Properties

Label 4.24.a.a
Level $4$
Weight $24$
Character orbit 4.a
Self dual yes
Analytic conductor $13.408$
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] = [4,24,Mod(1,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(13.4081614938\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 618312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(\beta = 192\sqrt{2473249}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 85260) q^{3} + (540 \beta - 46133010) q^{5} + ( - 2106 \beta + 96041720) q^{7} + ( - 170520 \beta + 4299939909) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 85260) q^{3} + (540 \beta - 46133010) q^{5} + ( - 2106 \beta + 96041720) q^{7} + ( - 170520 \beta + 4299939909) q^{9} + (2983365 \beta - 623587347900) q^{11} + ( - 22978404 \beta - 3730149990490) q^{13} + (92173410 \beta - 53167180046040) q^{15} + ( - 148205592 \beta - 187448673951630) q^{17} + ( - 153252405 \beta - 420180139606276) q^{19} + ( - 275599280 \beta + 200200647539616) q^{21} + (11230584498 \beta + 32\!\cdots\!60) q^{23}+ \cdots + (11\!\cdots\!85 \beta - 49\!\cdots\!00) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9} - 1247174695800 q^{11} - 7460299980980 q^{13} - 106334360092080 q^{15} - 374897347903260 q^{17} - 840360279212552 q^{19} + 400401295079232 q^{21} + 64\!\cdots\!20 q^{23}+ \cdots - 98\!\cdots\!00 q^{99}+O(q^{100}) \) 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
786.828
−785.828
0 −216690. 0 1.16920e8 0 −5.39865e8 0 −4.71886e10 0
1.2 0 387210. 0 −2.09186e8 0 7.31949e8 0 5.57885e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4.24.a.a 2
3.b odd 2 1 36.24.a.e 2
4.b odd 2 1 16.24.a.c 2
8.b even 2 1 64.24.a.e 2
8.d odd 2 1 64.24.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.24.a.a 2 1.a even 1 1 trivial
16.24.a.c 2 4.b odd 2 1
36.24.a.e 2 3.b odd 2 1
64.24.a.e 2 8.b even 2 1
64.24.a.f 2 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 170520 T - 83904583536 \) Copy content Toggle raw display
$5$ \( T^{2} + 92266020 T - 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} - 192083440 T - 39\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{2} + 1247174695800 T - 42\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{2} + 7460299980980 T - 34\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + 374897347903260 T + 33\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} + 840360279212552 T + 17\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 31\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 30\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 67\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 31\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 69\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 41\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
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