Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1122,2,Mod(1121,1122)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1122.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1122.h (of order \(2\), degree \(1\), 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: | \(8.95921510679\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1121.1 | −1.00000 | −1.63364 | − | 0.575506i | 1.00000 | 2.19878 | 1.63364 | + | 0.575506i | −4.54668 | −1.00000 | 2.33759 | + | 1.88034i | −2.19878 | ||||||||||||
1121.2 | −1.00000 | −1.63364 | + | 0.575506i | 1.00000 | 2.19878 | 1.63364 | − | 0.575506i | −4.54668 | −1.00000 | 2.33759 | − | 1.88034i | −2.19878 | ||||||||||||
1121.3 | −1.00000 | −1.60123 | − | 0.660342i | 1.00000 | −1.88112 | 1.60123 | + | 0.660342i | −2.39717 | −1.00000 | 2.12790 | + | 2.11472i | 1.88112 | ||||||||||||
1121.4 | −1.00000 | −1.60123 | + | 0.660342i | 1.00000 | −1.88112 | 1.60123 | − | 0.660342i | −2.39717 | −1.00000 | 2.12790 | − | 2.11472i | 1.88112 | ||||||||||||
1121.5 | −1.00000 | −1.57348 | − | 0.723996i | 1.00000 | 3.66340 | 1.57348 | + | 0.723996i | 1.80951 | −1.00000 | 1.95166 | + | 2.27838i | −3.66340 | ||||||||||||
1121.6 | −1.00000 | −1.57348 | + | 0.723996i | 1.00000 | 3.66340 | 1.57348 | − | 0.723996i | 1.80951 | −1.00000 | 1.95166 | − | 2.27838i | −3.66340 | ||||||||||||
1121.7 | −1.00000 | −1.46442 | − | 0.924915i | 1.00000 | −3.53846 | 1.46442 | + | 0.924915i | −1.65251 | −1.00000 | 1.28906 | + | 2.70893i | 3.53846 | ||||||||||||
1121.8 | −1.00000 | −1.46442 | + | 0.924915i | 1.00000 | −3.53846 | 1.46442 | − | 0.924915i | −1.65251 | −1.00000 | 1.28906 | − | 2.70893i | 3.53846 | ||||||||||||
1121.9 | −1.00000 | −0.744620 | − | 1.56382i | 1.00000 | −2.04802 | 0.744620 | + | 1.56382i | 2.83134 | −1.00000 | −1.89108 | + | 2.32891i | 2.04802 | ||||||||||||
1121.10 | −1.00000 | −0.744620 | + | 1.56382i | 1.00000 | −2.04802 | 0.744620 | − | 1.56382i | 2.83134 | −1.00000 | −1.89108 | − | 2.32891i | 2.04802 | ||||||||||||
1121.11 | −1.00000 | −0.524078 | − | 1.65086i | 1.00000 | 3.81008 | 0.524078 | + | 1.65086i | 0.269313 | −1.00000 | −2.45068 | + | 1.73036i | −3.81008 | ||||||||||||
1121.12 | −1.00000 | −0.524078 | + | 1.65086i | 1.00000 | 3.81008 | 0.524078 | − | 1.65086i | 0.269313 | −1.00000 | −2.45068 | − | 1.73036i | −3.81008 | ||||||||||||
1121.13 | −1.00000 | −0.260343 | − | 1.71237i | 1.00000 | 1.40515 | 0.260343 | + | 1.71237i | −1.57706 | −1.00000 | −2.86444 | + | 0.891608i | −1.40515 | ||||||||||||
1121.14 | −1.00000 | −0.260343 | + | 1.71237i | 1.00000 | 1.40515 | 0.260343 | − | 1.71237i | −1.57706 | −1.00000 | −2.86444 | − | 0.891608i | −1.40515 | ||||||||||||
1121.15 | −1.00000 | 0.260343 | − | 1.71237i | 1.00000 | −1.40515 | −0.260343 | + | 1.71237i | 1.57706 | −1.00000 | −2.86444 | − | 0.891608i | 1.40515 | ||||||||||||
1121.16 | −1.00000 | 0.260343 | + | 1.71237i | 1.00000 | −1.40515 | −0.260343 | − | 1.71237i | 1.57706 | −1.00000 | −2.86444 | + | 0.891608i | 1.40515 | ||||||||||||
1121.17 | −1.00000 | 0.524078 | − | 1.65086i | 1.00000 | −3.81008 | −0.524078 | + | 1.65086i | −0.269313 | −1.00000 | −2.45068 | − | 1.73036i | 3.81008 | ||||||||||||
1121.18 | −1.00000 | 0.524078 | + | 1.65086i | 1.00000 | −3.81008 | −0.524078 | − | 1.65086i | −0.269313 | −1.00000 | −2.45068 | + | 1.73036i | 3.81008 | ||||||||||||
1121.19 | −1.00000 | 0.744620 | − | 1.56382i | 1.00000 | 2.04802 | −0.744620 | + | 1.56382i | −2.83134 | −1.00000 | −1.89108 | − | 2.32891i | −2.04802 | ||||||||||||
1121.20 | −1.00000 | 0.744620 | + | 1.56382i | 1.00000 | 2.04802 | −0.744620 | − | 1.56382i | −2.83134 | −1.00000 | −1.89108 | + | 2.32891i | −2.04802 | ||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
561.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1122.2.h.g | ✓ | 28 |
3.b | odd | 2 | 1 | 1122.2.h.h | yes | 28 | |
11.b | odd | 2 | 1 | 1122.2.h.h | yes | 28 | |
17.b | even | 2 | 1 | inner | 1122.2.h.g | ✓ | 28 |
33.d | even | 2 | 1 | inner | 1122.2.h.g | ✓ | 28 |
51.c | odd | 2 | 1 | 1122.2.h.h | yes | 28 | |
187.b | odd | 2 | 1 | 1122.2.h.h | yes | 28 | |
561.h | even | 2 | 1 | inner | 1122.2.h.g | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1122.2.h.g | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
1122.2.h.g | ✓ | 28 | 17.b | even | 2 | 1 | inner |
1122.2.h.g | ✓ | 28 | 33.d | even | 2 | 1 | inner |
1122.2.h.g | ✓ | 28 | 561.h | even | 2 | 1 | inner |
1122.2.h.h | yes | 28 | 3.b | odd | 2 | 1 | |
1122.2.h.h | yes | 28 | 11.b | odd | 2 | 1 | |
1122.2.h.h | yes | 28 | 51.c | odd | 2 | 1 | |
1122.2.h.h | yes | 28 | 187.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1122, [\chi])\):
\( T_{5}^{14} - 55T_{5}^{12} + 1210T_{5}^{10} - 13651T_{5}^{8} + 84648T_{5}^{6} - 288904T_{5}^{4} + 503744T_{5}^{2} - 345600 \) |
\( T_{7}^{14} - 43T_{7}^{12} + 650T_{7}^{10} - 4651T_{7}^{8} + 17080T_{7}^{6} - 31304T_{7}^{4} + 23360T_{7}^{2} - 1536 \) |
\( T_{23}^{14} - 160 T_{23}^{12} + 8616 T_{23}^{10} - 217024 T_{23}^{8} + 2785664 T_{23}^{6} - 17878528 T_{23}^{4} + 51064832 T_{23}^{2} - 52002816 \) |
\( T_{83}^{7} - 3T_{83}^{6} - 246T_{83}^{5} - 841T_{83}^{4} + 9288T_{83}^{3} + 65152T_{83}^{2} + 141184T_{83} + 100608 \) |