Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(9,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([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("605.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 605.j (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: | \(4.83094932229\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −2.48013 | − | 0.805844i | −1.25775 | − | 1.73114i | 3.88364 | + | 2.82163i | 0.0494116 | − | 2.23552i | 1.72435 | + | 5.30700i | −0.581248 | + | 0.800020i | −4.29253 | − | 5.90817i | −0.487865 | + | 1.50149i | −1.92403 | + | 5.50457i |
9.2 | −2.48013 | − | 0.805844i | 1.25775 | + | 1.73114i | 3.88364 | + | 2.82163i | −1.35398 | − | 1.77953i | −1.72435 | − | 5.30700i | −0.581248 | + | 0.800020i | −4.29253 | − | 5.90817i | −0.487865 | + | 1.50149i | 1.92403 | + | 5.50457i |
9.3 | −1.92380 | − | 0.625081i | −1.71325 | − | 2.35809i | 1.69226 | + | 1.22950i | 2.16467 | + | 0.560538i | 1.82196 | + | 5.60741i | −1.88754 | + | 2.59798i | −0.109082 | − | 0.150138i | −1.69829 | + | 5.22681i | −3.81402 | − | 2.43146i |
9.4 | −1.92380 | − | 0.625081i | 1.71325 | + | 2.35809i | 1.69226 | + | 1.22950i | −1.42178 | + | 1.72585i | −1.82196 | − | 5.60741i | −1.88754 | + | 2.59798i | −0.109082 | − | 0.150138i | −1.69829 | + | 5.22681i | 3.81402 | − | 2.43146i |
9.5 | −0.312280 | − | 0.101466i | −0.565450 | − | 0.778275i | −1.53081 | − | 1.11220i | −2.23364 | + | 0.104146i | 0.0976102 | + | 0.300413i | 1.92358 | − | 2.64758i | 0.751190 | + | 1.03393i | 0.641073 | − | 1.97302i | 0.708089 | + | 0.194116i |
9.6 | −0.312280 | − | 0.101466i | 0.565450 | + | 0.778275i | −1.53081 | − | 1.11220i | 1.86827 | − | 1.22865i | −0.0976102 | − | 0.300413i | 1.92358 | − | 2.64758i | 0.751190 | + | 1.03393i | 0.641073 | − | 1.97302i | −0.708089 | + | 0.194116i |
9.7 | 0.312280 | + | 0.101466i | −0.565450 | − | 0.778275i | −1.53081 | − | 1.11220i | −2.23364 | + | 0.104146i | −0.0976102 | − | 0.300413i | −1.92358 | + | 2.64758i | −0.751190 | − | 1.03393i | 0.641073 | − | 1.97302i | −0.708089 | − | 0.194116i |
9.8 | 0.312280 | + | 0.101466i | 0.565450 | + | 0.778275i | −1.53081 | − | 1.11220i | 1.86827 | − | 1.22865i | 0.0976102 | + | 0.300413i | −1.92358 | + | 2.64758i | −0.751190 | − | 1.03393i | 0.641073 | − | 1.97302i | 0.708089 | − | 0.194116i |
9.9 | 1.92380 | + | 0.625081i | −1.71325 | − | 2.35809i | 1.69226 | + | 1.22950i | 2.16467 | + | 0.560538i | −1.82196 | − | 5.60741i | 1.88754 | − | 2.59798i | 0.109082 | + | 0.150138i | −1.69829 | + | 5.22681i | 3.81402 | + | 2.43146i |
9.10 | 1.92380 | + | 0.625081i | 1.71325 | + | 2.35809i | 1.69226 | + | 1.22950i | −1.42178 | + | 1.72585i | 1.82196 | + | 5.60741i | 1.88754 | − | 2.59798i | 0.109082 | + | 0.150138i | −1.69829 | + | 5.22681i | −3.81402 | + | 2.43146i |
9.11 | 2.48013 | + | 0.805844i | −1.25775 | − | 1.73114i | 3.88364 | + | 2.82163i | 0.0494116 | − | 2.23552i | −1.72435 | − | 5.30700i | 0.581248 | − | 0.800020i | 4.29253 | + | 5.90817i | −0.487865 | + | 1.50149i | 1.92403 | − | 5.50457i |
9.12 | 2.48013 | + | 0.805844i | 1.25775 | + | 1.73114i | 3.88364 | + | 2.82163i | −1.35398 | − | 1.77953i | 1.72435 | + | 5.30700i | 0.581248 | − | 0.800020i | 4.29253 | + | 5.90817i | −0.487865 | + | 1.50149i | −1.92403 | − | 5.50457i |
124.1 | −1.53281 | − | 2.10973i | −2.03508 | + | 0.661236i | −1.48342 | + | 4.56549i | 1.27403 | + | 1.83762i | 4.51440 | + | 3.27991i | 0.940480 | + | 0.305580i | 6.94547 | − | 2.25672i | 1.27725 | − | 0.927974i | 1.92403 | − | 5.50457i |
124.2 | −1.53281 | − | 2.10973i | 2.03508 | − | 0.661236i | −1.48342 | + | 4.56549i | 2.14138 | + | 0.643821i | −4.51440 | − | 3.27991i | 0.940480 | + | 0.305580i | 6.94547 | − | 2.25672i | 1.27725 | − | 0.927974i | −1.92403 | − | 5.50457i |
124.3 | −1.18898 | − | 1.63648i | −2.77210 | + | 0.900709i | −0.646385 | + | 1.98937i | −2.08073 | + | 0.818876i | 4.76995 | + | 3.46557i | 3.05411 | + | 0.992339i | 0.176498 | − | 0.0573478i | 4.44619 | − | 3.23035i | 3.81402 | + | 2.43146i |
124.4 | −1.18898 | − | 1.63648i | 2.77210 | − | 0.900709i | −0.646385 | + | 1.98937i | 0.135816 | − | 2.23194i | −4.76995 | − | 3.46557i | 3.05411 | + | 0.992339i | 0.176498 | − | 0.0573478i | 4.44619 | − | 3.23035i | −3.81402 | + | 2.43146i |
124.5 | −0.193000 | − | 0.265641i | −0.914917 | + | 0.297274i | 0.584718 | − | 1.79958i | 1.74584 | − | 1.39716i | 0.255547 | + | 0.185666i | −3.11241 | − | 1.01128i | −1.21545 | + | 0.394924i | −1.67835 | + | 1.21939i | −0.708089 | − | 0.194116i |
124.6 | −0.193000 | − | 0.265641i | 0.914917 | − | 0.297274i | 0.584718 | − | 1.79958i | −0.789281 | + | 2.09214i | −0.255547 | − | 0.185666i | −3.11241 | − | 1.01128i | −1.21545 | + | 0.394924i | −1.67835 | + | 1.21939i | 0.708089 | − | 0.194116i |
124.7 | 0.193000 | + | 0.265641i | −0.914917 | + | 0.297274i | 0.584718 | − | 1.79958i | 1.74584 | − | 1.39716i | −0.255547 | − | 0.185666i | 3.11241 | + | 1.01128i | 1.21545 | − | 0.394924i | −1.67835 | + | 1.21939i | 0.708089 | + | 0.194116i |
124.8 | 0.193000 | + | 0.265641i | 0.914917 | − | 0.297274i | 0.584718 | − | 1.79958i | −0.789281 | + | 2.09214i | 0.255547 | + | 0.185666i | 3.11241 | + | 1.01128i | 1.21545 | − | 0.394924i | −1.67835 | + | 1.21939i | −0.708089 | + | 0.194116i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
55.d | odd | 2 | 1 | inner |
55.h | odd | 10 | 3 | inner |
55.j | even | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 605.2.j.k | 48 | |
5.b | even | 2 | 1 | inner | 605.2.j.k | 48 | |
11.b | odd | 2 | 1 | inner | 605.2.j.k | 48 | |
11.c | even | 5 | 1 | 605.2.b.h | ✓ | 12 | |
11.c | even | 5 | 3 | inner | 605.2.j.k | 48 | |
11.d | odd | 10 | 1 | 605.2.b.h | ✓ | 12 | |
11.d | odd | 10 | 3 | inner | 605.2.j.k | 48 | |
55.d | odd | 2 | 1 | inner | 605.2.j.k | 48 | |
55.h | odd | 10 | 1 | 605.2.b.h | ✓ | 12 | |
55.h | odd | 10 | 3 | inner | 605.2.j.k | 48 | |
55.j | even | 10 | 1 | 605.2.b.h | ✓ | 12 | |
55.j | even | 10 | 3 | inner | 605.2.j.k | 48 | |
55.k | odd | 20 | 2 | 3025.2.a.bo | 12 | ||
55.l | even | 20 | 2 | 3025.2.a.bo | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
605.2.b.h | ✓ | 12 | 11.c | even | 5 | 1 | |
605.2.b.h | ✓ | 12 | 11.d | odd | 10 | 1 | |
605.2.b.h | ✓ | 12 | 55.h | odd | 10 | 1 | |
605.2.b.h | ✓ | 12 | 55.j | even | 10 | 1 | |
605.2.j.k | 48 | 1.a | even | 1 | 1 | trivial | |
605.2.j.k | 48 | 5.b | even | 2 | 1 | inner | |
605.2.j.k | 48 | 11.b | odd | 2 | 1 | inner | |
605.2.j.k | 48 | 11.c | even | 5 | 3 | inner | |
605.2.j.k | 48 | 11.d | odd | 10 | 3 | inner | |
605.2.j.k | 48 | 55.d | odd | 2 | 1 | inner | |
605.2.j.k | 48 | 55.h | odd | 10 | 3 | inner | |
605.2.j.k | 48 | 55.j | even | 10 | 3 | inner | |
3025.2.a.bo | 12 | 55.k | odd | 20 | 2 | ||
3025.2.a.bo | 12 | 55.l | even | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):
\( T_{2}^{24} - 11 T_{2}^{22} + 92 T_{2}^{20} - 696 T_{2}^{18} + 5021 T_{2}^{16} - 19632 T_{2}^{14} + \cdots + 81 \) |
\( T_{19}^{24} + 72 T_{19}^{22} + 3852 T_{19}^{20} + 185328 T_{19}^{18} + 8492688 T_{19}^{16} + \cdots + 228509902503936 \) |