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: |
\( x^{12} - x^{11} - 14 x^{10} + 13 x^{9} + 70 x^{8} - 61 x^{7} - 152 x^{6} + 127 x^{5} + 138 x^{4} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{12} - x^{11} - 14 x^{10} + 13 x^{9} + 70 x^{8} - 61 x^{7} - 152 x^{6} + 127 x^{5} + 138 x^{4} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11 \nu^{11} + 196 \nu^{10} + 267 \nu^{9} - 2571 \nu^{8} - 2231 \nu^{7} + 11170 \nu^{6} + 7389 \nu^{5} + \cdots - 321 ) / 463 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 24 \nu^{11} + 133 \nu^{10} + 330 \nu^{9} - 1695 \nu^{8} - 1795 \nu^{7} + 7282 \nu^{6} + 4799 \nu^{5} + \cdots - 532 ) / 463 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 31 \nu^{11} - 79 \nu^{10} + 542 \nu^{9} + 878 \nu^{8} - 3341 \nu^{7} - 2825 \nu^{6} + 8996 \nu^{5} + \cdots - 610 ) / 463 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 109 \nu^{11} - 6 \nu^{10} - 1383 \nu^{9} - 115 \nu^{8} + 5818 \nu^{7} + 1151 \nu^{6} - 9198 \nu^{5} + \cdots + 24 ) / 463 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 128 \nu^{11} + 92 \nu^{10} + 1760 \nu^{9} - 1169 \nu^{8} - 8493 \nu^{7} + 5347 \nu^{6} + \cdots - 1757 ) / 463 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 191 \nu^{11} - 36 \nu^{10} - 2742 \nu^{9} + 236 \nu^{8} + 14073 \nu^{7} - 39 \nu^{6} - 31112 \nu^{5} + \cdots + 607 ) / 463 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 199 \nu^{11} + 74 \nu^{10} - 2852 \nu^{9} - 1051 \nu^{8} + 14517 \nu^{7} + 4633 \nu^{6} - 31940 \nu^{5} + \cdots - 296 ) / 463 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 272 \nu^{11} - 36 \nu^{10} + 3740 \nu^{9} + 699 \nu^{8} - 17874 \nu^{7} - 3743 \nu^{6} + \cdots + 1070 ) / 463 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 286 \nu^{11} + 3 \nu^{10} + 4164 \nu^{9} + 289 \nu^{8} - 21892 \nu^{7} - 2659 \nu^{6} + 50899 \nu^{5} + \cdots + 1377 ) / 463 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 3\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 6\beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} + 7\beta_{10} + 8\beta_{9} + \beta_{8} - 6\beta_{7} + 7\beta_{5} + 6\beta_{4} + 8\beta_{2} + 12\beta _1 - 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{10} - 9 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} - 11 \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 31 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9 \beta_{11} + 42 \beta_{10} + 52 \beta_{9} + 9 \beta_{8} - 33 \beta_{7} - 2 \beta_{6} + 43 \beta_{5} + \cdots - 31 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{11} - 21 \beta_{10} - 59 \beta_{9} + 52 \beta_{8} - 52 \beta_{7} - 85 \beta_{6} + 11 \beta_{5} + \cdots + 153 \)
|
| \(\nu^{9}\) | \(=\) |
\( 63 \beta_{11} + 240 \beta_{10} + 315 \beta_{9} + 63 \beta_{8} - 183 \beta_{7} - 29 \beta_{6} + \cdots - 157 \)
|
| \(\nu^{10}\) | \(=\) |
\( 28 \beta_{11} - 156 \beta_{10} - 345 \beta_{9} + 320 \beta_{8} - 319 \beta_{7} - 579 \beta_{6} + \cdots + 796 \)
|
| \(\nu^{11}\) | \(=\) |
\( 407 \beta_{11} + 1343 \beta_{10} + 1849 \beta_{9} + 411 \beta_{8} - 1033 \beta_{7} - 282 \beta_{6} + \cdots - 799 \)
|
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 |
|
−2.34801 | −1.00000 | 3.51317 | 1.25493 | 2.34801 | −0.763138 | −3.55295 | 1.00000 | −2.94660 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.89872 | −1.00000 | 1.60515 | −1.52728 | 1.89872 | −1.22868 | 0.749713 | 1.00000 | 2.89988 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.65821 | −1.00000 | 0.749676 | −1.28282 | 1.65821 | −4.15353 | 2.07331 | 1.00000 | 2.12720 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.15642 | −1.00000 | −0.662692 | −1.96149 | 1.15642 | 2.38369 | 3.07919 | 1.00000 | 2.26831 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.679772 | −1.00000 | −1.53791 | 3.36496 | 0.679772 | −0.0622521 | 2.40497 | 1.00000 | −2.28741 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0331136 | −1.00000 | −1.99890 | 0.127095 | 0.0331136 | 4.18754 | 0.132418 | 1.00000 | −0.00420859 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.645300 | −1.00000 | −1.58359 | −1.52961 | −0.645300 | −1.04662 | −2.31249 | 1.00000 | −0.987060 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.849061 | −1.00000 | −1.27910 | 2.48531 | −0.849061 | −1.35029 | −2.78415 | 1.00000 | 2.11018 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.28801 | −1.00000 | −0.341031 | 1.67109 | −1.28801 | −2.27382 | −3.01527 | 1.00000 | 2.15238 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.40839 | −1.00000 | −0.0164266 | −3.77875 | −1.40839 | 0.976719 | −2.83992 | 1.00000 | −5.32197 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.13773 | −1.00000 | 2.56988 | −2.55253 | −2.13773 | 4.81735 | 1.21826 | 1.00000 | −5.45661 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.44576 | −1.00000 | 3.98177 | −2.27090 | −2.44576 | −2.48697 | 4.84693 | 1.00000 | −5.55410 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.bc | yes | 12 |
| 13.b | even | 2 | 1 | 5577.2.a.ba | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5577.2.a.ba | ✓ | 12 | 13.b | even | 2 | 1 | |
| 5577.2.a.bc | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):
|
\( T_{2}^{12} - T_{2}^{11} - 14 T_{2}^{10} + 13 T_{2}^{9} + 70 T_{2}^{8} - 61 T_{2}^{7} - 152 T_{2}^{6} + \cdots + 1 \)
|
|
\( T_{5}^{12} + 6 T_{5}^{11} - 11 T_{5}^{10} - 125 T_{5}^{9} - 89 T_{5}^{8} + 773 T_{5}^{7} + 1289 T_{5}^{6} + \cdots - 287 \)
|
|
\( T_{7}^{12} + T_{7}^{11} - 40 T_{7}^{10} - 72 T_{7}^{9} + 456 T_{7}^{8} + 1256 T_{7}^{7} - 839 T_{7}^{6} + \cdots + 91 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( T^{12} + 6 T^{11} + \cdots - 287 \)
|
| $7$ |
\( T^{12} + T^{11} + \cdots + 91 \)
|
| $11$ |
\( (T - 1)^{12} \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} - 3 T^{11} + \cdots - 10067 \)
|
| $19$ |
\( T^{12} - 6 T^{11} + \cdots - 21629 \)
|
| $23$ |
\( T^{12} + 22 T^{11} + \cdots + 637 \)
|
| $29$ |
\( T^{12} - 13 T^{11} + \cdots - 14491 \)
|
| $31$ |
\( T^{12} - 2 T^{11} + \cdots + 18719 \)
|
| $37$ |
\( T^{12} + \cdots - 511347187 \)
|
| $41$ |
\( T^{12} - 4 T^{11} + \cdots + 1876568 \)
|
| $43$ |
\( T^{12} + 26 T^{11} + \cdots - 17562791 \)
|
| $47$ |
\( T^{12} + \cdots + 9410965171 \)
|
| $53$ |
\( T^{12} + 5 T^{11} + \cdots - 25488217 \)
|
| $59$ |
\( T^{12} + \cdots - 14977982089 \)
|
| $61$ |
\( T^{12} + 11 T^{11} + \cdots + 5141479 \)
|
| $67$ |
\( T^{12} + 28 T^{11} + \cdots + 30662051 \)
|
| $71$ |
\( T^{12} + \cdots - 2261363483 \)
|
| $73$ |
\( T^{12} + 7 T^{11} + \cdots - 6902603 \)
|
| $79$ |
\( T^{12} + 40 T^{11} + \cdots + 242479 \)
|
| $83$ |
\( T^{12} - T^{11} + \cdots + 55403608 \)
|
| $89$ |
\( T^{12} - 11 T^{11} + \cdots + 19492871 \)
|
| $97$ |
\( T^{12} + \cdots - 1317493701 \)
|
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