Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(4.42514325043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{41}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). 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 |
|
−1.70156 | 3.00000 | −5.10469 | 0 | −5.10469 | 22.2094 | 22.2984 | 9.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 4.70156 | 3.00000 | 14.1047 | 0 | 14.1047 | −16.2094 | 28.7016 | 9.00000 | 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 | 75.4.a.f | 2 | |
| 3.b | odd | 2 | 1 | 225.4.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 1200.4.a.bn | 2 | ||
| 5.b | even | 2 | 1 | 75.4.a.c | 2 | ||
| 5.c | odd | 4 | 2 | 15.4.b.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 225.4.a.o | 2 | ||
| 15.e | even | 4 | 2 | 45.4.b.b | 4 | ||
| 20.d | odd | 2 | 1 | 1200.4.a.bt | 2 | ||
| 20.e | even | 4 | 2 | 240.4.f.f | 4 | ||
| 40.i | odd | 4 | 2 | 960.4.f.q | 4 | ||
| 40.k | even | 4 | 2 | 960.4.f.p | 4 | ||
| 60.l | odd | 4 | 2 | 720.4.f.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.4.b.a | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 45.4.b.b | 4 | 15.e | even | 4 | 2 | ||
| 75.4.a.c | 2 | 5.b | even | 2 | 1 | ||
| 75.4.a.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 225.4.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 225.4.a.o | 2 | 15.d | odd | 2 | 1 | ||
| 240.4.f.f | 4 | 20.e | even | 4 | 2 | ||
| 720.4.f.j | 4 | 60.l | odd | 4 | 2 | ||
| 960.4.f.p | 4 | 40.k | even | 4 | 2 | ||
| 960.4.f.q | 4 | 40.i | odd | 4 | 2 | ||
| 1200.4.a.bn | 2 | 4.b | odd | 2 | 1 | ||
| 1200.4.a.bt | 2 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} - 8 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3T - 8 \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 6T - 360 \)
|
| $11$ |
\( T^{2} + 42T + 72 \)
|
| $13$ |
\( T^{2} - 78T + 1152 \)
|
| $17$ |
\( T^{2} - 102T + 1576 \)
|
| $19$ |
\( T^{2} - 56T - 5120 \)
|
| $23$ |
\( T^{2} + 48T - 80 \)
|
| $29$ |
\( T^{2} + 318T + 7200 \)
|
| $31$ |
\( T^{2} - 52T - 800 \)
|
| $37$ |
\( T^{2} - 306T - 6480 \)
|
| $41$ |
\( T^{2} + 408T + 40140 \)
|
| $43$ |
\( T^{2} - 120T - 90864 \)
|
| $47$ |
\( T^{2} - 180T - 78656 \)
|
| $53$ |
\( T^{2} - 402T - 28520 \)
|
| $59$ |
\( T^{2} + 186T + 8280 \)
|
| $61$ |
\( T^{2} - 340T - 65564 \)
|
| $67$ |
\( T^{2} - 732T + 97056 \)
|
| $71$ |
\( T^{2} + 36T - 331776 \)
|
| $73$ |
\( T^{2} - 1332 T + 324000 \)
|
| $79$ |
\( T^{2} - 380T - 886400 \)
|
| $83$ |
\( T^{2} + 984T - 201392 \)
|
| $89$ |
\( T^{2} - 1116T + 98820 \)
|
| $97$ |
\( T^{2} + 768T - 442944 \)
|
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