Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,2,Mod(7,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 5, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 825.cw (of order \(20\), degree \(8\), 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: | \(6.58765816676\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 165) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −2.46605 | + | 0.390584i | −0.453990 | − | 0.891007i | 4.02673 | − | 1.30836i | 0 | 1.46758 | + | 2.01994i | 3.26542 | + | 1.66382i | −4.96977 | + | 2.53222i | −0.587785 | + | 0.809017i | 0 | ||||
7.2 | −2.13992 | + | 0.338930i | 0.453990 | + | 0.891007i | 2.56227 | − | 0.832532i | 0 | −1.27349 | − | 1.75281i | 0.770101 | + | 0.392386i | −1.33999 | + | 0.682757i | −0.587785 | + | 0.809017i | 0 | ||||
7.3 | −1.86197 | + | 0.294907i | 0.453990 | + | 0.891007i | 1.47784 | − | 0.480180i | 0 | −1.10808 | − | 1.52514i | 0.414728 | + | 0.211314i | 0.749326 | − | 0.381801i | −0.587785 | + | 0.809017i | 0 | ||||
7.4 | −1.73704 | + | 0.275120i | −0.453990 | − | 0.891007i | 1.03949 | − | 0.337751i | 0 | 1.03373 | + | 1.42281i | −3.16835 | − | 1.61435i | 1.42129 | − | 0.724185i | −0.587785 | + | 0.809017i | 0 | ||||
7.5 | −0.779339 | + | 0.123435i | −0.453990 | − | 0.891007i | −1.30998 | + | 0.425638i | 0 | 0.463794 | + | 0.638358i | −0.804274 | − | 0.409798i | 2.37448 | − | 1.20986i | −0.587785 | + | 0.809017i | 0 | ||||
7.6 | −0.139396 | + | 0.0220781i | 0.453990 | + | 0.891007i | −1.88317 | + | 0.611879i | 0 | −0.0829560 | − | 0.114179i | 1.56314 | + | 0.796462i | 0.500497 | − | 0.255016i | −0.587785 | + | 0.809017i | 0 | ||||
7.7 | 0.329725 | − | 0.0522232i | 0.453990 | + | 0.891007i | −1.79612 | + | 0.583595i | 0 | 0.196223 | + | 0.270078i | −2.61195 | − | 1.33085i | −1.15665 | + | 0.589341i | −0.587785 | + | 0.809017i | 0 | ||||
7.8 | 1.12702 | − | 0.178502i | −0.453990 | − | 0.891007i | −0.663811 | + | 0.215685i | 0 | −0.670701 | − | 0.923140i | −1.36271 | − | 0.694338i | −2.74302 | + | 1.39764i | −0.587785 | + | 0.809017i | 0 | ||||
7.9 | 1.27330 | − | 0.201671i | −0.453990 | − | 0.891007i | −0.321488 | + | 0.104458i | 0 | −0.757757 | − | 1.04296i | 4.03315 | + | 2.05499i | −2.68561 | + | 1.36839i | −0.587785 | + | 0.809017i | 0 | ||||
7.10 | 1.44976 | − | 0.229620i | 0.453990 | + | 0.891007i | 0.146976 | − | 0.0477555i | 0 | 0.862772 | + | 1.18750i | 3.00927 | + | 1.53330i | −2.41359 | + | 1.22978i | −0.587785 | + | 0.809017i | 0 | ||||
7.11 | 2.36180 | − | 0.374072i | 0.453990 | + | 0.891007i | 3.53603 | − | 1.14893i | 0 | 1.40553 | + | 1.93455i | 0.732807 | + | 0.373384i | 3.66039 | − | 1.86506i | −0.587785 | + | 0.809017i | 0 | ||||
7.12 | 2.58211 | − | 0.408966i | −0.453990 | − | 0.891007i | 4.59791 | − | 1.49395i | 0 | −1.53664 | − | 2.11501i | 1.91487 | + | 0.975676i | 6.60261 | − | 3.36420i | −0.587785 | + | 0.809017i | 0 | ||||
118.1 | −2.46605 | − | 0.390584i | −0.453990 | + | 0.891007i | 4.02673 | + | 1.30836i | 0 | 1.46758 | − | 2.01994i | 3.26542 | − | 1.66382i | −4.96977 | − | 2.53222i | −0.587785 | − | 0.809017i | 0 | ||||
118.2 | −2.13992 | − | 0.338930i | 0.453990 | − | 0.891007i | 2.56227 | + | 0.832532i | 0 | −1.27349 | + | 1.75281i | 0.770101 | − | 0.392386i | −1.33999 | − | 0.682757i | −0.587785 | − | 0.809017i | 0 | ||||
118.3 | −1.86197 | − | 0.294907i | 0.453990 | − | 0.891007i | 1.47784 | + | 0.480180i | 0 | −1.10808 | + | 1.52514i | 0.414728 | − | 0.211314i | 0.749326 | + | 0.381801i | −0.587785 | − | 0.809017i | 0 | ||||
118.4 | −1.73704 | − | 0.275120i | −0.453990 | + | 0.891007i | 1.03949 | + | 0.337751i | 0 | 1.03373 | − | 1.42281i | −3.16835 | + | 1.61435i | 1.42129 | + | 0.724185i | −0.587785 | − | 0.809017i | 0 | ||||
118.5 | −0.779339 | − | 0.123435i | −0.453990 | + | 0.891007i | −1.30998 | − | 0.425638i | 0 | 0.463794 | − | 0.638358i | −0.804274 | + | 0.409798i | 2.37448 | + | 1.20986i | −0.587785 | − | 0.809017i | 0 | ||||
118.6 | −0.139396 | − | 0.0220781i | 0.453990 | − | 0.891007i | −1.88317 | − | 0.611879i | 0 | −0.0829560 | + | 0.114179i | 1.56314 | − | 0.796462i | 0.500497 | + | 0.255016i | −0.587785 | − | 0.809017i | 0 | ||||
118.7 | 0.329725 | + | 0.0522232i | 0.453990 | − | 0.891007i | −1.79612 | − | 0.583595i | 0 | 0.196223 | − | 0.270078i | −2.61195 | + | 1.33085i | −1.15665 | − | 0.589341i | −0.587785 | − | 0.809017i | 0 | ||||
118.8 | 1.12702 | + | 0.178502i | −0.453990 | + | 0.891007i | −0.663811 | − | 0.215685i | 0 | −0.670701 | + | 0.923140i | −1.36271 | + | 0.694338i | −2.74302 | − | 1.39764i | −0.587785 | − | 0.809017i | 0 | ||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 825.2.cw.b | 96 | |
5.b | even | 2 | 1 | 165.2.w.a | ✓ | 96 | |
5.c | odd | 4 | 1 | 165.2.w.a | ✓ | 96 | |
5.c | odd | 4 | 1 | inner | 825.2.cw.b | 96 | |
11.d | odd | 10 | 1 | inner | 825.2.cw.b | 96 | |
15.d | odd | 2 | 1 | 495.2.bj.c | 96 | ||
15.e | even | 4 | 1 | 495.2.bj.c | 96 | ||
55.h | odd | 10 | 1 | 165.2.w.a | ✓ | 96 | |
55.l | even | 20 | 1 | 165.2.w.a | ✓ | 96 | |
55.l | even | 20 | 1 | inner | 825.2.cw.b | 96 | |
165.r | even | 10 | 1 | 495.2.bj.c | 96 | ||
165.u | odd | 20 | 1 | 495.2.bj.c | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.2.w.a | ✓ | 96 | 5.b | even | 2 | 1 | |
165.2.w.a | ✓ | 96 | 5.c | odd | 4 | 1 | |
165.2.w.a | ✓ | 96 | 55.h | odd | 10 | 1 | |
165.2.w.a | ✓ | 96 | 55.l | even | 20 | 1 | |
495.2.bj.c | 96 | 15.d | odd | 2 | 1 | ||
495.2.bj.c | 96 | 15.e | even | 4 | 1 | ||
495.2.bj.c | 96 | 165.r | even | 10 | 1 | ||
495.2.bj.c | 96 | 165.u | odd | 20 | 1 | ||
825.2.cw.b | 96 | 1.a | even | 1 | 1 | trivial | |
825.2.cw.b | 96 | 5.c | odd | 4 | 1 | inner | |
825.2.cw.b | 96 | 11.d | odd | 10 | 1 | inner | |
825.2.cw.b | 96 | 55.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 86 T_{2}^{92} - 180 T_{2}^{89} + 4943 T_{2}^{88} - 20 T_{2}^{87} + 15800 T_{2}^{85} + \cdots + 43046721 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\).