Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(251,600)]
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, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.251");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.b (of order \(2\), degree \(1\), 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: | \(8\) |
| Coefficient field: | 8.0.1649659456.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 4\nu^{2} - 8\nu - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 4\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 4\nu^{4} + 4\nu^{3} + 4\nu^{2} - 16 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 4\nu^{3} - 4\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - 2\nu^{5} + 4\nu^{4} + 4\nu^{2} - 8\nu + 8 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2\beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} + 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + 5\beta_{3} - 2\beta_{2} + 4\beta _1 + 5 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 251.1 |
|
−1.13622 | − | 0.842022i | 1.71822 | − | 0.218455i | 0.581998 | + | 1.91345i | 0 | −2.13622 | − | 1.19857i | − | 3.64426i | 0.949886 | − | 2.66415i | 2.90455 | − | 0.750707i | 0 | |||||||||||||||||||||||||||||
| 251.2 | −1.13622 | + | 0.842022i | 1.71822 | + | 0.218455i | 0.581998 | − | 1.91345i | 0 | −2.13622 | + | 1.19857i | 3.64426i | 0.949886 | + | 2.66415i | 2.90455 | + | 0.750707i | 0 | |||||||||||||||||||||||||||||||
| 251.3 | −0.578647 | − | 1.29041i | −0.751690 | − | 1.56044i | −1.33034 | + | 1.49339i | 0 | −1.57865 | + | 1.87293i | 4.28591i | 2.69688 | + | 0.852541i | −1.86993 | + | 2.34593i | 0 | |||||||||||||||||||||||||||||||
| 251.4 | −0.578647 | + | 1.29041i | −0.751690 | + | 1.56044i | −1.33034 | − | 1.49339i | 0 | −1.57865 | − | 1.87293i | − | 4.28591i | 2.69688 | − | 0.852541i | −1.86993 | − | 2.34593i | 0 | ||||||||||||||||||||||||||||||
| 251.5 | 0.814732 | − | 1.15595i | −1.48716 | + | 0.887900i | −0.672424 | − | 1.88357i | 0 | −0.185268 | + | 2.44247i | 0.797253i | −2.72515 | − | 0.757320i | 1.42327 | − | 2.64089i | 0 | |||||||||||||||||||||||||||||||
| 251.6 | 0.814732 | + | 1.15595i | −1.48716 | − | 0.887900i | −0.672424 | + | 1.88357i | 0 | −0.185268 | − | 2.44247i | − | 0.797253i | −2.72515 | + | 0.757320i | 1.42327 | + | 2.64089i | 0 | ||||||||||||||||||||||||||||||
| 251.7 | 1.40014 | − | 0.199044i | 0.520627 | − | 1.65195i | 1.92076 | − | 0.557378i | 0 | 0.400136 | − | 2.41659i | − | 1.92736i | 2.57839 | − | 1.16272i | −2.45790 | − | 1.72010i | 0 | ||||||||||||||||||||||||||||||
| 251.8 | 1.40014 | + | 0.199044i | 0.520627 | + | 1.65195i | 1.92076 | + | 0.557378i | 0 | 0.400136 | + | 2.41659i | 1.92736i | 2.57839 | + | 1.16272i | −2.45790 | + | 1.72010i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.f | 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.b.f | 8 | |
| 3.b | odd | 2 | 1 | 600.2.b.e | 8 | ||
| 4.b | odd | 2 | 1 | 2400.2.b.f | 8 | ||
| 5.b | even | 2 | 1 | 120.2.b.a | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 600.2.m.c | 16 | ||
| 8.b | even | 2 | 1 | 2400.2.b.e | 8 | ||
| 8.d | odd | 2 | 1 | 600.2.b.e | 8 | ||
| 12.b | even | 2 | 1 | 2400.2.b.e | 8 | ||
| 15.d | odd | 2 | 1 | 120.2.b.b | yes | 8 | |
| 15.e | even | 4 | 2 | 600.2.m.d | 16 | ||
| 20.d | odd | 2 | 1 | 480.2.b.b | 8 | ||
| 20.e | even | 4 | 2 | 2400.2.m.d | 16 | ||
| 24.f | even | 2 | 1 | inner | 600.2.b.f | 8 | |
| 24.h | odd | 2 | 1 | 2400.2.b.f | 8 | ||
| 40.e | odd | 2 | 1 | 120.2.b.b | yes | 8 | |
| 40.f | even | 2 | 1 | 480.2.b.a | 8 | ||
| 40.i | odd | 4 | 2 | 2400.2.m.c | 16 | ||
| 40.k | even | 4 | 2 | 600.2.m.d | 16 | ||
| 60.h | even | 2 | 1 | 480.2.b.a | 8 | ||
| 60.l | odd | 4 | 2 | 2400.2.m.c | 16 | ||
| 120.i | odd | 2 | 1 | 480.2.b.b | 8 | ||
| 120.m | even | 2 | 1 | 120.2.b.a | ✓ | 8 | |
| 120.q | odd | 4 | 2 | 600.2.m.c | 16 | ||
| 120.w | even | 4 | 2 | 2400.2.m.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.2.b.a | ✓ | 8 | 5.b | even | 2 | 1 | |
| 120.2.b.a | ✓ | 8 | 120.m | even | 2 | 1 | |
| 120.2.b.b | yes | 8 | 15.d | odd | 2 | 1 | |
| 120.2.b.b | yes | 8 | 40.e | odd | 2 | 1 | |
| 480.2.b.a | 8 | 40.f | even | 2 | 1 | ||
| 480.2.b.a | 8 | 60.h | even | 2 | 1 | ||
| 480.2.b.b | 8 | 20.d | odd | 2 | 1 | ||
| 480.2.b.b | 8 | 120.i | odd | 2 | 1 | ||
| 600.2.b.e | 8 | 3.b | odd | 2 | 1 | ||
| 600.2.b.e | 8 | 8.d | odd | 2 | 1 | ||
| 600.2.b.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 600.2.b.f | 8 | 24.f | even | 2 | 1 | inner | |
| 600.2.m.c | 16 | 5.c | odd | 4 | 2 | ||
| 600.2.m.c | 16 | 120.q | odd | 4 | 2 | ||
| 600.2.m.d | 16 | 15.e | even | 4 | 2 | ||
| 600.2.m.d | 16 | 40.k | even | 4 | 2 | ||
| 2400.2.b.e | 8 | 8.b | even | 2 | 1 | ||
| 2400.2.b.e | 8 | 12.b | even | 2 | 1 | ||
| 2400.2.b.f | 8 | 4.b | odd | 2 | 1 | ||
| 2400.2.b.f | 8 | 24.h | odd | 2 | 1 | ||
| 2400.2.m.c | 16 | 40.i | odd | 4 | 2 | ||
| 2400.2.m.c | 16 | 60.l | odd | 4 | 2 | ||
| 2400.2.m.d | 16 | 20.e | even | 4 | 2 | ||
| 2400.2.m.d | 16 | 120.w | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):
|
\( T_{7}^{8} + 36T_{7}^{6} + 384T_{7}^{4} + 1136T_{7}^{2} + 576 \)
|
|
\( T_{11}^{8} + 48T_{11}^{6} + 672T_{11}^{4} + 2560T_{11}^{2} + 256 \)
|
|
\( T_{23}^{4} + 2T_{23}^{3} - 44T_{23}^{2} - 188T_{23} - 192 \)
|
|
\( T_{43}^{4} - 12T_{43}^{2} + 4T_{43} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} - 4 T^{5} + \cdots + 81 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 36 T^{6} + \cdots + 576 \)
|
| $11$ |
\( T^{8} + 48 T^{6} + \cdots + 256 \)
|
| $13$ |
\( T^{8} + 52 T^{6} + \cdots + 9216 \)
|
| $17$ |
\( T^{8} + 52 T^{6} + \cdots + 4096 \)
|
| $19$ |
\( (T^{4} + 2 T^{3} - 36 T^{2} + \cdots - 32)^{2} \)
|
| $23$ |
\( (T^{4} + 2 T^{3} + \cdots - 192)^{2} \)
|
| $29$ |
\( (T^{4} - 64 T^{2} + \cdots - 48)^{2} \)
|
| $31$ |
\( T^{8} + 140 T^{6} + \cdots + 9216 \)
|
| $37$ |
\( T^{8} + 228 T^{6} + \cdots + 746496 \)
|
| $41$ |
\( T^{8} + 64 T^{6} + \cdots + 16384 \)
|
| $43$ |
\( (T^{4} - 12 T^{2} + \cdots + 16)^{2} \)
|
| $47$ |
\( (T^{4} - 14 T^{3} + \cdots - 2208)^{2} \)
|
| $53$ |
\( (T^{4} - 8 T^{3} + \cdots + 144)^{2} \)
|
| $59$ |
\( T^{8} + 160 T^{6} + \cdots + 256 \)
|
| $61$ |
\( T^{8} + 208 T^{6} + \cdots + 36864 \)
|
| $67$ |
\( (T^{4} - 16 T^{3} + \cdots + 2896)^{2} \)
|
| $71$ |
\( (T^{4} - 12 T^{3} + \cdots - 2304)^{2} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots - 656)^{2} \)
|
| $79$ |
\( T^{8} + 108 T^{6} + \cdots + 9216 \)
|
| $83$ |
\( T^{8} + 152 T^{6} + \cdots + 1032256 \)
|
| $89$ |
\( T^{8} + 384 T^{6} + \cdots + 1048576 \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots - 784)^{2} \)
|
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