Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,4,Mod(73,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("123.73");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 123.e (of order \(4\), 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: | \(7.25723493071\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 | − | 5.61902i | −2.12132 | + | 2.12132i | −23.5733 | − | 13.3921i | 11.9197 | + | 11.9197i | −10.6525 | + | 10.6525i | 87.5068i | − | 9.00000i | −75.2505 | |||||||||
73.2 | − | 5.25757i | 2.12132 | − | 2.12132i | −19.6420 | 21.1469i | −11.1530 | − | 11.1530i | −8.38626 | + | 8.38626i | 61.2088i | − | 9.00000i | 111.181 | ||||||||||
73.3 | − | 4.46476i | 2.12132 | − | 2.12132i | −11.9341 | − | 8.04569i | −9.47119 | − | 9.47119i | 16.4598 | − | 16.4598i | 17.5647i | − | 9.00000i | −35.9221 | |||||||||
73.4 | − | 4.24235i | −2.12132 | + | 2.12132i | −9.99752 | 16.7806i | 8.99938 | + | 8.99938i | 8.84825 | − | 8.84825i | 8.47416i | − | 9.00000i | 71.1889 | ||||||||||
73.5 | − | 4.04342i | 2.12132 | − | 2.12132i | −8.34925 | − | 13.7007i | −8.57739 | − | 8.57739i | −23.0314 | + | 23.0314i | 1.41217i | − | 9.00000i | −55.3977 | |||||||||
73.6 | − | 3.81117i | −2.12132 | + | 2.12132i | −6.52505 | − | 7.20336i | 8.08472 | + | 8.08472i | 18.1690 | − | 18.1690i | − | 5.62130i | − | 9.00000i | −27.4532 | ||||||||
73.7 | − | 2.63096i | 2.12132 | − | 2.12132i | 1.07806 | 9.61057i | −5.58111 | − | 5.58111i | 6.96584 | − | 6.96584i | − | 23.8840i | − | 9.00000i | 25.2850 | |||||||||
73.8 | − | 2.16950i | −2.12132 | + | 2.12132i | 3.29326 | − | 7.68303i | 4.60221 | + | 4.60221i | −9.55652 | + | 9.55652i | − | 24.5008i | − | 9.00000i | −16.6683 | ||||||||
73.9 | − | 1.23251i | −2.12132 | + | 2.12132i | 6.48091 | 10.4273i | 2.61456 | + | 2.61456i | −9.33905 | + | 9.33905i | − | 17.8479i | − | 9.00000i | 12.8517 | |||||||||
73.10 | − | 0.702755i | 2.12132 | − | 2.12132i | 7.50614 | − | 13.0908i | −1.49077 | − | 1.49077i | 4.57630 | − | 4.57630i | − | 10.8970i | − | 9.00000i | −9.19959 | ||||||||
73.11 | 0.335020i | −2.12132 | + | 2.12132i | 7.88776 | − | 3.72745i | −0.710685 | − | 0.710685i | 11.7913 | − | 11.7913i | 5.32272i | − | 9.00000i | 1.24877 | ||||||||||
73.12 | 0.406748i | 2.12132 | − | 2.12132i | 7.83456 | − | 5.41007i | 0.862843 | + | 0.862843i | 4.20418 | − | 4.20418i | 6.44068i | − | 9.00000i | 2.20054 | ||||||||||
73.13 | 0.926424i | 2.12132 | − | 2.12132i | 7.14174 | 15.2132i | 1.96524 | + | 1.96524i | −14.5651 | + | 14.5651i | 14.0277i | − | 9.00000i | −14.0939 | |||||||||||
73.14 | 1.95804i | −2.12132 | + | 2.12132i | 4.16607 | − | 19.9285i | −4.15363 | − | 4.15363i | −25.5597 | + | 25.5597i | 23.8217i | − | 9.00000i | 39.0208 | ||||||||||
73.15 | 2.33516i | −2.12132 | + | 2.12132i | 2.54705 | 14.1999i | −4.95361 | − | 4.95361i | −0.414133 | + | 0.414133i | 24.6290i | − | 9.00000i | −33.1590 | |||||||||||
73.16 | 2.68829i | 2.12132 | − | 2.12132i | 0.773103 | 10.5013i | 5.70272 | + | 5.70272i | 23.9191 | − | 23.9191i | 23.5846i | − | 9.00000i | −28.2306 | |||||||||||
73.17 | 3.35016i | 2.12132 | − | 2.12132i | −3.22354 | − | 17.1459i | 7.10675 | + | 7.10675i | −8.67318 | + | 8.67318i | 16.0019i | − | 9.00000i | 57.4414 | ||||||||||
73.18 | 3.47014i | −2.12132 | + | 2.12132i | −4.04185 | 12.6171i | −7.36127 | − | 7.36127i | 1.00923 | − | 1.00923i | 13.7353i | − | 9.00000i | −43.7829 | |||||||||||
73.19 | 3.73883i | −2.12132 | + | 2.12132i | −5.97887 | − | 16.5635i | −7.93126 | − | 7.93126i | 18.0717 | − | 18.0717i | 7.55667i | − | 9.00000i | 61.9282 | ||||||||||
73.20 | 4.27333i | 2.12132 | − | 2.12132i | −10.2613 | 0.477644i | 9.06509 | + | 9.06509i | −14.6412 | + | 14.6412i | − | 9.66333i | − | 9.00000i | −2.04113 | ||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.c | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 123.4.e.a | ✓ | 44 |
41.c | even | 4 | 1 | inner | 123.4.e.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
123.4.e.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
123.4.e.a | ✓ | 44 | 41.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(123, [\chi])\).