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Properties

Label 59.10.a.b
Level $59$
Weight $10$
Character orbit 59.a
Self dual yes
Analytic conductor $30.387$
Analytic rank $0$
Dimension $25$
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] = [59,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(30.3871143337\)
Analytic rank: \(0\)
Dimension: \(25\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 15 q^{2} + 81 q^{3} + 7423 q^{4} + 1703 q^{5} + 5990 q^{6} + 16882 q^{7} + 7593 q^{8} + 227698 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 15 q^{2} + 81 q^{3} + 7423 q^{4} + 1703 q^{5} + 5990 q^{6} + 16882 q^{7} + 7593 q^{8} + 227698 q^{9} + 17038 q^{10} + 94489 q^{11} + 41472 q^{12} + 309221 q^{13} + 39531 q^{14} + 1008725 q^{15} + 2887675 q^{16} - 36043 q^{17} - 1346603 q^{18} + 932847 q^{19} - 5025268 q^{20} + 235613 q^{21} - 1731787 q^{22} - 1747188 q^{23} + 87564 q^{24} + 14196732 q^{25} + 4273623 q^{26} + 6592881 q^{27} + 21099627 q^{28} + 13182063 q^{29} + 40270506 q^{30} + 20491716 q^{31} + 37684841 q^{32} + 15727984 q^{33} + 47107877 q^{34} + 52349169 q^{35} + 195820409 q^{36} + 56189773 q^{37} + 67191736 q^{38} + 31403498 q^{39} + 90203660 q^{40} + 57151896 q^{41} + 250951804 q^{42} + 83080613 q^{43} + 237200901 q^{44} + 113457950 q^{45} + 139036904 q^{46} + 85823874 q^{47} + 36387896 q^{48} + 225021845 q^{49} + 252441241 q^{50} + 227077524 q^{51} + 343029307 q^{52} + 124600417 q^{53} + 343135378 q^{54} - 11315000 q^{55} + 131080793 q^{56} + 95960121 q^{57} + 19572234 q^{58} + 302934025 q^{59} + 1212736898 q^{60} + 811979190 q^{61} - 47110788 q^{62} + 1093221239 q^{63} + 1333663903 q^{64} + 381782134 q^{65} - 109428490 q^{66} + 232095070 q^{67} - 326675337 q^{68} - 395146192 q^{69} - 352003222 q^{70} - 55277439 q^{71} - 1652988929 q^{72} + 132989682 q^{73} - 874079725 q^{74} - 2954317522 q^{75} - 327507070 q^{76} - 814522869 q^{77} - 3669872822 q^{78} - 545015554 q^{79} - 5311522264 q^{80} + 903860957 q^{81} - 109368843 q^{82} - 3205511241 q^{83} - 4880027048 q^{84} + 780231248 q^{85} - 839600933 q^{86} - 3197959141 q^{87} - 2621094753 q^{88} - 1927173410 q^{89} - 6947033562 q^{90} - 2106515643 q^{91} - 5308609636 q^{92} - 562682096 q^{93} + 753865542 q^{94} - 2452647879 q^{95} - 6255393908 q^{96} - 1405215706 q^{97} - 4377004476 q^{98} - 211950437 q^{99}+O(q^{100}) \) 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 −43.1917 −262.518 1353.53 −1398.92 11338.6 7208.53 −36346.9 49232.6 60421.8
1.2 −41.6271 273.600 1220.81 568.812 −11389.1 1824.48 −29505.8 55173.7 −23678.0
1.3 −40.9131 88.3231 1161.88 −2400.23 −3613.57 −5332.18 −26588.5 −11882.0 98200.7
1.4 −30.9462 −138.877 445.664 −742.017 4297.70 4800.13 2052.83 −396.247 22962.6
1.5 −30.6808 154.610 429.309 61.7713 −4743.57 −11142.6 2537.01 4221.38 −1895.19
1.6 −29.9930 71.1031 387.582 458.178 −2132.60 7540.82 3731.68 −14627.4 −13742.1
1.7 −27.6051 −170.917 250.039 1096.72 4718.19 −4819.97 7231.44 9529.78 −30274.9
1.8 −26.6874 −240.252 200.219 2475.57 6411.72 9478.20 8320.64 38038.2 −66066.5
1.9 −12.3104 60.2261 −360.455 −1055.16 −741.405 −7636.39 10740.2 −16055.8 12989.5
1.10 −10.2385 −63.1656 −407.173 460.040 646.723 7185.23 9410.97 −15693.1 −4710.13
1.11 −10.2026 174.240 −407.906 2060.53 −1777.71 10809.5 9385.48 10676.6 −21022.9
1.12 0.233284 244.013 −511.946 1466.25 56.9244 −5823.87 −238.870 39859.3 342.052
1.13 3.08905 −13.8824 −502.458 −1654.12 −42.8835 −6990.08 −3133.71 −19490.3 −5109.67
1.14 5.46466 −182.902 −482.137 2280.50 −999.495 −9561.98 −5432.63 13770.0 12462.2
1.15 7.77520 126.911 −451.546 −1972.67 986.759 7737.58 −7491.77 −3576.60 −15337.9
1.16 16.8669 −55.1151 −227.506 2433.32 −929.622 4126.80 −12473.2 −16645.3 41042.6
1.17 17.4444 −84.5320 −207.694 −880.143 −1474.61 −10420.6 −12554.6 −12537.3 −15353.6
1.18 18.2061 −184.115 −180.539 −2221.80 −3352.01 144.979 −12608.4 14215.4 −40450.3
1.19 25.1449 −168.323 120.268 −441.786 −4232.48 −439.905 −9850.09 8649.77 −11108.7
1.20 28.7615 267.188 315.225 654.448 7684.74 12237.7 −5659.56 51706.6 18822.9
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
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 59.10.a.b ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.10.a.b ✓ 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 15 T_{2}^{24} - 9999 T_{2}^{23} + 143459 T_{2}^{22} + 43251396 T_{2}^{21} + \cdots + 21\!\cdots\!76 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display