Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(10,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([11, 45]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.x (of order \(66\), degree \(20\), 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: | \(6.76332905120\) |
Analytic rank: | \(0\) |
Dimension: | \(1720\) |
Relative dimension: | \(86\) over \(\Q(\zeta_{66})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{66}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −2.81634 | − | 0.134159i | −0.868367 | − | 0.501352i | 5.92282 | + | 0.565561i | 0.682147 | + | 0.867421i | 2.37835 | + | 1.52848i | −2.60519 | − | 0.461517i | −11.0231 | − | 1.58489i | −0.997293 | − | 1.72736i | −1.80478 | − | 2.53447i |
10.2 | −2.75726 | − | 0.131345i | −2.59140 | − | 1.49614i | 5.59430 | + | 0.534191i | −1.91302 | − | 2.43261i | 6.94865 | + | 4.46563i | 2.39759 | + | 1.11874i | −9.89022 | − | 1.42200i | 2.97689 | + | 5.15612i | 4.95520 | + | 6.95860i |
10.3 | −2.66188 | − | 0.126801i | 2.74138 | + | 1.58273i | 5.07858 | + | 0.484946i | −2.17682 | − | 2.76805i | −7.09652 | − | 4.56066i | −2.23027 | − | 1.42333i | −8.18154 | − | 1.17633i | 3.51009 | + | 6.07966i | 5.44344 | + | 7.64425i |
10.4 | −2.60759 | − | 0.124215i | −0.156257 | − | 0.0902149i | 4.79315 | + | 0.457690i | 1.55659 | + | 1.97937i | 0.396247 | + | 0.254653i | 2.62842 | − | 0.302344i | −7.27376 | − | 1.04581i | −1.48372 | − | 2.56988i | −3.81308 | − | 5.35472i |
10.5 | −2.53212 | − | 0.120620i | 2.12284 | + | 1.22562i | 4.40615 | + | 0.420736i | 0.698990 | + | 0.888838i | −5.22747 | − | 3.35949i | 1.99423 | − | 1.73870i | −6.08778 | − | 0.875291i | 1.50431 | + | 2.60554i | −1.66272 | − | 2.33496i |
10.6 | −2.51092 | − | 0.119610i | −0.674633 | − | 0.389500i | 4.29946 | + | 0.410548i | −1.02626 | − | 1.30499i | 1.64736 | + | 1.05869i | 0.282243 | − | 2.63065i | −5.77012 | − | 0.829619i | −1.19658 | − | 2.07254i | 2.42076 | + | 3.39948i |
10.7 | −2.46426 | − | 0.117387i | 2.31172 | + | 1.33467i | 4.06785 | + | 0.388432i | 2.54618 | + | 3.23774i | −5.54001 | − | 3.56035i | −0.750402 | + | 2.53710i | −5.09475 | − | 0.732514i | 2.06271 | + | 3.57272i | −5.89439 | − | 8.27751i |
10.8 | −2.46098 | − | 0.117231i | 0.813611 | + | 0.469739i | 4.05173 | + | 0.386893i | −2.32023 | − | 2.95041i | −1.94721 | − | 1.25140i | 2.58436 | + | 0.566636i | −5.04849 | − | 0.725863i | −1.05869 | − | 1.83371i | 5.36415 | + | 7.53290i |
10.9 | −2.38715 | − | 0.113714i | −2.68556 | − | 1.55051i | 3.69459 | + | 0.352791i | 1.10017 | + | 1.39899i | 6.23451 | + | 4.00668i | −2.64121 | − | 0.154983i | −4.04836 | − | 0.582067i | 3.30816 | + | 5.72990i | −2.46719 | − | 3.46469i |
10.10 | −2.32487 | − | 0.110747i | −1.14180 | − | 0.659217i | 3.40182 | + | 0.324834i | 1.50586 | + | 1.91486i | 2.58153 | + | 1.65905i | 1.11205 | + | 2.40070i | −3.26519 | − | 0.469463i | −0.630865 | − | 1.09269i | −3.28887 | − | 4.61857i |
10.11 | −2.32005 | − | 0.110518i | 0.889184 | + | 0.513371i | 3.37949 | + | 0.322702i | 1.20915 | + | 1.53756i | −2.00622 | − | 1.28932i | −2.10097 | − | 1.60808i | −3.20685 | − | 0.461075i | −0.972901 | − | 1.68511i | −2.63537 | − | 3.70087i |
10.12 | −2.27557 | − | 0.108399i | −1.58135 | − | 0.912990i | 3.17553 | + | 0.303227i | −0.952228 | − | 1.21086i | 3.49950 | + | 2.24899i | −1.38677 | + | 2.25319i | −2.68336 | − | 0.385809i | 0.167102 | + | 0.289429i | 2.03561 | + | 2.85861i |
10.13 | −2.15759 | − | 0.102778i | 0.0221198 | + | 0.0127709i | 2.65367 | + | 0.253394i | −1.64064 | − | 2.08625i | −0.0464127 | − | 0.0298277i | −1.90987 | + | 1.83096i | −1.42338 | − | 0.204651i | −1.49967 | − | 2.59751i | 3.32541 | + | 4.66988i |
10.14 | −2.14570 | − | 0.102212i | 1.52425 | + | 0.880026i | 2.60265 | + | 0.248523i | −0.493336 | − | 0.627327i | −3.18064 | − | 2.04407i | −2.59427 | + | 0.519385i | −1.30657 | − | 0.187856i | 0.0488912 | + | 0.0846820i | 0.994431 | + | 1.39648i |
10.15 | −1.96582 | − | 0.0936434i | −1.98627 | − | 1.14678i | 1.86472 | + | 0.178059i | −0.904452 | − | 1.15010i | 3.79726 | + | 2.44035i | 0.573602 | − | 2.58282i | 0.246999 | + | 0.0355131i | 1.13019 | + | 1.95755i | 1.67029 | + | 2.34559i |
10.16 | −1.94545 | − | 0.0926733i | 2.50563 | + | 1.44662i | 1.78525 | + | 0.170471i | −0.415136 | − | 0.527888i | −4.74051 | − | 3.04654i | 1.93603 | + | 1.80327i | 0.398343 | + | 0.0572730i | 2.68544 | + | 4.65132i | 0.758705 | + | 1.06545i |
10.17 | −1.86475 | − | 0.0888291i | −0.0252474 | − | 0.0145766i | 1.47847 | + | 0.141177i | −0.105696 | − | 0.134404i | 0.0457854 | + | 0.0294245i | 1.60285 | + | 2.10496i | 0.951292 | + | 0.136775i | −1.49958 | − | 2.59734i | 0.185158 | + | 0.260019i |
10.18 | −1.80792 | − | 0.0861219i | −1.72811 | − | 0.997726i | 1.27021 | + | 0.121291i | −0.508859 | − | 0.647067i | 3.03836 | + | 1.95264i | 2.51479 | − | 0.822081i | 1.29709 | + | 0.186494i | 0.490914 | + | 0.850289i | 0.864250 | + | 1.21367i |
10.19 | −1.79179 | − | 0.0853534i | −1.73157 | − | 0.999725i | 1.21228 | + | 0.115759i | 1.61018 | + | 2.04751i | 3.01729 | + | 1.93909i | −0.898356 | − | 2.48857i | 1.38886 | + | 0.199688i | 0.498900 | + | 0.864120i | −2.71034 | − | 3.80614i |
10.20 | −1.78916 | − | 0.0852280i | 1.65195 | + | 0.953752i | 1.20287 | + | 0.114861i | 0.349108 | + | 0.443927i | −2.87431 | − | 1.84720i | −0.100171 | − | 2.64385i | 1.40357 | + | 0.201802i | 0.319287 | + | 0.553021i | −0.586773 | − | 0.824008i |
See next 80 embeddings (of 1720 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
121.f | odd | 22 | 1 | inner |
847.x | even | 66 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.x.a | ✓ | 1720 |
7.d | odd | 6 | 1 | inner | 847.2.x.a | ✓ | 1720 |
121.f | odd | 22 | 1 | inner | 847.2.x.a | ✓ | 1720 |
847.x | even | 66 | 1 | inner | 847.2.x.a | ✓ | 1720 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
847.2.x.a | ✓ | 1720 | 1.a | even | 1 | 1 | trivial |
847.2.x.a | ✓ | 1720 | 7.d | odd | 6 | 1 | inner |
847.2.x.a | ✓ | 1720 | 121.f | odd | 22 | 1 | inner |
847.2.x.a | ✓ | 1720 | 847.x | even | 66 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).