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Properties

Label 57.5.k.a
Level $57$
Weight $5$
Character orbit 57.k
Analytic conductor $5.892$
Analytic rank $0$
Dimension $36$
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] = [57,5,Mod(10,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.10");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 57.k (of order \(18\), degree \(6\), 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: \(5.89208789578\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 9 q^{2} + 21 q^{4} - 27 q^{6} + 45 q^{7} - 297 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 9 q^{2} + 21 q^{4} - 27 q^{6} + 45 q^{7} - 297 q^{8} + 510 q^{10} + 45 q^{11} + 864 q^{12} - 9 q^{13} + 1017 q^{14} + 378 q^{15} + 1029 q^{16} + 630 q^{17} - 1251 q^{19} + 1098 q^{20} - 540 q^{21} - 1644 q^{22} + 945 q^{23} - 864 q^{24} - 2838 q^{25} + 576 q^{26} - 4374 q^{27} - 2994 q^{28} + 2070 q^{29} - 3600 q^{31} - 333 q^{32} + 1539 q^{33} + 5019 q^{34} + 4086 q^{35} - 567 q^{36} - 2016 q^{38} + 1350 q^{39} - 3891 q^{40} - 8910 q^{41} - 2943 q^{42} + 3135 q^{43} - 24903 q^{44} + 729 q^{45} + 6417 q^{46} - 3051 q^{47} + 6156 q^{48} + 159 q^{49} + 10395 q^{50} - 1755 q^{51} + 12189 q^{52} + 21186 q^{53} + 729 q^{54} + 10239 q^{55} + 4509 q^{57} + 5376 q^{58} + 9054 q^{59} - 20601 q^{60} - 30495 q^{61} + 7947 q^{62} - 3321 q^{63} - 5631 q^{64} + 34857 q^{65} + 13608 q^{66} + 27651 q^{67} - 3123 q^{68} + 4536 q^{69} + 1752 q^{70} + 7884 q^{71} + 7290 q^{72} - 22182 q^{73} + 9801 q^{74} + 29454 q^{76} - 23652 q^{77} - 15849 q^{78} - 43986 q^{79} - 67518 q^{80} - 55635 q^{82} + 4455 q^{83} - 10287 q^{84} + 38661 q^{85} + 40869 q^{86} - 6021 q^{87} + 9252 q^{88} - 41841 q^{89} + 4293 q^{90} + 58788 q^{91} + 59751 q^{92} + 24246 q^{93} - 40896 q^{95} - 11502 q^{96} + 40263 q^{97} - 34965 q^{98} - 2187 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −7.15677 + 1.26193i 3.34002 + 3.98048i 34.5918 − 12.5904i −19.4332 − 7.07309i −28.9269 − 24.2725i −22.2023 + 38.4555i −130.980 + 75.6216i −4.68850 + 26.5898i 148.004 + 26.0972i
10.2 −4.82934 + 0.851543i 3.34002 + 3.98048i 7.56232 − 2.75246i 18.7560 + 6.82662i −19.5197 − 16.3789i 2.38242 − 4.12646i 33.7724 − 19.4985i −4.68850 + 26.5898i −96.3922 − 16.9966i
10.3 −2.31993 + 0.409066i 3.34002 + 3.98048i −9.82034 + 3.57431i −18.4695 − 6.72233i −9.37690 − 7.86815i 31.3304 − 54.2658i 53.9622 − 31.1551i −4.68850 + 26.5898i 45.5977 + 8.04011i
10.4 1.60227 − 0.282524i 3.34002 + 3.98048i −12.5476 + 4.56696i −18.2953 − 6.65896i 6.47621 + 5.43419i −27.4476 + 47.5407i −41.3587 + 23.8785i −4.68850 + 26.5898i −31.1954 − 5.50060i
10.5 2.88568 − 0.508823i 3.34002 + 3.98048i −6.96685 + 2.53573i 33.4576 + 12.1776i 11.6636 + 9.78691i 9.10553 − 15.7712i −59.4158 + 34.3037i −4.68850 + 26.5898i 102.744 + 18.1165i
10.6 6.61235 − 1.16594i 3.34002 + 3.98048i 27.3287 − 9.94684i 3.32053 + 1.20858i 26.7264 + 22.4261i −0.815626 + 1.41271i 76.0728 − 43.9207i −4.68850 + 26.5898i 23.3657 + 4.12000i
13.1 −3.35778 − 4.00165i 1.77719 − 4.88279i −1.96012 + 11.1164i 5.60180 + 31.7694i −25.5066 + 9.28365i −24.9227 + 43.1673i −21.3172 + 12.3075i −20.6832 − 17.3553i 108.320 − 129.091i
13.2 −3.15949 − 3.76533i 1.77719 − 4.88279i −1.41699 + 8.03614i −3.21717 − 18.2455i −24.0003 + 8.73540i 27.5292 − 47.6819i −33.3726 + 19.2677i −20.6832 − 17.3553i −58.5357 + 69.7602i
13.3 −0.579973 − 0.691185i 1.77719 − 4.88279i 2.63700 − 14.9552i −3.89866 − 22.1104i −4.40563 + 1.60352i −31.6609 + 54.8382i −24.3685 + 14.0692i −20.6832 − 17.3553i −13.0213 + 15.5181i
13.4 −0.251105 − 0.299255i 1.77719 − 4.88279i 2.75187 − 15.6066i 6.64623 + 37.6926i −1.90746 + 0.694258i 42.0673 − 72.8627i −10.7744 + 6.22059i −20.6832 − 17.3553i 9.61081 − 11.4537i
13.5 2.59715 + 3.09516i 1.77719 − 4.88279i −0.0564709 + 0.320263i −3.93843 − 22.3359i 19.7286 − 7.18064i 12.7272 − 22.0442i 54.8481 − 31.6666i −20.6832 − 17.3553i 58.9046 − 70.1998i
13.6 4.36454 + 5.20146i 1.77719 − 4.88279i −5.22757 + 29.6470i 4.51584 + 25.6106i 33.1542 − 12.0671i 1.31972 − 2.28583i −82.9385 + 47.8845i −20.6832 − 17.3553i −113.503 + 135.267i
22.1 −3.35778 + 4.00165i 1.77719 + 4.88279i −1.96012 − 11.1164i 5.60180 − 31.7694i −25.5066 − 9.28365i −24.9227 − 43.1673i −21.3172 − 12.3075i −20.6832 + 17.3553i 108.320 + 129.091i
22.2 −3.15949 + 3.76533i 1.77719 + 4.88279i −1.41699 − 8.03614i −3.21717 + 18.2455i −24.0003 − 8.73540i 27.5292 + 47.6819i −33.3726 − 19.2677i −20.6832 + 17.3553i −58.5357 − 69.7602i
22.3 −0.579973 + 0.691185i 1.77719 + 4.88279i 2.63700 + 14.9552i −3.89866 + 22.1104i −4.40563 − 1.60352i −31.6609 − 54.8382i −24.3685 − 14.0692i −20.6832 + 17.3553i −13.0213 − 15.5181i
22.4 −0.251105 + 0.299255i 1.77719 + 4.88279i 2.75187 + 15.6066i 6.64623 − 37.6926i −1.90746 − 0.694258i 42.0673 + 72.8627i −10.7744 − 6.22059i −20.6832 + 17.3553i 9.61081 + 11.4537i
22.5 2.59715 − 3.09516i 1.77719 + 4.88279i −0.0564709 − 0.320263i −3.93843 + 22.3359i 19.7286 + 7.18064i 12.7272 + 22.0442i 54.8481 + 31.6666i −20.6832 + 17.3553i 58.9046 + 70.1998i
22.6 4.36454 − 5.20146i 1.77719 + 4.88279i −5.22757 − 29.6470i 4.51584 − 25.6106i 33.1542 + 12.0671i 1.31972 + 2.28583i −82.9385 − 47.8845i −20.6832 + 17.3553i −113.503 − 135.267i
34.1 −2.37627 + 6.52876i −5.11721 + 0.902302i −24.7213 − 20.7436i −18.0749 + 15.1666i 6.26898 − 35.5531i 16.6664 − 28.8671i 97.9036 − 56.5247i 25.3717 − 9.23454i −56.0684 − 154.047i
34.2 −1.42734 + 3.92158i −5.11721 + 0.902302i −1.08480 − 0.910254i 35.3362 − 29.6506i 3.76554 − 21.3555i 27.5078 − 47.6448i −52.7084 + 30.4312i 25.3717 − 9.23454i 65.8405 + 180.895i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.5.k.a ✓ 36
19.f odd 18 1 inner 57.5.k.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.5.k.a ✓ 36 1.a even 1 1 trivial
57.5.k.a ✓ 36 19.f odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 9 T_{2}^{35} + 30 T_{2}^{34} + 126 T_{2}^{33} + 33 T_{2}^{32} + 1773 T_{2}^{31} + \cdots + 55\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display