Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(48,833)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.48");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 833.t (of order \(16\), degree \(8\), 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: | \(6.65153848837\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
48.1 | −2.14596 | + | 0.888885i | −0.289938 | − | 0.0576723i | 2.40081 | − | 2.40081i | 0.677905 | + | 1.01456i | 0.673459 | − | 0.133959i | 0 | −1.24022 | + | 2.99415i | −2.69090 | − | 1.11461i | −2.35658 | − | 1.57462i | ||
48.2 | −1.48449 | + | 0.614895i | −2.76433 | − | 0.549859i | 0.411394 | − | 0.411394i | −1.24665 | − | 1.86574i | 4.44172 | − | 0.883512i | 0 | 0.872044 | − | 2.10530i | 4.56752 | + | 1.89193i | 2.99787 | + | 2.00311i | ||
48.3 | −1.00261 | + | 0.415295i | 1.56628 | + | 0.311553i | −0.581456 | + | 0.581456i | −2.30140 | − | 3.44429i | −1.69976 | + | 0.338102i | 0 | 1.17209 | − | 2.82967i | −0.415468 | − | 0.172092i | 3.73780 | + | 2.49752i | ||
48.4 | −0.636489 | + | 0.263642i | 0.868736 | + | 0.172802i | −1.07860 | + | 1.07860i | 1.49239 | + | 2.23353i | −0.598499 | + | 0.119049i | 0 | 0.929438 | − | 2.24386i | −2.04680 | − | 0.847811i | −1.53875 | − | 1.02816i | ||
48.5 | 0.356088 | − | 0.147496i | −2.74793 | − | 0.546598i | −1.30917 | + | 1.30917i | 1.94799 | + | 2.91537i | −1.05913 | + | 0.210673i | 0 | −0.568075 | + | 1.37145i | 4.48072 | + | 1.85598i | 1.12366 | + | 0.750808i | ||
48.6 | 0.633297 | − | 0.262320i | −0.123072 | − | 0.0244806i | −1.08196 | + | 1.08196i | −0.572504 | − | 0.856813i | −0.0843631 | + | 0.0167809i | 0 | −0.926022 | + | 2.23562i | −2.75709 | − | 1.14202i | −0.587324 | − | 0.392437i | ||
48.7 | 1.01653 | − | 0.421062i | 3.20556 | + | 0.637626i | −0.558167 | + | 0.558167i | 0.201698 | + | 0.301862i | 3.52704 | − | 0.701572i | 0 | −1.17450 | + | 2.83548i | 7.09743 | + | 2.93985i | 0.332135 | + | 0.221925i | ||
48.8 | 1.95763 | − | 0.810875i | −2.27107 | − | 0.451743i | 1.76057 | − | 1.76057i | −0.941797 | − | 1.40950i | −4.81221 | + | 0.957207i | 0 | 0.397183 | − | 0.958885i | 2.18203 | + | 0.903827i | −2.98661 | − | 1.99559i | ||
48.9 | 2.22988 | − | 0.923647i | 0.542085 | + | 0.107827i | 2.70503 | − | 2.70503i | 1.99806 | + | 2.99031i | 1.30838 | − | 0.260253i | 0 | 1.68611 | − | 4.07063i | −2.48941 | − | 1.03115i | 7.21743 | + | 4.82253i | ||
97.1 | −0.906232 | − | 2.18784i | −0.875783 | − | 1.31070i | −2.55116 | + | 2.55116i | 0.0733090 | − | 0.368549i | −2.07394 | + | 3.10387i | 0 | 3.51781 | + | 1.45712i | 0.197106 | − | 0.475856i | −0.872761 | + | 0.173603i | ||
97.2 | −0.863214 | − | 2.08398i | 1.49306 | + | 2.23452i | −2.18363 | + | 2.18363i | 0.770889 | − | 3.87552i | 3.36787 | − | 5.04037i | 0 | 2.26763 | + | 0.939281i | −1.61580 | + | 3.90089i | −8.74195 | + | 1.73888i | ||
97.3 | −0.658869 | − | 1.59065i | 1.09570 | + | 1.63982i | −0.681850 | + | 0.681850i | −0.769298 | + | 3.86752i | 1.88647 | − | 2.82330i | 0 | −1.64747 | − | 0.682403i | −0.340426 | + | 0.821860i | 6.65874 | − | 1.32451i | ||
97.4 | −0.164464 | − | 0.397051i | 0.267880 | + | 0.400911i | 1.28361 | − | 1.28361i | 0.127352 | − | 0.640239i | 0.115125 | − | 0.172297i | 0 | −1.51487 | − | 0.627480i | 1.05908 | − | 2.55685i | −0.275152 | + | 0.0547312i | ||
97.5 | −0.0706467 | − | 0.170556i | −1.80391 | − | 2.69974i | 1.39012 | − | 1.39012i | 0.687402 | − | 3.45580i | −0.333017 | + | 0.498395i | 0 | −0.676412 | − | 0.280179i | −2.88645 | + | 6.96851i | −0.637971 | + | 0.126900i | ||
97.6 | 0.272544 | + | 0.657980i | −0.907192 | − | 1.35771i | 1.05556 | − | 1.05556i | −0.616776 | + | 3.10074i | 0.646095 | − | 0.966950i | 0 | 2.29818 | + | 0.951937i | 0.127674 | − | 0.308232i | −2.20832 | + | 0.439263i | ||
97.7 | 0.351866 | + | 0.849480i | 1.69837 | + | 2.54179i | 0.816407 | − | 0.816407i | 0.186515 | − | 0.937676i | −1.56160 | + | 2.33711i | 0 | 2.67975 | + | 1.10999i | −2.42820 | + | 5.86219i | 0.862166 | − | 0.171495i | ||
97.8 | 0.696482 | + | 1.68146i | 0.342709 | + | 0.512900i | −0.927993 | + | 0.927993i | 0.237708 | − | 1.19504i | −0.623729 | + | 0.933476i | 0 | 1.15620 | + | 0.478915i | 1.00243 | − | 2.42009i | 2.17497 | − | 0.432628i | ||
97.9 | 0.959851 | + | 2.31728i | −1.14492 | − | 1.71350i | −3.03428 | + | 3.03428i | 0.119128 | − | 0.598894i | 2.87171 | − | 4.29781i | 0 | −5.30917 | − | 2.19913i | −0.477176 | + | 1.15200i | 1.50215 | − | 0.298797i | ||
146.1 | −0.906232 | + | 2.18784i | −0.875783 | + | 1.31070i | −2.55116 | − | 2.55116i | 0.0733090 | + | 0.368549i | −2.07394 | − | 3.10387i | 0 | 3.51781 | − | 1.45712i | 0.197106 | + | 0.475856i | −0.872761 | − | 0.173603i | ||
146.2 | −0.863214 | + | 2.08398i | 1.49306 | − | 2.23452i | −2.18363 | − | 2.18363i | 0.770889 | + | 3.87552i | 3.36787 | + | 5.04037i | 0 | 2.26763 | − | 0.939281i | −1.61580 | − | 3.90089i | −8.74195 | − | 1.73888i | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
119.p | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 833.2.t.b | ✓ | 72 |
7.b | odd | 2 | 1 | 833.2.t.c | yes | 72 | |
7.c | even | 3 | 2 | 833.2.bc.c | 144 | ||
7.d | odd | 6 | 2 | 833.2.bc.b | 144 | ||
17.e | odd | 16 | 1 | 833.2.t.c | yes | 72 | |
119.p | even | 16 | 1 | inner | 833.2.t.b | ✓ | 72 |
119.s | even | 48 | 2 | 833.2.bc.c | 144 | ||
119.t | odd | 48 | 2 | 833.2.bc.b | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
833.2.t.b | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
833.2.t.b | ✓ | 72 | 119.p | even | 16 | 1 | inner |
833.2.t.c | yes | 72 | 7.b | odd | 2 | 1 | |
833.2.t.c | yes | 72 | 17.e | odd | 16 | 1 | |
833.2.bc.b | 144 | 7.d | odd | 6 | 2 | ||
833.2.bc.b | 144 | 119.t | odd | 48 | 2 | ||
833.2.bc.c | 144 | 7.c | even | 3 | 2 | ||
833.2.bc.c | 144 | 119.s | even | 48 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):
\( T_{2}^{72} - 8 T_{2}^{67} + 72 T_{2}^{65} + 4695 T_{2}^{64} + 544 T_{2}^{63} + 48 T_{2}^{62} + \cdots + 638401 \) |
\( T_{3}^{72} + 12 T_{3}^{70} + 24 T_{3}^{69} + 16 T_{3}^{68} + 216 T_{3}^{67} - 632 T_{3}^{66} + \cdots + 4817408 \) |