Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [736,4,Mod(1,736)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("736.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 736 = 2^{5} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 736.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: | \(43.4254057642\) |
| 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: |
\( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 93 \nu^{7} + 1879 \nu^{6} - 4689 \nu^{5} - 118958 \nu^{4} + 640515 \nu^{3} + 1659151 \nu^{2} + \cdots + 4336164 ) / 751446 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 247 \nu^{7} + 2746 \nu^{6} + 18520 \nu^{5} - 233346 \nu^{4} - 353877 \nu^{3} + 5171924 \nu^{2} + \cdots - 8344320 ) / 250482 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 487 \nu^{7} - 3555 \nu^{6} - 45473 \nu^{5} + 283119 \nu^{4} + 1155928 \nu^{3} - 5738121 \nu^{2} + \cdots + 10385325 ) / 375723 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2746 \nu^{7} + 26303 \nu^{6} + 222458 \nu^{5} - 2104732 \nu^{4} - 4630204 \nu^{3} + \cdots - 16939116 ) / 751446 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3383 \nu^{7} - 35582 \nu^{6} - 248248 \nu^{5} + 2904268 \nu^{4} + 3464273 \nu^{3} - 58336076 \nu^{2} + \cdots + 24827184 ) / 751446 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 35 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{5} - 3\beta_{4} - 2\beta_{3} + 4\beta_{2} + 58\beta _1 + 57 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 12\beta_{6} + 14\beta_{5} - 21\beta_{4} - 4\beta_{3} + 78\beta_{2} + 248\beta _1 + 1904 \)
|
| \(\nu^{5}\) | \(=\) |
\( -68\beta_{7} + 78\beta_{6} + 188\beta_{5} - 318\beta_{4} - 274\beta_{3} + 500\beta_{2} + 3913\beta _1 + 7296 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14 \beta_{7} + 1518 \beta_{6} + 2194 \beta_{5} - 2526 \beta_{4} - 1208 \beta_{3} + 6449 \beta_{2} + \cdots + 121649 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4455 \beta_{7} + 11388 \beta_{6} + 23829 \beta_{5} - 28803 \beta_{4} - 27330 \beta_{3} + \cdots + 718143 \)
|
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 | −7.42564 | 0 | 4.36302 | 0 | 9.40153 | 0 | 28.1401 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.48973 | 0 | −7.85529 | 0 | 7.65099 | 0 | 3.13714 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.24371 | 0 | −5.01937 | 0 | −22.4881 | 0 | −16.4784 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.963239 | 0 | 5.79749 | 0 | 15.5937 | 0 | −26.0722 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.55781 | 0 | −15.7437 | 0 | −25.1863 | 0 | −20.4576 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.87424 | 0 | 13.0818 | 0 | −35.1038 | 0 | 35.0036 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 8.05402 | 0 | 11.6466 | 0 | 24.4224 | 0 | 37.8672 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 8.70977 | 0 | −18.2706 | 0 | 11.7096 | 0 | 48.8601 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 736.4.a.f | yes | 8 |
| 4.b | odd | 2 | 1 | 736.4.a.e | ✓ | 8 | |
| 8.b | even | 2 | 1 | 1472.4.a.bg | 8 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.bh | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 736.4.a.e | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 736.4.a.f | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 1472.4.a.bg | 8 | 8.b | even | 2 | 1 | ||
| 1472.4.a.bh | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 12T_{3}^{7} - 81T_{3}^{6} + 1188T_{3}^{5} + 1395T_{3}^{4} - 32868T_{3}^{3} + 4757T_{3}^{2} + 210540T_{3} - 179952 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(736))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 12 T^{7} + \cdots - 179952 \)
|
| $5$ |
\( T^{8} + 12 T^{7} + \cdots + 43708192 \)
|
| $7$ |
\( T^{8} + \cdots - 6377755008 \)
|
| $11$ |
\( T^{8} + \cdots + 8187544416 \)
|
| $13$ |
\( T^{8} + \cdots + 3821115836436 \)
|
| $17$ |
\( T^{8} + \cdots - 18868965322784 \)
|
| $19$ |
\( T^{8} + \cdots - 349561386749184 \)
|
| $23$ |
\( (T + 23)^{8} \)
|
| $29$ |
\( T^{8} + \cdots - 46\!\cdots\!32 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!92 \)
|
| $37$ |
\( T^{8} + \cdots - 180113889841504 \)
|
| $41$ |
\( T^{8} + \cdots + 15\!\cdots\!96 \)
|
| $43$ |
\( T^{8} + \cdots - 13\!\cdots\!64 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots - 49\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots - 74\!\cdots\!24 \)
|
| $61$ |
\( T^{8} + \cdots + 17\!\cdots\!16 \)
|
| $67$ |
\( T^{8} + \cdots + 78\!\cdots\!32 \)
|
| $71$ |
\( T^{8} + \cdots - 76\!\cdots\!68 \)
|
| $73$ |
\( T^{8} + \cdots + 43\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!68 \)
|
| $83$ |
\( T^{8} + \cdots - 50\!\cdots\!36 \)
|
| $89$ |
\( T^{8} + \cdots + 11\!\cdots\!84 \)
|
| $97$ |
\( T^{8} + \cdots + 55\!\cdots\!08 \)
|
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