Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(243,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.243");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.c (of order \(2\), degree \(1\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
243.1 | − | 2.63297i | −2.36979 | −4.93252 | −2.17160 | 6.23957i | 0.610382i | 7.72123i | 2.61589 | 5.71774i | |||||||||||||||||
243.2 | − | 2.62169i | 2.11987 | −4.87326 | −3.20754 | − | 5.55765i | 3.91928i | 7.53280i | 1.49386 | 8.40917i | ||||||||||||||||
243.3 | − | 2.61102i | 0.235196 | −4.81744 | 1.96208 | − | 0.614102i | − | 1.03120i | 7.35640i | −2.94468 | − | 5.12303i | ||||||||||||||
243.4 | − | 2.59649i | −0.654344 | −4.74178 | 3.42486 | 1.69900i | 3.69426i | 7.11902i | −2.57183 | − | 8.89263i | ||||||||||||||||
243.5 | − | 2.55622i | 3.17337 | −4.53424 | 2.75037 | − | 8.11182i | − | 1.58272i | 6.47805i | 7.07029 | − | 7.03053i | ||||||||||||||
243.6 | − | 2.25711i | 0.492207 | −3.09455 | −0.807025 | − | 1.11097i | − | 3.09615i | 2.47053i | −2.75773 | 1.82155i | |||||||||||||||
243.7 | − | 2.24927i | 2.53871 | −3.05919 | 0.673508 | − | 5.71023i | 2.16122i | 2.38241i | 3.44505 | − | 1.51490i | |||||||||||||||
243.8 | − | 2.11254i | −0.282536 | −2.46283 | −0.276653 | 0.596869i | 3.62481i | 0.977749i | −2.92017 | 0.584442i | |||||||||||||||||
243.9 | − | 1.99163i | 2.57151 | −1.96657 | −3.32751 | − | 5.12149i | − | 2.02836i | − | 0.0665784i | 3.61268 | 6.62716i | ||||||||||||||
243.10 | − | 1.91000i | −2.76324 | −1.64808 | −1.89976 | 5.27777i | 4.14984i | − | 0.672161i | 4.63547 | 3.62854i | ||||||||||||||||
243.11 | − | 1.84971i | −2.33988 | −1.42141 | 1.46414 | 4.32808i | 0.831776i | − | 1.07022i | 2.47502 | − | 2.70823i | |||||||||||||||
243.12 | − | 1.79229i | −0.857853 | −1.21231 | −4.20126 | 1.53752i | − | 2.69374i | − | 1.41177i | −2.26409 | 7.52988i | |||||||||||||||
243.13 | − | 1.45158i | 2.14579 | −0.107079 | 1.18384 | − | 3.11478i | − | 1.24262i | − | 2.74772i | 1.60440 | − | 1.71843i | |||||||||||||
243.14 | − | 1.40832i | −1.33929 | 0.0166459 | 0.666179 | 1.88614i | 1.13912i | − | 2.84007i | −1.20632 | − | 0.938191i | |||||||||||||||
243.15 | − | 1.36842i | −0.881379 | 0.127429 | 4.27697 | 1.20610i | − | 1.78127i | − | 2.91121i | −2.22317 | − | 5.85269i | ||||||||||||||
243.16 | − | 1.26529i | 1.92995 | 0.399048 | 4.00045 | − | 2.44194i | 4.74746i | − | 3.03548i | 0.724700 | − | 5.06172i | ||||||||||||||
243.17 | − | 1.15791i | −0.124357 | 0.659243 | 0.927728 | 0.143994i | − | 4.45290i | − | 3.07917i | −2.98454 | − | 1.07423i | ||||||||||||||
243.18 | − | 0.992653i | −3.29818 | 1.01464 | −2.08442 | 3.27394i | − | 2.40684i | − | 2.99249i | 7.87797 | 2.06910i | |||||||||||||||
243.19 | − | 0.915688i | 0.547336 | 1.16152 | −3.25559 | − | 0.501189i | 1.57444i | − | 2.89496i | −2.70042 | 2.98110i | |||||||||||||||
243.20 | − | 0.765140i | 3.12278 | 1.41456 | −0.808157 | − | 2.38936i | − | 0.856237i | − | 2.61262i | 6.75173 | 0.618354i | ||||||||||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.c.a | ✓ | 52 |
61.b | even | 2 | 1 | inner | 671.2.c.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.c.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
671.2.c.a | ✓ | 52 | 61.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).