Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1341,2,Mod(1,1341)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1341, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1341.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1341 = 3^{2} \cdot 149 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1341.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: | \(10.7079389111\) |
| 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: |
\( x^{10} - 3x^{9} - 12x^{8} + 37x^{7} + 44x^{6} - 142x^{5} - 50x^{4} + 181x^{3} - 5x^{2} - 30x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 447) |
| 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
\( x^{10} - 3x^{9} - 12x^{8} + 37x^{7} + 44x^{6} - 142x^{5} - 50x^{4} + 181x^{3} - 5x^{2} - 30x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13 \nu^{9} + 47 \nu^{8} - 293 \nu^{7} - 661 \nu^{6} + 2072 \nu^{5} + 3251 \nu^{4} - 5322 \nu^{3} + \cdots + 1119 ) / 647 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 29 \nu^{9} + 144 \nu^{8} + 355 \nu^{7} - 1860 \nu^{6} - 1636 \nu^{5} + 7579 \nu^{4} + 4357 \nu^{3} + \cdots + 2630 ) / 647 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 69 \nu^{9} - 298 \nu^{8} - 510 \nu^{7} + 3310 \nu^{6} - 101 \nu^{5} - 10715 \nu^{4} + 4202 \nu^{3} + \cdots - 1974 ) / 647 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73 \nu^{9} - 184 \nu^{8} - 849 \nu^{7} + 2161 \nu^{6} + 3025 \nu^{5} - 7923 \nu^{4} - 3806 \nu^{3} + \cdots - 3073 ) / 647 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 95 \nu^{9} - 204 \nu^{8} - 1096 \nu^{7} + 1988 \nu^{6} + 4043 \nu^{5} - 4860 \nu^{4} - 6442 \nu^{3} + \cdots - 383 ) / 647 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 179 \nu^{9} + 398 \nu^{8} + 2392 \nu^{7} - 5033 \nu^{6} - 10165 \nu^{5} + 19986 \nu^{4} + \cdots + 2111 ) / 647 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 204 \nu^{9} + 656 \nu^{8} + 2408 \nu^{7} - 8042 \nu^{6} - 8675 \nu^{5} + 30357 \nu^{4} + \cdots + 4289 ) / 647 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + 2\beta_{3} + 6\beta_{2} + \beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 9 \beta_{9} + 10 \beta_{8} - \beta_{7} + 12 \beta_{6} - 9 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} + \cdots + 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} - 13\beta_{7} + 6\beta_{6} + 10\beta_{5} + 22\beta_{3} + 34\beta_{2} + 11\beta _1 + 112 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 64 \beta_{9} + 77 \beta_{8} - 17 \beta_{7} + 107 \beta_{6} - 61 \beta_{5} + 56 \beta_{4} + 28 \beta_{3} + \cdots + 137 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{9} + 16 \beta_{8} - 123 \beta_{7} + 95 \beta_{6} + 79 \beta_{5} + 4 \beta_{4} + 191 \beta_{3} + \cdots + 787 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 421 \beta_{9} + 544 \beta_{8} - 190 \beta_{7} + 870 \beta_{6} - 377 \beta_{5} + 382 \beta_{4} + \cdots + 1192 \)
|
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 |
|
−2.77402 | 0 | 5.69517 | −2.50038 | 0 | −2.37841 | −10.2505 | 0 | 6.93609 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.55905 | 0 | 4.54875 | 2.49465 | 0 | 3.88385 | −6.52240 | 0 | −6.38395 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.06206 | 0 | 2.25211 | −1.47857 | 0 | −0.281819 | −0.519863 | 0 | 3.04891 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.41389 | 0 | −0.000922380 | 0 | −3.18405 | 0 | 4.73714 | 2.82908 | 0 | 4.50188 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.482231 | 0 | −1.76745 | 3.08943 | 0 | 3.63159 | 1.81678 | 0 | −1.48982 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0333697 | 0 | −1.99889 | −4.39978 | 0 | −2.75944 | 0.133442 | 0 | 0.146819 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.415912 | 0 | −1.82702 | 1.25623 | 0 | −3.67648 | −1.59170 | 0 | 0.522481 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.58830 | 0 | 0.522693 | 3.92990 | 0 | −0.0296253 | −2.34641 | 0 | 6.24185 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.81194 | 0 | 1.28314 | −3.62012 | 0 | 1.49239 | −1.29890 | 0 | −6.55946 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.50847 | 0 | 4.29241 | 0.412683 | 0 | 4.38080 | 5.75043 | 0 | 1.03520 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(149\) | \(1\) |
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 | 1341.2.a.f | 10 | |
| 3.b | odd | 2 | 1 | 447.2.a.d | ✓ | 10 | |
| 12.b | even | 2 | 1 | 7152.2.a.bc | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 447.2.a.d | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 1341.2.a.f | 10 | 1.a | even | 1 | 1 | trivial | |
| 7152.2.a.bc | 10 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1341))\):
|
\( T_{2}^{10} + 3T_{2}^{9} - 12T_{2}^{8} - 37T_{2}^{7} + 44T_{2}^{6} + 142T_{2}^{5} - 50T_{2}^{4} - 181T_{2}^{3} - 5T_{2}^{2} + 30T_{2} + 1 \)
|
|
\( T_{5}^{10} + 4 T_{5}^{9} - 34 T_{5}^{8} - 132 T_{5}^{7} + 392 T_{5}^{6} + 1440 T_{5}^{5} - 1848 T_{5}^{4} + \cdots - 2944 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 3 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + 4 T^{9} + \cdots - 2944 \)
|
| $7$ |
\( T^{10} - 9 T^{9} + \cdots - 88 \)
|
| $11$ |
\( T^{10} - 7 T^{9} + \cdots - 8900 \)
|
| $13$ |
\( T^{10} - 4 T^{9} + \cdots - 25216 \)
|
| $17$ |
\( T^{10} + 10 T^{9} + \cdots + 640 \)
|
| $19$ |
\( T^{10} - 9 T^{9} + \cdots + 300140 \)
|
| $23$ |
\( T^{10} + 9 T^{9} + \cdots + 3930304 \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots - 484480 \)
|
| $31$ |
\( T^{10} - 15 T^{9} + \cdots - 122816 \)
|
| $37$ |
\( T^{10} - 31 T^{9} + \cdots + 470822 \)
|
| $41$ |
\( T^{10} - 5 T^{9} + \cdots - 862 \)
|
| $43$ |
\( T^{10} - 8 T^{9} + \cdots - 287488 \)
|
| $47$ |
\( T^{10} + 2 T^{9} + \cdots + 4361216 \)
|
| $53$ |
\( T^{10} + 4 T^{9} + \cdots + 1969024 \)
|
| $59$ |
\( T^{10} - 59 T^{9} + \cdots - 522884 \)
|
| $61$ |
\( T^{10} + \cdots - 965807038 \)
|
| $67$ |
\( T^{10} - 5 T^{9} + \cdots - 6152884 \)
|
| $71$ |
\( T^{10} + \cdots + 352744592 \)
|
| $73$ |
\( T^{10} + \cdots - 195280966 \)
|
| $79$ |
\( T^{10} + 6 T^{9} + \cdots + 14112256 \)
|
| $83$ |
\( T^{10} + \cdots + 110712236 \)
|
| $89$ |
\( T^{10} - 5 T^{9} + \cdots + 5946530 \)
|
| $97$ |
\( T^{10} - 32 T^{9} + \cdots - 66617216 \)
|
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