Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2768,2,Mod(1,2768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2768, 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("2768.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2768 = 2^{4} \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2768.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: | \(22.1025912795\) |
| 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: |
\( x^{10} - 2x^{9} - 21x^{8} + 43x^{7} + 135x^{6} - 283x^{5} - 250x^{4} + 588x^{3} - 132x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 692) |
| 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
\( x^{10} - 2x^{9} - 21x^{8} + 43x^{7} + 135x^{6} - 283x^{5} - 250x^{4} + 588x^{3} - 132x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 47 \nu^{9} + 85 \nu^{8} + 1010 \nu^{7} - 1836 \nu^{6} - 6826 \nu^{5} + 12125 \nu^{4} + 14766 \nu^{3} + \cdots + 500 ) / 79 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 56 \nu^{9} + 108 \nu^{8} + 1195 \nu^{7} - 2317 \nu^{6} - 7923 \nu^{5} + 15220 \nu^{4} + 16069 \nu^{3} + \cdots + 231 ) / 79 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 278 \nu^{9} - 491 \nu^{8} - 5969 \nu^{7} + 10653 \nu^{6} + 40170 \nu^{5} - 70760 \nu^{4} - 85129 \nu^{3} + \cdots - 1364 ) / 237 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 211 \nu^{9} - 390 \nu^{8} - 4504 \nu^{7} + 8424 \nu^{6} + 29967 \nu^{5} - 55716 \nu^{4} - 61718 \nu^{3} + \cdots - 1137 ) / 79 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 667 \nu^{9} + 1213 \nu^{8} + 14167 \nu^{7} - 26106 \nu^{6} - 93738 \nu^{5} + 171502 \nu^{4} + \cdots + 3097 ) / 237 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 812 \nu^{9} + 1487 \nu^{8} + 17288 \nu^{7} - 31977 \nu^{6} - 114765 \nu^{5} + 210104 \nu^{4} + \cdots + 4811 ) / 237 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1090 \nu^{9} - 1978 \nu^{8} - 23257 \nu^{7} + 42630 \nu^{6} + 154935 \nu^{5} - 280864 \nu^{4} + \cdots - 7360 ) / 237 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 596 \nu^{9} + 1093 \nu^{8} + 12690 \nu^{7} - 23514 \nu^{6} - 84250 \nu^{5} + 154547 \nu^{4} + \cdots + 3446 ) / 79 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 7\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{9} + 9\beta_{8} + 13\beta_{7} - \beta_{5} - 8\beta_{4} - \beta_{3} + \beta_{2} + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15 \beta_{9} - 3 \beta_{8} - 19 \beta_{7} - 14 \beta_{6} + \beta_{5} + \beta_{4} - 12 \beta_{3} + \cdots - 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 33 \beta_{9} + 83 \beta_{8} + 147 \beta_{7} + 3 \beta_{6} - 12 \beta_{5} - 70 \beta_{4} - 8 \beta_{3} + \cdots + 334 \)
|
| \(\nu^{7}\) | \(=\) |
\( 181 \beta_{9} - 60 \beta_{8} - 261 \beta_{7} - 162 \beta_{6} + 21 \beta_{5} + 21 \beta_{4} - 120 \beta_{3} + \cdots - 234 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 421 \beta_{9} + 796 \beta_{8} + 1590 \beta_{7} + 66 \beta_{6} - 121 \beta_{5} - 647 \beta_{4} + \cdots + 3052 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2037 \beta_{9} - 873 \beta_{8} - 3215 \beta_{7} - 1760 \beta_{6} + 298 \beta_{5} + 345 \beta_{4} + \cdots - 3167 \)
|
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 | −2.95912 | 0 | −0.806298 | 0 | −0.910981 | 0 | 5.75638 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.81477 | 0 | 1.26218 | 0 | −1.17420 | 0 | 4.92296 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.38624 | 0 | 4.21401 | 0 | 2.08839 | 0 | 2.69413 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.47069 | 0 | −0.385862 | 0 | −4.40277 | 0 | −0.837076 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.171204 | 0 | −4.14907 | 0 | −1.26435 | 0 | −2.97069 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.158391 | 0 | 1.59626 | 0 | 2.20221 | 0 | −2.97491 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.0726101 | 0 | 3.86015 | 0 | −4.89468 | 0 | −2.99473 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.25989 | 0 | −3.12977 | 0 | −4.62591 | 0 | 2.10710 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.33431 | 0 | −3.35337 | 0 | 4.25271 | 0 | 2.44900 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.29361 | 0 | 3.89176 | 0 | −2.27042 | 0 | 7.84784 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(173\) | \( +1 \) |
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 | 2768.2.a.m | 10 | |
| 4.b | odd | 2 | 1 | 692.2.a.c | ✓ | 10 | |
| 12.b | even | 2 | 1 | 6228.2.a.n | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 692.2.a.c | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 2768.2.a.m | 10 | 1.a | even | 1 | 1 | trivial | |
| 6228.2.a.n | 10 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 2T_{3}^{9} - 21T_{3}^{8} - 43T_{3}^{7} + 135T_{3}^{6} + 283T_{3}^{5} - 250T_{3}^{4} - 588T_{3}^{3} - 132T_{3}^{2} - T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2768))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $5$ |
\( T^{10} - 3 T^{9} + \cdots - 1728 \)
|
| $7$ |
\( T^{10} + 11 T^{9} + \cdots - 5987 \)
|
| $11$ |
\( T^{10} + T^{9} + \cdots + 1143 \)
|
| $13$ |
\( T^{10} - 5 T^{9} + \cdots + 3081 \)
|
| $17$ |
\( T^{10} - 6 T^{9} + \cdots + 217152 \)
|
| $19$ |
\( T^{10} - 3 T^{9} + \cdots + 48519 \)
|
| $23$ |
\( T^{10} + 15 T^{9} + \cdots + 4809408 \)
|
| $29$ |
\( T^{10} - 11 T^{9} + \cdots - 69471 \)
|
| $31$ |
\( T^{10} + 3 T^{9} + \cdots - 855616 \)
|
| $37$ |
\( T^{10} - 40 T^{9} + \cdots + 3786569 \)
|
| $41$ |
\( T^{10} + T^{9} + \cdots - 5069523 \)
|
| $43$ |
\( T^{10} - 6 T^{9} + \cdots + 2240832 \)
|
| $47$ |
\( T^{10} - 14 T^{9} + \cdots - 1728 \)
|
| $53$ |
\( T^{10} + \cdots - 332502336 \)
|
| $59$ |
\( T^{10} - 30 T^{9} + \cdots + 1296 \)
|
| $61$ |
\( T^{10} - 25 T^{9} + \cdots + 565056 \)
|
| $67$ |
\( T^{10} - 12 T^{9} + \cdots + 21598016 \)
|
| $71$ |
\( T^{10} + 6 T^{9} + \cdots - 14746581 \)
|
| $73$ |
\( T^{10} - 15 T^{9} + \cdots - 546691 \)
|
| $79$ |
\( T^{10} + 7 T^{9} + \cdots + 757717 \)
|
| $83$ |
\( T^{10} + \cdots - 163446336 \)
|
| $89$ |
\( T^{10} - 443 T^{8} + \cdots + 60064173 \)
|
| $97$ |
\( T^{10} - 15 T^{9} + \cdots + 379968 \)
|
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