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Properties

Label 88.5.f.a
Level $88$
Weight $5$
Character orbit 88.f
Analytic conductor $9.097$
Analytic rank $0$
Dimension $40$
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] = [88,5,Mod(67,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("88.67");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 88.f (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: \(9.09655675138\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 28 q^{4} + 102 q^{6} - 216 q^{8} + 1080 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 28 q^{4} + 102 q^{6} - 216 q^{8} + 1080 q^{9} - 298 q^{10} + 260 q^{12} + 588 q^{14} - 752 q^{16} + 240 q^{17} - 486 q^{18} - 1408 q^{19} + 1380 q^{20} + 2728 q^{24} - 4664 q^{25} + 1140 q^{26} + 3648 q^{27} + 940 q^{28} + 2550 q^{30} + 180 q^{32} + 5168 q^{34} - 5184 q^{35} - 4200 q^{36} - 3480 q^{38} + 6292 q^{40} + 720 q^{41} + 7420 q^{42} - 1452 q^{44} - 6522 q^{46} - 3184 q^{48} - 10712 q^{49} - 17682 q^{50} - 21824 q^{51} - 716 q^{52} + 6374 q^{54} + 1512 q^{56} + 7488 q^{57} + 3196 q^{58} + 14016 q^{59} + 13092 q^{60} - 5814 q^{62} + 1888 q^{64} - 1344 q^{65} - 1210 q^{66} - 2880 q^{67} + 15324 q^{68} + 21396 q^{70} - 32860 q^{72} - 20560 q^{73} - 21882 q^{74} + 30016 q^{75} - 9020 q^{76} - 6216 q^{78} - 2568 q^{80} + 34216 q^{81} - 8428 q^{82} + 10560 q^{83} - 51372 q^{84} + 12276 q^{86} - 7260 q^{88} + 29232 q^{89} + 54876 q^{90} - 7296 q^{91} + 30756 q^{92} + 11968 q^{94} + 25732 q^{96} - 8912 q^{97} - 3480 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −3.92348 − 0.778674i −5.81110 14.7873 + 6.11021i − 22.9760i 22.7997 + 4.52495i 73.9439i −53.2599 − 35.4878i −47.2312 −17.8908 + 90.1458i
67.2 −3.92348 + 0.778674i −5.81110 14.7873 − 6.11021i 22.9760i 22.7997 − 4.52495i − 73.9439i −53.2599 + 35.4878i −47.2312 −17.8908 − 90.1458i
67.3 −3.87415 − 0.995483i 5.25802 14.0180 + 7.71329i 33.5663i −20.3703 − 5.23427i 28.7745i −46.6294 − 43.8371i −53.3532 33.4146 − 130.041i
67.4 −3.87415 + 0.995483i 5.25802 14.0180 − 7.71329i − 33.5663i −20.3703 + 5.23427i − 28.7745i −46.6294 + 43.8371i −53.3532 33.4146 + 130.041i
67.5 −3.81523 − 1.20167i −15.9902 13.1120 + 9.16930i − 25.4726i 61.0062 + 19.2149i − 75.3551i −39.0067 − 50.7393i 174.686 −30.6096 + 97.1837i
67.6 −3.81523 + 1.20167i −15.9902 13.1120 − 9.16930i 25.4726i 61.0062 − 19.2149i 75.3551i −39.0067 + 50.7393i 174.686 −30.6096 − 97.1837i
67.7 −3.69277 − 1.53735i 13.1783 11.2731 + 11.3541i − 5.19751i −48.6644 − 20.2596i − 60.7809i −24.1738 − 59.2590i 92.6675 −7.99037 + 19.1932i
67.8 −3.69277 + 1.53735i 13.1783 11.2731 − 11.3541i 5.19751i −48.6644 + 20.2596i 60.7809i −24.1738 + 59.2590i 92.6675 −7.99037 − 19.1932i
67.9 −3.05538 − 2.58160i −1.68035 2.67066 + 15.7755i − 15.4106i 5.13412 + 4.33801i 2.17941i 32.5663 − 55.0948i −78.1764 −39.7841 + 47.0853i
67.10 −3.05538 + 2.58160i −1.68035 2.67066 − 15.7755i 15.4106i 5.13412 − 4.33801i − 2.17941i 32.5663 + 55.0948i −78.1764 −39.7841 − 47.0853i
67.11 −2.87498 − 2.78109i −10.1167 0.531058 + 15.9912i 31.1293i 29.0852 + 28.1353i − 7.91102i 42.9462 − 47.4513i 21.3466 86.5735 − 89.4963i
67.12 −2.87498 + 2.78109i −10.1167 0.531058 − 15.9912i − 31.1293i 29.0852 − 28.1353i 7.91102i 42.9462 + 47.4513i 21.3466 86.5735 + 89.4963i
67.13 −2.12885 − 3.38644i 16.0567 −6.93600 + 14.4185i 9.28725i −34.1823 − 54.3751i 65.5554i 63.5930 − 7.20637i 176.818 31.4508 − 19.7712i
67.14 −2.12885 + 3.38644i 16.0567 −6.93600 − 14.4185i − 9.28725i −34.1823 + 54.3751i − 65.5554i 63.5930 + 7.20637i 176.818 31.4508 + 19.7712i
67.15 −2.05516 − 3.43166i 5.56322 −7.55263 + 14.1052i − 35.1489i −11.4333 − 19.0911i 10.8414i 63.9263 − 3.07044i −50.0506 −120.619 + 72.2367i
67.16 −2.05516 + 3.43166i 5.56322 −7.55263 − 14.1052i 35.1489i −11.4333 + 19.0911i − 10.8414i 63.9263 + 3.07044i −50.0506 −120.619 − 72.2367i
67.17 −0.960550 − 3.88296i −11.4422 −14.1547 + 7.45955i − 0.413725i 10.9908 + 44.4295i 34.7719i 42.5614 + 47.7968i 49.9239 −1.60648 + 0.397404i
67.18 −0.960550 + 3.88296i −11.4422 −14.1547 − 7.45955i 0.413725i 10.9908 − 44.4295i − 34.7719i 42.5614 − 47.7968i 49.9239 −1.60648 − 0.397404i
67.19 −0.0167110 − 3.99997i −9.42555 −15.9994 + 0.133687i − 37.4632i 0.157511 + 37.7019i − 74.0303i 0.802111 + 63.9950i 7.84099 −149.852 + 0.626050i
67.20 −0.0167110 + 3.99997i −9.42555 −15.9994 − 0.133687i 37.4632i 0.157511 − 37.7019i 74.0303i 0.802111 − 63.9950i 7.84099 −149.852 − 0.626050i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.5.f.a ✓ 40
4.b odd 2 1 352.5.f.a 40
8.b even 2 1 352.5.f.a 40
8.d odd 2 1 inner 88.5.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.5.f.a ✓ 40 1.a even 1 1 trivial
88.5.f.a ✓ 40 8.d odd 2 1 inner
352.5.f.a 40 4.b odd 2 1
352.5.f.a 40 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(88, [\chi])\).