Properties

Label 1.36.a.a
Level $1$
Weight $36$
Character orbit 1.a
Self dual yes
Analytic conductor $7.760$
Analytic rank $0$
Dimension $3$
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] = [1,36,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, 36, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 36);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 36 \)
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: \(7.75951306336\)
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: \( x^{3} - 12422194x - 2645665785 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 46552) q^{2} + ( - \beta_{2} + \beta_1 - 34958436) q^{3} + (72 \beta_{2} + 194112 \beta_1 + 11613754048) q^{4} + ( - 2484 \beta_{2} + 7444916 \beta_1 + 297550684670) q^{5} + (54528 \beta_{2} - 197319012 \beta_1 - 1595510188128) q^{6} + ( - 852426 \beta_{2} + 723239370 \beta_1 + 292807383115352) q^{7} + (10055232 \beta_{2} + 17597768192 \beta_1 + 74\!\cdots\!00) q^{8}+ \cdots + ( - 92389176 \beta_{2} - 228209975496 \beta_1 + 50\!\cdots\!17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 46552) q^{2} + ( - \beta_{2} + \beta_1 - 34958436) q^{3} + (72 \beta_{2} + 194112 \beta_1 + 11613754048) q^{4} + ( - 2484 \beta_{2} + 7444916 \beta_1 + 297550684670) q^{5} + (54528 \beta_{2} - 197319012 \beta_1 - 1595510188128) q^{6} + ( - 852426 \beta_{2} + 723239370 \beta_1 + 292807383115352) q^{7} + (10055232 \beta_{2} + 17597768192 \beta_1 + 74\!\cdots\!00) q^{8}+ \cdots + ( - 30\!\cdots\!87 \beta_{2} + \cdots + 10\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 139656 q^{2} - 104875308 q^{3} + 34841262144 q^{4} + 892652054010 q^{5} - 4786530564384 q^{6} + 878422149346056 q^{7} + 22\!\cdots\!00 q^{8}+ \cdots + 15\!\cdots\!51 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 139656 q^{2} - 104875308 q^{3} + 34841262144 q^{4} + 892652054010 q^{5} - 4786530564384 q^{6} + 878422149346056 q^{7} + 22\!\cdots\!00 q^{8}+ \cdots + 30\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 12422194x - 2645665785 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 24\nu^{2} + 44712\nu - 198755104 ) / 979 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -39144\nu^{2} + 84967848\nu + 324169574624 ) / 979 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 1631\beta_1 ) / 161280 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -621\beta_{2} + 1180109\beta _1 + 445211432960 ) / 53760 \) 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
−213.765
−3412.77
3626.53
−165109. −3.45913e8 −7.09870e9 −2.05014e12 5.71135e13 −1.25160e14 6.84517e15 6.96245e16 3.38496e17
1.2 −26808.0 3.95729e8 −3.36411e10 8.21401e11 −1.06087e13 6.06942e14 1.82297e15 1.06570e17 −2.20201e16
1.3 331573. −1.54691e8 7.55810e10 2.12139e12 −5.12913e13 3.96640e14 1.36679e16 −2.61023e16 7.03395e17
\(n\): 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 1.36.a.a 3
3.b odd 2 1 9.36.a.b 3
4.b odd 2 1 16.36.a.d 3
5.b even 2 1 25.36.a.a 3
5.c odd 4 2 25.36.b.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.36.a.a 3 1.a even 1 1 trivial
9.36.a.b 3 3.b odd 2 1
16.36.a.d 3 4.b odd 2 1
25.36.a.a 3 5.b even 2 1
25.36.b.a 6 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 139656 T^{2} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{3} + 104875308 T^{2} + \cdots - 21\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{3} - 892652054010 T^{2} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} - 878422149346056 T^{2} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 49\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 68\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 20\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 44\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 57\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 33\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 88\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 11\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 83\!\cdots\!44 \) Copy content Toggle raw display
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