Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,4,Mod(100,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("143.100");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 143.e (of order \(3\), 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: | \(8.43727313082\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Relative dimension: | \(17\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
100.1 | −2.72586 | + | 4.72134i | −3.35776 | + | 5.81581i | −10.8607 | − | 18.8112i | 3.97680 | −18.3056 | − | 31.7062i | −5.48891 | − | 9.50707i | 74.8051 | −9.04908 | − | 15.6735i | −10.8402 | + | 18.7758i | ||||
100.2 | −2.65028 | + | 4.59042i | 1.29711 | − | 2.24666i | −10.0480 | − | 17.4036i | −4.49343 | 6.87541 | + | 11.9086i | 16.6322 | + | 28.8078i | 64.1151 | 10.1350 | + | 17.5543i | 11.9088 | − | 20.6267i | ||||
100.3 | −2.35333 | + | 4.07609i | 5.09818 | − | 8.83032i | −7.07634 | − | 12.2566i | 19.7628 | 23.9954 | + | 41.5613i | 1.43260 | + | 2.48133i | 28.9586 | −38.4830 | − | 66.6545i | −46.5084 | + | 80.5548i | ||||
100.4 | −1.99496 | + | 3.45536i | −0.837037 | + | 1.44979i | −3.95969 | − | 6.85839i | −20.8270 | −3.33970 | − | 5.78454i | −4.98381 | − | 8.63221i | −0.321630 | 12.0987 | + | 20.9556i | 41.5489 | − | 71.9649i | ||||
100.5 | −1.39751 | + | 2.42056i | −5.07280 | + | 8.78635i | 0.0939301 | + | 0.162692i | 5.45712 | −14.1786 | − | 24.5580i | −3.86797 | − | 6.69952i | −22.8852 | −37.9666 | − | 65.7601i | −7.62638 | + | 13.2093i | ||||
100.6 | −1.37242 | + | 2.37710i | 2.09604 | − | 3.63045i | 0.232916 | + | 0.403423i | 5.57379 | 5.75330 | + | 9.96502i | 3.44074 | + | 5.95954i | −23.2374 | 4.71323 | + | 8.16355i | −7.64959 | + | 13.2495i | ||||
100.7 | −1.01173 | + | 1.75236i | 3.73606 | − | 6.47104i | 1.95282 | + | 3.38238i | −6.22433 | 7.55973 | + | 13.0938i | −14.2839 | − | 24.7405i | −24.0905 | −14.4162 | − | 24.9696i | 6.29732 | − | 10.9073i | ||||
100.8 | −0.875485 | + | 1.51638i | −1.73778 | + | 3.00992i | 2.46705 | + | 4.27306i | −8.23074 | −3.04279 | − | 5.27028i | 8.98252 | + | 15.5582i | −22.6472 | 7.46026 | + | 12.9216i | 7.20589 | − | 12.4810i | ||||
100.9 | 0.0572303 | − | 0.0991258i | −2.90301 | + | 5.02816i | 3.99345 | + | 6.91686i | 11.2238 | 0.332281 | + | 0.575527i | 7.98278 | + | 13.8266i | 1.82987 | −3.35494 | − | 5.81093i | 0.642343 | − | 1.11257i | ||||
100.10 | 0.274609 | − | 0.475636i | 0.280984 | − | 0.486679i | 3.84918 | + | 6.66698i | 16.1600 | −0.154321 | − | 0.267292i | −15.5667 | − | 26.9624i | 8.62181 | 13.3421 | + | 23.1092i | 4.43769 | − | 7.68630i | ||||
100.11 | 0.535504 | − | 0.927520i | 2.99596 | − | 5.18916i | 3.42647 | + | 5.93482i | 0.973634 | −3.20870 | − | 5.55764i | 14.1596 | + | 24.5251i | 15.9076 | −4.45160 | − | 7.71041i | 0.521385 | − | 0.903066i | ||||
100.12 | 1.23624 | − | 2.14123i | −3.24665 | + | 5.62336i | 0.943432 | + | 1.63407i | −10.7576 | 8.02726 | + | 13.9036i | −12.4880 | − | 21.6299i | 24.4450 | −7.58147 | − | 13.1315i | −13.2990 | + | 23.0345i | ||||
100.13 | 1.58741 | − | 2.74948i | −1.39720 | + | 2.42003i | −1.03977 | − | 1.80093i | −8.68483 | 4.43589 | + | 7.68318i | 3.36772 | + | 5.83306i | 18.7965 | 9.59564 | + | 16.6201i | −13.7864 | + | 23.8788i | ||||
100.14 | 1.60580 | − | 2.78133i | 3.36970 | − | 5.83650i | −1.15719 | − | 2.00431i | 9.24473 | −10.8221 | − | 18.7445i | −4.28214 | − | 7.41689i | 18.2600 | −9.20981 | − | 15.9519i | 14.8452 | − | 25.7126i | ||||
100.15 | 2.09463 | − | 3.62801i | 4.10387 | − | 7.10811i | −4.77499 | − | 8.27053i | −18.8743 | −17.1922 | − | 29.7778i | 7.40609 | + | 12.8277i | −6.49330 | −20.1835 | − | 34.9588i | −39.5349 | + | 68.4764i | ||||
100.16 | 2.30726 | − | 3.99630i | −1.84379 | + | 3.19354i | −6.64694 | − | 11.5128i | 19.7910 | 8.50823 | + | 14.7367i | 9.39345 | + | 16.2699i | −24.4287 | 6.70087 | + | 11.6062i | 45.6630 | − | 79.0906i | ||||
100.17 | 2.68288 | − | 4.64689i | 0.418116 | − | 0.724197i | −10.3957 | − | 18.0059i | 1.92861 | −2.24351 | − | 3.88587i | −8.83613 | − | 15.3046i | −68.6355 | 13.1504 | + | 22.7771i | 5.17423 | − | 8.96203i | ||||
133.1 | −2.72586 | − | 4.72134i | −3.35776 | − | 5.81581i | −10.8607 | + | 18.8112i | 3.97680 | −18.3056 | + | 31.7062i | −5.48891 | + | 9.50707i | 74.8051 | −9.04908 | + | 15.6735i | −10.8402 | − | 18.7758i | ||||
133.2 | −2.65028 | − | 4.59042i | 1.29711 | + | 2.24666i | −10.0480 | + | 17.4036i | −4.49343 | 6.87541 | − | 11.9086i | 16.6322 | − | 28.8078i | 64.1151 | 10.1350 | − | 17.5543i | 11.9088 | + | 20.6267i | ||||
133.3 | −2.35333 | − | 4.07609i | 5.09818 | + | 8.83032i | −7.07634 | + | 12.2566i | 19.7628 | 23.9954 | − | 41.5613i | 1.43260 | − | 2.48133i | 28.9586 | −38.4830 | + | 66.6545i | −46.5084 | − | 80.5548i | ||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 143.4.e.a | ✓ | 34 |
13.c | even | 3 | 1 | inner | 143.4.e.a | ✓ | 34 |
13.c | even | 3 | 1 | 1859.4.a.i | 17 | ||
13.e | even | 6 | 1 | 1859.4.a.f | 17 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
143.4.e.a | ✓ | 34 | 1.a | even | 1 | 1 | trivial |
143.4.e.a | ✓ | 34 | 13.c | even | 3 | 1 | inner |
1859.4.a.f | 17 | 13.e | even | 6 | 1 | ||
1859.4.a.i | 17 | 13.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + 4 T_{2}^{33} + 115 T_{2}^{32} + 354 T_{2}^{31} + 7352 T_{2}^{30} + 19531 T_{2}^{29} + \cdots + 6744367448064 \) acting on \(S_{4}^{\mathrm{new}}(143, [\chi])\).