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Properties

Label 6018.2.a.z
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4 x^{10} - 27 x^{9} + 117 x^{8} + 200 x^{7} - 1023 x^{6} - 484 x^{5} + 3403 x^{4} + 562 x^{3} + \cdots + 1200 \) 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 \(1,\beta_1,\ldots,\beta_{10}\) 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 \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} - \beta_{2} q^{7} + q^{8} + q^{9} + \beta_1 q^{10} + (\beta_{4} + 1) q^{11} - q^{12} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 3 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 3 q^{7} + 11 q^{8} + 11 q^{9} + 4 q^{10} + 9 q^{11} - 11 q^{12} + 6 q^{13} + 3 q^{14} - 4 q^{15} + 11 q^{16} - 11 q^{17} + 11 q^{18} - q^{19} + 4 q^{20} - 3 q^{21} + 9 q^{22} + 10 q^{23} - 11 q^{24} + 15 q^{25} + 6 q^{26} - 11 q^{27} + 3 q^{28} + 14 q^{29} - 4 q^{30} + 17 q^{31} + 11 q^{32} - 9 q^{33} - 11 q^{34} + 8 q^{35} + 11 q^{36} + 30 q^{37} - q^{38} - 6 q^{39} + 4 q^{40} + 10 q^{41} - 3 q^{42} + 11 q^{43} + 9 q^{44} + 4 q^{45} + 10 q^{46} - 6 q^{47} - 11 q^{48} + 18 q^{49} + 15 q^{50} + 11 q^{51} + 6 q^{52} + 10 q^{53} - 11 q^{54} - 11 q^{55} + 3 q^{56} + q^{57} + 14 q^{58} - 11 q^{59} - 4 q^{60} + 13 q^{61} + 17 q^{62} + 3 q^{63} + 11 q^{64} + 32 q^{65} - 9 q^{66} + 26 q^{67} - 11 q^{68} - 10 q^{69} + 8 q^{70} + 14 q^{71} + 11 q^{72} + 20 q^{73} + 30 q^{74} - 15 q^{75} - q^{76} + 26 q^{77} - 6 q^{78} + 15 q^{79} + 4 q^{80} + 11 q^{81} + 10 q^{82} + 2 q^{83} - 3 q^{84} - 4 q^{85} + 11 q^{86} - 14 q^{87} + 9 q^{88} + q^{89} + 4 q^{90} + 17 q^{91} + 10 q^{92} - 17 q^{93} - 6 q^{94} + 3 q^{95} - 11 q^{96} + 33 q^{97} + 18 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4 x^{10} - 27 x^{9} + 117 x^{8} + 200 x^{7} - 1023 x^{6} - 484 x^{5} + 3403 x^{4} + 562 x^{3} + \cdots + 1200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 68377 \nu^{10} + 100902 \nu^{9} + 1972995 \nu^{8} - 2438247 \nu^{7} - 16451050 \nu^{6} + \cdots - 6530016 ) / 9881348 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 478220 \nu^{10} - 806963 \nu^{9} - 14837488 \nu^{8} + 21958815 \nu^{7} + 147883587 \nu^{6} + \cdots - 277474058 ) / 4940674 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 690399 \nu^{10} + 1061578 \nu^{9} + 21837033 \nu^{8} - 28769331 \nu^{7} - 224589340 \nu^{6} + \cdots + 437533206 ) / 4940674 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3156869 \nu^{10} + 5380560 \nu^{9} + 98175115 \nu^{8} - 145767653 \nu^{7} - 981973340 \nu^{6} + \cdots + 1824506128 ) / 9881348 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1183567 \nu^{10} - 2051153 \nu^{9} - 36518360 \nu^{8} + 55439219 \nu^{7} + 359948269 \nu^{6} + \cdots - 599619469 ) / 2470337 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4973375 \nu^{10} - 8328764 \nu^{9} - 154431069 \nu^{8} + 224826723 \nu^{7} + 1539701372 \nu^{6} + \cdots - 2734225624 ) / 9881348 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1707617 \nu^{10} - 2986333 \nu^{9} - 52746178 \nu^{8} + 80817374 \nu^{7} + 521178581 \nu^{6} + \cdots - 894863368 ) / 2470337 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10702739 \nu^{10} - 18943456 \nu^{9} - 330397033 \nu^{8} + 513011107 \nu^{7} + 3262430232 \nu^{6} + \cdots - 5650580696 ) / 9881348 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20411677 \nu^{10} + 35419276 \nu^{9} + 632398343 \nu^{8} - 959470865 \nu^{7} + \cdots + 10921660816 ) / 9881348 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2\beta_{9} - \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} + \beta_{2} + 10\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{10} - 20 \beta_{9} + 36 \beta_{8} + \beta_{7} - 16 \beta_{6} - 16 \beta_{5} + 20 \beta_{4} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{10} + 44 \beta_{9} - 26 \beta_{8} - 18 \beta_{7} + 37 \beta_{6} + 29 \beta_{5} - 83 \beta_{4} + \cdots - 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 25 \beta_{10} - 337 \beta_{9} + 572 \beta_{8} + 32 \beta_{7} - 251 \beta_{6} - 253 \beta_{5} + \cdots + 901 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 312 \beta_{10} + 824 \beta_{9} - 587 \beta_{8} - 300 \beta_{7} + 643 \beta_{6} + 596 \beta_{5} + \cdots - 937 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 512 \beta_{10} - 5508 \beta_{9} + 8945 \beta_{8} + 708 \beta_{7} - 4004 \beta_{6} - 4024 \beta_{5} + \cdots + 13390 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4996 \beta_{10} + 14746 \beta_{9} - 12143 \beta_{8} - 4849 \beta_{7} + 11135 \beta_{6} + 11092 \beta_{5} + \cdots - 19052 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9750 \beta_{10} - 89608 \beta_{9} + 140335 \beta_{8} + 13689 \beta_{7} - 64588 \beta_{6} + \cdots + 206442 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−4.08681
−2.64826
−1.55818
−1.33314
−0.760310
0.504544
2.03178
2.25177
2.25950
3.59322
3.74589
1.00000 −1.00000 1.00000 −4.08681 −1.00000 −0.153752 1.00000 1.00000 −4.08681
1.2 1.00000 −1.00000 1.00000 −2.64826 −1.00000 3.55508 1.00000 1.00000 −2.64826
1.3 1.00000 −1.00000 1.00000 −1.55818 −1.00000 −1.66703 1.00000 1.00000 −1.55818
1.4 1.00000 −1.00000 1.00000 −1.33314 −1.00000 −3.20872 1.00000 1.00000 −1.33314
1.5 1.00000 −1.00000 1.00000 −0.760310 −1.00000 −1.62737 1.00000 1.00000 −0.760310
1.6 1.00000 −1.00000 1.00000 0.504544 −1.00000 2.39214 1.00000 1.00000 0.504544
1.7 1.00000 −1.00000 1.00000 2.03178 −1.00000 −2.78135 1.00000 1.00000 2.03178
1.8 1.00000 −1.00000 1.00000 2.25177 −1.00000 3.43625 1.00000 1.00000 2.25177
1.9 1.00000 −1.00000 1.00000 2.25950 −1.00000 3.73494 1.00000 1.00000 2.25950
1.10 1.00000 −1.00000 1.00000 3.59322 −1.00000 3.34551 1.00000 1.00000 3.59322
1.11 1.00000 −1.00000 1.00000 3.74589 −1.00000 −4.02570 1.00000 1.00000 3.74589
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.z ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.z ✓ 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{11} - 4 T_{5}^{10} - 27 T_{5}^{9} + 117 T_{5}^{8} + 200 T_{5}^{7} - 1023 T_{5}^{6} - 484 T_{5}^{5} + \cdots + 1200 \) Copy content Toggle raw display
\( T_{7}^{11} - 3 T_{7}^{10} - 43 T_{7}^{9} + 118 T_{7}^{8} + 704 T_{7}^{7} - 1628 T_{7}^{6} - 5627 T_{7}^{5} + \cdots - 5472 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{11} \) Copy content Toggle raw display
$3$ \( (T + 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} - 4 T^{10} + \cdots + 1200 \) Copy content Toggle raw display
$7$ \( T^{11} - 3 T^{10} + \cdots - 5472 \) Copy content Toggle raw display
$11$ \( T^{11} - 9 T^{10} + \cdots - 113760 \) Copy content Toggle raw display
$13$ \( T^{11} - 6 T^{10} + \cdots - 2304 \) Copy content Toggle raw display
$17$ \( (T + 1)^{11} \) Copy content Toggle raw display
$19$ \( T^{11} + T^{10} + \cdots - 9376 \) Copy content Toggle raw display
$23$ \( T^{11} - 10 T^{10} + \cdots - 30452 \) Copy content Toggle raw display
$29$ \( T^{11} - 14 T^{10} + \cdots + 5632272 \) Copy content Toggle raw display
$31$ \( T^{11} - 17 T^{10} + \cdots + 46863360 \) Copy content Toggle raw display
$37$ \( T^{11} - 30 T^{10} + \cdots + 2962096 \) Copy content Toggle raw display
$41$ \( T^{11} - 10 T^{10} + \cdots + 263898 \) Copy content Toggle raw display
$43$ \( T^{11} - 11 T^{10} + \cdots + 797536 \) Copy content Toggle raw display
$47$ \( T^{11} + 6 T^{10} + \cdots - 1941504 \) Copy content Toggle raw display
$53$ \( T^{11} - 10 T^{10} + \cdots + 192960 \) Copy content Toggle raw display
$59$ \( (T + 1)^{11} \) Copy content Toggle raw display
$61$ \( T^{11} - 13 T^{10} + \cdots + 908064 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 125692576 \) Copy content Toggle raw display
$71$ \( T^{11} - 14 T^{10} + \cdots + 13393728 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots - 508429750 \) Copy content Toggle raw display
$79$ \( T^{11} - 15 T^{10} + \cdots + 2641824 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 381243936 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 202792054 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 800138736 \) Copy content Toggle raw display
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