Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,2,Mod(5,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([55, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 207.o (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: | \(1.65290332184\) |
Analytic rank: | \(0\) |
Dimension: | \(440\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{66})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{66}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −2.62689 | + | 0.637277i | 1.63894 | − | 0.560241i | 4.71676 | − | 2.43166i | 2.75970 | + | 2.17025i | −3.94829 | + | 2.51615i | 1.24506 | + | 0.886604i | −6.75505 | + | 5.85328i | 2.37226 | − | 1.83641i | −8.63247 | − | 3.94232i |
5.2 | −2.45913 | + | 0.596579i | 0.0507650 | − | 1.73131i | 3.91376 | − | 2.01768i | −2.75910 | − | 2.16978i | 0.908024 | + | 4.28780i | −2.46994 | − | 1.75883i | −4.59596 | + | 3.98242i | −2.99485 | − | 0.175779i | 8.07946 | + | 3.68976i |
5.3 | −2.43968 | + | 0.591859i | −0.572964 | + | 1.63454i | 3.82406 | − | 1.97144i | −0.573627 | − | 0.451105i | 0.430433 | − | 4.32686i | −0.847518 | − | 0.603515i | −4.36812 | + | 3.78500i | −2.34342 | − | 1.87306i | 1.66646 | + | 0.761045i |
5.4 | −1.96150 | + | 0.475854i | −1.72655 | − | 0.137906i | 1.84336 | − | 0.950318i | 2.46398 | + | 1.93770i | 3.45225 | − | 0.551085i | −3.19678 | − | 2.27642i | −0.112732 | + | 0.0976829i | 2.96196 | + | 0.476202i | −5.75516 | − | 2.62829i |
5.5 | −1.90183 | + | 0.461379i | 1.48247 | + | 0.895699i | 1.62642 | − | 0.838476i | −1.80924 | − | 1.42280i | −3.23267 | − | 1.01949i | 2.18638 | + | 1.55692i | 0.251684 | − | 0.218086i | 1.39545 | + | 2.65570i | 4.09731 | + | 1.87118i |
5.6 | −1.87165 | + | 0.454057i | −1.24149 | − | 1.20777i | 1.51923 | − | 0.783220i | −0.386973 | − | 0.304319i | 2.87203 | + | 1.69682i | 2.77945 | + | 1.97924i | 0.423207 | − | 0.366711i | 0.0825754 | + | 2.99886i | 0.862456 | + | 0.393871i |
5.7 | −1.19377 | + | 0.289604i | 0.337452 | − | 1.69886i | −0.436467 | + | 0.225014i | 1.95562 | + | 1.53792i | 0.0891586 | + | 2.12577i | 0.435801 | + | 0.310333i | 2.31259 | − | 2.00387i | −2.77225 | − | 1.14657i | −2.77994 | − | 1.26955i |
5.8 | −1.07455 | + | 0.260682i | −1.64011 | + | 0.556821i | −0.690978 | + | 0.356224i | −1.83293 | − | 1.44144i | 1.61722 | − | 1.02588i | 0.303689 | + | 0.216256i | 2.32091 | − | 2.01108i | 2.37990 | − | 1.82649i | 2.34533 | + | 1.07108i |
5.9 | −0.942830 | + | 0.228728i | 0.860190 | + | 1.50335i | −0.941060 | + | 0.485150i | 2.69473 | + | 2.11916i | −1.15487 | − | 1.22066i | 0.176132 | + | 0.125423i | 2.24271 | − | 1.94332i | −1.52015 | + | 2.58634i | −3.02538 | − | 1.38165i |
5.10 | −0.565298 | + | 0.137140i | 0.121015 | + | 1.72782i | −1.47692 | + | 0.761403i | −1.35334 | − | 1.06428i | −0.305362 | − | 0.960136i | −2.96537 | − | 2.11163i | 1.60971 | − | 1.39482i | −2.97071 | + | 0.418184i | 0.910994 | + | 0.416037i |
5.11 | −0.158710 | + | 0.0385026i | 1.72620 | − | 0.142203i | −1.75396 | + | 0.904232i | 0.845243 | + | 0.664707i | −0.268490 | + | 0.0890323i | 0.602097 | + | 0.428751i | 0.490404 | − | 0.424938i | 2.95956 | − | 0.490942i | −0.159741 | − | 0.0729514i |
5.12 | −0.156007 | + | 0.0378470i | 1.02630 | − | 1.39524i | −1.75476 | + | 0.904644i | −3.16235 | − | 2.48690i | −0.107306 | + | 0.256511i | 1.51977 | + | 1.08222i | 0.482163 | − | 0.417797i | −0.893396 | − | 2.86389i | 0.587472 | + | 0.268290i |
5.13 | 0.101474 | − | 0.0246174i | −0.566247 | − | 1.63688i | −1.76798 | + | 0.911457i | 0.359165 | + | 0.282450i | −0.0977550 | − | 0.152161i | −3.37292 | − | 2.40185i | −0.314794 | + | 0.272770i | −2.35873 | + | 1.85375i | 0.0433991 | + | 0.0198197i |
5.14 | 0.554186 | − | 0.134444i | −1.59629 | − | 0.672214i | −1.48862 | + | 0.767439i | 1.47451 | + | 1.15957i | −0.975014 | − | 0.157920i | 2.15103 | + | 1.53174i | −1.58375 | + | 1.37232i | 2.09626 | + | 2.14609i | 0.973049 | + | 0.444377i |
5.15 | 0.905370 | − | 0.219640i | 0.0930743 | + | 1.72955i | −1.00622 | + | 0.518741i | −1.25721 | − | 0.988678i | 0.464145 | + | 1.54544i | 4.26264 | + | 3.03541i | −2.20522 | + | 1.91084i | −2.98267 | + | 0.321953i | −1.35539 | − | 0.618986i |
5.16 | 1.28415 | − | 0.311532i | −1.70297 | − | 0.316062i | −0.225678 | + | 0.116345i | −2.59594 | − | 2.04147i | −2.28533 | + | 0.124657i | −2.08508 | − | 1.48478i | −2.25085 | + | 1.95038i | 2.80021 | + | 1.07649i | −3.96956 | − | 1.81284i |
5.17 | 1.29057 | − | 0.313088i | −0.828584 | + | 1.52100i | −0.210130 | + | 0.108329i | 3.10558 | + | 2.44225i | −0.593136 | + | 2.22238i | −2.63088 | − | 1.87344i | −2.24454 | + | 1.94491i | −1.62690 | − | 2.52056i | 4.77260 | + | 2.17957i |
5.18 | 1.41889 | − | 0.344219i | 1.58943 | + | 0.688262i | 0.117094 | − | 0.0603664i | 0.676831 | + | 0.532265i | 2.49214 | + | 0.429455i | −0.142829 | − | 0.101708i | −2.06150 | + | 1.78630i | 2.05259 | + | 2.18789i | 1.14356 | + | 0.522249i |
5.19 | 1.65908 | − | 0.402488i | 1.26516 | − | 1.18295i | 0.812874 | − | 0.419065i | 0.0437969 | + | 0.0344423i | 1.62288 | − | 2.47181i | −1.22353 | − | 0.871275i | −1.40048 | + | 1.21352i | 0.201279 | − | 2.99324i | 0.0865251 | + | 0.0395147i |
5.20 | 2.17585 | − | 0.527855i | −0.508290 | − | 1.65579i | 2.67802 | − | 1.38061i | −2.04245 | − | 1.60620i | −1.97998 | − | 3.33445i | 2.45761 | + | 1.75005i | 1.71401 | − | 1.48520i | −2.48328 | + | 1.68324i | −5.29190 | − | 2.41673i |
See next 80 embeddings (of 440 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
23.d | odd | 22 | 1 | inner |
207.o | even | 66 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 207.2.o.a | ✓ | 440 |
3.b | odd | 2 | 1 | 621.2.s.a | 440 | ||
9.c | even | 3 | 1 | 621.2.s.a | 440 | ||
9.d | odd | 6 | 1 | inner | 207.2.o.a | ✓ | 440 |
23.d | odd | 22 | 1 | inner | 207.2.o.a | ✓ | 440 |
69.g | even | 22 | 1 | 621.2.s.a | 440 | ||
207.o | even | 66 | 1 | inner | 207.2.o.a | ✓ | 440 |
207.p | odd | 66 | 1 | 621.2.s.a | 440 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
207.2.o.a | ✓ | 440 | 1.a | even | 1 | 1 | trivial |
207.2.o.a | ✓ | 440 | 9.d | odd | 6 | 1 | inner |
207.2.o.a | ✓ | 440 | 23.d | odd | 22 | 1 | inner |
207.2.o.a | ✓ | 440 | 207.o | even | 66 | 1 | inner |
621.2.s.a | 440 | 3.b | odd | 2 | 1 | ||
621.2.s.a | 440 | 9.c | even | 3 | 1 | ||
621.2.s.a | 440 | 69.g | even | 22 | 1 | ||
621.2.s.a | 440 | 207.p | odd | 66 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(207, [\chi])\).