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Properties

Label 59.18.a.b
Level $59$
Weight $18$
Character orbit 59.a
Self dual yes
Analytic conductor $108.101$
Analytic rank $0$
Dimension $44$
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] = [59,18,Mod(1,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 59.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: \(108.101031533\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 783 q^{2} + 13122 q^{3} + 3063551 q^{4} + 2062778 q^{5} + 9982022 q^{6} + 13722970 q^{7} + 168939849 q^{8} + 2182805218 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 783 q^{2} + 13122 q^{3} + 3063551 q^{4} + 2062778 q^{5} + 9982022 q^{6} + 13722970 q^{7} + 168939849 q^{8} + 2182805218 q^{9} + 246619918 q^{10} + 896501218 q^{11} + 2579890176 q^{12} + 5139901152 q^{13} - 1065727269 q^{14} + 52608606500 q^{15} + 170254553019 q^{16} + 31152575372 q^{17} + 371065029637 q^{18} + 222944612638 q^{19} + 632345964652 q^{20} + 518768862104 q^{21} - 1188197076091 q^{22} - 314739342184 q^{23} - 1830638682468 q^{24} + 8265149117122 q^{25} - 1422210694649 q^{26} + 1055641354104 q^{27} + 4733828767179 q^{28} + 7952343701542 q^{29} + 33815332595226 q^{30} + 22703202725740 q^{31} + 51508227606921 q^{32} + 39808250652964 q^{33} + 42559210973877 q^{34} + 53084789167044 q^{35} + 286899333699545 q^{36} + 70719636063816 q^{37} + 70760432282360 q^{38} + 89621954178128 q^{39} + 176727288274300 q^{40} + 77283001373080 q^{41} - 142968851337140 q^{42} + 218112956325030 q^{43} - 146577440549739 q^{44} + 156445227241670 q^{45} - 436430382603480 q^{46} - 206155334901712 q^{47} - 694457384549320 q^{48} + 17\!\cdots\!92 q^{49}+ \cdots - 33\!\cdots\!26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −676.925 7668.70 327156. −1.44413e6 −5.19114e6 1.63809e7 −1.32734e8 −7.03312e7 9.77567e8
1.2 −649.114 −10250.6 290278. 97139.6 6.65380e6 2.38362e7 −1.03343e8 −2.40659e7 −6.30547e7
1.3 −648.649 −13388.7 289673. 688083. 8.68455e6 −5.33195e6 −1.02876e8 5.01164e7 −4.46324e8
1.4 −604.538 −2312.95 234394. 1.70501e6 1.39826e6 1.94776e7 −6.24624e7 −1.23790e8 −1.03075e9
1.5 −601.953 18973.7 231275. 77857.7 −1.14213e7 −7.81843e6 −6.03174e7 2.30861e8 −4.68667e7
1.6 −564.875 18612.7 188012. −1.07262e6 −1.05138e7 −1.39668e7 −3.21639e7 2.17292e8 6.05898e8
1.7 −555.022 −16664.8 176977. 1.12157e6 9.24931e6 −2.22166e7 −2.54785e7 1.48574e8 −6.22494e8
1.8 −515.569 −3466.08 134740. −432270. 1.78701e6 −2.18683e6 −1.89095e6 −1.17126e8 2.22865e8
1.9 −497.349 8015.77 116284. 328518. −3.98664e6 −5.16891e6 7.35482e6 −6.48875e7 −1.63388e8
1.10 −483.168 18836.7 102380. 1.57853e6 −9.10130e6 1.70823e7 1.38632e7 2.25681e8 −7.62698e8
1.11 −468.643 −13841.8 88554.2 −1.14609e6 6.48686e6 −2.07367e7 1.99257e7 6.24551e7 5.37108e8
1.12 −359.808 −20613.5 −1609.92 −1.17284e6 7.41692e6 −1.26872e7 4.77401e7 2.95778e8 4.21997e8
1.13 −306.193 −3878.79 −37317.8 1.50317e6 1.18766e6 −7.27672e6 5.15598e7 −1.14095e8 −4.60261e8
1.14 −280.662 11335.8 −52300.8 169522. −3.18152e6 9.16077e6 5.14658e7 −640310. −4.75785e7
1.15 −272.365 9373.77 −56889.0 −610923. −2.55309e6 −2.57849e7 5.11941e7 −4.12726e7 1.66394e8
1.16 −253.334 3121.28 −66893.7 837990. −790727. 1.47093e7 5.01515e7 −1.19398e8 −2.12292e8
1.17 −200.828 −7142.34 −90740.1 −1.51564e6 1.43438e6 2.41453e7 4.45461e7 −7.81272e7 3.04383e8
1.18 −198.392 −7956.20 −91712.4 −305160. 1.57845e6 579402. 4.41987e7 −6.58390e7 6.05414e7
1.19 −70.2688 −18018.4 −126134. −230878. 1.26613e6 1.19779e7 1.80736e7 1.95522e8 1.62235e7
1.20 −54.5487 16104.4 −128096. 156135. −878473. −497697. 1.41373e7 1.30211e8 −8.51695e6
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \(-1\)

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 59.18.a.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.18.a.b ✓ 44 1.a even 1 1 trivial