Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(179,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.179");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.o (of order \(6\), degree \(2\), 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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 252) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
179.1 | −1.41419 | − | 0.00770275i | 0 | 1.99988 | + | 0.0217863i | − | 2.17323i | 0 | −0.191082 | + | 2.63884i | −2.82805 | − | 0.0462147i | 0 | −0.0167398 | + | 3.07336i | |||||||
179.2 | −1.41030 | − | 0.105074i | 0 | 1.97792 | + | 0.296374i | 4.26346i | 0 | −1.55034 | − | 2.14393i | −2.75833 | − | 0.625806i | 0 | 0.447980 | − | 6.01278i | ||||||||
179.3 | −1.40984 | + | 0.111168i | 0 | 1.97528 | − | 0.313458i | − | 3.13896i | 0 | 0.228884 | − | 2.63583i | −2.74998 | + | 0.661514i | 0 | 0.348953 | + | 4.42542i | |||||||
179.4 | −1.37735 | + | 0.320803i | 0 | 1.79417 | − | 0.883713i | 1.64200i | 0 | −1.79804 | + | 1.94089i | −2.18770 | + | 1.79275i | 0 | −0.526757 | − | 2.26160i | ||||||||
179.5 | −1.34745 | − | 0.429408i | 0 | 1.63122 | + | 1.15721i | − | 0.501474i | 0 | −2.46962 | − | 0.949199i | −1.70106 | − | 2.25973i | 0 | −0.215337 | + | 0.675709i | |||||||
179.6 | −1.32602 | − | 0.491613i | 0 | 1.51663 | + | 1.30377i | 0.834477i | 0 | 2.47783 | − | 0.927568i | −1.37012 | − | 2.47442i | 0 | 0.410240 | − | 1.10653i | ||||||||
179.7 | −1.20615 | + | 0.738385i | 0 | 0.909574 | − | 1.78120i | 1.66943i | 0 | 2.08401 | + | 1.63000i | 0.218133 | + | 2.82000i | 0 | −1.23268 | − | 2.01358i | ||||||||
179.8 | −1.17812 | + | 0.782325i | 0 | 0.775934 | − | 1.84335i | − | 3.27841i | 0 | 2.52907 | + | 0.777042i | 0.527954 | + | 2.77872i | 0 | 2.56478 | + | 3.86236i | |||||||
179.9 | −1.15931 | + | 0.809940i | 0 | 0.687993 | − | 1.87794i | − | 0.492858i | 0 | −1.41089 | − | 2.23817i | 0.723425 | + | 2.73435i | 0 | 0.399186 | + | 0.571375i | |||||||
179.10 | −1.08876 | − | 0.902556i | 0 | 0.370785 | + | 1.96533i | 0.834477i | 0 | −2.47783 | + | 0.927568i | 1.37012 | − | 2.47442i | 0 | 0.753162 | − | 0.908543i | ||||||||
179.11 | −1.04560 | − | 0.952218i | 0 | 0.186562 | + | 1.99128i | − | 0.501474i | 0 | 2.46962 | + | 0.949199i | 1.70106 | − | 2.25973i | 0 | −0.477513 | + | 0.524342i | |||||||
179.12 | −0.798016 | + | 1.16755i | 0 | −0.726340 | − | 1.86345i | 1.15027i | 0 | 0.422424 | − | 2.61181i | 2.75529 | + | 0.639023i | 0 | −1.34300 | − | 0.917936i | ||||||||
179.13 | −0.796149 | − | 1.16882i | 0 | −0.732292 | + | 1.86111i | 4.26346i | 0 | 1.55034 | + | 2.14393i | 2.75833 | − | 0.625806i | 0 | 4.98323 | − | 3.39435i | ||||||||
179.14 | −0.718821 | + | 1.21791i | 0 | −0.966593 | − | 1.75091i | 3.37581i | 0 | 2.01379 | + | 1.71600i | 2.82726 | + | 0.0813731i | 0 | −4.11142 | − | 2.42660i | ||||||||
179.15 | −0.713767 | − | 1.22088i | 0 | −0.981073 | + | 1.74284i | − | 2.17323i | 0 | 0.191082 | − | 2.63884i | 2.82805 | − | 0.0462147i | 0 | −2.65324 | + | 1.55118i | |||||||
179.16 | −0.608644 | − | 1.27654i | 0 | −1.25910 | + | 1.55392i | − | 3.13896i | 0 | −0.228884 | + | 2.63583i | 2.74998 | + | 0.661514i | 0 | −4.00700 | + | 1.91051i | |||||||
179.17 | −0.571896 | + | 1.29342i | 0 | −1.34587 | − | 1.47940i | 1.08703i | 0 | −2.19009 | + | 1.48443i | 2.68319 | − | 0.894712i | 0 | −1.40598 | − | 0.621665i | ||||||||
179.18 | −0.410851 | − | 1.35322i | 0 | −1.66240 | + | 1.11194i | 1.64200i | 0 | 1.79804 | − | 1.94089i | 2.18770 | + | 1.79275i | 0 | 2.22198 | − | 0.674616i | ||||||||
179.19 | −0.343960 | + | 1.37175i | 0 | −1.76338 | − | 0.943653i | − | 3.68619i | 0 | 2.28903 | − | 1.32677i | 1.90099 | − | 2.09434i | 0 | 5.05653 | + | 1.26790i | |||||||
179.20 | −0.177841 | + | 1.40299i | 0 | −1.93675 | − | 0.499017i | 1.62292i | 0 | −1.09603 | − | 2.40805i | 1.04455 | − | 2.62848i | 0 | −2.27693 | − | 0.288621i | ||||||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.n | odd | 6 | 1 | inner |
252.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 756.2.o.a | 88 | |
3.b | odd | 2 | 1 | 252.2.o.a | ✓ | 88 | |
4.b | odd | 2 | 1 | inner | 756.2.o.a | 88 | |
7.c | even | 3 | 1 | 756.2.bb.a | 88 | ||
9.c | even | 3 | 1 | 252.2.bb.a | yes | 88 | |
9.d | odd | 6 | 1 | 756.2.bb.a | 88 | ||
12.b | even | 2 | 1 | 252.2.o.a | ✓ | 88 | |
21.h | odd | 6 | 1 | 252.2.bb.a | yes | 88 | |
28.g | odd | 6 | 1 | 756.2.bb.a | 88 | ||
36.f | odd | 6 | 1 | 252.2.bb.a | yes | 88 | |
36.h | even | 6 | 1 | 756.2.bb.a | 88 | ||
63.g | even | 3 | 1 | 252.2.o.a | ✓ | 88 | |
63.n | odd | 6 | 1 | inner | 756.2.o.a | 88 | |
84.n | even | 6 | 1 | 252.2.bb.a | yes | 88 | |
252.o | even | 6 | 1 | inner | 756.2.o.a | 88 | |
252.bl | odd | 6 | 1 | 252.2.o.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.o.a | ✓ | 88 | 3.b | odd | 2 | 1 | |
252.2.o.a | ✓ | 88 | 12.b | even | 2 | 1 | |
252.2.o.a | ✓ | 88 | 63.g | even | 3 | 1 | |
252.2.o.a | ✓ | 88 | 252.bl | odd | 6 | 1 | |
252.2.bb.a | yes | 88 | 9.c | even | 3 | 1 | |
252.2.bb.a | yes | 88 | 21.h | odd | 6 | 1 | |
252.2.bb.a | yes | 88 | 36.f | odd | 6 | 1 | |
252.2.bb.a | yes | 88 | 84.n | even | 6 | 1 | |
756.2.o.a | 88 | 1.a | even | 1 | 1 | trivial | |
756.2.o.a | 88 | 4.b | odd | 2 | 1 | inner | |
756.2.o.a | 88 | 63.n | odd | 6 | 1 | inner | |
756.2.o.a | 88 | 252.o | even | 6 | 1 | inner | |
756.2.bb.a | 88 | 7.c | even | 3 | 1 | ||
756.2.bb.a | 88 | 9.d | odd | 6 | 1 | ||
756.2.bb.a | 88 | 28.g | odd | 6 | 1 | ||
756.2.bb.a | 88 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).