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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: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} - 3 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} - 3 \beta_{4} q^{9} + ( - 2 \beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{7} + \beta_{6} + 8 \beta_{5} + \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{6} - \beta_{3} + 3 \beta_{2}) q^{15} + ( - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 4) q^{17} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + 4 \beta_{2} - 16) q^{19} + ( - \beta_{7} + 2 \beta_{6} + 9 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{23} + ( - 4 \beta_{7} - \beta_{5} - 9 \beta_{4} - \beta_{2} - 2 \beta_1 - 9) q^{25} + ( - 6 \beta_{4} - 3) q^{27} + (\beta_{7} + 6 \beta_{6} - 6 \beta_{3} + 5 \beta_{2} - \beta_1 + 10) q^{29} + ( - 14 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 14 \beta_{2} - 3 \beta_1 + 4) q^{31} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{2}) q^{33} + ( - 6 \beta_{7} + 4 \beta_{6} - 9 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + \cdots - 12 \beta_1) q^{37}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} + 12 q^{9} + 48 q^{17} - 96 q^{19} + 8 q^{23} - 36 q^{25} + 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} + 48 q^{51} + 72 q^{53} - 192 q^{57} + 168 q^{59} - 144 q^{61} - 120 q^{65} + 32 q^{67} + 224 q^{71} + 336 q^{73} - 108 q^{75} + 216 q^{79} - 36 q^{81} - 96 q^{85} + 120 q^{87} + 96 q^{89} + 48 q^{93} + 136 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 76\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 27\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 35\nu^{5} - 112\nu^{3} + 64\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} - 42\nu^{5} + 154\nu^{3} - 88\nu ) / 14 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 6\beta_{6} + 6\beta_{3} + 5\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} + 22\beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -76\beta_{3} - 54\beta_1 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \( T_{5}^{8} - 32T_{5}^{6} + 1010T_{5}^{4} - 768T_{5}^{3} - 256T_{5}^{2} + 336T_{5} + 196 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 32 T^{6} + 1010 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 124 T^{6} + \cdots + 10837264 \) Copy content Toggle raw display
$13$ \( T^{8} + 712 T^{6} + 123284 T^{4} + \cdots + 2979076 \) Copy content Toggle raw display
$17$ \( T^{8} - 48 T^{7} + \cdots + 168428484 \) Copy content Toggle raw display
$19$ \( T^{8} + 96 T^{7} + \cdots + 285745216 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 1443088144 \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{3} - 1924 T^{2} + \cdots + 468892)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 48 T^{7} + \cdots + 2052452416 \) Copy content Toggle raw display
$37$ \( T^{8} + 64 T^{7} + \cdots + 298373767696 \) Copy content Toggle raw display
$41$ \( T^{8} + 9808 T^{6} + \cdots + 26697826996036 \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{3} - 3784 T^{2} + \cdots + 192784)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 264 T^{7} + \cdots + 8881401308224 \) Copy content Toggle raw display
$53$ \( T^{8} - 72 T^{7} + \cdots + 71641191948544 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 372481662446656 \) Copy content Toggle raw display
$61$ \( T^{8} + 144 T^{7} + 3772 T^{6} + \cdots + 454276 \) Copy content Toggle raw display
$67$ \( T^{8} - 32 T^{7} + \cdots + 306756468736 \) Copy content Toggle raw display
$71$ \( (T^{4} - 112 T^{3} - 6460 T^{2} + \cdots - 7722596)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 336 T^{7} + \cdots + 3712242251524 \) Copy content Toggle raw display
$79$ \( T^{8} - 216 T^{7} + \cdots + 49705658450176 \) Copy content Toggle raw display
$83$ \( T^{8} + 19712 T^{6} + \cdots + 61585579131904 \) Copy content Toggle raw display
$89$ \( T^{8} - 96 T^{7} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + 13640 T^{6} + \cdots + 5315948141956 \) Copy content Toggle raw display
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