Properties

Label 6006.2.a.bz
Level $6006$
Weight $2$
Character orbit 6006.a
Self dual yes
Analytic conductor $47.958$
Analytic rank $0$
Dimension $4$
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] = [6006,2,Mod(1,6006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6006, 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("6006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6006.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: \(47.9581514540\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.34196.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 8x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta _1 + 1 \) 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
2.86545
−2.57258
−0.897478
0.604608
−1.00000 1.00000 1.00000 −4.21081 −1.00000 1.00000 −1.00000 1.00000 4.21081
1.2 −1.00000 1.00000 1.00000 −2.61817 −1.00000 1.00000 −1.00000 1.00000 2.61817
1.3 −1.00000 1.00000 1.00000 3.19453 −1.00000 1.00000 −1.00000 1.00000 −3.19453
1.4 −1.00000 1.00000 1.00000 3.63445 −1.00000 1.00000 −1.00000 1.00000 −3.63445
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

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 6006.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6006.2.a.bz 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6006))\):

\( T_{5}^{4} - 24T_{5}^{2} + 4T_{5} + 128 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} - 32T_{17}^{2} + 32T_{17} + 224 \) Copy content Toggle raw display
\( T_{19}^{4} - 32T_{19}^{2} + 16T_{19} + 64 \) Copy content Toggle raw display
\( T_{23}^{4} - 2T_{23}^{3} - 32T_{23}^{2} + 64 \) Copy content Toggle raw display
\( T_{31}^{4} - 6T_{31}^{3} - 36T_{31}^{2} + 4T_{31} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 24 T^{2} + 4 T + 128 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 32 T^{2} + 32 T + 224 \) Copy content Toggle raw display
$19$ \( T^{4} - 32 T^{2} + 16 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} - 32 T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + 88 T^{2} - 64 T - 32 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} - 36 T^{2} + 4 T + 32 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 24 T^{2} + 76 T - 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} - 60 T^{2} + \cdots + 1792 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} - 36 T^{2} - 4 T + 32 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + 40 T^{2} + \cdots - 320 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} - 84 T^{2} + \cdots + 2144 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} - 92 T^{2} - 524 T - 512 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} - 128 T^{2} + \cdots + 496 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} - 36 T^{2} - 4 T + 32 \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} - 92 T^{2} - 524 T - 512 \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} - 188 T^{2} + \cdots - 3200 \) Copy content Toggle raw display
$83$ \( T^{4} + 22 T^{3} - 56 T^{2} + \cdots - 13184 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} - 200 T^{2} + \cdots - 4448 \) Copy content Toggle raw display
$97$ \( T^{4} - 24 T^{3} - 60 T^{2} + \cdots - 11560 \) Copy content Toggle raw display
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