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Properties

Label 126.4.g.f
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,4,Mod(37,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} - 8 q^{8} + ( - 2 \beta_{3} - 6 \beta_{2} + \cdots + 8) q^{10}+ \cdots + (4 \beta_{3} + 486 \beta_{2} + \cdots - 762) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 7 q^{5} + 6 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 7 q^{5} + 6 q^{7} - 32 q^{8} + 14 q^{10} + 25 q^{11} - 98 q^{13} + 18 q^{14} - 32 q^{16} - 98 q^{17} + 119 q^{19} + 56 q^{20} + 100 q^{22} - 122 q^{23} + 129 q^{25} - 98 q^{26} + 12 q^{28} + 146 q^{29} - 98 q^{31} + 64 q^{32} - 392 q^{34} - 128 q^{35} - 289 q^{37} - 238 q^{38} + 56 q^{40} + 672 q^{41} + 614 q^{43} + 100 q^{44} + 244 q^{46} - 672 q^{47} + 190 q^{49} + 516 q^{50} + 196 q^{52} + 375 q^{53} - 1526 q^{55} - 48 q^{56} + 146 q^{58} - 763 q^{59} + 406 q^{61} - 392 q^{62} + 256 q^{64} + 654 q^{65} + 1041 q^{67} - 392 q^{68} - 986 q^{70} + 3304 q^{71} + 189 q^{73} + 578 q^{74} - 952 q^{76} - 3415 q^{77} - 524 q^{79} - 112 q^{80} + 672 q^{82} + 574 q^{83} - 1244 q^{85} + 614 q^{86} - 200 q^{88} - 2394 q^{89} + 1783 q^{91} + 976 q^{92} + 1344 q^{94} + 706 q^{95} - 126 q^{97} - 2090 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 97 ) / 49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 48\beta_{2} + \beta _1 - 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 49\beta_{3} - 97 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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.

comment: embeddings in the coefficient field
 
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
3.72311 − 6.44862i
−3.22311 + 5.58259i
3.72311 + 6.44862i
−3.22311 − 5.58259i
1.00000 − 1.73205i 0 −2.00000 − 3.46410i −5.22311 + 9.04669i 0 −12.3924 + 13.7633i −8.00000 0 10.4462 + 18.0934i
37.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 1.72311 − 2.98452i 0 15.3924 − 10.2992i −8.00000 0 −3.44622 − 5.96903i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −5.22311 − 9.04669i 0 −12.3924 − 13.7633i −8.00000 0 10.4462 − 18.0934i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 1.72311 + 2.98452i 0 15.3924 + 10.2992i −8.00000 0 −3.44622 + 5.96903i
\(n\): e.g. 2-40 or 990-1000
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 126.4.g.f yes 4
3.b odd 2 1 126.4.g.e ✓ 4
7.b odd 2 1 882.4.g.bj 4
7.c even 3 1 inner 126.4.g.f yes 4
7.c even 3 1 882.4.a.ba 2
7.d odd 6 1 882.4.a.u 2
7.d odd 6 1 882.4.g.bj 4
21.c even 2 1 882.4.g.z 4
21.g even 6 1 882.4.a.bh 2
21.g even 6 1 882.4.g.z 4
21.h odd 6 1 126.4.g.e ✓ 4
21.h odd 6 1 882.4.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e ✓ 4 3.b odd 2 1
126.4.g.e ✓ 4 21.h odd 6 1
126.4.g.f yes 4 1.a even 1 1 trivial
126.4.g.f yes 4 7.c even 3 1 inner
882.4.a.u 2 7.d odd 6 1
882.4.a.ba 2 7.c even 3 1
882.4.a.bd 2 21.h odd 6 1
882.4.a.bh 2 21.g even 6 1
882.4.g.z 4 21.c even 2 1
882.4.g.z 4 21.g even 6 1
882.4.g.bj 4 7.b odd 2 1
882.4.g.bj 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 7T_{5}^{3} + 85T_{5}^{2} - 252T_{5} + 1296 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 7 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 25 T^{3} + \cdots + 4875264 \) Copy content Toggle raw display
$13$ \( (T^{2} + 49 T - 606)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 98 T^{3} + \cdots + 5875776 \) Copy content Toggle raw display
$19$ \( T^{4} - 119 T^{3} + \cdots + 9647236 \) Copy content Toggle raw display
$23$ \( T^{4} + 122 T^{3} + \cdots + 32901696 \) Copy content Toggle raw display
$29$ \( (T^{2} - 73 T - 1032)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1255072329 \) Copy content Toggle raw display
$37$ \( T^{4} + 289 T^{3} + \cdots + 158404 \) Copy content Toggle raw display
$41$ \( (T - 168)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 307 T + 2284)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 7242690816 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24444697104 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1155728016 \) Copy content Toggle raw display
$61$ \( T^{4} - 406 T^{3} + \cdots + 212226624 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 44865170596 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1652 T + 644448)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 198073062916 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 155826773001 \) Copy content Toggle raw display
$83$ \( (T^{2} - 287 T - 361596)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 1669553420544 \) Copy content Toggle raw display
$97$ \( (T^{2} + 63 T - 1158214)^{2} \) Copy content Toggle raw display
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