Properties

Label 126.2.k.a
Level $126$
Weight $2$
Character orbit 126.k
Analytic conductor $1.006$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,2,Mod(17,126)]
 
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([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.k (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} - \beta_{3} q^{8} + (\beta_{4} + \beta_{2} + 1) q^{10} - 3 \beta_1 q^{11} - \beta_{7} q^{13} + ( - \beta_{6} - \beta_{3}) q^{14} - \beta_{2} q^{16} + (\beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{17} + ( - \beta_{7} + \beta_{4} + 2 \beta_{2} - 4) q^{19} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{20} - 3 q^{22} + ( - 2 \beta_{6} + \beta_{5}) q^{23} + (4 \beta_{7} - 2 \beta_{4} + 4 \beta_{2} - 4) q^{25} - \beta_{5} q^{26} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{28} + (\beta_{6} - 2 \beta_{5} + 3 \beta_{3}) q^{29} + (3 \beta_{4} - \beta_{2} - 1) q^{31} - \beta_1 q^{32} + ( - \beta_{7} + 4 \beta_{2} - 2) q^{34} + (2 \beta_{6} - 2 \beta_{5} - 7 \beta_{3} + 2 \beta_1) q^{35} + ( - \beta_{7} - \beta_{4} - 4 \beta_{2}) q^{37} + (\beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{38} + ( - \beta_{7} + \beta_{4} - \beta_{2} + 2) q^{40} + ( - 2 \beta_{6} - 4 \beta_{3} + 8 \beta_1) q^{41} + ( - \beta_{7} + 2 \beta_{4} + 4) q^{43} + (3 \beta_{3} - 3 \beta_1) q^{44} + (2 \beta_{7} - \beta_{4}) q^{46} + (\beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{47} + (2 \beta_{7} - 2 \beta_{4} + 5 \beta_{2}) q^{49} + ( - 2 \beta_{6} + 4 \beta_{5} + 4 \beta_{3}) q^{50} - \beta_{4} q^{52} + (\beta_{6} + \beta_{5} - 3 \beta_1) q^{53} + ( - 3 \beta_{7} - 6 \beta_{2} + 3) q^{55} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{56} + ( - \beta_{7} - \beta_{4} + 3 \beta_{2}) q^{58} + ( - 4 \beta_{6} + 4 \beta_{5} + 2 \beta_{3} - \beta_1) q^{59} + ( - \beta_{7} + \beta_{4} - 2 \beta_{2} + 4) q^{61} + (3 \beta_{6} + \beta_{3} - 2 \beta_1) q^{62} - q^{64} + (2 \beta_{6} - \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{65} + ( - 10 \beta_{2} + 10) q^{67} + ( - \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{68} + ( - 2 \beta_{7} - 7 \beta_{2} + 2) q^{70} + (\beta_{6} - 2 \beta_{5} - 6 \beta_{3}) q^{71} + (2 \beta_{4} + 2 \beta_{2} + 2) q^{73} + ( - \beta_{6} - \beta_{5} - 4 \beta_1) q^{74} + ( - \beta_{7} + 4 \beta_{2} - 2) q^{76} + (3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{77} + (\beta_{7} + \beta_{4} + 7 \beta_{2}) q^{79} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{80} + (2 \beta_{7} - 2 \beta_{4} - 4 \beta_{2} + 8) q^{82} + ( - 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{83} + (\beta_{7} - 2 \beta_{4}) q^{85} + (2 \beta_{6} - \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{86} + (3 \beta_{2} - 3) q^{88} + (6 \beta_{3} + 6 \beta_1) q^{89} + ( - \beta_{4} + 6 \beta_{2} - 6) q^{91} + ( - \beta_{6} + 2 \beta_{5}) q^{92} + (\beta_{4} - 2 \beta_{2} - 2) q^{94} + ( - \beta_{6} - \beta_{5}) q^{95} + (2 \beta_{7} - 10 \beta_{2} + 5) q^{97} + ( - 2 \beta_{6} + 2 \beta_{5} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{7} + 12 q^{10} - 4 q^{16} - 24 q^{19} - 24 q^{22} - 16 q^{25} - 4 q^{28} - 12 q^{31} - 16 q^{37} + 12 q^{40} + 32 q^{43} + 20 q^{49} + 12 q^{58} + 24 q^{61} - 8 q^{64} + 40 q^{67} - 12 q^{70} + 24 q^{73} + 28 q^{79} + 48 q^{82} - 12 q^{88} - 24 q^{91} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.09077 3.62132i 0 −1.62132 2.09077i 1.00000i 0 3.62132 + 2.09077i
17.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.358719 + 0.621320i 0 2.62132 + 0.358719i 1.00000i 0 −0.621320 0.358719i
17.3 0.866025 0.500000i 0 0.500000 0.866025i −0.358719 0.621320i 0 2.62132 + 0.358719i 1.00000i 0 −0.621320 0.358719i
17.4 0.866025 0.500000i 0 0.500000 0.866025i 2.09077 + 3.62132i 0 −1.62132 2.09077i 1.00000i 0 3.62132 + 2.09077i
89.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.09077 + 3.62132i 0 −1.62132 + 2.09077i 1.00000i 0 3.62132 2.09077i
89.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.358719 0.621320i 0 2.62132 0.358719i 1.00000i 0 −0.621320 + 0.358719i
89.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.358719 + 0.621320i 0 2.62132 0.358719i 1.00000i 0 −0.621320 + 0.358719i
89.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.09077 3.62132i 0 −1.62132 + 2.09077i 1.00000i 0 3.62132 2.09077i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.k.a 8
3.b odd 2 1 inner 126.2.k.a 8
4.b odd 2 1 1008.2.bt.c 8
5.b even 2 1 3150.2.bf.a 8
5.c odd 4 1 3150.2.bp.b 8
5.c odd 4 1 3150.2.bp.e 8
7.b odd 2 1 882.2.k.a 8
7.c even 3 1 882.2.d.a 8
7.c even 3 1 882.2.k.a 8
7.d odd 6 1 inner 126.2.k.a 8
7.d odd 6 1 882.2.d.a 8
9.c even 3 1 1134.2.l.f 8
9.c even 3 1 1134.2.t.e 8
9.d odd 6 1 1134.2.l.f 8
9.d odd 6 1 1134.2.t.e 8
12.b even 2 1 1008.2.bt.c 8
15.d odd 2 1 3150.2.bf.a 8
15.e even 4 1 3150.2.bp.b 8
15.e even 4 1 3150.2.bp.e 8
21.c even 2 1 882.2.k.a 8
21.g even 6 1 inner 126.2.k.a 8
21.g even 6 1 882.2.d.a 8
21.h odd 6 1 882.2.d.a 8
21.h odd 6 1 882.2.k.a 8
28.f even 6 1 1008.2.bt.c 8
28.f even 6 1 7056.2.k.f 8
28.g odd 6 1 7056.2.k.f 8
35.i odd 6 1 3150.2.bf.a 8
35.k even 12 1 3150.2.bp.b 8
35.k even 12 1 3150.2.bp.e 8
63.i even 6 1 1134.2.t.e 8
63.k odd 6 1 1134.2.l.f 8
63.s even 6 1 1134.2.l.f 8
63.t odd 6 1 1134.2.t.e 8
84.j odd 6 1 1008.2.bt.c 8
84.j odd 6 1 7056.2.k.f 8
84.n even 6 1 7056.2.k.f 8
105.p even 6 1 3150.2.bf.a 8
105.w odd 12 1 3150.2.bp.b 8
105.w odd 12 1 3150.2.bp.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 1.a even 1 1 trivial
126.2.k.a 8 3.b odd 2 1 inner
126.2.k.a 8 7.d odd 6 1 inner
126.2.k.a 8 21.g even 6 1 inner
882.2.d.a 8 7.c even 3 1
882.2.d.a 8 7.d odd 6 1
882.2.d.a 8 21.g even 6 1
882.2.d.a 8 21.h odd 6 1
882.2.k.a 8 7.b odd 2 1
882.2.k.a 8 7.c even 3 1
882.2.k.a 8 21.c even 2 1
882.2.k.a 8 21.h odd 6 1
1008.2.bt.c 8 4.b odd 2 1
1008.2.bt.c 8 12.b even 2 1
1008.2.bt.c 8 28.f even 6 1
1008.2.bt.c 8 84.j odd 6 1
1134.2.l.f 8 9.c even 3 1
1134.2.l.f 8 9.d odd 6 1
1134.2.l.f 8 63.k odd 6 1
1134.2.l.f 8 63.s even 6 1
1134.2.t.e 8 9.c even 3 1
1134.2.t.e 8 9.d odd 6 1
1134.2.t.e 8 63.i even 6 1
1134.2.t.e 8 63.t odd 6 1
3150.2.bf.a 8 5.b even 2 1
3150.2.bf.a 8 15.d odd 2 1
3150.2.bf.a 8 35.i odd 6 1
3150.2.bf.a 8 105.p even 6 1
3150.2.bp.b 8 5.c odd 4 1
3150.2.bp.b 8 15.e even 4 1
3150.2.bp.b 8 35.k even 12 1
3150.2.bp.b 8 105.w odd 12 1
3150.2.bp.e 8 5.c odd 4 1
3150.2.bp.e 8 15.e even 4 1
3150.2.bp.e 8 35.k even 12 1
3150.2.bp.e 8 105.w odd 12 1
7056.2.k.f 8 28.f even 6 1
7056.2.k.f 8 28.g odd 6 1
7056.2.k.f 8 84.j odd 6 1
7056.2.k.f 8 84.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 18 T^{6} + 315 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 36 T^{6} + 1260 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + 54 T^{2} + 72 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 54 T^{2} + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} - 39 T^{2} - 306 T + 2601)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 144 T^{2} + 576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 36 T^{6} + 1260 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( T^{8} - 54 T^{6} + 2835 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} + 198 T^{6} + \cdots + 74805201 \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + 54 T^{2} - 72 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + 36 T^{2} + 144 T + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 14 T^{3} + 165 T^{2} - 434 T + 961)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 54 T^{2} + 441)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 108 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 198 T^{2} + 2601)^{2} \) Copy content Toggle raw display
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