Properties

Label 4830.2.a.cb
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 5 ) / 2 \) 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
2.11491
−1.86081
−0.254102
1.00000 −1.00000 1.00000 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 4830.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.cb 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{3} - 2T_{11}^{2} - 20T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 24T_{13} + 32 \) Copy content Toggle raw display
\( T_{17}^{3} - 16T_{17} - 8 \) Copy content Toggle raw display
\( T_{19}^{3} - 6T_{19}^{2} - 4T_{19} + 16 \) Copy content Toggle raw display
\( T_{29}^{3} - 4T_{29}^{2} - 16T_{29} + 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 20 T - 16 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 24 T + 32 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 4 T^{2} - 16 T + 56 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 4 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 52 T - 56 \) Copy content Toggle raw display
$43$ \( T^{3} - 22 T^{2} + 140 T - 208 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} - 4 T + 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$59$ \( T^{3} - 64T - 64 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 2 T^{2} - 20 T + 16 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 180 T + 1184 \) Copy content Toggle raw display
$73$ \( T^{3} - 14 T^{2} - 36 T + 248 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 156 T - 592 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} + 16 T - 224 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} - 200 T - 1184 \) Copy content Toggle raw display
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