Properties

Label 504.4.a.i
Level $504$
Weight $4$
Character orbit 504.a
Self dual yes
Analytic conductor $29.737$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,4,Mod(1,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

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: yes
Analytic conductor: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 11) q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 11) q^{5} + 7 q^{7} + ( - 6 \beta - 18) q^{11} + ( - 5 \beta + 21) q^{13} + (6 \beta - 4) q^{17} + (13 \beta - 59) q^{19} + ( - 4 \beta + 52) q^{23} + (22 \beta + 53) q^{25} + ( - 22 \beta + 28) q^{29} + ( - 14 \beta + 10) q^{31} + ( - 7 \beta - 77) q^{35} + ( - 10 \beta + 252) q^{37} + (18 \beta + 272) q^{41} + ( - 14 \beta + 206) q^{43} + ( - 14 \beta + 250) q^{47} + 49 q^{49} + (72 \beta - 134) q^{53} + (84 \beta + 540) q^{55} + ( - 13 \beta + 99) q^{59} + ( - 31 \beta - 173) q^{61} + (34 \beta + 54) q^{65} + ( - 68 \beta + 504) q^{67} + (44 \beta + 612) q^{71} + ( - 36 \beta + 358) q^{73} + ( - 42 \beta - 126) q^{77} + (36 \beta + 292) q^{79} + (47 \beta + 615) q^{83} + ( - 62 \beta - 298) q^{85} + ( - 160 \beta - 298) q^{89} + ( - 35 \beta + 147) q^{91} + ( - 84 \beta - 92) q^{95} + ( - 70 \beta - 428) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 22 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 22 q^{5} + 14 q^{7} - 36 q^{11} + 42 q^{13} - 8 q^{17} - 118 q^{19} + 104 q^{23} + 106 q^{25} + 56 q^{29} + 20 q^{31} - 154 q^{35} + 504 q^{37} + 544 q^{41} + 412 q^{43} + 500 q^{47} + 98 q^{49} - 268 q^{53} + 1080 q^{55} + 198 q^{59} - 346 q^{61} + 108 q^{65} + 1008 q^{67} + 1224 q^{71} + 716 q^{73} - 252 q^{77} + 584 q^{79} + 1230 q^{83} - 596 q^{85} - 596 q^{89} + 294 q^{91} - 184 q^{95} - 856 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.27492
−3.27492
0 0 0 −18.5498 0 7.00000 0 0 0
1.2 0 0 0 −3.45017 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.4.a.i 2
3.b odd 2 1 56.4.a.c 2
4.b odd 2 1 1008.4.a.x 2
12.b even 2 1 112.4.a.h 2
15.d odd 2 1 1400.4.a.i 2
15.e even 4 2 1400.4.g.h 4
21.c even 2 1 392.4.a.h 2
21.g even 6 2 392.4.i.i 4
21.h odd 6 2 392.4.i.l 4
24.f even 2 1 448.4.a.r 2
24.h odd 2 1 448.4.a.s 2
84.h odd 2 1 784.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.c 2 3.b odd 2 1
112.4.a.h 2 12.b even 2 1
392.4.a.h 2 21.c even 2 1
392.4.i.i 4 21.g even 6 2
392.4.i.l 4 21.h odd 6 2
448.4.a.r 2 24.f even 2 1
448.4.a.s 2 24.h odd 2 1
504.4.a.i 2 1.a even 1 1 trivial
784.4.a.t 2 84.h odd 2 1
1008.4.a.x 2 4.b odd 2 1
1400.4.a.i 2 15.d odd 2 1
1400.4.g.h 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5}^{2} + 22T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 36T_{11} - 1728 \) 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
$5$ \( T^{2} + 22T + 64 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36T - 1728 \) Copy content Toggle raw display
$13$ \( T^{2} - 42T - 984 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T - 2036 \) Copy content Toggle raw display
$19$ \( T^{2} + 118T - 6152 \) Copy content Toggle raw display
$23$ \( T^{2} - 104T + 1792 \) Copy content Toggle raw display
$29$ \( T^{2} - 56T - 26804 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 11072 \) Copy content Toggle raw display
$37$ \( T^{2} - 504T + 57804 \) Copy content Toggle raw display
$41$ \( T^{2} - 544T + 55516 \) Copy content Toggle raw display
$43$ \( T^{2} - 412T + 31264 \) Copy content Toggle raw display
$47$ \( T^{2} - 500T + 51328 \) Copy content Toggle raw display
$53$ \( T^{2} + 268T - 277532 \) Copy content Toggle raw display
$59$ \( T^{2} - 198T + 168 \) Copy content Toggle raw display
$61$ \( T^{2} + 346T - 24848 \) Copy content Toggle raw display
$67$ \( T^{2} - 1008T - 9552 \) Copy content Toggle raw display
$71$ \( T^{2} - 1224 T + 264192 \) Copy content Toggle raw display
$73$ \( T^{2} - 716T + 54292 \) Copy content Toggle raw display
$79$ \( T^{2} - 584T + 11392 \) Copy content Toggle raw display
$83$ \( T^{2} - 1230 T + 252312 \) Copy content Toggle raw display
$89$ \( T^{2} + 596 T - 1370396 \) Copy content Toggle raw display
$97$ \( T^{2} + 856T - 96116 \) Copy content Toggle raw display
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