Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(293,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.293");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.w (of order \(4\), degree \(2\), 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: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
293.1 | −1.41107 | − | 0.0941764i | 1.68122 | + | 0.416519i | 1.98226 | + | 0.265780i | 0 | −2.33310 | − | 0.746071i | 0.361989 | − | 0.361989i | −2.77209 | − | 0.561717i | 2.65302 | + | 1.40052i | 0 | ||||
293.2 | −1.39193 | − | 0.250043i | −1.01856 | + | 1.40091i | 1.87496 | + | 0.696087i | 0 | 1.76805 | − | 1.69529i | 2.29041 | − | 2.29041i | −2.43576 | − | 1.43773i | −0.925085 | − | 2.85381i | 0 | ||||
293.3 | −1.30986 | + | 0.533177i | 0.667305 | − | 1.59834i | 1.43144 | − | 1.39677i | 0 | −0.0218716 | + | 2.44939i | −0.582772 | + | 0.582772i | −1.13026 | + | 2.59278i | −2.10941 | − | 2.13317i | 0 | ||||
293.4 | −1.11940 | − | 0.864261i | −0.170116 | − | 1.72368i | 0.506107 | + | 1.93490i | 0 | −1.29928 | + | 2.07651i | −2.06963 | + | 2.06963i | 1.10573 | − | 2.60334i | −2.94212 | + | 0.586449i | 0 | ||||
293.5 | −0.864261 | − | 1.11940i | 0.170116 | + | 1.72368i | −0.506107 | + | 1.93490i | 0 | 1.78246 | − | 1.68013i | −2.06963 | + | 2.06963i | 2.60334 | − | 1.10573i | −2.94212 | + | 0.586449i | 0 | ||||
293.6 | −0.533177 | + | 1.30986i | −1.59834 | + | 0.667305i | −1.43144 | − | 1.39677i | 0 | −0.0218716 | − | 2.44939i | −0.582772 | + | 0.582772i | 2.59278 | − | 1.13026i | 2.10941 | − | 2.13317i | 0 | ||||
293.7 | −0.250043 | − | 1.39193i | 1.01856 | − | 1.40091i | −1.87496 | + | 0.696087i | 0 | −2.20465 | − | 1.06748i | 2.29041 | − | 2.29041i | 1.43773 | + | 2.43576i | −0.925085 | − | 2.85381i | 0 | ||||
293.8 | −0.0941764 | − | 1.41107i | −1.68122 | − | 0.416519i | −1.98226 | + | 0.265780i | 0 | −0.429408 | + | 2.41156i | 0.361989 | − | 0.361989i | 0.561717 | + | 2.77209i | 2.65302 | + | 1.40052i | 0 | ||||
293.9 | 0.0941764 | + | 1.41107i | 0.416519 | + | 1.68122i | −1.98226 | + | 0.265780i | 0 | −2.33310 | + | 0.746071i | 0.361989 | − | 0.361989i | −0.561717 | − | 2.77209i | −2.65302 | + | 1.40052i | 0 | ||||
293.10 | 0.250043 | + | 1.39193i | 1.40091 | − | 1.01856i | −1.87496 | + | 0.696087i | 0 | 1.76805 | + | 1.69529i | 2.29041 | − | 2.29041i | −1.43773 | − | 2.43576i | 0.925085 | − | 2.85381i | 0 | ||||
293.11 | 0.533177 | − | 1.30986i | −0.667305 | + | 1.59834i | −1.43144 | − | 1.39677i | 0 | 1.73781 | + | 1.72627i | −0.582772 | + | 0.582772i | −2.59278 | + | 1.13026i | −2.10941 | − | 2.13317i | 0 | ||||
293.12 | 0.864261 | + | 1.11940i | −1.72368 | − | 0.170116i | −0.506107 | + | 1.93490i | 0 | −1.29928 | − | 2.07651i | −2.06963 | + | 2.06963i | −2.60334 | + | 1.10573i | 2.94212 | + | 0.586449i | 0 | ||||
293.13 | 1.11940 | + | 0.864261i | 1.72368 | + | 0.170116i | 0.506107 | + | 1.93490i | 0 | 1.78246 | + | 1.68013i | −2.06963 | + | 2.06963i | −1.10573 | + | 2.60334i | 2.94212 | + | 0.586449i | 0 | ||||
293.14 | 1.30986 | − | 0.533177i | 1.59834 | − | 0.667305i | 1.43144 | − | 1.39677i | 0 | 1.73781 | − | 1.72627i | −0.582772 | + | 0.582772i | 1.13026 | − | 2.59278i | 2.10941 | − | 2.13317i | 0 | ||||
293.15 | 1.39193 | + | 0.250043i | −1.40091 | + | 1.01856i | 1.87496 | + | 0.696087i | 0 | −2.20465 | + | 1.06748i | 2.29041 | − | 2.29041i | 2.43576 | + | 1.43773i | 0.925085 | − | 2.85381i | 0 | ||||
293.16 | 1.41107 | + | 0.0941764i | −0.416519 | − | 1.68122i | 1.98226 | + | 0.265780i | 0 | −0.429408 | − | 2.41156i | 0.361989 | − | 0.361989i | 2.77209 | + | 0.561717i | −2.65302 | + | 1.40052i | 0 | ||||
557.1 | −1.41107 | + | 0.0941764i | 1.68122 | − | 0.416519i | 1.98226 | − | 0.265780i | 0 | −2.33310 | + | 0.746071i | 0.361989 | + | 0.361989i | −2.77209 | + | 0.561717i | 2.65302 | − | 1.40052i | 0 | ||||
557.2 | −1.39193 | + | 0.250043i | −1.01856 | − | 1.40091i | 1.87496 | − | 0.696087i | 0 | 1.76805 | + | 1.69529i | 2.29041 | + | 2.29041i | −2.43576 | + | 1.43773i | −0.925085 | + | 2.85381i | 0 | ||||
557.3 | −1.30986 | − | 0.533177i | 0.667305 | + | 1.59834i | 1.43144 | + | 1.39677i | 0 | −0.0218716 | − | 2.44939i | −0.582772 | − | 0.582772i | −1.13026 | − | 2.59278i | −2.10941 | + | 2.13317i | 0 | ||||
557.4 | −1.11940 | + | 0.864261i | −0.170116 | + | 1.72368i | 0.506107 | − | 1.93490i | 0 | −1.29928 | − | 2.07651i | −2.06963 | − | 2.06963i | 1.10573 | + | 2.60334i | −2.94212 | − | 0.586449i | 0 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
8.b | even | 2 | 1 | inner |
15.e | even | 4 | 1 | inner |
24.h | odd | 2 | 1 | inner |
40.i | odd | 4 | 1 | inner |
120.w | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.2.w.j | 32 | |
3.b | odd | 2 | 1 | inner | 600.2.w.j | 32 | |
5.b | even | 2 | 1 | 120.2.w.c | ✓ | 32 | |
5.c | odd | 4 | 1 | 120.2.w.c | ✓ | 32 | |
5.c | odd | 4 | 1 | inner | 600.2.w.j | 32 | |
8.b | even | 2 | 1 | inner | 600.2.w.j | 32 | |
15.d | odd | 2 | 1 | 120.2.w.c | ✓ | 32 | |
15.e | even | 4 | 1 | 120.2.w.c | ✓ | 32 | |
15.e | even | 4 | 1 | inner | 600.2.w.j | 32 | |
20.d | odd | 2 | 1 | 480.2.bi.c | 32 | ||
20.e | even | 4 | 1 | 480.2.bi.c | 32 | ||
24.h | odd | 2 | 1 | inner | 600.2.w.j | 32 | |
40.e | odd | 2 | 1 | 480.2.bi.c | 32 | ||
40.f | even | 2 | 1 | 120.2.w.c | ✓ | 32 | |
40.i | odd | 4 | 1 | 120.2.w.c | ✓ | 32 | |
40.i | odd | 4 | 1 | inner | 600.2.w.j | 32 | |
40.k | even | 4 | 1 | 480.2.bi.c | 32 | ||
60.h | even | 2 | 1 | 480.2.bi.c | 32 | ||
60.l | odd | 4 | 1 | 480.2.bi.c | 32 | ||
120.i | odd | 2 | 1 | 120.2.w.c | ✓ | 32 | |
120.m | even | 2 | 1 | 480.2.bi.c | 32 | ||
120.q | odd | 4 | 1 | 480.2.bi.c | 32 | ||
120.w | even | 4 | 1 | 120.2.w.c | ✓ | 32 | |
120.w | even | 4 | 1 | inner | 600.2.w.j | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.w.c | ✓ | 32 | 5.b | even | 2 | 1 | |
120.2.w.c | ✓ | 32 | 5.c | odd | 4 | 1 | |
120.2.w.c | ✓ | 32 | 15.d | odd | 2 | 1 | |
120.2.w.c | ✓ | 32 | 15.e | even | 4 | 1 | |
120.2.w.c | ✓ | 32 | 40.f | even | 2 | 1 | |
120.2.w.c | ✓ | 32 | 40.i | odd | 4 | 1 | |
120.2.w.c | ✓ | 32 | 120.i | odd | 2 | 1 | |
120.2.w.c | ✓ | 32 | 120.w | even | 4 | 1 | |
480.2.bi.c | 32 | 20.d | odd | 2 | 1 | ||
480.2.bi.c | 32 | 20.e | even | 4 | 1 | ||
480.2.bi.c | 32 | 40.e | odd | 2 | 1 | ||
480.2.bi.c | 32 | 40.k | even | 4 | 1 | ||
480.2.bi.c | 32 | 60.h | even | 2 | 1 | ||
480.2.bi.c | 32 | 60.l | odd | 4 | 1 | ||
480.2.bi.c | 32 | 120.m | even | 2 | 1 | ||
480.2.bi.c | 32 | 120.q | odd | 4 | 1 | ||
600.2.w.j | 32 | 1.a | even | 1 | 1 | trivial | |
600.2.w.j | 32 | 3.b | odd | 2 | 1 | inner | |
600.2.w.j | 32 | 5.c | odd | 4 | 1 | inner | |
600.2.w.j | 32 | 8.b | even | 2 | 1 | inner | |
600.2.w.j | 32 | 15.e | even | 4 | 1 | inner | |
600.2.w.j | 32 | 24.h | odd | 2 | 1 | inner | |
600.2.w.j | 32 | 40.i | odd | 4 | 1 | inner | |
600.2.w.j | 32 | 120.w | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):
\( T_{7}^{8} + 4T_{7}^{5} + 92T_{7}^{4} + 40T_{7}^{3} + 8T_{7}^{2} - 16T_{7} + 16 \) |
\( T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128 \) |
\( T_{17}^{16} + 1420T_{17}^{12} + 639792T_{17}^{8} + 93389888T_{17}^{4} + 220463104 \) |