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Properties

Label 57.8.j.a
Level $57$
Weight $8$
Character orbit 57.j
Analytic conductor $17.806$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [57,8,Mod(2,57)] 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("57.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 57.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.8059464526\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 54 \zeta_{18}^{5} + 27 \zeta_{18}^{2}) q^{3} + 128 \zeta_{18} q^{4} + (249 \zeta_{18}^{5} + \cdots + 249 \zeta_{18}) q^{7} - 2187 \zeta_{18}^{4} q^{9} + ( - 3456 \zeta_{18}^{3} + 6912) q^{12} + \cdots + ( - 9922251 \zeta_{18}^{5} + 19844502 \zeta_{18}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 31104 q^{12} - 37815 q^{13} - 129273 q^{19} - 183951 q^{21} - 531441 q^{27} - 481920 q^{28} - 1877187 q^{43} - 2470629 q^{49} + 5611776 q^{52} + 10606638 q^{61} - 3332988 q^{63} + 6291456 q^{64} + 13330581 q^{67}+ \cdots - 39192498 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(\zeta_{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.939693 + 0.342020i
−0.766044 + 0.642788i
0.939693 − 0.342020i
−0.173648 + 0.984808i
−0.173648 − 0.984808i
−0.766044 − 0.642788i
0 30.0602 − 35.8244i 120.281 + 43.7786i 0 0 −754.586 − 1306.98i 0 −379.769 − 2153.77i 0
14.1 0 −46.0549 − 8.12072i −98.0537 + 82.2768i 0 0 813.879 + 1409.68i 0 2055.11 + 747.998i 0
29.1 0 30.0602 + 35.8244i 120.281 − 43.7786i 0 0 −754.586 + 1306.98i 0 −379.769 + 2153.77i 0
32.1 0 15.9947 − 43.9451i −22.2270 + 126.055i 0 0 −59.2934 + 102.699i 0 −1675.34 − 1405.78i 0
41.1 0 15.9947 + 43.9451i −22.2270 − 126.055i 0 0 −59.2934 − 102.699i 0 −1675.34 + 1405.78i 0
53.1 0 −46.0549 + 8.12072i −98.0537 − 82.2768i 0 0 813.879 − 1409.68i 0 2055.11 − 747.998i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.8.j.a ✓ 6
3.b odd 2 1 CM 57.8.j.a ✓ 6
19.f odd 18 1 inner 57.8.j.a ✓ 6
57.j even 18 1 inner 57.8.j.a ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.8.j.a ✓ 6 1.a even 1 1 trivial
57.8.j.a ✓ 6 3.b odd 2 1 CM
57.8.j.a ✓ 6 19.f odd 18 1 inner
57.8.j.a ✓ 6 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{8}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 10460353203 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 84\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 19\!\cdots\!03 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} + 43091 T + 893871739)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 71\!\cdots\!03 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 20\!\cdots\!27 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 48\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 72\!\cdots\!87 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 22\!\cdots\!27 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 25\!\cdots\!27 \) Copy content Toggle raw display
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