Properties

Label 504.4.a.o
Level $504$
Weight $4$
Character orbit 504.a
Self dual yes
Analytic conductor $29.737$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 6) q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 6) q^{5} - 7 q^{7} + ( - \beta + 2) q^{11} + ( - 4 \beta - 30) q^{13} + ( - 9 \beta - 54) q^{17} + ( - 12 \beta - 4) q^{19} + (17 \beta - 26) q^{23} + (12 \beta - 1) q^{25} - 120 q^{29} + (20 \beta + 108) q^{31} + ( - 7 \beta - 42) q^{35} + ( - 20 \beta - 150) q^{37} + ( - 11 \beta - 306) q^{41} + (16 \beta + 196) q^{43} + ( - 10 \beta - 252) q^{47} + 49 q^{49} + (42 \beta - 12) q^{53} + ( - 4 \beta - 76) q^{55} + (10 \beta - 532) q^{59} + (68 \beta + 70) q^{61} + ( - 54 \beta - 532) q^{65} + ( - 8 \beta + 340) q^{67} + ( - 13 \beta - 270) q^{71} + ( - 8 \beta - 162) q^{73} + (7 \beta - 14) q^{77} + ( - 104 \beta - 168) q^{79} + (24 \beta - 176) q^{83} + ( - 108 \beta - 1116) q^{85} + (81 \beta - 426) q^{89} + (28 \beta + 210) q^{91} + ( - 76 \beta - 1080) q^{95} + ( - 8 \beta - 322) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{5} - 14 q^{7} + 4 q^{11} - 60 q^{13} - 108 q^{17} - 8 q^{19} - 52 q^{23} - 2 q^{25} - 240 q^{29} + 216 q^{31} - 84 q^{35} - 300 q^{37} - 612 q^{41} + 392 q^{43} - 504 q^{47} + 98 q^{49} - 24 q^{53} - 152 q^{55} - 1064 q^{59} + 140 q^{61} - 1064 q^{65} + 680 q^{67} - 540 q^{71} - 324 q^{73} - 28 q^{77} - 336 q^{79} - 352 q^{83} - 2232 q^{85} - 852 q^{89} + 420 q^{91} - 2160 q^{95} - 644 q^{97}+O(q^{100}) \) 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
−4.69042
4.69042
0 0 0 −3.38083 0 −7.00000 0 0 0
1.2 0 0 0 15.3808 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.o yes 2
3.b odd 2 1 504.4.a.k 2
4.b odd 2 1 1008.4.a.bh 2
12.b even 2 1 1008.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.4.a.k 2 3.b odd 2 1
504.4.a.o yes 2 1.a even 1 1 trivial
1008.4.a.z 2 12.b even 2 1
1008.4.a.bh 2 4.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} - 12T_{5} - 52 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 84 \) 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
$5$ \( T^{2} - 12T - 52 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 84 \) Copy content Toggle raw display
$13$ \( T^{2} + 60T - 508 \) Copy content Toggle raw display
$17$ \( T^{2} + 108T - 4212 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 12656 \) Copy content Toggle raw display
$23$ \( T^{2} + 52T - 24756 \) Copy content Toggle raw display
$29$ \( (T + 120)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 216T - 23536 \) Copy content Toggle raw display
$37$ \( T^{2} + 300T - 12700 \) Copy content Toggle raw display
$41$ \( T^{2} + 612T + 82988 \) Copy content Toggle raw display
$43$ \( T^{2} - 392T + 15888 \) Copy content Toggle raw display
$47$ \( T^{2} + 504T + 54704 \) Copy content Toggle raw display
$53$ \( T^{2} + 24T - 155088 \) Copy content Toggle raw display
$59$ \( T^{2} + 1064 T + 274224 \) Copy content Toggle raw display
$61$ \( T^{2} - 140T - 402012 \) Copy content Toggle raw display
$67$ \( T^{2} - 680T + 109968 \) Copy content Toggle raw display
$71$ \( T^{2} + 540T + 58028 \) Copy content Toggle raw display
$73$ \( T^{2} + 324T + 20612 \) Copy content Toggle raw display
$79$ \( T^{2} + 336T - 923584 \) Copy content Toggle raw display
$83$ \( T^{2} + 352T - 19712 \) Copy content Toggle raw display
$89$ \( T^{2} + 852T - 395892 \) Copy content Toggle raw display
$97$ \( T^{2} + 644T + 98052 \) Copy content Toggle raw display
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