Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(113,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 9, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 630.ca (of order \(12\), 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: | \(5.03057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
113.1 | −0.258819 | + | 0.965926i | −1.59099 | + | 0.684658i | −0.866025 | − | 0.500000i | 2.19432 | − | 0.430082i | −0.249551 | − | 1.71398i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 2.06249 | − | 2.17857i | −0.152503 | + | 2.23086i |
113.2 | −0.258819 | + | 0.965926i | −1.58094 | − | 0.707560i | −0.866025 | − | 0.500000i | −1.12367 | + | 1.93322i | 1.09263 | − | 1.34394i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 1.99872 | + | 2.23721i | −1.57652 | − | 1.58574i |
113.3 | −0.258819 | + | 0.965926i | −1.38247 | + | 1.04345i | −0.866025 | − | 0.500000i | −2.13456 | + | 0.666072i | −0.650081 | − | 1.60543i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 0.822445 | − | 2.88506i | −0.0909116 | − | 2.23422i |
113.4 | −0.258819 | + | 0.965926i | −0.970129 | − | 1.43487i | −0.866025 | − | 0.500000i | 2.21913 | + | 0.274732i | 1.63707 | − | 0.565701i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −1.11770 | + | 2.78402i | −0.839723 | + | 2.07241i |
113.5 | −0.258819 | + | 0.965926i | −0.680914 | − | 1.59259i | −0.866025 | − | 0.500000i | −1.99168 | − | 1.01646i | 1.71456 | − | 0.245518i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −2.07271 | + | 2.16884i | 1.49731 | − | 1.66074i |
113.6 | −0.258819 | + | 0.965926i | 0.0184135 | + | 1.73195i | −0.866025 | − | 0.500000i | 1.05084 | + | 1.97376i | −1.67770 | − | 0.430476i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −2.99932 | + | 0.0637827i | −2.17848 | + | 0.504186i |
113.7 | −0.258819 | + | 0.965926i | 0.995387 | + | 1.41746i | −0.866025 | − | 0.500000i | 0.0237855 | − | 2.23594i | −1.62679 | + | 0.594603i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −1.01841 | + | 2.82185i | 2.15360 | + | 0.601679i |
113.8 | −0.258819 | + | 0.965926i | 1.50085 | − | 0.864557i | −0.866025 | − | 0.500000i | −1.17878 | − | 1.90013i | 0.446650 | + | 1.67347i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 1.50508 | − | 2.59513i | 2.14047 | − | 0.646821i |
113.9 | −0.258819 | + | 0.965926i | 1.61766 | − | 0.619021i | −0.866025 | − | 0.500000i | 0.0745996 | + | 2.23482i | 0.179248 | + | 1.72275i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 2.23363 | − | 2.00273i | −2.17798 | − | 0.506357i |
113.10 | 0.258819 | − | 0.965926i | −1.70384 | − | 0.311322i | −0.866025 | − | 0.500000i | 1.82858 | + | 1.28698i | −0.741701 | + | 1.56521i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 2.80616 | + | 1.06089i | 1.71640 | − | 1.43317i |
113.11 | 0.258819 | − | 0.965926i | −1.32530 | + | 1.11516i | −0.866025 | − | 0.500000i | 0.902382 | − | 2.04590i | 0.734145 | + | 1.56877i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 0.512852 | − | 2.95584i | −1.74263 | − | 1.40115i |
113.12 | 0.258819 | − | 0.965926i | −0.955974 | + | 1.44434i | −0.866025 | − | 0.500000i | −2.22543 | − | 0.217885i | 1.14770 | + | 1.29722i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −1.17223 | − | 2.76150i | −0.786443 | + | 2.09320i |
113.13 | 0.258819 | − | 0.965926i | −0.718418 | − | 1.57603i | −0.866025 | − | 0.500000i | 0.920277 | + | 2.03791i | −1.70827 | + | 0.286032i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −1.96775 | + | 2.26450i | 2.20666 | − | 0.361468i |
113.14 | 0.258819 | − | 0.965926i | −0.221516 | + | 1.71783i | −0.866025 | − | 0.500000i | −0.743840 | + | 2.10872i | 1.60196 | + | 0.658574i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −2.90186 | − | 0.761052i | 1.84435 | + | 1.26427i |
113.15 | 0.258819 | − | 0.965926i | −0.137151 | − | 1.72661i | −0.866025 | − | 0.500000i | −2.16332 | − | 0.565716i | −1.70328 | − | 0.314402i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −2.96238 | + | 0.473615i | −1.10635 | + | 1.94319i |
113.16 | 0.258819 | − | 0.965926i | 1.15104 | + | 1.29426i | −0.866025 | − | 0.500000i | 2.23016 | + | 0.162443i | 1.54807 | − | 0.776846i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −0.350194 | + | 2.97949i | 0.734115 | − | 2.11213i |
113.17 | 0.258819 | − | 0.965926i | 1.52110 | − | 0.828409i | −0.866025 | − | 0.500000i | 0.418681 | − | 2.19652i | −0.406492 | − | 1.68368i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 1.62748 | − | 2.52018i | −2.01331 | − | 0.972916i |
113.18 | 0.258819 | − | 0.965926i | 1.73114 | − | 0.0560710i | −0.866025 | − | 0.500000i | −2.03351 | + | 0.929968i | 0.393892 | − | 1.68667i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 2.99371 | − | 0.194134i | 0.371969 | + | 2.20491i |
407.1 | −0.258819 | − | 0.965926i | −1.59099 | − | 0.684658i | −0.866025 | + | 0.500000i | 2.19432 | + | 0.430082i | −0.249551 | + | 1.71398i | −0.965926 | + | 0.258819i | 0.707107 | + | 0.707107i | 2.06249 | + | 2.17857i | −0.152503 | − | 2.23086i |
407.2 | −0.258819 | − | 0.965926i | −1.58094 | + | 0.707560i | −0.866025 | + | 0.500000i | −1.12367 | − | 1.93322i | 1.09263 | + | 1.34394i | −0.965926 | + | 0.258819i | 0.707107 | + | 0.707107i | 1.99872 | − | 2.23721i | −1.57652 | + | 1.58574i |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.2.ca.b | yes | 72 |
5.c | odd | 4 | 1 | 630.2.ca.a | ✓ | 72 | |
9.d | odd | 6 | 1 | 630.2.ca.a | ✓ | 72 | |
45.l | even | 12 | 1 | inner | 630.2.ca.b | yes | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.ca.a | ✓ | 72 | 5.c | odd | 4 | 1 | |
630.2.ca.a | ✓ | 72 | 9.d | odd | 6 | 1 | |
630.2.ca.b | yes | 72 | 1.a | even | 1 | 1 | trivial |
630.2.ca.b | yes | 72 | 45.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{72} + 144 T_{13}^{70} - 24 T_{13}^{69} + 7071 T_{13}^{68} - 4800 T_{13}^{67} + 23184 T_{13}^{66} - 246276 T_{13}^{65} - 8607399 T_{13}^{64} + 2669072 T_{13}^{63} - 143310792 T_{13}^{62} + \cdots + 19\!\cdots\!76 \)
acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\).