Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2668,2,Mod(1,2668)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2668, 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("2668.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2668 = 2^{2} \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2668.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: | \(21.3040872593\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 3x^{9} - 12x^{8} + 35x^{7} + 48x^{6} - 136x^{5} - 62x^{4} + 190x^{3} - 7x^{2} - 50x + 3 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3x^{9} - 12x^{8} + 35x^{7} + 48x^{6} - 136x^{5} - 62x^{4} + 190x^{3} - 7x^{2} - 50x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7 \nu^{9} - 63 \nu^{8} + 115 \nu^{7} + 450 \nu^{6} - 1469 \nu^{5} - 551 \nu^{4} + 4125 \nu^{3} + \cdots + 272 ) / 179 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25 \nu^{9} - 46 \nu^{8} - 382 \nu^{7} + 482 \nu^{6} + 2067 \nu^{5} - 1124 \nu^{4} - 4830 \nu^{3} + \cdots + 102 ) / 537 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26 \nu^{9} + 234 \nu^{8} - 197 \nu^{7} - 2234 \nu^{6} + 3206 \nu^{5} + 6496 \nu^{4} - 8903 \nu^{3} + \cdots + 882 ) / 537 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 36 \nu^{9} - 34 \nu^{8} + 994 \nu^{7} - 422 \nu^{6} - 6356 \nu^{5} + 4905 \nu^{4} + 12182 \nu^{3} + \cdots + 2130 ) / 537 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 68 \nu^{9} - 254 \nu^{8} - 545 \nu^{7} + 2428 \nu^{6} + 1047 \nu^{5} - 7117 \nu^{4} + 717 \nu^{3} + \cdots - 324 ) / 537 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 68 \nu^{9} + 254 \nu^{8} + 545 \nu^{7} - 2428 \nu^{6} - 1047 \nu^{5} + 7117 \nu^{4} - 180 \nu^{3} + \cdots + 1398 ) / 537 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 111 \nu^{9} + 283 \nu^{8} + 1424 \nu^{7} - 3121 \nu^{6} - 6292 \nu^{5} + 10962 \nu^{4} + \cdots + 1287 ) / 537 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + 2\beta_{8} + 4\beta_{7} - \beta_{6} + 2\beta_{4} + 6\beta_{2} + 11\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{9} + 8\beta_{8} + 19\beta_{7} - 3\beta_{6} + \beta_{5} + 4\beta_{4} - 2\beta_{3} + 9\beta_{2} + 45\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37 \beta_{9} + 21 \beta_{8} + 64 \beta_{7} - 18 \beta_{6} + 2 \beta_{5} + 26 \beta_{4} - 3 \beta_{3} + \cdots + 74 \)
|
| \(\nu^{7}\) | \(=\) |
\( 130 \beta_{9} + 63 \beta_{8} + 237 \beta_{7} - 54 \beta_{6} + 17 \beta_{5} + 64 \beta_{4} - 24 \beta_{3} + \cdots + 94 \)
|
| \(\nu^{8}\) | \(=\) |
\( 472 \beta_{9} + 184 \beta_{8} + 766 \beta_{7} - 225 \beta_{6} + 47 \beta_{5} + 277 \beta_{4} + \cdots + 433 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1570 \beta_{9} + 518 \beta_{8} + 2599 \beta_{7} - 689 \beta_{6} + 225 \beta_{5} + 767 \beta_{4} + \cdots + 749 \)
|
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 |
|
0 | −3.10558 | 0 | −0.251628 | 0 | −2.65136 | 0 | 6.64464 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.43474 | 0 | −0.328209 | 0 | 3.84402 | 0 | 2.92794 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.86748 | 0 | 3.88905 | 0 | −4.16941 | 0 | 0.487477 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.24390 | 0 | −3.60512 | 0 | −0.503042 | 0 | −1.45272 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.706586 | 0 | 1.36485 | 0 | 0.728719 | 0 | −2.50074 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.0603057 | 0 | −1.27674 | 0 | −0.888050 | 0 | −2.99636 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.536560 | 0 | 1.04904 | 0 | 2.64710 | 0 | −2.71210 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.78871 | 0 | 0.692349 | 0 | −1.82904 | 0 | 0.199493 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.93537 | 0 | 1.82730 | 0 | −3.27015 | 0 | 0.745647 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.15795 | 0 | −3.36089 | 0 | 1.09121 | 0 | 1.65673 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(23\) | \(-1\) |
| \(29\) | \(1\) |
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 | 2668.2.a.b | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2668.2.a.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 3T_{3}^{9} - 12T_{3}^{8} - 35T_{3}^{7} + 48T_{3}^{6} + 136T_{3}^{5} - 62T_{3}^{4} - 190T_{3}^{3} - 7T_{3}^{2} + 50T_{3} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2668))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots + 3 \)
|
| $5$ |
\( T^{10} - 24 T^{8} + \cdots - 9 \)
|
| $7$ |
\( T^{10} + 5 T^{9} + \cdots + 239 \)
|
| $11$ |
\( T^{10} + 8 T^{9} + \cdots - 1 \)
|
| $13$ |
\( T^{10} + 5 T^{9} + \cdots + 1995 \)
|
| $17$ |
\( T^{10} + 8 T^{9} + \cdots + 97 \)
|
| $19$ |
\( T^{10} - 2 T^{9} + \cdots - 84679 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( (T + 1)^{10} \)
|
| $31$ |
\( T^{10} + 8 T^{9} + \cdots - 769755 \)
|
| $37$ |
\( T^{10} - 11 T^{9} + \cdots + 23809 \)
|
| $41$ |
\( T^{10} + T^{9} + \cdots + 20005 \)
|
| $43$ |
\( T^{10} + T^{9} + \cdots - 24505 \)
|
| $47$ |
\( T^{10} + 39 T^{9} + \cdots - 6583109 \)
|
| $53$ |
\( T^{10} + 23 T^{9} + \cdots + 183955 \)
|
| $59$ |
\( T^{10} + \cdots + 239134565 \)
|
| $61$ |
\( T^{10} + 4 T^{9} + \cdots + 69140165 \)
|
| $67$ |
\( T^{10} + 21 T^{9} + \cdots + 701011 \)
|
| $71$ |
\( T^{10} + 15 T^{9} + \cdots + 13279291 \)
|
| $73$ |
\( T^{10} + 14 T^{9} + \cdots - 327285 \)
|
| $79$ |
\( T^{10} + \cdots + 469917917 \)
|
| $83$ |
\( T^{10} + 66 T^{9} + \cdots - 641307 \)
|
| $89$ |
\( T^{10} + \cdots + 274650687 \)
|
| $97$ |
\( T^{10} + 20 T^{9} + \cdots - 12391365 \)
|
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