Properties

Label 6018.2.a.r
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.18461324.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 4x^{4} + 12x^{3} + 3x^{2} - 6x - 2 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 4x^{4} + 12x^{3} + 3x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 10\nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 4\nu^{4} - \nu^{3} + 15\nu^{2} - 7\nu - 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{5} + 7\nu^{4} + 5\nu^{3} - 27\nu^{2} + 5\nu + 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + \beta_{4} + 3\beta_{3} + 9\beta_{2} + 9\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 6\beta_{4} + 13\beta_{3} + 23\beta_{2} + 33\beta _1 + 12 \) 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
2.94685
1.91714
0.908132
−0.380739
−0.558656
−1.83273
1.00000 1.00000 1.00000 −2.94685 1.00000 2.59293 1.00000 1.00000 −2.94685
1.2 1.00000 1.00000 1.00000 −1.91714 1.00000 −1.73720 1.00000 1.00000 −1.91714
1.3 1.00000 1.00000 1.00000 −0.908132 1.00000 −1.54786 1.00000 1.00000 −0.908132
1.4 1.00000 1.00000 1.00000 0.380739 1.00000 1.68872 1.00000 1.00000 0.380739
1.5 1.00000 1.00000 1.00000 0.558656 1.00000 −3.85965 1.00000 1.00000 0.558656
1.6 1.00000 1.00000 1.00000 1.83273 1.00000 −4.13695 1.00000 1.00000 1.83273
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

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 6018.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{6} + 3T_{5}^{5} - 4T_{5}^{4} - 12T_{5}^{3} + 3T_{5}^{2} + 6T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} + 7T_{7}^{5} + T_{7}^{4} - 69T_{7}^{3} - 77T_{7}^{2} + 140T_{7} + 188 \) 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
$5$ \( T^{6} + 3 T^{5} - 4 T^{4} - 12 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + T^{4} - 69 T^{3} + \cdots + 188 \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} - 2 T^{4} - 155 T^{3} + \cdots + 668 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} - 27 T^{4} - 188 T^{3} + \cdots - 394 \) Copy content Toggle raw display
$17$ \( (T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 14 T^{5} + 28 T^{4} + \cdots + 2524 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} - 80 T^{4} + \cdots - 5788 \) Copy content Toggle raw display
$29$ \( T^{6} + 21 T^{5} + 134 T^{4} + \cdots + 578 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} - 66 T^{4} + \cdots + 1642 \) Copy content Toggle raw display
$37$ \( T^{6} + 13 T^{5} - 48 T^{4} + \cdots + 6454 \) Copy content Toggle raw display
$41$ \( T^{6} + 18 T^{5} + 66 T^{4} + \cdots - 1772 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} - 176 T^{4} + \cdots + 58640 \) Copy content Toggle raw display
$47$ \( T^{6} + 4 T^{5} - 144 T^{4} + \cdots - 34088 \) Copy content Toggle raw display
$53$ \( T^{6} + 25 T^{5} + 167 T^{4} + \cdots - 15226 \) Copy content Toggle raw display
$59$ \( (T + 1)^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 58 T^{4} + 129 T^{3} + \cdots + 1346 \) Copy content Toggle raw display
$67$ \( T^{6} + 13 T^{5} - 49 T^{4} + \cdots + 18320 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} - 218 T^{4} + \cdots + 201298 \) Copy content Toggle raw display
$73$ \( T^{6} + 4 T^{5} - 354 T^{4} + \cdots + 825268 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} - 62 T^{4} + \cdots + 8932 \) Copy content Toggle raw display
$83$ \( T^{6} - 9 T^{5} - 182 T^{4} + \cdots - 29872 \) Copy content Toggle raw display
$89$ \( T^{6} + 11 T^{5} - 8 T^{4} - 288 T^{3} + \cdots - 56 \) Copy content Toggle raw display
$97$ \( T^{6} - 16 T^{5} - 291 T^{4} + \cdots - 1467212 \) Copy content Toggle raw display
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