Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [560,2,Mod(141,560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(560, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("560.141");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 560.bd (of order \(4\), degree \(2\), 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: | \(4.47162251319\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
141.1 | −1.41203 | − | 0.0784738i | 0.257753 | + | 0.257753i | 1.98768 | + | 0.221615i | −0.707107 | + | 0.707107i | −0.343730 | − | 0.384184i | − | 1.00000i | −2.78929 | − | 0.468910i | − | 2.86713i | 1.05395 | − | 0.942970i | ||
141.2 | −1.40016 | − | 0.198865i | 1.92081 | + | 1.92081i | 1.92091 | + | 0.556887i | 0.707107 | − | 0.707107i | −2.30746 | − | 3.07142i | − | 1.00000i | −2.57883 | − | 1.16173i | 4.37899i | −1.13068 | + | 0.849445i | |||
141.3 | −1.39986 | + | 0.200956i | −2.22143 | − | 2.22143i | 1.91923 | − | 0.562622i | −0.707107 | + | 0.707107i | 3.55611 | + | 2.66329i | − | 1.00000i | −2.57360 | + | 1.17328i | 6.86952i | 0.847755 | − | 1.13195i | |||
141.4 | −1.37680 | + | 0.323129i | −1.67958 | − | 1.67958i | 1.79118 | − | 0.889770i | 0.707107 | − | 0.707107i | 2.85517 | + | 1.76973i | − | 1.00000i | −2.17859 | + | 1.80382i | 2.64195i | −0.745060 | + | 1.20203i | |||
141.5 | −1.16029 | + | 0.808533i | 0.925827 | + | 0.925827i | 0.692549 | − | 1.87627i | −0.707107 | + | 0.707107i | −1.82279 | − | 0.325667i | − | 1.00000i | 0.713464 | + | 2.73696i | − | 1.28569i | 0.248730 | − | 1.39217i | ||
141.6 | −1.04918 | − | 0.948274i | 0.448521 | + | 0.448521i | 0.201553 | + | 1.98982i | 0.707107 | − | 0.707107i | −0.0452579 | − | 0.895900i | − | 1.00000i | 1.67543 | − | 2.27880i | − | 2.59766i | −1.41241 | + | 0.0713505i | ||
141.7 | −0.914710 | + | 1.07857i | −1.42818 | − | 1.42818i | −0.326610 | − | 1.97315i | 0.707107 | − | 0.707107i | 2.84676 | − | 0.234016i | − | 1.00000i | 2.42693 | + | 1.45259i | 1.07940i | 0.115864 | + | 1.40946i | |||
141.8 | −0.630992 | + | 1.26564i | 0.659301 | + | 0.659301i | −1.20370 | − | 1.59722i | 0.707107 | − | 0.707107i | −1.25045 | + | 0.418425i | − | 1.00000i | 2.78103 | − | 0.515617i | − | 2.13064i | 0.448765 | + | 1.34112i | ||
141.9 | −0.590325 | + | 1.28511i | −0.839605 | − | 0.839605i | −1.30303 | − | 1.51727i | −0.707107 | + | 0.707107i | 1.57463 | − | 0.583348i | − | 1.00000i | 2.71908 | − | 0.778863i | − | 1.59013i | −0.491290 | − | 1.32614i | ||
141.10 | −0.560807 | − | 1.29827i | 0.693100 | + | 0.693100i | −1.37099 | + | 1.45615i | −0.707107 | + | 0.707107i | 0.511134 | − | 1.28852i | − | 1.00000i | 2.65934 | + | 0.963293i | − | 2.03922i | 1.31456 | + | 0.521463i | ||
141.11 | −0.551277 | − | 1.30234i | −1.16279 | − | 1.16279i | −1.39219 | + | 1.43590i | 0.707107 | − | 0.707107i | −0.873327 | + | 2.15536i | − | 1.00000i | 2.63752 | + | 1.02152i | − | 0.295857i | −1.31071 | − | 0.531083i | ||
141.12 | 0.381101 | + | 1.36190i | −0.605289 | − | 0.605289i | −1.70952 | + | 1.03804i | −0.707107 | + | 0.707107i | 0.593665 | − | 1.05502i | − | 1.00000i | −2.06521 | − | 1.93260i | − | 2.26725i | −1.23249 | − | 0.693527i | ||
141.13 | 0.504607 | + | 1.32113i | 2.22016 | + | 2.22016i | −1.49074 | + | 1.33330i | 0.707107 | − | 0.707107i | −1.81280 | + | 4.05342i | − | 1.00000i | −2.51369 | − | 1.29667i | 6.85821i | 1.29099 | + | 0.577366i | |||
141.14 | 0.517296 | − | 1.31621i | −0.296675 | − | 0.296675i | −1.46481 | − | 1.36174i | −0.707107 | + | 0.707107i | −0.543956 | + | 0.237018i | − | 1.00000i | −2.55007 | + | 1.22357i | − | 2.82397i | 0.564917 | + | 1.29648i | ||
141.15 | 0.817344 | − | 1.15410i | 1.28029 | + | 1.28029i | −0.663897 | − | 1.88660i | 0.707107 | − | 0.707107i | 2.52403 | − | 0.431148i | − | 1.00000i | −2.71995 | − | 0.775794i | 0.278310i | −0.238123 | − | 1.39402i | |||
141.16 | 1.13939 | + | 0.837723i | −1.78962 | − | 1.78962i | 0.596439 | + | 1.90899i | −0.707107 | + | 0.707107i | −0.539875 | − | 3.53828i | − | 1.00000i | −0.919631 | + | 2.67475i | 3.40546i | −1.39803 | + | 0.213314i | |||
141.17 | 1.17062 | − | 0.793500i | −1.25769 | − | 1.25769i | 0.740715 | − | 1.85778i | 0.707107 | − | 0.707107i | −2.47025 | − | 0.474301i | − | 1.00000i | −0.607051 | − | 2.76252i | 0.163545i | 0.266666 | − | 1.38884i | |||
141.18 | 1.18248 | + | 0.775723i | 1.63220 | + | 1.63220i | 0.796507 | + | 1.83455i | −0.707107 | + | 0.707107i | 0.663904 | + | 3.19617i | − | 1.00000i | −0.481253 | + | 2.78718i | 2.32815i | −1.38466 | + | 0.287619i | |||
141.19 | 1.20137 | − | 0.746130i | 2.18833 | + | 2.18833i | 0.886580 | − | 1.79276i | −0.707107 | + | 0.707107i | 4.26177 | + | 0.996215i | − | 1.00000i | −0.272519 | − | 2.81527i | 6.57756i | −0.321903 | + | 1.37709i | |||
141.20 | 1.32415 | + | 0.496608i | −0.576404 | − | 0.576404i | 1.50676 | + | 1.31517i | 0.707107 | − | 0.707107i | −0.477000 | − | 1.04949i | − | 1.00000i | 1.34206 | + | 2.48976i | − | 2.33552i | 1.28747 | − | 0.585162i | ||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 560.2.bd.a | ✓ | 44 |
4.b | odd | 2 | 1 | 2240.2.bd.a | 44 | ||
16.e | even | 4 | 1 | inner | 560.2.bd.a | ✓ | 44 |
16.f | odd | 4 | 1 | 2240.2.bd.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
560.2.bd.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
560.2.bd.a | ✓ | 44 | 16.e | even | 4 | 1 | inner |
2240.2.bd.a | 44 | 4.b | odd | 2 | 1 | ||
2240.2.bd.a | 44 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 8 T_{3}^{41} + 256 T_{3}^{40} + 72 T_{3}^{39} + 32 T_{3}^{38} + 1480 T_{3}^{37} + \cdots + 10137856 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).