Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(24,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.24");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.u (of order \(18\), degree \(6\), 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: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 95) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24.1 | −0.691434 | − | 1.89970i | −1.87167 | − | 0.330026i | −1.59869 | + | 1.34146i | 0 | 0.667187 | + | 3.78381i | −2.67790 | + | 1.54609i | 0.152212 | + | 0.0878797i | 0.575162 | + | 0.209342i | 0 | ||||
24.2 | −0.575828 | − | 1.58207i | 3.20261 | + | 0.564707i | −0.639290 | + | 0.536428i | 0 | −0.950745 | − | 5.39194i | −0.474919 | + | 0.274194i | −1.69930 | − | 0.981094i | 7.11876 | + | 2.59101i | 0 | ||||
24.3 | −0.408399 | − | 1.12207i | 2.23864 | + | 0.394733i | 0.439845 | − | 0.369074i | 0 | −0.471342 | − | 2.67312i | 1.90222 | − | 1.09825i | −2.66196 | − | 1.53688i | 2.03663 | + | 0.741274i | 0 | ||||
24.4 | 0.408399 | + | 1.12207i | −2.23864 | − | 0.394733i | 0.439845 | − | 0.369074i | 0 | −0.471342 | − | 2.67312i | −1.90222 | + | 1.09825i | 2.66196 | + | 1.53688i | 2.03663 | + | 0.741274i | 0 | ||||
24.5 | 0.575828 | + | 1.58207i | −3.20261 | − | 0.564707i | −0.639290 | + | 0.536428i | 0 | −0.950745 | − | 5.39194i | 0.474919 | − | 0.274194i | 1.69930 | + | 0.981094i | 7.11876 | + | 2.59101i | 0 | ||||
24.6 | 0.691434 | + | 1.89970i | 1.87167 | + | 0.330026i | −1.59869 | + | 1.34146i | 0 | 0.667187 | + | 3.78381i | 2.67790 | − | 1.54609i | −0.152212 | − | 0.0878797i | 0.575162 | + | 0.209342i | 0 | ||||
74.1 | −2.58240 | − | 0.455347i | 0.597006 | − | 0.711484i | 4.58206 | + | 1.66773i | 0 | −1.86568 | + | 1.56549i | 3.31824 | − | 1.91579i | −6.53146 | − | 3.77094i | 0.371151 | + | 2.10490i | 0 | ||||
74.2 | −1.44009 | − | 0.253927i | 0.970838 | − | 1.15700i | 0.130002 | + | 0.0473169i | 0 | −1.69189 | + | 1.41966i | 3.52622 | − | 2.03586i | 2.35759 | + | 1.36116i | 0.124823 | + | 0.707907i | 0 | ||||
74.3 | −0.708527 | − | 0.124932i | 0.793382 | − | 0.945515i | −1.39298 | − | 0.507004i | 0 | −0.680257 | + | 0.570804i | −1.11885 | + | 0.645970i | 2.16976 | + | 1.25271i | 0.256400 | + | 1.45411i | 0 | ||||
74.4 | 0.708527 | + | 0.124932i | −0.793382 | + | 0.945515i | −1.39298 | − | 0.507004i | 0 | −0.680257 | + | 0.570804i | 1.11885 | − | 0.645970i | −2.16976 | − | 1.25271i | 0.256400 | + | 1.45411i | 0 | ||||
74.5 | 1.44009 | + | 0.253927i | −0.970838 | + | 1.15700i | 0.130002 | + | 0.0473169i | 0 | −1.69189 | + | 1.41966i | −3.52622 | + | 2.03586i | −2.35759 | − | 1.36116i | 0.124823 | + | 0.707907i | 0 | ||||
74.6 | 2.58240 | + | 0.455347i | −0.597006 | + | 0.711484i | 4.58206 | + | 1.66773i | 0 | −1.86568 | + | 1.56549i | −3.31824 | + | 1.91579i | 6.53146 | + | 3.77094i | 0.371151 | + | 2.10490i | 0 | ||||
99.1 | −0.691434 | + | 1.89970i | −1.87167 | + | 0.330026i | −1.59869 | − | 1.34146i | 0 | 0.667187 | − | 3.78381i | −2.67790 | − | 1.54609i | 0.152212 | − | 0.0878797i | 0.575162 | − | 0.209342i | 0 | ||||
99.2 | −0.575828 | + | 1.58207i | 3.20261 | − | 0.564707i | −0.639290 | − | 0.536428i | 0 | −0.950745 | + | 5.39194i | −0.474919 | − | 0.274194i | −1.69930 | + | 0.981094i | 7.11876 | − | 2.59101i | 0 | ||||
99.3 | −0.408399 | + | 1.12207i | 2.23864 | − | 0.394733i | 0.439845 | + | 0.369074i | 0 | −0.471342 | + | 2.67312i | 1.90222 | + | 1.09825i | −2.66196 | + | 1.53688i | 2.03663 | − | 0.741274i | 0 | ||||
99.4 | 0.408399 | − | 1.12207i | −2.23864 | + | 0.394733i | 0.439845 | + | 0.369074i | 0 | −0.471342 | + | 2.67312i | −1.90222 | − | 1.09825i | 2.66196 | − | 1.53688i | 2.03663 | − | 0.741274i | 0 | ||||
99.5 | 0.575828 | − | 1.58207i | −3.20261 | + | 0.564707i | −0.639290 | − | 0.536428i | 0 | −0.950745 | + | 5.39194i | 0.474919 | + | 0.274194i | 1.69930 | − | 0.981094i | 7.11876 | − | 2.59101i | 0 | ||||
99.6 | 0.691434 | − | 1.89970i | 1.87167 | − | 0.330026i | −1.59869 | − | 1.34146i | 0 | 0.667187 | − | 3.78381i | 2.67790 | + | 1.54609i | −0.152212 | + | 0.0878797i | 0.575162 | − | 0.209342i | 0 | ||||
149.1 | −1.16796 | + | 1.39192i | 0.0605732 | + | 0.166424i | −0.226016 | − | 1.28180i | 0 | −0.302396 | − | 0.110063i | 0.929646 | − | 0.536732i | −1.09903 | − | 0.634528i | 2.27411 | − | 1.90820i | 0 | ||||
149.2 | −1.00948 | + | 1.20305i | −0.781523 | − | 2.14722i | −0.0809872 | − | 0.459301i | 0 | 3.37214 | + | 1.22736i | 3.47568 | − | 2.00668i | −2.08582 | − | 1.20425i | −1.70162 | + | 1.42783i | 0 | ||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
95.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.u.c | 36 | |
5.b | even | 2 | 1 | inner | 475.2.u.c | 36 | |
5.c | odd | 4 | 1 | 95.2.k.b | ✓ | 18 | |
5.c | odd | 4 | 1 | 475.2.l.b | 18 | ||
15.e | even | 4 | 1 | 855.2.bs.b | 18 | ||
19.e | even | 9 | 1 | inner | 475.2.u.c | 36 | |
95.p | even | 18 | 1 | inner | 475.2.u.c | 36 | |
95.q | odd | 36 | 1 | 95.2.k.b | ✓ | 18 | |
95.q | odd | 36 | 1 | 475.2.l.b | 18 | ||
95.q | odd | 36 | 1 | 1805.2.a.t | 9 | ||
95.q | odd | 36 | 1 | 9025.2.a.ce | 9 | ||
95.r | even | 36 | 1 | 1805.2.a.u | 9 | ||
95.r | even | 36 | 1 | 9025.2.a.cd | 9 | ||
285.bi | even | 36 | 1 | 855.2.bs.b | 18 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.k.b | ✓ | 18 | 5.c | odd | 4 | 1 | |
95.2.k.b | ✓ | 18 | 95.q | odd | 36 | 1 | |
475.2.l.b | 18 | 5.c | odd | 4 | 1 | ||
475.2.l.b | 18 | 95.q | odd | 36 | 1 | ||
475.2.u.c | 36 | 1.a | even | 1 | 1 | trivial | |
475.2.u.c | 36 | 5.b | even | 2 | 1 | inner | |
475.2.u.c | 36 | 19.e | even | 9 | 1 | inner | |
475.2.u.c | 36 | 95.p | even | 18 | 1 | inner | |
855.2.bs.b | 18 | 15.e | even | 4 | 1 | ||
855.2.bs.b | 18 | 285.bi | even | 36 | 1 | ||
1805.2.a.t | 9 | 95.q | odd | 36 | 1 | ||
1805.2.a.u | 9 | 95.r | even | 36 | 1 | ||
9025.2.a.cd | 9 | 95.r | even | 36 | 1 | ||
9025.2.a.ce | 9 | 95.q | odd | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{34} - 24 T_{2}^{32} - 79 T_{2}^{30} + 669 T_{2}^{28} + 3744 T_{2}^{26} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).