Properties

Label 588.3.m.f
Level $588$
Weight $3$
Character orbit 588.m
Analytic conductor $16.022$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{4} ) q^{3} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} -3 \beta_{4} q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{4} ) q^{3} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} -3 \beta_{4} q^{9} + ( -\beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{7} ) q^{11} + ( -2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 8 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{13} + ( 3 \beta_{2} - \beta_{3} + \beta_{6} ) q^{15} + ( 4 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{17} + ( -16 + 4 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{19} + ( -2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{23} + ( -9 - 2 \beta_{1} - \beta_{2} - 9 \beta_{4} - \beta_{5} - 4 \beta_{7} ) q^{25} + ( -3 - 6 \beta_{4} ) q^{27} + ( 10 - \beta_{1} + 5 \beta_{2} - 6 \beta_{3} + 6 \beta_{6} + \beta_{7} ) q^{29} + ( 4 - 3 \beta_{1} + 14 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 14 \beta_{5} ) q^{31} + ( 2 \beta_{2} + \beta_{5} - 3 \beta_{7} ) q^{33} + ( -12 \beta_{1} + 8 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{37} + ( -2 \beta_{1} + 12 \beta_{2} + \beta_{3} + 12 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{39} + ( -14 + 5 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - 28 \beta_{4} + 26 \beta_{5} - 7 \beta_{6} + 5 \beta_{7} ) q^{41} + ( -14 - 2 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{45} + ( -44 + 4 \beta_{2} - 22 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} + 7 \beta_{7} ) q^{47} + ( 6 \beta_{3} - 12 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} ) q^{51} + ( 18 - 6 \beta_{1} - 26 \beta_{2} + 4 \beta_{3} + 18 \beta_{4} - 26 \beta_{5} + 8 \beta_{6} - 12 \beta_{7} ) q^{53} + ( -\beta_{1} + 10 \beta_{2} + 6 \beta_{3} + 20 \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{55} + ( -24 - \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{57} + ( 14 + 5 \beta_{1} + 22 \beta_{2} + 12 \beta_{3} - 14 \beta_{4} - 22 \beta_{5} ) q^{59} + ( -24 + 28 \beta_{2} - 12 \beta_{4} + 14 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{61} + ( -10 \beta_{1} + 8 \beta_{3} + 30 \beta_{4} + 19 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{65} + ( 8 - 6 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} - 8 \beta_{6} - 12 \beta_{7} ) q^{67} + ( -2 - 3 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{69} + ( 28 + \beta_{1} - 43 \beta_{2} - 8 \beta_{3} + 8 \beta_{6} - \beta_{7} ) q^{71} + ( 28 + 8 \beta_{1} + 6 \beta_{2} - 15 \beta_{3} - 28 \beta_{4} - 6 \beta_{5} ) q^{73} + ( -18 - 2 \beta_{2} - 9 \beta_{4} - \beta_{5} - 6 \beta_{7} ) q^{75} + ( 8 \beta_{1} - 54 \beta_{4} - 28 \beta_{5} + 4 \beta_{7} ) q^{79} + ( -9 - 9 \beta_{4} ) q^{81} + ( 16 + 14 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 32 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} + 14 \beta_{7} ) q^{83} + ( -12 + 15 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{85} + ( 10 - 3 \beta_{1} + 5 \beta_{2} - 18 \beta_{3} - 10 \beta_{4} - 5 \beta_{5} ) q^{87} + ( 16 + 46 \beta_{2} + 8 \beta_{4} + 23 \beta_{5} + 11 \beta_{6} + 16 \beta_{7} ) q^{89} + ( -6 \beta_{1} - 8 \beta_{3} - 12 \beta_{4} - 42 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{93} + ( 34 - 4 \beta_{1} - 36 \beta_{2} - 10 \beta_{3} + 34 \beta_{4} - 36 \beta_{5} - 20 \beta_{6} - 8 \beta_{7} ) q^{95} + ( -28 - 4 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} - 56 \beta_{4} + 20 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} + 12 q^{9} + O(q^{10}) \) \( 8 q + 12 q^{3} + 12 q^{9} + 48 q^{17} - 96 q^{19} + 8 q^{23} - 36 q^{25} + 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} + 48 q^{51} + 72 q^{53} - 192 q^{57} + 168 q^{59} - 144 q^{61} - 120 q^{65} + 32 q^{67} + 224 q^{71} + 336 q^{73} - 108 q^{75} + 216 q^{79} - 36 q^{81} - 96 q^{85} + 120 q^{87} + 96 q^{89} + 48 q^{93} + 136 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 76 \nu \)\()/14\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 20 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 27 \nu \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 2 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 21 \nu^{2} + 2 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 35 \nu^{5} - 112 \nu^{3} + 64 \nu \)\()/14\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{7} - 42 \nu^{5} + 154 \nu^{3} - 88 \nu \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{1}\)\()/7\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 2 \beta_{4}\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 6 \beta_{6} + 6 \beta_{3} + 5 \beta_{1}\)\()/7\)
\(\nu^{4}\)\(=\)\(4 \beta_{5} - 6 \beta_{4} + 4 \beta_{2} - 6\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{7} + 22 \beta_{6}\)\()/7\)
\(\nu^{6}\)\(=\)\(14 \beta_{2} - 20\)
\(\nu^{7}\)\(=\)\((\)\(-76 \beta_{3} - 54 \beta_{1}\)\()/7\)

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
313.1
1.60021 0.923880i
−0.662827 + 0.382683i
−1.60021 + 0.923880i
0.662827 0.382683i
1.60021 + 0.923880i
−0.662827 0.382683i
−1.60021 0.923880i
0.662827 + 0.382683i
0 1.50000 0.866025i 0 −5.04718 2.91399i 0 0 0 1.50000 2.59808i 0
313.2 0 1.50000 0.866025i 0 −0.416265 0.240331i 0 0 0 1.50000 2.59808i 0
313.3 0 1.50000 0.866025i 0 0.804540 + 0.464502i 0 0 0 1.50000 2.59808i 0
313.4 0 1.50000 0.866025i 0 4.65891 + 2.68982i 0 0 0 1.50000 2.59808i 0
325.1 0 1.50000 + 0.866025i 0 −5.04718 + 2.91399i 0 0 0 1.50000 + 2.59808i 0
325.2 0 1.50000 + 0.866025i 0 −0.416265 + 0.240331i 0 0 0 1.50000 + 2.59808i 0
325.3 0 1.50000 + 0.866025i 0 0.804540 0.464502i 0 0 0 1.50000 + 2.59808i 0
325.4 0 1.50000 + 0.866025i 0 4.65891 2.68982i 0 0 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.m.f 8
3.b odd 2 1 1764.3.z.l 8
7.b odd 2 1 588.3.m.e 8
7.c even 3 1 588.3.d.c 8
7.c even 3 1 588.3.m.e 8
7.d odd 6 1 588.3.d.c 8
7.d odd 6 1 inner 588.3.m.f 8
21.c even 2 1 1764.3.z.m 8
21.g even 6 1 1764.3.d.h 8
21.g even 6 1 1764.3.z.l 8
21.h odd 6 1 1764.3.d.h 8
21.h odd 6 1 1764.3.z.m 8
28.f even 6 1 2352.3.f.j 8
28.g odd 6 1 2352.3.f.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 7.c even 3 1
588.3.d.c 8 7.d odd 6 1
588.3.m.e 8 7.b odd 2 1
588.3.m.e 8 7.c even 3 1
588.3.m.f 8 1.a even 1 1 trivial
588.3.m.f 8 7.d odd 6 1 inner
1764.3.d.h 8 21.g even 6 1
1764.3.d.h 8 21.h odd 6 1
1764.3.z.l 8 3.b odd 2 1
1764.3.z.l 8 21.g even 6 1
1764.3.z.m 8 21.c even 2 1
1764.3.z.m 8 21.h odd 6 1
2352.3.f.j 8 28.f even 6 1
2352.3.f.j 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 32 T_{5}^{6} + 1010 T_{5}^{4} - 768 T_{5}^{3} - 256 T_{5}^{2} + 336 T_{5} + 196 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 3 - 3 T + T^{2} )^{4} \)
$5$ \( 196 + 336 T - 256 T^{2} - 768 T^{3} + 1010 T^{4} - 32 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 10837264 + 158016 T + 410512 T^{2} - 5952 T^{3} + 12084 T^{4} - 96 T^{5} + 124 T^{6} + T^{8} \)
$13$ \( 2979076 + 1723280 T^{2} + 123284 T^{4} + 712 T^{6} + T^{8} \)
$17$ \( 168428484 - 68212368 T + 9208512 T^{2} - 71118 T^{4} + 768 T^{6} - 48 T^{7} + T^{8} \)
$19$ \( 285745216 + 199602432 T + 62839360 T^{2} + 11430144 T^{3} + 1297976 T^{4} + 92928 T^{5} + 4040 T^{6} + 96 T^{7} + T^{8} \)
$23$ \( 1443088144 - 374409728 T + 73892080 T^{2} - 6639680 T^{3} + 491380 T^{4} - 14816 T^{5} + 676 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( ( 468892 + 65440 T - 1924 T^{2} - 40 T^{3} + T^{4} )^{2} \)
$31$ \( 2052452416 - 7467548928 T + 9185192768 T^{2} - 468122880 T^{3} + 5473592 T^{4} + 136320 T^{5} - 2072 T^{6} - 48 T^{7} + T^{8} \)
$37$ \( 298373767696 + 125188551424 T + 50617849744 T^{2} + 870228736 T^{3} + 27408076 T^{4} + 234880 T^{5} + 7588 T^{6} + 64 T^{7} + T^{8} \)
$41$ \( 26697826996036 + 51769047200 T^{2} + 34716572 T^{4} + 9808 T^{6} + T^{8} \)
$43$ \( ( 192784 - 203168 T - 3784 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$47$ \( 8881401308224 + 111863586048 T - 17029896064 T^{2} - 220411392 T^{3} + 34157384 T^{4} + 1550208 T^{5} + 29104 T^{6} + 264 T^{7} + T^{8} \)
$53$ \( 71641191948544 - 4130215804416 T + 191864861056 T^{2} - 3885089280 T^{3} + 73453104 T^{4} - 582528 T^{5} + 10648 T^{6} - 72 T^{7} + T^{8} \)
$59$ \( 372481662446656 - 28782925866240 T + 869535448960 T^{2} - 9902630400 T^{3} - 20126776 T^{4} + 1115520 T^{5} + 2768 T^{6} - 168 T^{7} + T^{8} \)
$61$ \( 454276 - 20414112 T + 303671288 T^{2} + 95104320 T^{3} + 8405102 T^{4} - 452160 T^{5} + 3772 T^{6} + 144 T^{7} + T^{8} \)
$67$ \( 306756468736 - 146466111488 T + 66583023616 T^{2} - 1634828288 T^{3} + 45594496 T^{4} - 335360 T^{5} + 7072 T^{6} - 32 T^{7} + T^{8} \)
$71$ \( ( -7722596 + 634384 T - 6460 T^{2} - 112 T^{3} + T^{4} )^{2} \)
$73$ \( 3712242251524 - 23767993248 T - 14553796808 T^{2} + 93506880 T^{3} + 58001486 T^{4} - 2546880 T^{5} + 45212 T^{6} - 336 T^{7} + T^{8} \)
$79$ \( 49705658450176 + 439257156096 T + 91586574976 T^{2} - 3820758528 T^{3} + 148346160 T^{4} - 2562432 T^{5} + 34216 T^{6} - 216 T^{7} + T^{8} \)
$83$ \( 61585579131904 + 132200402944 T^{2} + 89759168 T^{4} + 19712 T^{6} + T^{8} \)
$89$ \( 1748734585879876 - 136075027453008 T + 4335067503424 T^{2} - 62684901888 T^{3} + 308791826 T^{4} + 1849344 T^{5} - 16192 T^{6} - 96 T^{7} + T^{8} \)
$97$ \( 5315948141956 + 28759544464 T^{2} + 36017108 T^{4} + 13640 T^{6} + T^{8} \)
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