Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [200,14,Mod(49,200)]
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, 1]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("200.49");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(214.461857904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14243x^{6} + 56566105x^{4} + 60027860232x^{2} + 9346092751044 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{39}\cdot 3^{4}\cdot 5^{4}\cdot 17^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 14243x^{6} + 56566105x^{4} + 60027860232x^{2} + 9346092751044 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -1574008\nu^{6} - 22626436832\nu^{4} - 78205479551392\nu^{2} - 27637813138750668 ) / 9881993688525 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1019\nu^{7} - 14490631\nu^{5} - 54361931081\nu^{3} - 38917562330394\nu ) / 200041837890300 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -12951664\nu^{6} - 79139348576\nu^{4} + 384515923684064\nu^{2} + 1031900421080223576 ) / 1976398737705 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 138357511 \nu^{7} + 1532174768099 \nu^{5} + \cdots - 72\!\cdots\!74 \nu ) / 15\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 301588232\nu^{6} + 3705632505088\nu^{4} + 11261493860346368\nu^{2} + 6164712415971303012 ) / 1976398737705 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1921081909 \nu^{7} + 25457814561737 \nu^{5} + \cdots + 69\!\cdots\!18 \nu ) / 15\!\cdots\!71 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 30745622351 \nu^{7} + 482135146979659 \nu^{5} + \cdots + 30\!\cdots\!66 \nu ) / 16\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + 3\beta_{6} - 2211\beta_{4} - 8160\beta_{2} ) / 65280 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 111\beta_{5} + 653\beta_{3} + 79475\beta _1 - 464891520 ) / 130560 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 553\beta_{7} - 553\beta_{6} + 749923\beta_{4} + 195159360\beta_{2} ) / 3840 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -213209\beta_{5} - 772155\beta_{3} - 172491605\beta _1 + 585762916992 ) / 26112 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -143093877\beta_{7} + 27836321\beta_{6} - 190198561847\beta_{4} - 78512758345920\beta_{2} ) / 130560 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 18031889\beta_{5} + 42378937\beta_{3} + 14024681175\beta _1 - 39146933223480 ) / 240 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1108187111327 \beta_{7} + 378058683213 \beta_{6} + \cdots + 73\!\cdots\!40 \beta_{2} ) / 130560 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(177\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 2181.88i | 0 | 0 | 0 | − | 131313.i | 0 | −3.16628e6 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 1835.74i | 0 | 0 | 0 | − | 262440.i | 0 | −1.77560e6 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 931.336i | 0 | 0 | 0 | 500587.i | 0 | 726936. | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | − | 341.481i | 0 | 0 | 0 | 519291.i | 0 | 1.47771e6 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 341.481i | 0 | 0 | 0 | − | 519291.i | 0 | 1.47771e6 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 931.336i | 0 | 0 | 0 | − | 500587.i | 0 | 726936. | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 1835.74i | 0 | 0 | 0 | 262440.i | 0 | −1.77560e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 2181.88i | 0 | 0 | 0 | 131313.i | 0 | −3.16628e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 200.14.c.f | 8 | |
| 5.b | even | 2 | 1 | inner | 200.14.c.f | 8 | |
| 5.c | odd | 4 | 1 | 40.14.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 200.14.a.f | 4 | ||
| 20.e | even | 4 | 1 | 80.14.a.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.14.a.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 80.14.a.k | 4 | 20.e | even | 4 | 1 | ||
| 200.14.a.f | 4 | 5.c | odd | 4 | 1 | ||
| 200.14.c.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 200.14.c.f | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 9114528T_{3}^{6} + 24144446935296T_{3}^{4} + 16608508418396160000T_{3}^{2} + 1622666910746941194240000 \)
acting on \(S_{14}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 80\!\cdots\!96 \)
|
| $11$ |
\( (T^{4} + \cdots + 18\!\cdots\!96)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 28\!\cdots\!96 \)
|
| $17$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $19$ |
\( (T^{4} + \cdots + 62\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 67\!\cdots\!16 \)
|
| $29$ |
\( (T^{4} + \cdots + 19\!\cdots\!96)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 36\!\cdots\!76 \)
|
| $41$ |
\( (T^{4} + \cdots - 15\!\cdots\!56)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 40\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 28\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $59$ |
\( (T^{4} + \cdots + 11\!\cdots\!24)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 57\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 42\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + \cdots + 67\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 89\!\cdots\!56 \)
|
| $79$ |
\( (T^{4} + \cdots - 48\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 64\!\cdots\!96 \)
|
| $89$ |
\( (T^{4} + \cdots - 17\!\cdots\!84)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
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