Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(299,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.299");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.m (of order \(2\), degree \(1\), 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: | \(4.79102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{11} \) |
| Twist minimal: | no (minimal twist has level 120) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{15} + 3\nu^{13} + 10\nu^{11} + 8\nu^{9} + 8\nu^{7} + 64\nu^{5} + 64\nu ) / 192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} + \nu^{7} - 2\nu^{5} - 4\nu^{3} + 8\nu ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{15} - 3\nu^{13} + 2\nu^{11} - 8\nu^{9} + 4\nu^{7} + 8\nu^{5} - 48\nu^{3} + 32\nu ) / 192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{15} - 3\nu^{13} - 10\nu^{11} - 20\nu^{9} - 44\nu^{7} - 40\nu^{5} - 144\nu^{3} - 160\nu ) / 192 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{10} + \nu^{8} - 2\nu^{6} - 4\nu^{4} + 8\nu^{2} - 16 ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{14} - 3\nu^{12} + 4\nu^{10} + 32\nu^{8} + 32\nu^{6} + 112\nu^{4} + 192\nu^{2} + 256 ) / 192 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -\nu^{15} + 3\nu^{13} - 4\nu^{11} + 4\nu^{9} + 28\nu^{7} + 56\nu^{5} + 144\nu^{3} + 224\nu ) / 192 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{14} + 3\nu^{12} + 4\nu^{10} + 14\nu^{8} - 4\nu^{6} + 40\nu^{4} + 48\nu^{2} - 32 ) / 96 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( -\nu^{14} + 3\nu^{12} + 8\nu^{10} + 4\nu^{8} + 40\nu^{6} + 128\nu^{4} + 96\nu^{2} + 320 ) / 192 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( -\nu^{15} - \nu^{11} - 2\nu^{9} - 8\nu^{7} - 40\nu^{5} + 32\nu ) / 96 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( \nu^{14} - 3\nu^{12} - 2\nu^{10} - 10\nu^{8} - 28\nu^{6} - 8\nu^{4} - 48\nu^{2} - 128 ) / 96 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( \nu^{14} + \nu^{12} + 8\nu^{8} + 16\nu^{6} + 16\nu^{4} + 64\nu^{2} ) / 64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{14} + 2\beta_{12} + \beta_{8} - \beta_{2} - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{13} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{15} - \beta_{14} - 2\beta_{11} - \beta_{8} - \beta_{2} - 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{13} + \beta_{10} - 3\beta_{7} + \beta_{6} + 3\beta_{5} - 3\beta_{4} - 2\beta_{3} - 4\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -2\beta_{15} - 3\beta_{14} - 4\beta_{12} + 2\beta_{11} + 4\beta_{9} + \beta_{8} - 3\beta_{2} - 1 \)
|
| \(\nu^{9}\) | \(=\) |
\( \beta_{13} - \beta_{10} - 5\beta_{7} - \beta_{6} - 11\beta_{5} - 5\beta_{4} - 6\beta_{3} + 4\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -6\beta_{15} - 5\beta_{14} - 12\beta_{12} + 6\beta_{11} + 4\beta_{9} - 17\beta_{8} + 11\beta_{2} - 7 \)
|
| \(\nu^{11}\) | \(=\) |
\( 7\beta_{13} - 7\beta_{10} - 3\beta_{7} + 9\beta_{6} + 3\beta_{5} + 13\beta_{4} + 6\beta_{3} - 4\beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 6\beta_{15} - 3\beta_{14} + 12\beta_{12} + 10\beta_{11} - 20\beta_{9} + 9\beta_{8} - 3\beta_{2} + 15 \)
|
| \(\nu^{13}\) | \(=\) |
\( 17\beta_{13} + 15\beta_{10} + 11\beta_{7} - 33\beta_{6} + 21\beta_{5} + 27\beta_{4} - 6\beta_{3} - 28\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 42\beta_{15} + 27\beta_{14} - 12\beta_{12} + 6\beta_{11} - 12\beta_{9} - 17\beta_{8} - 5\beta_{2} + 57 \)
|
| \(\nu^{15}\) | \(=\) |
\( -57\beta_{13} - 39\beta_{10} - 3\beta_{7} - 55\beta_{6} + 35\beta_{5} - 19\beta_{4} + 22\beta_{3} + 60\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 299.1 |
|
−1.29041 | − | 0.578647i | 1.56044 | − | 0.751690i | 1.33034 | + | 1.49339i | 0 | −2.44857 | + | 0.0670494i | −4.28591 | −0.852541 | − | 2.69688i | 1.86993 | − | 2.34593i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.2 | −1.29041 | + | 0.578647i | 1.56044 | + | 0.751690i | 1.33034 | − | 1.49339i | 0 | −2.44857 | − | 0.0670494i | −4.28591 | −0.852541 | + | 2.69688i | 1.86993 | + | 2.34593i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.3 | −1.15595 | − | 0.814732i | −0.887900 | + | 1.48716i | 0.672424 | + | 1.88357i | 0 | 2.23800 | − | 0.995672i | −0.797253 | 0.757320 | − | 2.72515i | −1.42327 | − | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.4 | −1.15595 | + | 0.814732i | −0.887900 | − | 1.48716i | 0.672424 | − | 1.88357i | 0 | 2.23800 | + | 0.995672i | −0.797253 | 0.757320 | + | 2.72515i | −1.42327 | + | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.5 | −0.842022 | − | 1.13622i | 0.218455 | + | 1.71822i | −0.581998 | + | 1.91345i | 0 | 1.76833 | − | 1.69499i | 3.64426 | 2.66415 | − | 0.949886i | −2.90455 | + | 0.750707i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.6 | −0.842022 | + | 1.13622i | 0.218455 | − | 1.71822i | −0.581998 | − | 1.91345i | 0 | 1.76833 | + | 1.69499i | 3.64426 | 2.66415 | + | 0.949886i | −2.90455 | − | 0.750707i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.7 | −0.199044 | − | 1.40014i | 1.65195 | − | 0.520627i | −1.92076 | + | 0.557378i | 0 | −1.05776 | − | 2.20933i | 1.92736 | 1.16272 | + | 2.57839i | 2.45790 | − | 1.72010i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.8 | −0.199044 | + | 1.40014i | 1.65195 | + | 0.520627i | −1.92076 | − | 0.557378i | 0 | −1.05776 | + | 2.20933i | 1.92736 | 1.16272 | − | 2.57839i | 2.45790 | + | 1.72010i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.9 | 0.199044 | − | 1.40014i | −1.65195 | − | 0.520627i | −1.92076 | − | 0.557378i | 0 | −1.05776 | + | 2.20933i | −1.92736 | −1.16272 | + | 2.57839i | 2.45790 | + | 1.72010i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.10 | 0.199044 | + | 1.40014i | −1.65195 | + | 0.520627i | −1.92076 | + | 0.557378i | 0 | −1.05776 | − | 2.20933i | −1.92736 | −1.16272 | − | 2.57839i | 2.45790 | − | 1.72010i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.11 | 0.842022 | − | 1.13622i | −0.218455 | + | 1.71822i | −0.581998 | − | 1.91345i | 0 | 1.76833 | + | 1.69499i | −3.64426 | −2.66415 | − | 0.949886i | −2.90455 | − | 0.750707i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.12 | 0.842022 | + | 1.13622i | −0.218455 | − | 1.71822i | −0.581998 | + | 1.91345i | 0 | 1.76833 | − | 1.69499i | −3.64426 | −2.66415 | + | 0.949886i | −2.90455 | + | 0.750707i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.13 | 1.15595 | − | 0.814732i | 0.887900 | + | 1.48716i | 0.672424 | − | 1.88357i | 0 | 2.23800 | + | 0.995672i | 0.797253 | −0.757320 | − | 2.72515i | −1.42327 | + | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.14 | 1.15595 | + | 0.814732i | 0.887900 | − | 1.48716i | 0.672424 | + | 1.88357i | 0 | 2.23800 | − | 0.995672i | 0.797253 | −0.757320 | + | 2.72515i | −1.42327 | − | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.15 | 1.29041 | − | 0.578647i | −1.56044 | − | 0.751690i | 1.33034 | − | 1.49339i | 0 | −2.44857 | − | 0.0670494i | 4.28591 | 0.852541 | − | 2.69688i | 1.86993 | + | 2.34593i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 299.16 | 1.29041 | + | 0.578647i | −1.56044 | + | 0.751690i | 1.33034 | + | 1.49339i | 0 | −2.44857 | + | 0.0670494i | 4.28591 | 0.852541 | + | 2.69688i | 1.86993 | − | 2.34593i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 24.f | even | 2 | 1 | inner |
| 120.m | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 600.2.m.d | 16 | |
| 3.b | odd | 2 | 1 | 600.2.m.c | 16 | ||
| 4.b | odd | 2 | 1 | 2400.2.m.c | 16 | ||
| 5.b | even | 2 | 1 | inner | 600.2.m.d | 16 | |
| 5.c | odd | 4 | 1 | 120.2.b.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 600.2.b.e | 8 | ||
| 8.b | even | 2 | 1 | 2400.2.m.d | 16 | ||
| 8.d | odd | 2 | 1 | 600.2.m.c | 16 | ||
| 12.b | even | 2 | 1 | 2400.2.m.d | 16 | ||
| 15.d | odd | 2 | 1 | 600.2.m.c | 16 | ||
| 15.e | even | 4 | 1 | 120.2.b.a | ✓ | 8 | |
| 15.e | even | 4 | 1 | 600.2.b.f | 8 | ||
| 20.d | odd | 2 | 1 | 2400.2.m.c | 16 | ||
| 20.e | even | 4 | 1 | 480.2.b.a | 8 | ||
| 20.e | even | 4 | 1 | 2400.2.b.e | 8 | ||
| 24.f | even | 2 | 1 | inner | 600.2.m.d | 16 | |
| 24.h | odd | 2 | 1 | 2400.2.m.c | 16 | ||
| 40.e | odd | 2 | 1 | 600.2.m.c | 16 | ||
| 40.f | even | 2 | 1 | 2400.2.m.d | 16 | ||
| 40.i | odd | 4 | 1 | 480.2.b.b | 8 | ||
| 40.i | odd | 4 | 1 | 2400.2.b.f | 8 | ||
| 40.k | even | 4 | 1 | 120.2.b.a | ✓ | 8 | |
| 40.k | even | 4 | 1 | 600.2.b.f | 8 | ||
| 60.h | even | 2 | 1 | 2400.2.m.d | 16 | ||
| 60.l | odd | 4 | 1 | 480.2.b.b | 8 | ||
| 60.l | odd | 4 | 1 | 2400.2.b.f | 8 | ||
| 120.i | odd | 2 | 1 | 2400.2.m.c | 16 | ||
| 120.m | even | 2 | 1 | inner | 600.2.m.d | 16 | |
| 120.q | odd | 4 | 1 | 120.2.b.b | yes | 8 | |
| 120.q | odd | 4 | 1 | 600.2.b.e | 8 | ||
| 120.w | even | 4 | 1 | 480.2.b.a | 8 | ||
| 120.w | even | 4 | 1 | 2400.2.b.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.2.b.a | ✓ | 8 | 15.e | even | 4 | 1 | |
| 120.2.b.a | ✓ | 8 | 40.k | even | 4 | 1 | |
| 120.2.b.b | yes | 8 | 5.c | odd | 4 | 1 | |
| 120.2.b.b | yes | 8 | 120.q | odd | 4 | 1 | |
| 480.2.b.a | 8 | 20.e | even | 4 | 1 | ||
| 480.2.b.a | 8 | 120.w | even | 4 | 1 | ||
| 480.2.b.b | 8 | 40.i | odd | 4 | 1 | ||
| 480.2.b.b | 8 | 60.l | odd | 4 | 1 | ||
| 600.2.b.e | 8 | 5.c | odd | 4 | 1 | ||
| 600.2.b.e | 8 | 120.q | odd | 4 | 1 | ||
| 600.2.b.f | 8 | 15.e | even | 4 | 1 | ||
| 600.2.b.f | 8 | 40.k | even | 4 | 1 | ||
| 600.2.m.c | 16 | 3.b | odd | 2 | 1 | ||
| 600.2.m.c | 16 | 8.d | odd | 2 | 1 | ||
| 600.2.m.c | 16 | 15.d | odd | 2 | 1 | ||
| 600.2.m.c | 16 | 40.e | odd | 2 | 1 | ||
| 600.2.m.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 600.2.m.d | 16 | 5.b | even | 2 | 1 | inner | |
| 600.2.m.d | 16 | 24.f | even | 2 | 1 | inner | |
| 600.2.m.d | 16 | 120.m | even | 2 | 1 | inner | |
| 2400.2.b.e | 8 | 20.e | even | 4 | 1 | ||
| 2400.2.b.e | 8 | 120.w | even | 4 | 1 | ||
| 2400.2.b.f | 8 | 40.i | odd | 4 | 1 | ||
| 2400.2.b.f | 8 | 60.l | odd | 4 | 1 | ||
| 2400.2.m.c | 16 | 4.b | odd | 2 | 1 | ||
| 2400.2.m.c | 16 | 20.d | odd | 2 | 1 | ||
| 2400.2.m.c | 16 | 24.h | odd | 2 | 1 | ||
| 2400.2.m.c | 16 | 120.i | odd | 2 | 1 | ||
| 2400.2.m.d | 16 | 8.b | even | 2 | 1 | ||
| 2400.2.m.d | 16 | 12.b | even | 2 | 1 | ||
| 2400.2.m.d | 16 | 40.f | even | 2 | 1 | ||
| 2400.2.m.d | 16 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):
|
\( T_{7}^{8} - 36T_{7}^{6} + 384T_{7}^{4} - 1136T_{7}^{2} + 576 \)
|
|
\( T_{11}^{8} + 48T_{11}^{6} + 672T_{11}^{4} + 2560T_{11}^{2} + 256 \)
|
|
\( T_{29}^{4} - 64T_{29}^{2} + 112T_{29} - 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + T^{14} + \cdots + 256 \)
|
| $3$ |
\( T^{16} - 4 T^{12} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} - 36 T^{6} + \cdots + 576)^{2} \)
|
| $11$ |
\( (T^{8} + 48 T^{6} + \cdots + 256)^{2} \)
|
| $13$ |
\( (T^{8} - 52 T^{6} + \cdots + 9216)^{2} \)
|
| $17$ |
\( (T^{8} - 52 T^{6} + \cdots + 4096)^{2} \)
|
| $19$ |
\( (T^{4} - 2 T^{3} - 36 T^{2} + \cdots - 32)^{4} \)
|
| $23$ |
\( (T^{8} + 92 T^{6} + \cdots + 36864)^{2} \)
|
| $29$ |
\( (T^{4} - 64 T^{2} + \cdots - 48)^{4} \)
|
| $31$ |
\( (T^{8} + 140 T^{6} + \cdots + 9216)^{2} \)
|
| $37$ |
\( (T^{8} - 228 T^{6} + \cdots + 746496)^{2} \)
|
| $41$ |
\( (T^{8} + 64 T^{6} + \cdots + 16384)^{2} \)
|
| $43$ |
\( (T^{8} + 24 T^{6} + \cdots + 256)^{2} \)
|
| $47$ |
\( (T^{8} + 236 T^{6} + \cdots + 4875264)^{2} \)
|
| $53$ |
\( (T^{8} + 112 T^{6} + \cdots + 20736)^{2} \)
|
| $59$ |
\( (T^{8} + 160 T^{6} + \cdots + 256)^{2} \)
|
| $61$ |
\( (T^{8} + 208 T^{6} + \cdots + 36864)^{2} \)
|
| $67$ |
\( (T^{8} + 360 T^{6} + \cdots + 8386816)^{2} \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 2304)^{4} \)
|
| $73$ |
\( (T^{8} + 240 T^{6} + \cdots + 430336)^{2} \)
|
| $79$ |
\( (T^{8} + 108 T^{6} + \cdots + 9216)^{2} \)
|
| $83$ |
\( (T^{8} - 152 T^{6} + \cdots + 1032256)^{2} \)
|
| $89$ |
\( (T^{8} + 384 T^{6} + \cdots + 1048576)^{2} \)
|
| $97$ |
\( (T^{8} + 368 T^{6} + \cdots + 614656)^{2} \)
|
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