Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(8,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.bk (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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −1.27716 | + | 2.50657i | 1.72730 | − | 0.128199i | −3.47619 | − | 4.78457i | −0.295036 | + | 2.21652i | −1.88470 | + | 4.49333i | −0.707107 | + | 0.707107i | 10.8754 | − | 1.72250i | 2.96713 | − | 0.442878i | −5.17905 | − | 3.57038i |
8.2 | −1.24682 | + | 2.44703i | 0.890522 | + | 1.48559i | −3.25781 | − | 4.48399i | −1.63146 | − | 1.52917i | −4.74560 | + | 0.326866i | 0.707107 | − | 0.707107i | 9.60928 | − | 1.52196i | −1.41394 | + | 2.64590i | 5.77605 | − | 2.08562i |
8.3 | −1.21549 | + | 2.38554i | −0.0368220 | − | 1.73166i | −3.03780 | − | 4.18118i | 0.817695 | − | 2.08120i | 4.17570 | + | 2.01698i | −0.707107 | + | 0.707107i | 8.37801 | − | 1.32695i | −2.99729 | + | 0.127526i | 3.97087 | + | 4.48032i |
8.4 | −1.14547 | + | 2.24812i | −1.38991 | + | 1.03351i | −2.56637 | − | 3.53230i | 1.60960 | − | 1.55216i | −0.731341 | − | 4.30855i | 0.707107 | − | 0.707107i | 5.89662 | − | 0.933933i | 0.863721 | − | 2.87298i | 1.64569 | + | 5.39653i |
8.5 | −1.13379 | + | 2.22520i | −1.71426 | + | 0.247589i | −2.49044 | − | 3.42779i | 1.16615 | + | 1.90790i | 1.39269 | − | 4.09529i | −0.707107 | + | 0.707107i | 5.51785 | − | 0.873941i | 2.87740 | − | 0.848865i | −5.56763 | + | 0.431745i |
8.6 | −1.09213 | + | 2.14343i | −0.0951717 | − | 1.72943i | −2.22596 | − | 3.06377i | −1.39499 | + | 1.74757i | 3.81085 | + | 1.68477i | 0.707107 | − | 0.707107i | 4.24598 | − | 0.672496i | −2.98188 | + | 0.329187i | −2.22229 | − | 4.89863i |
8.7 | −1.04373 | + | 2.04843i | 1.54603 | − | 0.780894i | −1.93114 | − | 2.65798i | 1.20254 | − | 1.88518i | −0.0140247 | + | 3.98198i | 0.707107 | − | 0.707107i | 2.91887 | − | 0.462304i | 1.78041 | − | 2.41457i | 2.60653 | + | 4.43094i |
8.8 | −1.00540 | + | 1.97320i | −0.446385 | + | 1.67354i | −1.70714 | − | 2.34967i | −2.11390 | + | 0.728982i | −2.85344 | − | 2.56338i | −0.707107 | + | 0.707107i | 1.97810 | − | 0.313301i | −2.60148 | − | 1.49409i | 0.686881 | − | 4.90408i |
8.9 | −0.996070 | + | 1.95490i | 1.41904 | + | 0.993139i | −1.65390 | − | 2.27640i | 2.07107 | + | 0.843023i | −3.35495 | + | 1.78484i | 0.707107 | − | 0.707107i | 1.76348 | − | 0.279307i | 1.02735 | + | 2.81861i | −3.71095 | + | 3.20901i |
8.10 | −0.995192 | + | 1.95317i | −0.685670 | + | 1.59055i | −1.64891 | − | 2.26954i | 0.282336 | + | 2.21817i | −2.42425 | − | 2.92214i | 0.707107 | − | 0.707107i | 1.74356 | − | 0.276153i | −2.05971 | − | 2.18119i | −4.61346 | − | 1.65606i |
8.11 | −0.916279 | + | 1.79830i | 1.53212 | − | 0.807837i | −1.21874 | − | 1.67745i | −2.08851 | − | 0.798830i | 0.0488799 | + | 3.49542i | −0.707107 | + | 0.707107i | 0.146404 | − | 0.0231881i | 1.69480 | − | 2.47541i | 3.35019 | − | 3.02381i |
8.12 | −0.878575 | + | 1.72430i | 1.34260 | + | 1.09426i | −1.02575 | − | 1.41182i | −0.562535 | − | 2.16415i | −3.06642 | + | 1.35366i | −0.707107 | + | 0.707107i | −0.487197 | + | 0.0771645i | 0.605175 | + | 2.93833i | 4.22588 | + | 0.931390i |
8.13 | −0.851523 | + | 1.67121i | −0.524400 | − | 1.65076i | −0.892275 | − | 1.22811i | 2.19833 | + | 0.409071i | 3.20530 | + | 0.529279i | −0.707107 | + | 0.707107i | −0.892875 | + | 0.141417i | −2.45001 | + | 1.73131i | −2.55557 | + | 3.32554i |
8.14 | −0.764791 | + | 1.50099i | −1.73141 | + | 0.0470098i | −0.492485 | − | 0.677848i | 1.36230 | − | 1.77317i | 1.25361 | − | 2.63478i | −0.707107 | + | 0.707107i | −1.93362 | + | 0.306256i | 2.99558 | − | 0.162787i | 1.61963 | + | 3.40090i |
8.15 | −0.672489 | + | 1.31983i | −1.30188 | + | 1.14242i | −0.114150 | − | 0.157113i | −1.72356 | − | 1.42455i | −0.632301 | − | 2.48652i | 0.707107 | − | 0.707107i | −2.64197 | + | 0.418446i | 0.389772 | − | 2.97457i | 3.03924 | − | 1.31682i |
8.16 | −0.649632 | + | 1.27497i | 0.191343 | − | 1.72145i | −0.0279682 | − | 0.0384950i | −1.41872 | − | 1.72836i | 2.07050 | + | 1.36227i | 0.707107 | − | 0.707107i | −2.75939 | + | 0.437045i | −2.92678 | − | 0.658773i | 3.12526 | − | 0.686035i |
8.17 | −0.641887 | + | 1.25977i | 1.23644 | + | 1.21293i | 0.000559265 | 0 | 0.000769763i | 0.0490481 | + | 2.23553i | −2.32168 | + | 0.779075i | −0.707107 | + | 0.707107i | −2.79427 | + | 0.442569i | 0.0575908 | + | 2.99945i | −2.84775 | − | 1.37317i |
8.18 | −0.640516 | + | 1.25708i | −1.19705 | − | 1.25183i | 0.00557249 | + | 0.00766987i | 2.19052 | − | 0.449020i | 2.34038 | − | 0.702977i | 0.707107 | − | 0.707107i | −2.80019 | + | 0.443506i | −0.134142 | + | 2.99700i | −0.838608 | + | 3.04127i |
8.19 | −0.634690 | + | 1.24565i | −1.69875 | − | 0.338004i | 0.0267608 | + | 0.0368331i | 0.131414 | + | 2.23220i | 1.49921 | − | 1.90152i | 0.707107 | − | 0.707107i | −2.82449 | + | 0.447355i | 2.77151 | + | 1.14837i | −2.86395 | − | 1.25306i |
8.20 | −0.537657 | + | 1.05521i | −1.08000 | − | 1.35411i | 0.351174 | + | 0.483349i | −1.77494 | + | 1.35999i | 2.00954 | − | 0.411579i | −0.707107 | + | 0.707107i | −3.03827 | + | 0.481215i | −0.667215 | + | 2.92486i | −0.480765 | − | 2.60415i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.f | odd | 20 | 1 | inner |
75.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.bk.a | ✓ | 480 |
3.b | odd | 2 | 1 | inner | 525.2.bk.a | ✓ | 480 |
25.f | odd | 20 | 1 | inner | 525.2.bk.a | ✓ | 480 |
75.l | even | 20 | 1 | inner | 525.2.bk.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.bk.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
525.2.bk.a | ✓ | 480 | 3.b | odd | 2 | 1 | inner |
525.2.bk.a | ✓ | 480 | 25.f | odd | 20 | 1 | inner |
525.2.bk.a | ✓ | 480 | 75.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).