Properties

Label 483.2.a.h
Level $483$
Weight $2$
Character orbit 483.a
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_1 + 1) q^{5} + \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_1 + 1) q^{5} + \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{2} + \beta_1 - 4) q^{10} + (\beta_{2} - \beta_1 + 2) q^{11} + (\beta_{2} + 2) q^{12} + ( - \beta_{2} + 3) q^{13} - \beta_1 q^{14} + ( - \beta_1 + 1) q^{15} + (\beta_1 + 4) q^{16} + ( - \beta_{2} - \beta_1 + 2) q^{17} + \beta_1 q^{18} + ( - \beta_{2} + \beta_1 - 2) q^{19} + (\beta_{2} - 4 \beta_1 + 1) q^{20} - q^{21} + ( - \beta_{2} + 4 \beta_1 - 3) q^{22} - q^{23} + (2 \beta_1 + 1) q^{24} + (\beta_{2} - 2 \beta_1) q^{25} + (\beta_1 - 1) q^{26} + q^{27} + ( - \beta_{2} - 2) q^{28} + ( - \beta_{2} + 3 \beta_1 + 2) q^{29} + ( - \beta_{2} + \beta_1 - 4) q^{30} + ( - \beta_{2} + \beta_1 - 6) q^{31} + (\beta_{2} + 2) q^{32} + (\beta_{2} - \beta_1 + 2) q^{33} + ( - \beta_{2} - 5) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{2} - 3 \beta_1 + 2) q^{37} + (\beta_{2} - 4 \beta_1 + 3) q^{38} + ( - \beta_{2} + 3) q^{39} + ( - 2 \beta_{2} + \beta_1 - 7) q^{40} + ( - 3 \beta_{2} + \beta_1 - 2) q^{41} - \beta_1 q^{42} + (\beta_{2} - 2 \beta_1 - 3) q^{43} + (2 \beta_{2} - 3 \beta_1 + 11) q^{44} + ( - \beta_1 + 1) q^{45} - \beta_1 q^{46} + (2 \beta_{2} + 6) q^{47} + (\beta_1 + 4) q^{48} + q^{49} + ( - 2 \beta_{2} + 2 \beta_1 - 7) q^{50} + ( - \beta_{2} - \beta_1 + 2) q^{51} + (3 \beta_{2} - \beta_1 - 2) q^{52} + ( - \beta_1 + 1) q^{53} + \beta_1 q^{54} + (2 \beta_{2} - 5 \beta_1 + 5) q^{55} + ( - 2 \beta_1 - 1) q^{56} + ( - \beta_{2} + \beta_1 - 2) q^{57} + (3 \beta_{2} + 11) q^{58} + ( - 2 \beta_{2} - \beta_1 + 1) q^{59} + (\beta_{2} - 4 \beta_1 + 1) q^{60} + (\beta_{2} - 2 \beta_1 - 1) q^{61} + (\beta_{2} - 8 \beta_1 + 3) q^{62} - q^{63} + (2 \beta_1 - 7) q^{64} + ( - \beta_{2} - \beta_1 + 4) q^{65} + ( - \beta_{2} + 4 \beta_1 - 3) q^{66} + (2 \beta_{2} + 3 \beta_1 - 7) q^{67} + (2 \beta_{2} - 5 \beta_1 - 5) q^{68} - q^{69} + (\beta_{2} - \beta_1 + 4) q^{70} + (3 \beta_{2} + 2 \beta_1 + 3) q^{71} + (2 \beta_1 + 1) q^{72} + (\beta_{2} + 3 \beta_1 + 4) q^{73} + ( - 3 \beta_{2} + 4 \beta_1 - 11) q^{74} + (\beta_{2} - 2 \beta_1) q^{75} + ( - 2 \beta_{2} + 3 \beta_1 - 11) q^{76} + ( - \beta_{2} + \beta_1 - 2) q^{77} + (\beta_1 - 1) q^{78} + ( - \beta_{2} + 3 \beta_1 + 4) q^{79} + ( - \beta_{2} - 3 \beta_1) q^{80} + q^{81} + (\beta_{2} - 8 \beta_1 + 1) q^{82} + ( - \beta_{2} - \beta_1 + 4) q^{83} + ( - \beta_{2} - 2) q^{84} + ( - \beta_1 + 7) q^{85} + ( - 2 \beta_{2} - \beta_1 - 7) q^{86} + ( - \beta_{2} + 3 \beta_1 + 2) q^{87} + ( - \beta_{2} + 7 \beta_1 - 4) q^{88} + (\beta_{2} + 5) q^{89} + ( - \beta_{2} + \beta_1 - 4) q^{90} + (\beta_{2} - 3) q^{91} + ( - \beta_{2} - 2) q^{92} + ( - \beta_{2} + \beta_1 - 6) q^{93} + (10 \beta_1 + 2) q^{94} + ( - 2 \beta_{2} + 5 \beta_1 - 5) q^{95} + (\beta_{2} + 2) q^{96} + (\beta_{2} + 3 \beta_1 - 4) q^{97} + \beta_1 q^{98} + (\beta_{2} - \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 12 q^{10} + 6 q^{11} + 6 q^{12} + 9 q^{13} + 3 q^{15} + 12 q^{16} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{21} - 9 q^{22} - 3 q^{23} + 3 q^{24} - 3 q^{26} + 3 q^{27} - 6 q^{28} + 6 q^{29} - 12 q^{30} - 18 q^{31} + 6 q^{32} + 6 q^{33} - 15 q^{34} - 3 q^{35} + 6 q^{36} + 6 q^{37} + 9 q^{38} + 9 q^{39} - 21 q^{40} - 6 q^{41} - 9 q^{43} + 33 q^{44} + 3 q^{45} + 18 q^{47} + 12 q^{48} + 3 q^{49} - 21 q^{50} + 6 q^{51} - 6 q^{52} + 3 q^{53} + 15 q^{55} - 3 q^{56} - 6 q^{57} + 33 q^{58} + 3 q^{59} + 3 q^{60} - 3 q^{61} + 9 q^{62} - 3 q^{63} - 21 q^{64} + 12 q^{65} - 9 q^{66} - 21 q^{67} - 15 q^{68} - 3 q^{69} + 12 q^{70} + 9 q^{71} + 3 q^{72} + 12 q^{73} - 33 q^{74} - 33 q^{76} - 6 q^{77} - 3 q^{78} + 12 q^{79} + 3 q^{81} + 3 q^{82} + 12 q^{83} - 6 q^{84} + 21 q^{85} - 21 q^{86} + 6 q^{87} - 12 q^{88} + 15 q^{89} - 12 q^{90} - 9 q^{91} - 6 q^{92} - 18 q^{93} + 6 q^{94} - 15 q^{95} + 6 q^{96} - 12 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.36147
−0.167449
2.52892
−2.36147 1.00000 3.57653 3.36147 −2.36147 −1.00000 −3.72294 1.00000 −7.93800
1.2 −0.167449 1.00000 −1.97196 1.16745 −0.167449 −1.00000 0.665102 1.00000 −0.195488
1.3 2.52892 1.00000 4.39543 −1.52892 2.52892 −1.00000 6.05784 1.00000 −3.86651
\(n\): 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.h 3
3.b odd 2 1 1449.2.a.l 3
4.b odd 2 1 7728.2.a.bt 3
7.b odd 2 1 3381.2.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.h 3 1.a even 1 1 trivial
1449.2.a.l 3 3.b odd 2 1
3381.2.a.v 3 7.b odd 2 1
7728.2.a.bt 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(483))\):

\( T_{2}^{3} - 6T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5}^{2} - 3T_{5} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 6T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 3 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 9 T^{2} + \cdots - 6 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} + \cdots + 50 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} + \cdots + 262 \) Copy content Toggle raw display
$31$ \( T^{3} + 18 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} + \cdots - 50 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} + \cdots - 590 \) Copy content Toggle raw display
$43$ \( T^{3} + 9 T^{2} + \cdots - 124 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$59$ \( T^{3} - 3 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} + \cdots - 90 \) Copy content Toggle raw display
$67$ \( T^{3} + 21 T^{2} + \cdots - 908 \) Copy content Toggle raw display
$71$ \( T^{3} - 9 T^{2} + \cdots + 424 \) Copy content Toggle raw display
$73$ \( T^{3} - 12 T^{2} + \cdots - 10 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + \cdots + 320 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$89$ \( T^{3} - 15 T^{2} + \cdots - 50 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} + \cdots - 482 \) Copy content Toggle raw display
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