Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(29,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.y (of order \(10\), degree \(4\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −0.809017 | − | 0.587785i | −1.68775 | + | 0.389232i | 0.309017 | + | 0.951057i | −1.12975 | + | 1.92968i | 1.59420 | + | 0.677139i | 1.54573 | − | 0.502238i | 0.309017 | − | 0.951057i | 2.69700 | − | 1.31385i | 2.04823 | − | 0.897090i |
29.2 | −0.809017 | − | 0.587785i | −1.67374 | − | 0.445633i | 0.309017 | + | 0.951057i | −1.13834 | − | 1.92463i | 1.09215 | + | 1.34433i | −3.43231 | + | 1.11523i | 0.309017 | − | 0.951057i | 2.60282 | + | 1.49175i | −0.210331 | + | 2.22615i |
29.3 | −0.809017 | − | 0.587785i | −1.66935 | − | 0.461806i | 0.309017 | + | 0.951057i | 1.02596 | + | 1.98681i | 1.07909 | + | 1.35483i | −2.19882 | + | 0.714440i | 0.309017 | − | 0.951057i | 2.57347 | + | 1.54183i | 0.337800 | − | 2.21041i |
29.4 | −0.809017 | − | 0.587785i | −1.61743 | − | 0.619623i | 0.309017 | + | 0.951057i | −0.184552 | − | 2.22844i | 0.944320 | + | 1.45198i | 2.03784 | − | 0.662134i | 0.309017 | − | 0.951057i | 2.23213 | + | 2.00439i | −1.16054 | + | 1.91132i |
29.5 | −0.809017 | − | 0.587785i | −1.60626 | + | 0.648031i | 0.309017 | + | 0.951057i | 2.20040 | − | 0.397802i | 1.68039 | + | 0.419865i | 4.23373 | − | 1.37562i | 0.309017 | − | 0.951057i | 2.16011 | − | 2.08181i | −2.01398 | − | 0.971534i |
29.6 | −0.809017 | − | 0.587785i | −1.51893 | + | 0.832372i | 0.309017 | + | 0.951057i | 1.72517 | − | 1.42260i | 1.71810 | + | 0.219404i | 0.914445 | − | 0.297121i | 0.309017 | − | 0.951057i | 1.61431 | − | 2.52863i | −2.23187 | + | 0.136884i |
29.7 | −0.809017 | − | 0.587785i | −1.32727 | + | 1.11282i | 0.309017 | + | 0.951057i | −2.11788 | − | 0.717333i | 1.72788 | − | 0.120141i | 0.344014 | − | 0.111777i | 0.309017 | − | 0.951057i | 0.523274 | − | 2.95401i | 1.29177 | + | 1.82520i |
29.8 | −0.809017 | − | 0.587785i | −1.26585 | − | 1.18221i | 0.309017 | + | 0.951057i | 1.86663 | + | 1.23113i | 0.329206 | + | 1.70048i | 3.84247 | − | 1.24850i | 0.309017 | − | 0.951057i | 0.204746 | + | 2.99301i | −0.786495 | − | 2.09319i |
29.9 | −0.809017 | − | 0.587785i | −1.22753 | + | 1.22195i | 0.309017 | + | 0.951057i | 0.560987 | − | 2.16455i | 1.71134 | − | 0.267056i | −4.05181 | + | 1.31651i | 0.309017 | − | 0.951057i | 0.0136636 | − | 2.99997i | −1.72614 | + | 1.42142i |
29.10 | −0.809017 | − | 0.587785i | −1.02011 | − | 1.39978i | 0.309017 | + | 0.951057i | −2.08406 | + | 0.810354i | 0.00251367 | + | 1.73205i | −2.70101 | + | 0.877611i | 0.309017 | − | 0.951057i | −0.918766 | + | 2.85585i | 2.16236 | + | 0.569392i |
29.11 | −0.809017 | − | 0.587785i | −1.01604 | − | 1.40273i | 0.309017 | + | 0.951057i | −2.08406 | − | 0.810354i | −0.00251367 | + | 1.73205i | 2.70101 | − | 0.877611i | 0.309017 | − | 0.951057i | −0.935328 | + | 2.85047i | 1.20973 | + | 1.88057i |
29.12 | −0.809017 | − | 0.587785i | −0.733182 | − | 1.56922i | 0.309017 | + | 0.951057i | 1.86663 | − | 1.23113i | −0.329206 | + | 1.70048i | −3.84247 | + | 1.24850i | 0.309017 | − | 0.951057i | −1.92489 | + | 2.30105i | −2.23378 | − | 0.101171i |
29.13 | −0.809017 | − | 0.587785i | −0.515138 | + | 1.65367i | 0.309017 | + | 0.951057i | −1.43197 | + | 1.71740i | 1.38876 | − | 1.03506i | −4.13806 | + | 1.34454i | 0.309017 | − | 0.951057i | −2.46927 | − | 1.70374i | 2.16795 | − | 0.547714i |
29.14 | −0.809017 | − | 0.587785i | −0.292097 | + | 1.70724i | 0.309017 | + | 0.951057i | 1.90023 | + | 1.17861i | 1.23980 | − | 1.20950i | −0.119229 | + | 0.0387399i | 0.309017 | − | 0.951057i | −2.82936 | − | 0.997360i | −0.844545 | − | 2.07045i |
29.15 | −0.809017 | − | 0.587785i | −0.0894846 | − | 1.72974i | 0.309017 | + | 0.951057i | −0.184552 | + | 2.22844i | −0.944320 | + | 1.45198i | −2.03784 | + | 0.662134i | 0.309017 | − | 0.951057i | −2.98398 | + | 0.309570i | 1.45915 | − | 1.69437i |
29.16 | −0.809017 | − | 0.587785i | 0.0766541 | − | 1.73035i | 0.309017 | + | 0.951057i | 1.02596 | − | 1.98681i | −1.07909 | + | 1.35483i | 2.19882 | − | 0.714440i | 0.309017 | − | 0.951057i | −2.98825 | − | 0.265277i | −1.99783 | + | 1.00432i |
29.17 | −0.809017 | − | 0.587785i | 0.0933927 | − | 1.72953i | 0.309017 | + | 0.951057i | −1.13834 | + | 1.92463i | −1.09215 | + | 1.34433i | 3.43231 | − | 1.11523i | 0.309017 | − | 0.951057i | −2.98256 | − | 0.323051i | 2.05220 | − | 0.887956i |
29.18 | −0.809017 | − | 0.587785i | 0.116082 | + | 1.72816i | 0.309017 | + | 0.951057i | −1.22659 | + | 1.86962i | 0.921873 | − | 1.46634i | 3.97153 | − | 1.29043i | 0.309017 | − | 0.951057i | −2.97305 | + | 0.401214i | 2.09127 | − | 0.791586i |
29.19 | −0.809017 | − | 0.587785i | 0.195625 | + | 1.72097i | 0.309017 | + | 0.951057i | −0.323295 | − | 2.21257i | 0.853296 | − | 1.50728i | 1.33289 | − | 0.433081i | 0.309017 | − | 0.951057i | −2.92346 | + | 0.673329i | −1.03897 | + | 1.98004i |
29.20 | −0.809017 | − | 0.587785i | 0.317374 | + | 1.70273i | 0.309017 | + | 0.951057i | 2.12610 | − | 0.692603i | 0.744076 | − | 1.56408i | −0.896862 | + | 0.291408i | 0.309017 | − | 0.951057i | −2.79855 | + | 1.08080i | −2.12715 | − | 0.689362i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | inner |
31.f | odd | 10 | 1 | inner |
465.w | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.y.a | ✓ | 128 |
3.b | odd | 2 | 1 | 930.2.y.b | yes | 128 | |
5.b | even | 2 | 1 | 930.2.y.b | yes | 128 | |
15.d | odd | 2 | 1 | inner | 930.2.y.a | ✓ | 128 |
31.f | odd | 10 | 1 | inner | 930.2.y.a | ✓ | 128 |
93.k | even | 10 | 1 | 930.2.y.b | yes | 128 | |
155.m | odd | 10 | 1 | 930.2.y.b | yes | 128 | |
465.w | even | 10 | 1 | inner | 930.2.y.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.y.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
930.2.y.a | ✓ | 128 | 15.d | odd | 2 | 1 | inner |
930.2.y.a | ✓ | 128 | 31.f | odd | 10 | 1 | inner |
930.2.y.a | ✓ | 128 | 465.w | even | 10 | 1 | inner |
930.2.y.b | yes | 128 | 3.b | odd | 2 | 1 | |
930.2.y.b | yes | 128 | 5.b | even | 2 | 1 | |
930.2.y.b | yes | 128 | 93.k | even | 10 | 1 | |
930.2.y.b | yes | 128 | 155.m | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{64} - 98 T_{17}^{62} + 45 T_{17}^{61} + 8611 T_{17}^{60} - 4410 T_{17}^{59} + \cdots + 22\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).