Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [712,2,Mod(5,712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, base_ring=CyclotomicField(44))
chi = DirichletCharacter(H, H._module([0, 22, 35]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("712.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 712 = 2^{3} \cdot 89 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 712.z (of order \(44\), degree \(20\), 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: | \(5.68534862392\) |
Analytic rank: | \(0\) |
Dimension: | \(1760\) |
Relative dimension: | \(88\) over \(\Q(\zeta_{44})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{44}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.41269 | + | 0.0657255i | 0.185650 | + | 0.247999i | 1.99136 | − | 0.185699i | −3.57439 | − | 2.29712i | −0.278565 | − | 0.338143i | −0.707473 | − | 3.25220i | −2.80096 | + | 0.393217i | 0.818160 | − | 2.78640i | 5.20047 | + | 3.01018i |
5.2 | −1.41068 | − | 0.0998698i | −1.89369 | − | 2.52967i | 1.98005 | + | 0.281769i | 3.11427 | + | 2.00142i | 2.41876 | + | 3.75769i | −0.542982 | − | 2.49605i | −2.76509 | − | 0.595234i | −1.96799 | + | 6.70235i | −4.19337 | − | 3.13439i |
5.3 | −1.40734 | − | 0.139245i | −1.70529 | − | 2.27800i | 1.96122 | + | 0.391929i | −0.154355 | − | 0.0991976i | 2.08273 | + | 3.44338i | 0.915148 | + | 4.20687i | −2.70554 | − | 0.824668i | −1.43607 | + | 4.89082i | 0.203417 | + | 0.161098i |
5.4 | −1.40277 | − | 0.179535i | 1.63730 | + | 2.18718i | 1.93553 | + | 0.503693i | −3.30181 | − | 2.12195i | −1.90408 | − | 3.36206i | 0.968417 | + | 4.45174i | −2.62468 | − | 1.05406i | −1.25779 | + | 4.28365i | 4.25072 | + | 3.56939i |
5.5 | −1.40193 | + | 0.186027i | −0.728833 | − | 0.973607i | 1.93079 | − | 0.521591i | 1.76410 | + | 1.13372i | 1.20289 | + | 1.22934i | 0.251840 | + | 1.15769i | −2.60979 | + | 1.09041i | 0.428485 | − | 1.45929i | −2.68404 | − | 1.26122i |
5.6 | −1.39492 | + | 0.232776i | 0.670447 | + | 0.895612i | 1.89163 | − | 0.649409i | 0.314819 | + | 0.202322i | −1.14370 | − | 1.09325i | 0.423193 | + | 1.94539i | −2.48752 | + | 1.34620i | 0.492576 | − | 1.67756i | −0.486244 | − | 0.208942i |
5.7 | −1.39061 | + | 0.257303i | 1.20971 | + | 1.61598i | 1.86759 | − | 0.715615i | 0.0714897 | + | 0.0459436i | −2.09803 | − | 1.93594i | −0.988216 | − | 4.54275i | −2.41296 | + | 1.47568i | −0.302806 | + | 1.03126i | −0.111236 | − | 0.0454952i |
5.8 | −1.38396 | − | 0.290948i | 0.442165 | + | 0.590663i | 1.83070 | + | 0.805321i | −0.258658 | − | 0.166229i | −0.440087 | − | 0.946101i | −0.0445955 | − | 0.205002i | −2.29931 | − | 1.64717i | 0.691825 | − | 2.35614i | 0.309608 | + | 0.305311i |
5.9 | −1.38368 | − | 0.292279i | 1.56751 | + | 2.09395i | 1.82915 | + | 0.808841i | 3.28038 | + | 2.10817i | −1.55692 | − | 3.35552i | 0.105832 | + | 0.486500i | −2.29455 | − | 1.65380i | −1.08235 | + | 3.68613i | −3.92283 | − | 3.87583i |
5.10 | −1.36806 | − | 0.358343i | −0.960323 | − | 1.28284i | 1.74318 | + | 0.980470i | −1.65741 | − | 1.06515i | 0.854083 | + | 2.09913i | 0.00614263 | + | 0.0282372i | −2.03343 | − | 1.96600i | 0.121735 | − | 0.414593i | 1.88574 | + | 2.05111i |
5.11 | −1.33040 | + | 0.479618i | −0.516178 | − | 0.689534i | 1.53993 | − | 1.27617i | 2.44507 | + | 1.57135i | 1.01744 | + | 0.669788i | 0.00524368 | + | 0.0241048i | −1.43665 | + | 2.43640i | 0.636181 | − | 2.16663i | −4.00658 | − | 0.917828i |
5.12 | −1.29272 | + | 0.573470i | −1.14765 | − | 1.53308i | 1.34227 | − | 1.48267i | −2.62139 | − | 1.68467i | 2.36277 | + | 1.32371i | 0.505235 | + | 2.32253i | −0.884908 | + | 2.68644i | −0.188043 | + | 0.640416i | 4.35484 | + | 0.674517i |
5.13 | −1.25716 | + | 0.647723i | −0.954713 | − | 1.27535i | 1.16091 | − | 1.62858i | −0.543700 | − | 0.349415i | 2.02630 | + | 0.984928i | −0.755927 | − | 3.47494i | −0.404582 | + | 2.79934i | 0.130164 | − | 0.443299i | 0.909843 | + | 0.0871041i |
5.14 | −1.25708 | − | 0.647879i | −0.313795 | − | 0.419182i | 1.16051 | + | 1.62887i | 2.04508 | + | 1.31429i | 0.122887 | + | 0.730247i | −0.809786 | − | 3.72252i | −0.403539 | − | 2.79949i | 0.767952 | − | 2.61540i | −1.71933 | − | 2.97713i |
5.15 | −1.25579 | + | 0.650376i | 1.73223 | + | 2.31399i | 1.15402 | − | 1.63347i | 0.623593 | + | 0.400759i | −3.68028 | − | 1.77928i | 0.374270 | + | 1.72049i | −0.386839 | + | 2.80185i | −1.50872 | + | 5.13822i | −1.04375 | − | 0.0976995i |
5.16 | −1.17059 | − | 0.793547i | 1.70658 | + | 2.27973i | 0.740568 | + | 1.85784i | −0.719538 | − | 0.462419i | −0.188641 | − | 4.02288i | −0.573882 | − | 2.63809i | 0.607378 | − | 2.76244i | −1.43954 | + | 4.90261i | 0.475334 | + | 1.11229i |
5.17 | −1.16966 | − | 0.794916i | −0.464236 | − | 0.620146i | 0.736216 | + | 1.85957i | 2.27997 | + | 1.46525i | 0.0500343 | + | 1.09439i | 1.02206 | + | 4.69834i | 0.617075 | − | 2.76029i | 0.676131 | − | 2.30269i | −1.50204 | − | 3.52622i |
5.18 | −1.15621 | − | 0.814351i | −1.82803 | − | 2.44196i | 0.673665 | + | 1.88313i | −2.65192 | − | 1.70428i | 0.124981 | + | 4.31209i | −0.844517 | − | 3.88218i | 0.754627 | − | 2.72590i | −1.77629 | + | 6.04949i | 1.67830 | + | 4.13011i |
5.19 | −1.15244 | + | 0.819685i | −2.00041 | − | 2.67223i | 0.656232 | − | 1.88928i | −1.18847 | − | 0.763785i | 4.49573 | + | 1.43988i | −0.461078 | − | 2.11954i | 0.792344 | + | 2.71518i | −2.29399 | + | 7.81261i | 1.99571 | − | 0.0939580i |
5.20 | −1.08572 | − | 0.906209i | 0.903177 | + | 1.20650i | 0.357572 | + | 1.96778i | −1.55814 | − | 1.00135i | 0.112747 | − | 2.12839i | 0.175460 | + | 0.806576i | 1.39499 | − | 2.46049i | 0.205276 | − | 0.699106i | 0.784263 | + | 2.49918i |
See next 80 embeddings (of 1760 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
89.g | even | 44 | 1 | inner |
712.z | even | 44 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 712.2.z.a | ✓ | 1760 |
8.b | even | 2 | 1 | inner | 712.2.z.a | ✓ | 1760 |
89.g | even | 44 | 1 | inner | 712.2.z.a | ✓ | 1760 |
712.z | even | 44 | 1 | inner | 712.2.z.a | ✓ | 1760 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
712.2.z.a | ✓ | 1760 | 1.a | even | 1 | 1 | trivial |
712.2.z.a | ✓ | 1760 | 8.b | even | 2 | 1 | inner |
712.2.z.a | ✓ | 1760 | 89.g | even | 44 | 1 | inner |
712.2.z.a | ✓ | 1760 | 712.z | even | 44 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(712, [\chi])\).