Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5577,2,Mod(1,5577)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5577, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5577.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5577 = 3 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5577.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: | \(44.5325692073\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 55 x^{5} + 156 x^{4} + \cdots - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 55 x^{5} + 156 x^{4} + \cdots - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( - \nu^{11} + \nu^{10} + 14 \nu^{9} - 11 \nu^{8} - 72 \nu^{7} + 41 \nu^{6} + 164 \nu^{5} - 55 \nu^{4} + \cdots + 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( 3 \nu^{11} - 4 \nu^{10} - 40 \nu^{9} + 45 \nu^{8} + 195 \nu^{7} - 177 \nu^{6} - 416 \nu^{5} + 276 \nu^{4} + \cdots - 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( 3 \nu^{11} - 4 \nu^{10} - 41 \nu^{9} + 47 \nu^{8} + 204 \nu^{7} - 193 \nu^{6} - 442 \nu^{5} + 314 \nu^{4} + \cdots - 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( - 7 \nu^{11} + 9 \nu^{10} + 95 \nu^{9} - 103 \nu^{8} - 471 \nu^{7} + 412 \nu^{6} + 1021 \nu^{5} + \cdots + 24 \)
|
| \(\beta_{7}\) | \(=\) |
\( 9 \nu^{11} - 12 \nu^{10} - 122 \nu^{9} + 139 \nu^{8} + 603 \nu^{7} - 564 \nu^{6} - 1299 \nu^{5} + \cdots - 27 \)
|
| \(\beta_{8}\) | \(=\) |
\( 10 \nu^{11} - 13 \nu^{10} - 136 \nu^{9} + 151 \nu^{8} + 673 \nu^{7} - 614 \nu^{6} - 1447 \nu^{5} + \cdots - 30 \)
|
| \(\beta_{9}\) | \(=\) |
\( 10 \nu^{11} - 13 \nu^{10} - 136 \nu^{9} + 150 \nu^{8} + 675 \nu^{7} - 605 \nu^{6} - 1463 \nu^{5} + \cdots - 37 \)
|
| \(\beta_{10}\) | \(=\) |
\( 29 \nu^{11} - 38 \nu^{10} - 394 \nu^{9} + 440 \nu^{8} + 1951 \nu^{7} - 1783 \nu^{6} - 4208 \nu^{5} + \cdots - 94 \)
|
| \(\beta_{11}\) | \(=\) |
\( 48 \nu^{11} - 62 \nu^{10} - 654 \nu^{9} + 719 \nu^{8} + 3247 \nu^{7} - 2917 \nu^{6} - 7023 \nu^{5} + \cdots - 167 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{7} + \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 5\beta_{9} - \beta_{8} - 8\beta_{7} - \beta_{6} + \beta_{5} + 7\beta_{3} + 2\beta_{2} + 17\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} - \beta_{8} - 11\beta_{7} - 8\beta_{6} + 10\beta_{5} + \beta_{4} + 9\beta_{3} + 25\beta_{2} + 10\beta _1 + 29 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{11} + 9 \beta_{10} + 20 \beta_{9} - 12 \beta_{8} - 52 \beta_{7} - 11 \beta_{6} + 13 \beta_{5} + \cdots + 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} - 87 \beta_{7} - 52 \beta_{6} + 74 \beta_{5} + \cdots + 134 \)
|
| \(\nu^{9}\) | \(=\) |
\( 13 \beta_{11} + 61 \beta_{10} + 70 \beta_{9} - 98 \beta_{8} - 322 \beta_{7} - 87 \beta_{6} + 116 \beta_{5} + \cdots + 111 \)
|
| \(\nu^{10}\) | \(=\) |
\( 29 \beta_{11} + 87 \beta_{10} - 44 \beta_{9} - 158 \beta_{8} - 613 \beta_{7} - 322 \beta_{6} + \cdots + 672 \)
|
| \(\nu^{11}\) | \(=\) |
\( 117 \beta_{11} + 377 \beta_{10} + 193 \beta_{9} - 695 \beta_{8} - 1972 \beta_{7} - 613 \beta_{6} + \cdots + 812 \)
|
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 |
|
−2.05767 | 1.00000 | 2.23401 | 1.75904 | −2.05767 | −1.42137 | −0.481510 | 1.00000 | −3.61953 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.02006 | 1.00000 | 2.08064 | −0.0469821 | −2.02006 | 0.491367 | −0.162896 | 1.00000 | 0.0949067 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.75487 | 1.00000 | 1.07957 | −3.61281 | −1.75487 | 4.71827 | 1.61524 | 1.00000 | 6.34002 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.02189 | 1.00000 | −0.955744 | 1.45866 | −1.02189 | 0.540395 | 3.02044 | 1.00000 | −1.49058 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.301545 | 1.00000 | −1.90907 | 3.39054 | −0.301545 | −3.49618 | 1.17876 | 1.00000 | −1.02240 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.289762 | 1.00000 | −1.91604 | −3.46259 | −0.289762 | 3.21728 | 1.13472 | 1.00000 | 1.00333 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.141839 | 1.00000 | −1.97988 | −1.12217 | −0.141839 | −2.80636 | 0.564503 | 1.00000 | 0.159167 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.849459 | 1.00000 | −1.27842 | 0.956012 | 0.849459 | 1.65017 | −2.78488 | 1.00000 | 0.812093 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.36657 | 1.00000 | −0.132473 | 0.447803 | 1.36657 | 0.0498518 | −2.91418 | 1.00000 | 0.611957 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.79509 | 1.00000 | 1.22236 | −3.59289 | 1.79509 | 2.13252 | −1.39594 | 1.00000 | −6.44957 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.08473 | 1.00000 | 2.34610 | 0.0509140 | 2.08473 | −3.00207 | 0.721527 | 1.00000 | 0.106142 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.49178 | 1.00000 | 4.20896 | −2.22553 | 2.49178 | −3.07388 | 5.50423 | 1.00000 | −5.54553 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 5577.2.a.bd | yes | 12 |
| 13.b | even | 2 | 1 | 5577.2.a.bb | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5577.2.a.bb | ✓ | 12 | 13.b | even | 2 | 1 | |
| 5577.2.a.bd | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):
|
\( T_{2}^{12} - T_{2}^{11} - 14 T_{2}^{10} + 11 T_{2}^{9} + 72 T_{2}^{8} - 41 T_{2}^{7} - 164 T_{2}^{6} + \cdots - 1 \)
|
|
\( T_{5}^{12} + 6 T_{5}^{11} - 13 T_{5}^{10} - 117 T_{5}^{9} + 11 T_{5}^{8} + 681 T_{5}^{7} + 53 T_{5}^{6} + \cdots + 1 \)
|
|
\( T_{7}^{12} + T_{7}^{11} - 40 T_{7}^{10} - 50 T_{7}^{9} + 536 T_{7}^{8} + 642 T_{7}^{7} - 3053 T_{7}^{6} + \cdots - 91 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} + 6 T^{11} + \cdots + 1 \)
|
| $7$ |
\( T^{12} + T^{11} + \cdots - 91 \)
|
| $11$ |
\( (T - 1)^{12} \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} + 25 T^{11} + \cdots + 24767 \)
|
| $19$ |
\( T^{12} - 6 T^{11} + \cdots - 8023 \)
|
| $23$ |
\( T^{12} + 40 T^{11} + \cdots - 4523 \)
|
| $29$ |
\( T^{12} + 31 T^{11} + \cdots - 23019737 \)
|
| $31$ |
\( T^{12} + \cdots + 715345519 \)
|
| $37$ |
\( T^{12} + \cdots - 388359007 \)
|
| $41$ |
\( T^{12} - 10 T^{11} + \cdots - 13259336 \)
|
| $43$ |
\( T^{12} + 34 T^{11} + \cdots - 1096537 \)
|
| $47$ |
\( T^{12} + 3 T^{11} + \cdots - 659093 \)
|
| $53$ |
\( T^{12} + \cdots + 2066770411 \)
|
| $59$ |
\( T^{12} - 8 T^{11} + \cdots + 14865691 \)
|
| $61$ |
\( T^{12} + \cdots + 101529421 \)
|
| $67$ |
\( T^{12} + \cdots + 579985211 \)
|
| $71$ |
\( T^{12} + 16 T^{11} + \cdots - 9127027 \)
|
| $73$ |
\( T^{12} + \cdots - 70043184653 \)
|
| $79$ |
\( T^{12} + \cdots + 11266100761 \)
|
| $83$ |
\( T^{12} + \cdots - 3459821896 \)
|
| $89$ |
\( T^{12} + \cdots + 62628820351 \)
|
| $97$ |
\( T^{12} + \cdots + 406431099179 \)
|
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