Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,10,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(25.2367559720\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - x^{4} - 429x^{3} + 184x^{2} + 37472x + 27600 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{5} - x^{4} - 429x^{3} + 184x^{2} + 37472x + 27600 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 253\nu + 78 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 27\nu^{2} - 267\nu - 4731 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 337\nu^{2} + 288\nu + 14154 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 687 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 27\beta_{2} + 507\beta _1 + 375 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14\beta_{4} + 169\beta_{3} - 155\beta_{2} + 134\beta _1 + 87639 ) / 2 \)
|
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 |
|
−31.2370 | 113.534 | 463.747 | 479.511 | −3546.46 | 0 | 1507.26 | −6793.01 | −14978.5 | |||||||||||||||||||||||||||||||||
| 1.2 | −19.5965 | −209.955 | −127.978 | −1967.58 | 4114.37 | 0 | 12541.3 | 24397.9 | 38557.6 | ||||||||||||||||||||||||||||||||||
| 1.3 | 5.48794 | 3.40615 | −481.883 | 1657.85 | 18.6927 | 0 | −5454.36 | −19671.4 | 9098.16 | ||||||||||||||||||||||||||||||||||
| 1.4 | 24.4690 | 159.470 | 86.7323 | −2028.30 | 3902.06 | 0 | −10405.9 | 5747.57 | −49630.4 | ||||||||||||||||||||||||||||||||||
| 1.5 | 38.8765 | −227.455 | 999.381 | 325.520 | −8842.67 | 0 | 18947.7 | 32052.9 | 12655.1 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
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 | 49.10.a.e | 5 | |
| 7.b | odd | 2 | 1 | 49.10.a.f | 5 | ||
| 7.c | even | 3 | 2 | 7.10.c.a | ✓ | 10 | |
| 7.d | odd | 6 | 2 | 49.10.c.g | 10 | ||
| 21.h | odd | 6 | 2 | 63.10.e.b | 10 | ||
| 28.g | odd | 6 | 2 | 112.10.i.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.c.a | ✓ | 10 | 7.c | even | 3 | 2 | |
| 49.10.a.e | 5 | 1.a | even | 1 | 1 | trivial | |
| 49.10.a.f | 5 | 7.b | odd | 2 | 1 | ||
| 49.10.c.g | 10 | 7.d | odd | 6 | 2 | ||
| 63.10.e.b | 10 | 21.h | odd | 6 | 2 | ||
| 112.10.i.c | 10 | 28.g | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{5} - 18T_{2}^{4} - 1588T_{2}^{3} + 18672T_{2}^{2} + 529728T_{2} - 3195648 \)
|
|
\( T_{3}^{5} + 161T_{3}^{4} - 54114T_{3}^{3} - 4935546T_{3}^{2} + 882053361T_{3} - 2945025783 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 18 T^{4} + \cdots - 3195648 \)
|
| $3$ |
\( T^{5} + \cdots - 2945025783 \)
|
| $5$ |
\( T^{5} + \cdots - 10\!\cdots\!75 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 30\!\cdots\!95 \)
|
| $13$ |
\( T^{5} + \cdots + 79\!\cdots\!92 \)
|
| $17$ |
\( T^{5} + \cdots + 17\!\cdots\!41 \)
|
| $19$ |
\( T^{5} + \cdots + 62\!\cdots\!99 \)
|
| $23$ |
\( T^{5} + \cdots - 20\!\cdots\!49 \)
|
| $29$ |
\( T^{5} + \cdots + 73\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 12\!\cdots\!13 \)
|
| $37$ |
\( T^{5} + \cdots - 68\!\cdots\!75 \)
|
| $41$ |
\( T^{5} + \cdots - 20\!\cdots\!40 \)
|
| $43$ |
\( T^{5} + \cdots + 72\!\cdots\!36 \)
|
| $47$ |
\( T^{5} + \cdots - 59\!\cdots\!85 \)
|
| $53$ |
\( T^{5} + \cdots + 68\!\cdots\!61 \)
|
| $59$ |
\( T^{5} + \cdots + 88\!\cdots\!45 \)
|
| $61$ |
\( T^{5} + \cdots - 16\!\cdots\!79 \)
|
| $67$ |
\( T^{5} + \cdots - 14\!\cdots\!15 \)
|
| $71$ |
\( T^{5} + \cdots + 25\!\cdots\!92 \)
|
| $73$ |
\( T^{5} + \cdots + 29\!\cdots\!73 \)
|
| $79$ |
\( T^{5} + \cdots + 16\!\cdots\!11 \)
|
| $83$ |
\( T^{5} + \cdots + 39\!\cdots\!36 \)
|
| $89$ |
\( T^{5} + \cdots - 46\!\cdots\!79 \)
|
| $97$ |
\( T^{5} + \cdots - 84\!\cdots\!56 \)
|
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