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Properties

Label 58.5.f.a
Level $58$
Weight $5$
Character orbit 58.f
Analytic conductor $5.995$
Analytic rank $0$
Dimension $60$
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] = [58,5,Mod(3,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 58.f (of order \(28\), degree \(12\), 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: \(5.99545785886\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 20 q^{2} + 16 q^{3} - 56 q^{5} + 72 q^{7} + 160 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 20 q^{2} + 16 q^{3} - 56 q^{5} + 72 q^{7} + 160 q^{8} + 48 q^{10} - 716 q^{11} - 128 q^{12} + 368 q^{14} - 1630 q^{15} + 640 q^{16} + 66 q^{17} + 604 q^{18} + 1550 q^{19} - 928 q^{20} - 780 q^{21} + 672 q^{22} - 1662 q^{23} + 1280 q^{24} + 3434 q^{25} + 4068 q^{26} + 1180 q^{27} - 3982 q^{29} - 752 q^{30} - 1276 q^{31} + 1280 q^{32} - 13202 q^{33} - 5460 q^{34} - 4116 q^{35} + 1520 q^{36} + 14036 q^{37} - 280 q^{38} + 3054 q^{39} + 2400 q^{40} - 5366 q^{41} + 7672 q^{42} + 3244 q^{43} + 1248 q^{44} + 744 q^{45} + 568 q^{46} - 5028 q^{47} - 1024 q^{48} - 16766 q^{49} + 372 q^{50} + 12264 q^{51} - 1760 q^{52} + 25012 q^{53} - 7544 q^{54} + 18418 q^{55} - 2944 q^{56} - 18308 q^{58} + 24460 q^{59} - 13040 q^{60} + 3502 q^{61} + 952 q^{62} - 47656 q^{63} - 27084 q^{65} + 10800 q^{66} - 27160 q^{67} - 528 q^{68} + 17316 q^{69} - 31912 q^{70} - 53466 q^{71} + 12448 q^{72} + 24182 q^{73} + 368 q^{74} + 52284 q^{75} + 8880 q^{76} + 74022 q^{77} - 2568 q^{78} + 27216 q^{79} + 33578 q^{81} + 18296 q^{82} + 11404 q^{83} + 15760 q^{84} + 31828 q^{85} - 53722 q^{87} + 4992 q^{88} - 47844 q^{89} - 15928 q^{90} - 4620 q^{91} - 39536 q^{92} - 9632 q^{93} - 54272 q^{94} - 25772 q^{95} - 19848 q^{97} - 3292 q^{98} - 155412 q^{99}+O(q^{100}) \) 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 −2.39490 − 1.50481i −13.6394 + 4.77265i 3.47107 + 7.20775i −31.4707 + 7.18299i 39.8470 + 9.09483i 12.7522 + 6.14115i 2.53347 − 22.4851i 99.9280 − 79.6899i 86.1783 + 30.1551i
3.2 −2.39490 − 1.50481i −6.40518 + 2.24127i 3.47107 + 7.20775i 22.5956 − 5.15730i 18.7124 + 4.27099i −13.1290 − 6.32260i 2.53347 − 22.4851i −27.3254 + 21.7912i −61.8750 − 21.6510i
3.3 −2.39490 − 1.50481i −1.00910 + 0.353098i 3.47107 + 7.20775i 2.91059 − 0.664323i 2.94803 + 0.672869i −23.7329 − 11.4292i 2.53347 − 22.4851i −62.4348 + 49.7901i −7.97025 − 2.78891i
3.4 −2.39490 − 1.50481i 7.51044 − 2.62802i 3.47107 + 7.20775i 3.09043 − 0.705370i −21.9414 − 5.00798i 68.9386 + 33.1991i 2.53347 − 22.4851i −13.8281 + 11.0275i −8.46271 − 2.96123i
3.5 −2.39490 − 1.50481i 11.1670 − 3.90751i 3.47107 + 7.20775i −47.5872 + 10.8615i −32.6239 − 7.44620i −47.6764 − 22.9597i 2.53347 − 22.4851i 46.1053 − 36.7678i 130.311 + 45.5978i
11.1 −2.66971 + 0.934170i −1.88846 − 16.7606i 6.25465 − 4.98792i 4.28976 − 8.90779i 20.6989 + 42.9816i 42.6645 − 53.4996i −12.0385 + 19.1592i −198.381 + 45.2792i −3.13102 + 27.7885i
11.2 −2.66971 + 0.934170i −0.828413 − 7.35237i 6.25465 − 4.98792i 16.7220 − 34.7235i 9.07998 + 18.8548i −57.9910 + 72.7184i −12.0385 + 19.1592i 25.5981 − 5.84260i −12.2050 + 108.323i
11.3 −2.66971 + 0.934170i −0.456133 − 4.04829i 6.25465 − 4.98792i −10.3978 + 21.5912i 4.99954 + 10.3816i 19.8689 − 24.9149i −12.0385 + 19.1592i 62.7885 − 14.3311i 7.58914 − 67.3555i
11.4 −2.66971 + 0.934170i 1.00898 + 8.95496i 6.25465 − 4.98792i 14.7345 − 30.5965i −11.0591 − 22.9646i 18.3299 − 22.9850i −12.0385 + 19.1592i −0.204167 + 0.0465998i −10.7544 + 95.4483i
11.5 −2.66971 + 0.934170i 1.02274 + 9.07706i 6.25465 − 4.98792i −11.9766 + 24.8698i −11.2099 − 23.2777i −3.16211 + 3.96516i −12.0385 + 19.1592i −2.37789 + 0.542737i 8.74152 − 77.5832i
15.1 2.81064 − 0.316683i −12.5011 + 7.85497i 7.79942 − 1.78017i −14.0926 − 11.2385i −32.6486 + 26.0364i 18.6973 − 81.9182i 21.3576 − 7.47336i 59.4328 − 123.413i −43.1682 − 27.1244i
15.2 2.81064 − 0.316683i −6.07779 + 3.81893i 7.79942 − 1.78017i 14.9420 + 11.9158i −15.8731 + 12.6584i −8.34921 + 36.5803i 21.3576 − 7.47336i −12.7892 + 26.5571i 45.7700 + 28.7592i
15.3 2.81064 − 0.316683i 4.65774 − 2.92665i 7.79942 − 1.78017i 15.8635 + 12.6507i 12.1644 − 9.70080i 9.44200 − 41.3681i 21.3576 − 7.47336i −22.0153 + 45.7153i 48.5928 + 30.5329i
15.4 2.81064 − 0.316683i 7.72040 − 4.85105i 7.79942 − 1.78017i −37.7511 − 30.1055i 20.1630 − 16.0795i 11.0478 − 48.4035i 21.3576 − 7.47336i 0.927339 − 1.92564i −115.639 − 72.6607i
15.5 2.81064 − 0.316683i 12.1735 − 7.64915i 7.79942 − 1.78017i 11.0509 + 8.81276i 31.7931 − 25.3542i −10.3552 + 45.3689i 21.3576 − 7.47336i 54.5412 − 113.256i 33.8509 + 21.2699i
19.1 1.50481 + 2.39490i −4.59385 + 13.1285i −3.47107 + 7.20775i −24.2508 − 5.53509i −38.3542 + 8.75411i 27.9679 − 13.4686i −22.4851 + 2.53347i −87.9250 − 70.1179i −23.2370 − 66.4075i
19.2 1.50481 + 2.39490i −3.65838 + 10.4551i −3.47107 + 7.20775i 43.5958 + 9.95047i −30.5440 + 6.97147i −9.97605 + 4.80421i −22.4851 + 2.53347i −32.5962 − 25.9946i 41.7733 + 119.381i
19.3 1.50481 + 2.39490i 0.483256 − 1.38107i −3.47107 + 7.20775i −23.7716 − 5.42571i 4.03473 − 0.920900i −79.5231 + 38.2963i −22.4851 + 2.53347i 61.6545 + 49.1679i −22.7778 − 65.0952i
19.4 1.50481 + 2.39490i 1.76881 − 5.05496i −3.47107 + 7.20775i 18.1608 + 4.14509i 14.7678 − 3.37066i 21.7107 − 10.4553i −22.4851 + 2.53347i 40.9044 + 32.6202i 17.4016 + 49.7309i
19.5 1.50481 + 2.39490i 5.16868 − 14.7712i −3.47107 + 7.20775i −36.4102 − 8.31038i 43.1535 − 9.84951i 54.6680 − 26.3267i −22.4851 + 2.53347i −128.146 − 102.193i −34.8880 − 99.7042i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.5.f.a ✓ 60
29.f odd 28 1 inner 58.5.f.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.5.f.a ✓ 60 1.a even 1 1 trivial
58.5.f.a ✓ 60 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 16 T_{3}^{59} + 128 T_{3}^{58} - 3236 T_{3}^{57} - 15962 T_{3}^{56} + 428612 T_{3}^{55} + \cdots + 26\!\cdots\!56 \) acting on \(S_{5}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display