Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,2,Mod(13,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([19, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.s (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: | \(1.39738203537\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −2.43824 | + | 0.386179i | −0.183665 | − | 0.0935818i | 3.89376 | − | 1.26516i | 2.22917 | + | 0.175511i | 0.483957 | + | 0.157247i | −2.41722 | − | 1.07566i | −4.60620 | + | 2.34698i | −1.73838 | − | 2.39268i | −5.50302 | + | 0.432920i |
13.2 | −2.43824 | + | 0.386179i | 0.183665 | + | 0.0935818i | 3.89376 | − | 1.26516i | −2.22917 | − | 0.175511i | −0.483957 | − | 0.157247i | 1.07566 | + | 2.41722i | −4.60620 | + | 2.34698i | −1.73838 | − | 2.39268i | 5.50302 | − | 0.432920i |
13.3 | −2.26523 | + | 0.358778i | −2.54886 | − | 1.29871i | 3.10045 | − | 1.00740i | −0.855724 | − | 2.06585i | 6.23972 | + | 2.02741i | 0.506256 | − | 2.59686i | −2.57481 | + | 1.31193i | 3.04670 | + | 4.19343i | 2.67959 | + | 4.37262i |
13.4 | −2.26523 | + | 0.358778i | 2.54886 | + | 1.29871i | 3.10045 | − | 1.00740i | 0.855724 | + | 2.06585i | −6.23972 | − | 2.02741i | 2.59686 | − | 0.506256i | −2.57481 | + | 1.31193i | 3.04670 | + | 4.19343i | −2.67959 | − | 4.37262i |
13.5 | −1.23441 | + | 0.195511i | −0.157781 | − | 0.0803936i | −0.416571 | + | 0.135352i | 0.231923 | − | 2.22401i | 0.210485 | + | 0.0683906i | 0.122096 | + | 2.64293i | 2.71491 | − | 1.38332i | −1.74492 | − | 2.40168i | 0.148530 | + | 2.79068i |
13.6 | −1.23441 | + | 0.195511i | 0.157781 | + | 0.0803936i | −0.416571 | + | 0.135352i | −0.231923 | + | 2.22401i | −0.210485 | − | 0.0683906i | −2.64293 | − | 0.122096i | 2.71491 | − | 1.38332i | −1.74492 | − | 2.40168i | −0.148530 | − | 2.79068i |
13.7 | −1.05758 | + | 0.167504i | −1.81417 | − | 0.924365i | −0.811695 | + | 0.263736i | −1.08801 | + | 1.95352i | 2.07346 | + | 0.673709i | 2.20364 | − | 1.46423i | 2.72237 | − | 1.38712i | 0.673400 | + | 0.926855i | 0.823432 | − | 2.24825i |
13.8 | −1.05758 | + | 0.167504i | 1.81417 | + | 0.924365i | −0.811695 | + | 0.263736i | 1.08801 | − | 1.95352i | −2.07346 | − | 0.673709i | 1.46423 | − | 2.20364i | 2.72237 | − | 1.38712i | 0.673400 | + | 0.926855i | −0.823432 | + | 2.24825i |
13.9 | −0.437303 | + | 0.0692620i | −2.82799 | − | 1.44093i | −1.71568 | + | 0.557457i | 2.19504 | − | 0.426380i | 1.33649 | + | 0.434253i | −2.15975 | + | 1.52822i | 1.50065 | − | 0.764621i | 4.15790 | + | 5.72286i | −0.930365 | + | 0.338490i |
13.10 | −0.437303 | + | 0.0692620i | 2.82799 | + | 1.44093i | −1.71568 | + | 0.557457i | −2.19504 | + | 0.426380i | −1.33649 | − | 0.434253i | −1.52822 | + | 2.15975i | 1.50065 | − | 0.764621i | 4.15790 | + | 5.72286i | 0.930365 | − | 0.338490i |
13.11 | 0.401286 | − | 0.0635575i | −0.682337 | − | 0.347668i | −1.74512 | + | 0.567024i | −1.40273 | − | 1.74137i | −0.295909 | − | 0.0961468i | −1.74509 | − | 1.98864i | −1.38827 | + | 0.707357i | −1.41865 | − | 1.95260i | −0.673572 | − | 0.609633i |
13.12 | 0.401286 | − | 0.0635575i | 0.682337 | + | 0.347668i | −1.74512 | + | 0.567024i | 1.40273 | + | 1.74137i | 0.295909 | + | 0.0961468i | 1.98864 | + | 1.74509i | −1.38827 | + | 0.707357i | −1.41865 | − | 1.95260i | 0.673572 | + | 0.609633i |
13.13 | 1.00543 | − | 0.159244i | −2.24750 | − | 1.14516i | −0.916583 | + | 0.297816i | −2.16146 | + | 0.572778i | −2.44206 | − | 0.793475i | 1.00386 | + | 2.44791i | −2.68816 | + | 1.36968i | 1.97652 | + | 2.72044i | −2.08199 | + | 0.920089i |
13.14 | 1.00543 | − | 0.159244i | 2.24750 | + | 1.14516i | −0.916583 | + | 0.297816i | 2.16146 | − | 0.572778i | 2.44206 | + | 0.793475i | −2.44791 | − | 1.00386i | −2.68816 | + | 1.36968i | 1.97652 | + | 2.72044i | 2.08199 | − | 0.920089i |
13.15 | 1.78879 | − | 0.283316i | −1.39081 | − | 0.708653i | 1.21738 | − | 0.395551i | 1.39172 | − | 1.75018i | −2.68864 | − | 0.873591i | 2.56523 | − | 0.647770i | −1.16181 | + | 0.591970i | −0.331192 | − | 0.455847i | 1.99363 | − | 3.52499i |
13.16 | 1.78879 | − | 0.283316i | 1.39081 | + | 0.708653i | 1.21738 | − | 0.395551i | −1.39172 | + | 1.75018i | 2.68864 | + | 0.873591i | 0.647770 | − | 2.56523i | −1.16181 | + | 0.591970i | −0.331192 | − | 0.455847i | −1.99363 | + | 3.52499i |
13.17 | 2.28620 | − | 0.362099i | −0.873949 | − | 0.445299i | 3.19350 | − | 1.03763i | 1.80613 | + | 1.31829i | −2.15927 | − | 0.701589i | −1.94696 | + | 1.79146i | 2.80043 | − | 1.42689i | −1.19786 | − | 1.64871i | 4.60653 | + | 2.35989i |
13.18 | 2.28620 | − | 0.362099i | 0.873949 | + | 0.445299i | 3.19350 | − | 1.03763i | −1.80613 | − | 1.31829i | 2.15927 | + | 0.701589i | −1.79146 | + | 1.94696i | 2.80043 | − | 1.42689i | −1.19786 | − | 1.64871i | −4.60653 | − | 2.35989i |
27.1 | −2.43824 | − | 0.386179i | −0.183665 | + | 0.0935818i | 3.89376 | + | 1.26516i | 2.22917 | − | 0.175511i | 0.483957 | − | 0.157247i | −2.41722 | + | 1.07566i | −4.60620 | − | 2.34698i | −1.73838 | + | 2.39268i | −5.50302 | − | 0.432920i |
27.2 | −2.43824 | − | 0.386179i | 0.183665 | − | 0.0935818i | 3.89376 | + | 1.26516i | −2.22917 | + | 0.175511i | −0.483957 | + | 0.157247i | 1.07566 | − | 2.41722i | −4.60620 | − | 2.34698i | −1.73838 | + | 2.39268i | 5.50302 | + | 0.432920i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
25.f | odd | 20 | 1 | inner |
175.s | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.2.s.a | ✓ | 144 |
5.b | even | 2 | 1 | 875.2.s.c | 144 | ||
5.c | odd | 4 | 1 | 875.2.s.a | 144 | ||
5.c | odd | 4 | 1 | 875.2.s.b | 144 | ||
7.b | odd | 2 | 1 | inner | 175.2.s.a | ✓ | 144 |
25.d | even | 5 | 1 | 875.2.s.b | 144 | ||
25.e | even | 10 | 1 | 875.2.s.a | 144 | ||
25.f | odd | 20 | 1 | inner | 175.2.s.a | ✓ | 144 |
25.f | odd | 20 | 1 | 875.2.s.c | 144 | ||
35.c | odd | 2 | 1 | 875.2.s.c | 144 | ||
35.f | even | 4 | 1 | 875.2.s.a | 144 | ||
35.f | even | 4 | 1 | 875.2.s.b | 144 | ||
175.l | odd | 10 | 1 | 875.2.s.b | 144 | ||
175.m | odd | 10 | 1 | 875.2.s.a | 144 | ||
175.s | even | 20 | 1 | inner | 175.2.s.a | ✓ | 144 |
175.s | even | 20 | 1 | 875.2.s.c | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.2.s.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
175.2.s.a | ✓ | 144 | 7.b | odd | 2 | 1 | inner |
175.2.s.a | ✓ | 144 | 25.f | odd | 20 | 1 | inner |
175.2.s.a | ✓ | 144 | 175.s | even | 20 | 1 | inner |
875.2.s.a | 144 | 5.c | odd | 4 | 1 | ||
875.2.s.a | 144 | 25.e | even | 10 | 1 | ||
875.2.s.a | 144 | 35.f | even | 4 | 1 | ||
875.2.s.a | 144 | 175.m | odd | 10 | 1 | ||
875.2.s.b | 144 | 5.c | odd | 4 | 1 | ||
875.2.s.b | 144 | 25.d | even | 5 | 1 | ||
875.2.s.b | 144 | 35.f | even | 4 | 1 | ||
875.2.s.b | 144 | 175.l | odd | 10 | 1 | ||
875.2.s.c | 144 | 5.b | even | 2 | 1 | ||
875.2.s.c | 144 | 25.f | odd | 20 | 1 | ||
875.2.s.c | 144 | 35.c | odd | 2 | 1 | ||
875.2.s.c | 144 | 175.s | even | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).