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Properties

Label 126.3.b.a
Level $126$
Weight $3$
Character orbit 126.b
Analytic conductor $3.433$
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] = [126,3,Mod(71,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.b (of order \(2\), degree \(1\), 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: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} + (2 \beta_{2} - \beta_1) q^{5} - \beta_{3} q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} + (2 \beta_{2} - \beta_1) q^{5} - \beta_{3} q^{7} - 2 \beta_1 q^{8} + ( - 4 \beta_{3} + 2) q^{10} + (4 \beta_{2} - 2 \beta_1) q^{11} + ( - 4 \beta_{3} - 8) q^{13} - \beta_{2} q^{14} + 4 q^{16} + ( - 2 \beta_{2} + 13 \beta_1) q^{17} + 20 q^{19} + ( - 4 \beta_{2} + 2 \beta_1) q^{20} + ( - 8 \beta_{3} + 4) q^{22} + (4 \beta_{2} - 2 \beta_1) q^{23} + (8 \beta_{3} - 33) q^{25} + ( - 4 \beta_{2} - 8 \beta_1) q^{26} + 2 \beta_{3} q^{28} + ( - 4 \beta_{2} - 19 \beta_1) q^{29} + (8 \beta_{3} + 4) q^{31} + 4 \beta_1 q^{32} + (4 \beta_{3} - 26) q^{34} + (\beta_{2} - 14 \beta_1) q^{35} + 38 q^{37} + 20 \beta_1 q^{38} + (8 \beta_{3} - 4) q^{40} + (6 \beta_{2} + 27 \beta_1) q^{41} + (24 \beta_{3} + 20) q^{43} + ( - 8 \beta_{2} + 4 \beta_1) q^{44} + ( - 8 \beta_{3} + 4) q^{46} + 12 \beta_1 q^{47} + 7 q^{49} + (8 \beta_{2} - 33 \beta_1) q^{50} + (8 \beta_{3} + 16) q^{52} + ( - 24 \beta_{2} - 3 \beta_1) q^{53} + (16 \beta_{3} - 116) q^{55} + 2 \beta_{2} q^{56} + (8 \beta_{3} + 38) q^{58} + (8 \beta_{2} + 20 \beta_1) q^{59} + ( - 16 \beta_{3} + 58) q^{61} + (8 \beta_{2} + 4 \beta_1) q^{62} - 8 q^{64} + ( - 12 \beta_{2} - 48 \beta_1) q^{65} + ( - 32 \beta_{3} - 48) q^{67} + (4 \beta_{2} - 26 \beta_1) q^{68} + ( - 2 \beta_{3} + 28) q^{70} + (4 \beta_{2} - 2 \beta_1) q^{71} + ( - 20 \beta_{3} - 24) q^{73} + 38 \beta_1 q^{74} - 40 q^{76} + (2 \beta_{2} - 28 \beta_1) q^{77} + ( - 16 \beta_{3} + 76) q^{79} + (8 \beta_{2} - 4 \beta_1) q^{80} + ( - 12 \beta_{3} - 54) q^{82} + ( - 8 \beta_{2} + 64 \beta_1) q^{83} + ( - 56 \beta_{3} + 82) q^{85} + (24 \beta_{2} + 20 \beta_1) q^{86} + (16 \beta_{3} - 8) q^{88} + ( - 18 \beta_{2} + 51 \beta_1) q^{89} + (8 \beta_{3} + 28) q^{91} + ( - 8 \beta_{2} + 4 \beta_1) q^{92} - 24 q^{94} + (40 \beta_{2} - 20 \beta_1) q^{95} + ( - 44 \beta_{3} - 72) q^{97} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 8 q^{10} - 32 q^{13} + 16 q^{16} + 80 q^{19} + 16 q^{22} - 132 q^{25} + 16 q^{31} - 104 q^{34} + 152 q^{37} - 16 q^{40} + 80 q^{43} + 16 q^{46} + 28 q^{49} + 64 q^{52} - 464 q^{55} + 152 q^{58} + 232 q^{61} - 32 q^{64} - 192 q^{67} + 112 q^{70} - 96 q^{73} - 160 q^{76} + 304 q^{79} - 216 q^{82} + 328 q^{85} - 32 q^{88} + 112 q^{91} - 96 q^{94} - 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
− 1.16372i
2.57794i
− 2.57794i
1.16372i
− 1.41421i 0 −2.00000 − 6.06910i 0 −2.64575 2.82843i 0 −8.58301
71.2 − 1.41421i 0 −2.00000 8.89753i 0 2.64575 2.82843i 0 12.5830
71.3 1.41421i 0 −2.00000 − 8.89753i 0 2.64575 − 2.82843i 0 12.5830
71.4 1.41421i 0 −2.00000 6.06910i 0 −2.64575 − 2.82843i 0 −8.58301
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.b.a ✓ 4
3.b odd 2 1 inner 126.3.b.a ✓ 4
4.b odd 2 1 1008.3.d.a 4
5.b even 2 1 3150.3.e.e 4
5.c odd 4 2 3150.3.c.b 8
7.b odd 2 1 882.3.b.f 4
7.c even 3 2 882.3.s.e 8
7.d odd 6 2 882.3.s.i 8
8.b even 2 1 4032.3.d.i 4
8.d odd 2 1 4032.3.d.j 4
9.c even 3 2 1134.3.q.c 8
9.d odd 6 2 1134.3.q.c 8
12.b even 2 1 1008.3.d.a 4
15.d odd 2 1 3150.3.e.e 4
15.e even 4 2 3150.3.c.b 8
21.c even 2 1 882.3.b.f 4
21.g even 6 2 882.3.s.i 8
21.h odd 6 2 882.3.s.e 8
24.f even 2 1 4032.3.d.j 4
24.h odd 2 1 4032.3.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a ✓ 4 1.a even 1 1 trivial
126.3.b.a ✓ 4 3.b odd 2 1 inner
882.3.b.f 4 7.b odd 2 1
882.3.b.f 4 21.c even 2 1
882.3.s.e 8 7.c even 3 2
882.3.s.e 8 21.h odd 6 2
882.3.s.i 8 7.d odd 6 2
882.3.s.i 8 21.g even 6 2
1008.3.d.a 4 4.b odd 2 1
1008.3.d.a 4 12.b even 2 1
1134.3.q.c 8 9.c even 3 2
1134.3.q.c 8 9.d odd 6 2
3150.3.c.b 8 5.c odd 4 2
3150.3.c.b 8 15.e even 4 2
3150.3.e.e 4 5.b even 2 1
3150.3.e.e 4 15.d odd 2 1
4032.3.d.i 4 8.b even 2 1
4032.3.d.i 4 24.h odd 2 1
4032.3.d.j 4 8.d odd 2 1
4032.3.d.j 4 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 116T^{2} + 2916 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 464 T^{2} + 46656 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16 T - 48)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 788 T^{2} + 79524 \) Copy content Toggle raw display
$19$ \( (T - 20)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 464 T^{2} + 46656 \) Copy content Toggle raw display
$29$ \( T^{4} + 1892 T^{2} + 248004 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 432)^{2} \) Copy content Toggle raw display
$37$ \( (T - 38)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 3924 T^{2} + 910116 \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T - 3632)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 16164 T^{2} + 64738116 \) Copy content Toggle raw display
$59$ \( T^{4} + 3392 T^{2} + 9216 \) Copy content Toggle raw display
$61$ \( (T^{2} - 116 T + 1572)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 96 T - 4864)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 464 T^{2} + 46656 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48 T - 2224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 152 T + 3984)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 18176 T^{2} + 53231616 \) Copy content Toggle raw display
$89$ \( T^{4} + 19476 T^{2} + 443556 \) Copy content Toggle raw display
$97$ \( (T^{2} + 144 T - 8368)^{2} \) Copy content Toggle raw display
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