Properties

Label 69.4.a.a
Level $69$
Weight $4$
Character orbit 69.a
Self dual yes
Analytic conductor $4.071$
Analytic rank $1$
Dimension $2$
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] = [69,4,Mod(1,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(4.07113179040\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + 3 q^{3} + (4 \beta + 1) q^{4} + (\beta - 13) q^{5} + ( - 3 \beta - 6) q^{6} + (7 \beta - 5) q^{7} + ( - \beta - 6) q^{8} + 9 q^{9} + (11 \beta + 21) q^{10} + ( - 10 \beta - 30) q^{11}+ \cdots + ( - 90 \beta - 270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 26 q^{5} - 12 q^{6} - 10 q^{7} - 12 q^{8} + 18 q^{9} + 42 q^{10} - 60 q^{11} + 6 q^{12} - 24 q^{13} - 50 q^{14} - 78 q^{15} + 18 q^{16} - 150 q^{17} - 36 q^{18} - 46 q^{19}+ \cdots - 540 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−4.23607 3.00000 9.94427 −10.7639 −12.7082 10.6525 −8.23607 9.00000 45.5967
1.2 0.236068 3.00000 −7.94427 −15.2361 0.708204 −20.6525 −3.76393 9.00000 −3.59675
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(23\) \( +1 \)

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 69.4.a.a 2
3.b odd 2 1 207.4.a.c 2
4.b odd 2 1 1104.4.a.h 2
5.b even 2 1 1725.4.a.n 2
23.b odd 2 1 1587.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.a.a 2 1.a even 1 1 trivial
207.4.a.c 2 3.b odd 2 1
1104.4.a.h 2 4.b odd 2 1
1587.4.a.b 2 23.b odd 2 1
1725.4.a.n 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 26T + 164 \) Copy content Toggle raw display
$7$ \( T^{2} + 10T - 220 \) Copy content Toggle raw display
$11$ \( T^{2} + 60T + 400 \) Copy content Toggle raw display
$13$ \( T^{2} + 24T - 4356 \) Copy content Toggle raw display
$17$ \( T^{2} + 150T + 5620 \) Copy content Toggle raw display
$19$ \( T^{2} + 46T - 4916 \) Copy content Toggle raw display
$23$ \( (T + 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 216T - 2916 \) Copy content Toggle raw display
$31$ \( T^{2} - 324T - 4176 \) Copy content Toggle raw display
$37$ \( T^{2} - 140T - 15580 \) Copy content Toggle raw display
$41$ \( T^{2} + 364T + 12644 \) Copy content Toggle raw display
$43$ \( T^{2} + 126T - 51156 \) Copy content Toggle raw display
$47$ \( T^{2} - 120T - 303920 \) Copy content Toggle raw display
$53$ \( T^{2} - 490T + 59420 \) Copy content Toggle raw display
$59$ \( T^{2} + 32T - 268864 \) Copy content Toggle raw display
$61$ \( T^{2} + 908T + 58196 \) Copy content Toggle raw display
$67$ \( T^{2} + 874T + 190844 \) Copy content Toggle raw display
$71$ \( T^{2} - 488T - 88384 \) Copy content Toggle raw display
$73$ \( T^{2} - 652T - 252844 \) Copy content Toggle raw display
$79$ \( T^{2} + 1198 T + 225956 \) Copy content Toggle raw display
$83$ \( T^{2} + 100T - 31120 \) Copy content Toggle raw display
$89$ \( T^{2} - 102T - 685604 \) Copy content Toggle raw display
$97$ \( T^{2} - 1624T + 32764 \) Copy content Toggle raw display
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