Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [900,2,Mod(299,900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("900.299");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 900.o (of order \(6\), degree \(2\), not 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: | \(7.18653618192\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
299.1 | −1.41402 | + | 0.0231729i | 1.38433 | + | 1.04098i | 1.99893 | − | 0.0655341i | 0 | −1.98160 | − | 1.43989i | 2.09698 | − | 3.63208i | −2.82501 | + | 0.138988i | 0.832737 | + | 2.88211i | 0 | ||||
299.2 | −1.41203 | + | 0.0785846i | −0.883270 | − | 1.48991i | 1.98765 | − | 0.221927i | 0 | 1.36429 | + | 2.03438i | −0.753289 | + | 1.30473i | −2.78918 | + | 0.469566i | −1.43967 | + | 2.63199i | 0 | ||||
299.3 | −1.40223 | + | 0.183692i | −0.138348 | + | 1.72652i | 1.93251 | − | 0.515159i | 0 | −0.123151 | − | 2.44639i | 0.472281 | − | 0.818015i | −2.61520 | + | 1.07736i | −2.96172 | − | 0.477722i | 0 | ||||
299.4 | −1.38473 | − | 0.287286i | 1.71411 | + | 0.248628i | 1.83493 | + | 0.795625i | 0 | −2.30215 | − | 0.836722i | −0.859188 | + | 1.48816i | −2.31231 | − | 1.62887i | 2.87637 | + | 0.852352i | 0 | ||||
299.5 | −1.30270 | + | 0.550438i | −1.28923 | + | 1.15667i | 1.39404 | − | 1.43411i | 0 | 1.04280 | − | 2.21643i | 0.00927831 | − | 0.0160705i | −1.02662 | + | 2.63554i | 0.324236 | − | 2.98243i | 0 | ||||
299.6 | −1.29375 | − | 0.571155i | 0.626740 | − | 1.61468i | 1.34756 | + | 1.47786i | 0 | −1.73308 | + | 1.73102i | 1.75781 | − | 3.04461i | −0.899318 | − | 2.68165i | −2.21439 | − | 2.02397i | 0 | ||||
299.7 | −1.28841 | − | 0.583101i | −1.73149 | + | 0.0441645i | 1.31999 | + | 1.50254i | 0 | 2.25661 | + | 0.952730i | −0.535425 | + | 0.927384i | −0.824545 | − | 2.70557i | 2.99610 | − | 0.152941i | 0 | ||||
299.8 | −1.26730 | + | 0.627655i | −1.19759 | − | 1.25131i | 1.21210 | − | 1.59085i | 0 | 2.30310 | + | 0.834117i | 2.43888 | − | 4.22426i | −0.537583 | + | 2.77687i | −0.131568 | + | 2.99711i | 0 | ||||
299.9 | −1.22744 | + | 0.702415i | 0.688876 | − | 1.58917i | 1.01323 | − | 1.72435i | 0 | 0.270699 | + | 2.43449i | −2.03748 | + | 3.52902i | −0.0324711 | + | 2.82824i | −2.05090 | − | 2.18948i | 0 | ||||
299.10 | −1.14918 | − | 0.824243i | −1.73149 | + | 0.0441645i | 0.641247 | + | 1.89441i | 0 | 2.02620 | + | 1.37641i | −0.535425 | + | 0.927384i | 0.824545 | − | 2.70557i | 2.99610 | − | 0.152941i | 0 | ||||
299.11 | −1.14151 | − | 0.834840i | 0.626740 | − | 1.61468i | 0.606083 | + | 1.90595i | 0 | −2.06343 | + | 1.31994i | 1.75781 | − | 3.04461i | 0.899318 | − | 2.68165i | −2.21439 | − | 2.02397i | 0 | ||||
299.12 | −1.04909 | + | 0.948372i | 1.73081 | + | 0.0655074i | 0.201183 | − | 1.98986i | 0 | −1.87790 | + | 1.57273i | −1.64491 | + | 2.84907i | 1.67606 | + | 2.27834i | 2.99142 | + | 0.226762i | 0 | ||||
299.13 | −0.941160 | − | 1.05557i | 1.71411 | + | 0.248628i | −0.228435 | + | 1.98691i | 0 | −1.35081 | − | 2.04336i | −0.859188 | + | 1.48816i | 2.31231 | − | 1.62887i | 2.87637 | + | 0.852352i | 0 | ||||
299.14 | −0.912116 | + | 1.08076i | −1.57038 | + | 0.730690i | −0.336088 | − | 1.97156i | 0 | 0.642667 | − | 2.36368i | −1.37334 | + | 2.37870i | 2.43734 | + | 1.43506i | 1.93218 | − | 2.29492i | 0 | ||||
299.15 | −0.782320 | + | 1.17812i | −0.415454 | − | 1.68149i | −0.775952 | − | 1.84334i | 0 | 2.30602 | + | 0.826004i | 0.0777105 | − | 0.134598i | 2.77872 | + | 0.527913i | −2.65480 | + | 1.39716i | 0 | ||||
299.16 | −0.686944 | − | 1.23617i | 1.38433 | + | 1.04098i | −1.05622 | + | 1.69835i | 0 | 0.335864 | − | 2.42635i | 2.09698 | − | 3.63208i | 2.82501 | + | 0.138988i | 0.832737 | + | 2.88211i | 0 | ||||
299.17 | −0.637958 | − | 1.26214i | −0.883270 | − | 1.48991i | −1.18602 | + | 1.61039i | 0 | −1.31699 | + | 2.06532i | −0.753289 | + | 1.30473i | 2.78918 | + | 0.469566i | −1.43967 | + | 2.63199i | 0 | ||||
299.18 | −0.629125 | + | 1.26657i | 0.415454 | + | 1.68149i | −1.20840 | − | 1.59366i | 0 | −2.39109 | − | 0.531664i | −0.0777105 | + | 0.134598i | 2.77872 | − | 0.527913i | −2.65480 | + | 1.39716i | 0 | ||||
299.19 | −0.542034 | − | 1.30622i | −0.138348 | + | 1.72652i | −1.41240 | + | 1.41603i | 0 | 2.33019 | − | 0.755118i | 0.472281 | − | 0.818015i | 2.61520 | + | 1.07736i | −2.96172 | − | 0.477722i | 0 | ||||
299.20 | −0.479908 | + | 1.33030i | 1.57038 | − | 0.730690i | −1.53938 | − | 1.27684i | 0 | 0.218396 | + | 2.43973i | 1.37334 | − | 2.37870i | 2.43734 | − | 1.43506i | 1.93218 | − | 2.29492i | 0 | ||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
20.d | odd | 2 | 1 | inner |
36.h | even | 6 | 1 | inner |
45.h | odd | 6 | 1 | inner |
180.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 900.2.o.d | 96 | |
4.b | odd | 2 | 1 | inner | 900.2.o.d | 96 | |
5.b | even | 2 | 1 | inner | 900.2.o.d | 96 | |
5.c | odd | 4 | 1 | 900.2.r.d | ✓ | 48 | |
5.c | odd | 4 | 1 | 900.2.r.e | yes | 48 | |
9.d | odd | 6 | 1 | inner | 900.2.o.d | 96 | |
20.d | odd | 2 | 1 | inner | 900.2.o.d | 96 | |
20.e | even | 4 | 1 | 900.2.r.d | ✓ | 48 | |
20.e | even | 4 | 1 | 900.2.r.e | yes | 48 | |
36.h | even | 6 | 1 | inner | 900.2.o.d | 96 | |
45.h | odd | 6 | 1 | inner | 900.2.o.d | 96 | |
45.l | even | 12 | 1 | 900.2.r.d | ✓ | 48 | |
45.l | even | 12 | 1 | 900.2.r.e | yes | 48 | |
180.n | even | 6 | 1 | inner | 900.2.o.d | 96 | |
180.v | odd | 12 | 1 | 900.2.r.d | ✓ | 48 | |
180.v | odd | 12 | 1 | 900.2.r.e | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
900.2.o.d | 96 | 1.a | even | 1 | 1 | trivial | |
900.2.o.d | 96 | 4.b | odd | 2 | 1 | inner | |
900.2.o.d | 96 | 5.b | even | 2 | 1 | inner | |
900.2.o.d | 96 | 9.d | odd | 6 | 1 | inner | |
900.2.o.d | 96 | 20.d | odd | 2 | 1 | inner | |
900.2.o.d | 96 | 36.h | even | 6 | 1 | inner | |
900.2.o.d | 96 | 45.h | odd | 6 | 1 | inner | |
900.2.o.d | 96 | 180.n | even | 6 | 1 | inner | |
900.2.r.d | ✓ | 48 | 5.c | odd | 4 | 1 | |
900.2.r.d | ✓ | 48 | 20.e | even | 4 | 1 | |
900.2.r.d | ✓ | 48 | 45.l | even | 12 | 1 | |
900.2.r.d | ✓ | 48 | 180.v | odd | 12 | 1 | |
900.2.r.e | yes | 48 | 5.c | odd | 4 | 1 | |
900.2.r.e | yes | 48 | 20.e | even | 4 | 1 | |
900.2.r.e | yes | 48 | 45.l | even | 12 | 1 | |
900.2.r.e | yes | 48 | 180.v | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):
\( T_{7}^{48} + 96 T_{7}^{46} + 5355 T_{7}^{44} + 201092 T_{7}^{42} + 5657979 T_{7}^{40} + 123073686 T_{7}^{38} + 2133295130 T_{7}^{36} + 29672232036 T_{7}^{34} + 334230570957 T_{7}^{32} + \cdots + 160000 \) |
\( T_{13}^{48} - 168 T_{13}^{46} + 16536 T_{13}^{44} - 1088580 T_{13}^{42} + 53568642 T_{13}^{40} - 2029962192 T_{13}^{38} + 61142871718 T_{13}^{36} - 1471147003398 T_{13}^{34} + \cdots + 1600000000 \) |