Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,6,Mod(1,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.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: | \(128.307055850\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 82x^{4} - 112x^{3} + 1514x^{2} + 3695x - 767 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 82x^{4} - 112x^{3} + 1514x^{2} + 3695x - 767 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} - 10\nu^{4} - 124\nu^{3} + 272\nu^{2} + 1958\nu - 298 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -20\nu^{5} + 112\nu^{4} + 1132\nu^{3} - 2996\nu^{2} - 16952\nu + 4297 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 142\nu^{5} - 770\nu^{4} - 8192\nu^{3} + 20116\nu^{2} + 123430\nu - 18446 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -40\nu^{5} + 212\nu^{4} + 2372\nu^{3} - 5716\nu^{2} - 36172\nu + 7205 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1142\nu^{5} - 6010\nu^{4} - 67960\nu^{3} + 162140\nu^{2} + 1034462\nu - 211894 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 3\beta_{2} + 10\beta _1 + 8 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 12\beta_{4} - 2\beta_{3} - 21\beta_{2} + 27\beta _1 + 1111 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{5} + 130\beta_{4} - \beta_{3} - 195\beta_{2} + 451\beta _1 + 3995 ) / 40 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 95\beta_{5} + 1227\beta_{4} - 55\beta_{3} - 1491\beta_{2} + 2790\beta _1 + 55366 ) / 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 835\beta_{5} + 11584\beta_{4} - 65\beta_{3} - 13752\beta_{2} + 28630\beta _1 + 371552 ) / 40 \)
|
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 | −26.2718 | 0 | 0 | 0 | −23.9306 | 0 | 447.206 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −19.0114 | 0 | 0 | 0 | −91.0286 | 0 | 118.434 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.81633 | 0 | 0 | 0 | 222.821 | 0 | −146.640 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.81633 | 0 | 0 | 0 | −222.821 | 0 | −146.640 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 19.0114 | 0 | 0 | 0 | 91.0286 | 0 | 118.434 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 26.2718 | 0 | 0 | 0 | 23.9306 | 0 | 447.206 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 800.6.a.z | 6 | |
| 4.b | odd | 2 | 1 | inner | 800.6.a.z | 6 | |
| 5.b | even | 2 | 1 | 800.6.a.ba | 6 | ||
| 5.c | odd | 4 | 2 | 160.6.c.c | ✓ | 12 | |
| 20.d | odd | 2 | 1 | 800.6.a.ba | 6 | ||
| 20.e | even | 4 | 2 | 160.6.c.c | ✓ | 12 | |
| 40.i | odd | 4 | 2 | 320.6.c.k | 12 | ||
| 40.k | even | 4 | 2 | 320.6.c.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.6.c.c | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 160.6.c.c | ✓ | 12 | 20.e | even | 4 | 2 | |
| 320.6.c.k | 12 | 40.i | odd | 4 | 2 | ||
| 320.6.c.k | 12 | 40.k | even | 4 | 2 | ||
| 800.6.a.z | 6 | 1.a | even | 1 | 1 | trivial | |
| 800.6.a.z | 6 | 4.b | odd | 2 | 1 | inner | |
| 800.6.a.ba | 6 | 5.b | even | 2 | 1 | ||
| 800.6.a.ba | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):
|
\( T_{3}^{6} - 1148T_{3}^{4} + 350800T_{3}^{2} - 24038400 \)
|
|
\( T_{11}^{6} - 746112T_{11}^{4} + 138141388800T_{11}^{2} - 6940560693657600 \)
|
|
\( T_{13}^{3} + 684T_{13}^{2} - 419040T_{13} - 54165248 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 1148 T^{4} + \cdots - 24038400 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 235600358400 \)
|
| $11$ |
\( T^{6} + \cdots - 69\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + 684 T^{2} + \cdots - 54165248)^{2} \)
|
| $17$ |
\( (T^{3} + 872 T^{2} + \cdots - 58165248)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 73\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} - 5326 T^{2} + \cdots + 85122490200)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + 10036 T^{2} + \cdots - 426824175360)^{2} \)
|
| $41$ |
\( (T^{3} + 27418 T^{2} + \cdots + 49588433400)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 1026686935296)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 1837262215800)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 73\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 77616325963776)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 221085541919000)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 363291797547008)^{2} \)
|
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