Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,2,Mod(237,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 15, 5, 18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.237");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 880.cg (of order \(20\), 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: | \(7.02683537787\) |
Analytic rank: | \(0\) |
Dimension: | \(1120\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
237.1 | −1.41407 | − | 0.0198985i | −1.45345 | − | 1.05599i | 1.99921 | + | 0.0562758i | −0.196156 | + | 2.22745i | 2.03427 | + | 1.52217i | 1.02883 | + | 0.162951i | −2.82591 | − | 0.119359i | 0.0703399 | + | 0.216484i | 0.321702 | − | 3.14587i |
237.2 | −1.41096 | + | 0.0958386i | 1.41736 | + | 1.02977i | 1.98163 | − | 0.270449i | 1.47958 | + | 1.67656i | −2.09853 | − | 1.31713i | 4.63365 | + | 0.733898i | −2.77009 | + | 0.571511i | 0.0214232 | + | 0.0659338i | −2.24830 | − | 2.22376i |
237.3 | −1.40836 | − | 0.128540i | −0.487010 | − | 0.353834i | 1.96695 | + | 0.362061i | 2.23512 | − | 0.0651494i | 0.640404 | + | 0.560925i | −5.04818 | − | 0.799553i | −2.72364 | − | 0.762745i | −0.815070 | − | 2.50853i | −3.15623 | − | 0.195548i |
237.4 | −1.40170 | + | 0.187685i | −0.902146 | − | 0.655448i | 1.92955 | − | 0.526158i | −1.09161 | − | 1.95151i | 1.38756 | + | 0.749424i | −2.96306 | − | 0.469302i | −2.60590 | + | 1.09967i | −0.542795 | − | 1.67055i | 1.89638 | + | 2.53056i |
237.5 | −1.40154 | − | 0.188892i | 1.67325 | + | 1.21569i | 1.92864 | + | 0.529481i | 1.55118 | − | 1.61054i | −2.11550 | − | 2.01990i | 0.977944 | + | 0.154891i | −2.60305 | − | 1.10640i | 0.394824 | + | 1.21514i | −2.47826 | + | 1.96423i |
237.6 | −1.39666 | − | 0.222097i | −0.422825 | − | 0.307201i | 1.90135 | + | 0.620391i | −0.105986 | − | 2.23355i | 0.522317 | + | 0.522965i | 4.02525 | + | 0.637538i | −2.51775 | − | 1.28876i | −0.842642 | − | 2.59339i | −0.348040 | + | 3.14307i |
237.7 | −1.39433 | + | 0.236295i | −2.08493 | − | 1.51479i | 1.88833 | − | 0.658947i | −0.842305 | + | 2.07136i | 3.26503 | + | 1.61947i | −0.0369532 | − | 0.00585281i | −2.47725 | + | 1.36499i | 1.12529 | + | 3.46330i | 0.685003 | − | 3.08719i |
237.8 | −1.38816 | + | 0.270193i | −0.172111 | − | 0.125046i | 1.85399 | − | 0.750144i | 1.95249 | + | 1.08986i | 0.272704 | + | 0.127081i | 0.496601 | + | 0.0786539i | −2.37096 | + | 1.54226i | −0.913065 | − | 2.81013i | −3.00484 | − | 0.985354i |
237.9 | −1.38601 | − | 0.281012i | 2.10621 | + | 1.53025i | 1.84206 | + | 0.778972i | −1.47245 | + | 1.68282i | −2.48922 | − | 2.71282i | 2.08045 | + | 0.329510i | −2.33423 | − | 1.59731i | 1.16740 | + | 3.59290i | 2.51373 | − | 1.91864i |
237.10 | −1.38334 | − | 0.293885i | 2.51165 | + | 1.82482i | 1.82726 | + | 0.813087i | −1.94695 | − | 1.09971i | −2.93818 | − | 3.26249i | −2.63309 | − | 0.417041i | −2.28877 | − | 1.66178i | 2.05137 | + | 6.31348i | 2.37011 | + | 2.09346i |
237.11 | −1.36973 | + | 0.351921i | −2.62600 | − | 1.90790i | 1.75230 | − | 0.964072i | 1.57452 | − | 1.58773i | 4.26833 | + | 1.68916i | 0.0803937 | + | 0.0127331i | −2.06090 | + | 1.93719i | 2.32873 | + | 7.16710i | −1.59790 | + | 2.72887i |
237.12 | −1.36711 | + | 0.361964i | −1.06810 | − | 0.776017i | 1.73796 | − | 0.989688i | 1.47475 | − | 1.68081i | 1.74109 | + | 0.674286i | 2.78066 | + | 0.440414i | −2.01775 | + | 1.98209i | −0.388425 | − | 1.19545i | −1.40775 | + | 2.83165i |
237.13 | −1.36287 | + | 0.377621i | 0.965913 | + | 0.701777i | 1.71481 | − | 1.02929i | −2.19119 | − | 0.445737i | −1.58141 | − | 0.591679i | −0.698960 | − | 0.110704i | −1.94837 | + | 2.05033i | −0.486554 | − | 1.49746i | 3.15462 | − | 0.219960i |
237.14 | −1.35870 | − | 0.392342i | −2.14022 | − | 1.55496i | 1.69213 | + | 1.06615i | 2.23480 | + | 0.0754042i | 2.29784 | + | 2.95242i | 0.486486 | + | 0.0770518i | −1.88081 | − | 2.11248i | 1.23559 | + | 3.80274i | −3.00683 | − | 0.979257i |
237.15 | −1.35823 | + | 0.393971i | 0.454023 | + | 0.329867i | 1.68957 | − | 1.07021i | −2.18674 | − | 0.467100i | −0.746625 | − | 0.269163i | 1.07630 | + | 0.170470i | −1.87320 | + | 2.11923i | −0.829726 | − | 2.55364i | 3.15411 | − | 0.227083i |
237.16 | −1.35284 | − | 0.412093i | 0.989602 | + | 0.718988i | 1.66036 | + | 1.11499i | 0.266189 | − | 2.22017i | −1.04249 | − | 1.38048i | −3.25032 | − | 0.514800i | −1.78672 | − | 2.19263i | −0.464682 | − | 1.43014i | −1.27503 | + | 2.89384i |
237.17 | −1.33558 | − | 0.465005i | −0.657131 | − | 0.477434i | 1.56754 | + | 1.24210i | −2.22153 | + | 0.254585i | 0.655641 | + | 0.943219i | 1.88906 | + | 0.299198i | −1.51599 | − | 2.38784i | −0.723173 | − | 2.22570i | 3.08541 | + | 0.693003i |
237.18 | −1.32114 | + | 0.504581i | 1.26975 | + | 0.922525i | 1.49080 | − | 1.33324i | −0.0549814 | + | 2.23539i | −2.14299 | − | 0.578089i | −4.67509 | − | 0.740462i | −1.29681 | + | 2.51362i | −0.165847 | − | 0.510425i | −1.05530 | − | 2.98100i |
237.19 | −1.31878 | + | 0.510715i | 2.48263 | + | 1.80374i | 1.47834 | − | 1.34704i | 2.01503 | + | 0.969365i | −4.19523 | − | 1.11081i | −1.58504 | − | 0.251045i | −1.26165 | + | 2.53145i | 1.98294 | + | 6.10288i | −3.15244 | − | 0.249273i |
237.20 | −1.31197 | − | 0.527959i | −0.695819 | − | 0.505542i | 1.44252 | + | 1.38533i | −1.93533 | + | 1.12004i | 0.645987 | + | 1.03062i | −3.08510 | − | 0.488632i | −1.16114 | − | 2.57910i | −0.698459 | − | 2.14964i | 3.13043 | − | 0.447677i |
See next 80 embeddings (of 1120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
80.t | odd | 4 | 1 | inner |
880.cg | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 880.2.cg.a | ✓ | 1120 |
5.c | odd | 4 | 1 | 880.2.cy.a | yes | 1120 | |
11.d | odd | 10 | 1 | inner | 880.2.cg.a | ✓ | 1120 |
16.e | even | 4 | 1 | 880.2.cy.a | yes | 1120 | |
55.l | even | 20 | 1 | 880.2.cy.a | yes | 1120 | |
80.t | odd | 4 | 1 | inner | 880.2.cg.a | ✓ | 1120 |
176.u | odd | 20 | 1 | 880.2.cy.a | yes | 1120 | |
880.cg | even | 20 | 1 | inner | 880.2.cg.a | ✓ | 1120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
880.2.cg.a | ✓ | 1120 | 1.a | even | 1 | 1 | trivial |
880.2.cg.a | ✓ | 1120 | 11.d | odd | 10 | 1 | inner |
880.2.cg.a | ✓ | 1120 | 80.t | odd | 4 | 1 | inner |
880.2.cg.a | ✓ | 1120 | 880.cg | even | 20 | 1 | inner |
880.2.cy.a | yes | 1120 | 5.c | odd | 4 | 1 | |
880.2.cy.a | yes | 1120 | 16.e | even | 4 | 1 | |
880.2.cy.a | yes | 1120 | 55.l | even | 20 | 1 | |
880.2.cy.a | yes | 1120 | 176.u | odd | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).