Properties

Label 504.3.by.c
Level $504$
Weight $3$
Character orbit 504.by
Analytic conductor $13.733$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,3,Mod(73,504)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.by (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: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1 + 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1 + 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 6 \beta_1 + 5) q^{11} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_1 - 1) q^{13} + ( - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 2) q^{17} + ( - 5 \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 6) q^{19} + ( - 2 \beta_{7} - \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - 10 \beta_1 - 1) q^{23} + (\beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_1 + 10) q^{25} + ( - 3 \beta_{5} + \beta_{4} - 5 \beta_{2} - 2 \beta_1 - 9) q^{29} + ( - 4 \beta_{7} - 3 \beta_{5} + 6 \beta_{3} - 7 \beta_1 - 3) q^{31} + ( - 2 \beta_{7} + \beta_{6} + 15 \beta_{5} - \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + \cdots + 4) q^{35}+ \cdots + ( - 7 \beta_{7} - 7 \beta_{6} - 10 \beta_{5} - 12 \beta_{4} + 7 \beta_{3} + \cdots - 46) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} + 8 q^{7} + 22 q^{11} - 36 q^{17} + 42 q^{19} - 48 q^{23} + 42 q^{25} - 68 q^{29} - 60 q^{31} + 12 q^{35} - 118 q^{37} - 92 q^{43} + 12 q^{47} - 20 q^{49} - 10 q^{53} + 54 q^{59} + 24 q^{61} + 148 q^{65} + 22 q^{67} + 392 q^{71} - 138 q^{73} + 126 q^{77} + 164 q^{79} + 200 q^{85} + 60 q^{89} + 90 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} - 193196 \nu - 1034839 ) / 658490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3371 \nu^{7} + 6667 \nu^{6} - 38294 \nu^{5} - 96948 \nu^{4} + 94716 \nu^{3} + 623641 \nu^{2} + 796111 \nu - 1188686 ) / 329245 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6088 \nu^{7} - 4954 \nu^{6} - 27942 \nu^{5} - 30829 \nu^{4} + 324398 \nu^{3} - 31292 \nu^{2} + 378248 \nu - 816983 ) / 329245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6546 \nu^{7} - 1812 \nu^{6} + 70064 \nu^{5} + 173218 \nu^{4} - 443336 \nu^{3} - 974856 \nu^{2} + 215444 \nu + 3514676 ) / 329245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11533 \nu^{7} + 30909 \nu^{6} + 121832 \nu^{5} - 29696 \nu^{4} - 997968 \nu^{3} - 162453 \nu^{2} + 2146717 \nu + 2603393 ) / 329245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45244 \nu^{7} - 5583 \nu^{6} + 215876 \nu^{5} + 916372 \nu^{4} - 1365974 \nu^{3} - 3003654 \nu^{2} - 3864944 \nu + 10829159 ) / 658490 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} - 4\beta _1 + 6 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} - 2\beta_{4} - 5\beta_{3} + 3\beta_{2} + 32\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 7\beta_{6} - 18\beta_{5} + 27\beta_{4} - 6\beta_{3} - 9\beta_{2} + 9\beta _1 + 81 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{7} + 33\beta_{5} + 10\beta_{4} - 10\beta_{3} + 20\beta_{2} + 141\beta _1 - 131 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{6} + 75\beta_{5} + 66\beta_{4} - 108\beta_{3} - 99\beta_{2} + 736\beta _1 - 33 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 91\beta_{7} + 91\beta_{6} - 150\beta_{5} + 225\beta_{4} + 160\beta_{3} - 75\beta_{2} + 75\beta _1 - 494 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} + 199\beta_{5} - 8\beta_{4} + 8\beta_{3} - 16\beta_{2} + 718\beta _1 - 726 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−1.33172 1.34622i
2.40015 + 0.808379i
−1.90015 1.67440i
1.83172 + 0.480194i
−1.33172 + 1.34622i
2.40015 0.808379i
−1.90015 + 1.67440i
1.83172 0.480194i
0 0 0 −5.30550 + 3.06313i 0 0.664986 + 6.96834i 0 0 0
73.2 0 0 0 −3.18140 + 1.83678i 0 2.47188 6.54903i 0 0 0
73.3 0 0 0 4.68140 2.70281i 0 −6.12873 + 3.38210i 0 0 0
73.4 0 0 0 6.80550 3.92916i 0 6.99187 0.337312i 0 0 0
145.1 0 0 0 −5.30550 3.06313i 0 0.664986 6.96834i 0 0 0
145.2 0 0 0 −3.18140 1.83678i 0 2.47188 + 6.54903i 0 0 0
145.3 0 0 0 4.68140 + 2.70281i 0 −6.12873 3.38210i 0 0 0
145.4 0 0 0 6.80550 + 3.92916i 0 6.99187 + 0.337312i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.3.by.c 8
3.b odd 2 1 168.3.z.b 8
4.b odd 2 1 1008.3.cg.p 8
7.c even 3 1 3528.3.f.b 8
7.d odd 6 1 inner 504.3.by.c 8
7.d odd 6 1 3528.3.f.b 8
12.b even 2 1 336.3.bh.g 8
21.c even 2 1 1176.3.z.c 8
21.g even 6 1 168.3.z.b 8
21.g even 6 1 1176.3.f.c 8
21.h odd 6 1 1176.3.f.c 8
21.h odd 6 1 1176.3.z.c 8
28.f even 6 1 1008.3.cg.p 8
84.j odd 6 1 336.3.bh.g 8
84.j odd 6 1 2352.3.f.g 8
84.n even 6 1 2352.3.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b 8 3.b odd 2 1
168.3.z.b 8 21.g even 6 1
336.3.bh.g 8 12.b even 2 1
336.3.bh.g 8 84.j odd 6 1
504.3.by.c 8 1.a even 1 1 trivial
504.3.by.c 8 7.d odd 6 1 inner
1008.3.cg.p 8 4.b odd 2 1
1008.3.cg.p 8 28.f even 6 1
1176.3.f.c 8 21.g even 6 1
1176.3.f.c 8 21.h odd 6 1
1176.3.z.c 8 21.c even 2 1
1176.3.z.c 8 21.h odd 6 1
2352.3.f.g 8 84.j odd 6 1
2352.3.f.g 8 84.n even 6 1
3528.3.f.b 8 7.c even 3 1
3528.3.f.b 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 6T_{5}^{7} - 53T_{5}^{6} + 390T_{5}^{5} + 2861T_{5}^{4} - 13260T_{5}^{3} - 48268T_{5}^{2} + 195024T_{5} + 913936 \) acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} - 53 T^{6} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + 42 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{7} + 527 T^{6} + \cdots + 17875984 \) Copy content Toggle raw display
$13$ \( T^{8} + 262 T^{6} + 19817 T^{4} + \cdots + 2408704 \) Copy content Toggle raw display
$17$ \( T^{8} + 36 T^{7} + \cdots + 3470623744 \) Copy content Toggle raw display
$19$ \( T^{8} - 42 T^{7} + 43 T^{6} + \cdots + 17272336 \) Copy content Toggle raw display
$23$ \( T^{8} + 48 T^{7} + \cdots + 22620160000 \) Copy content Toggle raw display
$29$ \( (T^{4} + 34 T^{3} - 1063 T^{2} + \cdots + 224128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 60 T^{7} + \cdots + 25912950625 \) Copy content Toggle raw display
$37$ \( T^{8} + 118 T^{7} + \cdots + 17069945104 \) Copy content Toggle raw display
$41$ \( T^{8} + 7280 T^{6} + \cdots + 580010189056 \) Copy content Toggle raw display
$43$ \( (T^{4} + 46 T^{3} - 3723 T^{2} + \cdots + 1658308)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 52408029184 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 1491466217536 \) Copy content Toggle raw display
$59$ \( T^{8} - 54 T^{7} + \cdots + 1048985640000 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + \cdots + 43785853599744 \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{7} + \cdots + 95387087104 \) Copy content Toggle raw display
$71$ \( (T^{4} - 196 T^{3} + 2460 T^{2} + \cdots - 13209344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 622497709609216 \) Copy content Toggle raw display
$79$ \( T^{8} - 164 T^{7} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + 4430 T^{6} + \cdots + 839297841424 \) Copy content Toggle raw display
$89$ \( T^{8} - 60 T^{7} + \cdots + 723343446016 \) Copy content Toggle raw display
$97$ \( T^{8} + 65270 T^{6} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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