Properties

Label 6030.2.a.bo
Level $6030$
Weight $2$
Character orbit 6030.a
Self dual yes
Analytic conductor $48.150$
Analytic rank $1$
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] = [6030,2,Mod(1,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, 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("6030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.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: \(48.1497924188\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 670)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{7} - q^{8} - q^{10} - 3 q^{11} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_1 - 1) q^{14} + q^{16} + (2 \beta_{2} - \beta_1) q^{17} + 2 \beta_1 q^{19} + q^{20} + 3 q^{22} + ( - 2 \beta_1 - 2) q^{23} + q^{25} + (\beta_{2} - \beta_1) q^{26} + ( - \beta_1 + 1) q^{28} + ( - \beta_{2} + \beta_1 - 2) q^{29} + ( - \beta_{2} + 3 \beta_1) q^{31} - q^{32} + ( - 2 \beta_{2} + \beta_1) q^{34} + ( - \beta_1 + 1) q^{35} + (\beta_{2} - 3 \beta_1 + 1) q^{37} - 2 \beta_1 q^{38} - q^{40} + ( - 2 \beta_{2} + \beta_1 - 4) q^{41} + (2 \beta_{2} + \beta_1 + 2) q^{43} - 3 q^{44} + (2 \beta_1 + 2) q^{46} + ( - 3 \beta_{2} + 3 \beta_1) q^{47} + (\beta_{2} - 2 \beta_1 - 2) q^{49} - q^{50} + ( - \beta_{2} + \beta_1) q^{52} + (\beta_{2} - \beta_1) q^{53} - 3 q^{55} + (\beta_1 - 1) q^{56} + (\beta_{2} - \beta_1 + 2) q^{58} + (2 \beta_{2} - \beta_1 + 4) q^{59} + ( - \beta_{2} + 3 \beta_1 - 1) q^{61} + (\beta_{2} - 3 \beta_1) q^{62} + q^{64} + ( - \beta_{2} + \beta_1) q^{65} - q^{67} + (2 \beta_{2} - \beta_1) q^{68} + (\beta_1 - 1) q^{70} + (\beta_{2} - \beta_1 - 3) q^{71} + (\beta_1 - 8) q^{73} + ( - \beta_{2} + 3 \beta_1 - 1) q^{74} + 2 \beta_1 q^{76} + (3 \beta_1 - 3) q^{77} + (2 \beta_{2} - 2 \beta_1) q^{79} + q^{80} + (2 \beta_{2} - \beta_1 + 4) q^{82} + (\beta_{2} - 4 \beta_1 + 3) q^{83} + (2 \beta_{2} - \beta_1) q^{85} + ( - 2 \beta_{2} - \beta_1 - 2) q^{86} + 3 q^{88} + (2 \beta_{2} + 7) q^{89} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{91} + ( - 2 \beta_1 - 2) q^{92} + (3 \beta_{2} - 3 \beta_1) q^{94} + 2 \beta_1 q^{95} + (2 \beta_1 - 7) q^{97} + ( - \beta_{2} + 2 \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 9 q^{11} - 3 q^{14} + 3 q^{16} + 3 q^{20} + 9 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} - 3 q^{32} + 3 q^{35} + 3 q^{37} - 3 q^{40} - 12 q^{41} + 6 q^{43} - 9 q^{44} + 6 q^{46} - 6 q^{49} - 3 q^{50} - 9 q^{55} - 3 q^{56} + 6 q^{58} + 12 q^{59} - 3 q^{61} + 3 q^{64} - 3 q^{67} - 3 q^{70} - 9 q^{71} - 24 q^{73} - 3 q^{74} - 9 q^{77} + 3 q^{80} + 12 q^{82} + 9 q^{83} - 6 q^{86} + 9 q^{88} + 21 q^{89} - 6 q^{91} - 6 q^{92} - 21 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 2 \) : 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
2.60168
−0.339877
−2.26180
−1.00000 0 1.00000 1.00000 0 −1.60168 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 1.33988 −1.00000 0 −1.00000
1.3 −1.00000 0 1.00000 1.00000 0 3.26180 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

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 6030.2.a.bo 3
3.b odd 2 1 670.2.a.h 3
12.b even 2 1 5360.2.a.z 3
15.d odd 2 1 3350.2.a.k 3
15.e even 4 2 3350.2.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.a.h 3 3.b odd 2 1
3350.2.a.k 3 15.d odd 2 1
3350.2.c.j 6 15.e even 4 2
5360.2.a.z 3 12.b even 2 1
6030.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):

\( T_{7}^{3} - 3T_{7}^{2} - 3T_{7} + 7 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} - 12T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{3} - 42T_{17} + 98 \) Copy content Toggle raw display
\( T_{23}^{3} + 6T_{23}^{2} - 12T_{23} - 24 \) Copy content Toggle raw display
\( T_{29}^{3} + 6T_{29}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 3 T + 7 \) Copy content Toggle raw display
$11$ \( (T + 3)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 12T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 42T + 98 \) Copy content Toggle raw display
$19$ \( T^{3} - 24T - 16 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 12 T - 24 \) Copy content Toggle raw display
$29$ \( T^{3} + 6T^{2} - 18 \) Copy content Toggle raw display
$31$ \( T^{3} - 48T + 114 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 45 T - 67 \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} + 6 T - 202 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 54 T + 122 \) Copy content Toggle raw display
$47$ \( T^{3} - 108T - 54 \) Copy content Toggle raw display
$53$ \( T^{3} - 12T + 2 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + 6 T + 202 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 45 T + 67 \) Copy content Toggle raw display
$67$ \( (T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + 9 T^{2} + 15 T - 7 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 186 T + 462 \) Copy content Toggle raw display
$79$ \( T^{3} - 48T + 16 \) Copy content Toggle raw display
$83$ \( T^{3} - 9 T^{2} - 57 T + 29 \) Copy content Toggle raw display
$89$ \( T^{3} - 21 T^{2} + 99 T + 89 \) Copy content Toggle raw display
$97$ \( T^{3} + 21 T^{2} + 123 T + 159 \) Copy content Toggle raw display
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