Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2499,4,Mod(1,2499)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2499, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2499.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2499.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: | \(147.445773104\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 51) |
| 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 \(\beta = 3\sqrt{2}\). 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 |
|
−4.24264 | 3.00000 | 10.0000 | 13.9706 | −12.7279 | 0 | −8.48528 | 9.00000 | −59.2721 | ||||||||||||||||||||||||
| 1.2 | 4.24264 | 3.00000 | 10.0000 | −19.9706 | 12.7279 | 0 | 8.48528 | 9.00000 | −84.7279 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 2499.4.a.l | 2 | |
| 7.b | odd | 2 | 1 | 51.4.a.d | ✓ | 2 | |
| 21.c | even | 2 | 1 | 153.4.a.e | 2 | ||
| 28.d | even | 2 | 1 | 816.4.a.o | 2 | ||
| 35.c | odd | 2 | 1 | 1275.4.a.m | 2 | ||
| 84.h | odd | 2 | 1 | 2448.4.a.v | 2 | ||
| 119.d | odd | 2 | 1 | 867.4.a.j | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.4.a.d | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 153.4.a.e | 2 | 21.c | even | 2 | 1 | ||
| 816.4.a.o | 2 | 28.d | even | 2 | 1 | ||
| 867.4.a.j | 2 | 119.d | odd | 2 | 1 | ||
| 1275.4.a.m | 2 | 35.c | odd | 2 | 1 | ||
| 2448.4.a.v | 2 | 84.h | odd | 2 | 1 | ||
| 2499.4.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2499))\):
|
\( T_{2}^{2} - 18 \)
|
|
\( T_{5}^{2} + 6T_{5} - 279 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 18 \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} + 6T - 279 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 66T + 801 \)
|
| $13$ |
\( T^{2} - 2T - 1151 \)
|
| $17$ |
\( (T - 17)^{2} \)
|
| $19$ |
\( T^{2} - 26T - 4439 \)
|
| $23$ |
\( T^{2} + 198T + 9513 \)
|
| $29$ |
\( T^{2} - 444T + 44676 \)
|
| $31$ |
\( T^{2} + 532T + 69604 \)
|
| $37$ |
\( T^{2} - 88T - 71792 \)
|
| $41$ |
\( T^{2} + 570T + 57897 \)
|
| $43$ |
\( T^{2} + 182T + 3673 \)
|
| $47$ |
\( T^{2} + 420T + 41508 \)
|
| $53$ |
\( T^{2} + 300T - 254268 \)
|
| $59$ |
\( T^{2} + 444T - 116604 \)
|
| $61$ |
\( T^{2} + 400T - 398048 \)
|
| $67$ |
\( T^{2} + 968T + 140944 \)
|
| $71$ |
\( T^{2} + 1848 T + 797328 \)
|
| $73$ |
\( T^{2} + 868T + 185764 \)
|
| $79$ |
\( (T - 254)^{2} \)
|
| $83$ |
\( T^{2} + 996T + 245412 \)
|
| $89$ |
\( T^{2} - 288 T - 1122336 \)
|
| $97$ |
\( T^{2} + 1024 T + 213472 \)
|
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