.280683	+	0.648187i
275.8	0.430790	−	1.34700i	−1.66539	−	0.475894i	−1.62884	−	1.16055i	2.36229	+	1.36387i	−1.35846	+	2.03828i		0		−2.26495	+	1.69410i	2.54705	+	1.58510i	2.85479	−	2.59447i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.c	even	3	1	inner
12.b	even	2	1	inner
21.h	odd	6	1	inner
28.g	odd	6	1	inner
84.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.2.n.e		24
3.b	odd	2	1	inner	588.2.n.e		24
4.b	odd	2	1	inner	588.2.n.e		24
7.b	odd	2	1		84.2.n.a	✓	24
7.c	even	3	1		588.2.e.d		12
7.c	even	3	1	inner	588.2.n.e		24
7.d	odd	6	1		84.2.n.a	✓	24
7.d	odd	6	1		588.2.e.e		12
12.b	even	2	1	inner	588.2.n.e		24
21.c	even	2	1		84.2.n.a	✓	24
21.g	even	6	1		84.2.n.a	✓	24
21.g	even	6	1		588.2.e.e		12
21.h	odd	6	1		588.2.e.d		12
21.h	odd	6	1	inner	588.2.n.e		24
28.d	even	2	1		84.2.n.a	✓	24
28.f	even	6	1		84.2.n.a	✓	24
28.f	even	6	1		588.2.e.e		12
28.g	odd	6	1		588.2.e.d		12
28.g	odd	6	1	inner	588.2.n.e		24
84.h	odd	2	1		84.2.n.a	✓	24
84.j	odd	6	1		84.2.n.a	✓	24
84.j	odd	6	1		588.2.e.e		12
84.n	even	6	1		588.2.e.d		12
84.n	even	6	1	inner	588.2.n.e		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.n.a	✓	24	7.b	odd	2	1
84.2.n.a	✓	24	7.d	odd	6	1
84.2.n.a	✓	24	21.c	even	2	1
84.2.n.a	✓	24	21.g	even	6	1
84.2.n.a	✓	24	28.d	even	2	1
84.2.n.a	✓	24	28.f	even	6	1
84.2.n.a	✓	24	84.h	odd	2	1
84.2.n.a	✓	24	84.j	odd	6	1
588.2.e.d		12	7.c	even	3	1
588.2.e.d		12	21.h	odd	6	1
588.2.e.d		12	28.g	odd	6	1
588.2.e.d		12	84.n	even	6	1
588.2.e.e		12	7.d	odd	6	1
588.2.e.e		12	21.g	even	6	1
588.2.e.e		12	28.f	even	6	1
588.2.e.e		12	84.j	odd	6	1
588.2.n.e		24	1.a	even	1	1	trivial
588.2.n.e		24	3.b	odd	2	1	inner
588.2.n.e		24	4.b	odd	2	1	inner
588.2.n.e		24	7.c	even	3	1	inner
588.2.n.e		24	12.b	even	2	1	inner
588.2.n.e		24	21.h	odd	6	1	inner
588.2.n.e		24	28.g	odd	6	1	inner
588.2.n.e		24	84.n	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{12} - 12T_{5}^{10} + 109T_{5}^{8} - 404T_{5}^{6} + 1129T_{5}^{4} - 280T_{5}^{2} + 64 \)

\( T_{11}^{12} + 22T_{11}^{10} + 379T_{11}^{8} + 2054T_{11}^{6} + 8209T_{11}^{4} + 13440T_{11}^{2} + 16384 \)

\( T_{13}^{3} - 3T_{13}^{2} - 10T_{13} + 16 \)

\( T_{19}^{12} - 16T_{19}^{10} + 185T_{19}^{8} - 1008T_{19}^{6} + 4017T_{19}^{4} - 4544T_{19}^{2} + 4096 \)

\( T_{67}^{12} - 144T_{67}^{10} + 17305T_{67}^{8} - 463312T_{67}^{6} + 9557617T_{67}^{4} - 52755056T_{67}^{2} + 236421376 \)