Properties

Label 58.7.f.b
Level $58$
Weight $7$
Character orbit 58.f
Analytic conductor $13.343$
Analytic rank $0$
Dimension $96$
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] = [58,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(13.3431368500\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 64 q^{2} - 28 q^{3} - 588 q^{5} - 610 q^{7} + 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 64 q^{2} - 28 q^{3} - 588 q^{5} - 610 q^{7} + 2048 q^{8} - 1744 q^{10} + 1752 q^{11} + 896 q^{12} - 2608 q^{14} - 3048 q^{15} + 16384 q^{16} - 2144 q^{17} + 24368 q^{18} - 20742 q^{19} + 16064 q^{20} + 80556 q^{21} + 35280 q^{22} + 24730 q^{23} - 17920 q^{24} + 9556 q^{25} - 45072 q^{26} - 196048 q^{27} + 55514 q^{29} + 54464 q^{30} + 130770 q^{31} + 65536 q^{32} + 469616 q^{33} - 65184 q^{34} - 153664 q^{35} + 56960 q^{36} - 361120 q^{37} - 229040 q^{38} + 16520 q^{39} + 19456 q^{40} + 267404 q^{41} - 288064 q^{42} - 104504 q^{43} - 18432 q^{44} + 17462 q^{45} - 12912 q^{46} - 278966 q^{47} + 28672 q^{48} - 59808 q^{49} + 882256 q^{50} + 465150 q^{51} + 125952 q^{52} - 1150130 q^{53} + 31312 q^{54} - 2377762 q^{55} + 83456 q^{56} + 392288 q^{58} - 2028140 q^{59} - 97536 q^{60} + 875164 q^{61} + 404880 q^{62} + 4558036 q^{63} - 731620 q^{65} - 978576 q^{66} - 394170 q^{67} + 68608 q^{68} - 2223748 q^{69} - 1125008 q^{70} - 3274600 q^{71} + 34304 q^{72} + 2623288 q^{73} + 1590592 q^{74} + 3502816 q^{75} + 1044096 q^{76} - 196188 q^{77} + 988064 q^{78} + 854806 q^{79} + 4049800 q^{81} - 3418208 q^{82} - 2509204 q^{83} - 3344768 q^{84} - 4533520 q^{85} + 1785564 q^{87} - 147456 q^{88} + 5032828 q^{89} + 757472 q^{90} + 5996522 q^{91} + 6471360 q^{92} - 1360380 q^{93} + 4358912 q^{94} - 3508130 q^{95} + 6610532 q^{97} - 8015840 q^{98} - 833400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −4.78980 3.00963i −39.6453 + 13.8725i 13.8843 + 28.8310i 0.654668 0.149424i 231.644 + 52.8713i 415.647 + 200.165i 20.2677 179.881i 809.351 645.436i −3.58543 1.25460i
3.2 −4.78980 3.00963i −36.0200 + 12.6040i 13.8843 + 28.8310i 154.337 35.2264i 210.462 + 48.0365i −363.728 175.162i 20.2677 179.881i 568.627 453.465i −845.261 295.770i
3.3 −4.78980 3.00963i −23.8056 + 8.32995i 13.8843 + 28.8310i −75.3301 + 17.1936i 139.094 + 31.7473i −529.438 254.964i 20.2677 179.881i −72.6354 + 57.9248i 412.562 + 144.362i
3.4 −4.78980 3.00963i −7.40636 + 2.59160i 13.8843 + 28.8310i −219.003 + 49.9861i 43.2747 + 9.87716i 150.194 + 72.3295i 20.2677 179.881i −521.817 + 416.135i 1199.42 + 419.695i
3.5 −4.78980 3.00963i 11.1155 3.88948i 13.8843 + 28.8310i −22.9295 + 5.23352i −64.9469 14.8237i 63.4923 + 30.5763i 20.2677 179.881i −461.529 + 368.057i 125.579 + 43.9419i
3.6 −4.78980 3.00963i 13.6302 4.76940i 13.8843 + 28.8310i 229.696 52.4266i −79.6398 18.1773i 200.345 + 96.4812i 20.2677 179.881i −406.921 + 324.509i −1257.98 440.187i
3.7 −4.78980 3.00963i 37.4065 13.0891i 13.8843 + 28.8310i 47.7836 10.9063i −218.563 49.8855i −542.272 261.144i 20.2677 179.881i 657.967 524.711i −261.698 91.5720i
3.8 −4.78980 3.00963i 48.8836 17.1051i 13.8843 + 28.8310i −93.7078 + 21.3882i −285.623 65.1915i 519.716 + 250.282i 20.2677 179.881i 1527.07 1217.80i 513.212 + 179.581i
11.1 −5.33941 + 1.86834i −5.49168 48.7400i 25.0186 19.9517i −33.3230 + 69.1959i 120.385 + 249.983i −207.373 + 260.037i −96.3081 + 153.273i −1634.71 + 373.111i 48.6436 431.724i
11.2 −5.33941 + 1.86834i −2.73962 24.3148i 25.0186 19.9517i 84.6403 175.757i 60.0564 + 124.708i 167.550 210.101i −96.3081 + 153.273i 127.017 28.9908i −123.555 + 1096.58i
11.3 −5.33941 + 1.86834i −2.36752 21.0123i 25.0186 19.9517i −105.239 + 218.531i 51.8992 + 107.770i 350.064 438.966i −96.3081 + 153.273i 274.811 62.7239i 153.624 1363.45i
11.4 −5.33941 + 1.86834i −1.90143 16.8756i 25.0186 19.9517i −12.5664 + 26.0943i 41.6820 + 86.5535i −167.248 + 209.722i −96.3081 + 153.273i 429.551 98.0421i 18.3439 162.807i
11.5 −5.33941 + 1.86834i 1.43761 + 12.7591i 25.0186 19.9517i 13.2903 27.5976i −31.5143 65.4402i −171.059 + 214.502i −96.3081 + 153.273i 549.994 125.533i −19.4006 + 172.186i
11.6 −5.33941 + 1.86834i 2.20529 + 19.5725i 25.0186 19.9517i −4.56263 + 9.47440i −48.3430 100.385i 151.066 189.431i −96.3081 + 153.273i 332.504 75.8920i 6.66035 59.1122i
11.7 −5.33941 + 1.86834i 5.12154 + 45.4549i 25.0186 19.9517i −100.885 + 209.491i −112.271 233.134i −148.380 + 186.062i −96.3081 + 153.273i −1329.20 + 303.381i 147.269 1307.04i
11.8 −5.33941 + 1.86834i 5.73307 + 50.8824i 25.0186 19.9517i 75.9075 157.623i −125.677 260.971i −99.3913 + 124.633i −96.3081 + 153.273i −1845.43 + 421.206i −110.807 + 983.438i
15.1 5.62129 0.633367i −42.6695 + 26.8110i 31.1977 7.12067i 24.9612 + 19.9059i −222.876 + 177.738i −109.411 + 479.360i 170.861 59.7869i 785.552 1631.21i 152.922 + 96.0870i
15.2 5.62129 0.633367i −28.3817 + 17.8334i 31.1977 7.12067i −50.5077 40.2785i −148.247 + 118.223i 72.6781 318.423i 170.861 59.7869i 171.190 355.480i −309.429 194.427i
15.3 5.62129 0.633367i −16.8789 + 10.6057i 31.1977 7.12067i 92.2719 + 73.5844i −88.1635 + 70.3081i 15.6373 68.5116i 170.861 59.7869i −143.886 + 298.783i 565.293 + 355.197i
15.4 5.62129 0.633367i −1.34237 + 0.843466i 31.1977 7.12067i −129.241 103.066i −7.01161 + 5.59157i −126.038 + 552.209i 170.861 59.7869i −315.211 + 654.542i −791.779 497.508i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

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 58.7.f.b 96
29.f odd 28 1 inner 58.7.f.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.7.f.b 96 1.a even 1 1 trivial
58.7.f.b 96 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + 28 T_{3}^{95} + 392 T_{3}^{94} + 103028 T_{3}^{93} - 5786827 T_{3}^{92} - 296330384 T_{3}^{91} + 4207616884 T_{3}^{90} - 412254793840 T_{3}^{89} + 17736027712794 T_{3}^{88} + \cdots + 47\!\cdots\!04 \) acting on \(S_{7}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display