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Properties

Label 126.4.g.a
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (15 \zeta_{6} - 15) q^{5} + (7 \zeta_{6} + 14) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (15 \zeta_{6} - 15) q^{5} + (7 \zeta_{6} + 14) q^{7} + 8 q^{8} - 30 \zeta_{6} q^{10} - 9 \zeta_{6} q^{11} - 88 q^{13} + (28 \zeta_{6} - 42) q^{14} + (16 \zeta_{6} - 16) q^{16} - 84 \zeta_{6} q^{17} + (104 \zeta_{6} - 104) q^{19} + 60 q^{20} + 18 q^{22} + (84 \zeta_{6} - 84) q^{23} - 100 \zeta_{6} q^{25} + ( - 176 \zeta_{6} + 176) q^{26} + ( - 84 \zeta_{6} + 28) q^{28} - 51 q^{29} - 185 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + 168 q^{34} + (210 \zeta_{6} - 315) q^{35} + (44 \zeta_{6} - 44) q^{37} - 208 \zeta_{6} q^{38} + (120 \zeta_{6} - 120) q^{40} + 168 q^{41} + 326 q^{43} + (36 \zeta_{6} - 36) q^{44} - 168 \zeta_{6} q^{46} + (138 \zeta_{6} - 138) q^{47} + (245 \zeta_{6} + 147) q^{49} + 200 q^{50} + 352 \zeta_{6} q^{52} + 639 \zeta_{6} q^{53} + 135 q^{55} + (56 \zeta_{6} + 112) q^{56} + ( - 102 \zeta_{6} + 102) q^{58} + 159 \zeta_{6} q^{59} + (722 \zeta_{6} - 722) q^{61} + 370 q^{62} + 64 q^{64} + ( - 1320 \zeta_{6} + 1320) q^{65} + 166 \zeta_{6} q^{67} + (336 \zeta_{6} - 336) q^{68} + ( - 630 \zeta_{6} + 210) q^{70} - 1086 q^{71} - 218 \zeta_{6} q^{73} - 88 \zeta_{6} q^{74} + 416 q^{76} + ( - 189 \zeta_{6} + 63) q^{77} + ( - 583 \zeta_{6} + 583) q^{79} - 240 \zeta_{6} q^{80} + (336 \zeta_{6} - 336) q^{82} + 597 q^{83} + 1260 q^{85} + (652 \zeta_{6} - 652) q^{86} - 72 \zeta_{6} q^{88} + (1038 \zeta_{6} - 1038) q^{89} + ( - 616 \zeta_{6} - 1232) q^{91} + 336 q^{92} - 276 \zeta_{6} q^{94} - 1560 \zeta_{6} q^{95} - 169 q^{97} + (294 \zeta_{6} - 784) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 15 q^{5} + 35 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 15 q^{5} + 35 q^{7} + 16 q^{8} - 30 q^{10} - 9 q^{11} - 176 q^{13} - 56 q^{14} - 16 q^{16} - 84 q^{17} - 104 q^{19} + 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} + 176 q^{26} - 28 q^{28} - 102 q^{29} - 185 q^{31} - 32 q^{32} + 336 q^{34} - 420 q^{35} - 44 q^{37} - 208 q^{38} - 120 q^{40} + 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} - 138 q^{47} + 539 q^{49} + 400 q^{50} + 352 q^{52} + 639 q^{53} + 270 q^{55} + 280 q^{56} + 102 q^{58} + 159 q^{59} - 722 q^{61} + 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} - 336 q^{68} - 210 q^{70} - 2172 q^{71} - 218 q^{73} - 88 q^{74} + 832 q^{76} - 63 q^{77} + 583 q^{79} - 240 q^{80} - 336 q^{82} + 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} - 1038 q^{89} - 3080 q^{91} + 672 q^{92} - 276 q^{94} - 1560 q^{95} - 338 q^{97} - 1274 q^{98}+O(q^{100}) \) 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 −7.50000 + 12.9904i 0 17.5000 + 6.06218i 8.00000 0 −15.0000 − 25.9808i
109.1 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i −7.50000 − 12.9904i 0 17.5000 − 6.06218i 8.00000 0 −15.0000 + 25.9808i
\(n\): 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.a 2
3.b odd 2 1 42.4.e.b ✓ 2
7.b odd 2 1 882.4.g.l 2
7.c even 3 1 inner 126.4.g.a 2
7.c even 3 1 882.4.a.r 1
7.d odd 6 1 882.4.a.h 1
7.d odd 6 1 882.4.g.l 2
12.b even 2 1 336.4.q.d 2
21.c even 2 1 294.4.e.e 2
21.g even 6 1 294.4.a.g 1
21.g even 6 1 294.4.e.e 2
21.h odd 6 1 42.4.e.b ✓ 2
21.h odd 6 1 294.4.a.a 1
84.j odd 6 1 2352.4.a.q 1
84.n even 6 1 336.4.q.d 2
84.n even 6 1 2352.4.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b ✓ 2 3.b odd 2 1
42.4.e.b ✓ 2 21.h odd 6 1
126.4.g.a 2 1.a even 1 1 trivial
126.4.g.a 2 7.c even 3 1 inner
294.4.a.a 1 21.h odd 6 1
294.4.a.g 1 21.g even 6 1
294.4.e.e 2 21.c even 2 1
294.4.e.e 2 21.g even 6 1
336.4.q.d 2 12.b even 2 1
336.4.q.d 2 84.n even 6 1
882.4.a.h 1 7.d odd 6 1
882.4.a.r 1 7.c even 3 1
882.4.g.l 2 7.b odd 2 1
882.4.g.l 2 7.d odd 6 1
2352.4.a.q 1 84.j odd 6 1
2352.4.a.u 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 15T_{5} + 225 \) 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
$5$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$7$ \( T^{2} - 35T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$13$ \( (T + 88)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} + 104T + 10816 \) Copy content Toggle raw display
$23$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T + 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 185T + 34225 \) Copy content Toggle raw display
$37$ \( T^{2} + 44T + 1936 \) Copy content Toggle raw display
$41$ \( (T - 168)^{2} \) Copy content Toggle raw display
$43$ \( (T - 326)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 138T + 19044 \) Copy content Toggle raw display
$53$ \( T^{2} - 639T + 408321 \) Copy content Toggle raw display
$59$ \( T^{2} - 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} + 722T + 521284 \) Copy content Toggle raw display
$67$ \( T^{2} - 166T + 27556 \) Copy content Toggle raw display
$71$ \( (T + 1086)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 218T + 47524 \) Copy content Toggle raw display
$79$ \( T^{2} - 583T + 339889 \) Copy content Toggle raw display
$83$ \( (T - 597)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1038 T + 1077444 \) Copy content Toggle raw display
$97$ \( (T + 169)^{2} \) Copy content Toggle raw display
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