Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [153,2,Mod(4,153)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([4, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("153.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 153 = 3^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 153.n (of order \(12\), degree \(4\), 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: | \(1.22171115093\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.25660 | + | 1.30285i | −1.06362 | + | 1.36701i | 2.39483 | − | 4.14796i | 0.847469 | + | 3.16280i | 0.619154 | − | 4.47053i | −0.996589 | + | 3.71932i | 7.26900i | −0.737432 | − | 2.90795i | −6.03304 | − | 6.03304i | ||
4.2 | −2.07111 | + | 1.19576i | 1.60904 | − | 0.641087i | 1.85966 | − | 3.22103i | 0.229055 | + | 0.854845i | −2.56591 | + | 3.25178i | 0.639316 | − | 2.38596i | 4.11179i | 2.17801 | − | 2.06307i | −1.49658 | − | 1.49658i | ||
4.3 | −1.84575 | + | 1.06565i | 0.00129728 | + | 1.73205i | 1.27121 | − | 2.20180i | −0.945773 | − | 3.52967i | −1.84815 | − | 3.19556i | 0.990725 | − | 3.69744i | 1.15604i | −3.00000 | + | 0.00449391i | 5.50705 | + | 5.50705i | ||
4.4 | −1.64395 | + | 0.949132i | −1.09575 | − | 1.34139i | 0.801704 | − | 1.38859i | 0.583765 | + | 2.17864i | 3.07451 | + | 1.16516i | 0.516120 | − | 1.92618i | − | 0.752835i | −0.598665 | + | 2.93966i | −3.02750 | − | 3.02750i | |
4.5 | −1.25944 | + | 0.727138i | −1.72945 | − | 0.0948539i | 0.0574603 | − | 0.0995242i | −0.384824 | − | 1.43618i | 2.24711 | − | 1.13809i | −0.382826 | + | 1.42872i | − | 2.74143i | 2.98201 | + | 0.328090i | 1.52897 | + | 1.52897i | |
4.6 | −1.15506 | + | 0.666877i | 1.50110 | + | 0.864115i | −0.110551 | + | 0.191481i | −0.0545182 | − | 0.203465i | −2.31013 | + | 0.00294161i | −0.965572 | + | 3.60357i | − | 2.96240i | 1.50661 | + | 2.59425i | 0.198658 | + | 0.198658i | |
4.7 | −0.429699 | + | 0.248087i | −0.0790082 | − | 1.73025i | −0.876906 | + | 1.51885i | −0.533502 | − | 1.99106i | 0.463201 | + | 0.723884i | 0.631823 | − | 2.35799i | − | 1.86254i | −2.98752 | + | 0.273408i | 0.723200 | + | 0.723200i | |
4.8 | −0.218953 | + | 0.126413i | 0.963283 | − | 1.43947i | −0.968040 | + | 1.67669i | 1.02659 | + | 3.83129i | −0.0289461 | + | 0.436949i | −0.551719 | + | 2.05904i | − | 0.995142i | −1.14417 | − | 2.77324i | −0.709099 | − | 0.709099i | |
4.9 | 0.145499 | − | 0.0840042i | −0.814046 | + | 1.52883i | −0.985887 | + | 1.70761i | −0.454855 | − | 1.69754i | 0.00998502 | + | 0.290828i | −0.714807 | + | 2.66770i | 0.667291i | −1.67466 | − | 2.48908i | −0.208782 | − | 0.208782i | ||
4.10 | 0.268917 | − | 0.155259i | 1.64363 | + | 0.546341i | −0.951789 | + | 1.64855i | 0.0951356 | + | 0.355051i | 0.526823 | − | 0.108268i | 0.984025 | − | 3.67243i | 1.21213i | 2.40302 | + | 1.79596i | 0.0807084 | + | 0.0807084i | ||
4.11 | 0.793288 | − | 0.458005i | −1.72440 | − | 0.162580i | −0.580463 | + | 1.00539i | 0.915201 | + | 3.41558i | −1.44241 | + | 0.660813i | 0.0913582 | − | 0.340953i | 2.89544i | 2.94714 | + | 0.560706i | 2.29037 | + | 2.29037i | ||
4.12 | 1.11815 | − | 0.645565i | 1.43984 | − | 0.962729i | −0.166493 | + | 0.288374i | −0.684695 | − | 2.55532i | 0.988459 | − | 2.00599i | −0.585596 | + | 2.18547i | 3.01219i | 1.14630 | − | 2.77236i | −2.41521 | − | 2.41521i | ||
4.13 | 1.19294 | − | 0.688745i | 0.266702 | + | 1.71139i | −0.0512619 | + | 0.0887882i | 0.193813 | + | 0.723321i | 1.49687 | + | 1.85790i | 0.137184 | − | 0.511976i | 2.89620i | −2.85774 | + | 0.912864i | 0.729391 | + | 0.729391i | ||
4.14 | 1.41154 | − | 0.814955i | −1.69753 | − | 0.344088i | 0.328304 | − | 0.568639i | −0.870254 | − | 3.24783i | −2.67655 | + | 0.897715i | 1.08515 | − | 4.04983i | 2.18961i | 2.76321 | + | 1.16820i | −3.87524 | − | 3.87524i | ||
4.15 | 1.92090 | − | 1.10903i | −0.421083 | − | 1.68009i | 1.45992 | − | 2.52865i | 0.182375 | + | 0.680632i | −2.67213 | − | 2.76029i | −0.556256 | + | 2.07597i | − | 2.04026i | −2.64538 | + | 1.41491i | 1.10517 | + | 1.10517i | |
4.16 | 2.29727 | − | 1.32633i | −1.16603 | + | 1.28077i | 2.51830 | − | 4.36183i | 0.221043 | + | 0.824944i | −0.979965 | + | 4.48882i | 0.0436914 | − | 0.163059i | − | 8.05510i | −0.280745 | − | 2.98683i | 1.60194 | + | 1.60194i | |
13.1 | −2.29727 | + | 1.32633i | −1.28077 | − | 1.16603i | 2.51830 | − | 4.36183i | −0.824944 | + | 0.221043i | 4.48882 | + | 0.979965i | −0.163059 | − | 0.0436914i | 8.05510i | 0.280745 | + | 2.98683i | 1.60194 | − | 1.60194i | ||
13.2 | −1.92090 | + | 1.10903i | 1.68009 | − | 0.421083i | 1.45992 | − | 2.52865i | −0.680632 | + | 0.182375i | −2.76029 | + | 2.67213i | 2.07597 | + | 0.556256i | 2.04026i | 2.64538 | − | 1.41491i | 1.10517 | − | 1.10517i | ||
13.3 | −1.41154 | + | 0.814955i | 0.344088 | − | 1.69753i | 0.328304 | − | 0.568639i | 3.24783 | − | 0.870254i | 0.897715 | + | 2.67655i | −4.04983 | − | 1.08515i | − | 2.18961i | −2.76321 | − | 1.16820i | −3.87524 | + | 3.87524i | |
13.4 | −1.19294 | + | 0.688745i | −1.71139 | + | 0.266702i | −0.0512619 | + | 0.0887882i | −0.723321 | + | 0.193813i | 1.85790 | − | 1.49687i | −0.511976 | − | 0.137184i | − | 2.89620i | 2.85774 | − | 0.912864i | 0.729391 | − | 0.729391i | |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
17.c | even | 4 | 1 | inner |
153.n | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 153.2.n.a | ✓ | 64 |
3.b | odd | 2 | 1 | 459.2.o.a | 64 | ||
9.c | even | 3 | 1 | inner | 153.2.n.a | ✓ | 64 |
9.d | odd | 6 | 1 | 459.2.o.a | 64 | ||
17.c | even | 4 | 1 | inner | 153.2.n.a | ✓ | 64 |
51.f | odd | 4 | 1 | 459.2.o.a | 64 | ||
153.m | odd | 12 | 1 | 459.2.o.a | 64 | ||
153.n | even | 12 | 1 | inner | 153.2.n.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
153.2.n.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
153.2.n.a | ✓ | 64 | 9.c | even | 3 | 1 | inner |
153.2.n.a | ✓ | 64 | 17.c | even | 4 | 1 | inner |
153.2.n.a | ✓ | 64 | 153.n | even | 12 | 1 | inner |
459.2.o.a | 64 | 3.b | odd | 2 | 1 | ||
459.2.o.a | 64 | 9.d | odd | 6 | 1 | ||
459.2.o.a | 64 | 51.f | odd | 4 | 1 | ||
459.2.o.a | 64 | 153.m | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(153, [\chi])\).