Properties

Label 126.6.g.b
Level $126$
Weight $6$
Character orbit 126.g
Analytic conductor $20.208$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(20.2083612964\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{2} - 16 \zeta_{6} q^{4} + ( - 86 \zeta_{6} + 86) q^{5} + (147 \zeta_{6} - 49) q^{7} + 64 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{2} - 16 \zeta_{6} q^{4} + ( - 86 \zeta_{6} + 86) q^{5} + (147 \zeta_{6} - 49) q^{7} + 64 q^{8} + 344 \zeta_{6} q^{10} + 34 \zeta_{6} q^{11} - 3 q^{13} + ( - 196 \zeta_{6} - 392) q^{14} + (256 \zeta_{6} - 256) q^{16} - 1904 \zeta_{6} q^{17} + ( - 1489 \zeta_{6} + 1489) q^{19} - 1376 q^{20} - 136 q^{22} + (224 \zeta_{6} - 224) q^{23} - 4271 \zeta_{6} q^{25} + ( - 12 \zeta_{6} + 12) q^{26} + ( - 1568 \zeta_{6} + 2352) q^{28} + 6508 q^{29} - 1731 \zeta_{6} q^{31} - 1024 \zeta_{6} q^{32} + 7616 q^{34} + (4214 \zeta_{6} + 8428) q^{35} + ( - 7633 \zeta_{6} + 7633) q^{37} + 5956 \zeta_{6} q^{38} + ( - 5504 \zeta_{6} + 5504) q^{40} - 15414 q^{41} + 18491 q^{43} + ( - 544 \zeta_{6} + 544) q^{44} - 896 \zeta_{6} q^{46} + ( - 18462 \zeta_{6} + 18462) q^{47} + (7203 \zeta_{6} - 19208) q^{49} + 17084 q^{50} + 48 \zeta_{6} q^{52} - 19956 \zeta_{6} q^{53} + 2924 q^{55} + (9408 \zeta_{6} - 3136) q^{56} + (26032 \zeta_{6} - 26032) q^{58} - 31828 \zeta_{6} q^{59} + ( - 57654 \zeta_{6} + 57654) q^{61} + 6924 q^{62} + 4096 q^{64} + (258 \zeta_{6} - 258) q^{65} + 60563 \zeta_{6} q^{67} + (30464 \zeta_{6} - 30464) q^{68} + (33712 \zeta_{6} - 50568) q^{70} + 44834 q^{71} - 20821 \zeta_{6} q^{73} + 30532 \zeta_{6} q^{74} - 23824 q^{76} + (3332 \zeta_{6} - 4998) q^{77} + ( - 30531 \zeta_{6} + 30531) q^{79} + 22016 \zeta_{6} q^{80} + ( - 61656 \zeta_{6} + 61656) q^{82} - 110602 q^{83} - 163744 q^{85} + (73964 \zeta_{6} - 73964) q^{86} + 2176 \zeta_{6} q^{88} + (58992 \zeta_{6} - 58992) q^{89} + ( - 441 \zeta_{6} + 147) q^{91} + 3584 q^{92} + 73848 \zeta_{6} q^{94} - 128054 \zeta_{6} q^{95} - 119846 q^{97} + ( - 76832 \zeta_{6} + 48020) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 16 q^{4} + 86 q^{5} + 49 q^{7} + 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 16 q^{4} + 86 q^{5} + 49 q^{7} + 128 q^{8} + 344 q^{10} + 34 q^{11} - 6 q^{13} - 980 q^{14} - 256 q^{16} - 1904 q^{17} + 1489 q^{19} - 2752 q^{20} - 272 q^{22} - 224 q^{23} - 4271 q^{25} + 12 q^{26} + 3136 q^{28} + 13016 q^{29} - 1731 q^{31} - 1024 q^{32} + 15232 q^{34} + 21070 q^{35} + 7633 q^{37} + 5956 q^{38} + 5504 q^{40} - 30828 q^{41} + 36982 q^{43} + 544 q^{44} - 896 q^{46} + 18462 q^{47} - 31213 q^{49} + 34168 q^{50} + 48 q^{52} - 19956 q^{53} + 5848 q^{55} + 3136 q^{56} - 26032 q^{58} - 31828 q^{59} + 57654 q^{61} + 13848 q^{62} + 8192 q^{64} - 258 q^{65} + 60563 q^{67} - 30464 q^{68} - 67424 q^{70} + 89668 q^{71} - 20821 q^{73} + 30532 q^{74} - 47648 q^{76} - 6664 q^{77} + 30531 q^{79} + 22016 q^{80} + 61656 q^{82} - 221204 q^{83} - 327488 q^{85} - 73964 q^{86} + 2176 q^{88} - 58992 q^{89} - 147 q^{91} + 7168 q^{92} + 73848 q^{94} - 128054 q^{95} - 239692 q^{97} + 19208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
0.500000 + 0.866025i
0.500000 0.866025i
−2.00000 + 3.46410i 0 −8.00000 13.8564i 43.0000 74.4782i 0 24.5000 + 127.306i 64.0000 0 172.000 + 297.913i
109.1 −2.00000 3.46410i 0 −8.00000 + 13.8564i 43.0000 + 74.4782i 0 24.5000 127.306i 64.0000 0 172.000 297.913i
\(n\): 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 126.6.g.b 2
3.b odd 2 1 42.6.e.b 2
7.c even 3 1 inner 126.6.g.b 2
7.c even 3 1 882.6.a.m 1
7.d odd 6 1 882.6.a.y 1
12.b even 2 1 336.6.q.a 2
21.c even 2 1 294.6.e.m 2
21.g even 6 1 294.6.a.e 1
21.g even 6 1 294.6.e.m 2
21.h odd 6 1 42.6.e.b 2
21.h odd 6 1 294.6.a.d 1
84.n even 6 1 336.6.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.b 2 3.b odd 2 1
42.6.e.b 2 21.h odd 6 1
126.6.g.b 2 1.a even 1 1 trivial
126.6.g.b 2 7.c even 3 1 inner
294.6.a.d 1 21.h odd 6 1
294.6.a.e 1 21.g even 6 1
294.6.e.m 2 21.c even 2 1
294.6.e.m 2 21.g even 6 1
336.6.q.a 2 12.b even 2 1
336.6.q.a 2 84.n even 6 1
882.6.a.m 1 7.c even 3 1
882.6.a.y 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 86T_{5} + 7396 \) acting on \(S_{6}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 86T + 7396 \) Copy content Toggle raw display
$7$ \( T^{2} - 49T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1904 T + 3625216 \) Copy content Toggle raw display
$19$ \( T^{2} - 1489 T + 2217121 \) Copy content Toggle raw display
$23$ \( T^{2} + 224T + 50176 \) Copy content Toggle raw display
$29$ \( (T - 6508)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1731 T + 2996361 \) Copy content Toggle raw display
$37$ \( T^{2} - 7633 T + 58262689 \) Copy content Toggle raw display
$41$ \( (T + 15414)^{2} \) Copy content Toggle raw display
$43$ \( (T - 18491)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 18462 T + 340845444 \) Copy content Toggle raw display
$53$ \( T^{2} + 19956 T + 398241936 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1013021584 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 3323983716 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 3667876969 \) Copy content Toggle raw display
$71$ \( (T - 44834)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 20821 T + 433514041 \) Copy content Toggle raw display
$79$ \( T^{2} - 30531 T + 932141961 \) Copy content Toggle raw display
$83$ \( (T + 110602)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3480056064 \) Copy content Toggle raw display
$97$ \( (T + 119846)^{2} \) Copy content Toggle raw display
show more
show less