Properties

Label 126.2.e.c
Level $126$
Weight $2$
Character orbit 126.e
Analytic conductor $1.006$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,2,Mod(25,126)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.e (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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{4} - \beta_{2}) q^{3} + q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_{4} - \beta_{2}) q^{3} + q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 4 \beta_{5} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 6 q^{8} + 8 q^{9} - q^{10} - q^{11} + 2 q^{12} + 8 q^{13} - 2 q^{14} + 12 q^{15} + 6 q^{16} - 4 q^{17} - 8 q^{18} - 3 q^{19} + q^{20} - 10 q^{21} + q^{22} - 7 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} - 7 q^{27} + 2 q^{28} - 5 q^{29} - 12 q^{30} - 40 q^{31} - 6 q^{32} - 3 q^{33} + 4 q^{34} - 13 q^{35} + 8 q^{36} + 3 q^{37} + 3 q^{38} - 5 q^{39} - q^{40} + 10 q^{42} - 6 q^{43} - q^{44} + 9 q^{45} + 7 q^{46} + 18 q^{47} + 2 q^{48} + 12 q^{49} - 2 q^{50} + 6 q^{51} + 8 q^{52} + 15 q^{53} + 7 q^{54} - 26 q^{55} - 2 q^{56} + 22 q^{57} + 5 q^{58} + 28 q^{59} + 12 q^{60} - 16 q^{61} + 40 q^{62} - 31 q^{63} + 6 q^{64} + 24 q^{65} + 3 q^{66} - 2 q^{67} - 4 q^{68} + 3 q^{69} + 13 q^{70} + 14 q^{71} - 8 q^{72} + 19 q^{73} - 3 q^{74} + 8 q^{75} - 3 q^{76} + 10 q^{77} + 5 q^{78} - 10 q^{79} + q^{80} + 8 q^{81} + 2 q^{83} - 10 q^{84} - 2 q^{85} + 6 q^{86} - 27 q^{87} + q^{88} - 9 q^{89} - 9 q^{90} - 46 q^{91} - 7 q^{92} - 38 q^{93} - 18 q^{94} + 8 q^{95} - 2 q^{96} + 28 q^{97} - 12 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
−1.00000 −1.64400 0.545231i 1.00000 −0.794182 + 1.37556i 1.64400 + 0.545231i 1.23855 + 2.33795i −1.00000 2.40545 + 1.79272i 0.794182 1.37556i
25.2 −1.00000 0.933463 1.45899i 1.00000 −0.296790 + 0.514055i −0.933463 + 1.45899i 2.32383 1.26483i −1.00000 −1.25729 2.72382i 0.296790 0.514055i
25.3 −1.00000 1.71053 + 0.272169i 1.00000 1.59097 2.75564i −1.71053 0.272169i −2.56238 + 0.658939i −1.00000 2.85185 + 0.931107i −1.59097 + 2.75564i
121.1 −1.00000 −1.64400 + 0.545231i 1.00000 −0.794182 1.37556i 1.64400 0.545231i 1.23855 2.33795i −1.00000 2.40545 1.79272i 0.794182 + 1.37556i
121.2 −1.00000 0.933463 + 1.45899i 1.00000 −0.296790 0.514055i −0.933463 1.45899i 2.32383 + 1.26483i −1.00000 −1.25729 + 2.72382i 0.296790 + 0.514055i
121.3 −1.00000 1.71053 0.272169i 1.00000 1.59097 + 2.75564i −1.71053 + 0.272169i −2.56238 0.658939i −1.00000 2.85185 0.931107i −1.59097 2.75564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.e.c 6
3.b odd 2 1 378.2.e.d 6
4.b odd 2 1 1008.2.q.g 6
7.b odd 2 1 882.2.e.o 6
7.c even 3 1 126.2.h.d yes 6
7.c even 3 1 882.2.f.n 6
7.d odd 6 1 882.2.f.o 6
7.d odd 6 1 882.2.h.p 6
9.c even 3 1 126.2.h.d yes 6
9.c even 3 1 1134.2.g.m 6
9.d odd 6 1 378.2.h.c 6
9.d odd 6 1 1134.2.g.l 6
12.b even 2 1 3024.2.q.g 6
21.c even 2 1 2646.2.e.p 6
21.g even 6 1 2646.2.f.m 6
21.g even 6 1 2646.2.h.o 6
21.h odd 6 1 378.2.h.c 6
21.h odd 6 1 2646.2.f.l 6
28.g odd 6 1 1008.2.t.h 6
36.f odd 6 1 1008.2.t.h 6
36.h even 6 1 3024.2.t.h 6
63.g even 3 1 882.2.f.n 6
63.g even 3 1 1134.2.g.m 6
63.h even 3 1 inner 126.2.e.c 6
63.h even 3 1 7938.2.a.bv 3
63.i even 6 1 2646.2.e.p 6
63.i even 6 1 7938.2.a.bz 3
63.j odd 6 1 378.2.e.d 6
63.j odd 6 1 7938.2.a.ca 3
63.k odd 6 1 882.2.f.o 6
63.l odd 6 1 882.2.h.p 6
63.n odd 6 1 1134.2.g.l 6
63.n odd 6 1 2646.2.f.l 6
63.o even 6 1 2646.2.h.o 6
63.s even 6 1 2646.2.f.m 6
63.t odd 6 1 882.2.e.o 6
63.t odd 6 1 7938.2.a.bw 3
84.n even 6 1 3024.2.t.h 6
252.u odd 6 1 1008.2.q.g 6
252.bb even 6 1 3024.2.q.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 1.a even 1 1 trivial
126.2.e.c 6 63.h even 3 1 inner
126.2.h.d yes 6 7.c even 3 1
126.2.h.d yes 6 9.c even 3 1
378.2.e.d 6 3.b odd 2 1
378.2.e.d 6 63.j odd 6 1
378.2.h.c 6 9.d odd 6 1
378.2.h.c 6 21.h odd 6 1
882.2.e.o 6 7.b odd 2 1
882.2.e.o 6 63.t odd 6 1
882.2.f.n 6 7.c even 3 1
882.2.f.n 6 63.g even 3 1
882.2.f.o 6 7.d odd 6 1
882.2.f.o 6 63.k odd 6 1
882.2.h.p 6 7.d odd 6 1
882.2.h.p 6 63.l odd 6 1
1008.2.q.g 6 4.b odd 2 1
1008.2.q.g 6 252.u odd 6 1
1008.2.t.h 6 28.g odd 6 1
1008.2.t.h 6 36.f odd 6 1
1134.2.g.l 6 9.d odd 6 1
1134.2.g.l 6 63.n odd 6 1
1134.2.g.m 6 9.c even 3 1
1134.2.g.m 6 63.g even 3 1
2646.2.e.p 6 21.c even 2 1
2646.2.e.p 6 63.i even 6 1
2646.2.f.l 6 21.h odd 6 1
2646.2.f.l 6 63.n odd 6 1
2646.2.f.m 6 21.g even 6 1
2646.2.f.m 6 63.s even 6 1
2646.2.h.o 6 21.g even 6 1
2646.2.h.o 6 63.o even 6 1
3024.2.q.g 6 12.b even 2 1
3024.2.q.g 6 252.bb even 6 1
3024.2.t.h 6 36.h even 6 1
3024.2.t.h 6 84.n even 6 1
7938.2.a.bv 3 63.h even 3 1
7938.2.a.bw 3 63.t odd 6 1
7938.2.a.bz 3 63.i even 6 1
7938.2.a.ca 3 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - T_{5}^{5} + 7T_{5}^{4} + 12T_{5}^{3} + 33T_{5}^{2} + 18T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + 7 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + 7 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + \cdots + 4761 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + \cdots + 131769 \) Copy content Toggle raw display
$31$ \( (T^{3} + 20 T^{2} + \cdots + 201)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$47$ \( (T^{3} - 9 T^{2} + \cdots + 189)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( (T^{3} - 14 T^{2} + \cdots + 63)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} - 5 T - 93)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + T^{2} - 112 T - 211)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} + \cdots + 1593)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 19 T^{5} + \cdots + 398161 \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} + \cdots - 321)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} + \cdots + 21609 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + \cdots + 61504 \) Copy content Toggle raw display
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