Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [270,10,Mod(1,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 270.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: | \(139.059675764\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{9569}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 2392 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{9569})\). 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 |
|
−16.0000 | 0 | 256.000 | −625.000 | 0 | −5138.01 | −4096.00 | 0 | 10000.0 | ||||||||||||||||||||||||
| 1.2 | −16.0000 | 0 | 256.000 | −625.000 | 0 | −1322.99 | −4096.00 | 0 | 10000.0 | |||||||||||||||||||||||||
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 | 270.10.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 270.10.a.b | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.10.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 270.10.a.b | yes | 2 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(270))\):
|
\( T_{7}^{2} + 6461T_{7} + 6797518 \)
|
|
\( T_{11}^{2} + 28533T_{11} - 683707050 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 625)^{2} \)
|
| $7$ |
\( T^{2} + 6461 T + 6797518 \)
|
| $11$ |
\( T^{2} + 28533 T - 683707050 \)
|
| $13$ |
\( T^{2} + \cdots - 5833626347 \)
|
| $17$ |
\( T^{2} + \cdots - 28210449888 \)
|
| $19$ |
\( T^{2} + \cdots - 723888544328 \)
|
| $23$ |
\( T^{2} + \cdots - 31338087822 \)
|
| $29$ |
\( T^{2} + \cdots + 11463639031674 \)
|
| $31$ |
\( T^{2} + \cdots - 24058563518240 \)
|
| $37$ |
\( T^{2} + \cdots - 73593367962134 \)
|
| $41$ |
\( T^{2} + \cdots - 130557815243784 \)
|
| $43$ |
\( T^{2} + \cdots - 8660758086452 \)
|
| $47$ |
\( T^{2} + \cdots - 22\!\cdots\!56 \)
|
| $53$ |
\( T^{2} + \cdots + 669926813760360 \)
|
| $59$ |
\( T^{2} + \cdots + 14\!\cdots\!88 \)
|
| $61$ |
\( T^{2} + \cdots - 32\!\cdots\!70 \)
|
| $67$ |
\( T^{2} + \cdots + 67\!\cdots\!36 \)
|
| $71$ |
\( T^{2} + \cdots - 54\!\cdots\!60 \)
|
| $73$ |
\( T^{2} + \cdots - 10\!\cdots\!34 \)
|
| $79$ |
\( T^{2} + \cdots - 97\!\cdots\!39 \)
|
| $83$ |
\( T^{2} + \cdots + 22\!\cdots\!92 \)
|
| $89$ |
\( T^{2} + \cdots - 33\!\cdots\!40 \)
|
| $97$ |
\( T^{2} + \cdots + 12\!\cdots\!10 \)
|
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