Properties

Label 6018.2.a.w
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 22x^{7} + 20x^{6} + 129x^{5} - 106x^{4} - 126x^{3} + 48x^{2} + 24x - 8 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 22x^{7} + 20x^{6} + 129x^{5} - 106x^{4} - 126x^{3} + 48x^{2} + 24x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{8} - 4\nu^{7} + 74\nu^{6} + 91\nu^{5} - 543\nu^{4} - 533\nu^{3} + 1171\nu^{2} + 504\nu - 300 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -45\nu^{8} + 5\nu^{7} + 1032\nu^{6} - 26\nu^{5} - 6637\nu^{4} - 234\nu^{3} + 9908\nu^{2} + 1840\nu - 1900 ) / 52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{8} - 28\nu^{7} + 518\nu^{6} + 637\nu^{5} - 3827\nu^{4} - 3653\nu^{3} + 8405\nu^{2} + 2774\nu - 1866 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\nu^{8} - 84\nu^{7} - 838\nu^{6} + 1729\nu^{5} + 3989\nu^{4} - 9139\nu^{3} + 1633\nu^{2} + 3434\nu - 1152 ) / 26 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 137 \nu^{8} + 125 \nu^{7} + 2998 \nu^{6} - 2444 \nu^{5} - 17309 \nu^{4} + 12350 \nu^{3} + 15182 \nu^{2} - 1944 \nu - 1740 ) / 52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{8} - 9\nu^{7} - 242\nu^{6} + 174\nu^{5} + 1415\nu^{4} - 868\nu^{3} - 1344\nu^{2} + 68\nu + 160 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 134 \nu^{8} + 129 \nu^{7} + 2924 \nu^{6} - 2535 \nu^{5} - 16766 \nu^{4} + 12857 \nu^{3} + 14011 \nu^{2} - 2162 \nu - 1414 ) / 26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 2\beta_{6} - \beta_{2} + 11\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} + 8\beta_{7} + 14\beta_{6} - \beta_{4} + 12\beta_{2} + 12\beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{8} + 3\beta_{7} + 29\beta_{6} - 7\beta_{5} + 4\beta_{4} + 8\beta_{3} - 15\beta_{2} + 125\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -61\beta_{8} + 58\beta_{7} + 186\beta_{6} + 9\beta_{5} - 34\beta_{4} + 10\beta_{3} + 146\beta_{2} + 150\beta _1 + 595 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 248 \beta_{8} + 66 \beta_{7} + 398 \beta_{6} - 139 \beta_{5} + 68 \beta_{4} + 184 \beta_{3} - 193 \beta_{2} + 1469 \beta _1 + 408 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 969 \beta_{8} + 376 \beta_{7} + 2438 \beta_{6} + 195 \beta_{5} - 627 \beta_{4} + 244 \beta_{3} + 1791 \beta_{2} + 1965 \beta _1 + 7062 \) 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
−3.44871
−2.94492
−0.637772
−0.621481
0.354931
0.374254
1.44979
2.83873
3.63517
−1.00000 1.00000 1.00000 −3.44871 −1.00000 −0.188516 −1.00000 1.00000 3.44871
1.2 −1.00000 1.00000 1.00000 −2.94492 −1.00000 −3.69752 −1.00000 1.00000 2.94492
1.3 −1.00000 1.00000 1.00000 −0.637772 −1.00000 −2.55112 −1.00000 1.00000 0.637772
1.4 −1.00000 1.00000 1.00000 −0.621481 −1.00000 2.55275 −1.00000 1.00000 0.621481
1.5 −1.00000 1.00000 1.00000 0.354931 −1.00000 1.61788 −1.00000 1.00000 −0.354931
1.6 −1.00000 1.00000 1.00000 0.374254 −1.00000 −3.42020 −1.00000 1.00000 −0.374254
1.7 −1.00000 1.00000 1.00000 1.44979 −1.00000 4.37806 −1.00000 1.00000 −1.44979
1.8 −1.00000 1.00000 1.00000 2.83873 −1.00000 −0.0272276 −1.00000 1.00000 −2.83873
1.9 −1.00000 1.00000 1.00000 3.63517 −1.00000 1.33590 −1.00000 1.00000 −3.63517
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.w 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.w 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{9} - T_{5}^{8} - 22T_{5}^{7} + 20T_{5}^{6} + 129T_{5}^{5} - 106T_{5}^{4} - 126T_{5}^{3} + 48T_{5}^{2} + 24T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{9} - 31T_{7}^{7} + 284T_{7}^{5} - 93T_{7}^{4} - 836T_{7}^{3} + 605T_{7}^{2} + 164T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} - 22 T^{7} + 20 T^{6} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{9} - 31 T^{7} + 284 T^{5} - 93 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{9} - 6 T^{8} - 14 T^{7} + 111 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{9} - 2 T^{8} - 49 T^{7} + \cdots + 1304 \) Copy content Toggle raw display
$17$ \( (T - 1)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + 5 T^{8} - 55 T^{7} + \cdots + 2116 \) Copy content Toggle raw display
$23$ \( T^{9} - 15 T^{8} + 47 T^{7} + \cdots + 484 \) Copy content Toggle raw display
$29$ \( T^{9} - 11 T^{8} - 22 T^{7} + \cdots - 91064 \) Copy content Toggle raw display
$31$ \( T^{9} + 5 T^{8} - 82 T^{7} + \cdots - 67888 \) Copy content Toggle raw display
$37$ \( T^{9} - 9 T^{8} - 112 T^{7} + \cdots - 429224 \) Copy content Toggle raw display
$41$ \( T^{9} - T^{8} - 177 T^{7} + 448 T^{6} + \cdots + 3124 \) Copy content Toggle raw display
$43$ \( T^{9} - 4 T^{8} - 226 T^{7} + \cdots - 2366464 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} - 142 T^{7} + \cdots - 6150496 \) Copy content Toggle raw display
$53$ \( T^{9} - 4 T^{8} - 195 T^{7} + \cdots + 305006 \) Copy content Toggle raw display
$59$ \( (T - 1)^{9} \) Copy content Toggle raw display
$61$ \( T^{9} - 10 T^{8} - 328 T^{7} + \cdots + 25785136 \) Copy content Toggle raw display
$67$ \( T^{9} + T^{8} - 183 T^{7} + \cdots + 110272 \) Copy content Toggle raw display
$71$ \( T^{9} - 14 T^{8} - 194 T^{7} + \cdots - 480152 \) Copy content Toggle raw display
$73$ \( T^{9} + T^{8} - 353 T^{7} + \cdots - 12012508 \) Copy content Toggle raw display
$79$ \( T^{9} - 4 T^{8} - 336 T^{7} + \cdots + 780848 \) Copy content Toggle raw display
$83$ \( T^{9} - 22 T^{8} - 138 T^{7} + \cdots + 16800752 \) Copy content Toggle raw display
$89$ \( T^{9} - 22 T^{8} - 118 T^{7} + \cdots + 522112 \) Copy content Toggle raw display
$97$ \( T^{9} + 15 T^{8} - 256 T^{7} + \cdots + 4888004 \) Copy content Toggle raw display
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