Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [473,2,Mod(76,473)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 31]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("473.76");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 473 = 11 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 473.w (of order \(42\), degree \(12\), 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: | \(3.77692401561\) |
Analytic rank: | \(0\) |
Dimension: | \(504\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 | −0.613285 | − | 2.68698i | 0.369676 | − | 0.398416i | −5.04180 | + | 2.42800i | 0.144296 | + | 0.957343i | −1.29725 | − | 0.748969i | 0.842602 | + | 1.45943i | 6.17928 | + | 7.74857i | 0.202115 | + | 2.69704i | 2.48387 | − | 0.974845i |
76.2 | −0.589874 | − | 2.58441i | 1.89592 | − | 2.04332i | −4.52927 | + | 2.18118i | −0.412899 | − | 2.73941i | −6.39911 | − | 3.69453i | 0.296478 | + | 0.513515i | 5.00319 | + | 6.27380i | −0.356436 | − | 4.75631i | −6.83619 | + | 2.68301i |
76.3 | −0.562314 | − | 2.46366i | −2.22656 | + | 2.39966i | −3.95148 | + | 1.90293i | −0.542591 | − | 3.59986i | 7.16395 | + | 4.13611i | −1.18655 | − | 2.05516i | 3.75900 | + | 4.71364i | −0.576610 | − | 7.69432i | −8.56371 | + | 3.36101i |
76.4 | −0.559146 | − | 2.44978i | −0.669393 | + | 0.721434i | −3.88683 | + | 1.87180i | 0.165193 | + | 1.09599i | 2.14164 | + | 1.23648i | −2.34418 | − | 4.06024i | 3.62542 | + | 4.54614i | 0.151810 | + | 2.02577i | 2.59256 | − | 1.01750i |
76.5 | −0.546308 | − | 2.39353i | −1.62751 | + | 1.75404i | −3.62860 | + | 1.74744i | 0.449946 | + | 2.98520i | 5.08747 | + | 2.93725i | 0.763291 | + | 1.32206i | 3.10346 | + | 3.89161i | −0.203673 | − | 2.71783i | 6.89936 | − | 2.70780i |
76.6 | −0.496652 | − | 2.17598i | −0.725880 | + | 0.782313i | −2.68627 | + | 1.29364i | −0.428723 | − | 2.84439i | 2.06280 | + | 1.19096i | 2.36941 | + | 4.10394i | 1.36589 | + | 1.71277i | 0.139079 | + | 1.85588i | −5.97640 | + | 2.34556i |
76.7 | −0.493610 | − | 2.16265i | 0.376719 | − | 0.406007i | −2.63145 | + | 1.26724i | −0.351823 | − | 2.33419i | −1.06400 | − | 0.614301i | −0.524871 | − | 0.909103i | 1.27337 | + | 1.59675i | 0.201266 | + | 2.68571i | −4.87437 | + | 1.91305i |
76.8 | −0.484408 | − | 2.12233i | 2.17672 | − | 2.34595i | −2.46769 | + | 1.18838i | 0.431771 | + | 2.86461i | −6.03330 | − | 3.48333i | −1.18160 | − | 2.04659i | 1.00294 | + | 1.25765i | −0.541166 | − | 7.22136i | 5.87050 | − | 2.30400i |
76.9 | −0.412201 | − | 1.80597i | 0.743758 | − | 0.801580i | −1.28968 | + | 0.621075i | 0.351427 | + | 2.33156i | −1.75421 | − | 1.01279i | 1.62205 | + | 2.80947i | −0.656673 | − | 0.823441i | 0.134835 | + | 1.79925i | 4.06587 | − | 1.59574i |
76.10 | −0.354968 | − | 1.55522i | −1.13957 | + | 1.22817i | −0.490756 | + | 0.236336i | 0.00101781 | + | 0.00675270i | 2.31457 | + | 1.33632i | 0.0793577 | + | 0.137452i | −1.44744 | − | 1.81503i | 0.0144220 | + | 0.192448i | 0.0101406 | − | 0.00397990i |
76.11 | −0.338969 | − | 1.48512i | 1.60762 | − | 1.73261i | −0.288748 | + | 0.139054i | −0.280400 | − | 1.86033i | −3.11806 | − | 1.80022i | −1.28427 | − | 2.22442i | −1.59515 | − | 2.00026i | −0.193280 | − | 2.57914i | −2.66777 | + | 1.04702i |
76.12 | −0.326253 | − | 1.42941i | −1.61194 | + | 1.73726i | −0.134823 | + | 0.0649274i | 0.428619 | + | 2.84370i | 3.00915 | + | 1.73733i | −0.179222 | − | 0.310421i | −1.69148 | − | 2.12106i | −0.195526 | − | 2.60911i | 3.92496 | − | 1.54043i |
76.13 | −0.322397 | − | 1.41251i | −1.08303 | + | 1.16722i | −0.0893135 | + | 0.0430111i | −0.325539 | − | 2.15981i | 1.99788 | + | 1.15348i | −1.17225 | − | 2.03039i | −1.71712 | − | 2.15320i | 0.0347228 | + | 0.463344i | −2.94580 | + | 1.15614i |
76.14 | −0.298715 | − | 1.30876i | 1.76185 | − | 1.89882i | 0.178324 | − | 0.0858764i | −0.152231 | − | 1.00999i | −3.01139 | − | 1.73863i | 1.59204 | + | 2.75750i | −1.83962 | − | 2.30681i | −0.277223 | − | 3.69929i | −1.27635 | + | 0.500931i |
76.15 | −0.231231 | − | 1.01309i | −0.0484510 | + | 0.0522177i | 0.829057 | − | 0.399253i | 0.598452 | + | 3.97047i | 0.0641046 | + | 0.0370108i | −0.0616950 | − | 0.106859i | −1.89197 | − | 2.37246i | 0.223811 | + | 2.98655i | 3.88406 | − | 1.52438i |
76.16 | −0.205438 | − | 0.900084i | 0.515876 | − | 0.555982i | 1.03399 | − | 0.497944i | −0.139820 | − | 0.927642i | −0.606412 | − | 0.350112i | −1.50664 | − | 2.60957i | −1.81186 | − | 2.27201i | 0.181202 | + | 2.41798i | −0.806232 | + | 0.316423i |
76.17 | −0.173875 | − | 0.761795i | −1.91643 | + | 2.06542i | 1.25184 | − | 0.602854i | −0.341326 | − | 2.26455i | 1.90664 | + | 1.10080i | 1.23866 | + | 2.14543i | −1.65129 | − | 2.07065i | −0.369065 | − | 4.92483i | −1.66577 | + | 0.653768i |
76.18 | −0.131533 | − | 0.576284i | 0.714374 | − | 0.769912i | 1.48714 | − | 0.716167i | −0.616223 | − | 4.08837i | −0.537652 | − | 0.310413i | 1.52888 | + | 2.64810i | −1.34542 | − | 1.68710i | 0.141756 | + | 1.89160i | −2.27501 | + | 0.892876i |
76.19 | −0.0350325 | − | 0.153487i | 1.67297 | − | 1.80304i | 1.77961 | − | 0.857013i | 0.369634 | + | 2.45236i | −0.335351 | − | 0.193615i | 1.06688 | + | 1.84789i | −0.390202 | − | 0.489298i | −0.227910 | − | 3.04125i | 0.363457 | − | 0.142647i |
76.20 | −0.0221475 | − | 0.0970344i | −1.96696 | + | 2.11988i | 1.79301 | − | 0.863469i | 0.245703 | + | 1.63013i | 0.249264 | + | 0.143913i | 2.53990 | + | 4.39924i | −0.247608 | − | 0.310491i | −0.400763 | − | 5.34781i | 0.152737 | − | 0.0599449i |
See next 80 embeddings (of 504 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
43.h | odd | 42 | 1 | inner |
473.w | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 473.2.w.a | ✓ | 504 |
11.b | odd | 2 | 1 | inner | 473.2.w.a | ✓ | 504 |
43.h | odd | 42 | 1 | inner | 473.2.w.a | ✓ | 504 |
473.w | even | 42 | 1 | inner | 473.2.w.a | ✓ | 504 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
473.2.w.a | ✓ | 504 | 1.a | even | 1 | 1 | trivial |
473.2.w.a | ✓ | 504 | 11.b | odd | 2 | 1 | inner |
473.2.w.a | ✓ | 504 | 43.h | odd | 42 | 1 | inner |
473.2.w.a | ✓ | 504 | 473.w | even | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(473, [\chi])\).