Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,2,Mod(149,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 975.bp (of order \(12\), 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: | \(7.78541419707\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 195) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 | −0.659148 | − | 2.45997i | −1.18458 | − | 1.26364i | −3.88494 | + | 2.24297i | 0 | −2.32769 | + | 3.74695i | 3.19491 | + | 0.856074i | 4.47674 | + | 4.47674i | −0.193547 | + | 2.99375i | 0 | ||||
149.2 | −0.614171 | − | 2.29212i | 1.72808 | − | 0.117173i | −3.14455 | + | 1.81551i | 0 | −1.32991 | − | 3.88901i | 4.37590 | + | 1.17252i | 2.73676 | + | 2.73676i | 2.97254 | − | 0.404970i | 0 | ||||
149.3 | −0.572069 | − | 2.13499i | −1.68142 | + | 0.415718i | −2.49887 | + | 1.44273i | 0 | 1.84944 | + | 3.35200i | −1.63647 | − | 0.438491i | 1.38389 | + | 1.38389i | 2.65436 | − | 1.39800i | 0 | ||||
149.4 | −0.553380 | − | 2.06524i | 1.36527 | − | 1.06585i | −2.22694 | + | 1.28572i | 0 | −2.95675 | − | 2.22980i | −3.20573 | − | 0.858972i | 0.863953 | + | 0.863953i | 0.727941 | − | 2.91034i | 0 | ||||
149.5 | −0.396555 | − | 1.47996i | 0.369915 | + | 1.69209i | −0.300984 | + | 0.173773i | 0 | 2.35754 | − | 1.21847i | −0.816332 | − | 0.218736i | −1.79028 | − | 1.79028i | −2.72633 | + | 1.25186i | 0 | ||||
149.6 | −0.297051 | − | 1.10861i | −0.959522 | + | 1.44198i | 0.591272 | − | 0.341371i | 0 | 1.88363 | + | 0.635393i | 0.219935 | + | 0.0589315i | −2.17721 | − | 2.17721i | −1.15864 | − | 2.76723i | 0 | ||||
149.7 | −0.156721 | − | 0.584891i | 1.71474 | − | 0.244293i | 1.41451 | − | 0.816670i | 0 | −0.411620 | − | 0.964648i | −4.30957 | − | 1.15475i | −1.55569 | − | 1.55569i | 2.88064 | − | 0.837797i | 0 | ||||
149.8 | −0.116204 | − | 0.433679i | 0.878333 | − | 1.49283i | 1.55748 | − | 0.899210i | 0 | −0.749473 | − | 0.207442i | 3.58664 | + | 0.961038i | −1.20590 | − | 1.20590i | −1.45706 | − | 2.62240i | 0 | ||||
149.9 | −0.113734 | − | 0.424461i | −1.71161 | − | 0.265309i | 1.56482 | − | 0.903449i | 0 | 0.0820551 | + | 0.756687i | −0.543270 | − | 0.145569i | −1.18291 | − | 1.18291i | 2.85922 | + | 0.908210i | 0 | ||||
149.10 | 0.113734 | + | 0.424461i | −1.08557 | − | 1.34964i | 1.56482 | − | 0.903449i | 0 | 0.449405 | − | 0.614282i | −0.543270 | − | 0.145569i | 1.18291 | + | 1.18291i | −0.643078 | + | 2.93026i | 0 | ||||
149.11 | 0.116204 | + | 0.433679i | −0.853659 | + | 1.50707i | 1.55748 | − | 0.899210i | 0 | −0.752784 | − | 0.195086i | 3.58664 | + | 0.961038i | 1.20590 | + | 1.20590i | −1.54253 | − | 2.57305i | 0 | ||||
149.12 | 0.156721 | + | 0.584891i | 0.645804 | + | 1.60715i | 1.41451 | − | 0.816670i | 0 | −0.838798 | + | 0.629600i | −4.30957 | − | 1.15475i | 1.55569 | + | 1.55569i | −2.16587 | + | 2.07581i | 0 | ||||
149.13 | 0.297051 | + | 1.10861i | 0.769034 | − | 1.55196i | 0.591272 | − | 0.341371i | 0 | 1.94897 | + | 0.391547i | 0.219935 | + | 0.0589315i | 2.17721 | + | 2.17721i | −1.81717 | − | 2.38702i | 0 | ||||
149.14 | 0.396555 | + | 1.47996i | 1.65035 | − | 0.525689i | −0.300984 | + | 0.173773i | 0 | 1.43245 | + | 2.23399i | −0.816332 | − | 0.218736i | 1.79028 | + | 1.79028i | 2.44730 | − | 1.73514i | 0 | ||||
149.15 | 0.553380 | + | 2.06524i | −0.240414 | + | 1.71528i | −2.22694 | + | 1.28572i | 0 | −3.67552 | + | 0.452691i | −3.20573 | − | 0.858972i | −0.863953 | − | 0.863953i | −2.88440 | − | 0.824757i | 0 | ||||
149.16 | 0.572069 | + | 2.13499i | −0.480688 | − | 1.66401i | −2.49887 | + | 1.44273i | 0 | 3.27767 | − | 1.97820i | −1.63647 | − | 0.438491i | −1.38389 | − | 1.38389i | −2.53788 | + | 1.59974i | 0 | ||||
149.17 | 0.614171 | + | 2.29212i | 0.762566 | + | 1.55515i | −3.14455 | + | 1.81551i | 0 | −3.09624 | + | 2.70302i | 4.37590 | + | 1.17252i | −2.73676 | − | 2.73676i | −1.83698 | + | 2.37181i | 0 | ||||
149.18 | 0.659148 | + | 2.45997i | −1.68663 | − | 0.394058i | −3.88494 | + | 2.24297i | 0 | −0.142367 | − | 4.40880i | 3.19491 | + | 0.856074i | −4.47674 | − | 4.47674i | 2.68944 | + | 1.32926i | 0 | ||||
449.1 | −2.60640 | + | 0.698381i | 1.68986 | + | 0.379961i | 4.57351 | − | 2.64052i | 0 | −4.66980 | + | 0.189838i | −0.724221 | + | 2.70283i | −6.26025 | + | 6.26025i | 2.71126 | + | 1.28416i | 0 | ||||
449.2 | −2.42430 | + | 0.649589i | 0.167016 | − | 1.72398i | 3.72321 | − | 2.14960i | 0 | 0.714980 | + | 4.28794i | 0.743154 | − | 2.77349i | −4.08041 | + | 4.08041i | −2.94421 | − | 0.575866i | 0 | ||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
65.s | odd | 12 | 1 | inner |
195.bh | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 975.2.bp.i | 72 | |
3.b | odd | 2 | 1 | inner | 975.2.bp.i | 72 | |
5.b | even | 2 | 1 | 975.2.bp.h | 72 | ||
5.c | odd | 4 | 1 | 195.2.bg.a | ✓ | 72 | |
5.c | odd | 4 | 1 | 975.2.bo.e | 72 | ||
13.f | odd | 12 | 1 | 975.2.bp.h | 72 | ||
15.d | odd | 2 | 1 | 975.2.bp.h | 72 | ||
15.e | even | 4 | 1 | 195.2.bg.a | ✓ | 72 | |
15.e | even | 4 | 1 | 975.2.bo.e | 72 | ||
39.k | even | 12 | 1 | 975.2.bp.h | 72 | ||
65.o | even | 12 | 1 | 975.2.bo.e | 72 | ||
65.s | odd | 12 | 1 | inner | 975.2.bp.i | 72 | |
65.t | even | 12 | 1 | 195.2.bg.a | ✓ | 72 | |
195.bc | odd | 12 | 1 | 195.2.bg.a | ✓ | 72 | |
195.bh | even | 12 | 1 | inner | 975.2.bp.i | 72 | |
195.bn | odd | 12 | 1 | 975.2.bo.e | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bg.a | ✓ | 72 | 5.c | odd | 4 | 1 | |
195.2.bg.a | ✓ | 72 | 15.e | even | 4 | 1 | |
195.2.bg.a | ✓ | 72 | 65.t | even | 12 | 1 | |
195.2.bg.a | ✓ | 72 | 195.bc | odd | 12 | 1 | |
975.2.bo.e | 72 | 5.c | odd | 4 | 1 | ||
975.2.bo.e | 72 | 15.e | even | 4 | 1 | ||
975.2.bo.e | 72 | 65.o | even | 12 | 1 | ||
975.2.bo.e | 72 | 195.bn | odd | 12 | 1 | ||
975.2.bp.h | 72 | 5.b | even | 2 | 1 | ||
975.2.bp.h | 72 | 13.f | odd | 12 | 1 | ||
975.2.bp.h | 72 | 15.d | odd | 2 | 1 | ||
975.2.bp.h | 72 | 39.k | even | 12 | 1 | ||
975.2.bp.i | 72 | 1.a | even | 1 | 1 | trivial | |
975.2.bp.i | 72 | 3.b | odd | 2 | 1 | inner | |
975.2.bp.i | 72 | 65.s | odd | 12 | 1 | inner | |
975.2.bp.i | 72 | 195.bh | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):
\( T_{2}^{72} - 126 T_{2}^{68} + 10081 T_{2}^{64} + 3396 T_{2}^{62} - 487974 T_{2}^{60} - 247176 T_{2}^{58} + \cdots + 1679616 \) |
\( T_{7}^{36} - 21 T_{7}^{34} - 50 T_{7}^{33} - 235 T_{7}^{32} + 1360 T_{7}^{31} + 9272 T_{7}^{30} + \cdots + 3833714889 \) |