Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,4,Mod(4,207)]
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([22, 12]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 207.m (of order \(33\), 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: | \(12.2133953712\) |
Analytic rank: | \(0\) |
Dimension: | \(1400\) |
Relative dimension: | \(70\) over \(\Q(\zeta_{33})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{33}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −3.26722 | + | 4.58817i | −1.71615 | + | 4.90457i | −7.76003 | − | 22.4211i | −14.9593 | + | 7.71206i | −16.8960 | − | 23.8983i | 12.7056 | − | 9.99183i | 84.9902 | + | 24.9554i | −21.1097 | − | 16.8339i | 13.4911 | − | 93.8328i |
4.2 | −3.11000 | + | 4.36738i | 0.691500 | − | 5.14993i | −6.78539 | − | 19.6051i | 6.99819 | − | 3.60782i | 20.3412 | + | 19.0363i | 28.6222 | − | 22.5088i | 65.5705 | + | 19.2532i | −26.0437 | − | 7.12236i | −6.00764 | + | 41.7841i |
4.3 | −3.09157 | + | 4.34150i | −4.81737 | − | 1.94756i | −6.67428 | − | 19.2841i | 6.06640 | − | 3.12745i | 23.3485 | − | 14.8936i | −10.1357 | + | 7.97080i | 63.4448 | + | 18.6291i | 19.4141 | + | 18.7642i | −5.17688 | + | 36.0060i |
4.4 | −2.95477 | + | 4.14939i | 4.94560 | + | 1.59406i | −5.87025 | − | 16.9610i | 0.0815901 | − | 0.0420626i | −21.2275 | + | 15.8112i | 1.55780 | − | 1.22507i | 48.6222 | + | 14.2768i | 21.9180 | + | 15.7672i | −0.0665456 | + | 0.462835i |
4.5 | −2.91301 | + | 4.09075i | 1.41219 | + | 5.00057i | −5.63207 | − | 16.2728i | 10.3413 | − | 5.33132i | −24.5698 | − | 8.78978i | −21.3348 | + | 16.7779i | 44.4262 | + | 13.0447i | −23.0114 | + | 14.1236i | −8.31527 | + | 57.8340i |
4.6 | −2.78793 | + | 3.91510i | −2.45036 | − | 4.58211i | −4.93890 | − | 14.2700i | −19.3913 | + | 9.99691i | 24.7708 | + | 3.18118i | −10.7080 | + | 8.42089i | 32.7449 | + | 9.61476i | −14.9914 | + | 22.4557i | 14.9227 | − | 103.790i |
4.7 | −2.76588 | + | 3.88413i | 2.37999 | − | 4.61906i | −4.81987 | − | 13.9261i | −2.51163 | + | 1.29484i | 11.3583 | + | 22.0199i | −13.8589 | + | 10.8987i | 30.8209 | + | 9.04983i | −15.6713 | − | 21.9866i | 1.91755 | − | 13.3369i |
4.8 | −2.70102 | + | 3.79305i | −3.62300 | + | 3.72477i | −4.47519 | − | 12.9302i | 11.5740 | − | 5.96683i | −4.34245 | − | 23.8029i | 15.3081 | − | 12.0384i | 25.3897 | + | 7.45510i | −0.747791 | − | 26.9896i | −8.62921 | + | 60.0174i |
4.9 | −2.69115 | + | 3.77920i | 4.57480 | − | 2.46398i | −4.42348 | − | 12.7808i | −10.4806 | + | 5.40314i | −2.99960 | + | 23.9200i | 4.10886 | − | 3.23125i | 24.5932 | + | 7.22121i | 14.8576 | − | 22.5445i | 7.78546 | − | 54.1490i |
4.10 | −2.46695 | + | 3.46434i | −4.27108 | + | 2.95937i | −3.29930 | − | 9.53272i | −2.65977 | + | 1.37121i | 0.284244 | − | 22.0971i | −16.6811 | + | 13.1182i | 8.51848 | + | 2.50125i | 9.48422 | − | 25.2794i | 1.81118 | − | 12.5970i |
4.11 | −2.41610 | + | 3.39293i | 4.94150 | − | 1.60674i | −3.05792 | − | 8.83529i | 19.5280 | − | 10.0674i | −6.48755 | + | 20.6482i | −0.748547 | + | 0.588664i | 5.39337 | + | 1.58363i | 21.8367 | − | 15.8794i | −13.0236 | + | 90.5811i |
4.12 | −2.33527 | + | 3.27943i | −5.18945 | − | 0.263780i | −2.68463 | − | 7.75672i | −8.48781 | + | 4.37577i | 12.9838 | − | 16.4025i | 13.6708 | − | 10.7509i | 0.804118 | + | 0.236110i | 26.8608 | + | 2.73775i | 5.47131 | − | 38.0538i |
4.13 | −2.19970 | + | 3.08904i | −1.31400 | − | 5.02727i | −2.08697 | − | 6.02992i | 7.73363 | − | 3.98697i | 18.4198 | + | 6.99946i | −12.7782 | + | 10.0489i | −5.89140 | − | 1.72987i | −23.5468 | + | 13.2117i | −4.69575 | + | 32.6597i |
4.14 | −2.19107 | + | 3.07693i | 0.578436 | + | 5.16386i | −2.05016 | − | 5.92354i | 10.6961 | − | 5.51421i | −17.1562 | − | 9.53457i | 16.0768 | − | 12.6430i | −6.27628 | − | 1.84288i | −26.3308 | + | 5.97392i | −6.46903 | + | 44.9931i |
4.15 | −2.18200 | + | 3.06419i | 3.37679 | + | 3.94934i | −2.01161 | − | 5.81215i | −5.70696 | + | 2.94214i | −19.4697 | + | 1.72968i | 19.1429 | − | 15.0541i | −6.67574 | − | 1.96018i | −4.19453 | + | 26.6722i | 3.43731 | − | 23.9070i |
4.16 | −2.16862 | + | 3.04540i | 3.13722 | + | 4.14221i | −1.95501 | − | 5.64863i | −16.6844 | + | 8.60138i | −19.4181 | + | 0.571212i | −19.7783 | + | 15.5539i | −7.25554 | − | 2.13042i | −7.31574 | + | 25.9900i | 9.98737 | − | 69.4637i |
4.17 | −1.98001 | + | 2.78054i | −4.08591 | − | 3.21019i | −1.19440 | − | 3.45100i | 13.0530 | − | 6.72929i | 17.0162 | − | 5.00482i | 11.9856 | − | 9.42560i | −14.2411 | − | 4.18156i | 6.38935 | + | 26.2331i | −7.13406 | + | 49.6185i |
4.18 | −1.82839 | + | 2.56761i | −2.26950 | + | 4.67433i | −0.633090 | − | 1.82919i | −7.14677 | + | 3.68442i | −7.85234 | − | 14.3737i | −9.19892 | + | 7.23411i | −18.3410 | − | 5.38542i | −16.6987 | − | 21.2168i | 3.60692 | − | 25.0867i |
4.19 | −1.69742 | + | 2.38370i | 3.62079 | − | 3.72691i | −0.184220 | − | 0.532269i | −15.8263 | + | 8.15904i | 2.73782 | + | 14.9570i | 16.2228 | − | 12.7578i | −20.8807 | − | 6.13112i | −0.779765 | − | 26.9887i | 7.41529 | − | 51.5745i |
4.20 | −1.57756 | + | 2.21538i | −1.25640 | − | 5.04197i | 0.197356 | + | 0.570222i | −6.16483 | + | 3.17819i | 13.1519 | + | 5.17063i | 21.8224 | − | 17.1613i | −22.4506 | − | 6.59209i | −23.8429 | + | 12.6694i | 2.68452 | − | 18.6712i |
See next 80 embeddings (of 1400 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
23.c | even | 11 | 1 | inner |
207.m | even | 33 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 207.4.m.a | ✓ | 1400 |
9.c | even | 3 | 1 | inner | 207.4.m.a | ✓ | 1400 |
23.c | even | 11 | 1 | inner | 207.4.m.a | ✓ | 1400 |
207.m | even | 33 | 1 | inner | 207.4.m.a | ✓ | 1400 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
207.4.m.a | ✓ | 1400 | 1.a | even | 1 | 1 | trivial |
207.4.m.a | ✓ | 1400 | 9.c | even | 3 | 1 | inner |
207.4.m.a | ✓ | 1400 | 23.c | even | 11 | 1 | inner |
207.4.m.a | ✓ | 1400 | 207.m | even | 33 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(207, [\chi])\).