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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{31}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 31 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{31}\). 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 |
|
−7.13553 | −27.0000 | −77.0842 | 0 | 192.659 | −112.897 | 1463.38 | 729.000 | 0 | ||||||||||||||||||||||||
| 1.2 | 15.1355 | −27.0000 | 101.084 | 0 | −408.659 | 198.897 | −407.384 | 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.f | yes | 2 |
| 3.b | odd | 2 | 1 | 225.8.a.l | 2 | ||
| 5.b | even | 2 | 1 | 75.8.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 75.8.b.e | 4 | ||
| 15.d | odd | 2 | 1 | 225.8.a.u | 2 | ||
| 15.e | even | 4 | 2 | 225.8.b.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.8.a.d | ✓ | 2 | 5.b | even | 2 | 1 | |
| 75.8.a.f | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 75.8.b.e | 4 | 5.c | odd | 4 | 2 | ||
| 225.8.a.l | 2 | 3.b | odd | 2 | 1 | ||
| 225.8.a.u | 2 | 15.d | odd | 2 | 1 | ||
| 225.8.b.o | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 8T_{2} - 108 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 8T - 108 \)
|
| $3$ |
\( (T + 27)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 86T - 22455 \)
|
| $11$ |
\( T^{2} + 4612 T - 3456108 \)
|
| $13$ |
\( T^{2} - 9018 T + 9466697 \)
|
| $17$ |
\( T^{2} + 19948 T - 506147724 \)
|
| $19$ |
\( T^{2} + 23934 T - 791814895 \)
|
| $23$ |
\( T^{2} + \cdots - 1770541740 \)
|
| $29$ |
\( T^{2} + \cdots + 29988616500 \)
|
| $31$ |
\( T^{2} + \cdots - 25626949575 \)
|
| $37$ |
\( T^{2} + \cdots - 205209213980 \)
|
| $41$ |
\( T^{2} + \cdots - 186336929520 \)
|
| $43$ |
\( T^{2} + \cdots + 285507233321 \)
|
| $47$ |
\( T^{2} + \cdots + 19283408964 \)
|
| $53$ |
\( T^{2} + \cdots - 1554251453280 \)
|
| $59$ |
\( T^{2} + \cdots - 1475873866620 \)
|
| $61$ |
\( T^{2} + \cdots - 8423736487399 \)
|
| $67$ |
\( T^{2} + \cdots + 5094434196801 \)
|
| $71$ |
\( T^{2} + \cdots - 284467184496 \)
|
| $73$ |
\( T^{2} + \cdots + 3729052906820 \)
|
| $79$ |
\( T^{2} + \cdots + 9582807148800 \)
|
| $83$ |
\( T^{2} + \cdots - 12878345203452 \)
|
| $89$ |
\( T^{2} + \cdots + 45203910693120 \)
|
| $97$ |
\( T^{2} + \cdots + 42934761529441 \)
|
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