Properties

Label 576.8.a.bb
Level $576$
Weight $8$
Character orbit 576.a
Self dual yes
Analytic conductor $179.934$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,8,Mod(1,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(179.933774679\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{435}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 435 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 288)
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 \(\beta = 48\sqrt{435}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 140 q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 140 q^{5} + \beta q^{7} + 4 \beta q^{11} + 2238 q^{13} + 1144 q^{17} - 14 \beta q^{19} - 88 \beta q^{23} - 58525 q^{25} + 41324 q^{29} - 183 \beta q^{31} - 140 \beta q^{35} + 137594 q^{37} + 123848 q^{41} - 638 \beta q^{43} + 776 \beta q^{47} + 178697 q^{49} + 981068 q^{53} - 560 \beta q^{55} + 568 \beta q^{59} - 1004750 q^{61} - 313320 q^{65} + 1084 \beta q^{67} - 2336 \beta q^{71} + 603798 q^{73} + 4008960 q^{77} + 1833 \beta q^{79} + 2284 \beta q^{83} - 160160 q^{85} - 6185104 q^{89} + 2238 \beta q^{91} + 1960 \beta q^{95} - 6619986 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 280 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 280 q^{5} + 4476 q^{13} + 2288 q^{17} - 117050 q^{25} + 82648 q^{29} + 275188 q^{37} + 247696 q^{41} + 357394 q^{49} + 1962136 q^{53} - 2009500 q^{61} - 626640 q^{65} + 1207596 q^{73} + 8017920 q^{77} - 320320 q^{85} - 12370208 q^{89} - 13239972 q^{97}+O(q^{100}) \) 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
−20.8567
20.8567
0 0 0 −140.000 0 −1001.12 0 0 0
1.2 0 0 0 −140.000 0 1001.12 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.8.a.bb 2
3.b odd 2 1 576.8.a.br 2
4.b odd 2 1 inner 576.8.a.bb 2
8.b even 2 1 288.8.a.p yes 2
8.d odd 2 1 288.8.a.p yes 2
12.b even 2 1 576.8.a.br 2
24.f even 2 1 288.8.a.f 2
24.h odd 2 1 288.8.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.8.a.f 2 24.f even 2 1
288.8.a.f 2 24.h odd 2 1
288.8.a.p yes 2 8.b even 2 1
288.8.a.p yes 2 8.d odd 2 1
576.8.a.bb 2 1.a even 1 1 trivial
576.8.a.bb 2 4.b odd 2 1 inner
576.8.a.br 2 3.b odd 2 1
576.8.a.br 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} + 140 \) Copy content Toggle raw display
\( T_{7}^{2} - 1002240 \) Copy content Toggle raw display
\( T_{11}^{2} - 16035840 \) 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
$5$ \( (T + 140)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1002240 \) Copy content Toggle raw display
$11$ \( T^{2} - 16035840 \) Copy content Toggle raw display
$13$ \( (T - 2238)^{2} \) Copy content Toggle raw display
$17$ \( (T - 1144)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 196439040 \) Copy content Toggle raw display
$23$ \( T^{2} - 7761346560 \) Copy content Toggle raw display
$29$ \( (T - 41324)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 33564015360 \) Copy content Toggle raw display
$37$ \( (T - 137594)^{2} \) Copy content Toggle raw display
$41$ \( (T - 123848)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 407955778560 \) Copy content Toggle raw display
$47$ \( T^{2} - 603524874240 \) Copy content Toggle raw display
$53$ \( (T - 981068)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 323346677760 \) Copy content Toggle raw display
$61$ \( (T + 1004750)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 1177688125440 \) Copy content Toggle raw display
$71$ \( T^{2} - 5469119447040 \) Copy content Toggle raw display
$73$ \( (T - 603798)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 3367415151360 \) Copy content Toggle raw display
$83$ \( T^{2} - 5228341309440 \) Copy content Toggle raw display
$89$ \( (T + 6185104)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6619986)^{2} \) Copy content Toggle raw display
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