Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1840,4,Mod(1,1840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1840.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1840.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: | \(108.563514411\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 460) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 61\nu^{2} + 58\nu + 348 ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} - 7\nu^{3} - 107\nu^{2} + 361\nu + 306 ) / 30 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 51\nu^{2} + 68\nu + 118 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} - 6\beta_{2} - \beta _1 + 45 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} - 6\beta_{3} + 3\beta_{2} + 23\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 128\beta_{4} - 12\beta_{3} - 300\beta_{2} - 73\beta _1 + 2019 ) / 2 \)
|
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 |
|
0 | −7.44221 | 0 | 5.00000 | 0 | 23.3042 | 0 | 28.3866 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.380607 | 0 | 5.00000 | 0 | −24.3526 | 0 | −26.8551 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.0785019 | 0 | 5.00000 | 0 | 6.13060 | 0 | −26.9938 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.75371 | 0 | 5.00000 | 0 | 19.0133 | 0 | 18.6126 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.99061 | 0 | 5.00000 | 0 | −4.09549 | 0 | 36.8498 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 1840.4.a.q | 5 | |
| 4.b | odd | 2 | 1 | 460.4.a.a | ✓ | 5 | |
| 20.d | odd | 2 | 1 | 2300.4.a.d | 5 | ||
| 20.e | even | 4 | 2 | 2300.4.c.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.4.a.a | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1840.4.a.q | 5 | 1.a | even | 1 | 1 | trivial | |
| 2300.4.a.d | 5 | 20.d | odd | 2 | 1 | ||
| 2300.4.c.c | 10 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):
|
\( T_{3}^{5} - 7T_{3}^{4} - 58T_{3}^{3} + 385T_{3}^{2} + 123T_{3} - 12 \)
|
|
\( T_{7}^{5} - 20T_{7}^{4} - 576T_{7}^{3} + 12437T_{7}^{2} - 7210T_{7} - 270924 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 7 T^{4} + \cdots - 12 \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 20 T^{4} + \cdots - 270924 \)
|
| $11$ |
\( T^{5} - 63 T^{4} + \cdots - 72719208 \)
|
| $13$ |
\( T^{5} + 99 T^{4} + \cdots - 14392890 \)
|
| $17$ |
\( T^{5} + 44 T^{4} + \cdots + 451144080 \)
|
| $19$ |
\( T^{5} - 199 T^{4} + \cdots + 800499864 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 43256062404 \)
|
| $31$ |
\( T^{5} + \cdots + 9011869285 \)
|
| $37$ |
\( T^{5} + \cdots - 409221703936 \)
|
| $41$ |
\( T^{5} + \cdots + 12760146261 \)
|
| $43$ |
\( T^{5} + \cdots - 803496865792 \)
|
| $47$ |
\( T^{5} + \cdots - 13556115821808 \)
|
| $53$ |
\( T^{5} + \cdots - 4075860543408 \)
|
| $59$ |
\( T^{5} + \cdots + 897857553600 \)
|
| $61$ |
\( T^{5} + \cdots + 310836562128 \)
|
| $67$ |
\( T^{5} + \cdots - 137714931264960 \)
|
| $71$ |
\( T^{5} + \cdots + 108175489939041 \)
|
| $73$ |
\( T^{5} + \cdots + 13803900176376 \)
|
| $79$ |
\( T^{5} + \cdots + 8954177264640 \)
|
| $83$ |
\( T^{5} + \cdots + 29112932977344 \)
|
| $89$ |
\( T^{5} + \cdots + 180704830832640 \)
|
| $97$ |
\( T^{5} + \cdots + 347840565067560 \)
|
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