Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,2,Mod(76,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([14, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.76");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.l (of order \(9\), degree \(6\), 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: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 135) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 | −1.62376 | + | 1.36249i | 0.235856 | − | 1.71592i | 0.432902 | − | 2.45511i | 0 | 1.95495 | + | 3.10759i | 0.0109747 | + | 0.0622407i | 0.522478 | + | 0.904959i | −2.88874 | − | 0.809420i | 0 | ||||
76.2 | −1.25101 | + | 1.04972i | 0.905836 | + | 1.47630i | 0.115814 | − | 0.656812i | 0 | −2.68292 | − | 0.895989i | 0.576430 | + | 3.26909i | −1.08849 | − | 1.88532i | −1.35892 | + | 2.67457i | 0 | ||||
76.3 | −0.186537 | + | 0.156523i | 1.71306 | + | 0.255766i | −0.337000 | + | 1.91122i | 0 | −0.359583 | + | 0.220424i | −0.688903 | − | 3.90696i | −0.479795 | − | 0.831029i | 2.86917 | + | 0.876286i | 0 | ||||
76.4 | 0.531925 | − | 0.446338i | −0.942993 | − | 1.45285i | −0.263570 | + | 1.49478i | 0 | −1.15006 | − | 0.351912i | 0.0506352 | + | 0.287167i | 1.22136 | + | 2.11545i | −1.22153 | + | 2.74005i | 0 | ||||
76.5 | 1.76334 | − | 1.47962i | −1.47207 | + | 0.912694i | 0.572799 | − | 3.24850i | 0 | −1.24532 | + | 3.78748i | 0.745456 | + | 4.22769i | −1.49463 | − | 2.58877i | 1.33398 | − | 2.68710i | 0 | ||||
151.1 | −1.62376 | − | 1.36249i | 0.235856 | + | 1.71592i | 0.432902 | + | 2.45511i | 0 | 1.95495 | − | 3.10759i | 0.0109747 | − | 0.0622407i | 0.522478 | − | 0.904959i | −2.88874 | + | 0.809420i | 0 | ||||
151.2 | −1.25101 | − | 1.04972i | 0.905836 | − | 1.47630i | 0.115814 | + | 0.656812i | 0 | −2.68292 | + | 0.895989i | 0.576430 | − | 3.26909i | −1.08849 | + | 1.88532i | −1.35892 | − | 2.67457i | 0 | ||||
151.3 | −0.186537 | − | 0.156523i | 1.71306 | − | 0.255766i | −0.337000 | − | 1.91122i | 0 | −0.359583 | − | 0.220424i | −0.688903 | + | 3.90696i | −0.479795 | + | 0.831029i | 2.86917 | − | 0.876286i | 0 | ||||
151.4 | 0.531925 | + | 0.446338i | −0.942993 | + | 1.45285i | −0.263570 | − | 1.49478i | 0 | −1.15006 | + | 0.351912i | 0.0506352 | − | 0.287167i | 1.22136 | − | 2.11545i | −1.22153 | − | 2.74005i | 0 | ||||
151.5 | 1.76334 | + | 1.47962i | −1.47207 | − | 0.912694i | 0.572799 | + | 3.24850i | 0 | −1.24532 | − | 3.78748i | 0.745456 | − | 4.22769i | −1.49463 | + | 2.58877i | 1.33398 | + | 2.68710i | 0 | ||||
301.1 | −1.73836 | − | 0.632710i | −0.550980 | + | 1.64208i | 1.08947 | + | 0.914177i | 0 | 1.99676 | − | 2.50591i | 0.872404 | − | 0.732034i | 0.534435 | + | 0.925669i | −2.39284 | − | 1.80950i | 0 | ||||
301.2 | −1.22047 | − | 0.444217i | −1.42893 | − | 0.978851i | −0.239858 | − | 0.201265i | 0 | 1.30916 | + | 1.82942i | 1.16252 | − | 0.975467i | 1.50214 | + | 2.60178i | 1.08370 | + | 2.79743i | 0 | ||||
301.3 | 0.282715 | + | 0.102900i | 0.864728 | − | 1.50075i | −1.46275 | − | 1.22739i | 0 | 0.398898 | − | 0.335304i | 3.28585 | − | 2.75715i | −0.588102 | − | 1.01862i | −1.50449 | − | 2.59548i | 0 | ||||
301.4 | 1.38905 | + | 0.505573i | −1.28409 | − | 1.16238i | 0.141770 | + | 0.118959i | 0 | −1.19600 | − | 2.26380i | −2.40771 | + | 2.02030i | −1.34141 | − | 2.32340i | 0.297765 | + | 2.98519i | 0 | ||||
301.5 | 2.22676 | + | 0.810474i | 1.72562 | + | 0.149064i | 2.76950 | + | 2.32388i | 0 | 3.72174 | + | 1.73050i | 0.151115 | − | 0.126801i | 1.91389 | + | 3.31495i | 2.95556 | + | 0.514456i | 0 | ||||
376.1 | −0.451738 | + | 2.56194i | 1.35188 | + | 1.08278i | −4.48006 | − | 1.63061i | 0 | −3.38471 | + | 2.97431i | −3.98979 | + | 1.45216i | 3.59987 | − | 6.23516i | 0.655179 | + | 2.92758i | 0 | ||||
376.2 | −0.226731 | + | 1.28586i | −1.31414 | − | 1.12828i | 0.277364 | + | 0.100952i | 0 | 1.74877 | − | 1.43398i | 0.822529 | − | 0.299376i | −1.49839 | + | 2.59529i | 0.453949 | + | 2.96546i | 0 | ||||
376.3 | 0.0302581 | − | 0.171602i | −0.306826 | + | 1.70466i | 1.85085 | + | 0.673656i | 0 | 0.283239 | + | 0.104232i | −1.63696 | + | 0.595804i | 0.345853 | − | 0.599035i | −2.81172 | − | 1.04607i | 0 | ||||
376.4 | 0.130038 | − | 0.737480i | 0.729834 | − | 1.57078i | 1.35242 | + | 0.492240i | 0 | −1.06351 | − | 0.742498i | −0.498180 | + | 0.181323i | 1.28774 | − | 2.23043i | −1.93468 | − | 2.29281i | 0 | ||||
376.5 | 0.344526 | − | 1.95390i | −1.72679 | + | 0.134863i | −1.81965 | − | 0.662300i | 0 | −0.331416 | + | 3.42045i | 1.54362 | − | 0.561833i | 0.0630587 | − | 0.109221i | 2.96362 | − | 0.465760i | 0 | ||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.l.d | 30 | |
5.b | even | 2 | 1 | 135.2.k.a | ✓ | 30 | |
5.c | odd | 4 | 2 | 675.2.u.c | 60 | ||
15.d | odd | 2 | 1 | 405.2.k.a | 30 | ||
27.e | even | 9 | 1 | inner | 675.2.l.d | 30 | |
135.n | odd | 18 | 1 | 405.2.k.a | 30 | ||
135.n | odd | 18 | 1 | 3645.2.a.g | 15 | ||
135.p | even | 18 | 1 | 135.2.k.a | ✓ | 30 | |
135.p | even | 18 | 1 | 3645.2.a.h | 15 | ||
135.r | odd | 36 | 2 | 675.2.u.c | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.k.a | ✓ | 30 | 5.b | even | 2 | 1 | |
135.2.k.a | ✓ | 30 | 135.p | even | 18 | 1 | |
405.2.k.a | 30 | 15.d | odd | 2 | 1 | ||
405.2.k.a | 30 | 135.n | odd | 18 | 1 | ||
675.2.l.d | 30 | 1.a | even | 1 | 1 | trivial | |
675.2.l.d | 30 | 27.e | even | 9 | 1 | inner | |
675.2.u.c | 60 | 5.c | odd | 4 | 2 | ||
675.2.u.c | 60 | 135.r | odd | 36 | 2 | ||
3645.2.a.g | 15 | 135.n | odd | 18 | 1 | ||
3645.2.a.h | 15 | 135.p | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 7 T_{2}^{27} - 3 T_{2}^{26} + 36 T_{2}^{25} + 169 T_{2}^{24} - 93 T_{2}^{23} - 315 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).