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.
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 |
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 \(S_{7}^{\mathrm{new}}(58, [\chi])\).