Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1100,6,Mod(1,1100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1100, 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("1100.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1100.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: | \(176.422201794\) |
| 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: |
\( x^{8} - 3x^{7} - 1111x^{6} + 2926x^{5} + 349410x^{4} - 822682x^{3} - 25635603x^{2} + 144346581x - 191266515 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 5^{2} \) |
| 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
\( x^{8} - 3x^{7} - 1111x^{6} + 2926x^{5} + 349410x^{4} - 822682x^{3} - 25635603x^{2} + 144346581x - 191266515 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 279 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 825345 \nu^{7} + 783568 \nu^{6} - 910438423 \nu^{5} - 1580309025 \nu^{4} + \cdots + 31044799336977 ) / 47468672334 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4095023 \nu^{7} + 12041912 \nu^{6} - 4641936961 \nu^{5} - 13781533885 \nu^{4} + \cdots + 195753145100283 ) / 94937344668 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6134627 \nu^{7} + 9108780 \nu^{6} - 6618941513 \nu^{5} - 10741456993 \nu^{4} + \cdots + 287736307934139 ) / 94937344668 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 6828901 \nu^{7} - 6879444 \nu^{6} + 7638226531 \nu^{5} + 11107254299 \nu^{4} + \cdots - 371975085413541 ) / 94937344668 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7269631 \nu^{7} - 25702260 \nu^{6} + 7909869757 \nu^{5} + 28732488365 \nu^{4} + \cdots - 212606712127179 ) / 94937344668 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 279 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - 2\beta_{6} - \beta_{5} - 2\beta_{4} - 4\beta_{3} + \beta_{2} + 481\beta _1 - 26 \)
|
| \(\nu^{4}\) | \(=\) |
\( 19\beta_{7} + 14\beta_{6} + 31\beta_{5} + 32\beta_{4} - 53\beta_{3} + 582\beta_{2} + 198\beta _1 + 134507 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 860 \beta_{7} - 1309 \beta_{6} - 587 \beta_{5} - 1810 \beta_{4} - 2531 \beta_{3} - 232 \beta_{2} + \cdots + 46988 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12050 \beta_{7} + 12901 \beta_{6} + 24968 \beta_{5} + 29995 \beta_{4} - 60763 \beta_{3} + \cdots + 71560444 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 589710 \beta_{7} - 761367 \beta_{6} - 277851 \beta_{5} - 1295784 \beta_{4} - 1442160 \beta_{3} + \cdots - 14616537 \)
|
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 | −21.8760 | 0 | 0 | 0 | −48.9354 | 0 | 235.558 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −14.8741 | 0 | 0 | 0 | 166.492 | 0 | −21.7626 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −10.1518 | 0 | 0 | 0 | −122.703 | 0 | −139.941 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.15294 | 0 | 0 | 0 | 47.7885 | 0 | −216.447 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.75672 | 0 | 0 | 0 | −86.6415 | 0 | −182.833 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 9.06056 | 0 | 0 | 0 | 199.386 | 0 | −160.906 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 25.3000 | 0 | 0 | 0 | 150.442 | 0 | 397.092 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 26.6316 | 0 | 0 | 0 | −210.828 | 0 | 466.240 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 1100.6.a.k | yes | 8 |
| 5.b | even | 2 | 1 | 1100.6.a.h | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 1100.6.b.i | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1100.6.a.h | ✓ | 8 | 5.b | even | 2 | 1 | |
| 1100.6.a.k | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 1100.6.b.i | 16 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 27 T_{3}^{7} - 796 T_{3}^{6} + 20845 T_{3}^{5} + 164040 T_{3}^{4} - 4174435 T_{3}^{3} + \cdots - 806019876 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 27 T^{7} + \cdots - 806019876 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
|
| $11$ |
\( (T - 121)^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 89\!\cdots\!25 \)
|
| $17$ |
\( T^{8} + \cdots + 75\!\cdots\!08 \)
|
| $19$ |
\( T^{8} + \cdots - 11\!\cdots\!89 \)
|
| $23$ |
\( T^{8} + \cdots - 35\!\cdots\!77 \)
|
| $29$ |
\( T^{8} + \cdots - 14\!\cdots\!19 \)
|
| $31$ |
\( T^{8} + \cdots + 62\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots - 30\!\cdots\!52 \)
|
| $41$ |
\( T^{8} + \cdots - 31\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots + 11\!\cdots\!68 \)
|
| $47$ |
\( T^{8} + \cdots + 96\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots + 75\!\cdots\!36 \)
|
| $61$ |
\( T^{8} + \cdots + 30\!\cdots\!04 \)
|
| $67$ |
\( T^{8} + \cdots - 39\!\cdots\!76 \)
|
| $71$ |
\( T^{8} + \cdots + 36\!\cdots\!12 \)
|
| $73$ |
\( T^{8} + \cdots + 41\!\cdots\!88 \)
|
| $79$ |
\( T^{8} + \cdots - 89\!\cdots\!08 \)
|
| $83$ |
\( T^{8} + \cdots + 80\!\cdots\!47 \)
|
| $89$ |
\( T^{8} + \cdots - 34\!\cdots\!19 \)
|
| $97$ |
\( T^{8} + \cdots - 23\!\cdots\!41 \)
|
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