Properties

Label 6018.2.a.x
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $10$
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 34x^{8} + 30x^{7} + 341x^{6} - 276x^{5} - 1032x^{4} + 1176x^{3} + 416x^{2} - 896x + 272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 34x^{8} + 30x^{7} + 341x^{6} - 276x^{5} - 1032x^{4} + 1176x^{3} + 416x^{2} - 896x + 272 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 158 \nu^{9} + 225 \nu^{8} - 4779 \nu^{7} - 5750 \nu^{6} + 32040 \nu^{5} + 12263 \nu^{4} + 67732 \nu^{3} + 300692 \nu^{2} - 399092 \nu - 66416 ) / 120304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1042 \nu^{9} - 3275 \nu^{8} - 35705 \nu^{7} + 105416 \nu^{6} + 375768 \nu^{5} - 987205 \nu^{4} - 1375194 \nu^{3} + 2975172 \nu^{2} + 843124 \nu - 1593080 ) / 120304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5903 \nu^{9} - 5918 \nu^{8} - 207243 \nu^{7} + 177972 \nu^{6} + 2223525 \nu^{5} - 1593723 \nu^{4} - 7953632 \nu^{3} + 6020464 \nu^{2} + 6322324 \nu - 4079568 ) / 120304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11493 \nu^{9} - 1955 \nu^{8} - 393074 \nu^{7} + 21556 \nu^{6} + 3954805 \nu^{5} + 26618 \nu^{4} - 11927502 \nu^{3} + 4205884 \nu^{2} + 8148160 \nu - 4432912 ) / 60152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24821 \nu^{9} + 10767 \nu^{8} + 848932 \nu^{7} - 255106 \nu^{6} - 8588857 \nu^{5} + 1739438 \nu^{4} + 26641132 \nu^{3} - 12453916 \nu^{2} + \cdots + 10265344 ) / 120304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20873 \nu^{9} - 4492 \nu^{8} - 711529 \nu^{7} + 62998 \nu^{6} + 7121451 \nu^{5} - 52371 \nu^{4} - 21305118 \nu^{3} + 7212112 \nu^{2} + 14231612 \nu - 7071528 ) / 60152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73925 \nu^{9} + 15269 \nu^{8} + 2527192 \nu^{7} - 221448 \nu^{6} - 25415913 \nu^{5} + 470848 \nu^{4} + 76622382 \nu^{3} - 27434996 \nu^{2} + \cdots + 26070664 ) / 120304 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 89793 \nu^{9} - 16752 \nu^{8} - 3064731 \nu^{7} + 197524 \nu^{6} + 30729695 \nu^{5} + 289317 \nu^{4} - 92137756 \nu^{3} + 30417112 \nu^{2} + \cdots - 30442128 ) / 120304 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_{2} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} - 30\beta_{7} - 22\beta_{6} + 13\beta_{5} - 22\beta_{4} + 2\beta_{3} - 22\beta_{2} + 95 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 38 \beta_{9} - 12 \beta_{8} + 26 \beta_{7} - 41 \beta_{6} + 24 \beta_{5} - 11 \beta_{4} - 72 \beta_{3} + 65 \beta_{2} + 250 \beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 177 \beta_{9} + 11 \beta_{8} - 659 \beta_{7} - 426 \beta_{6} + 189 \beta_{5} - 422 \beta_{4} + 63 \beta_{3} - 394 \beta_{2} - 27 \beta _1 + 1517 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 625 \beta_{9} - 112 \beta_{8} + 472 \beta_{7} - 790 \beta_{6} + 458 \beta_{5} - 127 \beta_{4} - 1430 \beta_{3} + 1522 \beta_{2} + 4330 \beta _1 + 367 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3987 \beta_{9} + 398 \beta_{8} - 13224 \beta_{7} - 8058 \beta_{6} + 3027 \beta_{5} - 7867 \beta_{4} + 1342 \beta_{3} - 6830 \beta_{2} - 996 \beta _1 + 25886 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10043 \beta_{9} - 692 \beta_{8} + 7656 \beta_{7} - 15141 \beta_{6} + 8213 \beta_{5} - 1726 \beta_{4} - 27140 \beta_{3} + 31757 \beta_{2} + 76804 \beta _1 + 5520 \) 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
−4.29748
−2.78078
−2.54849
−0.976160
0.580342
0.780661
0.931948
1.39621
3.58712
4.32662
−1.00000 −1.00000 1.00000 −4.29748 1.00000 0.914984 −1.00000 1.00000 4.29748
1.2 −1.00000 −1.00000 1.00000 −2.78078 1.00000 3.42043 −1.00000 1.00000 2.78078
1.3 −1.00000 −1.00000 1.00000 −2.54849 1.00000 4.63796 −1.00000 1.00000 2.54849
1.4 −1.00000 −1.00000 1.00000 −0.976160 1.00000 −3.87477 −1.00000 1.00000 0.976160
1.5 −1.00000 −1.00000 1.00000 0.580342 1.00000 4.77865 −1.00000 1.00000 −0.580342
1.6 −1.00000 −1.00000 1.00000 0.780661 1.00000 1.13694 −1.00000 1.00000 −0.780661
1.7 −1.00000 −1.00000 1.00000 0.931948 1.00000 −0.903548 −1.00000 1.00000 −0.931948
1.8 −1.00000 −1.00000 1.00000 1.39621 1.00000 −1.23174 −1.00000 1.00000 −1.39621
1.9 −1.00000 −1.00000 1.00000 3.58712 1.00000 −2.07087 −1.00000 1.00000 −3.58712
1.10 −1.00000 −1.00000 1.00000 4.32662 1.00000 3.19198 −1.00000 1.00000 −4.32662
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.x 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{10} - T_{5}^{9} - 34 T_{5}^{8} + 30 T_{5}^{7} + 341 T_{5}^{6} - 276 T_{5}^{5} - 1032 T_{5}^{4} + 1176 T_{5}^{3} + 416 T_{5}^{2} - 896 T_{5} + 272 \) Copy content Toggle raw display
\( T_{7}^{10} - 10 T_{7}^{9} + 5 T_{7}^{8} + 212 T_{7}^{7} - 476 T_{7}^{6} - 1035 T_{7}^{5} + 2952 T_{7}^{4} + 1663 T_{7}^{3} - 4842 T_{7}^{2} - 772 T_{7} + 2248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} - 34 T^{8} + 30 T^{7} + \cdots + 272 \) Copy content Toggle raw display
$7$ \( T^{10} - 10 T^{9} + 5 T^{8} + \cdots + 2248 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} - 64 T^{8} + 203 T^{7} + \cdots + 640 \) Copy content Toggle raw display
$13$ \( T^{10} - 91 T^{8} + 82 T^{7} + \cdots + 35312 \) Copy content Toggle raw display
$17$ \( (T - 1)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} - 15 T^{9} + 9 T^{8} + \cdots + 78064 \) Copy content Toggle raw display
$23$ \( T^{10} - 19 T^{9} + 69 T^{8} + 794 T^{7} + \cdots + 88 \) Copy content Toggle raw display
$29$ \( T^{10} + T^{9} - 138 T^{8} + \cdots - 511344 \) Copy content Toggle raw display
$31$ \( T^{10} - 15 T^{9} - 76 T^{8} + \cdots + 51744 \) Copy content Toggle raw display
$37$ \( T^{10} - T^{9} - 186 T^{8} + 184 T^{7} + \cdots - 1712 \) Copy content Toggle raw display
$41$ \( T^{10} + 5 T^{9} - 85 T^{8} + \cdots + 9896 \) Copy content Toggle raw display
$43$ \( T^{10} - 26 T^{9} + 82 T^{8} + \cdots - 21394944 \) Copy content Toggle raw display
$47$ \( T^{10} - 14 T^{9} - 122 T^{8} + \cdots + 6055808 \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} - 125 T^{8} + \cdots + 403604 \) Copy content Toggle raw display
$59$ \( (T + 1)^{10} \) Copy content Toggle raw display
$61$ \( T^{10} - 4 T^{9} - 132 T^{8} + \cdots - 457888 \) Copy content Toggle raw display
$67$ \( T^{10} - 15 T^{9} - 241 T^{8} + \cdots + 17017344 \) Copy content Toggle raw display
$71$ \( T^{10} - 14 T^{9} - 230 T^{8} + \cdots + 94416 \) Copy content Toggle raw display
$73$ \( T^{10} - 43 T^{9} + \cdots + 246308376 \) Copy content Toggle raw display
$79$ \( T^{10} - 362 T^{8} + \cdots + 42600416 \) Copy content Toggle raw display
$83$ \( T^{10} + 4 T^{9} - 376 T^{8} + \cdots + 148032 \) Copy content Toggle raw display
$89$ \( T^{10} + 22 T^{9} - 122 T^{8} + \cdots + 68934200 \) Copy content Toggle raw display
$97$ \( T^{10} - 37 T^{9} + 118 T^{8} + \cdots - 31502568 \) Copy content Toggle raw display
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