Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1000,2,Mod(101,1000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1000.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1000 = 2^{3} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1000.t (of order \(10\), degree \(4\), not 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: | \(7.98504020213\) |
Analytic rank: | \(0\) |
Dimension: | \(224\) |
Relative dimension: | \(56\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 200) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 | −1.41390 | − | 0.0299570i | 0.770982 | − | 1.06117i | 1.99821 | + | 0.0847122i | 0 | −1.12188 | + | 1.47728i | −2.31589 | −2.82272 | − | 0.179635i | 0.395392 | + | 1.21689i | 0 | ||||||
101.2 | −1.40768 | − | 0.135750i | −1.84742 | + | 2.54275i | 1.96314 | + | 0.382186i | 0 | 2.94576 | − | 3.32861i | −1.17589 | −2.71160 | − | 0.804493i | −2.12559 | − | 6.54190i | 0 | ||||||
101.3 | −1.40756 | + | 0.137018i | −0.793983 | + | 1.09282i | 1.96245 | − | 0.385721i | 0 | 0.967842 | − | 1.64700i | 0.296885 | −2.70942 | + | 0.811817i | 0.363197 | + | 1.11780i | 0 | ||||||
101.4 | −1.39393 | + | 0.238643i | 1.01126 | − | 1.39188i | 1.88610 | − | 0.665306i | 0 | −1.07746 | + | 2.18152i | 4.42380 | −2.47032 | + | 1.37750i | 0.0123697 | + | 0.0380700i | 0 | ||||||
101.5 | −1.37883 | − | 0.314378i | −0.0718391 | + | 0.0988780i | 1.80233 | + | 0.866946i | 0 | 0.130139 | − | 0.113751i | −4.12326 | −2.21256 | − | 1.76198i | 0.922435 | + | 2.83896i | 0 | ||||||
101.6 | −1.37312 | + | 0.338435i | 1.87526 | − | 2.58107i | 1.77092 | − | 0.929423i | 0 | −1.70143 | + | 4.17877i | −2.97769 | −2.11714 | + | 1.87555i | −2.21828 | − | 6.82716i | 0 | ||||||
101.7 | −1.31025 | + | 0.532208i | −0.171463 | + | 0.235999i | 1.43351 | − | 1.39465i | 0 | 0.0990593 | − | 0.400472i | 0.234809 | −1.13601 | + | 2.59027i | 0.900755 | + | 2.77224i | 0 | ||||||
101.8 | −1.30028 | − | 0.556118i | −0.0718391 | + | 0.0988780i | 1.38147 | + | 1.44622i | 0 | 0.148399 | − | 0.0886183i | 4.12326 | −0.992028 | − | 2.64875i | 0.922435 | + | 2.83896i | 0 | ||||||
101.9 | −1.25131 | + | 0.658957i | −1.48079 | + | 2.03813i | 1.13155 | − | 1.64912i | 0 | 0.509884 | − | 3.52610i | −3.58786 | −0.329223 | + | 2.80920i | −1.03418 | − | 3.18289i | 0 | ||||||
101.10 | −1.21863 | − | 0.717591i | −1.84742 | + | 2.54275i | 0.970125 | + | 1.74896i | 0 | 4.07598 | − | 1.77299i | 1.17589 | 0.0728129 | − | 2.82749i | −2.12559 | − | 6.54190i | 0 | ||||||
101.11 | −1.16147 | − | 0.806832i | 0.770982 | − | 1.06117i | 0.698045 | + | 1.87423i | 0 | −1.75166 | + | 0.610464i | 2.31589 | 0.701425 | − | 2.74007i | 0.395392 | + | 1.21689i | 0 | ||||||
101.12 | −1.10073 | + | 0.887910i | 0.604350 | − | 0.831817i | 0.423232 | − | 1.95471i | 0 | 0.0733494 | + | 1.45222i | 1.74290 | 1.26974 | + | 2.52740i | 0.600371 | + | 1.84775i | 0 | ||||||
101.13 | −1.08516 | + | 0.906879i | 1.50838 | − | 2.07611i | 0.355140 | − | 1.96822i | 0 | 0.245946 | + | 3.62082i | 2.17256 | 1.39955 | + | 2.45790i | −1.10796 | − | 3.40994i | 0 | ||||||
101.14 | −1.08405 | + | 0.908199i | −1.46605 | + | 2.01785i | 0.350349 | − | 1.96907i | 0 | −0.243326 | − | 3.51892i | 0.110917 | 1.40851 | + | 2.45277i | −0.995348 | − | 3.06337i | 0 | ||||||
101.15 | −1.05820 | − | 0.938193i | −0.793983 | + | 1.09282i | 0.239588 | + | 1.98560i | 0 | 1.86547 | − | 0.411520i | −0.296885 | 1.60934 | − | 2.32595i | 0.363197 | + | 1.11780i | 0 | ||||||
101.16 | −0.987445 | − | 1.01240i | 1.01126 | − | 1.39188i | −0.0499066 | + | 1.99938i | 0 | −2.40770 | + | 0.350605i | −4.42380 | 2.07345 | − | 1.92375i | 0.0123697 | + | 0.0380700i | 0 | ||||||
101.17 | −0.911952 | − | 1.08090i | 1.87526 | − | 2.58107i | −0.336688 | + | 1.97146i | 0 | −4.50002 | + | 0.326846i | 2.97769 | 2.43799 | − | 1.43395i | −2.21828 | − | 6.82716i | 0 | ||||||
101.18 | −0.747190 | − | 1.20071i | −0.171463 | + | 0.235999i | −0.883414 | + | 1.79432i | 0 | 0.411483 | + | 0.0295418i | −0.234809 | 2.81454 | − | 0.279973i | 0.900755 | + | 2.77224i | 0 | ||||||
101.19 | −0.747170 | + | 1.20072i | 0.893213 | − | 1.22940i | −0.883474 | − | 1.79429i | 0 | 0.808790 | + | 1.99107i | −3.99458 | 2.81455 | + | 0.279830i | 0.213451 | + | 0.656936i | 0 | ||||||
101.20 | −0.625006 | − | 1.26861i | −1.48079 | + | 2.03813i | −1.21874 | + | 1.58578i | 0 | 3.51109 | + | 0.604697i | 3.58786 | 2.77344 | + | 0.554981i | −1.03418 | − | 3.18289i | 0 | ||||||
See next 80 embeddings (of 224 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
40.f | even | 2 | 1 | inner |
200.o | even | 10 | 1 | inner |
200.t | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1000.2.t.b | 224 | |
5.b | even | 2 | 1 | inner | 1000.2.t.b | 224 | |
5.c | odd | 4 | 1 | 200.2.o.a | ✓ | 112 | |
5.c | odd | 4 | 1 | 1000.2.o.a | 112 | ||
8.b | even | 2 | 1 | inner | 1000.2.t.b | 224 | |
20.e | even | 4 | 1 | 800.2.be.a | 112 | ||
25.d | even | 5 | 1 | inner | 1000.2.t.b | 224 | |
25.e | even | 10 | 1 | inner | 1000.2.t.b | 224 | |
25.f | odd | 20 | 1 | 200.2.o.a | ✓ | 112 | |
25.f | odd | 20 | 1 | 1000.2.o.a | 112 | ||
40.f | even | 2 | 1 | inner | 1000.2.t.b | 224 | |
40.i | odd | 4 | 1 | 200.2.o.a | ✓ | 112 | |
40.i | odd | 4 | 1 | 1000.2.o.a | 112 | ||
40.k | even | 4 | 1 | 800.2.be.a | 112 | ||
100.l | even | 20 | 1 | 800.2.be.a | 112 | ||
200.o | even | 10 | 1 | inner | 1000.2.t.b | 224 | |
200.t | even | 10 | 1 | inner | 1000.2.t.b | 224 | |
200.v | even | 20 | 1 | 800.2.be.a | 112 | ||
200.x | odd | 20 | 1 | 200.2.o.a | ✓ | 112 | |
200.x | odd | 20 | 1 | 1000.2.o.a | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.2.o.a | ✓ | 112 | 5.c | odd | 4 | 1 | |
200.2.o.a | ✓ | 112 | 25.f | odd | 20 | 1 | |
200.2.o.a | ✓ | 112 | 40.i | odd | 4 | 1 | |
200.2.o.a | ✓ | 112 | 200.x | odd | 20 | 1 | |
800.2.be.a | 112 | 20.e | even | 4 | 1 | ||
800.2.be.a | 112 | 40.k | even | 4 | 1 | ||
800.2.be.a | 112 | 100.l | even | 20 | 1 | ||
800.2.be.a | 112 | 200.v | even | 20 | 1 | ||
1000.2.o.a | 112 | 5.c | odd | 4 | 1 | ||
1000.2.o.a | 112 | 25.f | odd | 20 | 1 | ||
1000.2.o.a | 112 | 40.i | odd | 4 | 1 | ||
1000.2.o.a | 112 | 200.x | odd | 20 | 1 | ||
1000.2.t.b | 224 | 1.a | even | 1 | 1 | trivial | |
1000.2.t.b | 224 | 5.b | even | 2 | 1 | inner | |
1000.2.t.b | 224 | 8.b | even | 2 | 1 | inner | |
1000.2.t.b | 224 | 25.d | even | 5 | 1 | inner | |
1000.2.t.b | 224 | 25.e | even | 10 | 1 | inner | |
1000.2.t.b | 224 | 40.f | even | 2 | 1 | inner | |
1000.2.t.b | 224 | 200.o | even | 10 | 1 | inner | |
1000.2.t.b | 224 | 200.t | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{112} - 57 T_{3}^{110} + 1797 T_{3}^{108} - 41631 T_{3}^{106} + 794837 T_{3}^{104} - 13058078 T_{3}^{102} + 188634219 T_{3}^{100} - 2435797056 T_{3}^{98} + 28488245305 T_{3}^{96} + \cdots + 70338682617856 \)
acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\).