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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,Mod(361,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.361"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,36,0,0,0,0,0,-324,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 91x^{6} - 2x^{5} + 5907x^{4} - 304x^{3} + 167650x^{2} + 161744x + 3378244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_{2}) q^{5} - 81 \beta_1 q^{9} + (8 \beta_{6} - 4 \beta_{4} + 11 \beta_{2}) q^{11} + ( - 9 \beta_{7} + 9 \beta_{6} + \cdots + 6 \beta_{3}) q^{13}+ \cdots + (648 \beta_{7} - 648 \beta_{6} + \cdots - 324 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 324 q^{9} - 1872 q^{17} - 1728 q^{19} + 3648 q^{23} + 3996 q^{25} - 5832 q^{27} - 2496 q^{29} - 3888 q^{31} + 12032 q^{37} - 18144 q^{41} - 4256 q^{43} - 19872 q^{47} + 16848 q^{51} + 22248 q^{53}+ \cdots + 641088 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 91x^{6} - 2x^{5} + 5907x^{4} - 304x^{3} + 167650x^{2} + 161744x + 3378244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 34362472 \nu^{7} - 540687503 \nu^{6} + 3136739218 \nu^{5} - 32091655815 \nu^{4} + \cdots - 14523443081156 ) / 64380213930250 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34362472 \nu^{7} - 540687503 \nu^{6} + 3136739218 \nu^{5} - 32091655815 \nu^{4} + \cdots - 78903657011406 ) / 32190106965125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 261 \nu^{7} + 6164 \nu^{6} - 17709 \nu^{5} - 34530 \nu^{4} + 40433 \nu^{3} - 1491132 \nu^{2} + \cdots - 633706972 ) / 36354250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 288888958 \nu^{7} + 6432294067 \nu^{6} - 37316248202 \nu^{5} + 767811292910 \nu^{4} + \cdots + 938678112667734 ) / 32190106965125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1027122 \nu^{7} - 21228 \nu^{6} + 69690818 \nu^{5} + 135887060 \nu^{4} + 6336175334 \nu^{3} + \cdots + 322687533894 ) / 35027319875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19694093334 \nu^{7} + 27832006759 \nu^{6} - 161464333154 \nu^{5} + 6371546717070 \nu^{4} + \cdots + 40\!\cdots\!18 ) / 225330748755875 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47876115417 \nu^{7} + 111999320058 \nu^{6} - 6243861714673 \nu^{5} + \cdots - 70\!\cdots\!84 ) / 450661497511750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 4\beta_{4} - 4\beta_{3} + \beta_{2} - 178\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} - 7\beta_{6} + 45\beta_{5} - 5\beta_{3} - 266 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -14\beta_{6} + 358\beta_{4} - 89\beta_{2} + 8490\beta _1 - 8490 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -637\beta_{7} - 2167\beta_{5} + 799\beta_{4} + 799\beta_{3} - 2167\beta_{2} + 20782\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1876\beta_{7} + 1876\beta_{6} - 6279\beta_{5} + 24952\beta_{3} + 429622 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 45073\beta_{6} - 78723\beta_{4} + 110545\beta_{2} - 1431250\beta _1 + 1431250 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
3.18047 − 5.50873i
−3.42932 + 5.93975i
3.92932 − 6.80578i
−2.68047 + 4.64270i
3.18047 + 5.50873i
−3.42932 − 5.93975i
3.92932 + 6.80578i
−2.68047 − 4.64270i
0 4.50000 − 7.79423i 0 −38.9057 − 67.3866i 0 0 0 −40.5000 − 70.1481i 0
361.2 0 4.50000 − 7.79423i 0 4.51051 + 7.81243i 0 0 0 −40.5000 − 70.1481i 0
361.3 0 4.50000 − 7.79423i 0 16.7027 + 28.9299i 0 0 0 −40.5000 − 70.1481i 0
361.4 0 4.50000 − 7.79423i 0 17.6925 + 30.6443i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −38.9057 + 67.3866i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 4.50000 + 7.79423i 0 4.51051 − 7.81243i 0 0 0 −40.5000 + 70.1481i 0
373.3 0 4.50000 + 7.79423i 0 16.7027 − 28.9299i 0 0 0 −40.5000 + 70.1481i 0
373.4 0 4.50000 + 7.79423i 0 17.6925 − 30.6443i 0 0 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 361.4
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 588.6.i.q 8
7.b odd 2 1 588.6.i.p 8
7.c even 3 1 588.6.a.m ✓ 4
7.c even 3 1 inner 588.6.i.q 8
7.d odd 6 1 588.6.a.o yes 4
7.d odd 6 1 588.6.i.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.6.a.m ✓ 4 7.c even 3 1
588.6.a.o yes 4 7.d odd 6 1
588.6.i.p 8 7.b odd 2 1
588.6.i.p 8 7.d odd 6 1
588.6.i.q 8 1.a even 1 1 trivial
588.6.i.q 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4252 T_{5}^{6} - 259200 T_{5}^{5} + 18909228 T_{5}^{4} - 551059200 T_{5}^{3} + \cdots + 688441916176 \) acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 688441916176 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} - 553428 T^{2} + \cdots + 9699012900)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 6692607711808)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 41\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 83\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 817160501542896)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 84\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
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