Properties

Label 504.4.s.e
Level $504$
Weight $4$
Character orbit 504.s
Analytic conductor $29.737$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,4,Mod(289,504)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), 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: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 7 \zeta_{6} + 7) q^{5} + ( - 7 \zeta_{6} - 14) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 7 \zeta_{6} + 7) q^{5} + ( - 7 \zeta_{6} - 14) q^{7} + 7 \zeta_{6} q^{11} - 52 q^{13} + 72 \zeta_{6} q^{17} + (20 \zeta_{6} - 20) q^{19} + (48 \zeta_{6} - 48) q^{23} + 76 \zeta_{6} q^{25} + 243 q^{29} - 95 \zeta_{6} q^{31} + (98 \zeta_{6} - 147) q^{35} + (352 \zeta_{6} - 352) q^{37} + 296 q^{41} + 158 q^{43} + (142 \zeta_{6} - 142) q^{47} + (245 \zeta_{6} + 147) q^{49} - 375 \zeta_{6} q^{53} + 49 q^{55} + 279 \zeta_{6} q^{59} + (246 \zeta_{6} - 246) q^{61} + (364 \zeta_{6} - 364) q^{65} + 730 \zeta_{6} q^{67} - 338 q^{71} + 542 \zeta_{6} q^{73} + ( - 147 \zeta_{6} + 49) q^{77} + ( - 305 \zeta_{6} + 305) q^{79} - 1123 q^{83} + 504 q^{85} + (426 \zeta_{6} - 426) q^{89} + (364 \zeta_{6} + 728) q^{91} + 140 \zeta_{6} q^{95} - 369 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{5} - 35 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{5} - 35 q^{7} + 7 q^{11} - 104 q^{13} + 72 q^{17} - 20 q^{19} - 48 q^{23} + 76 q^{25} + 486 q^{29} - 95 q^{31} - 196 q^{35} - 352 q^{37} + 592 q^{41} + 316 q^{43} - 142 q^{47} + 539 q^{49} - 375 q^{53} + 98 q^{55} + 279 q^{59} - 246 q^{61} - 364 q^{65} + 730 q^{67} - 676 q^{71} + 542 q^{73} - 49 q^{77} + 305 q^{79} - 2246 q^{83} + 1008 q^{85} - 426 q^{89} + 1820 q^{91} + 140 q^{95} - 738 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.50000 6.06218i 0 −17.5000 6.06218i 0 0 0
361.1 0 0 0 3.50000 + 6.06218i 0 −17.5000 + 6.06218i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.4.s.e 2
3.b odd 2 1 168.4.q.a 2
7.c even 3 1 inner 504.4.s.e 2
12.b even 2 1 336.4.q.a 2
21.g even 6 1 1176.4.a.i 1
21.h odd 6 1 168.4.q.a 2
21.h odd 6 1 1176.4.a.f 1
84.j odd 6 1 2352.4.a.c 1
84.n even 6 1 336.4.q.a 2
84.n even 6 1 2352.4.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.a 2 3.b odd 2 1
168.4.q.a 2 21.h odd 6 1
336.4.q.a 2 12.b even 2 1
336.4.q.a 2 84.n even 6 1
504.4.s.e 2 1.a even 1 1 trivial
504.4.s.e 2 7.c even 3 1 inner
1176.4.a.f 1 21.h odd 6 1
1176.4.a.i 1 21.g even 6 1
2352.4.a.c 1 84.j odd 6 1
2352.4.a.bg 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 7T_{5} + 49 \) acting on \(S_{4}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 35T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$13$ \( (T + 52)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$19$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 48T + 2304 \) Copy content Toggle raw display
$29$ \( (T - 243)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 95T + 9025 \) Copy content Toggle raw display
$37$ \( T^{2} + 352T + 123904 \) Copy content Toggle raw display
$41$ \( (T - 296)^{2} \) Copy content Toggle raw display
$43$ \( (T - 158)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 142T + 20164 \) Copy content Toggle raw display
$53$ \( T^{2} + 375T + 140625 \) Copy content Toggle raw display
$59$ \( T^{2} - 279T + 77841 \) Copy content Toggle raw display
$61$ \( T^{2} + 246T + 60516 \) Copy content Toggle raw display
$67$ \( T^{2} - 730T + 532900 \) Copy content Toggle raw display
$71$ \( (T + 338)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 542T + 293764 \) Copy content Toggle raw display
$79$ \( T^{2} - 305T + 93025 \) Copy content Toggle raw display
$83$ \( (T + 1123)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 426T + 181476 \) Copy content Toggle raw display
$97$ \( (T + 369)^{2} \) Copy content Toggle raw display
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