Properties

Label 126.2.f.a
Level $126$
Weight $2$
Character orbit 126.f
Analytic conductor $1.006$
Analytic rank $0$
Dimension $2$
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] = [126,2,Mod(43,126)]
 
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([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.f (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: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} + ( - \zeta_{6} + 2) q^{6} - \zeta_{6} q^{7} - q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} + ( - \zeta_{6} + 2) q^{6} - \zeta_{6} q^{7} - q^{8} - 3 q^{9} + 3 q^{10} + 6 \zeta_{6} q^{11} + (\zeta_{6} + 1) q^{12} + (2 \zeta_{6} - 2) q^{13} + ( - \zeta_{6} + 1) q^{14} + ( - 3 \zeta_{6} - 3) q^{15} - \zeta_{6} q^{16} + 6 q^{17} - 3 \zeta_{6} q^{18} - 7 q^{19} + 3 \zeta_{6} q^{20} + (\zeta_{6} - 2) q^{21} + (6 \zeta_{6} - 6) q^{22} + (3 \zeta_{6} - 3) q^{23} + (2 \zeta_{6} - 1) q^{24} - 4 \zeta_{6} q^{25} - 2 q^{26} + (6 \zeta_{6} - 3) q^{27} + q^{28} - 6 \zeta_{6} q^{29} + ( - 6 \zeta_{6} + 3) q^{30} + (2 \zeta_{6} - 2) q^{31} + ( - \zeta_{6} + 1) q^{32} + ( - 6 \zeta_{6} + 12) q^{33} + 6 \zeta_{6} q^{34} - 3 q^{35} + ( - 3 \zeta_{6} + 3) q^{36} + 2 q^{37} - 7 \zeta_{6} q^{38} + (2 \zeta_{6} + 2) q^{39} + (3 \zeta_{6} - 3) q^{40} + ( - \zeta_{6} - 1) q^{42} - 2 \zeta_{6} q^{43} - 6 q^{44} + (9 \zeta_{6} - 9) q^{45} - 3 q^{46} + (\zeta_{6} - 2) q^{48} + (\zeta_{6} - 1) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + ( - 12 \zeta_{6} + 6) q^{51} - 2 \zeta_{6} q^{52} + 6 q^{53} + (3 \zeta_{6} - 6) q^{54} + 18 q^{55} + \zeta_{6} q^{56} + (14 \zeta_{6} - 7) q^{57} + ( - 6 \zeta_{6} + 6) q^{58} + ( - 3 \zeta_{6} + 6) q^{60} - 5 \zeta_{6} q^{61} - 2 q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 6 \zeta_{6} q^{65} + (6 \zeta_{6} + 6) q^{66} + (8 \zeta_{6} - 8) q^{67} + (6 \zeta_{6} - 6) q^{68} + (3 \zeta_{6} + 3) q^{69} - 3 \zeta_{6} q^{70} + 3 q^{71} + 3 q^{72} + 2 q^{73} + 2 \zeta_{6} q^{74} + (4 \zeta_{6} - 8) q^{75} + ( - 7 \zeta_{6} + 7) q^{76} + ( - 6 \zeta_{6} + 6) q^{77} + (4 \zeta_{6} - 2) q^{78} - 5 \zeta_{6} q^{79} - 3 q^{80} + 9 q^{81} - 12 \zeta_{6} q^{83} + ( - 2 \zeta_{6} + 1) q^{84} + ( - 18 \zeta_{6} + 18) q^{85} + ( - 2 \zeta_{6} + 2) q^{86} + (6 \zeta_{6} - 12) q^{87} - 6 \zeta_{6} q^{88} - 9 q^{90} + 2 q^{91} - 3 \zeta_{6} q^{92} + (2 \zeta_{6} + 2) q^{93} + (21 \zeta_{6} - 21) q^{95} + ( - \zeta_{6} - 1) q^{96} - 2 \zeta_{6} q^{97} - q^{98} - 18 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 3 q^{5} + 3 q^{6} - q^{7} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 3 q^{5} + 3 q^{6} - q^{7} - 2 q^{8} - 6 q^{9} + 6 q^{10} + 6 q^{11} + 3 q^{12} - 2 q^{13} + q^{14} - 9 q^{15} - q^{16} + 12 q^{17} - 3 q^{18} - 14 q^{19} + 3 q^{20} - 3 q^{21} - 6 q^{22} - 3 q^{23} - 4 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29} - 2 q^{31} + q^{32} + 18 q^{33} + 6 q^{34} - 6 q^{35} + 3 q^{36} + 4 q^{37} - 7 q^{38} + 6 q^{39} - 3 q^{40} - 3 q^{42} - 2 q^{43} - 12 q^{44} - 9 q^{45} - 6 q^{46} - 3 q^{48} - q^{49} + 4 q^{50} - 2 q^{52} + 12 q^{53} - 9 q^{54} + 36 q^{55} + q^{56} + 6 q^{58} + 9 q^{60} - 5 q^{61} - 4 q^{62} + 3 q^{63} + 2 q^{64} + 6 q^{65} + 18 q^{66} - 8 q^{67} - 6 q^{68} + 9 q^{69} - 3 q^{70} + 6 q^{71} + 6 q^{72} + 4 q^{73} + 2 q^{74} - 12 q^{75} + 7 q^{76} + 6 q^{77} - 5 q^{79} - 6 q^{80} + 18 q^{81} - 12 q^{83} + 18 q^{85} + 2 q^{86} - 18 q^{87} - 6 q^{88} - 18 q^{90} + 4 q^{91} - 3 q^{92} + 6 q^{93} - 21 q^{95} - 3 q^{96} - 2 q^{97} - 2 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 1.50000 + 2.59808i 1.50000 + 0.866025i −0.500000 + 0.866025i −1.00000 −3.00000 3.00000
85.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 1.50000 2.59808i 1.50000 0.866025i −0.500000 0.866025i −1.00000 −3.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.f.a 2
3.b odd 2 1 378.2.f.a 2
4.b odd 2 1 1008.2.r.d 2
7.b odd 2 1 882.2.f.h 2
7.c even 3 1 882.2.e.b 2
7.c even 3 1 882.2.h.j 2
7.d odd 6 1 882.2.e.d 2
7.d odd 6 1 882.2.h.f 2
9.c even 3 1 inner 126.2.f.a 2
9.c even 3 1 1134.2.a.a 1
9.d odd 6 1 378.2.f.a 2
9.d odd 6 1 1134.2.a.h 1
12.b even 2 1 3024.2.r.a 2
21.c even 2 1 2646.2.f.c 2
21.g even 6 1 2646.2.e.j 2
21.g even 6 1 2646.2.h.a 2
21.h odd 6 1 2646.2.e.f 2
21.h odd 6 1 2646.2.h.e 2
36.f odd 6 1 1008.2.r.d 2
36.f odd 6 1 9072.2.a.c 1
36.h even 6 1 3024.2.r.a 2
36.h even 6 1 9072.2.a.w 1
63.g even 3 1 882.2.e.b 2
63.h even 3 1 882.2.h.j 2
63.i even 6 1 2646.2.h.a 2
63.j odd 6 1 2646.2.h.e 2
63.k odd 6 1 882.2.e.d 2
63.l odd 6 1 882.2.f.h 2
63.l odd 6 1 7938.2.a.l 1
63.n odd 6 1 2646.2.e.f 2
63.o even 6 1 2646.2.f.c 2
63.o even 6 1 7938.2.a.u 1
63.s even 6 1 2646.2.e.j 2
63.t odd 6 1 882.2.h.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 1.a even 1 1 trivial
126.2.f.a 2 9.c even 3 1 inner
378.2.f.a 2 3.b odd 2 1
378.2.f.a 2 9.d odd 6 1
882.2.e.b 2 7.c even 3 1
882.2.e.b 2 63.g even 3 1
882.2.e.d 2 7.d odd 6 1
882.2.e.d 2 63.k odd 6 1
882.2.f.h 2 7.b odd 2 1
882.2.f.h 2 63.l odd 6 1
882.2.h.f 2 7.d odd 6 1
882.2.h.f 2 63.t odd 6 1
882.2.h.j 2 7.c even 3 1
882.2.h.j 2 63.h even 3 1
1008.2.r.d 2 4.b odd 2 1
1008.2.r.d 2 36.f odd 6 1
1134.2.a.a 1 9.c even 3 1
1134.2.a.h 1 9.d odd 6 1
2646.2.e.f 2 21.h odd 6 1
2646.2.e.f 2 63.n odd 6 1
2646.2.e.j 2 21.g even 6 1
2646.2.e.j 2 63.s even 6 1
2646.2.f.c 2 21.c even 2 1
2646.2.f.c 2 63.o even 6 1
2646.2.h.a 2 21.g even 6 1
2646.2.h.a 2 63.i even 6 1
2646.2.h.e 2 21.h odd 6 1
2646.2.h.e 2 63.j odd 6 1
3024.2.r.a 2 12.b even 2 1
3024.2.r.a 2 36.h even 6 1
7938.2.a.l 1 63.l odd 6 1
7938.2.a.u 1 63.o even 6 1
9072.2.a.c 1 36.f odd 6 1
9072.2.a.w 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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