Properties

Label 504.4.s.a
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + (11 \zeta_{6} - 11) q^{5} + ( - 21 \zeta_{6} + 14) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (11 \zeta_{6} - 11) q^{5} + ( - 21 \zeta_{6} + 14) q^{7} + 39 \zeta_{6} q^{11} - 32 q^{13} + 12 \zeta_{6} q^{17} + ( - 88 \zeta_{6} + 88) q^{19} + (92 \zeta_{6} - 92) q^{23} + 4 \zeta_{6} q^{25} - 255 q^{29} + 35 \zeta_{6} q^{31} + (154 \zeta_{6} + 77) q^{35} + ( - 4 \zeta_{6} + 4) q^{37} - 16 q^{41} - 330 q^{43} + (298 \zeta_{6} - 298) q^{47} + ( - 147 \zeta_{6} - 245) q^{49} - 717 \zeta_{6} q^{53} - 429 q^{55} - 217 \zeta_{6} q^{59} + (386 \zeta_{6} - 386) q^{61} + ( - 352 \zeta_{6} + 352) q^{65} - 906 \zeta_{6} q^{67} + 34 q^{71} + 838 \zeta_{6} q^{73} + ( - 273 \zeta_{6} + 819) q^{77} + (1325 \zeta_{6} - 1325) q^{79} - 1163 q^{83} - 132 q^{85} + (54 \zeta_{6} - 54) q^{89} + (672 \zeta_{6} - 448) q^{91} + 968 \zeta_{6} q^{95} + 7 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 11 q^{5} + 7 q^{7} + 39 q^{11} - 64 q^{13} + 12 q^{17} + 88 q^{19} - 92 q^{23} + 4 q^{25} - 510 q^{29} + 35 q^{31} + 308 q^{35} + 4 q^{37} - 32 q^{41} - 660 q^{43} - 298 q^{47} - 637 q^{49} - 717 q^{53} - 858 q^{55} - 217 q^{59} - 386 q^{61} + 352 q^{65} - 906 q^{67} + 68 q^{71} + 838 q^{73} + 1365 q^{77} - 1325 q^{79} - 2326 q^{83} - 264 q^{85} - 54 q^{89} - 224 q^{91} + 968 q^{95} + 14 q^{97}+O(q^{100}) \) 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 −5.50000 + 9.52628i 0 3.50000 18.1865i 0 0 0
361.1 0 0 0 −5.50000 9.52628i 0 3.50000 + 18.1865i 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.a 2
3.b odd 2 1 168.4.q.c 2
7.c even 3 1 inner 504.4.s.a 2
12.b even 2 1 336.4.q.c 2
21.g even 6 1 1176.4.a.n 1
21.h odd 6 1 168.4.q.c 2
21.h odd 6 1 1176.4.a.c 1
84.j odd 6 1 2352.4.a.o 1
84.n even 6 1 336.4.q.c 2
84.n even 6 1 2352.4.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.c 2 3.b odd 2 1
168.4.q.c 2 21.h odd 6 1
336.4.q.c 2 12.b even 2 1
336.4.q.c 2 84.n even 6 1
504.4.s.a 2 1.a even 1 1 trivial
504.4.s.a 2 7.c even 3 1 inner
1176.4.a.c 1 21.h odd 6 1
1176.4.a.n 1 21.g even 6 1
2352.4.a.o 1 84.j odd 6 1
2352.4.a.y 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 11T_{5} + 121 \) 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
$5$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 39T + 1521 \) Copy content Toggle raw display
$13$ \( (T + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$19$ \( T^{2} - 88T + 7744 \) Copy content Toggle raw display
$23$ \( T^{2} + 92T + 8464 \) Copy content Toggle raw display
$29$ \( (T + 255)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$41$ \( (T + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T + 330)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 298T + 88804 \) Copy content Toggle raw display
$53$ \( T^{2} + 717T + 514089 \) Copy content Toggle raw display
$59$ \( T^{2} + 217T + 47089 \) Copy content Toggle raw display
$61$ \( T^{2} + 386T + 148996 \) Copy content Toggle raw display
$67$ \( T^{2} + 906T + 820836 \) Copy content Toggle raw display
$71$ \( (T - 34)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 838T + 702244 \) Copy content Toggle raw display
$79$ \( T^{2} + 1325 T + 1755625 \) Copy content Toggle raw display
$83$ \( (T + 1163)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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