Properties

Label 7569.2.a.h
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 261)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 1) q^{4} + 2 q^{5} + (2 \beta - 1) q^{7} + (2 \beta - 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 1) q^{4} + 2 q^{5} + (2 \beta - 1) q^{7} + (2 \beta - 1) q^{8} - 2 \beta q^{10} + (2 \beta - 5) q^{11} + ( - 4 \beta + 1) q^{13} + ( - \beta - 2) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 1) q^{17} + (2 \beta - 2) q^{20} + (3 \beta - 2) q^{22} + ( - 4 \beta + 6) q^{23} - q^{25} + (3 \beta + 4) q^{26} + ( - \beta + 3) q^{28} + (8 \beta - 4) q^{31} + ( - \beta + 5) q^{32} + (3 \beta + 4) q^{34} + (4 \beta - 2) q^{35} + 4 q^{37} + (4 \beta - 2) q^{40} + 2 q^{41} + ( - 4 \beta + 6) q^{43} + ( - 5 \beta + 7) q^{44} + ( - 2 \beta + 4) q^{46} + (6 \beta - 1) q^{47} - 2 q^{49} + \beta q^{50} + (\beta - 5) q^{52} - 8 q^{53} + (4 \beta - 10) q^{55} + 5 q^{56} + (4 \beta + 6) q^{59} + ( - 8 \beta + 6) q^{61} + ( - 4 \beta - 8) q^{62} + (2 \beta + 1) q^{64} + ( - 8 \beta + 2) q^{65} + (2 \beta + 5) q^{67} + (\beta - 5) q^{68} + ( - 2 \beta - 4) q^{70} + (8 \beta - 4) q^{71} + 2 q^{73} - 4 \beta q^{74} + ( - 8 \beta + 9) q^{77} + ( - 4 \beta - 6) q^{79} - 6 \beta q^{80} - 2 \beta q^{82} + (8 \beta + 4) q^{83} + ( - 8 \beta + 2) q^{85} + ( - 2 \beta + 4) q^{86} + ( - 8 \beta + 9) q^{88} + (12 \beta - 5) q^{89} + ( - 2 \beta - 9) q^{91} + (6 \beta - 10) q^{92} + ( - 5 \beta - 6) q^{94} - 8 q^{97} + 2 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 4 q^{5} - 2 q^{10} - 8 q^{11} - 2 q^{13} - 5 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{20} - q^{22} + 8 q^{23} - 2 q^{25} + 11 q^{26} + 5 q^{28} + 9 q^{32} + 11 q^{34} + 8 q^{37} + 4 q^{41} + 8 q^{43} + 9 q^{44} + 6 q^{46} + 4 q^{47} - 4 q^{49} + q^{50} - 9 q^{52} - 16 q^{53} - 16 q^{55} + 10 q^{56} + 16 q^{59} + 4 q^{61} - 20 q^{62} + 4 q^{64} - 4 q^{65} + 12 q^{67} - 9 q^{68} - 10 q^{70} + 4 q^{73} - 4 q^{74} + 10 q^{77} - 16 q^{79} - 6 q^{80} - 2 q^{82} + 16 q^{83} - 4 q^{85} + 6 q^{86} + 10 q^{88} + 2 q^{89} - 20 q^{91} - 14 q^{92} - 17 q^{94} - 16 q^{97} + 2 q^{98}+O(q^{100}) \) 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
−1.61803 0 0.618034 2.00000 0 2.23607 2.23607 0 −3.23607
1.2 0.618034 0 −1.61803 2.00000 0 −2.23607 −2.23607 0 1.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(29\) \( +1 \)

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 7569.2.a.h 2
3.b odd 2 1 7569.2.a.j 2
29.b even 2 1 261.2.a.c yes 2
87.d odd 2 1 261.2.a.a 2
116.d odd 2 1 4176.2.a.bt 2
145.d even 2 1 6525.2.a.q 2
348.b even 2 1 4176.2.a.bm 2
435.b odd 2 1 6525.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.a.a 2 87.d odd 2 1
261.2.a.c yes 2 29.b even 2 1
4176.2.a.bm 2 348.b even 2 1
4176.2.a.bt 2 116.d odd 2 1
6525.2.a.q 2 145.d even 2 1
6525.2.a.z 2 435.b odd 2 1
7569.2.a.h 2 1.a even 1 1 trivial
7569.2.a.j 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 5 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 80 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$53$ \( (T + 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 179 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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