Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(82,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.cc (of order \(12\), 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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −2.70653 | + | 0.725214i | 0 | 5.06734 | − | 2.92563i | −0.582598 | − | 2.15884i | 0 | −1.87525 | − | 1.86640i | −7.63059 | + | 7.63059i | 0 | 3.14244 | + | 5.42046i | ||||||
82.2 | −2.42255 | + | 0.649122i | 0 | 3.71536 | − | 2.14507i | −0.767700 | + | 2.10015i | 0 | 2.52094 | + | 0.803028i | −4.06139 | + | 4.06139i | 0 | 0.496542 | − | 5.58606i | ||||||
82.3 | −2.36642 | + | 0.634081i | 0 | 3.46585 | − | 2.00101i | −2.22777 | + | 0.192431i | 0 | −0.800970 | + | 2.52160i | −3.46818 | + | 3.46818i | 0 | 5.14984 | − | 1.86796i | ||||||
82.4 | −2.31911 | + | 0.621403i | 0 | 3.26006 | − | 1.88220i | 1.91020 | + | 1.16238i | 0 | 1.84774 | − | 1.89364i | −2.99543 | + | 2.99543i | 0 | −5.15227 | − | 1.50867i | ||||||
82.5 | −2.10293 | + | 0.563479i | 0 | 2.37276 | − | 1.36992i | −0.789671 | − | 2.09199i | 0 | 2.63835 | − | 0.197733i | −1.13894 | + | 1.13894i | 0 | 2.83942 | + | 3.95435i | ||||||
82.6 | −1.97779 | + | 0.529947i | 0 | 1.89875 | − | 1.09625i | 1.71761 | + | 1.43172i | 0 | −2.61323 | − | 0.413543i | −0.278697 | + | 0.278697i | 0 | −4.15581 | − | 1.92139i | ||||||
82.7 | −1.78707 | + | 0.478845i | 0 | 1.23228 | − | 0.711459i | 1.42046 | − | 1.72693i | 0 | −2.30923 | − | 1.29130i | 0.754954 | − | 0.754954i | 0 | −1.71153 | + | 3.76633i | ||||||
82.8 | −1.54962 | + | 0.415220i | 0 | 0.496873 | − | 0.286870i | −2.23119 | + | 0.147569i | 0 | 0.0784759 | − | 2.64459i | 1.61795 | − | 1.61795i | 0 | 3.39624 | − | 1.15511i | ||||||
82.9 | −1.39331 | + | 0.373336i | 0 | 0.0698789 | − | 0.0403446i | 0.709966 | + | 2.12037i | 0 | −1.34070 | + | 2.28090i | 1.95764 | − | 1.95764i | 0 | −1.78081 | − | 2.68927i | ||||||
82.10 | −1.34647 | + | 0.360785i | 0 | −0.0492386 | + | 0.0284279i | −1.34313 | − | 1.78774i | 0 | −0.423172 | + | 2.61169i | 2.02741 | − | 2.02741i | 0 | 2.45347 | + | 1.92256i | ||||||
82.11 | −1.17569 | + | 0.315026i | 0 | −0.449041 | + | 0.259254i | −1.79423 | + | 1.33444i | 0 | −1.33548 | − | 2.28397i | 2.16759 | − | 2.16759i | 0 | 1.68908 | − | 2.13412i | ||||||
82.12 | −0.772202 | + | 0.206911i | 0 | −1.17857 | + | 0.680446i | 2.09761 | + | 0.774613i | 0 | 2.11780 | − | 1.58585i | 1.89988 | − | 1.89988i | 0 | −1.78006 | − | 0.164139i | ||||||
82.13 | −0.678049 | + | 0.181683i | 0 | −1.30531 | + | 0.753621i | 1.99409 | − | 1.01174i | 0 | 1.52686 | + | 2.16071i | 1.74088 | − | 1.74088i | 0 | −1.16827 | + | 1.04830i | ||||||
82.14 | −0.646431 | + | 0.173211i | 0 | −1.34418 | + | 0.776063i | −0.270716 | + | 2.21962i | 0 | 2.23956 | + | 1.40868i | 1.68094 | − | 1.68094i | 0 | −0.209463 | − | 1.48172i | ||||||
82.15 | −0.240990 | + | 0.0645732i | 0 | −1.67814 | + | 0.968877i | 1.83527 | − | 1.27741i | 0 | −2.64502 | + | 0.0621414i | 0.694688 | − | 0.694688i | 0 | −0.359797 | + | 0.426352i | ||||||
82.16 | −0.121772 | + | 0.0326286i | 0 | −1.71829 | + | 0.992054i | −0.668137 | − | 2.13391i | 0 | 1.10538 | − | 2.40378i | 0.355155 | − | 0.355155i | 0 | 0.150987 | + | 0.238050i | ||||||
82.17 | 0.121772 | − | 0.0326286i | 0 | −1.71829 | + | 0.992054i | 0.668137 | + | 2.13391i | 0 | 1.10538 | − | 2.40378i | −0.355155 | + | 0.355155i | 0 | 0.150987 | + | 0.238050i | ||||||
82.18 | 0.240990 | − | 0.0645732i | 0 | −1.67814 | + | 0.968877i | −1.83527 | + | 1.27741i | 0 | −2.64502 | + | 0.0621414i | −0.694688 | + | 0.694688i | 0 | −0.359797 | + | 0.426352i | ||||||
82.19 | 0.646431 | − | 0.173211i | 0 | −1.34418 | + | 0.776063i | 0.270716 | − | 2.21962i | 0 | 2.23956 | + | 1.40868i | −1.68094 | + | 1.68094i | 0 | −0.209463 | − | 1.48172i | ||||||
82.20 | 0.678049 | − | 0.181683i | 0 | −1.30531 | + | 0.753621i | −1.99409 | + | 1.01174i | 0 | 1.52686 | + | 2.16071i | −1.74088 | + | 1.74088i | 0 | −1.16827 | + | 1.04830i | ||||||
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
7.d | odd | 6 | 1 | inner |
15.e | even | 4 | 1 | inner |
21.g | even | 6 | 1 | inner |
35.k | even | 12 | 1 | inner |
105.w | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.cc.a | ✓ | 128 |
3.b | odd | 2 | 1 | inner | 945.2.cc.a | ✓ | 128 |
5.c | odd | 4 | 1 | inner | 945.2.cc.a | ✓ | 128 |
7.d | odd | 6 | 1 | inner | 945.2.cc.a | ✓ | 128 |
15.e | even | 4 | 1 | inner | 945.2.cc.a | ✓ | 128 |
21.g | even | 6 | 1 | inner | 945.2.cc.a | ✓ | 128 |
35.k | even | 12 | 1 | inner | 945.2.cc.a | ✓ | 128 |
105.w | odd | 12 | 1 | inner | 945.2.cc.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
945.2.cc.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
945.2.cc.a | ✓ | 128 | 3.b | odd | 2 | 1 | inner |
945.2.cc.a | ✓ | 128 | 5.c | odd | 4 | 1 | inner |
945.2.cc.a | ✓ | 128 | 7.d | odd | 6 | 1 | inner |
945.2.cc.a | ✓ | 128 | 15.e | even | 4 | 1 | inner |
945.2.cc.a | ✓ | 128 | 21.g | even | 6 | 1 | inner |
945.2.cc.a | ✓ | 128 | 35.k | even | 12 | 1 | inner |
945.2.cc.a | ✓ | 128 | 105.w | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 240 T_{2}^{124} + 33200 T_{2}^{120} - 3094388 T_{2}^{116} + 215934052 T_{2}^{112} + \cdots + 3906250000 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).