Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1205,2,Mod(724,1205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1205.724");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1205 = 5 \cdot 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1205.b (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: | \(9.62197344356\) |
Analytic rank: | \(0\) |
Dimension: | \(46\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
724.1 | − | 2.68673i | − | 2.58072i | −5.21850 | 1.90521 | + | 1.17055i | −6.93370 | 1.24945i | 8.64724i | −3.66013 | 3.14494 | − | 5.11877i | ||||||||||||
724.2 | − | 2.67602i | 0.199289i | −5.16108 | −2.13713 | + | 0.657778i | 0.533301 | 2.87374i | 8.45912i | 2.96028 | 1.76023 | + | 5.71900i | |||||||||||||
724.3 | − | 2.49572i | 1.66439i | −4.22863 | 2.11925 | − | 0.713293i | 4.15385 | 4.24911i | 5.56203i | 0.229807 | −1.78018 | − | 5.28905i | |||||||||||||
724.4 | − | 2.44838i | − | 0.992209i | −3.99457 | 0.553587 | − | 2.16646i | −2.42931 | − | 0.744704i | 4.88345i | 2.01552 | −5.30431 | − | 1.35539i | |||||||||||
724.5 | − | 2.36215i | − | 2.05253i | −3.57974 | −0.618559 | + | 2.14881i | −4.84838 | 3.07728i | 3.73158i | −1.21289 | 5.07581 | + | 1.46113i | ||||||||||||
724.6 | − | 1.95888i | − | 0.350465i | −1.83721 | −1.30176 | − | 1.81808i | −0.686519 | 1.14869i | − | 0.318879i | 2.87717 | −3.56141 | + | 2.54999i | |||||||||||
724.7 | − | 1.92263i | − | 0.557567i | −1.69651 | −2.01334 | + | 0.972856i | −1.07200 | − | 0.976687i | − | 0.583491i | 2.68912 | 1.87044 | + | 3.87092i | ||||||||||
724.8 | − | 1.88286i | 2.55671i | −1.54517 | −0.979915 | − | 2.00992i | 4.81392 | 3.51799i | − | 0.856390i | −3.53675 | −3.78439 | + | 1.84504i | ||||||||||||
724.9 | − | 1.84209i | 1.69681i | −1.39329 | 1.66065 | + | 1.49741i | 3.12567 | 1.54055i | − | 1.11761i | 0.120844 | 2.75836 | − | 3.05907i | ||||||||||||
724.10 | − | 1.83307i | 2.40621i | −1.36013 | −1.46796 | + | 1.68674i | 4.41073 | − | 0.302396i | − | 1.17293i | −2.78983 | 3.09191 | + | 2.69086i | |||||||||||
724.11 | − | 1.70963i | 1.38336i | −0.922819 | −1.53605 | − | 1.62498i | 2.36503 | − | 5.05937i | − | 1.84158i | 1.08631 | −2.77811 | + | 2.62606i | |||||||||||
724.12 | − | 1.49203i | − | 3.12646i | −0.226139 | 0.643201 | − | 2.14156i | −4.66475 | 4.66024i | − | 2.64665i | −6.77474 | −3.19527 | − | 0.959673i | |||||||||||
724.13 | − | 1.45267i | − | 2.41257i | −0.110263 | −1.02000 | − | 1.98987i | −3.50468 | − | 1.25057i | − | 2.74517i | −2.82049 | −2.89064 | + | 1.48173i | ||||||||||
724.14 | − | 1.30874i | 1.88426i | 0.287190 | 2.21635 | − | 0.296297i | 2.46601 | 1.28779i | − | 2.99335i | −0.550432 | −0.387776 | − | 2.90063i | ||||||||||||
724.15 | − | 1.09514i | 0.223927i | 0.800662 | −1.50518 | + | 1.65361i | 0.245232 | − | 2.26365i | − | 3.06713i | 2.94986 | 1.81094 | + | 1.64839i | |||||||||||
724.16 | − | 0.930197i | − | 1.32621i | 1.13473 | −2.23573 | − | 0.0386721i | −1.23364 | − | 4.33497i | − | 2.91592i | 1.24116 | −0.0359727 | + | 2.07967i | ||||||||||
724.17 | − | 0.779313i | 2.96208i | 1.39267 | 2.14053 | − | 0.646643i | 2.30839 | − | 0.937135i | − | 2.64395i | −5.77395 | −0.503937 | − | 1.66814i | |||||||||||
724.18 | − | 0.760721i | − | 1.12938i | 1.42130 | 1.88699 | + | 1.19969i | −0.859145 | 2.53853i | − | 2.60266i | 1.72450 | 0.912633 | − | 1.43547i | |||||||||||
724.19 | − | 0.636788i | − | 2.24729i | 1.59450 | −1.29399 | + | 1.82362i | −1.43105 | 1.14493i | − | 2.28894i | −2.05030 | 1.16126 | + | 0.824000i | |||||||||||
724.20 | − | 0.434323i | 1.75836i | 1.81136 | 0.971846 | − | 2.01383i | 0.763695 | 2.56007i | − | 1.65536i | −0.0918227 | −0.874653 | − | 0.422095i | ||||||||||||
See all 46 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1205.2.b.c | ✓ | 46 |
5.b | even | 2 | 1 | inner | 1205.2.b.c | ✓ | 46 |
5.c | odd | 4 | 2 | 6025.2.a.p | 46 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1205.2.b.c | ✓ | 46 | 1.a | even | 1 | 1 | trivial |
1205.2.b.c | ✓ | 46 | 5.b | even | 2 | 1 | inner |
6025.2.a.p | 46 | 5.c | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} + 63 T_{2}^{44} + 1836 T_{2}^{42} + 32871 T_{2}^{40} + 404962 T_{2}^{38} + 3644147 T_{2}^{36} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\).