Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,2,Mod(69,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 5, 10, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.69");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 880.cl (of order \(20\), 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: | \(7.02683537787\) |
Analytic rank: | \(0\) |
Dimension: | \(1120\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
69.1 | −1.41410 | − | 0.0182095i | 0.519642 | − | 1.01985i | 1.99934 | + | 0.0515000i | 0.598527 | + | 2.15448i | −0.753395 | + | 1.43271i | −0.828964 | + | 2.55129i | −2.82632 | − | 0.109233i | 0.993280 | + | 1.36713i | −0.807142 | − | 3.05754i |
69.2 | −1.41376 | − | 0.0359969i | 1.29983 | − | 2.55107i | 1.99741 | + | 0.101782i | 0.705695 | + | 2.12179i | −1.92948 | + | 3.55979i | 1.21258 | − | 3.73193i | −2.82018 | − | 0.215795i | −3.05502 | − | 4.20487i | −0.921302 | − | 3.02510i |
69.3 | −1.41320 | + | 0.0534818i | 0.992902 | − | 1.94868i | 1.99428 | − | 0.151161i | −2.21272 | + | 0.322262i | −1.29895 | + | 2.80698i | 1.01028 | − | 3.10932i | −2.81024 | + | 0.320279i | −1.04814 | − | 1.44264i | 3.10979 | − | 0.573762i |
69.4 | −1.40924 | + | 0.118487i | 0.0780044 | − | 0.153092i | 1.97192 | − | 0.333953i | 1.99339 | − | 1.01311i | −0.0917875 | + | 0.224986i | 1.27618 | − | 3.92767i | −2.73934 | + | 0.704268i | 1.74600 | + | 2.40317i | −2.68913 | + | 1.66391i |
69.5 | −1.40868 | + | 0.124994i | −1.15920 | + | 2.27506i | 1.96875 | − | 0.352153i | −0.861575 | + | 2.06342i | 1.34857 | − | 3.34972i | −0.139307 | + | 0.428744i | −2.72932 | + | 0.742152i | −2.06879 | − | 2.84744i | 0.955769 | − | 3.01438i |
69.6 | −1.40837 | + | 0.128402i | 0.325764 | − | 0.639347i | 1.96703 | − | 0.361676i | −1.22240 | − | 1.87236i | −0.376703 | + | 0.942268i | −0.728483 | + | 2.24204i | −2.72386 | + | 0.761945i | 1.46071 | + | 2.01050i | 1.96201 | + | 2.48002i |
69.7 | −1.40125 | + | 0.191080i | −1.06155 | + | 2.08341i | 1.92698 | − | 0.535499i | −0.115276 | − | 2.23309i | 1.08940 | − | 3.12221i | 0.969208 | − | 2.98291i | −2.59785 | + | 1.11857i | −1.45035 | − | 1.99624i | 0.588229 | + | 3.10709i |
69.8 | −1.39937 | − | 0.204351i | 1.47694 | − | 2.89867i | 1.91648 | + | 0.571925i | −1.23044 | − | 1.86709i | −2.65914 | + | 3.75450i | −1.38653 | + | 4.26730i | −2.56500 | − | 1.19197i | −4.45755 | − | 6.13529i | 1.34029 | + | 2.86419i |
69.9 | −1.39897 | − | 0.207055i | −1.22930 | + | 2.41263i | 1.91426 | + | 0.579330i | −1.90794 | − | 1.16610i | 2.21930 | − | 3.12068i | 0.116717 | − | 0.359217i | −2.55804 | − | 1.20683i | −2.54626 | − | 3.50463i | 2.42771 | + | 2.02639i |
69.10 | −1.38353 | + | 0.293000i | −0.630219 | + | 1.23687i | 1.82830 | − | 0.810748i | 1.78995 | + | 1.34018i | 0.509521 | − | 1.89590i | 0.715623 | − | 2.20246i | −2.29196 | + | 1.65739i | 0.630674 | + | 0.868048i | −2.86912 | − | 1.32972i |
69.11 | −1.37063 | − | 0.348388i | −0.457643 | + | 0.898176i | 1.75725 | + | 0.955021i | 1.98781 | − | 1.02402i | 0.940173 | − | 1.07163i | −0.891410 | + | 2.74348i | −2.07582 | − | 1.92119i | 1.16607 | + | 1.60496i | −3.08130 | + | 0.711028i |
69.12 | −1.36915 | − | 0.354156i | −1.47318 | + | 2.89128i | 1.74915 | + | 0.969787i | 2.22551 | − | 0.217049i | 3.04097 | − | 3.43686i | −0.722522 | + | 2.22369i | −2.05139 | − | 1.94726i | −4.42589 | − | 6.09171i | −3.12393 | − | 0.491005i |
69.13 | −1.36740 | + | 0.360857i | 0.198607 | − | 0.389788i | 1.73956 | − | 0.986872i | −1.80717 | + | 1.31687i | −0.130917 | + | 0.604664i | −0.812900 | + | 2.50185i | −2.02256 | + | 1.97718i | 1.65087 | + | 2.27222i | 1.99592 | − | 2.45282i |
69.14 | −1.36193 | − | 0.380967i | 0.825023 | − | 1.61920i | 1.70973 | + | 1.03770i | 2.23309 | + | 0.115335i | −1.74049 | + | 1.89093i | −0.130304 | + | 0.401035i | −1.93320 | − | 2.06464i | −0.177784 | − | 0.244698i | −2.99738 | − | 1.00781i |
69.15 | −1.34173 | − | 0.446957i | −0.587275 | + | 1.15259i | 1.60046 | + | 1.19939i | −1.04830 | + | 1.97511i | 1.30312 | − | 1.28398i | 0.778431 | − | 2.39576i | −1.61130 | − | 2.32459i | 0.779780 | + | 1.07328i | 2.28932 | − | 2.18151i |
69.16 | −1.32327 | + | 0.498959i | 1.01937 | − | 2.00062i | 1.50208 | − | 1.32051i | 1.53011 | − | 1.63057i | −0.350669 | + | 3.15598i | 0.0518810 | − | 0.159673i | −1.32877 | + | 2.49687i | −1.20002 | − | 1.65169i | −1.21116 | + | 2.92114i |
69.17 | −1.30926 | + | 0.534642i | −0.752040 | + | 1.47596i | 1.42832 | − | 1.39997i | 0.990806 | − | 2.00457i | 0.195503 | − | 2.33449i | −1.27730 | + | 3.93112i | −1.12155 | + | 2.59656i | 0.150458 | + | 0.207088i | −0.225493 | + | 3.15423i |
69.18 | −1.30917 | − | 0.534848i | −0.250760 | + | 0.492144i | 1.42787 | + | 1.40042i | 0.339996 | − | 2.21007i | 0.591511 | − | 0.510184i | 0.413479 | − | 1.27256i | −1.12032 | − | 2.59709i | 1.58403 | + | 2.18023i | −1.62717 | + | 2.71152i |
69.19 | −1.28530 | − | 0.589926i | −0.507149 | + | 0.995335i | 1.30397 | + | 1.51646i | −2.19950 | − | 0.402737i | 1.23901 | − | 0.980121i | −0.969160 | + | 2.98277i | −0.781394 | − | 2.71835i | 1.02986 | + | 1.41749i | 2.58943 | + | 1.81518i |
69.20 | −1.26834 | − | 0.625557i | 0.750300 | − | 1.47255i | 1.21736 | + | 1.58683i | −1.25626 | − | 1.84981i | −1.87280 | + | 1.39833i | 1.11355 | − | 3.42715i | −0.551362 | − | 2.77417i | 0.157911 | + | 0.217346i | 0.436198 | + | 3.13205i |
See next 80 embeddings (of 1120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
16.e | even | 4 | 1 | inner |
55.j | even | 10 | 1 | inner |
80.q | even | 4 | 1 | inner |
176.w | even | 20 | 1 | inner |
880.cl | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 880.2.cl.a | ✓ | 1120 |
5.b | even | 2 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
11.c | even | 5 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
16.e | even | 4 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
55.j | even | 10 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
80.q | even | 4 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
176.w | even | 20 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
880.cl | even | 20 | 1 | inner | 880.2.cl.a | ✓ | 1120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
880.2.cl.a | ✓ | 1120 | 1.a | even | 1 | 1 | trivial |
880.2.cl.a | ✓ | 1120 | 5.b | even | 2 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 11.c | even | 5 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 16.e | even | 4 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 55.j | even | 10 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 80.q | even | 4 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 176.w | even | 20 | 1 | inner |
880.2.cl.a | ✓ | 1120 | 880.cl | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).