Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(12.0287864860\) |
| Analytic rank: | \(0\) |
| 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: | \( 1 \) |
| 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 \(\beta = \sqrt{31}\). 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 |
|
−2.56776 | 9.00000 | −25.4066 | 0 | −23.1099 | −82.6263 | 147.407 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 8.56776 | 9.00000 | 41.4066 | 0 | 77.1099 | 184.626 | 80.5934 | 81.0000 | 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 | 75.6.a.i | yes | 2 |
| 3.b | odd | 2 | 1 | 225.6.a.j | 2 | ||
| 5.b | even | 2 | 1 | 75.6.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 75.6.b.f | 4 | ||
| 15.d | odd | 2 | 1 | 225.6.a.t | 2 | ||
| 15.e | even | 4 | 2 | 225.6.b.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.6.a.g | ✓ | 2 | 5.b | even | 2 | 1 | |
| 75.6.a.i | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 75.6.b.f | 4 | 5.c | odd | 4 | 2 | ||
| 225.6.a.j | 2 | 3.b | odd | 2 | 1 | ||
| 225.6.a.t | 2 | 15.d | odd | 2 | 1 | ||
| 225.6.b.l | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 6T_{2} - 22 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 6T - 22 \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 102T - 15255 \)
|
| $11$ |
\( T^{2} + 12T - 240028 \)
|
| $13$ |
\( T^{2} - 1054T - 7967 \)
|
| $17$ |
\( T^{2} + 1716 T + 686564 \)
|
| $19$ |
\( T^{2} - 4214 T + 3564505 \)
|
| $23$ |
\( T^{2} + 444 T - 12967740 \)
|
| $29$ |
\( T^{2} - 4068 T + 4039940 \)
|
| $31$ |
\( T^{2} + 2598 T - 25471575 \)
|
| $37$ |
\( T^{2} - 4412 T - 92913020 \)
|
| $41$ |
\( T^{2} - 11232 T + 28823360 \)
|
| $43$ |
\( T^{2} + 8450 T + 16404289 \)
|
| $47$ |
\( T^{2} - 2460 T - 341830204 \)
|
| $53$ |
\( T^{2} + \cdots + 1056424400 \)
|
| $59$ |
\( T^{2} + 63924 T + 741079460 \)
|
| $61$ |
\( T^{2} - 7310 T - 18138959 \)
|
| $67$ |
\( T^{2} - 61734 T + 127378089 \)
|
| $71$ |
\( T^{2} + \cdots + 1224279104 \)
|
| $73$ |
\( T^{2} + 26564 T - 183636860 \)
|
| $79$ |
\( T^{2} + \cdots + 1649721600 \)
|
| $83$ |
\( T^{2} + \cdots + 1063204452 \)
|
| $89$ |
\( T^{2} + \cdots + 2346485760 \)
|
| $97$ |
\( T^{2} + \cdots - 13910130431 \)
|
| show more | |
| show less |