Properties

Label 69.4.a.b
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: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - 3 q^{3} + ( - 2 \beta + 1) q^{4} + ( - 4 \beta + 4) q^{5} + ( - 3 \beta + 3) q^{6} + (\beta - 14) q^{7} + ( - 5 \beta - 9) q^{8} + 9 q^{9} + (8 \beta - 36) q^{10} + (7 \beta - 36) q^{11}+ \cdots + (63 \beta - 324) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 8 q^{5} + 6 q^{6} - 28 q^{7} - 18 q^{8} + 18 q^{9} - 72 q^{10} - 72 q^{11} - 6 q^{12} - 60 q^{13} + 44 q^{14} - 24 q^{15} - 78 q^{16} + 96 q^{17} - 18 q^{18} - 148 q^{19}+ \cdots - 648 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−3.82843 −3.00000 6.65685 15.3137 11.4853 −16.8284 5.14214 9.00000 −58.6274
1.2 1.82843 −3.00000 −4.65685 −7.31371 −5.48528 −11.1716 −23.1421 9.00000 −13.3726
\(n\): 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.b 2
3.b odd 2 1 207.4.a.b 2
4.b odd 2 1 1104.4.a.q 2
5.b even 2 1 1725.4.a.m 2
23.b odd 2 1 1587.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.a.b 2 1.a even 1 1 trivial
207.4.a.b 2 3.b odd 2 1
1104.4.a.q 2 4.b odd 2 1
1587.4.a.c 2 23.b odd 2 1
1725.4.a.m 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 188 \) Copy content Toggle raw display
$11$ \( T^{2} + 72T + 904 \) Copy content Toggle raw display
$13$ \( T^{2} + 60T + 612 \) Copy content Toggle raw display
$17$ \( T^{2} - 96T + 2104 \) Copy content Toggle raw display
$19$ \( T^{2} + 148T + 868 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 204T - 24444 \) Copy content Toggle raw display
$31$ \( T^{2} + 168T - 16272 \) Copy content Toggle raw display
$37$ \( T^{2} - 116T - 11428 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 5404 \) Copy content Toggle raw display
$43$ \( T^{2} + 420T + 41508 \) Copy content Toggle raw display
$47$ \( T^{2} + 48T - 61376 \) Copy content Toggle raw display
$53$ \( T^{2} - 32T - 544 \) Copy content Toggle raw display
$59$ \( T^{2} + 40T - 199312 \) Copy content Toggle raw display
$61$ \( T^{2} + 764T + 126716 \) Copy content Toggle raw display
$67$ \( T^{2} + 988T + 73508 \) Copy content Toggle raw display
$71$ \( T^{2} + 224T - 641824 \) Copy content Toggle raw display
$73$ \( T^{2} - 820T + 143012 \) Copy content Toggle raw display
$79$ \( T^{2} - 1772 T + 757148 \) Copy content Toggle raw display
$83$ \( T^{2} - 1480 T + 191432 \) Copy content Toggle raw display
$89$ \( T^{2} - 744T - 365624 \) Copy content Toggle raw display
$97$ \( T^{2} + 260 T - 1646588 \) Copy content Toggle raw display
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