Properties

Label 504.2.cj.a
Level $504$
Weight $2$
Character orbit 504.cj
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(37,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cj (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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4}) q^{2} + \beta_{6} q^{4} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{5} + (3 \beta_{2} + 1) q^{7} + ( - 2 \beta_{7} + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{4}) q^{2} + \beta_{6} q^{4} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{5} + (3 \beta_{2} + 1) q^{7} + ( - 2 \beta_{7} + \beta_{4}) q^{8} + (\beta_{6} + 2 \beta_{2} + 2) q^{10} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{11} + ( - 2 \beta_{5} + \beta_{4}) q^{14} + ( - 4 \beta_{2} - \beta_1) q^{16} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{17} + (2 \beta_{2} + 4 \beta_1) q^{19} + ( - 2 \beta_{7} + 3 \beta_{4}) q^{20} + ( - \beta_{6} - \beta_1 - 2) q^{22} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3}) q^{23} + ( - 2 \beta_{6} - 3 \beta_1) q^{28} + ( - \beta_{7} - \beta_{4}) q^{29} + (\beta_{2} + 1) q^{31} + ( - 2 \beta_{7} + 3 \beta_{5} + 2 \beta_{3}) q^{32} + ( - 2 \beta_{6} - 2 \beta_1 + 4) q^{34} + (3 \beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{35} + (2 \beta_{2} + 4 \beta_1) q^{37} + (8 \beta_{7} + 2 \beta_{5} - 8 \beta_{3}) q^{38} + ( - 4 \beta_{2} - 3 \beta_1) q^{40} + ( - 6 \beta_{7} + 6 \beta_{4}) q^{41} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3}) q^{44} + (4 \beta_{6} - 8 \beta_{2} - 8) q^{46} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{47} + ( - 3 \beta_{2} - 8) q^{49} + ( - 5 \beta_{7} + 5 \beta_{5} + 5 \beta_{3}) q^{53} - 5 q^{55} + ( - 2 \beta_{7} - 3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3}) q^{56} + ( - 2 \beta_{2} + \beta_1) q^{58} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{59} + (2 \beta_{2} + 4 \beta_1) q^{61} + \beta_{4} q^{62} + (3 \beta_{6} + 3 \beta_1 + 4) q^{64} + ( - 4 \beta_{6} + 2 \beta_{2} + 2) q^{67} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{68} + ( - 2 \beta_{6} + 2 \beta_{2} - 3 \beta_1 - 4) q^{70} + (6 \beta_{7} - 6 \beta_{4}) q^{71} + (10 \beta_{2} + 10) q^{73} + (8 \beta_{7} + 2 \beta_{5} - 8 \beta_{3}) q^{74} + (2 \beta_{6} + 2 \beta_1 - 16) q^{76} + ( - 2 \beta_{7} - \beta_{5} - 3 \beta_{4} - \beta_{3}) q^{77} - 13 \beta_{2} q^{79} + ( - 6 \beta_{7} + \beta_{5} + 6 \beta_{3}) q^{80} + ( - 12 \beta_{2} - 6 \beta_1) q^{82} + (5 \beta_{7} + 5 \beta_{4}) q^{83} + ( - 4 \beta_{6} - 4 \beta_1 + 2) q^{85} + ( - 3 \beta_{6} + 4 \beta_{2} + 4) q^{88} + ( - 8 \beta_{5} - 8 \beta_{4} + 8 \beta_{3}) q^{89} + ( - 8 \beta_{7} - 4 \beta_{4}) q^{92} + ( - 2 \beta_{6} + 4 \beta_{2} + 4) q^{94} + (10 \beta_{7} + 10 \beta_{5} - 10 \beta_{3}) q^{95} - q^{97} + ( - 5 \beta_{5} - 8 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 4 q^{7} + 10 q^{10} + 14 q^{16} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 24 q^{34} + 10 q^{40} - 24 q^{46} - 52 q^{49} - 40 q^{55} + 10 q^{58} + 44 q^{64} - 50 q^{70} + 40 q^{73} - 120 q^{76} + 52 q^{79} + 36 q^{82} + 10 q^{88} + 12 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} - 16\nu^{2} + 19 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 12\nu^{3} + 11\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 6\nu^{3} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 23\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} + 16\nu^{4} - 40\nu^{2} - 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 3\nu^{5} - 8\nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 5\beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{5} - \beta_{4} - 4\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{2} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} - 2\beta_{5} - 13\beta_{4} - 13\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{6} + 8\beta_{2} + 8\beta _1 - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34\beta_{7} - 34\beta_{5} - 29\beta_{4} - 5\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\) \(1\)

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
−1.40126 + 0.809017i
−0.535233 + 0.309017i
0.535233 0.309017i
1.40126 0.809017i
−1.40126 0.809017i
−0.535233 0.309017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−1.40126 + 0.190983i 0 1.92705 0.535233i −1.93649 1.11803i 0 −0.500000 + 2.59808i −2.59808 + 1.11803i 0 2.92705 + 1.19682i
37.2 −0.535233 1.30902i 0 −1.42705 + 1.40126i −1.93649 1.11803i 0 −0.500000 + 2.59808i 2.59808 + 1.11803i 0 −0.427051 + 3.13331i
37.3 0.535233 + 1.30902i 0 −1.42705 + 1.40126i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i −2.59808 1.11803i 0 −0.427051 + 3.13331i
37.4 1.40126 0.190983i 0 1.92705 0.535233i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i 2.59808 1.11803i 0 2.92705 + 1.19682i
109.1 −1.40126 0.190983i 0 1.92705 + 0.535233i −1.93649 + 1.11803i 0 −0.500000 2.59808i −2.59808 1.11803i 0 2.92705 1.19682i
109.2 −0.535233 + 1.30902i 0 −1.42705 1.40126i −1.93649 + 1.11803i 0 −0.500000 2.59808i 2.59808 1.11803i 0 −0.427051 3.13331i
109.3 0.535233 1.30902i 0 −1.42705 1.40126i 1.93649 1.11803i 0 −0.500000 2.59808i −2.59808 + 1.11803i 0 −0.427051 3.13331i
109.4 1.40126 + 0.190983i 0 1.92705 + 0.535233i 1.93649 1.11803i 0 −0.500000 2.59808i 2.59808 + 1.11803i 0 2.92705 1.19682i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.a 8
3.b odd 2 1 inner 504.2.cj.a 8
4.b odd 2 1 2016.2.cr.a 8
7.c even 3 1 inner 504.2.cj.a 8
8.b even 2 1 inner 504.2.cj.a 8
8.d odd 2 1 2016.2.cr.a 8
12.b even 2 1 2016.2.cr.a 8
21.h odd 6 1 inner 504.2.cj.a 8
24.f even 2 1 2016.2.cr.a 8
24.h odd 2 1 inner 504.2.cj.a 8
28.g odd 6 1 2016.2.cr.a 8
56.k odd 6 1 2016.2.cr.a 8
56.p even 6 1 inner 504.2.cj.a 8
84.n even 6 1 2016.2.cr.a 8
168.s odd 6 1 inner 504.2.cj.a 8
168.v even 6 1 2016.2.cr.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.a 8 1.a even 1 1 trivial
504.2.cj.a 8 3.b odd 2 1 inner
504.2.cj.a 8 7.c even 3 1 inner
504.2.cj.a 8 8.b even 2 1 inner
504.2.cj.a 8 21.h odd 6 1 inner
504.2.cj.a 8 24.h odd 2 1 inner
504.2.cj.a 8 56.p even 6 1 inner
504.2.cj.a 8 168.s odd 6 1 inner
2016.2.cr.a 8 4.b odd 2 1
2016.2.cr.a 8 8.d odd 2 1
2016.2.cr.a 8 12.b even 2 1
2016.2.cr.a 8 24.f even 2 1
2016.2.cr.a 8 28.g odd 6 1
2016.2.cr.a 8 56.k odd 6 1
2016.2.cr.a 8 84.n even 6 1
2016.2.cr.a 8 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5T_{5}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} - 3 T^{4} - 4 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 125 T^{2} + 15625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 125)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 192 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$97$ \( (T + 1)^{8} \) Copy content Toggle raw display
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