Properties

Label 483.2.a.f
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: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + (\beta + 1) q^{4} + (\beta - 3) q^{5} - \beta q^{6} - q^{7} - 3 q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + (\beta + 1) q^{4} + (\beta - 3) q^{5} - \beta q^{6} - q^{7} - 3 q^{8} + q^{9} + (2 \beta - 3) q^{10} - 5 q^{11} + (\beta + 1) q^{12} + \beta q^{13} + \beta q^{14} + (\beta - 3) q^{15} + (\beta - 2) q^{16} + ( - 2 \beta - 1) q^{17} - \beta q^{18} + ( - 2 \beta + 3) q^{19} - \beta q^{20} - q^{21} + 5 \beta q^{22} + q^{23} - 3 q^{24} + ( - 5 \beta + 7) q^{25} + ( - \beta - 3) q^{26} + q^{27} + ( - \beta - 1) q^{28} + (4 \beta - 3) q^{29} + (2 \beta - 3) q^{30} + 3 q^{31} + (\beta + 3) q^{32} - 5 q^{33} + (3 \beta + 6) q^{34} + ( - \beta + 3) q^{35} + (\beta + 1) q^{36} - 9 q^{37} + ( - \beta + 6) q^{38} + \beta q^{39} + ( - 3 \beta + 9) q^{40} + ( - 4 \beta - 3) q^{41} + \beta q^{42} + (5 \beta - 6) q^{43} + ( - 5 \beta - 5) q^{44} + (\beta - 3) q^{45} - \beta q^{46} + ( - 2 \beta - 4) q^{47} + (\beta - 2) q^{48} + q^{49} + ( - 2 \beta + 15) q^{50} + ( - 2 \beta - 1) q^{51} + (2 \beta + 3) q^{52} + ( - 5 \beta - 1) q^{53} - \beta q^{54} + ( - 5 \beta + 15) q^{55} + 3 q^{56} + ( - 2 \beta + 3) q^{57} + ( - \beta - 12) q^{58} + (3 \beta - 3) q^{59} - \beta q^{60} + (3 \beta - 8) q^{61} - 3 \beta q^{62} - q^{63} + ( - 6 \beta + 1) q^{64} + ( - 2 \beta + 3) q^{65} + 5 \beta q^{66} + ( - 3 \beta - 5) q^{67} + ( - 5 \beta - 7) q^{68} + q^{69} + ( - 2 \beta + 3) q^{70} + (3 \beta - 6) q^{71} - 3 q^{72} + ( - 4 \beta + 7) q^{73} + 9 \beta q^{74} + ( - 5 \beta + 7) q^{75} + ( - \beta - 3) q^{76} + 5 q^{77} + ( - \beta - 3) q^{78} - q^{79} + ( - 4 \beta + 9) q^{80} + q^{81} + (7 \beta + 12) q^{82} + (2 \beta + 1) q^{83} + ( - \beta - 1) q^{84} + (3 \beta - 3) q^{85} + (\beta - 15) q^{86} + (4 \beta - 3) q^{87} + 15 q^{88} + (3 \beta + 4) q^{89} + (2 \beta - 3) q^{90} - \beta q^{91} + (\beta + 1) q^{92} + 3 q^{93} + (6 \beta + 6) q^{94} + (7 \beta - 15) q^{95} + (\beta + 3) q^{96} + ( - 2 \beta - 13) q^{97} - \beta q^{98} - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 5 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 5 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} - 4 q^{10} - 10 q^{11} + 3 q^{12} + q^{13} + q^{14} - 5 q^{15} - 3 q^{16} - 4 q^{17} - q^{18} + 4 q^{19} - q^{20} - 2 q^{21} + 5 q^{22} + 2 q^{23} - 6 q^{24} + 9 q^{25} - 7 q^{26} + 2 q^{27} - 3 q^{28} - 2 q^{29} - 4 q^{30} + 6 q^{31} + 7 q^{32} - 10 q^{33} + 15 q^{34} + 5 q^{35} + 3 q^{36} - 18 q^{37} + 11 q^{38} + q^{39} + 15 q^{40} - 10 q^{41} + q^{42} - 7 q^{43} - 15 q^{44} - 5 q^{45} - q^{46} - 10 q^{47} - 3 q^{48} + 2 q^{49} + 28 q^{50} - 4 q^{51} + 8 q^{52} - 7 q^{53} - q^{54} + 25 q^{55} + 6 q^{56} + 4 q^{57} - 25 q^{58} - 3 q^{59} - q^{60} - 13 q^{61} - 3 q^{62} - 2 q^{63} - 4 q^{64} + 4 q^{65} + 5 q^{66} - 13 q^{67} - 19 q^{68} + 2 q^{69} + 4 q^{70} - 9 q^{71} - 6 q^{72} + 10 q^{73} + 9 q^{74} + 9 q^{75} - 7 q^{76} + 10 q^{77} - 7 q^{78} - 2 q^{79} + 14 q^{80} + 2 q^{81} + 31 q^{82} + 4 q^{83} - 3 q^{84} - 3 q^{85} - 29 q^{86} - 2 q^{87} + 30 q^{88} + 11 q^{89} - 4 q^{90} - q^{91} + 3 q^{92} + 6 q^{93} + 18 q^{94} - 23 q^{95} + 7 q^{96} - 28 q^{97} - q^{98} - 10 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 1.00000 3.30278 −0.697224 −2.30278 −1.00000 −3.00000 1.00000 1.60555
1.2 1.30278 1.00000 −0.302776 −4.30278 1.30278 −1.00000 −3.00000 1.00000 −5.60555
\(n\): 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.f 2
3.b odd 2 1 1449.2.a.j 2
4.b odd 2 1 7728.2.a.x 2
7.b odd 2 1 3381.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 1.a even 1 1 trivial
1449.2.a.j 2 3.b odd 2 1
3381.2.a.p 2 7.b odd 2 1
7728.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(483))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} + 5T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$31$ \( (T - 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T - 69 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 7T - 69 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 13 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 13 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 27 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 11T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} + 28T + 183 \) Copy content Toggle raw display
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