Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2541,2,Mod(1,2541)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, 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("2541.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2541 = 3 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2541.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: | \(20.2899871536\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} + \cdots + 385 ) / 2024 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + \cdots - 2409 ) / 2024 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + \cdots + 275 ) / 2024 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + \cdots - 5995 ) / 2024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + \cdots - 7403 ) / 2024 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + \cdots + 9449 ) / 2024 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} + \cdots - 1331 ) / 506 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8\beta_{2} + 2\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} + 9\beta_{4} + 11\beta_{3} + 39\beta _1 + 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + \cdots + 158 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + \cdots + 157 \)
|
| \(\nu^{8}\) | \(=\) |
\( 18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + \cdots + 1095 \)
|
| \(\nu^{9}\) | \(=\) |
\( 255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + \cdots + 1444 \)
|
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 |
|
−2.79866 | 1.00000 | 5.83249 | −1.67767 | −2.79866 | 1.00000 | −10.7258 | 1.00000 | 4.69523 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.63994 | 1.00000 | 4.96928 | 3.08369 | −2.63994 | 1.00000 | −7.83871 | 1.00000 | −8.14075 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.80545 | 1.00000 | 1.25966 | 2.77715 | −1.80545 | 1.00000 | 1.33666 | 1.00000 | −5.01400 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.871604 | 1.00000 | −1.24031 | −4.06436 | −0.871604 | 1.00000 | 2.82426 | 1.00000 | 3.54252 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.473713 | 1.00000 | −1.77560 | 3.75881 | −0.473713 | 1.00000 | 1.78855 | 1.00000 | −1.78060 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.112481 | 1.00000 | −1.98735 | 1.06131 | 0.112481 | 1.00000 | −0.448501 | 1.00000 | 0.119378 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.33330 | 1.00000 | −0.222305 | −0.873210 | 1.33330 | 1.00000 | −2.96300 | 1.00000 | −1.16425 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.09767 | 1.00000 | 2.40021 | 3.15947 | 2.09767 | 1.00000 | 0.839503 | 1.00000 | 6.62751 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.39396 | 1.00000 | 3.73106 | −3.93829 | 2.39396 | 1.00000 | 4.14409 | 1.00000 | −9.42812 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.65195 | 1.00000 | 5.03286 | 1.71311 | 2.65195 | 1.00000 | 8.04301 | 1.00000 | 4.54308 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 2541.2.a.br | 10 | |
| 3.b | odd | 2 | 1 | 7623.2.a.cy | 10 | ||
| 11.b | odd | 2 | 1 | 2541.2.a.bq | 10 | ||
| 11.d | odd | 10 | 2 | 231.2.j.g | ✓ | 20 | |
| 33.d | even | 2 | 1 | 7623.2.a.cx | 10 | ||
| 33.f | even | 10 | 2 | 693.2.m.j | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.j.g | ✓ | 20 | 11.d | odd | 10 | 2 | |
| 693.2.m.j | 20 | 33.f | even | 10 | 2 | ||
| 2541.2.a.bq | 10 | 11.b | odd | 2 | 1 | ||
| 2541.2.a.br | 10 | 1.a | even | 1 | 1 | trivial | |
| 7623.2.a.cx | 10 | 33.d | even | 2 | 1 | ||
| 7623.2.a.cy | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):
|
\( T_{2}^{10} - 19T_{2}^{8} + T_{2}^{7} + 124T_{2}^{6} - 6T_{2}^{5} - 316T_{2}^{4} - 17T_{2}^{3} + 253T_{2}^{2} + 70T_{2} - 11 \)
|
|
\( T_{5}^{10} - 5 T_{5}^{9} - 28 T_{5}^{8} + 179 T_{5}^{7} + 108 T_{5}^{6} - 1873 T_{5}^{5} + 1751 T_{5}^{4} + \cdots + 4336 \)
|
|
\( T_{13}^{10} - 6 T_{13}^{9} - 65 T_{13}^{8} + 334 T_{13}^{7} + 1540 T_{13}^{6} - 4808 T_{13}^{5} + \cdots + 2816 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 19 T^{8} + \cdots - 11 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} - 5 T^{9} + \cdots + 4336 \)
|
| $7$ |
\( (T - 1)^{10} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} - 6 T^{9} + \cdots + 2816 \)
|
| $17$ |
\( T^{10} - 8 T^{9} + \cdots - 8336 \)
|
| $19$ |
\( T^{10} - 121 T^{8} + \cdots - 50944 \)
|
| $23$ |
\( T^{10} - 106 T^{8} + \cdots - 600380 \)
|
| $29$ |
\( T^{10} + 14 T^{9} + \cdots - 3572144 \)
|
| $31$ |
\( T^{10} - 26 T^{9} + \cdots + 532400 \)
|
| $37$ |
\( T^{10} - 24 T^{9} + \cdots - 1675684 \)
|
| $41$ |
\( T^{10} - 19 T^{9} + \cdots - 10096 \)
|
| $43$ |
\( T^{10} + 6 T^{9} + \cdots - 21296 \)
|
| $47$ |
\( T^{10} - 15 T^{9} + \cdots - 2694400 \)
|
| $53$ |
\( T^{10} + T^{9} + \cdots - 240496 \)
|
| $59$ |
\( T^{10} + \cdots - 159106816 \)
|
| $61$ |
\( T^{10} - 395 T^{8} + \cdots - 87904256 \)
|
| $67$ |
\( T^{10} - 38 T^{9} + \cdots + 52960256 \)
|
| $71$ |
\( T^{10} - 26 T^{9} + \cdots + 17073920 \)
|
| $73$ |
\( T^{10} + T^{9} + \cdots - 7615744 \)
|
| $79$ |
\( T^{10} - 5 T^{9} + \cdots + 1785296 \)
|
| $83$ |
\( T^{10} - 6 T^{9} + \cdots - 779264 \)
|
| $89$ |
\( T^{10} + 9 T^{9} + \cdots + 106384 \)
|
| $97$ |
\( T^{10} - 24 T^{9} + \cdots + 481024 \)
|
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