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Properties

Label 504.2.cj.d
Level $504$
Weight $2$
Character orbit 504.cj
Analytic conductor $4.024$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 4x^{14} + 6x^{12} + 8x^{10} + 20x^{8} + 32x^{6} + 96x^{4} + 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{13} + \beta_{5} - \beta_{4} + \cdots - 1) q^{4}+ \cdots + (\beta_{14} + \beta_{12} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{13} + \beta_{5} - \beta_{4} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 8 q^{7} - 8 q^{10} + 8 q^{16} + 32 q^{22} - 8 q^{25} - 16 q^{28} + 40 q^{31} - 80 q^{34} - 32 q^{40} - 8 q^{46} + 40 q^{49} + 8 q^{52} - 32 q^{64} - 16 q^{70} + 24 q^{73} + 32 q^{76} - 24 q^{79} - 16 q^{82} - 32 q^{88} - 24 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 4x^{14} + 6x^{12} + 8x^{10} + 20x^{8} + 32x^{6} + 96x^{4} + 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{15} - 44\nu^{13} - 6\nu^{11} - 24\nu^{9} - 148\nu^{7} - 64\nu^{5} - 384\nu^{3} - 2048\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{14} - 44\nu^{12} - 6\nu^{10} - 24\nu^{8} - 148\nu^{6} - 64\nu^{4} - 384\nu^{2} - 2496 ) / 448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 4\nu^{13} + 6\nu^{11} + 8\nu^{9} + 20\nu^{7} + 32\nu^{5} + 96\nu^{3} + 256\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{14} + 3\nu^{12} + 2\nu^{10} - 6\nu^{8} + 12\nu^{6} + 12\nu^{4} - 96\nu^{2} + 160 ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{14} - 3\nu^{12} - 2\nu^{10} + 6\nu^{8} - 12\nu^{6} - 12\nu^{4} + 320\nu^{2} - 160 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{15} + 30\nu^{13} + 6\nu^{11} + 52\nu^{9} + 148\nu^{7} + 8\nu^{5} + 608\nu^{3} + 2048\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{15} + 12\nu^{13} - 6\nu^{11} - 24\nu^{9} + 76\nu^{7} - 64\nu^{5} - 384\nu^{3} + 1088\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{14} + 24\nu^{12} + 2\nu^{10} + 64\nu^{8} + 124\nu^{6} + 208\nu^{4} + 800\nu^{2} + 1280 ) / 448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\nu^{15} + 20\nu^{13} + 18\nu^{11} + 72\nu^{9} - 4\nu^{7} + 192\nu^{5} + 1152\nu^{3} + 1664\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\nu^{14} + 20\nu^{12} + 18\nu^{10} + 72\nu^{8} - 4\nu^{6} + 192\nu^{4} + 1152\nu^{2} + 768 ) / 448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{14} - 9\nu^{12} - 6\nu^{10} - 38\nu^{8} - 36\nu^{6} - 36\nu^{4} - 384\nu^{2} - 480 ) / 112 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -11\nu^{15} - 24\nu^{13} - 2\nu^{11} - 64\nu^{9} - 124\nu^{7} - 208\nu^{5} - 800\nu^{3} - 1280\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -6\nu^{14} - 15\nu^{12} - 10\nu^{10} - 26\nu^{8} - 60\nu^{6} - 60\nu^{4} - 472\nu^{2} - 800 ) / 112 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65\nu^{15} + 124\nu^{13} + 78\nu^{11} + 312\nu^{9} + 580\nu^{7} + 832\nu^{5} + 4992\nu^{3} + 6912\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\nu^{15} + 39\nu^{13} + 26\nu^{11} + 90\nu^{9} + 156\nu^{7} + 156\nu^{5} + 1216\nu^{3} + 2080\nu ) / 224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{7} + \beta_{6} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + 2\beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - 2\beta_{12} + \beta_{9} - \beta_{6} + 2\beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{13} - 4\beta_{10} + 2\beta_{8} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{14} - 6\beta_{9} + 2\beta_{7} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4\beta_{13} - 6\beta_{11} - 2\beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6\beta_{15} - 12\beta_{14} - 12\beta_{12} - 6\beta_{7} + 6\beta_{6} + 4\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -12\beta_{13} - 12\beta_{8} - 12\beta_{5} + 4\beta_{4} + 12\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4\beta_{15} + 12\beta_{12} - 4\beta_{9} - 4\beta_{6} + 16\beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8\beta_{13} + 12\beta_{10} - 8\beta_{8} - 20\beta_{2} - 52 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -8\beta_{14} - 4\beta_{9} - 28\beta_{7} - 60\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -16\beta_{13} + 16\beta_{11} - 40\beta_{5} - 104\beta_{4} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 56\beta_{15} - 8\beta_{14} - 8\beta_{12} + 88\beta_{7} - 88\beta_{6} - 64\beta_{3} + 72\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
−1.33546 − 0.465333i
−1.01214 − 0.987711i
−0.349313 − 1.37039i
−0.264742 + 1.38921i
0.264742 − 1.38921i
0.349313 + 1.37039i
1.01214 + 0.987711i
1.33546 + 0.465333i
−1.33546 + 0.465333i
−1.01214 + 0.987711i
−0.349313 + 1.37039i
−0.264742 − 1.38921i
0.264742 + 1.38921i
0.349313 − 1.37039i
1.01214 − 0.987711i
1.33546 − 0.465333i
−1.33546 + 0.465333i 0 1.56693 − 1.24287i 0.937379 + 0.541196i 0 −1.62132 − 2.09077i −1.51423 + 2.38896i 0 −1.50367 − 0.286555i
37.2 −1.01214 + 0.987711i 0 0.0488544 − 1.99940i 2.26303 + 1.30656i 0 2.62132 + 0.358719i 1.92538 + 2.07193i 0 −3.58101 + 0.912798i
37.3 −0.349313 + 1.37039i 0 −1.75596 − 0.957392i −2.26303 − 1.30656i 0 2.62132 + 0.358719i 1.92538 − 2.07193i 0 2.58101 − 2.64485i
37.4 −0.264742 − 1.38921i 0 −1.85982 + 0.735566i 0.937379 + 0.541196i 0 −1.62132 − 2.09077i 1.51423 + 2.38896i 0 0.503673 − 1.44550i
37.5 0.264742 + 1.38921i 0 −1.85982 + 0.735566i −0.937379 − 0.541196i 0 −1.62132 − 2.09077i −1.51423 − 2.38896i 0 0.503673 − 1.44550i
37.6 0.349313 − 1.37039i 0 −1.75596 − 0.957392i 2.26303 + 1.30656i 0 2.62132 + 0.358719i −1.92538 + 2.07193i 0 2.58101 − 2.64485i
37.7 1.01214 − 0.987711i 0 0.0488544 − 1.99940i −2.26303 − 1.30656i 0 2.62132 + 0.358719i −1.92538 − 2.07193i 0 −3.58101 + 0.912798i
37.8 1.33546 − 0.465333i 0 1.56693 − 1.24287i −0.937379 − 0.541196i 0 −1.62132 − 2.09077i 1.51423 − 2.38896i 0 −1.50367 − 0.286555i
109.1 −1.33546 − 0.465333i 0 1.56693 + 1.24287i 0.937379 − 0.541196i 0 −1.62132 + 2.09077i −1.51423 − 2.38896i 0 −1.50367 + 0.286555i
109.2 −1.01214 − 0.987711i 0 0.0488544 + 1.99940i 2.26303 − 1.30656i 0 2.62132 − 0.358719i 1.92538 − 2.07193i 0 −3.58101 − 0.912798i
109.3 −0.349313 − 1.37039i 0 −1.75596 + 0.957392i −2.26303 + 1.30656i 0 2.62132 − 0.358719i 1.92538 + 2.07193i 0 2.58101 + 2.64485i
109.4 −0.264742 + 1.38921i 0 −1.85982 − 0.735566i 0.937379 − 0.541196i 0 −1.62132 + 2.09077i 1.51423 − 2.38896i 0 0.503673 + 1.44550i
109.5 0.264742 − 1.38921i 0 −1.85982 − 0.735566i −0.937379 + 0.541196i 0 −1.62132 + 2.09077i −1.51423 + 2.38896i 0 0.503673 + 1.44550i
109.6 0.349313 + 1.37039i 0 −1.75596 + 0.957392i 2.26303 − 1.30656i 0 2.62132 − 0.358719i −1.92538 − 2.07193i 0 2.58101 + 2.64485i
109.7 1.01214 + 0.987711i 0 0.0488544 + 1.99940i −2.26303 + 1.30656i 0 2.62132 − 0.358719i −1.92538 + 2.07193i 0 −3.58101 − 0.912798i
109.8 1.33546 + 0.465333i 0 1.56693 + 1.24287i −0.937379 + 0.541196i 0 −1.62132 + 2.09077i 1.51423 + 2.38896i 0 −1.50367 + 0.286555i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
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.d ✓ 16
3.b odd 2 1 inner 504.2.cj.d ✓ 16
4.b odd 2 1 2016.2.cr.d 16
7.c even 3 1 inner 504.2.cj.d ✓ 16
8.b even 2 1 inner 504.2.cj.d ✓ 16
8.d odd 2 1 2016.2.cr.d 16
12.b even 2 1 2016.2.cr.d 16
21.h odd 6 1 inner 504.2.cj.d ✓ 16
24.f even 2 1 2016.2.cr.d 16
24.h odd 2 1 inner 504.2.cj.d ✓ 16
28.g odd 6 1 2016.2.cr.d 16
56.k odd 6 1 2016.2.cr.d 16
56.p even 6 1 inner 504.2.cj.d ✓ 16
84.n even 6 1 2016.2.cr.d 16
168.s odd 6 1 inner 504.2.cj.d ✓ 16
168.v even 6 1 2016.2.cr.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.d ✓ 16 1.a even 1 1 trivial
504.2.cj.d ✓ 16 3.b odd 2 1 inner
504.2.cj.d ✓ 16 7.c even 3 1 inner
504.2.cj.d ✓ 16 8.b even 2 1 inner
504.2.cj.d ✓ 16 21.h odd 6 1 inner
504.2.cj.d ✓ 16 24.h odd 2 1 inner
504.2.cj.d ✓ 16 56.p even 6 1 inner
504.2.cj.d ✓ 16 168.s odd 6 1 inner
2016.2.cr.d 16 4.b odd 2 1
2016.2.cr.d 16 8.d odd 2 1
2016.2.cr.d 16 12.b even 2 1
2016.2.cr.d 16 24.f even 2 1
2016.2.cr.d 16 28.g odd 6 1
2016.2.cr.d 16 56.k odd 6 1
2016.2.cr.d 16 84.n even 6 1
2016.2.cr.d 16 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 8T_{5}^{6} + 56T_{5}^{4} - 64T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 8 T^{6} + 56 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 3 T^{2} + \cdots + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 32 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 62 T^{2} + 833)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 56 T^{6} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 10 T^{6} + \cdots + 289)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 24 T^{6} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{2} + 512)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots + 529)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 142 T^{6} + \cdots + 24137569)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 96 T^{2} + 2176)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 122 T^{2} + 833)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 216 T^{6} + \cdots + 121352256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 104 T^{6} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 136 T^{6} + \cdots + 17909824)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 80 T^{6} + \cdots + 1183744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 170 T^{6} + \cdots + 24137569)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 248 T^{2} + 6664)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3 T + 9)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 81)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 200 T^{2} + 7688)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 96 T^{6} + \cdots + 4734976)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 4)^{8} \) Copy content Toggle raw display
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