Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,8,Mod(1,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(23.4288769113\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 81x^{2} - 150x + 400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 81x^{2} - 150x + 400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 61\nu - 110 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + 333\nu + 335 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -49\nu^{3} + 300\nu^{2} + 2689\nu - 6640 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 6\beta _1 - 1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 55\beta _1 + 1202 ) / 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 61\beta_{2} + 666\beta _1 + 3239 ) / 30 \)
|
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 |
|
−21.0486 | −27.0000 | 315.042 | 0 | 568.311 | −923.886 | −3936.97 | 729.000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −8.75511 | −27.0000 | −51.3480 | 0 | 236.388 | −536.160 | 1570.21 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 3.02250 | −27.0000 | −118.865 | 0 | −81.6074 | 1505.28 | −746.147 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 17.7812 | −27.0000 | 188.170 | 0 | −480.092 | −1233.23 | 1069.90 | 729.000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 75.8.a.i | 4 | |
| 3.b | odd | 2 | 1 | 225.8.a.bb | 4 | ||
| 5.b | even | 2 | 1 | 75.8.a.j | 4 | ||
| 5.c | odd | 4 | 2 | 15.8.b.a | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 225.8.a.z | 4 | ||
| 15.e | even | 4 | 2 | 45.8.b.d | 8 | ||
| 20.e | even | 4 | 2 | 240.8.f.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.8.b.a | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 45.8.b.d | 8 | 15.e | even | 4 | 2 | ||
| 75.8.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.8.a.j | 4 | 5.b | even | 2 | 1 | ||
| 225.8.a.z | 4 | 15.d | odd | 2 | 1 | ||
| 225.8.a.bb | 4 | 3.b | odd | 2 | 1 | ||
| 240.8.f.e | 8 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 9T_{2}^{3} - 382T_{2}^{2} - 2232T_{2} + 9904 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 9 T^{3} + \cdots + 9904 \)
|
| $3$ |
\( (T + 27)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 919546603200 \)
|
| $11$ |
\( T^{4} + \cdots - 36924996384576 \)
|
| $13$ |
\( T^{4} + \cdots + 750457045131264 \)
|
| $17$ |
\( T^{4} + \cdots - 22\!\cdots\!24 \)
|
| $19$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 98\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 58\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 63\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 76\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 55\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots + 64\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots - 45\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 93\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots - 29\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 36\!\cdots\!36 \)
|
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