Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,3,Mod(2,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(54))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 81.h (of order \(54\), degree \(18\), 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: | \(2.20709014132\) |
Analytic rank: | \(0\) |
Dimension: | \(306\) |
Relative dimension: | \(17\) over \(\Q(\zeta_{54})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{54}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.74703 | − | 0.218240i | −2.98307 | − | 0.318284i | 10.0197 | + | 1.17113i | 1.51414 | + | 6.38868i | 11.1082 | + | 1.84364i | −3.22727 | − | 4.33498i | −22.5029 | − | 3.96787i | 8.79739 | + | 1.89893i | −4.27928 | − | 24.2690i |
2.2 | −3.51578 | − | 0.204771i | 1.87859 | + | 2.33900i | 8.34586 | + | 0.975491i | −0.560130 | − | 2.36337i | −6.12575 | − | 8.60809i | 0.853586 | + | 1.14656i | −15.2695 | − | 2.69243i | −1.94182 | + | 8.78802i | 1.48535 | + | 8.42381i |
2.3 | −3.19159 | − | 0.185889i | 0.779433 | − | 2.89698i | 6.17872 | + | 0.722189i | −1.14257 | − | 4.82088i | −3.02614 | + | 9.10107i | 2.64358 | + | 3.55094i | −6.99198 | − | 1.23288i | −7.78497 | − | 4.51600i | 2.75046 | + | 15.5986i |
2.4 | −2.38296 | − | 0.138791i | 2.64624 | − | 1.41330i | 1.68627 | + | 0.197097i | 1.89333 | + | 7.98858i | −6.50203 | + | 3.00056i | −1.92617 | − | 2.58729i | 5.41197 | + | 0.954277i | 5.00517 | − | 7.47986i | −3.40297 | − | 19.2992i |
2.5 | −2.16212 | − | 0.125929i | −2.73727 | + | 1.22773i | 0.685955 | + | 0.0801767i | −1.17077 | − | 4.93986i | 6.07293 | − | 2.30981i | 5.31774 | + | 7.14296i | 7.05851 | + | 1.24461i | 5.98534 | − | 6.72129i | 1.90927 | + | 10.8280i |
2.6 | −1.71539 | − | 0.0999103i | −0.611041 | + | 2.93711i | −1.04036 | − | 0.121601i | 0.258193 | + | 1.08940i | 1.34162 | − | 4.97725i | −6.70340 | − | 9.00422i | 8.54126 | + | 1.50605i | −8.25326 | − | 3.58939i | −0.334060 | − | 1.89455i |
2.7 | −1.04279 | − | 0.0607357i | −2.04391 | − | 2.19600i | −2.88923 | − | 0.337702i | 1.21177 | + | 5.11287i | 1.99800 | + | 2.41411i | 4.42126 | + | 5.93878i | 7.10711 | + | 1.25318i | −0.644854 | + | 8.97687i | −0.953092 | − | 5.40525i |
2.8 | −0.614022 | − | 0.0357627i | 2.99545 | − | 0.165205i | −3.59721 | − | 0.420453i | −2.20305 | − | 9.29539i | −1.84518 | − | 0.00568553i | −3.03111 | − | 4.07148i | 4.61661 | + | 0.814032i | 8.94541 | − | 0.989729i | 1.02029 | + | 5.78637i |
2.9 | −0.272030 | − | 0.0158440i | 2.02197 | + | 2.21622i | −3.89920 | − | 0.455752i | 0.613475 | + | 2.58845i | −0.514925 | − | 0.634914i | 5.77313 | + | 7.75466i | 2.12689 | + | 0.375028i | −0.823240 | + | 8.96227i | −0.125872 | − | 0.713857i |
2.10 | −0.218910 | − | 0.0127500i | 0.445650 | − | 2.96671i | −3.92519 | − | 0.458789i | −0.196368 | − | 0.828540i | −0.135383 | + | 0.643760i | −2.81710 | − | 3.78402i | 1.71721 | + | 0.302791i | −8.60279 | − | 2.64424i | 0.0324228 | + | 0.183879i |
2.11 | 1.00396 | + | 0.0584742i | −2.94698 | − | 0.561535i | −2.96843 | − | 0.346960i | −0.829724 | − | 3.50088i | −2.92582 | − | 0.736083i | −3.90171 | − | 5.24091i | −6.92145 | − | 1.22044i | 8.36936 | + | 3.30966i | −0.628301 | − | 3.56327i |
2.12 | 1.44133 | + | 0.0839482i | −1.99809 | + | 2.23778i | −1.90256 | − | 0.222377i | 1.78114 | + | 7.51523i | −3.06777 | + | 3.05765i | 0.363641 | + | 0.488455i | −8.41092 | − | 1.48307i | −1.01527 | − | 8.94255i | 1.93633 | + | 10.9815i |
2.13 | 2.03769 | + | 0.118682i | 2.77384 | − | 1.14271i | 0.165158 | + | 0.0193041i | 0.612887 | + | 2.58597i | 5.78787 | − | 1.99929i | 1.33468 | + | 1.79278i | −7.70630 | − | 1.35883i | 6.38843 | − | 6.33940i | 0.941967 | + | 5.34216i |
2.14 | 2.62917 | + | 0.153131i | −0.579386 | − | 2.94352i | 2.91611 | + | 0.340844i | −1.61217 | − | 6.80228i | −1.07256 | − | 7.82773i | 6.82619 | + | 9.16917i | −2.75970 | − | 0.486609i | −8.32862 | + | 3.41087i | −3.19702 | − | 18.1312i |
2.15 | 2.79319 | + | 0.162685i | 1.52144 | + | 2.58558i | 3.80251 | + | 0.444450i | −0.618939 | − | 2.61151i | 3.82905 | + | 7.46953i | −3.52360 | − | 4.73302i | −0.472829 | − | 0.0833725i | −4.37041 | + | 7.86762i | −1.30396 | − | 7.39515i |
2.16 | 3.40512 | + | 0.198326i | −1.03823 | − | 2.81462i | 7.58258 | + | 0.886276i | 1.79915 | + | 7.59120i | −2.97709 | − | 9.79003i | −7.23042 | − | 9.71214i | 12.2075 | + | 2.15252i | −6.84415 | + | 5.84445i | 4.62078 | + | 26.2058i |
2.17 | 3.55891 | + | 0.207283i | −2.59188 | + | 1.51068i | 8.64995 | + | 1.01103i | −0.346846 | − | 1.46346i | −9.53743 | + | 4.83912i | 4.11378 | + | 5.52576i | 16.5317 | + | 2.91499i | 4.43571 | − | 7.83100i | −0.931045 | − | 5.28022i |
5.1 | −0.860368 | − | 3.63018i | 0.315927 | + | 2.98332i | −8.86342 | + | 4.45138i | −2.70864 | + | 2.01651i | 10.5582 | − | 3.71362i | −9.77832 | + | 6.43130i | 14.1928 | + | 16.9143i | −8.80038 | + | 1.88502i | 9.65071 | + | 8.09791i |
5.2 | −0.769839 | − | 3.24821i | 2.36730 | − | 1.84279i | −6.38367 | + | 3.20600i | 5.00097 | − | 3.72308i | −7.80821 | − | 6.27084i | −2.54834 | + | 1.67607i | 6.74517 | + | 8.03858i | 2.20825 | − | 8.72489i | −15.9433 | − | 13.3780i |
5.3 | −0.723056 | − | 3.05081i | −0.981828 | − | 2.83479i | −5.21012 | + | 2.61662i | −4.98049 | + | 3.70784i | −7.93849 | + | 5.04508i | 4.70676 | − | 3.09568i | 3.68862 | + | 4.39593i | −7.07203 | + | 5.56655i | 14.9131 | + | 12.5136i |
See next 80 embeddings (of 306 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
81.h | odd | 54 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 81.3.h.a | ✓ | 306 |
3.b | odd | 2 | 1 | 243.3.h.a | 306 | ||
81.g | even | 27 | 1 | 243.3.h.a | 306 | ||
81.h | odd | 54 | 1 | inner | 81.3.h.a | ✓ | 306 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
81.3.h.a | ✓ | 306 | 1.a | even | 1 | 1 | trivial |
81.3.h.a | ✓ | 306 | 81.h | odd | 54 | 1 | inner |
243.3.h.a | 306 | 3.b | odd | 2 | 1 | ||
243.3.h.a | 306 | 81.g | even | 27 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(81, [\chi])\).