Properties

Label 1.52.a.a
Level $1$
Weight $52$
Character orbit 1.a
Self dual yes
Analytic conductor $16.473$
Analytic rank $0$
Dimension $4$
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] = [1,52,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 52, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 52);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 52 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(16.4731353414\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 495735060514x^{2} - 23954614981416598x + 48979992255622025570313 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{23}\cdot 3^{10}\cdot 5^{3}\cdot 7^{2}\cdot 13\cdot 17 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 8189010) q^{2} + ( - \beta_{2} - 6677 \beta_1 + 100965943260) q^{3} + ( - \beta_{3} + 511 \beta_{2} - 7825327 \beta_1 + 19\!\cdots\!28) q^{4}+ \cdots + ( - 386651232 \beta_{3} - 666471696408 \beta_{2} + \cdots + 12\!\cdots\!57) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 8189010) q^{2} + ( - \beta_{2} - 6677 \beta_1 + 100965943260) q^{3} + ( - \beta_{3} + 511 \beta_{2} - 7825327 \beta_1 + 19\!\cdots\!28) q^{4}+ \cdots + (90\!\cdots\!56 \beta_{3} + \cdots - 42\!\cdots\!16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32756040 q^{2} + 403863773040 q^{3} + 79\!\cdots\!12 q^{4}+ \cdots + 50\!\cdots\!28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32756040 q^{2} + 403863773040 q^{3} + 79\!\cdots\!12 q^{4}+ \cdots - 17\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 495735060514x^{2} - 23954614981416598x + 48979992255622025570313 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -252006\nu^{3} + 93063836718\nu^{2} + 70872137420108946\nu - 18539821380123502730388 ) / 77619756584545 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3755577054 \nu^{3} + \cdots + 52\!\cdots\!92 ) / 77619756584545 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 334830962412 \nu^{3} + \cdots - 85\!\cdots\!76 ) / 11088536654935 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 31\beta_{3} + 53711\beta_{2} + 512120665\beta _1 + 134730086400 ) / 269460172800 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4857031\beta_{3} - 2148411769\beta_{2} - 77190687886655\beta _1 + 33395213767414954291200 ) / 134730086400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 251133001843 \beta_{3} + 275887056853763 \beta_{2} + \cdots + 98\!\cdots\!00 ) / 5499187200 \) 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
−457245.
644100.
−511801.
324949.
−8.71140e7 6.49653e11 5.33705e15 8.70295e17 −5.65939e19 3.14283e21 −2.68769e23 −1.73165e24 −7.58149e25
1.2 −1.27049e7 −5.15198e11 −2.09038e15 −3.83602e17 6.54557e18 −2.02239e21 5.51672e22 −1.88826e24 4.87365e24
1.3 5.13381e7 2.68083e12 3.83803e14 4.84667e17 1.37629e20 2.59004e21 −9.58994e22 5.03313e24 2.48819e25
1.4 8.12369e7 −2.41142e12 4.34763e15 2.42754e17 −1.95896e20 2.85586e21 1.70259e23 3.66124e24 1.97206e25
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.52.a.a 4
3.b odd 2 1 9.52.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.52.a.a 4 1.a even 1 1 trivial
9.52.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 32756040 T^{3} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{4} - 403863773040 T^{3} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 54\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 95\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 65\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 61\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 95\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 37\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 18\!\cdots\!44 \) Copy content Toggle raw display
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