Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [946,2,Mod(85,946)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(946, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("946.85");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 946 = 2 \cdot 11 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 946.l (of order \(10\), degree \(4\), 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: | \(7.55384803121\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
85.1 | −0.809017 | + | 0.587785i | −2.99176 | + | 0.972082i | 0.309017 | − | 0.951057i | 0.0144910 | − | 0.0199451i | 1.84901 | − | 2.54494i | −0.635199 | + | 1.95494i | 0.309017 | + | 0.951057i | 5.57863 | − | 4.05311i | 0.0246536i | ||
85.2 | −0.809017 | + | 0.587785i | −2.64506 | + | 0.859431i | 0.309017 | − | 0.951057i | 2.19210 | − | 3.01717i | 1.63473 | − | 2.25002i | 0.978717 | − | 3.01218i | 0.309017 | + | 0.951057i | 3.83065 | − | 2.78313i | 3.72943i | ||
85.3 | −0.809017 | + | 0.587785i | −2.53894 | + | 0.824950i | 0.309017 | − | 0.951057i | −2.24874 | + | 3.09512i | 1.56915 | − | 2.15975i | 0.718873 | − | 2.21246i | 0.309017 | + | 0.951057i | 3.33860 | − | 2.42564i | − | 3.82578i | |
85.4 | −0.809017 | + | 0.587785i | −2.22721 | + | 0.723664i | 0.309017 | − | 0.951057i | 0.743511 | − | 1.02336i | 1.37649 | − | 1.89458i | −0.765347 | + | 2.35549i | 0.309017 | + | 0.951057i | 2.00972 | − | 1.46015i | 1.26494i | ||
85.5 | −0.809017 | + | 0.587785i | −2.14623 | + | 0.697353i | 0.309017 | − | 0.951057i | −0.292520 | + | 0.402619i | 1.32644 | − | 1.82569i | −0.706871 | + | 2.17552i | 0.309017 | + | 0.951057i | 1.69296 | − | 1.23001i | − | 0.497664i | |
85.6 | −0.809017 | + | 0.587785i | −2.00563 | + | 0.651669i | 0.309017 | − | 0.951057i | −1.09227 | + | 1.50338i | 1.23955 | − | 1.70609i | 0.593774 | − | 1.82745i | 0.309017 | + | 0.951057i | 1.17083 | − | 0.850656i | − | 1.85828i | |
85.7 | −0.809017 | + | 0.587785i | −1.17692 | + | 0.382406i | 0.309017 | − | 0.951057i | 1.85246 | − | 2.54969i | 0.727379 | − | 1.00115i | 0.358232 | − | 1.10253i | 0.309017 | + | 0.951057i | −1.18814 | + | 0.863232i | 3.15160i | ||
85.8 | −0.809017 | + | 0.587785i | −0.875069 | + | 0.284327i | 0.309017 | − | 0.951057i | −2.08664 | + | 2.87201i | 0.540822 | − | 0.744378i | −0.991886 | + | 3.05271i | 0.309017 | + | 0.951057i | −1.74215 | + | 1.26574i | − | 3.55000i | |
85.9 | −0.809017 | + | 0.587785i | −0.872172 | + | 0.283386i | 0.309017 | − | 0.951057i | 0.425915 | − | 0.586221i | 0.539032 | − | 0.741914i | 1.32297 | − | 4.07168i | 0.309017 | + | 0.951057i | −1.74667 | + | 1.26903i | 0.724609i | ||
85.10 | −0.809017 | + | 0.587785i | −0.441079 | + | 0.143315i | 0.309017 | − | 0.951057i | −0.893157 | + | 1.22933i | 0.272602 | − | 0.375205i | −1.18417 | + | 3.64449i | 0.309017 | + | 0.951057i | −2.25304 | + | 1.63693i | − | 1.51953i | |
85.11 | −0.809017 | + | 0.587785i | −0.298204 | + | 0.0968924i | 0.309017 | − | 0.951057i | 1.80281 | − | 2.48135i | 0.184300 | − | 0.253668i | 0.489048 | − | 1.50513i | 0.309017 | + | 0.951057i | −2.34751 | + | 1.70557i | 3.06712i | ||
85.12 | −0.809017 | + | 0.587785i | −0.124065 | + | 0.0403112i | 0.309017 | − | 0.951057i | −0.207165 | + | 0.285138i | 0.0766765 | − | 0.105536i | 1.05333 | − | 3.24181i | 0.309017 | + | 0.951057i | −2.41328 | + | 1.75335i | − | 0.352450i | |
85.13 | −0.809017 | + | 0.587785i | −0.0407586 | + | 0.0132433i | 0.309017 | − | 0.951057i | 2.35495 | − | 3.24131i | 0.0251902 | − | 0.0346714i | −1.44902 | + | 4.45963i | 0.309017 | + | 0.951057i | −2.42557 | + | 1.76228i | 4.00648i | ||
85.14 | −0.809017 | + | 0.587785i | 0.619989 | − | 0.201447i | 0.309017 | − | 0.951057i | −2.02952 | + | 2.79339i | −0.383174 | + | 0.527394i | 0.687411 | − | 2.11564i | 0.309017 | + | 0.951057i | −2.08325 | + | 1.51357i | − | 3.45282i | |
85.15 | −0.809017 | + | 0.587785i | 0.917762 | − | 0.298199i | 0.309017 | − | 0.951057i | 1.35569 | − | 1.86595i | −0.567208 | + | 0.780695i | −0.732700 | + | 2.25502i | 0.309017 | + | 0.951057i | −1.67369 | + | 1.21600i | 2.30644i | ||
85.16 | −0.809017 | + | 0.587785i | 1.27926 | − | 0.415656i | 0.309017 | − | 0.951057i | −0.441745 | + | 0.608010i | −0.790625 | + | 1.08820i | 0.155387 | − | 0.478232i | 0.309017 | + | 0.951057i | −0.963319 | + | 0.699892i | − | 0.751542i | |
85.17 | −0.809017 | + | 0.587785i | 1.34123 | − | 0.435794i | 0.309017 | − | 0.951057i | −0.610055 | + | 0.839668i | −0.828929 | + | 1.14092i | 0.0795635 | − | 0.244871i | 0.309017 | + | 0.951057i | −0.818056 | + | 0.594352i | − | 1.03789i | |
85.18 | −0.809017 | + | 0.587785i | 2.15707 | − | 0.700873i | 0.309017 | − | 0.951057i | −1.57860 | + | 2.17275i | −1.33314 | + | 1.83491i | −1.12581 | + | 3.46488i | 0.309017 | + | 0.951057i | 1.73466 | − | 1.26030i | − | 2.68567i | |
85.19 | −0.809017 | + | 0.587785i | 2.30619 | − | 0.749325i | 0.309017 | − | 0.951057i | 1.40834 | − | 1.93841i | −1.42530 | + | 1.96176i | −0.0303287 | + | 0.0933420i | 0.309017 | + | 0.951057i | 2.32996 | − | 1.69281i | 2.39600i | ||
85.20 | −0.809017 | + | 0.587785i | 2.43207 | − | 0.790226i | 0.309017 | − | 0.951057i | −0.118871 | + | 0.163612i | −1.50310 | + | 2.06884i | 0.609357 | − | 1.87541i | 0.309017 | + | 0.951057i | 2.86344 | − | 2.08041i | − | 0.202236i | |
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
473.k | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 946.2.l.a | ✓ | 88 |
11.d | odd | 10 | 1 | 946.2.l.b | yes | 88 | |
43.b | odd | 2 | 1 | 946.2.l.b | yes | 88 | |
473.k | even | 10 | 1 | inner | 946.2.l.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
946.2.l.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
946.2.l.a | ✓ | 88 | 473.k | even | 10 | 1 | inner |
946.2.l.b | yes | 88 | 11.d | odd | 10 | 1 | |
946.2.l.b | yes | 88 | 43.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{88} - 43 T_{3}^{86} + 5 T_{3}^{85} + 1086 T_{3}^{84} - 215 T_{3}^{83} - 21099 T_{3}^{82} + 5945 T_{3}^{81} + 349477 T_{3}^{80} - 119510 T_{3}^{79} - 4973995 T_{3}^{78} + 2030250 T_{3}^{77} + \cdots + 25825775616 \)
acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\).