Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,6,Mod(1,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 300.a (trivial) |
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: | yes |
| Analytic conductor: | \(48.1151459439\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{7}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 60\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 9.00000 | 0 | 0 | 0 | −147.745 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 0 | 0 | 169.745 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 300.6.a.h | yes | 2 |
| 3.b | odd | 2 | 1 | 900.6.a.r | 2 | ||
| 5.b | even | 2 | 1 | 300.6.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 300.6.d.f | 4 | ||
| 15.d | odd | 2 | 1 | 900.6.a.n | 2 | ||
| 15.e | even | 4 | 2 | 900.6.d.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 300.6.a.g | ✓ | 2 | 5.b | even | 2 | 1 | |
| 300.6.a.h | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 300.6.d.f | 4 | 5.c | odd | 4 | 2 | ||
| 900.6.a.n | 2 | 15.d | odd | 2 | 1 | ||
| 900.6.a.r | 2 | 3.b | odd | 2 | 1 | ||
| 900.6.d.j | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 22T_{7} - 25079 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(300))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 22T - 25079 \)
|
| $11$ |
\( T^{2} - 372T - 192204 \)
|
| $13$ |
\( T^{2} - 10T - 403175 \)
|
| $17$ |
\( T^{2} - 1212 T + 140436 \)
|
| $19$ |
\( T^{2} + 1550 T - 2448575 \)
|
| $23$ |
\( T^{2} - 2652 T + 1531476 \)
|
| $29$ |
\( T^{2} + 1812 T - 50209164 \)
|
| $31$ |
\( T^{2} - 10918 T + 29775481 \)
|
| $37$ |
\( T^{2} - 3316 T - 5415836 \)
|
| $41$ |
\( T^{2} - 19080 T + 90784800 \)
|
| $43$ |
\( T^{2} - 12262 T + 16395961 \)
|
| $47$ |
\( T^{2} + 15252 T - 72480924 \)
|
| $53$ |
\( T^{2} - 20784 T - 578076336 \)
|
| $59$ |
\( T^{2} - 18708 T + 29436516 \)
|
| $61$ |
\( T^{2} - 35734 T + 274776889 \)
|
| $67$ |
\( T^{2} + \cdots + 2408717761 \)
|
| $71$ |
\( T^{2} + \cdots + 3114093600 \)
|
| $73$ |
\( T^{2} + \cdots + 2356835044 \)
|
| $79$ |
\( T^{2} + \cdots + 1106050624 \)
|
| $83$ |
\( T^{2} + \cdots - 7021874844 \)
|
| $89$ |
\( T^{2} + \cdots + 1751155200 \)
|
| $97$ |
\( T^{2} + \cdots - 1011621551 \)
|
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