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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,Mod(37,126)] 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("126.37"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128,0,-4096,1420] 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: \(96.8112407505\)
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: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q - 32 \beta_1 q^{2} + (1024 \beta_1 - 1024) q^{4} + (\beta_{3} + 355 \beta_1) q^{5} + (2 \beta_{7} - 3 \beta_{6} + \cdots - 9073) q^{7} + 32768 q^{8} + ( - 32 \beta_{5} - 11360 \beta_1 + 11360) q^{10}+ \cdots + (2814752 \beta_{7} + \cdots + 15875963360) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 4096 q^{4} + 1420 q^{5} - 66362 q^{7} + 262144 q^{8} + 45440 q^{10} + 861962 q^{11} + 836748 q^{13} + 1361216 q^{14} - 4194304 q^{16} - 5647852 q^{17} + 16668730 q^{19} - 2908160 q^{20}+ \cdots + 80526496576 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\!\cdots\!09 \nu^{7} + \cdots + 29\!\cdots\!80 ) / 21\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21\!\cdots\!41 \nu^{7} + \cdots + 10\!\cdots\!40 ) / 77\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!87 \nu^{7} + \cdots + 26\!\cdots\!20 ) / 22\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93\!\cdots\!99 \nu^{7} + \cdots + 14\!\cdots\!40 ) / 77\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36\!\cdots\!81 \nu^{7} + \cdots + 16\!\cdots\!50 ) / 22\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 64\!\cdots\!67 \nu^{7} + \cdots + 85\!\cdots\!30 ) / 38\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!02 \nu^{7} + \cdots + 23\!\cdots\!15 ) / 55\!\cdots\!95 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 15\beta_{7} + 4\beta_{6} - 20\beta_{5} - 14\beta_{4} - 3\beta_{3} - 21\beta_{2} - 586\beta _1 + 583 ) / 1176 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 130\beta_{7} - 26\beta_{6} - 19\beta_{5} + 33\beta_{4} - 169\beta_{3} - 66\beta_{2} - 1840859\beta _1 + 45 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 301471 \beta_{7} - 215538 \beta_{6} + 1682665 \beta_{5} + 1485519 \beta_{4} - 1574896 \beta_{3} + \cdots - 9944720524 ) / 588 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 45665593 \beta_{7} + 11008259 \beta_{6} + 62094934 \beta_{5} + 28353066 \beta_{4} + \cdots - 470313624937 ) / 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 96839960170 \beta_{7} + 19367992034 \beta_{6} - 3301296147 \beta_{5} - 42037280215 \beta_{4} + \cdots - 16066695887 ) / 294 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 201077819895 \beta_{7} - 10034029932 \beta_{6} - 1629215759604 \beta_{5} - 1221517964268 \beta_{4} + \cdots + 11\!\cdots\!97 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11\!\cdots\!72 \beta_{7} + \cdots + 10\!\cdots\!27 ) / 147 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−73.4634 − 127.242i
−110.360 − 191.150i
−32.2642 − 55.8832i
217.088 + 376.008i
−73.4634 + 127.242i
−110.360 + 191.150i
−32.2642 + 55.8832i
217.088 − 376.008i
−16.0000 + 27.7128i 0 −512.000 − 886.810i −6385.96 + 11060.8i 0 −43747.2 + 7969.08i 32768.0 0 −204351. − 353946.i
37.2 −16.0000 + 27.7128i 0 −512.000 − 886.810i 1017.54 − 1762.43i 0 42078.2 − 14378.9i 32768.0 0 32561.3 + 56397.9i
37.3 −16.0000 + 27.7128i 0 −512.000 − 886.810i 1639.98 − 2840.53i 0 −17084.7 − 41054.1i 32768.0 0 52479.3 + 90896.9i
37.4 −16.0000 + 27.7128i 0 −512.000 − 886.810i 4438.44 − 7687.60i 0 −14427.2 + 42061.6i 32768.0 0 142030. + 246003.i
109.1 −16.0000 − 27.7128i 0 −512.000 + 886.810i −6385.96 − 11060.8i 0 −43747.2 − 7969.08i 32768.0 0 −204351. + 353946.i
109.2 −16.0000 − 27.7128i 0 −512.000 + 886.810i 1017.54 + 1762.43i 0 42078.2 + 14378.9i 32768.0 0 32561.3 − 56397.9i
109.3 −16.0000 − 27.7128i 0 −512.000 + 886.810i 1639.98 + 2840.53i 0 −17084.7 + 41054.1i 32768.0 0 52479.3 − 90896.9i
109.4 −16.0000 − 27.7128i 0 −512.000 + 886.810i 4438.44 + 7687.60i 0 −14427.2 − 42061.6i 32768.0 0 142030. − 246003.i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.4
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 126.12.g.d 8
3.b odd 2 1 42.12.e.d ✓ 8
7.c even 3 1 inner 126.12.g.d 8
21.h odd 6 1 42.12.e.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.12.e.d ✓ 8 3.b odd 2 1
42.12.e.d ✓ 8 21.h odd 6 1
126.12.g.d 8 1.a even 1 1 trivial
126.12.g.d 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 1420 T_{5}^{7} + 129418525 T_{5}^{6} - 1076271246300 T_{5}^{5} + \cdots + 57\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 93\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 20\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 52\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 53\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 33\!\cdots\!48)^{2} \) Copy content Toggle raw display
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