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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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,-108,0,196] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(183.682394985\)
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: \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + 27 \beta_{3} q^{3} + ( - \beta_{6} + 49 \beta_{3} + 49) q^{5} + ( - 729 \beta_{3} - 729) q^{9} + ( - 7 \beta_{7} + 4 \beta_{6} + \cdots + 4 \beta_1) q^{11} + ( - 13 \beta_{4} + 2 \beta_1 - 500) q^{13}+ \cdots + ( - 5103 \beta_{4} - 4374 \beta_{2} + \cdots - 76545) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 108 q^{3} + 196 q^{5} - 2916 q^{9} + 406 q^{11} - 3948 q^{13} - 10584 q^{15} + 7436 q^{17} - 15874 q^{19} - 6788 q^{23} + 69898 q^{25} + 157464 q^{27} - 189088 q^{29} + 55890 q^{31} + 10962 q^{33}+ \cdots - 591948 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37941104905 \nu^{7} + 1267919624362 \nu^{6} - 25668866266087 \nu^{5} + \cdots - 80\!\cdots\!92 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 317083894595 \nu^{7} - 5064341721014 \nu^{6} + 214521535571213 \nu^{5} + \cdots + 20\!\cdots\!08 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 134367782106 \nu^{7} - 270767347357 \nu^{6} + 88569667465984 \nu^{5} + \cdots + 108893146008780 ) / 64\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 66893433245 \nu^{7} - 810012927629 \nu^{6} + 45256420347923 \nu^{5} + 466843245101494 \nu^{4} + \cdots - 70\!\cdots\!80 ) / 16\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42447498334567 \nu^{7} + 90187524766174 \nu^{6} + \cdots - 20\!\cdots\!08 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 168638720921033 \nu^{7} - 341067259110176 \nu^{6} + \cdots + 80\!\cdots\!52 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14700551927453 \nu^{7} + 28926059685436 \nu^{6} + \cdots - 70\!\cdots\!80 ) / 16\!\cdots\!21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 6\beta_{6} + 3\beta_{5} - 2\beta_{4} - 62\beta_{3} + 3\beta_{2} + 6\beta_1 ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} + 90\beta_{6} + 3\beta_{5} - 20698\beta_{3} - 20698 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 554\beta_{4} - 1713\beta_{2} - 5190\beta _1 - 662798 ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2983 \beta_{7} - 81252 \beta_{6} - 12234 \beta_{5} + 2983 \beta_{4} + 14402104 \beta_{3} + \cdots - 81252 \beta_1 ) / 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -317930\beta_{7} - 4751682\beta_{6} - 1205427\beta_{5} + 727869098\beta_{3} + 727869098 ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3851383\beta_{4} + 14104425\beta_{2} + 72781182\beta _1 + 12053285746 ) / 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 257726282 \beta_{7} + 4330835214 \beta_{6} + 985681749 \beta_{5} - 257726282 \beta_{4} + \cdots + 4330835214 \beta_1 ) / 126 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{3}\)

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
−0.00737575 + 0.0127752i
−5.63535 + 9.76071i
−8.39785 + 14.5455i
15.0406 − 26.0510i
−0.00737575 − 0.0127752i
−5.63535 − 9.76071i
−8.39785 − 14.5455i
15.0406 + 26.0510i
0 −13.5000 + 23.3827i 0 −80.0854 − 138.712i 0 0 0 −364.500 − 631.333i 0
361.2 0 −13.5000 + 23.3827i 0 −64.7849 − 112.211i 0 0 0 −364.500 − 631.333i 0
361.3 0 −13.5000 + 23.3827i 0 20.0775 + 34.7753i 0 0 0 −364.500 − 631.333i 0
361.4 0 −13.5000 + 23.3827i 0 222.793 + 385.888i 0 0 0 −364.500 − 631.333i 0
373.1 0 −13.5000 − 23.3827i 0 −80.0854 + 138.712i 0 0 0 −364.500 + 631.333i 0
373.2 0 −13.5000 − 23.3827i 0 −64.7849 + 112.211i 0 0 0 −364.500 + 631.333i 0
373.3 0 −13.5000 − 23.3827i 0 20.0775 − 34.7753i 0 0 0 −364.500 + 631.333i 0
373.4 0 −13.5000 − 23.3827i 0 222.793 − 385.888i 0 0 0 −364.500 + 631.333i 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 588.8.i.o 8
7.b odd 2 1 84.8.i.a ✓ 8
7.c even 3 1 588.8.a.j 4
7.c even 3 1 inner 588.8.i.o 8
7.d odd 6 1 84.8.i.a ✓ 8
7.d odd 6 1 588.8.a.i 4
21.c even 2 1 252.8.k.b 8
21.g even 6 1 252.8.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.i.a ✓ 8 7.b odd 2 1
84.8.i.a ✓ 8 7.d odd 6 1
252.8.k.b 8 21.c even 2 1
252.8.k.b 8 21.g even 6 1
588.8.a.i 4 7.d odd 6 1
588.8.a.j 4 7.c even 3 1
588.8.i.o 8 1.a even 1 1 trivial
588.8.i.o 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 196 T_{5}^{7} + 140509 T_{5}^{6} + 29803308 T_{5}^{5} + 9091930509 T_{5}^{4} + \cdots + 13\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 31670239466400)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 97\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 81\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 74\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 28\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 95\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 45\!\cdots\!56)^{2} \) Copy content Toggle raw display
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