Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, 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("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 258 x^{7} - 126 x^{6} + 567 x^{5} - 397 x^{3} + \cdots + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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
\( x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 258 x^{7} - 126 x^{6} + 567 x^{5} - 397 x^{3} + \cdots + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{11} + 37 \nu^{10} - 2 \nu^{9} - 540 \nu^{8} + 340 \nu^{7} + 2732 \nu^{6} - 1791 \nu^{5} + \cdots - 444 ) / 179 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3 \nu^{11} - 37 \nu^{10} + 2 \nu^{9} + 540 \nu^{8} - 340 \nu^{7} - 2732 \nu^{6} + 1791 \nu^{5} + \cdots + 444 ) / 179 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25 \nu^{11} + 189 \nu^{10} + 222 \nu^{9} - 3068 \nu^{8} + 387 \nu^{7} + 17337 \nu^{6} - 8481 \nu^{5} + \cdots - 2805 ) / 537 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 38 \nu^{11} + 51 \nu^{10} + 631 \nu^{9} - 754 \nu^{8} - 3629 \nu^{7} + 3877 \nu^{6} + 8102 \nu^{5} + \cdots - 75 ) / 179 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 173 \nu^{11} - 642 \nu^{10} - 2331 \nu^{9} + 10018 \nu^{8} + 9093 \nu^{7} - 54501 \nu^{6} + \cdots + 8778 ) / 537 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 65 \nu^{11} + 205 \nu^{10} + 971 \nu^{9} - 3287 \nu^{8} - 4686 \nu^{7} + 18441 \nu^{6} + \cdots - 3355 ) / 179 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 88 \nu^{11} + 250 \nu^{10} + 1254 \nu^{9} - 3847 \nu^{8} - 5540 \nu^{7} + 20651 \nu^{6} + \cdots - 2463 ) / 179 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 120 \nu^{11} + 406 \nu^{10} + 1710 \nu^{9} - 6385 \nu^{8} - 7522 \nu^{7} + 34995 \nu^{6} + \cdots - 5409 ) / 179 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 234 \nu^{11} + 738 \nu^{10} + 3424 \nu^{9} - 11511 \nu^{8} - 16082 \nu^{7} + 62378 \nu^{6} + \cdots - 8677 ) / 179 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{3} + 6\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 4\beta_{7} + 2\beta_{5} + 7\beta_{4} + 9\beta_{3} + 3\beta_{2} + 28\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{11} + 3 \beta_{10} + 13 \beta_{7} + 11 \beta_{6} + 20 \beta_{5} + 9 \beta_{4} + 22 \beta_{3} + \cdots + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{11} + 17 \beta_{10} + 12 \beta_{9} - 2 \beta_{8} + 54 \beta_{7} + 2 \beta_{6} + 27 \beta_{5} + \cdots + 45 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 13 \beta_{11} + 47 \beta_{10} + 3 \beta_{9} - 4 \beta_{8} + 137 \beta_{7} + 93 \beta_{6} + 166 \beta_{5} + \cdots + 518 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 16 \beta_{11} + 191 \beta_{10} + 106 \beta_{9} - 31 \beta_{8} + 540 \beta_{7} + 40 \beta_{6} + \cdots + 467 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 122 \beta_{11} + 515 \beta_{10} + 56 \beta_{9} - 69 \beta_{8} + 1308 \beta_{7} + 721 \beta_{6} + \cdots + 3332 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 178 \beta_{11} + 1826 \beta_{10} + 843 \beta_{9} - 337 \beta_{8} + 4826 \beta_{7} + 513 \beta_{6} + \cdots + 4217 \)
|
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 |
|
−2.79524 | −1.00000 | 5.81337 | −1.00000 | 2.79524 | 2.95801 | −10.6593 | 1.00000 | 2.79524 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.40220 | −1.00000 | 3.77057 | −1.00000 | 2.40220 | 3.65207 | −4.25327 | 1.00000 | 2.40220 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.34870 | −1.00000 | 3.51639 | −1.00000 | 2.34870 | 3.56064 | −3.56155 | 1.00000 | 2.34870 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.51031 | −1.00000 | 0.281047 | −1.00000 | 1.51031 | −3.67192 | 2.59616 | 1.00000 | 1.51031 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.29625 | −1.00000 | −0.319727 | −1.00000 | 1.29625 | −2.32093 | 3.00695 | 1.00000 | 1.29625 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.601546 | −1.00000 | −1.63814 | −1.00000 | 0.601546 | −3.57330 | 2.18851 | 1.00000 | 0.601546 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.210345 | −1.00000 | −1.95575 | −1.00000 | −0.210345 | 1.78503 | −0.832073 | 1.00000 | −0.210345 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.248828 | −1.00000 | −1.93808 | −1.00000 | −0.248828 | 2.67441 | −0.979905 | 1.00000 | −0.248828 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.861696 | −1.00000 | −1.25748 | −1.00000 | −0.861696 | 0.469073 | −2.80696 | 1.00000 | −0.861696 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.05082 | −1.00000 | 2.20586 | −1.00000 | −2.05082 | −3.39279 | 0.422183 | 1.00000 | −2.05082 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.18644 | −1.00000 | 2.78053 | −1.00000 | −2.18644 | 4.93633 | 1.70658 | 1.00000 | −2.18644 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.39613 | −1.00000 | 3.74142 | −1.00000 | −2.39613 | −1.07661 | 4.17266 | 1.00000 | −2.39613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5415.2.a.bo | 12 | |
| 19.b | odd | 2 | 1 | 5415.2.a.bp | 12 | ||
| 19.f | odd | 18 | 2 | 285.2.u.c | ✓ | 24 | |
| 57.j | even | 18 | 2 | 855.2.bs.e | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.u.c | ✓ | 24 | 19.f | odd | 18 | 2 | |
| 855.2.bs.e | 24 | 57.j | even | 18 | 2 | ||
| 5415.2.a.bo | 12 | 1.a | even | 1 | 1 | trivial | |
| 5415.2.a.bp | 12 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 47 T_{2}^{9} + 75 T_{2}^{8} + 258 T_{2}^{7} - 126 T_{2}^{6} + \cdots + 9 \)
|
|
\( T_{7}^{12} - 6 T_{7}^{11} - 39 T_{7}^{10} + 267 T_{7}^{9} + 492 T_{7}^{8} - 4425 T_{7}^{7} + \cdots - 47296 \)
|
|
\( T_{11}^{12} - 75 T_{11}^{10} - T_{11}^{9} + 2100 T_{11}^{8} + 357 T_{11}^{7} - 27311 T_{11}^{6} + \cdots - 219456 \)
|
|
\( T_{13}^{12} - 15 T_{13}^{11} + 15 T_{13}^{10} + 766 T_{13}^{9} - 3561 T_{13}^{8} - 5808 T_{13}^{7} + \cdots - 24384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 3 T^{11} + \cdots + 9 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots - 47296 \)
|
| $11$ |
\( T^{12} - 75 T^{10} + \cdots - 219456 \)
|
| $13$ |
\( T^{12} - 15 T^{11} + \cdots - 24384 \)
|
| $17$ |
\( T^{12} - 18 T^{11} + \cdots - 729 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} - 15 T^{11} + \cdots + 114201 \)
|
| $29$ |
\( T^{12} + 15 T^{11} + \cdots - 3644352 \)
|
| $31$ |
\( T^{12} - 120 T^{10} + \cdots + 1677987 \)
|
| $37$ |
\( T^{12} + 6 T^{11} + \cdots - 82855232 \)
|
| $41$ |
\( T^{12} + \cdots + 154274112 \)
|
| $43$ |
\( T^{12} - 174 T^{10} + \cdots - 4293568 \)
|
| $47$ |
\( T^{12} + \cdots - 545443947 \)
|
| $53$ |
\( T^{12} + 6 T^{11} + \cdots - 18684423 \)
|
| $59$ |
\( T^{12} + \cdots - 2204407872 \)
|
| $61$ |
\( T^{12} + \cdots - 22398582439 \)
|
| $67$ |
\( T^{12} - 12 T^{11} + \cdots - 380096 \)
|
| $71$ |
\( T^{12} + 6 T^{11} + \cdots - 4325184 \)
|
| $73$ |
\( T^{12} - 63 T^{11} + \cdots - 13570048 \)
|
| $79$ |
\( T^{12} + \cdots + 26361529461 \)
|
| $83$ |
\( T^{12} - 33 T^{11} + \cdots - 87144201 \)
|
| $89$ |
\( T^{12} - 6 T^{11} + \cdots - 55518912 \)
|
| $97$ |
\( T^{12} + \cdots - 5002751808 \)
|
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