Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,2,Mod(41,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 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("435.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 435.q (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: | \(3.47349248793\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | −1.88783 | + | 1.88783i | 1.11368 | + | 1.32654i | − | 5.12777i | −1.00000 | −4.60672 | − | 0.401834i | −1.70742 | 5.90468 | + | 5.90468i | −0.519415 | + | 2.95469i | 1.88783 | − | 1.88783i | |||||
41.2 | −1.85345 | + | 1.85345i | −0.633081 | − | 1.61221i | − | 4.87058i | −1.00000 | 4.16154 | + | 1.81476i | −4.54343 | 5.32050 | + | 5.32050i | −2.19842 | + | 2.04131i | 1.85345 | − | 1.85345i | |||||
41.3 | −1.48848 | + | 1.48848i | −1.36422 | + | 1.06719i | − | 2.43117i | −1.00000 | 0.442117 | − | 3.61912i | −1.31013 | 0.641785 | + | 0.641785i | 0.722191 | − | 2.91178i | 1.48848 | − | 1.48848i | |||||
41.4 | −1.48187 | + | 1.48187i | 1.72513 | + | 0.154682i | − | 2.39186i | −1.00000 | −2.78563 | + | 2.32719i | 2.93783 | 0.580678 | + | 0.580678i | 2.95215 | + | 0.533693i | 1.48187 | − | 1.48187i | |||||
41.5 | −1.39967 | + | 1.39967i | 1.02821 | − | 1.39384i | − | 1.91814i | −1.00000 | 0.511758 | + | 3.39006i | −0.338767 | −0.114580 | − | 0.114580i | −0.885569 | − | 2.86632i | 1.39967 | − | 1.39967i | |||||
41.6 | −0.998333 | + | 0.998333i | −1.12890 | − | 1.31362i | 0.00666265i | −1.00000 | 2.43844 | + | 0.184414i | 2.30961 | −2.00332 | − | 2.00332i | −0.451185 | + | 2.96588i | 0.998333 | − | 0.998333i | ||||||
41.7 | −0.602660 | + | 0.602660i | 0.639395 | − | 1.60971i | 1.27360i | −1.00000 | 0.584771 | + | 1.35545i | −0.244261 | −1.97287 | − | 1.97287i | −2.18235 | − | 2.05849i | 0.602660 | − | 0.602660i | ||||||
41.8 | −0.398500 | + | 0.398500i | 1.60347 | + | 0.654884i | 1.68239i | −1.00000 | −0.899957 | + | 0.378013i | −4.31678 | −1.46744 | − | 1.46744i | 2.14225 | + | 2.10018i | 0.398500 | − | 0.398500i | ||||||
41.9 | −0.264256 | + | 0.264256i | −0.639569 | + | 1.60964i | 1.86034i | −1.00000 | −0.256348 | − | 0.594368i | −1.21376 | −1.02012 | − | 1.02012i | −2.18190 | − | 2.05896i | 0.264256 | − | 0.264256i | ||||||
41.10 | −0.203088 | + | 0.203088i | −1.68346 | + | 0.407389i | 1.91751i | −1.00000 | 0.259155 | − | 0.424627i | 4.71926 | −0.795601 | − | 0.795601i | 2.66807 | − | 1.37164i | 0.203088 | − | 0.203088i | ||||||
41.11 | 0.0236294 | − | 0.0236294i | 1.39231 | + | 1.03028i | 1.99888i | −1.00000 | 0.0572444 | − | 0.00855463i | 3.42666 | 0.0944913 | + | 0.0944913i | 0.877056 | + | 2.86893i | −0.0236294 | + | 0.0236294i | ||||||
41.12 | 0.602690 | − | 0.602690i | −1.24496 | − | 1.20419i | 1.27353i | −1.00000 | −1.47608 | + | 0.0245761i | 0.173231 | 1.97292 | + | 1.97292i | 0.0998697 | + | 2.99834i | −0.602690 | + | 0.602690i | ||||||
41.13 | 0.780332 | − | 0.780332i | 1.66535 | − | 0.476047i | 0.782165i | −1.00000 | 0.928048 | − | 1.67100i | 0.394466 | 2.17101 | + | 2.17101i | 2.54676 | − | 1.58557i | −0.780332 | + | 0.780332i | ||||||
41.14 | 1.00351 | − | 1.00351i | 0.602757 | − | 1.62379i | − | 0.0140454i | −1.00000 | −1.02461 | − | 2.23435i | 4.09600 | 1.99292 | + | 1.99292i | −2.27337 | − | 1.95750i | −1.00351 | + | 1.00351i | |||||
41.15 | 1.05880 | − | 1.05880i | 0.369716 | + | 1.69213i | − | 0.242102i | −1.00000 | 2.18308 | + | 1.40017i | 1.34177 | 1.86126 | + | 1.86126i | −2.72662 | + | 1.25122i | −1.05880 | + | 1.05880i | |||||
41.16 | 1.34488 | − | 1.34488i | −1.26804 | + | 1.17986i | − | 1.61741i | −1.00000 | −0.118585 | + | 3.29214i | −4.79407 | 0.514537 | + | 0.514537i | 0.215844 | − | 2.99223i | −1.34488 | + | 1.34488i | |||||
41.17 | 1.82725 | − | 1.82725i | 1.68768 | − | 0.389527i | − | 4.67770i | −1.00000 | 2.37206 | − | 3.79558i | 1.47752 | −4.89284 | − | 4.89284i | 2.69654 | − | 1.31480i | −1.82725 | + | 1.82725i | |||||
41.18 | 1.93705 | − | 1.93705i | −0.865476 | + | 1.50032i | − | 5.50431i | −1.00000 | 1.22972 | + | 4.58265i | 1.59227 | −6.78801 | − | 6.78801i | −1.50190 | − | 2.59698i | −1.93705 | + | 1.93705i | |||||
191.1 | −1.88783 | − | 1.88783i | 1.11368 | − | 1.32654i | 5.12777i | −1.00000 | −4.60672 | + | 0.401834i | −1.70742 | 5.90468 | − | 5.90468i | −0.519415 | − | 2.95469i | 1.88783 | + | 1.88783i | ||||||
191.2 | −1.85345 | − | 1.85345i | −0.633081 | + | 1.61221i | 4.87058i | −1.00000 | 4.16154 | − | 1.81476i | −4.54343 | 5.32050 | − | 5.32050i | −2.19842 | − | 2.04131i | 1.85345 | + | 1.85345i | ||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
87.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 435.2.q.c | ✓ | 36 |
3.b | odd | 2 | 1 | 435.2.q.d | yes | 36 | |
29.c | odd | 4 | 1 | 435.2.q.d | yes | 36 | |
87.f | even | 4 | 1 | inner | 435.2.q.c | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
435.2.q.c | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
435.2.q.c | ✓ | 36 | 87.f | even | 4 | 1 | inner |
435.2.q.d | yes | 36 | 3.b | odd | 2 | 1 | |
435.2.q.d | yes | 36 | 29.c | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 4 T_{2}^{35} + 8 T_{2}^{34} + 4 T_{2}^{33} + 125 T_{2}^{32} + 500 T_{2}^{31} + 1008 T_{2}^{30} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\).