Properties

Label 588.2.i.f
Level 588
Weight 2
Character orbit 588.i
Analytic conductor 4.695
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{9} + ( 6 - 6 \zeta_{6} ) q^{11} -2 q^{13} -4 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} - q^{27} + 6 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{33} -2 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{39} -12 q^{41} -4 q^{43} + 12 \zeta_{6} q^{47} + ( 6 - 6 \zeta_{6} ) q^{53} -4 q^{57} -10 \zeta_{6} q^{61} + ( -8 + 8 \zeta_{6} ) q^{67} + 6 q^{69} + 6 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -5 \zeta_{6} q^{75} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + ( 6 - 6 \zeta_{6} ) q^{87} + 12 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + 10 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{9} + 6q^{11} - 4q^{13} - 4q^{19} + 6q^{23} + 5q^{25} - 2q^{27} + 12q^{29} + 8q^{31} - 6q^{33} - 2q^{37} - 2q^{39} - 24q^{41} - 8q^{43} + 12q^{47} + 6q^{53} - 8q^{57} - 10q^{61} - 8q^{67} + 12q^{69} + 12q^{71} - 10q^{73} - 5q^{75} + 4q^{79} - q^{81} + 24q^{83} + 6q^{87} + 12q^{89} - 8q^{93} + 20q^{97} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
373.1 0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.i.f 2
3.b odd 2 1 1764.2.k.d 2
4.b odd 2 1 2352.2.q.g 2
7.b odd 2 1 588.2.i.c 2
7.c even 3 1 588.2.a.c 1
7.c even 3 1 inner 588.2.i.f 2
7.d odd 6 1 84.2.a.b 1
7.d odd 6 1 588.2.i.c 2
21.c even 2 1 1764.2.k.e 2
21.g even 6 1 252.2.a.b 1
21.g even 6 1 1764.2.k.e 2
21.h odd 6 1 1764.2.a.g 1
21.h odd 6 1 1764.2.k.d 2
28.d even 2 1 2352.2.q.s 2
28.f even 6 1 336.2.a.b 1
28.f even 6 1 2352.2.q.s 2
28.g odd 6 1 2352.2.a.s 1
28.g odd 6 1 2352.2.q.g 2
35.i odd 6 1 2100.2.a.a 1
35.k even 12 2 2100.2.k.a 2
56.j odd 6 1 1344.2.a.f 1
56.k odd 6 1 9408.2.a.r 1
56.m even 6 1 1344.2.a.o 1
56.p even 6 1 9408.2.a.co 1
63.i even 6 1 2268.2.j.f 2
63.k odd 6 1 2268.2.j.i 2
63.s even 6 1 2268.2.j.f 2
63.t odd 6 1 2268.2.j.i 2
84.j odd 6 1 1008.2.a.g 1
84.n even 6 1 7056.2.a.x 1
105.p even 6 1 6300.2.a.p 1
105.w odd 12 2 6300.2.k.r 2
112.v even 12 2 5376.2.c.x 2
112.x odd 12 2 5376.2.c.i 2
140.s even 6 1 8400.2.a.ct 1
168.ba even 6 1 4032.2.a.u 1
168.be odd 6 1 4032.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 7.d odd 6 1
252.2.a.b 1 21.g even 6 1
336.2.a.b 1 28.f even 6 1
588.2.a.c 1 7.c even 3 1
588.2.i.c 2 7.b odd 2 1
588.2.i.c 2 7.d odd 6 1
588.2.i.f 2 1.a even 1 1 trivial
588.2.i.f 2 7.c even 3 1 inner
1008.2.a.g 1 84.j odd 6 1
1344.2.a.f 1 56.j odd 6 1
1344.2.a.o 1 56.m even 6 1
1764.2.a.g 1 21.h odd 6 1
1764.2.k.d 2 3.b odd 2 1
1764.2.k.d 2 21.h odd 6 1
1764.2.k.e 2 21.c even 2 1
1764.2.k.e 2 21.g even 6 1
2100.2.a.a 1 35.i odd 6 1
2100.2.k.a 2 35.k even 12 2
2268.2.j.f 2 63.i even 6 1
2268.2.j.f 2 63.s even 6 1
2268.2.j.i 2 63.k odd 6 1
2268.2.j.i 2 63.t odd 6 1
2352.2.a.s 1 28.g odd 6 1
2352.2.q.g 2 4.b odd 2 1
2352.2.q.g 2 28.g odd 6 1
2352.2.q.s 2 28.d even 2 1
2352.2.q.s 2 28.f even 6 1
4032.2.a.t 1 168.be odd 6 1
4032.2.a.u 1 168.ba even 6 1
5376.2.c.i 2 112.x odd 12 2
5376.2.c.x 2 112.v even 12 2
6300.2.a.p 1 105.p even 6 1
6300.2.k.r 2 105.w odd 12 2
7056.2.a.x 1 84.n even 6 1
8400.2.a.ct 1 140.s even 6 1
9408.2.a.r 1 56.k odd 6 1
9408.2.a.co 1 56.p even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{13} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 12 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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