Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,7,Mod(3,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.3");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 50.f (of order \(20\), degree \(8\), 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: | \(11.5027041810\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | −2.56816 | − | 5.04029i | −7.53135 | − | 47.5511i | −18.8091 | + | 25.8885i | 56.7542 | + | 111.373i | −220.330 | + | 160.079i | −158.728 | + | 158.728i | 178.791 | + | 28.3177i | −1511.06 | + | 490.974i | 415.599 | − | 572.082i |
3.2 | −2.56816 | − | 5.04029i | −5.66017 | − | 35.7369i | −18.8091 | + | 25.8885i | −83.9876 | − | 92.5801i | −165.588 | + | 120.307i | −288.370 | + | 288.370i | 178.791 | + | 28.3177i | −551.767 | + | 179.280i | −250.937 | + | 661.083i |
3.3 | −2.56816 | − | 5.04029i | −4.62916 | − | 29.2273i | −18.8091 | + | 25.8885i | 117.480 | − | 42.7023i | −135.426 | + | 98.3928i | 437.022 | − | 437.022i | 178.791 | + | 28.3177i | −139.489 | + | 45.3226i | −516.939 | − | 482.467i |
3.4 | −2.56816 | − | 5.04029i | −0.867997 | − | 5.48032i | −18.8091 | + | 25.8885i | −62.4621 | + | 108.275i | −25.3933 | + | 18.4493i | 128.992 | − | 128.992i | 178.791 | + | 28.3177i | 664.040 | − | 215.760i | 706.151 | + | 36.7598i |
3.5 | −2.56816 | − | 5.04029i | 0.756144 | + | 4.77411i | −18.8091 | + | 25.8885i | 86.6763 | − | 90.0679i | 22.1210 | − | 16.0718i | −203.996 | + | 203.996i | 178.791 | + | 28.3177i | 671.100 | − | 218.054i | −676.567 | − | 205.566i |
3.6 | −2.56816 | − | 5.04029i | 2.78783 | + | 17.6017i | −18.8091 | + | 25.8885i | −73.1723 | − | 101.345i | 81.5581 | − | 59.2554i | 55.6165 | − | 55.6165i | 178.791 | + | 28.3177i | 391.273 | − | 127.132i | −322.891 | + | 629.080i |
3.7 | −2.56816 | − | 5.04029i | 6.17165 | + | 38.9663i | −18.8091 | + | 25.8885i | 103.570 | + | 69.9869i | 180.552 | − | 131.178i | 178.415 | − | 178.415i | 178.791 | + | 28.3177i | −786.961 | + | 255.699i | 86.7692 | − | 701.763i |
3.8 | −2.56816 | − | 5.04029i | 6.70041 | + | 42.3047i | −18.8091 | + | 25.8885i | −114.140 | + | 50.9609i | 196.020 | − | 142.417i | −217.111 | + | 217.111i | 178.791 | + | 28.3177i | −1051.47 | + | 341.644i | 549.988 | + | 444.425i |
13.1 | −5.58721 | + | 0.884927i | −23.3017 | + | 45.7322i | 30.4338 | − | 9.88854i | −53.9211 | − | 112.772i | 89.7220 | − | 276.136i | −444.049 | + | 444.049i | −161.289 | + | 82.1811i | −1119.97 | − | 1541.51i | 401.063 | + | 582.364i |
13.2 | −5.58721 | + | 0.884927i | −20.0992 | + | 39.4470i | 30.4338 | − | 9.88854i | 120.893 | + | 31.7777i | 77.3910 | − | 238.185i | 452.271 | − | 452.271i | −161.289 | + | 82.1811i | −723.590 | − | 995.936i | −703.577 | − | 70.5669i |
13.3 | −5.58721 | + | 0.884927i | −8.42605 | + | 16.5371i | 30.4338 | − | 9.88854i | −50.0727 | + | 114.533i | 32.4440 | − | 99.8524i | −109.038 | + | 109.038i | −161.289 | + | 82.1811i | 226.020 | + | 311.089i | 178.414 | − | 684.228i |
13.4 | −5.58721 | + | 0.884927i | −7.43527 | + | 14.5925i | 30.4338 | − | 9.88854i | −119.974 | − | 35.0902i | 28.6291 | − | 88.1112i | 339.866 | − | 339.866i | −161.289 | + | 82.1811i | 270.837 | + | 372.775i | 701.370 | + | 89.8882i |
13.5 | −5.58721 | + | 0.884927i | 0.641223 | − | 1.25847i | 30.4338 | − | 9.88854i | 38.7204 | − | 118.852i | −2.46899 | + | 7.59878i | −71.7898 | + | 71.7898i | −161.289 | + | 82.1811i | 427.323 | + | 588.159i | −111.164 | + | 698.314i |
13.6 | −5.58721 | + | 0.884927i | 10.1653 | − | 19.9505i | 30.4338 | − | 9.88854i | 98.5109 | + | 76.9455i | −39.1408 | + | 120.463i | −104.115 | + | 104.115i | −161.289 | + | 82.1811i | 133.807 | + | 184.170i | −618.492 | − | 342.736i |
13.7 | −5.58721 | + | 0.884927i | 16.9188 | − | 33.2050i | 30.4338 | − | 9.88854i | 33.0662 | − | 120.547i | −65.1448 | + | 200.495i | 363.736 | − | 363.736i | −161.289 | + | 82.1811i | −387.830 | − | 533.803i | −78.0721 | + | 702.784i |
13.8 | −5.58721 | + | 0.884927i | 17.1882 | − | 33.7337i | 30.4338 | − | 9.88854i | −124.892 | − | 5.19064i | −66.1821 | + | 203.688i | −345.465 | + | 345.465i | −161.289 | + | 82.1811i | −414.034 | − | 569.869i | 702.392 | − | 81.5193i |
17.1 | −2.56816 | + | 5.04029i | −7.53135 | + | 47.5511i | −18.8091 | − | 25.8885i | 56.7542 | − | 111.373i | −220.330 | − | 160.079i | −158.728 | − | 158.728i | 178.791 | − | 28.3177i | −1511.06 | − | 490.974i | 415.599 | + | 572.082i |
17.2 | −2.56816 | + | 5.04029i | −5.66017 | + | 35.7369i | −18.8091 | − | 25.8885i | −83.9876 | + | 92.5801i | −165.588 | − | 120.307i | −288.370 | − | 288.370i | 178.791 | − | 28.3177i | −551.767 | − | 179.280i | −250.937 | − | 661.083i |
17.3 | −2.56816 | + | 5.04029i | −4.62916 | + | 29.2273i | −18.8091 | − | 25.8885i | 117.480 | + | 42.7023i | −135.426 | − | 98.3928i | 437.022 | + | 437.022i | 178.791 | − | 28.3177i | −139.489 | − | 45.3226i | −516.939 | + | 482.467i |
17.4 | −2.56816 | + | 5.04029i | −0.867997 | + | 5.48032i | −18.8091 | − | 25.8885i | −62.4621 | − | 108.275i | −25.3933 | − | 18.4493i | 128.992 | + | 128.992i | 178.791 | − | 28.3177i | 664.040 | + | 215.760i | 706.151 | − | 36.7598i |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 50.7.f.b | ✓ | 64 |
25.f | odd | 20 | 1 | inner | 50.7.f.b | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
50.7.f.b | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
50.7.f.b | ✓ | 64 | 25.f | odd | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 32 T_{3}^{63} + 892 T_{3}^{62} - 147166 T_{3}^{61} - 3588791 T_{3}^{60} + \cdots + 21\!\cdots\!36 \) acting on \(S_{7}^{\mathrm{new}}(50, [\chi])\).