Properties

Label 4.34.a.a
Level $4$
Weight $34$
Character orbit 4.a
Self dual yes
Analytic conductor $27.593$
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] = [4,34,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, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 34 \)
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: \(27.5931315524\)
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: \( x^{3} - x^{2} - 65185566x - 173679864984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 30830596) q^{3} + (\beta_{2} - 1535 \beta_1 - 17960227962) q^{5} + (180 \beta_{2} - 122346 \beta_1 + 1513669971464) q^{7} + ( - 10530 \beta_{2} + 75248670 \beta_1 + 20\!\cdots\!41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 30830596) q^{3} + (\beta_{2} - 1535 \beta_1 - 17960227962) q^{5} + (180 \beta_{2} - 122346 \beta_1 + 1513669971464) q^{7} + ( - 10530 \beta_{2} + 75248670 \beta_1 + 20\!\cdots\!41) q^{9}+ \cdots + ( - 27\!\cdots\!60 \beta_{2} + \cdots + 75\!\cdots\!00) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots + 22\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 65185566x - 173679864984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -128\nu^{2} + 542720\nu + 5562320768 ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 343168\nu^{2} + 6776698880\nu - 14915325889408 ) / 125 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 5\beta_{2} + 2681\beta _1 + 109756416 ) / 329269248 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 1325\beta_{2} - 3308935\beta _1 + 894316769246208 ) / 20579328 \) 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
−6032.20
9172.25
−3139.05
0 −6.39317e7 0 −2.18954e11 0 −4.92542e13 0 −1.47179e15 0
1.2 0 2.16957e7 0 6.04966e11 0 1.12234e14 0 −5.08836e15 0
1.3 0 1.34728e8 0 −4.39893e11 0 −5.84388e13 0 1.25925e16 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 4.34.a.a 3
4.b odd 2 1 16.34.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.34.a.a 3 1.a even 1 1 trivial
16.34.a.d 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{34}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 92491788 T^{2} + \cdots + 18\!\cdots\!12 \) Copy content Toggle raw display
$5$ \( T^{3} + 53880683886 T^{2} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} - 4541009914392 T^{2} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 28\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 57\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 14\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 11\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 71\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 85\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 73\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 93\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 60\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 38\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 63\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 77\!\cdots\!88 \) Copy content Toggle raw display
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