Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,10,Mod(1,200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(200, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.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: | \(103.007167233\) |
| 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: |
\( x^{5} - 2x^{4} - 764x^{3} + 4237x^{2} + 111148x - 635000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3^{3}\cdot 5^{3} \) |
| 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 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
\( x^{5} - 2x^{4} - 764x^{3} + 4237x^{2} + 111148x - 635000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{4} - 38\nu^{3} + 2304\nu^{2} + 15848\nu - 212885 ) / 945 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 232\nu^{4} + 4304\nu^{3} - 87432\nu^{2} - 1154384\nu + 1707050 ) / 945 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 172\nu^{4} + 434\nu^{3} - 103872\nu^{2} + 295336\nu + 8293595 ) / 135 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -92\nu^{4} - 1574\nu^{3} + 54392\nu^{2} + 493304\nu - 6517800 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + \beta_{2} + 566\beta _1 + 2191 ) / 5760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_{4} - \beta_{3} + 35\beta_{2} - 962\beta _1 + 588461 ) / 1920 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -485\beta_{4} + 89\beta_{3} + 197\beta_{2} + 138610\beta _1 - 4704133 ) / 2880 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9239\beta_{4} + 181\beta_{3} + 20233\beta_{2} - 1138078\beta _1 + 269455159 ) / 1920 \)
|
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 | −236.819 | 0 | 0 | 0 | −5046.95 | 0 | 36400.2 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −61.8022 | 0 | 0 | 0 | 6789.55 | 0 | −15863.5 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −49.1879 | 0 | 0 | 0 | 1436.67 | 0 | −17263.5 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 126.650 | 0 | 0 | 0 | −9447.80 | 0 | −3642.88 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 222.159 | 0 | 0 | 0 | 7886.53 | 0 | 29671.8 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 200.10.a.j | yes | 5 |
| 4.b | odd | 2 | 1 | 400.10.a.bd | 5 | ||
| 5.b | even | 2 | 1 | 200.10.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 200.10.c.h | 10 | ||
| 20.d | odd | 2 | 1 | 400.10.a.be | 5 | ||
| 20.e | even | 4 | 2 | 400.10.c.u | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.10.a.i | ✓ | 5 | 5.b | even | 2 | 1 | |
| 200.10.a.j | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 200.10.c.h | 10 | 5.c | odd | 4 | 2 | ||
| 400.10.a.bd | 5 | 4.b | odd | 2 | 1 | ||
| 400.10.a.be | 5 | 20.d | odd | 2 | 1 | ||
| 400.10.c.u | 10 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - T_{3}^{4} - 63858T_{3}^{3} + 277362T_{3}^{2} + 573973425T_{3} + 20255684175 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 20255684175 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 36\!\cdots\!08 \)
|
| $11$ |
\( T^{5} + \cdots - 38\!\cdots\!81 \)
|
| $13$ |
\( T^{5} + \cdots - 37\!\cdots\!96 \)
|
| $17$ |
\( T^{5} + \cdots + 23\!\cdots\!75 \)
|
| $19$ |
\( T^{5} + \cdots - 97\!\cdots\!01 \)
|
| $23$ |
\( T^{5} + \cdots - 51\!\cdots\!52 \)
|
| $29$ |
\( T^{5} + \cdots + 16\!\cdots\!88 \)
|
| $31$ |
\( T^{5} + \cdots + 30\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 39\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots - 26\!\cdots\!13 \)
|
| $43$ |
\( T^{5} + \cdots + 31\!\cdots\!64 \)
|
| $47$ |
\( T^{5} + \cdots + 20\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots - 47\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots - 38\!\cdots\!92 \)
|
| $61$ |
\( T^{5} + \cdots - 70\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 55\!\cdots\!31 \)
|
| $71$ |
\( T^{5} + \cdots - 93\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 13\!\cdots\!27 \)
|
| $79$ |
\( T^{5} + \cdots - 81\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 42\!\cdots\!03 \)
|
| $89$ |
\( T^{5} + \cdots + 49\!\cdots\!09 \)
|
| $97$ |
\( T^{5} + \cdots + 67\!\cdots\!24 \)
|
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