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Properties

Label 87.5.b.a
Level $87$
Weight $5$
Character orbit 87.b
Analytic conductor $8.993$
Analytic rank $0$
Dimension $38$
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] = [87,5,Mod(59,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.59");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 87.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: \(8.99318678829\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{3} - 292 q^{4} + 32 q^{6} + 76 q^{7} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{3} - 292 q^{4} + 32 q^{6} + 76 q^{7} + 98 q^{9} + 252 q^{10} - 102 q^{12} - 424 q^{13} + 234 q^{15} + 1612 q^{16} - 26 q^{18} + 256 q^{19} - 832 q^{21} + 682 q^{22} - 1450 q^{24} - 3574 q^{25} + 1480 q^{27} - 3618 q^{28} - 76 q^{30} + 2200 q^{31} + 844 q^{33} + 4622 q^{34} - 192 q^{36} + 2480 q^{37} + 3794 q^{39} - 6544 q^{40} + 2690 q^{42} - 1268 q^{43} + 1736 q^{45} - 2916 q^{46} - 8186 q^{48} + 3458 q^{49} - 10380 q^{51} + 11406 q^{52} + 7804 q^{54} - 1004 q^{55} + 4612 q^{57} - 9378 q^{60} + 6384 q^{61} - 1276 q^{63} - 12714 q^{64} - 10262 q^{66} - 14160 q^{67} + 6868 q^{69} + 12348 q^{70} + 22990 q^{72} + 13380 q^{73} - 7064 q^{75} - 24152 q^{76} + 20670 q^{78} + 4528 q^{79} - 24234 q^{81} + 27788 q^{82} + 17720 q^{84} - 21136 q^{85} - 24864 q^{88} - 25910 q^{90} - 32068 q^{91} - 1648 q^{93} + 15450 q^{94} + 42528 q^{96} + 34588 q^{97} - 10602 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 − 7.92467i −6.29793 + 6.42932i −46.8004 10.5251i 50.9502 + 49.9090i 48.2175 244.083i −1.67225 − 80.9827i 83.4079
59.2 − 7.33412i 8.11678 − 3.88817i −37.7894 − 9.20856i −28.5163 − 59.5295i −46.2561 159.806i 50.7643 − 63.1188i −67.5367
59.3 − 6.76375i −7.01287 − 5.64089i −29.7484 − 16.8546i −38.1536 + 47.4333i −33.6601 92.9906i 17.3607 + 79.1177i −114.001
59.4 − 6.50065i −2.84064 − 8.53995i −26.2584 45.1455i −55.5152 + 18.4660i 30.4267 66.6864i −64.8616 + 48.5178i 293.475
59.5 − 6.48550i 4.82556 + 7.59697i −26.0617 18.5694i 49.2702 − 31.2962i −50.3702 65.2555i −34.4280 + 73.3192i 120.432
59.6 − 6.30012i 6.49961 + 6.22536i −23.6915 − 29.9408i 39.2205 − 40.9483i 86.8973 48.4575i 3.48984 + 80.9248i −188.631
59.7 − 5.39098i 8.89074 − 1.39810i −13.0626 43.1161i −7.53711 − 47.9298i 34.2188 − 15.8353i 77.0906 − 24.8603i 232.438
59.8 − 5.20653i 2.66863 − 8.59526i −11.1079 − 26.0029i −44.7514 − 13.8943i 47.2252 − 25.4708i −66.7569 − 45.8751i −135.385
59.9 − 4.92533i −7.98159 + 4.15864i −8.25887 − 41.0533i 20.4827 + 39.3120i −14.0469 − 38.1276i 46.4115 − 66.3850i −202.201
59.10 − 4.89446i −1.68913 + 8.84007i −7.95570 5.86159i 43.2673 + 8.26739i −37.3975 − 39.3725i −75.2936 − 29.8641i 28.6893
59.11 − 4.65178i −8.62915 + 2.55689i −5.63902 22.0427i 11.8941 + 40.1409i 11.4434 − 48.1969i 67.9246 − 44.1276i 102.538
59.12 − 3.22491i 1.64106 − 8.84912i 5.59996 12.8489i −28.5376 − 5.29226i −74.9970 − 69.6579i −75.6139 − 29.0438i 41.4366
59.13 − 2.83371i −7.92666 − 4.26240i 7.97006 3.00478i −12.0784 + 22.4619i 90.4328 − 67.9243i 44.6638 + 67.5732i 8.51470
59.14 − 2.77393i 8.60516 + 2.63651i 8.30534 − 37.6207i 7.31348 − 23.8701i −76.5256 − 67.4212i 67.0977 + 45.3752i −104.357
59.15 − 2.17836i 8.16635 − 3.78296i 11.2547 1.44760i −8.24066 − 17.7893i 27.7369 − 59.3707i 52.3785 − 61.7859i 3.15340
59.16 − 1.84726i 6.24711 + 6.47871i 12.5876 24.1141i 11.9678 − 11.5400i 17.1205 − 52.8087i −2.94730 + 80.9464i 44.5449
59.17 − 1.28857i −8.23003 − 3.64233i 14.3396 15.5544i −4.69340 + 10.6050i −57.7913 − 39.0947i 54.4669 + 59.9529i 20.0430
59.18 − 1.27672i −1.13663 + 8.92794i 14.3700 − 23.2747i 11.3984 + 1.45116i 39.1494 − 38.7738i −78.4161 − 20.2956i −29.7151
59.19 − 0.230949i −4.91636 − 7.53853i 15.9467 − 46.9631i −1.74101 + 1.13543i −3.82369 − 7.37804i −32.6589 + 74.1242i −10.8461
59.20 0.230949i −4.91636 + 7.53853i 15.9467 46.9631i −1.74101 − 1.13543i −3.82369 7.37804i −32.6589 − 74.1242i −10.8461
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.38
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.5.b.a ✓ 38
3.b odd 2 1 inner 87.5.b.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.5.b.a ✓ 38 1.a even 1 1 trivial
87.5.b.a ✓ 38 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(87, [\chi])\).