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Properties

Label 239.4.a.b
Level $239$
Weight $4$
Character orbit 239.a
Self dual yes
Analytic conductor $14.101$
Analytic rank $0$
Dimension $37$
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] = [239,4,Mod(1,239)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(239, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("239.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 239.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: \(14.1014564914\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} + 11 q^{3} + 170 q^{4} + 43 q^{5} + 54 q^{6} + 60 q^{7} + 27 q^{8} + 474 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} + 11 q^{3} + 170 q^{4} + 43 q^{5} + 54 q^{6} + 60 q^{7} + 27 q^{8} + 474 q^{9} + 147 q^{10} + 55 q^{11} + 136 q^{12} + 250 q^{13} + 169 q^{14} + 218 q^{15} + 918 q^{16} + 189 q^{17} + 69 q^{18} + 550 q^{19} + 486 q^{20} + 366 q^{21} + 226 q^{22} + 74 q^{23} + 655 q^{24} + 1604 q^{25} + 560 q^{26} + 620 q^{27} + 829 q^{28} + 389 q^{29} + 417 q^{30} + 1107 q^{31} + 125 q^{32} + 1392 q^{33} + 1423 q^{34} + 270 q^{35} + 2624 q^{36} + 1002 q^{37} + 1037 q^{38} + 550 q^{39} + 1536 q^{40} + 1518 q^{41} + 1234 q^{42} + 1098 q^{43} + 1037 q^{44} + 2031 q^{45} + 1030 q^{46} + 1214 q^{47} + 1479 q^{48} + 4663 q^{49} + 929 q^{50} + 1206 q^{51} + 2895 q^{52} + 904 q^{53} + 2184 q^{54} + 1350 q^{55} + 2556 q^{56} + 2158 q^{57} + 1396 q^{58} + 1658 q^{59} + 2124 q^{60} + 2313 q^{61} - 4519 q^{62} - 462 q^{63} + 3807 q^{64} - 56 q^{65} - 6012 q^{66} + 1535 q^{67} - 6526 q^{68} + 1676 q^{69} - 4099 q^{70} - 3255 q^{71} - 12691 q^{72} + 3154 q^{73} - 2629 q^{74} - 2557 q^{75} + 1981 q^{76} - 3734 q^{77} - 9079 q^{78} + 2260 q^{79} - 8242 q^{80} + 5581 q^{81} - 9898 q^{82} - 939 q^{83} - 13824 q^{84} + 1272 q^{85} - 3457 q^{86} - 5292 q^{87} - 1808 q^{88} + 1486 q^{89} - 10014 q^{90} + 174 q^{91} - 14076 q^{92} + 2652 q^{93} - 984 q^{94} - 1828 q^{95} - 14687 q^{96} + 6148 q^{97} - 6243 q^{98} - 3947 q^{99}+O(q^{100}) \) 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 −5.56532 9.56504 22.9727 8.90443 −53.2325 −11.2801 −83.3280 64.4899 −49.5560
1.2 −5.30310 −10.0457 20.1229 −9.35421 53.2736 16.9697 −64.2888 73.9171 49.6063
1.3 −5.19331 −0.852071 18.9705 −14.4938 4.42507 −33.8516 −56.9730 −26.2740 75.2706
1.4 −5.02414 4.31973 17.2420 −11.3995 −21.7029 16.2461 −46.4330 −8.33993 57.2725
1.5 −4.59397 −3.54103 13.1045 4.72841 16.2674 31.8111 −23.4500 −14.4611 −21.7221
1.6 −4.20811 −8.89126 9.70818 14.3148 37.4154 −14.1622 −7.18821 52.0545 −60.2384
1.7 −3.82561 4.58903 6.63530 13.2443 −17.5559 −25.5656 5.22079 −5.94079 −50.6676
1.8 −3.50254 −2.72284 4.26778 −11.8178 9.53686 22.4937 13.0723 −19.5861 41.3925
1.9 −3.37354 7.98211 3.38078 13.8556 −26.9280 24.5625 15.5831 36.7140 −46.7423
1.10 −3.12953 −6.58952 1.79393 20.5928 20.6221 28.8946 19.4220 16.4217 −64.4456
1.11 −2.91546 −4.96191 0.499907 3.11303 14.4663 −17.8402 21.8662 −2.37941 −9.07590
1.12 −2.86154 3.04040 0.188430 −5.35419 −8.70025 −31.0877 22.3531 −17.7559 15.3212
1.13 −1.94502 7.25713 −4.21689 3.86550 −14.1153 19.7745 23.7621 25.6659 −7.51848
1.14 −1.82758 2.53373 −4.65994 −21.6433 −4.63060 −18.0702 23.1371 −20.5802 39.5549
1.15 −1.50268 −0.901740 −5.74194 13.8014 1.35503 16.9859 20.6498 −26.1869 −20.7392
1.16 −0.483342 8.62567 −7.76638 −20.7756 −4.16915 33.2708 7.62055 47.4022 10.0417
1.17 −0.104490 −3.03341 −7.98908 −1.78609 0.316959 −23.0255 1.67069 −17.7985 0.186628
1.18 0.123948 9.22900 −7.98464 3.85596 1.14391 −18.4037 −1.98126 58.1744 0.477937
1.19 0.271729 −5.48441 −7.92616 −11.3325 −1.49027 −27.2237 −4.32760 3.07872 −3.07937
1.20 0.412743 −3.80494 −7.82964 −15.2997 −1.57046 4.25811 −6.53358 −12.5225 −6.31484
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(239\) \(1\)

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 239.4.a.b ✓ 37
3.b odd 2 1 2151.4.a.f 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.4.a.b ✓ 37 1.a even 1 1 trivial
2151.4.a.f 37 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 4 T_{2}^{36} - 225 T_{2}^{35} + 891 T_{2}^{34} + 22979 T_{2}^{33} - 89996 T_{2}^{32} + \cdots + 8610703477760 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(239))\). Copy content Toggle raw display