Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(251,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.251");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.bl (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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 315) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | −2.32000 | − | 1.33945i | 0 | 2.58825 | + | 4.48298i | −0.500000 | − | 0.866025i | 0 | −1.06513 | − | 2.42188i | − | 8.50953i | 0 | 2.67890i | |||||||||
251.2 | −1.99878 | − | 1.15400i | 0 | 1.66342 | + | 2.88113i | −0.500000 | − | 0.866025i | 0 | 2.12864 | + | 1.57127i | − | 3.06234i | 0 | 2.30799i | |||||||||
251.3 | −1.94805 | − | 1.12471i | 0 | 1.52994 | + | 2.64993i | −0.500000 | − | 0.866025i | 0 | −2.43271 | + | 1.04016i | − | 2.38412i | 0 | 2.24942i | |||||||||
251.4 | −1.58089 | − | 0.912729i | 0 | 0.666147 | + | 1.15380i | −0.500000 | − | 0.866025i | 0 | 2.11583 | − | 1.58847i | 1.21887i | 0 | 1.82546i | ||||||||||
251.5 | −1.02342 | − | 0.590871i | 0 | −0.301744 | − | 0.522636i | −0.500000 | − | 0.866025i | 0 | −2.07834 | − | 1.63723i | 3.07665i | 0 | 1.18174i | ||||||||||
251.6 | −0.963349 | − | 0.556190i | 0 | −0.381305 | − | 0.660440i | −0.500000 | − | 0.866025i | 0 | −1.36360 | − | 2.26729i | 3.07307i | 0 | 1.11238i | ||||||||||
251.7 | −0.552767 | − | 0.319140i | 0 | −0.796299 | − | 1.37923i | −0.500000 | − | 0.866025i | 0 | 2.58723 | + | 0.553395i | 2.29308i | 0 | 0.638280i | ||||||||||
251.8 | 0.334847 | + | 0.193324i | 0 | −0.925251 | − | 1.60258i | −0.500000 | − | 0.866025i | 0 | 1.56056 | − | 2.13651i | − | 1.48879i | 0 | − | 0.386648i | ||||||||
251.9 | 1.13192 | + | 0.653515i | 0 | −0.145837 | − | 0.252598i | −0.500000 | − | 0.866025i | 0 | 0.709992 | + | 2.54871i | − | 2.99529i | 0 | − | 1.30703i | ||||||||
251.10 | 1.41561 | + | 0.817305i | 0 | 0.335974 | + | 0.581925i | −0.500000 | − | 0.866025i | 0 | −2.43457 | − | 1.03581i | − | 2.17085i | 0 | − | 1.63461i | ||||||||
251.11 | 2.21303 | + | 1.27769i | 0 | 2.26501 | + | 3.92311i | −0.500000 | − | 0.866025i | 0 | 1.57885 | − | 2.12303i | 6.46517i | 0 | − | 2.55539i | |||||||||
251.12 | 2.29184 | + | 1.32320i | 0 | 2.50170 | + | 4.33307i | −0.500000 | − | 0.866025i | 0 | −1.30676 | + | 2.30052i | 7.94816i | 0 | − | 2.64639i | |||||||||
881.1 | −2.32000 | + | 1.33945i | 0 | 2.58825 | − | 4.48298i | −0.500000 | + | 0.866025i | 0 | −1.06513 | + | 2.42188i | 8.50953i | 0 | − | 2.67890i | |||||||||
881.2 | −1.99878 | + | 1.15400i | 0 | 1.66342 | − | 2.88113i | −0.500000 | + | 0.866025i | 0 | 2.12864 | − | 1.57127i | 3.06234i | 0 | − | 2.30799i | |||||||||
881.3 | −1.94805 | + | 1.12471i | 0 | 1.52994 | − | 2.64993i | −0.500000 | + | 0.866025i | 0 | −2.43271 | − | 1.04016i | 2.38412i | 0 | − | 2.24942i | |||||||||
881.4 | −1.58089 | + | 0.912729i | 0 | 0.666147 | − | 1.15380i | −0.500000 | + | 0.866025i | 0 | 2.11583 | + | 1.58847i | − | 1.21887i | 0 | − | 1.82546i | ||||||||
881.5 | −1.02342 | + | 0.590871i | 0 | −0.301744 | + | 0.522636i | −0.500000 | + | 0.866025i | 0 | −2.07834 | + | 1.63723i | − | 3.07665i | 0 | − | 1.18174i | ||||||||
881.6 | −0.963349 | + | 0.556190i | 0 | −0.381305 | + | 0.660440i | −0.500000 | + | 0.866025i | 0 | −1.36360 | + | 2.26729i | − | 3.07307i | 0 | − | 1.11238i | ||||||||
881.7 | −0.552767 | + | 0.319140i | 0 | −0.796299 | + | 1.37923i | −0.500000 | + | 0.866025i | 0 | 2.58723 | − | 0.553395i | − | 2.29308i | 0 | − | 0.638280i | ||||||||
881.8 | 0.334847 | − | 0.193324i | 0 | −0.925251 | + | 1.60258i | −0.500000 | + | 0.866025i | 0 | 1.56056 | + | 2.13651i | 1.48879i | 0 | 0.386648i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.bl.i | 24 | |
3.b | odd | 2 | 1 | 315.2.bl.i | ✓ | 24 | |
7.b | odd | 2 | 1 | 945.2.bl.j | 24 | ||
9.c | even | 3 | 1 | 315.2.bl.j | yes | 24 | |
9.d | odd | 6 | 1 | 945.2.bl.j | 24 | ||
21.c | even | 2 | 1 | 315.2.bl.j | yes | 24 | |
63.l | odd | 6 | 1 | 315.2.bl.i | ✓ | 24 | |
63.o | even | 6 | 1 | inner | 945.2.bl.i | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bl.i | ✓ | 24 | 3.b | odd | 2 | 1 | |
315.2.bl.i | ✓ | 24 | 63.l | odd | 6 | 1 | |
315.2.bl.j | yes | 24 | 9.c | even | 3 | 1 | |
315.2.bl.j | yes | 24 | 21.c | even | 2 | 1 | |
945.2.bl.i | 24 | 1.a | even | 1 | 1 | trivial | |
945.2.bl.i | 24 | 63.o | even | 6 | 1 | inner | |
945.2.bl.j | 24 | 7.b | odd | 2 | 1 | ||
945.2.bl.j | 24 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):
\( T_{2}^{24} + 6 T_{2}^{23} - 3 T_{2}^{22} - 90 T_{2}^{21} - 48 T_{2}^{20} + 954 T_{2}^{19} + 1571 T_{2}^{18} + \cdots + 14161 \) |
\( T_{11}^{24} + 9 T_{11}^{23} - 36 T_{11}^{22} - 567 T_{11}^{21} + 1074 T_{11}^{20} + 21777 T_{11}^{19} + \cdots + 207475216 \) |
\( T_{13}^{24} + 3 T_{13}^{23} - 69 T_{13}^{22} - 216 T_{13}^{21} + 3150 T_{13}^{20} + 11916 T_{13}^{19} + \cdots + 10732176 \) |
\( T_{17}^{12} - 9 T_{17}^{11} - 57 T_{17}^{10} + 681 T_{17}^{9} + 174 T_{17}^{8} - 15678 T_{17}^{7} + \cdots + 410004 \) |