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Properties

Label 126.4.g.b
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
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] = [126,4,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: \(2\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (6 \zeta_{6} - 6) q^{5} + ( - 21 \zeta_{6} + 7) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (6 \zeta_{6} - 6) q^{5} + ( - 21 \zeta_{6} + 7) q^{7} + 8 q^{8} - 12 \zeta_{6} q^{10} - 30 \zeta_{6} q^{11} + 53 q^{13} + (14 \zeta_{6} + 28) q^{14} + (16 \zeta_{6} - 16) q^{16} - 84 \zeta_{6} q^{17} + ( - 97 \zeta_{6} + 97) q^{19} + 24 q^{20} + 60 q^{22} + ( - 84 \zeta_{6} + 84) q^{23} + 89 \zeta_{6} q^{25} + (106 \zeta_{6} - 106) q^{26} + (56 \zeta_{6} - 84) q^{28} + 180 q^{29} - 179 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + 168 q^{34} + (42 \zeta_{6} + 84) q^{35} + ( - 145 \zeta_{6} + 145) q^{37} + 194 \zeta_{6} q^{38} + (48 \zeta_{6} - 48) q^{40} - 126 q^{41} - 325 q^{43} + (120 \zeta_{6} - 120) q^{44} + 168 \zeta_{6} q^{46} + (366 \zeta_{6} - 366) q^{47} + (147 \zeta_{6} - 392) q^{49} - 178 q^{50} - 212 \zeta_{6} q^{52} - 768 \zeta_{6} q^{53} + 180 q^{55} + ( - 168 \zeta_{6} + 56) q^{56} + (360 \zeta_{6} - 360) q^{58} - 264 \zeta_{6} q^{59} + (818 \zeta_{6} - 818) q^{61} + 358 q^{62} + 64 q^{64} + (318 \zeta_{6} - 318) q^{65} + 523 \zeta_{6} q^{67} + (336 \zeta_{6} - 336) q^{68} + (168 \zeta_{6} - 252) q^{70} + 342 q^{71} + 43 \zeta_{6} q^{73} + 290 \zeta_{6} q^{74} - 388 q^{76} + (420 \zeta_{6} - 630) q^{77} + ( - 1171 \zeta_{6} + 1171) q^{79} - 96 \zeta_{6} q^{80} + ( - 252 \zeta_{6} + 252) q^{82} + 810 q^{83} + 504 q^{85} + ( - 650 \zeta_{6} + 650) q^{86} - 240 \zeta_{6} q^{88} + (600 \zeta_{6} - 600) q^{89} + ( - 1113 \zeta_{6} + 371) q^{91} - 336 q^{92} - 732 \zeta_{6} q^{94} + 582 \zeta_{6} q^{95} + 386 q^{97} + ( - 784 \zeta_{6} + 490) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} - 7 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} - 7 q^{7} + 16 q^{8} - 12 q^{10} - 30 q^{11} + 106 q^{13} + 70 q^{14} - 16 q^{16} - 84 q^{17} + 97 q^{19} + 48 q^{20} + 120 q^{22} + 84 q^{23} + 89 q^{25} - 106 q^{26} - 112 q^{28} + 360 q^{29} - 179 q^{31} - 32 q^{32} + 336 q^{34} + 210 q^{35} + 145 q^{37} + 194 q^{38} - 48 q^{40} - 252 q^{41} - 650 q^{43} - 120 q^{44} + 168 q^{46} - 366 q^{47} - 637 q^{49} - 356 q^{50} - 212 q^{52} - 768 q^{53} + 360 q^{55} - 56 q^{56} - 360 q^{58} - 264 q^{59} - 818 q^{61} + 716 q^{62} + 128 q^{64} - 318 q^{65} + 523 q^{67} - 336 q^{68} - 336 q^{70} + 684 q^{71} + 43 q^{73} + 290 q^{74} - 776 q^{76} - 840 q^{77} + 1171 q^{79} - 96 q^{80} + 252 q^{82} + 1620 q^{83} + 1008 q^{85} + 650 q^{86} - 240 q^{88} - 600 q^{89} - 371 q^{91} - 672 q^{92} - 732 q^{94} + 582 q^{95} + 772 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
−1.00000 + 1.73205i 0 −2.00000 − 3.46410i −3.00000 + 5.19615i 0 −3.50000 − 18.1865i 8.00000 0 −6.00000 − 10.3923i
109.1 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i −3.00000 − 5.19615i 0 −3.50000 + 18.1865i 8.00000 0 −6.00000 + 10.3923i
\(n\): 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.b 2
3.b odd 2 1 42.4.e.a ✓ 2
7.b odd 2 1 882.4.g.g 2
7.c even 3 1 inner 126.4.g.b 2
7.c even 3 1 882.4.a.o 1
7.d odd 6 1 882.4.a.l 1
7.d odd 6 1 882.4.g.g 2
12.b even 2 1 336.4.q.f 2
21.c even 2 1 294.4.e.i 2
21.g even 6 1 294.4.a.c 1
21.g even 6 1 294.4.e.i 2
21.h odd 6 1 42.4.e.a ✓ 2
21.h odd 6 1 294.4.a.d 1
84.j odd 6 1 2352.4.a.bf 1
84.n even 6 1 336.4.q.f 2
84.n even 6 1 2352.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a ✓ 2 3.b odd 2 1
42.4.e.a ✓ 2 21.h odd 6 1
126.4.g.b 2 1.a even 1 1 trivial
126.4.g.b 2 7.c even 3 1 inner
294.4.a.c 1 21.g even 6 1
294.4.a.d 1 21.h odd 6 1
294.4.e.i 2 21.c even 2 1
294.4.e.i 2 21.g even 6 1
336.4.q.f 2 12.b even 2 1
336.4.q.f 2 84.n even 6 1
882.4.a.l 1 7.d odd 6 1
882.4.a.o 1 7.c even 3 1
882.4.g.g 2 7.b odd 2 1
882.4.g.g 2 7.d odd 6 1
2352.4.a.f 1 84.n even 6 1
2352.4.a.bf 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} + 36 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} + 7T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$13$ \( (T - 53)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} - 97T + 9409 \) Copy content Toggle raw display
$23$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T - 180)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 179T + 32041 \) Copy content Toggle raw display
$37$ \( T^{2} - 145T + 21025 \) Copy content Toggle raw display
$41$ \( (T + 126)^{2} \) Copy content Toggle raw display
$43$ \( (T + 325)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 366T + 133956 \) Copy content Toggle raw display
$53$ \( T^{2} + 768T + 589824 \) Copy content Toggle raw display
$59$ \( T^{2} + 264T + 69696 \) Copy content Toggle raw display
$61$ \( T^{2} + 818T + 669124 \) Copy content Toggle raw display
$67$ \( T^{2} - 523T + 273529 \) Copy content Toggle raw display
$71$ \( (T - 342)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 43T + 1849 \) Copy content Toggle raw display
$79$ \( T^{2} - 1171 T + 1371241 \) Copy content Toggle raw display
$83$ \( (T - 810)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$97$ \( (T - 386)^{2} \) Copy content Toggle raw display
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