Properties

Label 1.44.a.a
Level $1$
Weight $44$
Character orbit 1.a
Self dual yes
Analytic conductor $11.711$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,44,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, 44, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 44);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 44 \)
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: \(11.7110395346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11258260111x - 264759545317170 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5\cdot 7 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 736648) q^{2} + ( - \beta_{2} + 2135 \beta_1 + 8133812604) q^{3} + ( - 408 \beta_{2} + 810816 \beta_1 + 3274206140608) q^{4} + ( - 29844 \beta_{2} - 23658932 \beta_1 + 178401793591390) q^{5} + (2270208 \beta_{2} - 21343496316 \beta_1 - 30\!\cdots\!08) q^{6}+ \cdots + ( - 7904835576 \beta_{2} + \cdots + 15\!\cdots\!77) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 736648) q^{2} + ( - \beta_{2} + 2135 \beta_1 + 8133812604) q^{3} + ( - 408 \beta_{2} + 810816 \beta_1 + 3274206140608) q^{4} + ( - 29844 \beta_{2} - 23658932 \beta_1 + 178401793591390) q^{5} + (2270208 \beta_{2} - 21343496316 \beta_1 - 30\!\cdots\!08) q^{6}+ \cdots + (17\!\cdots\!13 \beta_{2} + \cdots - 47\!\cdots\!36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2209944 q^{2} + 24401437812 q^{3} + 9822618421824 q^{4} + 535205380774170 q^{5} - 91\!\cdots\!24 q^{6}+ \cdots + 47\!\cdots\!31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2209944 q^{2} + 24401437812 q^{3} + 9822618421824 q^{4} + 535205380774170 q^{5} - 91\!\cdots\!24 q^{6}+ \cdots - 14\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 11258260111x - 264759545317170 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{2} + 343074\nu + 45033040444 ) / 8369 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14838\nu^{2} + 1176206478\nu - 111366709018012 ) / 8369 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2473\beta_1 ) / 241920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 57179\beta_{2} - 196034413\beta _1 + 1815732190702080 ) / 241920 \) 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
−24885.9
116336.
−91450.2
−4.65343e6 3.22027e10 1.28583e13 5.54482e14 −1.49853e17 −5.57604e17 −1.89031e19 7.08758e20 −2.58024e21
1.2 −1.18359e6 −1.79507e10 −7.39521e12 −6.39116e14 2.12463e16 −1.72730e18 1.91639e19 −6.02939e18 7.56451e20
1.3 3.62707e6 1.01494e10 4.35955e12 6.19839e14 3.68127e16 2.58688e18 −1.60917e19 −2.25246e20 2.24820e21
\(n\): 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 1.44.a.a 3
3.b odd 2 1 9.44.a.b 3
4.b odd 2 1 16.44.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.44.a.a 3 1.a even 1 1 trivial
9.44.a.b 3 3.b odd 2 1
16.44.a.c 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{44}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2209944 T^{2} + \cdots - 19\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{3} - 24401437812 T^{2} + \cdots + 58\!\cdots\!08 \) Copy content Toggle raw display
$5$ \( T^{3} - 535205380774170 T^{2} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 50\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 36\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 27\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 49\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 90\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 87\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 31\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 17\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 26\!\cdots\!44 \) Copy content Toggle raw display
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