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Properties

Label 126.4.g.g
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: \(\Q(\sqrt{-3}, \sqrt{1345})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + 1) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + 1) q^{7} - 8 q^{8} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{10} + (\beta_{3} - 34 \beta_{2} + \beta_1 + 33) q^{11} + (\beta_{3} + 20) q^{13} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{14} - 16 \beta_{2} q^{16} + (4 \beta_{3} + 44 \beta_{2} + \cdots - 48) q^{17}+ \cdots + (12 \beta_{3} + 656 \beta_{2} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 32 q^{8} - 10 q^{10} + 67 q^{11} + 82 q^{13} + 18 q^{14} - 32 q^{16} - 92 q^{17} - 43 q^{19} - 40 q^{20} + 268 q^{22} + 148 q^{23} - 435 q^{25} + 82 q^{26} + 36 q^{28} - 154 q^{29} + 520 q^{31} + 64 q^{32} - 368 q^{34} - 650 q^{35} - 7 q^{37} + 86 q^{38} - 40 q^{40} + 852 q^{41} - 214 q^{43} + 268 q^{44} - 296 q^{46} + 576 q^{47} + 1318 q^{49} - 1740 q^{50} - 164 q^{52} - 243 q^{53} - 1010 q^{55} - 154 q^{58} - 7 q^{59} - 224 q^{61} + 2080 q^{62} + 256 q^{64} - 570 q^{65} - 687 q^{67} - 368 q^{68} + 1390 q^{70} - 944 q^{71} + 921 q^{73} + 14 q^{74} + 344 q^{76} + 371 q^{77} + 526 q^{79} + 80 q^{80} + 852 q^{82} + 442 q^{83} - 5840 q^{85} - 214 q^{86} - 536 q^{88} + 774 q^{89} + 1345 q^{91} - 1184 q^{92} - 1152 q^{94} - 1910 q^{95} + 3906 q^{97} + 1318 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 673 ) / 337 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 336\beta_{2} + \beta _1 - 337 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 337\beta_{3} - 673 \) 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
−8.91856 + 15.4474i
9.41856 − 16.3134i
−8.91856 − 15.4474i
9.41856 + 16.3134i
1.00000 − 1.73205i 0 −2.00000 − 3.46410i −7.91856 + 13.7153i 0 18.3371 + 2.59808i −8.00000 0 15.8371 + 27.4307i
37.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 10.4186 − 18.0455i 0 −18.3371 + 2.59808i −8.00000 0 −20.8371 − 36.0910i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −7.91856 − 13.7153i 0 18.3371 − 2.59808i −8.00000 0 15.8371 − 27.4307i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 10.4186 + 18.0455i 0 −18.3371 − 2.59808i −8.00000 0 −20.8371 + 36.0910i
\(n\): 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.g 4
3.b odd 2 1 42.4.e.c ✓ 4
7.b odd 2 1 882.4.g.bf 4
7.c even 3 1 inner 126.4.g.g 4
7.c even 3 1 882.4.a.v 2
7.d odd 6 1 882.4.a.z 2
7.d odd 6 1 882.4.g.bf 4
12.b even 2 1 336.4.q.j 4
21.c even 2 1 294.4.e.l 4
21.g even 6 1 294.4.a.m 2
21.g even 6 1 294.4.e.l 4
21.h odd 6 1 42.4.e.c ✓ 4
21.h odd 6 1 294.4.a.n 2
84.j odd 6 1 2352.4.a.ca 2
84.n even 6 1 336.4.q.j 4
84.n even 6 1 2352.4.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c ✓ 4 3.b odd 2 1
42.4.e.c ✓ 4 21.h odd 6 1
126.4.g.g 4 1.a even 1 1 trivial
126.4.g.g 4 7.c even 3 1 inner
294.4.a.m 2 21.g even 6 1
294.4.a.n 2 21.h odd 6 1
294.4.e.l 4 21.c even 2 1
294.4.e.l 4 21.g even 6 1
336.4.q.j 4 12.b even 2 1
336.4.q.j 4 84.n even 6 1
882.4.a.v 2 7.c even 3 1
882.4.a.z 2 7.d odd 6 1
882.4.g.bf 4 7.b odd 2 1
882.4.g.bf 4 7.d odd 6 1
2352.4.a.bq 2 84.n even 6 1
2352.4.a.ca 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5T_{5}^{3} + 355T_{5}^{2} + 1650T_{5} + 108900 \) 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
$5$ \( T^{4} - 5 T^{3} + \cdots + 108900 \) Copy content Toggle raw display
$7$ \( T^{4} - 659 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 67 T^{3} + \cdots + 617796 \) Copy content Toggle raw display
$13$ \( (T^{2} - 41 T + 84)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 92 T^{3} + \cdots + 10653696 \) Copy content Toggle raw display
$19$ \( T^{4} + 43 T^{3} + \cdots + 6574096 \) Copy content Toggle raw display
$23$ \( T^{4} - 148 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( (T^{2} + 77 T - 39204)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 4389725025 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + \cdots + 741146176 \) Copy content Toggle raw display
$41$ \( (T^{2} - 426 T + 33264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 107 T - 72794)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 1191906576 \) Copy content Toggle raw display
$53$ \( T^{4} + 243 T^{3} + \cdots + 137733696 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 44160500736 \) Copy content Toggle raw display
$61$ \( T^{4} + 224 T^{3} + \cdots + 51322896 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 5976217636 \) Copy content Toggle raw display
$71$ \( (T^{2} + 472 T - 78804)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 39938423716 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 2270427201 \) Copy content Toggle raw display
$83$ \( (T^{2} - 221 T - 197946)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 196582277376 \) Copy content Toggle raw display
$97$ \( (T^{2} - 1953 T + 541646)^{2} \) Copy content Toggle raw display
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