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Properties

Label 239.6.a.b
Level $239$
Weight $6$
Character orbit 239.a
Self dual yes
Analytic conductor $38.332$
Analytic rank $0$
Dimension $57$
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] = [239,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("239.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 239 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(38.3317329353\)
Analytic rank: \(0\)
Dimension: \(57\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 57 q + 8 q^{2} + 35 q^{3} + 1034 q^{4} + 188 q^{5} + 198 q^{6} + 401 q^{7} + 507 q^{8} + 5640 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 57 q + 8 q^{2} + 35 q^{3} + 1034 q^{4} + 188 q^{5} + 198 q^{6} + 401 q^{7} + 507 q^{8} + 5640 q^{9} + 883 q^{10} + 1207 q^{11} + 1696 q^{12} + 2973 q^{13} + 1413 q^{14} + 2219 q^{15} + 20518 q^{16} + 4446 q^{17} + 6273 q^{18} + 9024 q^{19} + 4534 q^{20} + 9711 q^{21} + 9538 q^{22} + 5531 q^{23} + 9415 q^{24} + 53289 q^{25} + 5132 q^{26} + 11330 q^{27} + 18469 q^{28} + 6680 q^{29} + 11973 q^{30} + 30536 q^{31} + 22917 q^{32} + 35730 q^{33} + 52359 q^{34} + 12752 q^{35} + 115400 q^{36} + 46313 q^{37} + 6709 q^{38} + 26572 q^{39} + 41256 q^{40} + 53568 q^{41} + 12874 q^{42} + 5831 q^{43} + 47765 q^{44} + 72705 q^{45} + 68326 q^{46} + 24630 q^{47} + 68847 q^{48} + 269890 q^{49} + 56997 q^{50} + 78915 q^{51} + 89391 q^{52} + 43880 q^{53} + 70584 q^{54} + 130646 q^{55} + 82116 q^{56} + 65854 q^{57} + 55888 q^{58} + 85865 q^{59} + 308580 q^{60} + 239750 q^{61} + 464989 q^{62} + 293160 q^{63} + 697135 q^{64} + 272554 q^{65} + 718788 q^{66} + 225596 q^{67} + 719354 q^{68} + 418973 q^{69} + 685957 q^{70} + 361440 q^{71} + 1093925 q^{72} + 372684 q^{73} + 372515 q^{74} + 472319 q^{75} + 643421 q^{76} + 279303 q^{77} + 585965 q^{78} + 267070 q^{79} + 705174 q^{80} + 880513 q^{81} + 649878 q^{82} + 134360 q^{83} + 852552 q^{84} + 276838 q^{85} + 369443 q^{86} + 207891 q^{87} + 321552 q^{88} + 601073 q^{89} + 398814 q^{90} + 506679 q^{91} + 261404 q^{92} + 279468 q^{93} + 577312 q^{94} + 38303 q^{95} + 114697 q^{96} + 724535 q^{97} - 81759 q^{98} + 167479 q^{99}+O(q^{100}) \) 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 −11.1985 7.51378 93.4066 −22.8984 −84.1432 25.3988 −687.663 −186.543 256.428
1.2 −11.0471 −22.1893 90.0382 −27.3213 245.128 −44.1493 −641.153 249.367 301.820
1.3 −10.6556 10.4106 81.5410 101.386 −110.931 162.118 −527.888 −134.619 −1080.33
1.4 −10.0912 −2.13938 69.8332 76.6362 21.5890 −250.447 −381.784 −238.423 −773.354
1.5 −10.0342 −7.87781 68.6853 −106.098 79.0476 190.037 −368.108 −180.940 1064.61
1.6 −9.97538 −4.34390 67.5082 36.3517 43.3320 −173.021 −354.207 −224.131 −362.622
1.7 −9.18970 30.4910 52.4505 −46.5500 −280.203 231.734 −187.934 686.704 427.781
1.8 −8.97350 24.3902 48.5237 85.1425 −218.866 −67.3939 −148.275 351.882 −764.026
1.9 −8.40026 3.79344 38.5644 13.5865 −31.8659 16.0201 −55.1425 −228.610 −114.130
1.10 −7.76338 17.8079 28.2701 −101.941 −138.249 −67.9719 28.9565 74.1210 791.408
1.11 −7.44229 −28.7754 23.3877 32.7159 214.155 −220.390 64.0955 585.025 −243.481
1.12 −7.23577 −29.6723 20.3564 −50.0661 214.702 187.758 84.2507 637.443 362.266
1.13 −7.04818 −0.0273427 17.6768 −20.2639 0.192716 −79.8849 100.953 −242.999 142.824
1.14 −6.73383 −21.5411 13.3445 −46.9852 145.054 −14.8041 125.623 221.017 316.390
1.15 −6.09616 −18.0531 5.16319 −30.2643 110.055 86.9538 163.602 82.9145 184.496
1.16 −5.98164 20.5752 3.78004 45.3663 −123.074 100.615 168.802 180.339 −271.365
1.17 −5.69360 −14.6448 0.417082 55.0811 83.3816 72.5236 179.821 −28.5300 −313.610
1.18 −5.62027 14.0292 −0.412588 −77.5335 −78.8476 172.216 182.167 −46.1828 435.759
1.19 −4.67146 29.8440 −10.1775 −59.1247 −139.415 −101.298 197.030 647.661 276.198
1.20 −4.57888 5.04668 −11.0338 −27.6863 −23.1081 −221.674 197.047 −217.531 126.772
See all 57 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.57
Significant digits:
Format:

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 239.6.a.b ✓ 57
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.6.a.b ✓ 57 1.a even 1 1 trivial