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Properties

Label 126.8.g.f
Level $126$
Weight $8$
Character orbit 126.g
Analytic conductor $39.361$
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] = [126,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(39.3605132110\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2881})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 721x^{2} + 720x + 518400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + 8 \beta_1 q^{2} + (64 \beta_1 - 64) q^{4} + (3 \beta_{3} - 3 \beta_{2} + 156 \beta_1) q^{5} + ( - 14 \beta_{3} - 21 \beta_{2} + \cdots + 126) q^{7}+ \cdots - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_1 q^{2} + (64 \beta_1 - 64) q^{4} + (3 \beta_{3} - 3 \beta_{2} + 156 \beta_1) q^{5} + ( - 14 \beta_{3} - 21 \beta_{2} + \cdots + 126) q^{7}+ \cdots + ( - 60760 \beta_{3} + 36456 \beta_{2} + \cdots + 6034840) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 128 q^{4} + 309 q^{5} + 868 q^{7} - 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{2} - 128 q^{4} + 309 q^{5} + 868 q^{7} - 2048 q^{8} - 2472 q^{10} - 6747 q^{11} - 14278 q^{13} - 1736 q^{14} - 8192 q^{16} + 27456 q^{17} + 263 q^{19} - 39552 q^{20} - 107952 q^{22} - 27576 q^{23} - 8171 q^{25} - 57112 q^{26} - 69440 q^{28} + 788718 q^{29} + 174212 q^{31} + 65536 q^{32} + 439296 q^{34} - 850290 q^{35} + 106589 q^{37} - 2104 q^{38} - 158208 q^{40} - 1134012 q^{41} - 531010 q^{43} - 431808 q^{44} + 220608 q^{46} + 520092 q^{47} - 376418 q^{49} - 130736 q^{50} + 456896 q^{52} - 366747 q^{53} - 606870 q^{55} - 444416 q^{56} + 3154872 q^{58} + 1131963 q^{59} - 614488 q^{61} + 2787392 q^{62} + 1048576 q^{64} - 3242118 q^{65} + 4777883 q^{67} + 1757184 q^{68} - 3519096 q^{70} + 13272456 q^{71} - 4647343 q^{73} - 852712 q^{74} - 33664 q^{76} - 1935969 q^{77} + 1334750 q^{79} + 1265664 q^{80} - 4536048 q^{82} + 2434494 q^{83} + 4750128 q^{85} - 2124040 q^{86} + 3454464 q^{88} - 13466826 q^{89} + 6884339 q^{91} + 3529728 q^{92} - 4160736 q^{94} - 25982598 q^{95} - 33410062 q^{97} + 16562392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 721x^{2} + 720x + 518400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 721\nu^{2} - 721\nu + 518400 ) / 519120 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} + 1441\nu + 720 ) / 720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 721\nu + 1441 ) / 721 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2162\beta _1 - 2163 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1442\beta_{3} + 721\beta_{2} + 721\beta _1 - 4323 ) / 3 \) 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.

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
13.6687 − 23.6749i
−13.1687 + 22.8089i
13.6687 + 23.6749i
−13.1687 − 22.8089i
4.00000 − 6.92820i 0 −32.0000 − 55.4256i −43.5186 + 75.3765i 0 780.587 + 462.847i −512.000 0 348.149 + 603.012i
37.2 4.00000 − 6.92820i 0 −32.0000 − 55.4256i 198.019 − 342.978i 0 −346.587 − 838.702i −512.000 0 −1584.15 − 2743.83i
109.1 4.00000 + 6.92820i 0 −32.0000 + 55.4256i −43.5186 − 75.3765i 0 780.587 − 462.847i −512.000 0 348.149 − 603.012i
109.2 4.00000 + 6.92820i 0 −32.0000 + 55.4256i 198.019 + 342.978i 0 −346.587 + 838.702i −512.000 0 −1584.15 + 2743.83i
\(n\): 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 126.8.g.f 4
3.b odd 2 1 42.8.e.c ✓ 4
7.c even 3 1 inner 126.8.g.f 4
21.c even 2 1 294.8.e.t 4
21.g even 6 1 294.8.a.t 2
21.g even 6 1 294.8.e.t 4
21.h odd 6 1 42.8.e.c ✓ 4
21.h odd 6 1 294.8.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.8.e.c ✓ 4 3.b odd 2 1
42.8.e.c ✓ 4 21.h odd 6 1
126.8.g.f 4 1.a even 1 1 trivial
126.8.g.f 4 7.c even 3 1 inner
294.8.a.s 2 21.h odd 6 1
294.8.a.t 2 21.g even 6 1
294.8.e.t 4 21.c even 2 1
294.8.e.t 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 309T_{5}^{3} + 129951T_{5}^{2} + 10651230T_{5} + 1188180900 \) acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 1188180900 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 678223072849 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 81729012968100 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7139 T - 6867476)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 83\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} - 394359 T + 38382378660)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 82\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 567006 T + 80065888560)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 265505 T - 568015812050)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots + 8314342748532)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 47\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots - 13321578284910)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 68727373321558)^{2} \) Copy content Toggle raw display
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