Properties

Label 588.2.o.a
Level $588$
Weight $2$
Character orbit 588.o
Analytic conductor $4.695$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.432972864.2
Defining polynomial: \(x^{8} - x^{7} + x^{6} + 4 x^{5} - 6 x^{4} + 8 x^{3} + 4 x^{2} - 8 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{6} ) q^{2} + \beta_{2} q^{3} + \beta_{5} q^{4} + ( \beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} + \beta_{1} q^{6} + ( -2 - \beta_{7} ) q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{6} ) q^{2} + \beta_{2} q^{3} + \beta_{5} q^{4} + ( \beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} + \beta_{1} q^{6} + ( -2 - \beta_{7} ) q^{8} + ( -1 - \beta_{2} ) q^{9} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{5} - \beta_{6} ) q^{11} + \beta_{3} q^{12} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{7} ) q^{15} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{16} + ( -\beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{17} + \beta_{6} q^{18} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( 2 \beta_{1} + 2 \beta_{7} ) q^{20} + ( 2 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{22} + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{23} + ( -2 \beta_{2} - \beta_{4} ) q^{24} + ( -\beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{25} + ( 4 + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{26} + q^{27} -2 q^{29} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{30} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{32} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{33} + ( 2 + \beta_{3} + \beta_{5} + 3 \beta_{7} ) q^{34} + ( -\beta_{3} - \beta_{5} ) q^{36} + ( 4 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{39} + ( -4 - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{40} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{41} + ( 4 \beta_{1} - \beta_{3} - \beta_{5} - 3 \beta_{7} ) q^{43} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{6} ) q^{44} + ( \beta_{4} - \beta_{6} ) q^{45} + ( -2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{46} + ( -4 - 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -2 - 2 \beta_{1} + \beta_{7} ) q^{48} + ( 4 - \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{50} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{51} + ( -8 \beta_{2} - 4 \beta_{6} ) q^{52} + ( 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{53} + ( -\beta_{1} - \beta_{6} ) q^{54} + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{55} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{58} -4 \beta_{2} q^{59} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{60} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{7} ) q^{64} + ( -4 - 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{66} + ( 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( -8 - 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{68} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{69} + ( -5 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} ) q^{71} + ( 2 + 2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{72} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{74} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{75} + ( -4 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{76} + ( -4 - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{78} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{79} + ( -8 \beta_{2} + 4 \beta_{6} ) q^{80} + \beta_{2} q^{81} + ( 6 + 6 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{7} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 2 - 6 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{7} ) q^{85} + ( 8 + 8 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{86} -2 \beta_{2} q^{87} + ( 2 \beta_{5} + 4 \beta_{6} ) q^{88} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{89} + ( 2 + \beta_{3} + \beta_{5} - \beta_{7} ) q^{90} + ( -4 - 2 \beta_{1} ) q^{92} + ( 8 \beta_{2} + 4 \beta_{5} + 4 \beta_{6} ) q^{94} + ( -6 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{95} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{96} + ( -2 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - 2 \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} - q^{4} - 2 q^{6} - 14 q^{8} - 4 q^{9} + O(q^{10}) \) \( 8 q + q^{2} - 4 q^{3} - q^{4} - 2 q^{6} - 14 q^{8} - 4 q^{9} - 8 q^{10} - q^{12} + 7 q^{16} + q^{18} + 12 q^{19} - 8 q^{20} + 12 q^{22} + 7 q^{24} + 8 q^{25} + 12 q^{26} + 8 q^{27} - 16 q^{29} - 8 q^{30} - 9 q^{32} + 8 q^{34} + 2 q^{36} + 12 q^{37} - 20 q^{38} - 20 q^{40} - 14 q^{44} + 6 q^{46} - 8 q^{47} - 14 q^{48} + 38 q^{50} + 28 q^{52} - 16 q^{53} + q^{54} - 8 q^{55} - 24 q^{57} - 2 q^{58} + 16 q^{59} + 4 q^{60} + 2 q^{64} - 8 q^{65} - 6 q^{66} - 32 q^{68} + 7 q^{72} + 14 q^{74} + 8 q^{75} - 40 q^{76} - 24 q^{78} + 36 q^{80} - 4 q^{81} + 20 q^{82} + 16 q^{83} + 40 q^{85} + 30 q^{86} + 8 q^{87} + 2 q^{88} + 16 q^{90} - 28 q^{92} - 32 q^{94} - 9 q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + x^{6} + 4 x^{5} - 6 x^{4} + 8 x^{3} + 4 x^{2} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - \nu^{5} + \nu^{4} - 2 \nu^{2} + 4 \nu - 4 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + \nu^{5} - 2 \nu^{3} + 4 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 5 \nu^{5} - 2 \nu^{3} + 8 \nu^{2} - 4 \nu + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + \nu^{5} - 6 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + \nu^{5} - 6 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 20 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} - 6 \nu^{3} + 12 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - \nu^{4} + 6 \nu^{3} + 2 \nu^{2} - 4 \nu + 12 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{1} - 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{2}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{6} - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_{1} - 4\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{5} + 3 \beta_{3} + 10 \beta_{1} + 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-6 \beta_{6} - 3 \beta_{5} + \beta_{4} + 20 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(-1\) \(1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.41156 0.0865986i
0.121053 + 1.40902i
0.630783 1.26575i
1.15972 + 0.809347i
−1.41156 + 0.0865986i
0.121053 1.40902i
0.630783 + 1.26575i
1.15972 0.809347i
−1.41156 + 0.0865986i −0.500000 0.866025i 1.98500 0.244478i 1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 + 0.866025i −2.14688 1.06990i
19.2 0.121053 1.40902i −0.500000 0.866025i −1.97069 0.341134i 2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 + 0.866025i 2.69928 3.86783i
19.3 0.630783 + 1.26575i −0.500000 0.866025i −1.20422 + 1.59682i −1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 + 0.866025i 0.146883 2.39420i
19.4 1.15972 0.809347i −0.500000 0.866025i 0.689916 1.87724i −2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 + 0.866025i −4.69928 + 0.403728i
31.1 −1.41156 0.0865986i −0.500000 + 0.866025i 1.98500 + 0.244478i 1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 0.866025i −2.14688 + 1.06990i
31.2 0.121053 + 1.40902i −0.500000 + 0.866025i −1.97069 + 0.341134i 2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 0.866025i 2.69928 + 3.86783i
31.3 0.630783 1.26575i −0.500000 + 0.866025i −1.20422 1.59682i −1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 0.866025i 0.146883 + 2.39420i
31.4 1.15972 + 0.809347i −0.500000 + 0.866025i 0.689916 + 1.87724i −2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 0.866025i −4.69928 0.403728i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
28.d even 2 1 inner
28.f 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.o.a 8
4.b odd 2 1 588.2.o.c 8
7.b odd 2 1 588.2.o.c 8
7.c even 3 1 84.2.b.b yes 4
7.c even 3 1 inner 588.2.o.a 8
7.d odd 6 1 84.2.b.a 4
7.d odd 6 1 588.2.o.c 8
21.g even 6 1 252.2.b.e 4
21.h odd 6 1 252.2.b.d 4
28.d even 2 1 inner 588.2.o.a 8
28.f even 6 1 84.2.b.b yes 4
28.f even 6 1 inner 588.2.o.a 8
28.g odd 6 1 84.2.b.a 4
28.g odd 6 1 588.2.o.c 8
56.j odd 6 1 1344.2.b.f 4
56.k odd 6 1 1344.2.b.f 4
56.m even 6 1 1344.2.b.e 4
56.p even 6 1 1344.2.b.e 4
84.j odd 6 1 252.2.b.d 4
84.n even 6 1 252.2.b.e 4
168.s odd 6 1 4032.2.b.j 4
168.v even 6 1 4032.2.b.n 4
168.ba even 6 1 4032.2.b.n 4
168.be odd 6 1 4032.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 7.d odd 6 1
84.2.b.a 4 28.g odd 6 1
84.2.b.b yes 4 7.c even 3 1
84.2.b.b yes 4 28.f even 6 1
252.2.b.d 4 21.h odd 6 1
252.2.b.d 4 84.j odd 6 1
252.2.b.e 4 21.g even 6 1
252.2.b.e 4 84.n even 6 1
588.2.o.a 8 1.a even 1 1 trivial
588.2.o.a 8 7.c even 3 1 inner
588.2.o.a 8 28.d even 2 1 inner
588.2.o.a 8 28.f even 6 1 inner
588.2.o.c 8 4.b odd 2 1
588.2.o.c 8 7.b odd 2 1
588.2.o.c 8 7.d odd 6 1
588.2.o.c 8 28.g odd 6 1
1344.2.b.e 4 56.m even 6 1
1344.2.b.e 4 56.p even 6 1
1344.2.b.f 4 56.j odd 6 1
1344.2.b.f 4 56.k odd 6 1
4032.2.b.j 4 168.s odd 6 1
4032.2.b.j 4 168.be odd 6 1
4032.2.b.n 4 168.v even 6 1
4032.2.b.n 4 168.ba even 6 1

Hecke kernels

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

\( T_{5}^{8} - 14 T_{5}^{6} + 164 T_{5}^{4} - 448 T_{5}^{2} + 1024 \)
\( T_{11}^{8} - 10 T_{11}^{6} + 92 T_{11}^{4} - 80 T_{11}^{2} + 64 \)
\( T_{19}^{4} - 6 T_{19}^{3} + 44 T_{19}^{2} + 48 T_{19} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 8 T + 4 T^{2} + 8 T^{3} - 6 T^{4} + 4 T^{5} + T^{6} - T^{7} + T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 1024 - 448 T^{2} + 164 T^{4} - 14 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 64 - 80 T^{2} + 92 T^{4} - 10 T^{6} + T^{8} \)
$13$ \( ( 128 + 40 T^{2} + T^{4} )^{2} \)
$17$ \( 262144 - 23552 T^{2} + 1604 T^{4} - 46 T^{6} + T^{8} \)
$19$ \( ( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$23$ \( 64 - 80 T^{2} + 92 T^{4} - 10 T^{6} + T^{8} \)
$29$ \( ( 2 + T )^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$41$ \( ( 128 + 62 T^{2} + T^{4} )^{2} \)
$43$ \( ( 5408 + 148 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$53$ \( ( 2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$59$ \( ( 16 - 4 T + T^{2} )^{4} \)
$61$ \( 4194304 - 229376 T^{2} + 10496 T^{4} - 112 T^{6} + T^{8} \)
$67$ \( 262144 - 63488 T^{2} + 14864 T^{4} - 124 T^{6} + T^{8} \)
$71$ \( ( 2312 + 170 T^{2} + T^{4} )^{2} \)
$73$ \( 262144 - 28672 T^{2} + 2624 T^{4} - 56 T^{6} + T^{8} \)
$79$ \( 16384 - 3584 T^{2} + 656 T^{4} - 28 T^{6} + T^{8} \)
$83$ \( ( -64 - 4 T + T^{2} )^{4} \)
$89$ \( 16384 - 7936 T^{2} + 3716 T^{4} - 62 T^{6} + T^{8} \)
$97$ \( ( 8192 + 184 T^{2} + T^{4} )^{2} \)
show more
show less