Properties

Label 2523.2.a.b
Level $2523$
Weight $2$
Character orbit 2523.a
Self dual yes
Analytic conductor $20.146$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2523,2,Mod(1,2523)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2523, 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("2523.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2523.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: \(20.1462564300\)
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: 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{5} + \beta q^{6} + ( - 2 \beta + 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{5} + \beta q^{6} + ( - 2 \beta + 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} - q^{10} + (4 \beta - 1) q^{11} + ( - \beta + 1) q^{12} + (\beta - 3) q^{13} + 2 q^{14} + ( - \beta + 1) q^{15} - 3 \beta q^{16} + ( - 3 \beta + 6) q^{17} - \beta q^{18} + ( - 5 \beta + 3) q^{19} + ( - \beta + 2) q^{20} + (2 \beta - 2) q^{21} + ( - 3 \beta - 4) q^{22} + ( - 3 \beta - 1) q^{23} + ( - 2 \beta + 1) q^{24} + ( - \beta - 3) q^{25} + (2 \beta - 1) q^{26} - q^{27} + (2 \beta - 4) q^{28} + q^{30} + (6 \beta - 3) q^{31} + ( - \beta + 5) q^{32} + ( - 4 \beta + 1) q^{33} + ( - 3 \beta + 3) q^{34} + (2 \beta - 4) q^{35} + (\beta - 1) q^{36} + ( - \beta + 2) q^{37} + (2 \beta + 5) q^{38} + ( - \beta + 3) q^{39} + ( - \beta + 3) q^{40} - 5 q^{41} - 2 q^{42} + 5 q^{43} + ( - \beta + 5) q^{44} + (\beta - 1) q^{45} + (4 \beta + 3) q^{46} - 7 q^{47} + 3 \beta q^{48} + ( - 4 \beta + 1) q^{49} + (4 \beta + 1) q^{50} + (3 \beta - 6) q^{51} + ( - 3 \beta + 4) q^{52} + (4 \beta - 5) q^{53} + \beta q^{54} + ( - \beta + 5) q^{55} + (2 \beta - 6) q^{56} + (5 \beta - 3) q^{57} + (5 \beta + 5) q^{59} + (\beta - 2) q^{60} + (10 \beta - 2) q^{61} + ( - 3 \beta - 6) q^{62} + ( - 2 \beta + 2) q^{63} + (2 \beta + 1) q^{64} + ( - 3 \beta + 4) q^{65} + (3 \beta + 4) q^{66} + (7 \beta + 2) q^{67} + (6 \beta - 9) q^{68} + (3 \beta + 1) q^{69} + (2 \beta - 2) q^{70} + ( - 7 \beta - 4) q^{71} + (2 \beta - 1) q^{72} + ( - 8 \beta + 10) q^{73} + ( - \beta + 1) q^{74} + (\beta + 3) q^{75} + (3 \beta - 8) q^{76} + (2 \beta - 10) q^{77} + ( - 2 \beta + 1) q^{78} + ( - 4 \beta - 1) q^{79} - 3 q^{80} + q^{81} + 5 \beta q^{82} + (\beta - 14) q^{83} + ( - 2 \beta + 4) q^{84} + (6 \beta - 9) q^{85} - 5 \beta q^{86} + (2 \beta + 9) q^{88} + (6 \beta - 8) q^{89} - q^{90} + (6 \beta - 8) q^{91} + ( - \beta - 2) q^{92} + ( - 6 \beta + 3) q^{93} + 7 \beta q^{94} + (3 \beta - 8) q^{95} + (\beta - 5) q^{96} + ( - \beta + 11) q^{97} + (3 \beta + 4) q^{98} + (4 \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} - q^{5} + q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} - q^{5} + q^{6} + 2 q^{7} + 2 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} - 5 q^{13} + 4 q^{14} + q^{15} - 3 q^{16} + 9 q^{17} - q^{18} + q^{19} + 3 q^{20} - 2 q^{21} - 11 q^{22} - 5 q^{23} - 7 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{30} + 9 q^{32} - 2 q^{33} + 3 q^{34} - 6 q^{35} - q^{36} + 3 q^{37} + 12 q^{38} + 5 q^{39} + 5 q^{40} - 10 q^{41} - 4 q^{42} + 10 q^{43} + 9 q^{44} - q^{45} + 10 q^{46} - 14 q^{47} + 3 q^{48} - 2 q^{49} + 6 q^{50} - 9 q^{51} + 5 q^{52} - 6 q^{53} + q^{54} + 9 q^{55} - 10 q^{56} - q^{57} + 15 q^{59} - 3 q^{60} + 6 q^{61} - 15 q^{62} + 2 q^{63} + 4 q^{64} + 5 q^{65} + 11 q^{66} + 11 q^{67} - 12 q^{68} + 5 q^{69} - 2 q^{70} - 15 q^{71} + 12 q^{73} + q^{74} + 7 q^{75} - 13 q^{76} - 18 q^{77} - 6 q^{79} - 6 q^{80} + 2 q^{81} + 5 q^{82} - 27 q^{83} + 6 q^{84} - 12 q^{85} - 5 q^{86} + 20 q^{88} - 10 q^{89} - 2 q^{90} - 10 q^{91} - 5 q^{92} + 7 q^{94} - 13 q^{95} - 9 q^{96} + 21 q^{97} + 11 q^{98} + 2 q^{99}+O(q^{100}) \) 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
−1.61803 −1.00000 0.618034 0.618034 1.61803 −1.23607 2.23607 1.00000 −1.00000
1.2 0.618034 −1.00000 −1.61803 −1.61803 −0.618034 3.23607 −2.23607 1.00000 −1.00000
\(n\): 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 2523.2.a.b 2
3.b odd 2 1 7569.2.a.m 2
29.b even 2 1 2523.2.a.g yes 2
87.d odd 2 1 7569.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.2.a.b 2 1.a even 1 1 trivial
2523.2.a.g yes 2 29.b even 2 1
7569.2.a.e 2 87.d odd 2 1
7569.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2523))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$23$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 45 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$41$ \( (T + 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( (T + 7)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
$67$ \( T^{2} - 11T - 31 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T - 5 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$83$ \( T^{2} + 27T + 181 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$97$ \( T^{2} - 21T + 109 \) Copy content Toggle raw display
show more
show less