Properties

Label 6042.2.a.u
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $5$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.14377697.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 5x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 9x^{3} + 5x^{2} + 19x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 3\nu + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 6\nu^{2} + 10\nu + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 9\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{3} + 15\beta_{2} + 29\beta _1 + 36 \) 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
−1.84112
−1.28745
−0.375482
1.83975
3.66430
1.00000 −1.00000 1.00000 −1.84112 −1.00000 −3.07196 1.00000 1.00000 −1.84112
1.2 1.00000 −1.00000 1.00000 −1.28745 −1.00000 −0.232422 1.00000 1.00000 −1.28745
1.3 1.00000 −1.00000 1.00000 −0.375482 −1.00000 3.10805 1.00000 1.00000 −0.375482
1.4 1.00000 −1.00000 1.00000 1.83975 −1.00000 4.29482 1.00000 1.00000 1.83975
1.5 1.00000 −1.00000 1.00000 3.66430 −1.00000 −2.09848 1.00000 1.00000 3.66430
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)
\(53\) \(-1\)

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 6042.2.a.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{5} - 2T_{5}^{4} - 9T_{5}^{3} + 5T_{5}^{2} + 19T_{5} + 6 \) Copy content Toggle raw display
\( T_{7}^{5} - 2T_{7}^{4} - 19T_{7}^{3} + 17T_{7}^{2} + 91T_{7} + 20 \) Copy content Toggle raw display
\( T_{11}^{5} + 3T_{11}^{4} - 24T_{11}^{3} - 77T_{11}^{2} + 44T_{11} + 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 9 T^{3} + 5 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$7$ \( T^{5} - 2 T^{4} - 19 T^{3} + 17 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$11$ \( T^{5} + 3 T^{4} - 24 T^{3} - 77 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$13$ \( T^{5} - 7 T^{4} - 4 T^{3} + 85 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$17$ \( T^{5} - 11 T^{4} + 4 T^{3} + 239 T^{2} + \cdots - 536 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} - 77 T^{3} + 263 T^{2} + \cdots + 84 \) Copy content Toggle raw display
$29$ \( T^{5} - 5 T^{4} - 39 T^{3} + 110 T^{2} + \cdots + 254 \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} - 20 T^{3} + 124 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$37$ \( T^{5} + 3 T^{4} - 24 T^{3} - 77 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$41$ \( T^{5} - 24 T^{4} + 195 T^{3} + \cdots + 860 \) Copy content Toggle raw display
$43$ \( T^{5} - 6 T^{4} - 89 T^{3} + \cdots + 2580 \) Copy content Toggle raw display
$47$ \( T^{5} + 23 T^{4} + 143 T^{3} + \cdots + 192 \) Copy content Toggle raw display
$53$ \( (T - 1)^{5} \) Copy content Toggle raw display
$59$ \( T^{5} - 11 T^{4} - 25 T^{3} + \cdots - 394 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} - 17 T^{3} + \cdots - 3000 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} - 89 T^{3} + \cdots - 4364 \) Copy content Toggle raw display
$71$ \( T^{5} + 10 T^{4} - 164 T^{3} + \cdots - 896 \) Copy content Toggle raw display
$73$ \( T^{5} - 9 T^{4} - 114 T^{3} + \cdots + 4560 \) Copy content Toggle raw display
$79$ \( T^{5} + 17 T^{4} + 20 T^{3} + \cdots + 3840 \) Copy content Toggle raw display
$83$ \( T^{5} - 9 T^{4} - 86 T^{3} + \cdots - 2280 \) Copy content Toggle raw display
$89$ \( T^{5} + 5 T^{4} - 65 T^{3} - 294 T^{2} + \cdots + 120 \) Copy content Toggle raw display
$97$ \( T^{5} - 13 T^{4} - 50 T^{3} + \cdots + 8388 \) Copy content Toggle raw display
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