Properties

Label 6042.2.a.v
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.21848308.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 4x^{4} + 9x^{3} + 6x^{2} - 4x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - 4\nu^{4} + 9\nu^{2} - 3\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 9\nu^{2} + 5\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + 4\nu^{4} + \nu^{3} - 12\nu^{2} + \nu + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 7\nu^{4} - 4\nu^{3} + 19\nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} + \beta_{4} - 3\beta_{3} - 2\beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{5} + 4\beta_{4} - 11\beta_{3} - 9\beta_{2} + 12\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{5} + 16\beta_{4} - 35\beta_{3} - 26\beta_{2} + 42\beta _1 + 20 \) 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
1.73472
−0.810057
−1.45853
3.24366
0.712460
−0.422254
−1.00000 1.00000 1.00000 −2.64265 −1.00000 −1.21612 −1.00000 1.00000 2.64265
1.2 −1.00000 1.00000 1.00000 −1.14477 −1.00000 2.60585 −1.00000 1.00000 1.14477
1.3 −1.00000 1.00000 1.00000 0.613278 −1.00000 −3.09360 −1.00000 1.00000 −0.613278
1.4 −1.00000 1.00000 1.00000 0.840706 −1.00000 −3.39143 −1.00000 1.00000 −0.840706
1.5 −1.00000 1.00000 1.00000 1.82997 −1.00000 1.67867 −1.00000 1.00000 −1.82997
1.6 −1.00000 1.00000 1.00000 3.50346 −1.00000 −3.58336 −1.00000 1.00000 −3.50346
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.v 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{6} - 3T_{5}^{5} - 8T_{5}^{4} + 23T_{5}^{3} + 2T_{5}^{2} - 24T_{5} + 10 \) Copy content Toggle raw display
\( T_{7}^{6} + 7T_{7}^{5} + 2T_{7}^{4} - 69T_{7}^{3} - 90T_{7}^{2} + 148T_{7} + 200 \) Copy content Toggle raw display
\( T_{11}^{6} - T_{11}^{5} - 28T_{11}^{4} - 31T_{11}^{3} + 102T_{11}^{2} + 142T_{11} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} - 8 T^{4} + 23 T^{3} + \cdots + 10 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + 2 T^{4} - 69 T^{3} + \cdots + 200 \) Copy content Toggle raw display
$11$ \( T^{6} - T^{5} - 28 T^{4} - 31 T^{3} + \cdots - 18 \) Copy content Toggle raw display
$13$ \( T^{6} + 13 T^{5} + 56 T^{4} + 75 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{5} - 42 T^{4} + \cdots - 1576 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 9 T^{5} - 14 T^{4} + \cdots - 1156 \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} - 5 T^{4} + \cdots - 2556 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} - 184 T^{4} + \cdots + 44832 \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{5} - 144 T^{4} + \cdots + 74236 \) Copy content Toggle raw display
$41$ \( T^{6} + 18 T^{5} + 89 T^{4} + \cdots - 2400 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} - 86 T^{4} + \cdots - 29816 \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{5} - 53 T^{4} - 387 T^{3} + \cdots - 134 \) Copy content Toggle raw display
$53$ \( (T + 1)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} - 153 T^{4} + \cdots - 4624 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} - 159 T^{4} + \cdots - 8200 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} - 106 T^{4} + \cdots + 5024 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 80 T^{4} + \cdots + 7200 \) Copy content Toggle raw display
$73$ \( T^{6} + 31 T^{5} + 308 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$79$ \( T^{6} + 17 T^{5} - 34 T^{4} + \cdots + 18572 \) Copy content Toggle raw display
$83$ \( T^{6} + 9 T^{5} - 246 T^{4} + \cdots - 137440 \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} - 111 T^{4} + \cdots - 2120 \) Copy content Toggle raw display
$97$ \( T^{6} + 7 T^{5} - 376 T^{4} + \cdots - 899480 \) Copy content Toggle raw display
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