↑

Properties

Label 504.2.cj.c
Level $504$
Weight $2$
Character orbit 504.cj
Analytic conductor $4.024$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.951588245534976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{3} q^{4} + ( - \beta_{11} - \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{3} q^{4} + ( - \beta_{11} - \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 4 q^{7} - 4 q^{8} - 8 q^{10} + 16 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{20} + 12 q^{22} - 2 q^{23} - 4 q^{25} + 2 q^{26} + 26 q^{28} + 10 q^{31} + 12 q^{32} + 32 q^{34} - 18 q^{38} + 10 q^{40} + 8 q^{41} + 30 q^{44} - 4 q^{46} - 30 q^{47} - 12 q^{49} + 16 q^{50} - 32 q^{52} + 4 q^{55} + 40 q^{56} - 22 q^{58} + 28 q^{62} + 24 q^{64} - 8 q^{65} - 4 q^{68} - 48 q^{70} - 32 q^{71} - 10 q^{73} - 18 q^{74} + 52 q^{76} - 22 q^{79} - 36 q^{80} - 26 q^{82} - 40 q^{86} - 14 q^{88} + 10 q^{89} + 20 q^{92} + 42 q^{94} + 34 q^{95} + 40 q^{97} - 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} + 8 \nu^{9} - 15 \nu^{8} - 6 \nu^{7} - 31 \nu^{6} + 51 \nu^{5} + 12 \nu^{4} + \cdots - 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{11} + \nu^{10} + 7 \nu^{9} - 10 \nu^{8} + \nu^{7} - 32 \nu^{6} + 39 \nu^{5} - \nu^{4} + \cdots - 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{11} - \nu^{10} - 3 \nu^{9} + 10 \nu^{8} + 3 \nu^{7} + 12 \nu^{6} - 27 \nu^{5} - 11 \nu^{4} + \cdots + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{11} - \nu^{10} - 9 \nu^{9} + 20 \nu^{8} - 3 \nu^{7} + 42 \nu^{6} - 63 \nu^{5} + 7 \nu^{4} + \cdots + 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9 \nu^{11} - 3 \nu^{10} - 18 \nu^{9} + 45 \nu^{8} + 4 \nu^{7} + 81 \nu^{6} - 135 \nu^{5} + \cdots + 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} + \cdots + 416 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9 \nu^{11} - \nu^{10} - 20 \nu^{9} + 49 \nu^{8} - 14 \nu^{7} + 97 \nu^{6} - 161 \nu^{5} + 40 \nu^{4} + \cdots + 320 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{11} + 7 \nu^{10} + 28 \nu^{9} - 55 \nu^{8} - 10 \nu^{7} - 135 \nu^{6} + 183 \nu^{5} + \cdots - 544 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} - \nu^{10} - 16 \nu^{9} + 29 \nu^{8} - 10 \nu^{7} + 77 \nu^{6} - 97 \nu^{5} + 28 \nu^{4} + \cdots + 288 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13 \nu^{11} - 11 \nu^{10} - 38 \nu^{9} + 65 \nu^{8} + 16 \nu^{7} + 189 \nu^{6} - 231 \nu^{5} + \cdots + 800 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} - 5 \nu^{7} + 68 \nu^{6} - 94 \nu^{5} + 16 \nu^{4} + \cdots + 240 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{10} - \beta_{8} - 3\beta_{6} - 5\beta_{5} + 3\beta_{4} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{11} + \beta_{9} - 5\beta_{7} - \beta_{6} - \beta_{5} + 5\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{11} - \beta_{10} - 3\beta_{9} - 3\beta_{8} - 3\beta_{7} - \beta_{2} + 9\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{10} - 15\beta_{8} - 9\beta_{6} - 5\beta_{5} - 3\beta_{4} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{11} + 9\beta_{9} - 5\beta_{7} - \beta_{6} - 3\beta_{5} - 7\beta_{4} + 9\beta_{3} + 9\beta_{2} + 3\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3\beta_{11} + 13\beta_{10} - 9\beta_{9} + 9\beta_{8} + 9\beta_{7} + 13\beta_{2} + 23\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -11\beta_{10} - 37\beta_{8} - 27\beta_{6} + 19\beta_{5} - 13\beta_{4} - 11\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15 \beta_{11} + 17 \beta_{9} + \beta_{7} - 15 \beta_{6} + 9 \beta_{5} - 77 \beta_{4} + 17 \beta_{3} + \cdots + 77 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(-1\) \(1\)

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
1.26950 + 0.623187i
−0.981777 − 1.01790i
1.41417 − 0.0105323i
−0.0950561 + 1.41102i
−0.716208 + 1.21944i
−0.390636 − 1.35919i
1.26950 − 0.623187i
−0.981777 + 1.01790i
1.41417 + 0.0105323i
−0.0950561 − 1.41102i
−0.716208 − 1.21944i
−0.390636 + 1.35919i
−1.38687 + 0.276739i 0 1.84683 − 0.767603i 0.476087 + 0.274869i 0 −2.60755 − 0.447998i −2.34889 + 1.57566i 0 −0.736339 − 0.249456i
37.2 −0.804171 − 1.16332i 0 −0.706619 + 1.87101i 2.80486 + 1.61939i 0 1.47779 − 2.19457i 2.74483 − 0.682591i 0 −0.371724 − 4.56521i
37.3 0.242117 − 1.39333i 0 −1.88276 − 0.674701i −1.28690 − 0.742990i 0 0.129755 + 2.64257i −1.39593 + 2.45995i 0 −1.34681 + 1.61319i
37.4 0.453773 + 1.33944i 0 −1.58818 + 1.21560i −0.476087 − 0.274869i 0 −2.60755 − 0.447998i −2.34889 − 1.57566i 0 0.152134 − 0.762416i
37.5 1.08560 − 0.906347i 0 0.357071 − 1.96787i 1.28690 + 0.742990i 0 0.129755 + 2.64257i −1.39593 − 2.45995i 0 2.07047 − 0.359782i
37.6 1.40955 + 0.114773i 0 1.97365 + 0.323556i −2.80486 − 1.61939i 0 1.47779 − 2.19457i 2.74483 + 0.682591i 0 −3.76772 − 2.60453i
109.1 −1.38687 − 0.276739i 0 1.84683 + 0.767603i 0.476087 − 0.274869i 0 −2.60755 + 0.447998i −2.34889 − 1.57566i 0 −0.736339 + 0.249456i
109.2 −0.804171 + 1.16332i 0 −0.706619 − 1.87101i 2.80486 − 1.61939i 0 1.47779 + 2.19457i 2.74483 + 0.682591i 0 −0.371724 + 4.56521i
109.3 0.242117 + 1.39333i 0 −1.88276 + 0.674701i −1.28690 + 0.742990i 0 0.129755 − 2.64257i −1.39593 − 2.45995i 0 −1.34681 − 1.61319i
109.4 0.453773 − 1.33944i 0 −1.58818 − 1.21560i −0.476087 + 0.274869i 0 −2.60755 + 0.447998i −2.34889 + 1.57566i 0 0.152134 + 0.762416i
109.5 1.08560 + 0.906347i 0 0.357071 + 1.96787i 1.28690 − 0.742990i 0 0.129755 − 2.64257i −1.39593 + 2.45995i 0 2.07047 + 0.359782i
109.6 1.40955 − 0.114773i 0 1.97365 − 0.323556i −2.80486 + 1.61939i 0 1.47779 + 2.19457i 2.74483 − 0.682591i 0 −3.76772 + 2.60453i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 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.c 12
3.b odd 2 1 56.2.p.a ✓ 12
4.b odd 2 1 2016.2.cr.c 12
7.c even 3 1 inner 504.2.cj.c 12
8.b even 2 1 inner 504.2.cj.c 12
8.d odd 2 1 2016.2.cr.c 12
12.b even 2 1 224.2.t.a 12
21.c even 2 1 392.2.p.g 12
21.g even 6 1 392.2.b.f 6
21.g even 6 1 392.2.p.g 12
21.h odd 6 1 56.2.p.a ✓ 12
21.h odd 6 1 392.2.b.e 6
24.f even 2 1 224.2.t.a 12
24.h odd 2 1 56.2.p.a ✓ 12
28.g odd 6 1 2016.2.cr.c 12
56.k odd 6 1 2016.2.cr.c 12
56.p even 6 1 inner 504.2.cj.c 12
84.h odd 2 1 1568.2.t.g 12
84.j odd 6 1 1568.2.b.e 6
84.j odd 6 1 1568.2.t.g 12
84.n even 6 1 224.2.t.a 12
84.n even 6 1 1568.2.b.f 6
168.e odd 2 1 1568.2.t.g 12
168.i even 2 1 392.2.p.g 12
168.s odd 6 1 56.2.p.a ✓ 12
168.s odd 6 1 392.2.b.e 6
168.v even 6 1 224.2.t.a 12
168.v even 6 1 1568.2.b.f 6
168.ba even 6 1 392.2.b.f 6
168.ba even 6 1 392.2.p.g 12
168.be odd 6 1 1568.2.b.e 6
168.be odd 6 1 1568.2.t.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a ✓ 12 3.b odd 2 1
56.2.p.a ✓ 12 21.h odd 6 1
56.2.p.a ✓ 12 24.h odd 2 1
56.2.p.a ✓ 12 168.s odd 6 1
224.2.t.a 12 12.b even 2 1
224.2.t.a 12 24.f even 2 1
224.2.t.a 12 84.n even 6 1
224.2.t.a 12 168.v even 6 1
392.2.b.e 6 21.h odd 6 1
392.2.b.e 6 168.s odd 6 1
392.2.b.f 6 21.g even 6 1
392.2.b.f 6 168.ba even 6 1
392.2.p.g 12 21.c even 2 1
392.2.p.g 12 21.g even 6 1
392.2.p.g 12 168.i even 2 1
392.2.p.g 12 168.ba even 6 1
504.2.cj.c 12 1.a even 1 1 trivial
504.2.cj.c 12 7.c even 3 1 inner
504.2.cj.c 12 8.b even 2 1 inner
504.2.cj.c 12 56.p even 6 1 inner
1568.2.b.e 6 84.j odd 6 1
1568.2.b.e 6 168.be odd 6 1
1568.2.b.f 6 84.n even 6 1
1568.2.b.f 6 168.v even 6 1
1568.2.t.g 12 84.h odd 2 1
1568.2.t.g 12 84.j odd 6 1
1568.2.t.g 12 168.e odd 2 1
1568.2.t.g 12 168.be odd 6 1
2016.2.cr.c 12 4.b odd 2 1
2016.2.cr.c 12 8.d odd 2 1
2016.2.cr.c 12 28.g odd 6 1
2016.2.cr.c 12 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 13T_{5}^{10} + 142T_{5}^{8} - 337T_{5}^{6} + 638T_{5}^{4} - 189T_{5}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 13 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 37 T^{10} + \cdots + 717409 \) Copy content Toggle raw display
$13$ \( (T^{6} + 32 T^{4} + \cdots + 1008)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - T^{5} + 18 T^{4} + \cdots + 441)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 61 T^{10} + \cdots + 9529569 \) Copy content Toggle raw display
$23$ \( (T^{6} + T^{5} + 8 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 72 T^{4} + \cdots + 112)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 5 T^{5} + 22 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 61 T^{10} + \cdots + 3969 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 40 T + 84)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 164 T^{4} + \cdots + 4032)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 15 T^{5} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 678446209 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 678446209 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 413015731569 \) Copy content Toggle raw display
$67$ \( T^{12} - 41 T^{10} + \cdots + 3969 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 432)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 5 T^{5} + \cdots + 194481)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 11 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 52 T^{4} + \cdots + 448)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 5 T^{5} + \cdots + 53361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 10 T^{2} + \cdots - 28)^{4} \) Copy content Toggle raw display
show more
show less