↑

Properties

Label 37.6.h.a
Level $37$
Weight $6$
Character orbit 37.h
Analytic conductor $5.934$
Analytic rank $0$
Dimension $90$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,6,Mod(3,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([13]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.h (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: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 90 q - 3 q^{2} - 9 q^{3} - 39 q^{4} - 153 q^{5} + 201 q^{7} + 792 q^{8} + 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q - 3 q^{2} - 9 q^{3} - 39 q^{4} - 153 q^{5} + 201 q^{7} + 792 q^{8} + 87 q^{9} - 603 q^{10} + 1449 q^{11} - 1257 q^{12} + 639 q^{13} - 3969 q^{14} + 4125 q^{15} + 4089 q^{16} + 615 q^{17} - 5454 q^{18} - 2802 q^{19} - 8532 q^{20} + 12828 q^{21} + 8205 q^{22} - 9 q^{23} - 15435 q^{24} - 15249 q^{25} - 13635 q^{26} + 4437 q^{27} + 40743 q^{28} + 9180 q^{29} + 38640 q^{30} + 15924 q^{32} - 38376 q^{33} - 42087 q^{34} - 57348 q^{35} - 93324 q^{36} - 16902 q^{37} + 95826 q^{38} - 29862 q^{39} - 2355 q^{40} + 27972 q^{41} + 72225 q^{42} + 118974 q^{44} + 37314 q^{45} + 2934 q^{46} - 65640 q^{47} + 118149 q^{48} - 93189 q^{49} + 55059 q^{50} + 36801 q^{51} - 271701 q^{52} + 109515 q^{53} - 165339 q^{54} - 36696 q^{55} + 92985 q^{56} + 160737 q^{57} - 109593 q^{58} - 42150 q^{59} - 39708 q^{60} - 191838 q^{61} + 193536 q^{62} + 47181 q^{63} + 201096 q^{64} + 184650 q^{65} + 441045 q^{66} - 174480 q^{67} + 93258 q^{69} - 176232 q^{70} - 111909 q^{71} - 674283 q^{72} + 85110 q^{73} - 518259 q^{74} + 211026 q^{75} - 511263 q^{76} - 347595 q^{77} + 340701 q^{78} - 291678 q^{79} + 268803 q^{81} + 466515 q^{82} + 255264 q^{83} + 428478 q^{84} + 189744 q^{85} + 183918 q^{86} + 410685 q^{87} + 454194 q^{88} - 542610 q^{89} + 675654 q^{90} - 801819 q^{91} - 1156026 q^{92} - 599226 q^{93} + 808032 q^{94} - 433731 q^{95} - 1058913 q^{96} + 419679 q^{97} + 685731 q^{98} + 1661748 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −6.88438 + 8.20449i −15.0305 + 12.6121i −14.3622 − 81.4518i −8.32738 + 22.8793i − 210.144i −131.906 − 48.0097i 470.335 + 271.548i 24.6549 − 139.825i −130.384 − 225.832i
3.2 −5.87863 + 7.00588i 11.4348 − 9.59493i −8.96732 − 50.8562i −11.3458 + 31.1724i 136.516i 187.288 + 68.1671i 155.560 + 89.8124i −3.50469 + 19.8761i −151.692 − 262.739i
3.3 −5.03691 + 6.00276i 9.01055 − 7.56075i −5.10589 − 28.9569i −6.94659 + 19.0856i 92.1710i −209.145 − 76.1225i −17.6194 − 10.1726i −18.1714 + 103.055i −79.5769 − 137.831i
3.4 −4.98431 + 5.94006i −6.24284 + 5.23836i −4.88432 − 27.7003i 35.0478 − 96.2930i − 63.1925i 87.0891 + 31.6978i −26.0040 − 15.0134i −30.6639 + 173.904i 397.298 + 688.140i
3.5 −2.99853 + 3.57350i −15.8714 + 13.3177i 1.77797 + 10.0834i −18.9349 + 52.0233i − 96.6499i 69.5153 + 25.3015i −170.641 − 98.5197i 32.3441 − 183.433i −129.129 − 223.657i
3.6 −2.40929 + 2.87128i 21.9339 − 18.4047i 3.11718 + 17.6784i 22.1706 − 60.9132i 107.320i −26.8299 − 9.76528i −162.142 − 93.6130i 100.165 − 568.064i 121.483 + 210.415i
3.7 −1.18489 + 1.41210i −0.120961 + 0.101498i 4.96669 + 28.1675i 2.33039 − 6.40269i − 0.291072i −65.8661 − 23.9733i −96.7449 − 55.8557i −42.1922 + 239.284i 6.27997 + 10.8772i
3.8 0.288267 − 0.343543i 15.5889 − 13.0806i 5.52182 + 31.3158i −34.3386 + 94.3445i − 9.12615i 86.7891 + 31.5887i 24.7783 + 14.3057i 29.7137 − 168.515i 22.5127 + 38.9932i
3.9 1.07511 − 1.28126i −23.3637 + 19.6045i 5.07096 + 28.7589i 29.2139 − 80.2646i 51.0120i −156.090 − 56.8119i 88.6512 + 51.1828i 119.331 − 676.759i −71.4320 − 123.724i
3.10 1.88509 − 2.24656i −2.60021 + 2.18183i 4.06327 + 23.0439i 12.5997 − 34.6173i 9.95447i 147.180 + 53.5692i 140.702 + 81.2343i −40.1958 + 227.962i −54.0184 − 93.5626i
3.11 3.69031 − 4.39794i 13.0876 − 10.9818i −0.166753 − 0.945704i 17.4711 − 48.0016i − 98.0846i −45.2616 − 16.4739i 154.328 + 89.1011i 8.48868 − 48.1417i −146.634 − 253.978i
3.12 3.72083 − 4.43431i −7.17971 + 6.02449i −0.261811 − 1.48480i −28.1480 + 77.3360i 54.2532i −209.218 − 76.1490i 152.860 + 88.2536i −26.9428 + 152.800i 238.198 + 412.571i
3.13 4.72436 − 5.63027i −16.1623 + 13.5618i −3.82366 − 21.6851i −11.7610 + 32.3130i 155.069i 201.974 + 73.5126i 63.5265 + 36.6770i 35.1013 − 199.070i 126.368 + 218.876i
3.14 6.11144 − 7.28333i 14.5967 − 12.2481i −10.1405 − 57.5094i −12.4485 + 34.2020i − 181.166i 57.0807 + 20.7757i −217.347 − 125.485i 20.8513 − 118.254i 173.026 + 299.690i
3.15 6.61549 − 7.88403i −8.58843 + 7.20655i −12.8365 − 72.7996i 17.5488 − 48.2149i 115.386i −76.1853 − 27.7292i −373.658 − 215.731i −20.3697 + 115.522i −264.034 − 457.320i
4.1 −10.5241 − 1.85569i 1.06669 + 6.04949i 77.2432 + 28.1142i −63.7210 − 75.9397i − 65.6449i 22.3577 − 18.7603i −464.593 − 268.233i 192.887 − 70.2051i 529.686 + 917.444i
4.2 −9.17284 − 1.61742i −2.60160 − 14.7544i 51.4549 + 18.7280i 38.7903 + 46.2285i 139.548i 17.4601 − 14.6507i −183.570 − 105.984i 17.4210 − 6.34074i −281.047 − 486.787i
4.3 −7.93594 − 1.39932i 4.19449 + 23.7882i 30.9508 + 11.2652i 37.4730 + 44.6585i − 194.651i 12.2949 − 10.3166i −6.54031 − 3.77605i −319.937 + 116.448i −234.891 − 406.844i
4.4 −5.98410 − 1.05516i 1.39884 + 7.93321i 4.62591 + 1.68369i −13.8446 − 16.4994i − 48.9491i −94.1712 + 79.0191i 142.489 + 82.2662i 167.366 − 60.9163i 65.4382 + 113.342i
4.5 −5.56701 − 0.981614i −4.83207 − 27.4040i −0.0421302 − 0.0153341i −54.5859 − 65.0529i 157.302i 117.413 − 98.5210i 156.877 + 90.5730i −499.286 + 181.725i 240.023 + 415.732i
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.6.h.a ✓ 90
37.h even 18 1 inner 37.6.h.a ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.h.a ✓ 90 1.a even 1 1 trivial
37.6.h.a ✓ 90 37.h even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).