Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2415,2,Mod(1,2415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2415.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: | \(19.2838720881\) |
| 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: |
\( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 83x^{6} - 137x^{5} - 164x^{4} + 208x^{3} + 108x^{2} - 83x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 83x^{6} - 137x^{5} - 164x^{4} + 208x^{3} + 108x^{2} - 83x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9 \nu^{9} + 21 \nu^{8} + 137 \nu^{7} - 272 \nu^{6} - 700 \nu^{5} + 855 \nu^{4} + 1584 \nu^{3} + \cdots - 331 ) / 131 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11 \nu^{9} + 18 \nu^{8} - 182 \nu^{7} - 308 \nu^{6} + 972 \nu^{5} + 1706 \nu^{4} - 1805 \nu^{3} + \cdots + 1438 ) / 131 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31 \nu^{9} - 15 \nu^{8} + 501 \nu^{7} + 213 \nu^{6} - 2644 \nu^{5} - 723 \nu^{4} + 5325 \nu^{3} + \cdots + 1116 ) / 262 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 45 \nu^{9} + 105 \nu^{8} + 685 \nu^{7} - 1491 \nu^{6} - 3238 \nu^{5} + 6109 \nu^{4} + 5431 \nu^{3} + \cdots + 1358 ) / 262 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29 \nu^{9} - 24 \nu^{8} - 456 \nu^{7} + 367 \nu^{6} + 2241 \nu^{5} - 1707 \nu^{4} - 3794 \nu^{3} + \cdots - 520 ) / 131 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 61 \nu^{9} + 55 \nu^{8} + 1045 \nu^{7} - 781 \nu^{6} - 5938 \nu^{5} + 3175 \nu^{4} + 12701 \nu^{3} + \cdots + 886 ) / 262 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 8\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 10\beta_{3} - 2\beta_{2} + 41\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{8} + 14\beta_{7} + 12\beta_{6} - 2\beta_{5} - 13\beta_{4} + \beta_{3} + 60\beta_{2} - 2\beta _1 + 161 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3 \beta_{9} + 2 \beta_{8} + 13 \beta_{7} - 14 \beta_{6} - 13 \beta_{5} - 28 \beta_{4} + 87 \beta_{3} + \cdots - 46 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{9} + 143 \beta_{8} + 144 \beta_{7} + 111 \beta_{6} - 28 \beta_{5} - 128 \beta_{4} + 14 \beta_{3} + \cdots + 1141 \)
|
| \(\nu^{9}\) | \(=\) |
\( 48 \beta_{9} + 36 \beta_{8} + 128 \beta_{7} - 144 \beta_{6} - 125 \beta_{5} - 286 \beta_{4} + 725 \beta_{3} + \cdots - 504 \)
|
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 |
|
−2.81587 | 1.00000 | 5.92915 | 1.00000 | −2.81587 | −1.00000 | −11.0640 | 1.00000 | −2.81587 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.13650 | 1.00000 | 2.56465 | 1.00000 | −2.13650 | −1.00000 | −1.20638 | 1.00000 | −2.13650 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.42324 | 1.00000 | 0.0256085 | 1.00000 | −1.42324 | −1.00000 | 2.81003 | 1.00000 | −1.42324 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.866400 | 1.00000 | −1.24935 | 1.00000 | −0.866400 | −1.00000 | 2.81524 | 1.00000 | −0.866400 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.128823 | 1.00000 | −1.98340 | 1.00000 | −0.128823 | −1.00000 | 0.513155 | 1.00000 | −0.128823 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.663795 | 1.00000 | −1.55938 | 1.00000 | 0.663795 | −1.00000 | −2.36270 | 1.00000 | 0.663795 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.24544 | 1.00000 | −0.448869 | 1.00000 | 1.24544 | −1.00000 | −3.04993 | 1.00000 | 1.24544 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.24070 | 1.00000 | 3.02074 | 1.00000 | 2.24070 | −1.00000 | 2.28717 | 1.00000 | 2.24070 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.42079 | 1.00000 | 3.86022 | 1.00000 | 2.42079 | −1.00000 | 4.50319 | 1.00000 | 2.42079 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.80011 | 1.00000 | 5.84064 | 1.00000 | 2.80011 | −1.00000 | 10.7542 | 1.00000 | 2.80011 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 2415.2.a.v | ✓ | 10 |
| 3.b | odd | 2 | 1 | 7245.2.a.bu | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2415.2.a.v | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 7245.2.a.bu | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):
|
\( T_{2}^{10} - 2 T_{2}^{9} - 16 T_{2}^{8} + 30 T_{2}^{7} + 83 T_{2}^{6} - 137 T_{2}^{5} - 164 T_{2}^{4} + \cdots - 12 \)
|
|
\( T_{11}^{10} + T_{11}^{9} - 75 T_{11}^{8} + 29 T_{11}^{7} + 1877 T_{11}^{6} - 3407 T_{11}^{5} + \cdots + 15712 \)
|
|
\( T_{13}^{10} + 2 T_{13}^{9} - 97 T_{13}^{8} - 72 T_{13}^{7} + 3625 T_{13}^{6} - 1938 T_{13}^{5} + \cdots + 801312 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 12 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( (T + 1)^{10} \)
|
| $11$ |
\( T^{10} + T^{9} + \cdots + 15712 \)
|
| $13$ |
\( T^{10} + 2 T^{9} + \cdots + 801312 \)
|
| $17$ |
\( T^{10} - 8 T^{9} + \cdots - 96704 \)
|
| $19$ |
\( T^{10} - 11 T^{9} + \cdots - 857856 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( T^{10} - 6 T^{9} + \cdots + 769536 \)
|
| $31$ |
\( T^{10} - 16 T^{9} + \cdots - 5701632 \)
|
| $37$ |
\( T^{10} - 215 T^{8} + \cdots + 16464432 \)
|
| $41$ |
\( T^{10} + \cdots + 117176544 \)
|
| $43$ |
\( T^{10} - 4 T^{9} + \cdots - 377088 \)
|
| $47$ |
\( T^{10} - 21 T^{9} + \cdots + 5128192 \)
|
| $53$ |
\( T^{10} + \cdots + 193211776 \)
|
| $59$ |
\( T^{10} + \cdots + 1190323968 \)
|
| $61$ |
\( T^{10} - 29 T^{9} + \cdots - 33034000 \)
|
| $67$ |
\( T^{10} - 32 T^{9} + \cdots - 63414272 \)
|
| $71$ |
\( T^{10} + \cdots - 19522609152 \)
|
| $73$ |
\( T^{10} + \cdots + 313816688 \)
|
| $79$ |
\( T^{10} - 8 T^{9} + \cdots + 28602368 \)
|
| $83$ |
\( T^{10} - 2 T^{9} + \cdots + 1263616 \)
|
| $89$ |
\( T^{10} + \cdots - 819561472 \)
|
| $97$ |
\( T^{10} + 2 T^{9} + \cdots - 436736 \)
|
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