Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(214.461857904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} - 2792237 x^{8} + 94050736 x^{7} + 2615193085270 x^{6} - 30081688120700 x^{5} + \cdots + 14\!\cdots\!75 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{55}\cdot 3^{5}\cdot 5^{16} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} - 2792237 x^{8} + 94050736 x^{7} + 2615193085270 x^{6} - 30081688120700 x^{5} + \cdots + 14\!\cdots\!75 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} + 190\nu - 2233829 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 49\!\cdots\!04 \nu^{9} + \cdots + 16\!\cdots\!00 ) / 97\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 53\!\cdots\!66 \nu^{9} + \cdots + 17\!\cdots\!25 ) / 97\!\cdots\!25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!16 \nu^{9} + \cdots - 33\!\cdots\!75 ) / 10\!\cdots\!25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 72\!\cdots\!78 \nu^{9} + \cdots + 12\!\cdots\!00 ) / 32\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 29\!\cdots\!88 \nu^{9} + \cdots + 63\!\cdots\!25 ) / 97\!\cdots\!25 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 14\!\cdots\!58 \nu^{9} + \cdots + 29\!\cdots\!00 ) / 36\!\cdots\!75 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 32\!\cdots\!16 \nu^{9} + \cdots - 16\!\cdots\!75 ) / 38\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 95\beta _1 + 2233829 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} - 15 \beta_{6} + 57 \beta_{5} + 26 \beta_{4} - 115 \beta_{3} + \cdots - 106894476 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1348 \beta_{9} + 744 \beta_{8} - 44965 \beta_{7} + 6996 \beta_{6} - 38976 \beta_{5} + \cdots + 4105673536454 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12348258 \beta_{9} + 12138255 \beta_{8} + 43673604 \beta_{7} - 67642173 \beta_{6} + \cdots - 17\!\cdots\!88 ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3305106452 \beta_{9} + 1603646412 \beta_{8} - 79360526147 \beta_{7} + 10996288584 \beta_{6} + \cdots + 42\!\cdots\!28 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 16659377679934 \beta_{9} + 9315916577625 \beta_{8} + 122397067399180 \beta_{7} - 60769517110527 \beta_{6} + \cdots - 32\!\cdots\!52 ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 59\!\cdots\!49 \beta_{9} + \cdots + 45\!\cdots\!38 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 55\!\cdots\!99 \beta_{9} + \cdots - 12\!\cdots\!32 ) / 4 \)
|
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 | −1746.58 | 0 | 0 | 0 | 553578. | 0 | 1.45620e6 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1702.22 | 0 | 0 | 0 | −330830. | 0 | 1.30322e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1688.24 | 0 | 0 | 0 | −194262. | 0 | 1.25582e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −924.496 | 0 | 0 | 0 | 237934. | 0 | −739630. | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 312.525 | 0 | 0 | 0 | −246082. | 0 | −1.49665e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 499.184 | 0 | 0 | 0 | −40217.7 | 0 | −1.34514e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 853.661 | 0 | 0 | 0 | −61860.4 | 0 | −865586. | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1060.19 | 0 | 0 | 0 | 591800. | 0 | −470331. | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2225.01 | 0 | 0 | 0 | −497533. | 0 | 3.35636e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2386.96 | 0 | 0 | 0 | 328541. | 0 | 4.10323e6 | 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 | 200.14.a.l | 10 | |
| 5.b | even | 2 | 1 | 200.14.a.k | 10 | ||
| 5.c | odd | 4 | 2 | 40.14.c.a | ✓ | 20 | |
| 20.e | even | 4 | 2 | 80.14.c.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.14.c.a | ✓ | 20 | 5.c | odd | 4 | 2 | |
| 80.14.c.d | 20 | 20.e | even | 4 | 2 | ||
| 200.14.a.k | 10 | 5.b | even | 2 | 1 | ||
| 200.14.a.l | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 1276 T_{3}^{9} - 10436276 T_{3}^{8} + 10435297920 T_{3}^{7} + 37449134609760 T_{3}^{6} + \cdots + 34\!\cdots\!00 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots + 34\!\cdots\!00 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 50\!\cdots\!88 \)
|
| $11$ |
\( T^{10} + \cdots - 14\!\cdots\!76 \)
|
| $13$ |
\( T^{10} + \cdots + 27\!\cdots\!88 \)
|
| $17$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots + 34\!\cdots\!64 \)
|
| $23$ |
\( T^{10} + \cdots + 28\!\cdots\!48 \)
|
| $29$ |
\( T^{10} + \cdots - 22\!\cdots\!04 \)
|
| $31$ |
\( T^{10} + \cdots - 42\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots - 21\!\cdots\!56 \)
|
| $41$ |
\( T^{10} + \cdots + 15\!\cdots\!92 \)
|
| $43$ |
\( T^{10} + \cdots - 74\!\cdots\!08 \)
|
| $47$ |
\( T^{10} + \cdots - 13\!\cdots\!84 \)
|
| $53$ |
\( T^{10} + \cdots - 71\!\cdots\!64 \)
|
| $59$ |
\( T^{10} + \cdots - 68\!\cdots\!48 \)
|
| $61$ |
\( T^{10} + \cdots + 67\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots - 48\!\cdots\!32 \)
|
| $71$ |
\( T^{10} + \cdots + 39\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots - 17\!\cdots\!92 \)
|
| $79$ |
\( T^{10} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots - 34\!\cdots\!68 \)
|
| $89$ |
\( T^{10} + \cdots + 27\!\cdots\!84 \)
|
| $97$ |
\( T^{10} + \cdots - 62\!\cdots\!84 \)
|
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