Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,2,Mod(27,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([7, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("430.27");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 430.r (of order \(28\), 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: | \(3.43356728692\) |
Analytic rank: | \(0\) |
Dimension: | \(264\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 | −0.846724 | − | 0.532032i | −2.43675 | + | 1.53111i | 0.433884 | + | 0.900969i | 0.235258 | − | 2.22366i | 2.87785 | −1.03661 | + | 1.03661i | 0.111964 | − | 0.993712i | 2.29179 | − | 4.75895i | −1.38226 | + | 1.75766i | ||
27.2 | −0.846724 | − | 0.532032i | −2.36912 | + | 1.48862i | 0.433884 | + | 0.900969i | 1.52080 | + | 1.63926i | 2.79798 | 2.44635 | − | 2.44635i | 0.111964 | − | 0.993712i | 2.09509 | − | 4.35051i | −0.415555 | − | 2.19711i | ||
27.3 | −0.846724 | − | 0.532032i | −1.58096 | + | 0.993385i | 0.433884 | + | 0.900969i | −2.11822 | + | 0.716352i | 1.86715 | −0.600259 | + | 0.600259i | 0.111964 | − | 0.993712i | 0.210980 | − | 0.438105i | 2.17467 | + | 0.520406i | ||
27.4 | −0.846724 | − | 0.532032i | −1.25844 | + | 0.790731i | 0.433884 | + | 0.900969i | 1.73201 | − | 1.41427i | 1.48625 | 1.74142 | − | 1.74142i | 0.111964 | − | 0.993712i | −0.343232 | + | 0.712728i | −2.21897 | + | 0.276012i | ||
27.5 | −0.846724 | − | 0.532032i | −0.0459382 | + | 0.0288649i | 0.433884 | + | 0.900969i | −1.90179 | − | 1.17609i | 0.0542540 | −0.499604 | + | 0.499604i | 0.111964 | − | 0.993712i | −1.30037 | + | 2.70025i | 0.984578 | + | 2.00764i | ||
27.6 | −0.846724 | − | 0.532032i | 0.296314 | − | 0.186186i | 0.433884 | + | 0.900969i | −0.486637 | + | 2.18247i | −0.349954 | 0.0102848 | − | 0.0102848i | 0.111964 | − | 0.993712i | −1.24851 | + | 2.59257i | 1.57319 | − | 1.58905i | ||
27.7 | −0.846724 | − | 0.532032i | 0.306612 | − | 0.192657i | 0.433884 | + | 0.900969i | 1.17132 | − | 1.90473i | −0.362116 | −3.59081 | + | 3.59081i | 0.111964 | − | 0.993712i | −1.24476 | + | 2.58476i | −2.00516 | + | 0.989606i | ||
27.8 | −0.846724 | − | 0.532032i | 1.28668 | − | 0.808477i | 0.433884 | + | 0.900969i | 2.21966 | + | 0.270414i | −1.51960 | 2.16897 | − | 2.16897i | 0.111964 | − | 0.993712i | −0.299731 | + | 0.622397i | −1.73557 | − | 1.40989i | ||
27.9 | −0.846724 | − | 0.532032i | 1.93498 | − | 1.21583i | 0.433884 | + | 0.900969i | 0.121846 | − | 2.23275i | −2.28525 | 1.40881 | − | 1.40881i | 0.111964 | − | 0.993712i | 0.964257 | − | 2.00230i | −1.29106 | + | 1.82569i | ||
27.10 | −0.846724 | − | 0.532032i | 1.99342 | − | 1.25255i | 0.433884 | + | 0.900969i | 1.05240 | + | 1.97293i | −2.35427 | −2.68327 | + | 2.68327i | 0.111964 | − | 0.993712i | 1.10318 | − | 2.29079i | 0.158565 | − | 2.23044i | ||
27.11 | −0.846724 | − | 0.532032i | 2.25003 | − | 1.41379i | 0.433884 | + | 0.900969i | −2.02218 | + | 0.954350i | −2.65734 | 2.89927 | − | 2.89927i | 0.111964 | − | 0.993712i | 1.76219 | − | 3.65923i | 2.21997 | + | 0.267794i | ||
27.12 | 0.846724 | + | 0.532032i | −2.83739 | + | 1.78285i | 0.433884 | + | 0.900969i | −0.608113 | − | 2.15179i | −3.35102 | 1.61190 | − | 1.61190i | −0.111964 | + | 0.993712i | 3.57059 | − | 7.41440i | 0.629917 | − | 2.14551i | ||
27.13 | 0.846724 | + | 0.532032i | −1.90831 | + | 1.19907i | 0.433884 | + | 0.900969i | 2.05629 | + | 0.878446i | −2.25376 | 2.27342 | − | 2.27342i | −0.111964 | + | 0.993712i | 0.902233 | − | 1.87351i | 1.27375 | + | 1.83781i | ||
27.14 | 0.846724 | + | 0.532032i | −1.65275 | + | 1.03849i | 0.433884 | + | 0.900969i | −1.53740 | + | 1.62370i | −1.95193 | 0.349245 | − | 0.349245i | −0.111964 | + | 0.993712i | 0.351458 | − | 0.729810i | −2.16561 | + | 0.556884i | ||
27.15 | 0.846724 | + | 0.532032i | −1.05896 | + | 0.665389i | 0.433884 | + | 0.900969i | −1.40404 | − | 1.74031i | −1.25066 | −1.58379 | + | 1.58379i | −0.111964 | + | 0.993712i | −0.622995 | + | 1.29366i | −0.262931 | − | 2.22056i | ||
27.16 | 0.846724 | + | 0.532032i | −0.527328 | + | 0.331342i | 0.433884 | + | 0.900969i | 1.70699 | + | 1.44436i | −0.622786 | −1.95690 | + | 1.95690i | −0.111964 | + | 0.993712i | −1.13336 | + | 2.35345i | 0.676901 | + | 2.13115i | ||
27.17 | 0.846724 | + | 0.532032i | −0.126989 | + | 0.0797926i | 0.433884 | + | 0.900969i | 1.11185 | − | 1.94005i | −0.149977 | −0.830650 | + | 0.830650i | −0.111964 | + | 0.993712i | −1.29189 | + | 2.68264i | 1.97360 | − | 1.05114i | ||
27.18 | 0.846724 | + | 0.532032i | 0.919405 | − | 0.577701i | 0.433884 | + | 0.900969i | −1.69116 | + | 1.46286i | 1.08584 | −2.01843 | + | 2.01843i | −0.111964 | + | 0.993712i | −0.790083 | + | 1.64062i | −2.21024 | + | 0.338890i | ||
27.19 | 0.846724 | + | 0.532032i | 0.969909 | − | 0.609434i | 0.433884 | + | 0.900969i | 1.59901 | − | 1.56306i | 1.14548 | 2.66216 | − | 2.66216i | −0.111964 | + | 0.993712i | −0.732338 | + | 1.52072i | 2.18552 | − | 0.472759i | ||
27.20 | 0.846724 | + | 0.532032i | 1.04232 | − | 0.654933i | 0.433884 | + | 0.900969i | −0.0867089 | + | 2.23439i | 1.23100 | 2.75433 | − | 2.75433i | −0.111964 | + | 0.993712i | −0.644158 | + | 1.33761i | −1.26218 | + | 1.84578i | ||
See next 80 embeddings (of 264 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
43.f | odd | 14 | 1 | inner |
215.r | even | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 430.2.r.a | ✓ | 264 |
5.c | odd | 4 | 1 | inner | 430.2.r.a | ✓ | 264 |
43.f | odd | 14 | 1 | inner | 430.2.r.a | ✓ | 264 |
215.r | even | 28 | 1 | inner | 430.2.r.a | ✓ | 264 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
430.2.r.a | ✓ | 264 | 1.a | even | 1 | 1 | trivial |
430.2.r.a | ✓ | 264 | 5.c | odd | 4 | 1 | inner |
430.2.r.a | ✓ | 264 | 43.f | odd | 14 | 1 | inner |
430.2.r.a | ✓ | 264 | 215.r | even | 28 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).