Properties

Label 483.2.a.d
Level $483$
Weight $2$
Character orbit 483.a
Self dual yes
Analytic conductor $3.857$
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] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} - q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} - q^{7} + (2 \beta - 1) q^{8} + q^{9} + q^{10} + (2 \beta - 1) q^{11} + ( - \beta + 1) q^{12} + (\beta - 4) q^{13} + \beta q^{14} + (\beta - 1) q^{15} - 3 \beta q^{16} + (6 \beta - 3) q^{17} - \beta q^{18} + ( - 4 \beta + 3) q^{19} + (\beta - 2) q^{20} + q^{21} + ( - \beta - 2) q^{22} - q^{23} + ( - 2 \beta + 1) q^{24} + ( - \beta - 3) q^{25} + (3 \beta - 1) q^{26} - q^{27} + ( - \beta + 1) q^{28} + ( - 2 \beta - 5) q^{29} - q^{30} + (6 \beta - 3) q^{31} + ( - \beta + 5) q^{32} + ( - 2 \beta + 1) q^{33} + ( - 3 \beta - 6) q^{34} + (\beta - 1) q^{35} + (\beta - 1) q^{36} - 11 q^{37} + (\beta + 4) q^{38} + ( - \beta + 4) q^{39} + (\beta - 3) q^{40} + ( - 4 \beta + 5) q^{41} - \beta q^{42} - \beta q^{43} + ( - \beta + 3) q^{44} + ( - \beta + 1) q^{45} + \beta q^{46} + ( - 2 \beta - 4) q^{47} + 3 \beta q^{48} + q^{49} + (4 \beta + 1) q^{50} + ( - 6 \beta + 3) q^{51} + ( - 4 \beta + 5) q^{52} + ( - 5 \beta - 5) q^{53} + \beta q^{54} + (\beta - 3) q^{55} + ( - 2 \beta + 1) q^{56} + (4 \beta - 3) q^{57} + (7 \beta + 2) q^{58} + ( - \beta + 11) q^{59} + ( - \beta + 2) q^{60} - 3 \beta q^{61} + ( - 3 \beta - 6) q^{62} - q^{63} + (2 \beta + 1) q^{64} + (4 \beta - 5) q^{65} + (\beta + 2) q^{66} + ( - 5 \beta + 3) q^{67} + ( - 3 \beta + 9) q^{68} + q^{69} - q^{70} + ( - \beta + 6) q^{71} + (2 \beta - 1) q^{72} + ( - 6 \beta - 3) q^{73} + 11 \beta q^{74} + (\beta + 3) q^{75} + (3 \beta - 7) q^{76} + ( - 2 \beta + 1) q^{77} + ( - 3 \beta + 1) q^{78} + ( - 4 \beta - 3) q^{79} + 3 q^{80} + q^{81} + ( - \beta + 4) q^{82} + ( - 10 \beta + 7) q^{83} + (\beta - 1) q^{84} + (3 \beta - 9) q^{85} + (\beta + 1) q^{86} + (2 \beta + 5) q^{87} + 5 q^{88} + (\beta + 10) q^{89} + q^{90} + ( - \beta + 4) q^{91} + ( - \beta + 1) q^{92} + ( - 6 \beta + 3) q^{93} + (6 \beta + 2) q^{94} + ( - 3 \beta + 7) q^{95} + (\beta - 5) q^{96} + (12 \beta - 9) q^{97} - \beta q^{98} + (2 \beta - 1) q^{99} +O(q^{100}) \) 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 \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} + q^{12} - 7 q^{13} + q^{14} - q^{15} - 3 q^{16} - q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} - 5 q^{22} - 2 q^{23} - 7 q^{25} + q^{26} - 2 q^{27} + q^{28} - 12 q^{29} - 2 q^{30} + 9 q^{32} - 15 q^{34} - q^{35} - q^{36} - 22 q^{37} + 9 q^{38} + 7 q^{39} - 5 q^{40} + 6 q^{41} - q^{42} - q^{43} + 5 q^{44} + q^{45} + q^{46} - 10 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{50} + 6 q^{52} - 15 q^{53} + q^{54} - 5 q^{55} - 2 q^{57} + 11 q^{58} + 21 q^{59} + 3 q^{60} - 3 q^{61} - 15 q^{62} - 2 q^{63} + 4 q^{64} - 6 q^{65} + 5 q^{66} + q^{67} + 15 q^{68} + 2 q^{69} - 2 q^{70} + 11 q^{71} - 12 q^{73} + 11 q^{74} + 7 q^{75} - 11 q^{76} - q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} + 7 q^{82} + 4 q^{83} - q^{84} - 15 q^{85} + 3 q^{86} + 12 q^{87} + 10 q^{88} + 21 q^{89} + 2 q^{90} + 7 q^{91} + q^{92} + 10 q^{94} + 11 q^{95} - 9 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) \) 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 −1.00000 0.618034 −0.618034 1.61803 −1.00000 2.23607 1.00000 1.00000
1.2 0.618034 −1.00000 −1.61803 1.61803 −0.618034 −1.00000 −2.23607 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(23\) \( +1 \)

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 483.2.a.d 2
3.b odd 2 1 1449.2.a.h 2
4.b odd 2 1 7728.2.a.bn 2
7.b odd 2 1 3381.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.d 2 1.a even 1 1 trivial
1449.2.a.h 2 3.b odd 2 1
3381.2.a.r 2 7.b odd 2 1
7728.2.a.bn 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(483))\):

\( T_{2}^{2} + T_{2} - 1 \) 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
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 45 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$31$ \( T^{2} - 45 \) Copy content Toggle raw display
$37$ \( (T + 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$53$ \( T^{2} + 15T + 25 \) Copy content Toggle raw display
$59$ \( T^{2} - 21T + 109 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$71$ \( T^{2} - 11T + 29 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 121 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 109 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 171 \) Copy content Toggle raw display
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