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Properties

Label 59.15.b.b
Level $59$
Weight $15$
Character orbit 59.b
Analytic conductor $73.354$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,15,Mod(58,59)]
 
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([1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 59.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(73.3540912096\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 76 q^{3} - 606210 q^{4} + 71992 q^{5} - 1016394 q^{7} + 87762314 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 76 q^{3} - 606210 q^{4} + 71992 q^{5} - 1016394 q^{7} + 87762314 q^{9} - 2949860 q^{12} - 317515428 q^{15} + 3151063870 q^{16} + 1245261006 q^{17} - 1607375696 q^{19} - 361196016 q^{20} + 1326013972 q^{21} + 59664998 q^{22} + 58936179842 q^{25} - 19991606094 q^{26} - 54906244124 q^{27} + 55780277978 q^{28} - 12431425028 q^{29} + 163534730728 q^{35} - 1154285760786 q^{36} - 487614654858 q^{41} + 2881141613836 q^{45} - 903831645684 q^{46} - 339522016556 q^{48} + 187515933940 q^{49} - 4249332110720 q^{51} - 2815395069036 q^{53} - 11630537040732 q^{57} + 6375989743888 q^{59} - 723172920464 q^{60} + 184749032540 q^{62} - 30655984458434 q^{63} - 37878414427846 q^{64} + 23424093700016 q^{66} + 5852454721054 q^{68} - 31468301709790 q^{71} + 46958000668342 q^{74} + 85680485467696 q^{75} - 18991856218520 q^{76} + 82909595404520 q^{78} - 14661749773742 q^{79} + 33656055695064 q^{80} - 27106331582022 q^{81} - 69097030759736 q^{84} - 26808134650876 q^{85} + 20456922190694 q^{86} - 263402627573540 q^{87} + 299238556385642 q^{88} + 9128164555864 q^{94} - 634504794451740 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 − 247.058i 1853.18 −44653.5 −101323. − 457842.i 349494. 6.98419e6i −1.34870e6 2.50325e7i
58.2 − 246.807i −2676.03 −44529.9 23017.5 660464.i −1.25438e6 6.94663e6i 2.37816e6 − 5.68089e6i
58.3 − 239.568i 2613.78 −41008.9 125143. − 626178.i −914075. 5.89936e6i 2.04887e6 − 2.99803e7i
58.4 − 231.232i 526.470 −37084.3 61096.1 − 121737.i 1.02868e6 4.78656e6i −4.50580e6 − 1.41274e7i
58.5 − 230.882i −2775.11 −36922.6 −91080.3 640723.i −81193.6 4.74199e6i 2.91826e6 2.10288e7i
58.6 − 226.612i −2594.40 −34968.9 33394.6 587922.i 953602. 4.21157e6i 1.94795e6 − 7.56761e6i
58.7 − 213.796i 3406.69 −29324.7 −11342.2 − 728337.i 449529. 2.76667e6i 6.82258e6 2.42491e6i
58.8 − 206.427i 639.934 −26228.0 −46402.0 − 132100.i −1.52320e6 2.03208e6i −4.37345e6 9.57863e6i
58.9 − 204.065i −1161.88 −25258.6 92833.4 237098.i 26794.6 1.81100e6i −3.43301e6 − 1.89441e7i
58.10 − 201.294i 4114.02 −24135.3 −41286.8 − 828128.i −582130. 1.56030e6i 1.21422e7 8.31079e6i
58.11 − 188.277i 106.898 −19064.1 −83503.3 − 20126.4i 199101. 504606.i −4.77154e6 1.57217e7i
58.12 − 178.157i −4138.54 −15355.8 −24699.8 737309.i 905976. − 183172.i 1.23445e7 4.40044e6i
58.13 − 176.772i −1497.49 −14864.3 −139253. 264714.i 255943. − 268642.i −2.54049e6 2.46159e7i
58.14 − 175.639i −3816.16 −14465.0 144105. 670265.i −789341. − 337049.i 9.78008e6 − 2.53104e7i
58.15 − 162.367i 3193.15 −9979.20 86093.6 − 518465.i 1.15148e6 − 1.03993e6i 5.41327e6 − 1.39788e7i
58.16 − 156.998i 2120.38 −8264.45 −8287.20 − 332896.i −452412. − 1.27475e6i −286955. 1.30108e6i
58.17 − 150.968i 857.252 −6407.34 102104. − 129418.i −625077. − 1.50616e6i −4.04809e6 − 1.54144e7i
58.18 − 149.217i 1661.05 −5881.60 −64270.0 − 247856.i 1.54346e6 − 1.56713e6i −2.02390e6 9.59015e6i
58.19 − 142.778i −2735.68 −4001.50 −9784.29 390595.i −932191. − 1.76795e6i 2.70100e6 1.39698e6i
58.20 − 128.862i −671.216 −221.330 33878.8 86494.0i −79399.6 − 2.08275e6i −4.33244e6 − 4.36567e6i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.15.b.b ✓ 66
59.b odd 2 1 inner 59.15.b.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.15.b.b ✓ 66 1.a even 1 1 trivial
59.15.b.b ✓ 66 59.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 843777 T_{2}^{64} + 338884568369 T_{2}^{62} + \cdots + 23\!\cdots\!00 \) acting on \(S_{15}^{\mathrm{new}}(59, [\chi])\). Copy content Toggle raw display