Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(179,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 15, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.179");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bo (of order \(30\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(256\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
179.1 | 0.309017 | + | 0.951057i | −1.73157 | − | 0.0407018i | −0.809017 | + | 0.587785i | 0.493029 | + | 2.18104i | −0.496376 | − | 1.65940i | −0.863747 | + | 1.94001i | −0.809017 | − | 0.587785i | 2.99669 | + | 0.140956i | −1.92193 | + | 1.14288i |
179.2 | 0.309017 | + | 0.951057i | −1.69787 | − | 0.342394i | −0.809017 | + | 0.587785i | −1.92997 | − | 1.12925i | −0.199035 | − | 1.72058i | −0.934663 | + | 2.09929i | −0.809017 | − | 0.587785i | 2.76553 | + | 1.16268i | 0.477584 | − | 2.18447i |
179.3 | 0.309017 | + | 0.951057i | −1.66354 | − | 0.482315i | −0.809017 | + | 0.587785i | 0.575623 | − | 2.16071i | −0.0553539 | − | 1.73117i | −1.66082 | + | 3.73026i | −0.809017 | − | 0.587785i | 2.53474 | + | 1.60470i | 2.23283 | − | 0.120245i |
179.4 | 0.309017 | + | 0.951057i | −1.64568 | + | 0.540140i | −0.809017 | + | 0.587785i | 2.22856 | − | 0.183115i | −1.02225 | − | 1.39822i | 0.318388 | − | 0.715111i | −0.809017 | − | 0.587785i | 2.41650 | − | 1.77779i | 0.862815 | + | 2.06290i |
179.5 | 0.309017 | + | 0.951057i | −1.59415 | − | 0.677259i | −0.809017 | + | 0.587785i | −2.09420 | + | 0.783790i | 0.151491 | − | 1.72541i | 1.79370 | − | 4.02872i | −0.809017 | − | 0.587785i | 2.08264 | + | 2.15931i | −1.39257 | − | 1.74950i |
179.6 | 0.309017 | + | 0.951057i | −1.46052 | − | 0.931061i | −0.809017 | + | 0.587785i | 1.43804 | + | 1.71232i | 0.434165 | − | 1.67675i | 0.429946 | − | 0.965674i | −0.809017 | − | 0.587785i | 1.26625 | + | 2.71967i | −1.18414 | + | 1.89679i |
179.7 | 0.309017 | + | 0.951057i | −1.41936 | + | 0.992677i | −0.809017 | + | 0.587785i | −1.50750 | + | 1.65150i | −1.38270 | − | 1.04314i | 0.732525 | − | 1.64528i | −0.809017 | − | 0.587785i | 1.02919 | − | 2.81794i | −2.03651 | − | 0.923374i |
179.8 | 0.309017 | + | 0.951057i | −1.39110 | + | 1.03191i | −0.809017 | + | 0.587785i | −1.53101 | − | 1.62973i | −1.41128 | − | 1.00414i | 0.325961 | − | 0.732121i | −0.809017 | − | 0.587785i | 0.870333 | − | 2.87098i | 1.07685 | − | 1.95969i |
179.9 | 0.309017 | + | 0.951057i | −1.21541 | − | 1.23401i | −0.809017 | + | 0.587785i | 1.34040 | − | 1.78978i | 0.798025 | − | 1.53726i | 1.34591 | − | 3.02296i | −0.809017 | − | 0.587785i | −0.0455387 | + | 2.99965i | 2.11639 | + | 0.721724i |
179.10 | 0.309017 | + | 0.951057i | −1.03655 | + | 1.38764i | −0.809017 | + | 0.587785i | −1.97712 | + | 1.04451i | −1.64004 | − | 0.557016i | −1.83029 | + | 4.11091i | −0.809017 | − | 0.587785i | −0.851111 | − | 2.87674i | −1.60435 | − | 1.55758i |
179.11 | 0.309017 | + | 0.951057i | −0.835073 | − | 1.51745i | −0.809017 | + | 0.587785i | 2.23544 | + | 0.0527903i | 1.18513 | − | 1.26312i | −0.915394 | + | 2.05601i | −0.809017 | − | 0.587785i | −1.60531 | + | 2.53436i | 0.640584 | + | 2.14235i |
179.12 | 0.309017 | + | 0.951057i | −0.732396 | + | 1.56958i | −0.809017 | + | 0.587785i | 0.719639 | + | 2.11710i | −1.71909 | − | 0.211521i | 1.13296 | − | 2.54466i | −0.809017 | − | 0.587785i | −1.92719 | − | 2.29911i | −1.79110 | + | 1.33864i |
179.13 | 0.309017 | + | 0.951057i | −0.696767 | + | 1.58572i | −0.809017 | + | 0.587785i | 1.79380 | − | 1.33503i | −1.72342 | − | 0.172650i | 0.308507 | − | 0.692917i | −0.809017 | − | 0.587785i | −2.02903 | − | 2.20976i | 1.82400 | + | 1.29346i |
179.14 | 0.309017 | + | 0.951057i | −0.584760 | − | 1.63035i | −0.809017 | + | 0.587785i | −2.12123 | + | 0.707392i | 1.36986 | − | 1.05995i | −1.37162 | + | 3.08072i | −0.809017 | − | 0.587785i | −2.31611 | + | 1.90673i | −1.32826 | − | 1.79881i |
179.15 | 0.309017 | + | 0.951057i | −0.128921 | − | 1.72725i | −0.809017 | + | 0.587785i | 0.447993 | + | 2.19073i | 1.60287 | − | 0.656359i | 1.37162 | − | 3.08072i | −0.809017 | − | 0.587785i | −2.96676 | + | 0.445355i | −1.94507 | + | 1.10304i |
179.16 | 0.309017 | + | 0.951057i | 0.0869307 | + | 1.72987i | −0.809017 | + | 0.587785i | −2.09670 | − | 0.777090i | −1.61834 | + | 0.617235i | 1.84852 | − | 4.15185i | −0.809017 | − | 0.587785i | −2.98489 | + | 0.300757i | 0.0911422 | − | 2.23421i |
179.17 | 0.309017 | + | 0.951057i | 0.145674 | − | 1.72591i | −0.809017 | + | 0.587785i | −1.16344 | − | 1.90956i | 1.68646 | − | 0.394792i | 0.915394 | − | 2.05601i | −0.809017 | − | 0.587785i | −2.95756 | − | 0.502843i | 1.45657 | − | 1.69658i |
179.18 | 0.309017 | + | 0.951057i | 0.198005 | + | 1.72070i | −0.809017 | + | 0.587785i | 2.18990 | − | 0.452039i | −1.57529 | + | 0.720038i | 0.478693 | − | 1.07516i | −0.809017 | − | 0.587785i | −2.92159 | + | 0.681412i | 1.10663 | + | 1.94303i |
179.19 | 0.309017 | + | 0.951057i | 0.518984 | + | 1.65247i | −0.809017 | + | 0.587785i | −0.703473 | − | 2.12253i | −1.41122 | + | 1.00422i | −0.478693 | + | 1.07516i | −0.809017 | − | 0.587785i | −2.46131 | + | 1.71521i | 1.80126 | − | 1.32494i |
179.20 | 0.309017 | + | 0.951057i | 0.608421 | − | 1.62167i | −0.809017 | + | 0.587785i | 0.879797 | − | 2.05571i | 1.73032 | + | 0.0775177i | −1.34591 | + | 3.02296i | −0.809017 | − | 0.587785i | −2.25965 | − | 1.97332i | 2.22697 | + | 0.201486i |
See next 80 embeddings (of 256 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | inner |
31.h | odd | 30 | 1 | inner |
465.bm | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bo.a | ✓ | 256 |
3.b | odd | 2 | 1 | 930.2.bo.b | yes | 256 | |
5.b | even | 2 | 1 | 930.2.bo.b | yes | 256 | |
15.d | odd | 2 | 1 | inner | 930.2.bo.a | ✓ | 256 |
31.h | odd | 30 | 1 | inner | 930.2.bo.a | ✓ | 256 |
93.p | even | 30 | 1 | 930.2.bo.b | yes | 256 | |
155.v | odd | 30 | 1 | 930.2.bo.b | yes | 256 | |
465.bm | even | 30 | 1 | inner | 930.2.bo.a | ✓ | 256 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bo.a | ✓ | 256 | 1.a | even | 1 | 1 | trivial |
930.2.bo.a | ✓ | 256 | 15.d | odd | 2 | 1 | inner |
930.2.bo.a | ✓ | 256 | 31.h | odd | 30 | 1 | inner |
930.2.bo.a | ✓ | 256 | 465.bm | even | 30 | 1 | inner |
930.2.bo.b | yes | 256 | 3.b | odd | 2 | 1 | |
930.2.bo.b | yes | 256 | 5.b | even | 2 | 1 | |
930.2.bo.b | yes | 256 | 93.p | even | 30 | 1 | |
930.2.bo.b | yes | 256 | 155.v | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{128} + 3 T_{17}^{127} + 248 T_{17}^{126} + 420 T_{17}^{125} + 28137 T_{17}^{124} + \cdots + 12\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).