Properties

Label 10.34.a.c
Level $10$
Weight $34$
Character orbit 10.a
Self dual yes
Analytic conductor $68.983$
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] = [10,34,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(68.9828288810\)
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: \( x^{3} - x^{2} - 1666412968752x + 770804720224644780 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{3}\cdot 5^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q - 65536 q^{2} + ( - \beta_1 + 52174456) q^{3} + 4294967296 q^{4} + 152587890625 q^{5} + (65536 \beta_1 - 3419305148416) q^{6} + (61 \beta_{2} - 1026619 \beta_1 - 2886784941628) q^{7} - 281474976710656 q^{8}+ \cdots + ( - 47\!\cdots\!21 \beta_{2} + \cdots - 23\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 196608 q^{2} + 156523368 q^{3} + 12884901888 q^{4} + 457763671875 q^{5} - 10257915445248 q^{6} - 8660354824884 q^{7} - 844424930131968 q^{8} + 34\!\cdots\!39 q^{9} - 30\!\cdots\!00 q^{10} - 13\!\cdots\!64 q^{11}+ \cdots - 71\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 1666412968752x + 770804720224644780 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 60\nu - 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\nu^{2} + 34691530\nu - 55547110522260 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 20 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 54\beta_{2} - 3469153\beta _1 + 333282593750500 ) / 300 \) 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
899958.
579084.
−1.47904e6
−65536.0 −1.82299e6 4.29497e9 1.52588e11 1.19472e11 5.12751e13 −2.81475e14 −5.55574e15 −1.00000e16
1.2 −65536.0 1.74294e7 4.29497e9 1.52588e11 −1.14225e12 −1.65239e14 −2.81475e14 −5.25528e15 −1.00000e16
1.3 −65536.0 1.40917e8 4.29497e9 1.52588e11 −9.23513e12 1.05304e14 −2.81475e14 1.42985e16 −1.00000e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

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 10.34.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.34.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 156523368T_{3}^{2} + 2167434889179408T_{3} + 4477444605487021699584 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 65536)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( (T - 152587890625)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 34\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 49\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 52\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 71\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 83\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 84\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 33\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 70\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 32\!\cdots\!88 \) Copy content Toggle raw display
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