Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,4,Mod(3,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([24, 10]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("143.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 143.q (of order \(15\), 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: | \(8.43727313082\) |
Analytic rank: | \(0\) |
Dimension: | \(320\) |
Relative dimension: | \(40\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −5.10212 | − | 2.27161i | 1.38169 | − | 1.53452i | 15.5183 | + | 17.2349i | −0.783047 | − | 0.568917i | −10.5353 | + | 4.69064i | −13.2389 | − | 14.7033i | −26.2187 | − | 80.6929i | 2.37658 | + | 22.6117i | 2.70284 | + | 4.68145i |
3.2 | −4.67876 | − | 2.08312i | 6.53114 | − | 7.25356i | 12.1984 | + | 13.5477i | −0.321594 | − | 0.233652i | −45.6677 | + | 20.3325i | 19.0762 | + | 21.1863i | −16.1907 | − | 49.8299i | −7.13615 | − | 67.8960i | 1.01794 | + | 1.76312i |
3.3 | −4.47044 | − | 1.99037i | −5.27677 | + | 5.86045i | 10.6702 | + | 11.8505i | 11.7028 | + | 8.50257i | 35.2539 | − | 15.6961i | 10.5278 | + | 11.6923i | −12.0164 | − | 36.9826i | −3.67827 | − | 34.9964i | −35.3934 | − | 61.3031i |
3.4 | −4.29137 | − | 1.91064i | −0.857939 | + | 0.952838i | 9.41224 | + | 10.4533i | −6.11754 | − | 4.44465i | 5.50226 | − | 2.44976i | 3.47282 | + | 3.85696i | −8.80595 | − | 27.1019i | 2.65043 | + | 25.2171i | 17.7605 | + | 30.7620i |
3.5 | −4.23464 | − | 1.88538i | 0.592838 | − | 0.658414i | 9.02446 | + | 10.0227i | 14.0266 | + | 10.1910i | −3.75182 | + | 1.67042i | 6.56606 | + | 7.29235i | −7.85943 | − | 24.1888i | 2.74022 | + | 26.0714i | −40.1839 | − | 69.6006i |
3.6 | −4.06062 | − | 1.80790i | −3.81539 | + | 4.23742i | 7.86706 | + | 8.73726i | −15.6413 | − | 11.3641i | 23.1537 | − | 10.3087i | 19.4235 | + | 21.5720i | −5.16062 | − | 15.8827i | −0.576259 | − | 5.48274i | 42.9683 | + | 74.4232i |
3.7 | −3.96685 | − | 1.76615i | −6.00386 | + | 6.66796i | 7.26353 | + | 8.06697i | −4.50154 | − | 3.27056i | 35.5930 | − | 15.8470i | −18.3573 | − | 20.3879i | −3.83117 | − | 11.7911i | −5.59310 | − | 53.2148i | 12.0806 | + | 20.9242i |
3.8 | −3.83132 | − | 1.70581i | 3.43033 | − | 3.80977i | 6.41615 | + | 7.12585i | −11.3449 | − | 8.24252i | −19.6414 | + | 8.74492i | −6.22750 | − | 6.91634i | −2.05903 | − | 6.33706i | 0.0751056 | + | 0.714582i | 29.4055 | + | 50.9319i |
3.9 | −3.42768 | − | 1.52610i | 5.60413 | − | 6.22401i | 4.06695 | + | 4.51680i | 12.9835 | + | 9.43308i | −28.7076 | + | 12.7815i | −18.5875 | − | 20.6435i | 2.22852 | + | 6.85867i | −4.50984 | − | 42.9083i | −30.1075 | − | 52.1477i |
3.10 | −3.20952 | − | 1.42897i | −1.99079 | + | 2.21099i | 2.90603 | + | 3.22747i | 4.33588 | + | 3.15020i | 9.54892 | − | 4.25145i | −17.9899 | − | 19.9798i | 3.97026 | + | 12.2192i | 1.89701 | + | 18.0489i | −9.41455 | − | 16.3065i |
3.11 | −2.70930 | − | 1.20626i | 3.54938 | − | 3.94198i | 0.532200 | + | 0.591068i | 2.89861 | + | 2.10596i | −14.3714 | + | 6.39855i | 13.2416 | + | 14.7062i | 6.60270 | + | 20.3210i | −0.118883 | − | 1.13109i | −5.31287 | − | 9.20216i |
3.12 | −2.22111 | − | 0.988904i | −0.129116 | + | 0.143398i | −1.39763 | − | 1.55222i | −4.65106 | − | 3.37919i | 0.428589 | − | 0.190820i | 13.4245 | + | 14.9094i | 7.57983 | + | 23.3283i | 2.81838 | + | 26.8151i | 6.98884 | + | 12.1050i |
3.13 | −2.18439 | − | 0.972553i | −3.69451 | + | 4.10317i | −1.52734 | − | 1.69629i | 5.70526 | + | 4.14511i | 12.0608 | − | 5.36982i | −1.74948 | − | 1.94300i | 7.59774 | + | 23.3834i | −0.364326 | − | 3.46633i | −8.43117 | − | 14.6032i |
3.14 | −2.05383 | − | 0.914426i | 4.63025 | − | 5.14241i | −1.97098 | − | 2.18900i | −5.67493 | − | 4.12307i | −14.2121 | + | 6.32764i | −11.4861 | − | 12.7566i | 7.60426 | + | 23.4035i | −2.18293 | − | 20.7692i | 7.88511 | + | 13.6574i |
3.15 | −1.47923 | − | 0.658595i | −5.15589 | + | 5.72620i | −3.59867 | − | 3.99673i | 6.90200 | + | 5.01460i | 11.3980 | − | 5.07471i | 17.3187 | + | 19.2343i | 6.69396 | + | 20.6019i | −3.38386 | − | 32.1953i | −6.90705 | − | 11.9634i |
3.16 | −1.34373 | − | 0.598266i | −0.820388 | + | 0.911134i | −3.90536 | − | 4.33735i | −16.3994 | − | 11.9149i | 1.64748 | − | 0.733505i | −14.7799 | − | 16.4148i | 6.28910 | + | 19.3559i | 2.66514 | + | 25.3571i | 14.9081 | + | 25.8216i |
3.17 | −0.978817 | − | 0.435797i | −5.19277 | + | 5.76716i | −4.58488 | − | 5.09203i | −11.4291 | − | 8.30375i | 7.59609 | − | 3.38200i | −1.86862 | − | 2.07532i | 4.91743 | + | 15.1343i | −3.47296 | − | 33.0430i | 7.56827 | + | 13.1086i |
3.18 | −0.868356 | − | 0.386617i | 2.30554 | − | 2.56056i | −4.74848 | − | 5.27372i | 13.8209 | + | 10.0415i | −2.99199 | + | 1.33212i | 9.48972 | + | 10.5394i | 4.43431 | + | 13.6474i | 1.58131 | + | 15.0451i | −8.11928 | − | 14.0630i |
3.19 | −0.437809 | − | 0.194925i | −1.51751 | + | 1.68536i | −5.19936 | − | 5.77448i | 16.7692 | + | 12.1836i | 0.992898 | − | 0.442067i | −15.7902 | − | 17.5368i | 2.33549 | + | 7.18789i | 2.28465 | + | 21.7370i | −4.96684 | − | 8.60281i |
3.20 | −0.257195 | − | 0.114511i | 6.30663 | − | 7.00422i | −5.30001 | − | 5.88626i | −15.3363 | − | 11.1425i | −2.42409 | + | 1.07927i | 15.1685 | + | 16.8463i | 1.38509 | + | 4.26287i | −6.46326 | − | 61.4939i | 2.66849 | + | 4.62197i |
See next 80 embeddings (of 320 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
13.c | even | 3 | 1 | inner |
143.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 143.4.q.a | ✓ | 320 |
11.c | even | 5 | 1 | inner | 143.4.q.a | ✓ | 320 |
13.c | even | 3 | 1 | inner | 143.4.q.a | ✓ | 320 |
143.q | even | 15 | 1 | inner | 143.4.q.a | ✓ | 320 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
143.4.q.a | ✓ | 320 | 1.a | even | 1 | 1 | trivial |
143.4.q.a | ✓ | 320 | 11.c | even | 5 | 1 | inner |
143.4.q.a | ✓ | 320 | 13.c | even | 3 | 1 | inner |
143.4.q.a | ✓ | 320 | 143.q | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(143, [\chi])\).