Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1000,2,Mod(149,1000)]
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, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1000.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1000 = 2^{3} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1000.o (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: | \(112\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 | −1.37152 | + | 0.344862i | 2.07611 | + | 1.50838i | 1.76214 | − | 0.945971i | 0 | −3.36760 | − | 1.35280i | 2.17256i | −2.09058 | + | 1.90511i | 1.10796 | + | 3.40994i | 0 | ||||||
149.2 | −1.36533 | + | 0.368612i | 0.831817 | + | 0.604350i | 1.72825 | − | 1.00655i | 0 | −1.35847 | − | 0.518519i | 1.74290i | −1.98860 | + | 2.01133i | −0.600371 | − | 1.84775i | 0 | ||||||
149.3 | −1.36479 | − | 0.370599i | 0.457893 | + | 0.332679i | 1.72531 | + | 1.01158i | 0 | −0.501638 | − | 0.623732i | − | 1.85821i | −1.97980 | − | 2.02000i | −0.828060 | − | 2.54851i | 0 | |||||
149.4 | −1.32197 | − | 0.502384i | −0.457893 | − | 0.332679i | 1.49522 | + | 1.32827i | 0 | 0.438189 | + | 0.669830i | − | 1.85821i | −1.30934 | − | 2.50712i | −0.828060 | − | 2.54851i | 0 | |||||
149.5 | −1.26861 | + | 0.625006i | −2.03813 | − | 1.48079i | 1.21874 | − | 1.58578i | 0 | 3.51109 | + | 0.604697i | − | 3.58786i | −0.554981 | + | 2.77344i | 1.03418 | + | 3.18289i | 0 | |||||
149.6 | −0.938193 | + | 1.05820i | −1.09282 | − | 0.793983i | −0.239588 | − | 1.98560i | 0 | 1.86547 | − | 0.411520i | 0.296885i | 2.32595 | + | 1.60934i | −0.363197 | − | 1.11780i | 0 | ||||||
149.7 | −0.906879 | − | 1.08516i | −2.07611 | − | 1.50838i | −0.355140 | + | 1.96822i | 0 | 0.245946 | + | 3.62082i | 2.17256i | 2.45790 | − | 1.39955i | 1.10796 | + | 3.40994i | 0 | ||||||
149.8 | −0.887910 | − | 1.10073i | −0.831817 | − | 0.604350i | −0.423232 | + | 1.95471i | 0 | 0.0733494 | + | 1.45222i | 1.74290i | 2.52740 | − | 1.26974i | −0.600371 | − | 1.84775i | 0 | ||||||
149.9 | −0.806832 | + | 1.16147i | 1.06117 | + | 0.770982i | −0.698045 | − | 1.87423i | 0 | −1.75166 | + | 0.610464i | − | 2.31589i | 2.74007 | + | 0.701425i | −0.395392 | − | 1.21689i | 0 | |||||
149.10 | −0.658957 | − | 1.25131i | 2.03813 | + | 1.48079i | −1.13155 | + | 1.64912i | 0 | 0.509884 | − | 3.52610i | − | 3.58786i | 2.80920 | + | 0.329223i | 1.03418 | + | 3.18289i | 0 | |||||
149.11 | −0.314378 | + | 1.37883i | −0.0988780 | − | 0.0718391i | −1.80233 | − | 0.866946i | 0 | 0.130139 | − | 0.113751i | 4.12326i | 1.76198 | − | 2.21256i | −0.922435 | − | 2.83896i | 0 | ||||||
149.12 | −0.137018 | − | 1.40756i | 1.09282 | + | 0.793983i | −1.96245 | + | 0.385721i | 0 | 0.967842 | − | 1.64700i | 0.296885i | 0.811817 | + | 2.70942i | −0.363197 | − | 1.11780i | 0 | ||||||
149.13 | −0.135750 | + | 1.40768i | −2.54275 | − | 1.84742i | −1.96314 | − | 0.382186i | 0 | 2.94576 | − | 3.32861i | 1.17589i | 0.804493 | − | 2.71160i | 2.12559 | + | 6.54190i | 0 | ||||||
149.14 | 0.0299570 | − | 1.41390i | −1.06117 | − | 0.770982i | −1.99821 | − | 0.0847122i | 0 | −1.12188 | + | 1.47728i | − | 2.31589i | −0.179635 | + | 2.82272i | −0.395392 | − | 1.21689i | 0 | |||||
149.15 | 0.238643 | + | 1.39393i | 1.39188 | + | 1.01126i | −1.88610 | + | 0.665306i | 0 | −1.07746 | + | 2.18152i | − | 4.42380i | −1.37750 | − | 2.47032i | −0.0123697 | − | 0.0380700i | 0 | |||||
149.16 | 0.338435 | + | 1.37312i | 2.58107 | + | 1.87526i | −1.77092 | + | 0.929423i | 0 | −1.70143 | + | 4.17877i | 2.97769i | −1.87555 | − | 2.11714i | 2.21828 | + | 6.82716i | 0 | ||||||
149.17 | 0.532208 | + | 1.31025i | −0.235999 | − | 0.171463i | −1.43351 | + | 1.39465i | 0 | 0.0990593 | − | 0.400472i | − | 0.234809i | −2.59027 | − | 1.13601i | −0.900755 | − | 2.77224i | 0 | |||||
149.18 | 0.556118 | − | 1.30028i | 0.0988780 | + | 0.0718391i | −1.38147 | − | 1.44622i | 0 | 0.148399 | − | 0.0886183i | 4.12326i | −2.64875 | + | 0.992028i | −0.922435 | − | 2.83896i | 0 | ||||||
149.19 | 0.717591 | − | 1.21863i | 2.54275 | + | 1.84742i | −0.970125 | − | 1.74896i | 0 | 4.07598 | − | 1.77299i | 1.17589i | −2.82749 | − | 0.0728129i | 2.12559 | + | 6.54190i | 0 | ||||||
149.20 | 0.908199 | + | 1.08405i | −2.01785 | − | 1.46605i | −0.350349 | + | 1.96907i | 0 | −0.243326 | − | 3.51892i | − | 0.110917i | −2.45277 | + | 1.40851i | 0.995348 | + | 3.06337i | 0 | |||||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
25.e | even | 10 | 1 | inner |
200.o | 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.o.a | 112 | |
5.b | even | 2 | 1 | 200.2.o.a | ✓ | 112 | |
5.c | odd | 4 | 2 | 1000.2.t.b | 224 | ||
8.b | even | 2 | 1 | inner | 1000.2.o.a | 112 | |
20.d | odd | 2 | 1 | 800.2.be.a | 112 | ||
25.d | even | 5 | 1 | 200.2.o.a | ✓ | 112 | |
25.e | even | 10 | 1 | inner | 1000.2.o.a | 112 | |
25.f | odd | 20 | 2 | 1000.2.t.b | 224 | ||
40.e | odd | 2 | 1 | 800.2.be.a | 112 | ||
40.f | even | 2 | 1 | 200.2.o.a | ✓ | 112 | |
40.i | odd | 4 | 2 | 1000.2.t.b | 224 | ||
100.j | odd | 10 | 1 | 800.2.be.a | 112 | ||
200.n | odd | 10 | 1 | 800.2.be.a | 112 | ||
200.o | even | 10 | 1 | inner | 1000.2.o.a | 112 | |
200.t | even | 10 | 1 | 200.2.o.a | ✓ | 112 | |
200.x | odd | 20 | 2 | 1000.2.t.b | 224 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.2.o.a | ✓ | 112 | 5.b | even | 2 | 1 | |
200.2.o.a | ✓ | 112 | 25.d | even | 5 | 1 | |
200.2.o.a | ✓ | 112 | 40.f | even | 2 | 1 | |
200.2.o.a | ✓ | 112 | 200.t | even | 10 | 1 | |
800.2.be.a | 112 | 20.d | odd | 2 | 1 | ||
800.2.be.a | 112 | 40.e | odd | 2 | 1 | ||
800.2.be.a | 112 | 100.j | odd | 10 | 1 | ||
800.2.be.a | 112 | 200.n | odd | 10 | 1 | ||
1000.2.o.a | 112 | 1.a | even | 1 | 1 | trivial | |
1000.2.o.a | 112 | 8.b | even | 2 | 1 | inner | |
1000.2.o.a | 112 | 25.e | even | 10 | 1 | inner | |
1000.2.o.a | 112 | 200.o | even | 10 | 1 | inner | |
1000.2.t.b | 224 | 5.c | odd | 4 | 2 | ||
1000.2.t.b | 224 | 25.f | odd | 20 | 2 | ||
1000.2.t.b | 224 | 40.i | odd | 4 | 2 | ||
1000.2.t.b | 224 | 200.x | odd | 20 | 2 |
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 \(S_{2}^{\mathrm{new}}(1000, [\chi])\).