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: | \(\Q(\sqrt{19}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 19 \)
|
| 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 \(\beta = \sqrt{19}\). 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 |
|
−5.35890 | 3.00000 | 20.7178 | 0 | −16.0767 | −4.43560 | −68.1534 | 9.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 3.35890 | 3.00000 | 3.28220 | 0 | 10.0767 | 30.4356 | −15.8466 | 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.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 225.4.a.n | 2 | ||
| 4.b | odd | 2 | 1 | 1200.4.a.bl | 2 | ||
| 5.b | even | 2 | 1 | 75.4.a.e | yes | 2 | |
| 5.c | odd | 4 | 2 | 75.4.b.c | 4 | ||
| 15.d | odd | 2 | 1 | 225.4.a.j | 2 | ||
| 15.e | even | 4 | 2 | 225.4.b.h | 4 | ||
| 20.d | odd | 2 | 1 | 1200.4.a.bu | 2 | ||
| 20.e | even | 4 | 2 | 1200.4.f.v | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.4.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 75.4.a.e | yes | 2 | 5.b | even | 2 | 1 | |
| 75.4.b.c | 4 | 5.c | odd | 4 | 2 | ||
| 225.4.a.j | 2 | 15.d | odd | 2 | 1 | ||
| 225.4.a.n | 2 | 3.b | odd | 2 | 1 | ||
| 225.4.b.h | 4 | 15.e | even | 4 | 2 | ||
| 1200.4.a.bl | 2 | 4.b | odd | 2 | 1 | ||
| 1200.4.a.bu | 2 | 20.d | odd | 2 | 1 | ||
| 1200.4.f.v | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 18 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 18 \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 26T - 135 \)
|
| $11$ |
\( T^{2} - 28T - 108 \)
|
| $13$ |
\( T^{2} - 18T - 4783 \)
|
| $17$ |
\( T^{2} + 68T - 6444 \)
|
| $19$ |
\( T^{2} - 6T - 295 \)
|
| $23$ |
\( T^{2} - 132T + 1620 \)
|
| $29$ |
\( T^{2} - 92T - 12780 \)
|
| $31$ |
\( T^{2} - 122T - 11175 \)
|
| $37$ |
\( T^{2} + 284T + 9220 \)
|
| $41$ |
\( T^{2} - 392T - 12960 \)
|
| $43$ |
\( T^{2} - 690T + 118721 \)
|
| $47$ |
\( T^{2} + 620T + 76644 \)
|
| $53$ |
\( T^{2} + 848T + 164880 \)
|
| $59$ |
\( T^{2} - 124T - 73980 \)
|
| $61$ |
\( T^{2} - 750T + 81041 \)
|
| $67$ |
\( T^{2} + 358T - 157959 \)
|
| $71$ |
\( T^{2} - 824T + 162144 \)
|
| $73$ |
\( T^{2} + 108T + 1700 \)
|
| $79$ |
\( T^{2} + 880T - 292800 \)
|
| $83$ |
\( T^{2} - 156T - 694332 \)
|
| $89$ |
\( T^{2} + 864T - 207360 \)
|
| $97$ |
\( (T - 521)^{2} \)
|
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