Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(19,930)]
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([0, 15, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bn (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: | \(144\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −1.89205 | − | 1.19170i | −0.500000 | − | 0.866025i | −1.93366 | − | 1.74107i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | 2.07622 | − | 0.830237i |
19.2 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −1.39553 | + | 1.74714i | −0.500000 | − | 0.866025i | 1.99915 | + | 1.80004i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | −0.593198 | − | 2.15595i |
19.3 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −1.16157 | − | 1.91070i | −0.500000 | − | 0.866025i | 2.15311 | + | 1.93867i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | 2.22854 | + | 0.183353i |
19.4 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −0.978641 | + | 2.01054i | −0.500000 | − | 0.866025i | −2.38970 | − | 2.15169i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | −1.05133 | − | 1.97350i |
19.5 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −0.293219 | − | 2.21676i | −0.500000 | − | 0.866025i | 1.44492 | + | 1.30102i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | 1.96575 | + | 1.06576i |
19.6 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | 0.991216 | + | 2.00437i | −0.500000 | − | 0.866025i | 0.807100 | + | 0.726716i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | −2.20419 | − | 0.376228i |
19.7 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | 1.47304 | − | 1.68231i | −0.500000 | − | 0.866025i | −2.98158 | − | 2.68462i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | 0.495182 | + | 2.18055i |
19.8 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | 2.02048 | − | 0.957953i | −0.500000 | − | 0.866025i | 1.95080 | + | 1.75650i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | −0.412606 | + | 2.19767i |
19.9 | −0.587785 | + | 0.809017i | −0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | 2.14981 | + | 0.615060i | −0.500000 | − | 0.866025i | −2.84716 | − | 2.56360i | 0.951057 | + | 0.309017i | −0.669131 | − | 0.743145i | −1.76122 | + | 1.37771i |
19.10 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −2.23144 | + | 0.143766i | −0.500000 | − | 0.866025i | −0.807100 | − | 0.726716i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | −1.19530 | + | 1.88978i |
19.11 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −1.60756 | − | 1.55426i | −0.500000 | − | 0.866025i | 2.84716 | + | 2.56360i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | −2.20233 | + | 0.386974i |
19.12 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −1.25186 | + | 1.85280i | −0.500000 | − | 0.866025i | 2.38970 | + | 2.15169i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | 0.763121 | + | 2.10182i |
19.13 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −0.815306 | + | 2.08213i | −0.500000 | − | 0.866025i | −1.99915 | − | 1.80004i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | 1.20526 | + | 1.88344i |
19.14 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −0.180627 | − | 2.22876i | −0.500000 | − | 0.866025i | −1.95080 | − | 1.75650i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | −1.90927 | − | 1.16390i |
19.15 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | 0.720399 | − | 2.11684i | −0.500000 | − | 0.866025i | 2.98158 | + | 2.68462i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | −1.28912 | − | 1.82706i |
19.16 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | 1.97807 | + | 1.04271i | −0.500000 | − | 0.866025i | 1.93366 | + | 1.74107i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | 2.00625 | − | 0.987398i |
19.17 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | 2.06638 | − | 0.854445i | −0.500000 | − | 0.866025i | −1.44492 | − | 1.30102i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | 0.523327 | − | 2.17397i |
19.18 | 0.587785 | − | 0.809017i | 0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | 2.23550 | + | 0.0505968i | −0.500000 | − | 0.866025i | −2.15311 | − | 1.93867i | −0.951057 | − | 0.309017i | −0.669131 | − | 0.743145i | 1.35492 | − | 1.77881i |
49.1 | −0.587785 | − | 0.809017i | −0.406737 | − | 0.913545i | −0.309017 | + | 0.951057i | −1.89205 | + | 1.19170i | −0.500000 | + | 0.866025i | −1.93366 | + | 1.74107i | 0.951057 | − | 0.309017i | −0.669131 | + | 0.743145i | 2.07622 | + | 0.830237i |
49.2 | −0.587785 | − | 0.809017i | −0.406737 | − | 0.913545i | −0.309017 | + | 0.951057i | −1.39553 | − | 1.74714i | −0.500000 | + | 0.866025i | 1.99915 | − | 1.80004i | 0.951057 | − | 0.309017i | −0.669131 | + | 0.743145i | −0.593198 | + | 2.15595i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
31.g | even | 15 | 1 | inner |
155.u | 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.bn.b | ✓ | 144 |
5.b | even | 2 | 1 | inner | 930.2.bn.b | ✓ | 144 |
31.g | even | 15 | 1 | inner | 930.2.bn.b | ✓ | 144 |
155.u | even | 30 | 1 | inner | 930.2.bn.b | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bn.b | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
930.2.bn.b | ✓ | 144 | 5.b | even | 2 | 1 | inner |
930.2.bn.b | ✓ | 144 | 31.g | even | 15 | 1 | inner |
930.2.bn.b | ✓ | 144 | 155.u | even | 30 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{144} + 102 T_{7}^{142} + 4183 T_{7}^{140} + 53720 T_{7}^{138} - 1977413 T_{7}^{136} + \cdots + 11\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).