Properties

Label 6069.2.a.p
Level $6069$
Weight $2$
Character orbit 6069.a
Self dual yes
Analytic conductor $48.461$
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] = [6069,2,Mod(1,6069)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, 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("6069.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.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: \(48.4612089867\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.87939
−0.347296
−1.53209
−1.87939 1.00000 1.53209 −0.347296 −1.87939 −1.00000 0.879385 1.00000 0.652704
1.2 0.347296 1.00000 −1.87939 −1.53209 0.347296 −1.00000 −1.34730 1.00000 −0.532089
1.3 1.53209 1.00000 0.347296 1.87939 1.53209 −1.00000 −2.53209 1.00000 2.87939
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(17\) \(-1\)

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 6069.2.a.p yes 3
17.b even 2 1 6069.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6069.2.a.n 3 17.b even 2 1
6069.2.a.p yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):

\( T_{2}^{3} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} + 3T_{11} - 19 \) Copy content Toggle raw display
\( T_{23}^{3} - 27T_{23} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 3T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + 3 T - 19 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} - 6 T + 17 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 6 T + 17 \) Copy content Toggle raw display
$23$ \( T^{3} - 27T + 27 \) Copy content Toggle raw display
$29$ \( T^{3} - 3 T^{2} - 33 T - 37 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 9 T - 3 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 90 T + 111 \) Copy content Toggle raw display
$41$ \( T^{3} + 15 T^{2} + 36 T + 19 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 60 T - 71 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 45 T - 213 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} - 126 T + 57 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 144 T + 1143 \) Copy content Toggle raw display
$61$ \( T^{3} + 9 T^{2} - 30 T - 37 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} - 81 T - 379 \) Copy content Toggle raw display
$71$ \( T^{3} - 3 T^{2} - 81 T - 213 \) Copy content Toggle raw display
$73$ \( T^{3} + 15 T^{2} + 27 T - 51 \) Copy content Toggle raw display
$79$ \( (T + 7)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 9 T + 51 \) Copy content Toggle raw display
$89$ \( (T - 1)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 3 T^{2} - 108 T - 543 \) Copy content Toggle raw display
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