Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2695,2,Mod(1,2695)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2695, 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("2695.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2695 = 5 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2695.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: | \(21.5196833447\) |
| 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: |
\( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{9} - 3\nu^{8} + 27\nu^{7} + 35\nu^{6} - 116\nu^{5} - 130\nu^{4} + 177\nu^{3} + 169\nu^{2} - 80\nu - 29 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{9} - 4\nu^{8} - 56\nu^{7} + 39\nu^{6} + 198\nu^{5} - 112\nu^{4} - 247\nu^{3} + 49\nu^{2} + 85\nu + 61 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{9} + 4\nu^{8} + 56\nu^{7} - 39\nu^{6} - 175\nu^{5} + 112\nu^{4} + 63\nu^{3} - 72\nu^{2} + 214\nu - 15 ) / 23 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{9} + \nu^{8} + 83\nu^{7} - 4\nu^{6} - 314\nu^{5} + 5\nu^{4} + 401\nu^{3} - 18\nu^{2} - 73\nu + 2 ) / 23 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{9} + 17\nu^{8} + 54\nu^{7} - 183\nu^{6} - 255\nu^{5} + 591\nu^{4} + 446\nu^{3} - 582\nu^{2} - 206\nu + 57 ) / 23 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8\nu^{9} - 11\nu^{8} - 108\nu^{7} + 136\nu^{6} + 487\nu^{5} - 538\nu^{4} - 800\nu^{3} + 681\nu^{2} + 320\nu - 91 ) / 23 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 6 \nu^{9} + 14 \nu^{8} + 81 \nu^{7} - 171 \nu^{6} - 371 \nu^{5} + 668 \nu^{4} + 646 \nu^{3} + \cdots + 120 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{9} + 8\beta_{8} + \beta_{5} + \beta_{4} + 8\beta_{3} + \beta_{2} + 27\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{9} + 10\beta_{8} + \beta_{7} + 9\beta_{6} + 9\beta_{4} + 2\beta_{3} + 35\beta_{2} + 11\beta _1 + 73 \)
|
| \(\nu^{7}\) | \(=\) |
\( 52 \beta_{9} + 52 \beta_{8} + \beta_{7} + \beta_{6} + 9 \beta_{5} + 12 \beta_{4} + 54 \beta_{3} + \cdots + 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( 66 \beta_{9} + 77 \beta_{8} + 12 \beta_{7} + 62 \beta_{6} + \beta_{5} + 63 \beta_{4} + 24 \beta_{3} + \cdots + 401 \)
|
| \(\nu^{9}\) | \(=\) |
\( 320 \beta_{9} + 321 \beta_{8} + 13 \beta_{7} + 13 \beta_{6} + 62 \beta_{5} + 102 \beta_{4} + 341 \beta_{3} + \cdots + 163 \)
|
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 |
|
−2.29554 | −0.293572 | 3.26952 | 1.00000 | 0.673908 | 0 | −2.91425 | −2.91382 | −2.29554 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.15225 | −2.37643 | 2.63217 | 1.00000 | 5.11466 | 0 | −1.36058 | 2.64742 | −2.15225 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.47696 | 2.93275 | 0.181403 | 1.00000 | −4.33154 | 0 | 2.68599 | 5.60100 | −1.47696 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.568445 | 0.674153 | −1.67687 | 1.00000 | −0.383219 | 0 | 2.09010 | −2.54552 | −0.568445 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0495267 | −3.19919 | −1.99755 | 1.00000 | −0.158445 | 0 | −0.197985 | 7.23480 | 0.0495267 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.330732 | 2.43493 | −1.89062 | 1.00000 | 0.805311 | 0 | −1.28675 | 2.92889 | 0.330732 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.29753 | −1.54034 | −0.316404 | 1.00000 | −1.99865 | 0 | −3.00561 | −0.627346 | 1.29753 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.81378 | −1.20588 | 1.28980 | 1.00000 | −2.18721 | 0 | −1.28815 | −1.54584 | 1.81378 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.48590 | 0.309874 | 4.17970 | 1.00000 | 0.770315 | 0 | 5.41853 | −2.90398 | 2.48590 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.51572 | 2.26371 | 4.32884 | 1.00000 | 5.69486 | 0 | 5.85871 | 2.12439 | 2.51572 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 2695.2.a.x | yes | 10 |
| 7.b | odd | 2 | 1 | 2695.2.a.w | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2695.2.a.w | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 2695.2.a.x | yes | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 13T_{2}^{8} + 24T_{2}^{7} + 56T_{2}^{6} - 92T_{2}^{5} - 86T_{2}^{4} + 116T_{2}^{3} + 31T_{2}^{2} - 22T_{2} + 1 \)
|
|
\( T_{3}^{10} - 20T_{3}^{8} + 130T_{3}^{6} + 12T_{3}^{5} - 308T_{3}^{4} - 68T_{3}^{3} + 182T_{3}^{2} + 4T_{3} - 14 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} - 20 T^{8} + \cdots - 14 \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T + 1)^{10} \)
|
| $13$ |
\( T^{10} - 8 T^{9} + \cdots + 34 \)
|
| $17$ |
\( T^{10} - 28 T^{9} + \cdots - 9486 \)
|
| $19$ |
\( T^{10} - 8 T^{9} + \cdots + 14536 \)
|
| $23$ |
\( T^{10} + 8 T^{9} + \cdots + 58052 \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots - 17648 \)
|
| $31$ |
\( T^{10} + 4 T^{9} + \cdots + 106642 \)
|
| $37$ |
\( T^{10} - 28 T^{9} + \cdots - 65596 \)
|
| $41$ |
\( T^{10} - 44 T^{9} + \cdots - 1704254 \)
|
| $43$ |
\( T^{10} - 20 T^{9} + \cdots + 7268 \)
|
| $47$ |
\( T^{10} - 12 T^{9} + \cdots + 223634 \)
|
| $53$ |
\( T^{10} - 360 T^{8} + \cdots - 889916 \)
|
| $59$ |
\( T^{10} - 16 T^{9} + \cdots + 21298 \)
|
| $61$ |
\( T^{10} + \cdots + 127321954 \)
|
| $67$ |
\( T^{10} - 20 T^{9} + \cdots - 485852 \)
|
| $71$ |
\( T^{10} + 4 T^{9} + \cdots - 349808 \)
|
| $73$ |
\( T^{10} - 20 T^{9} + \cdots - 29120686 \)
|
| $79$ |
\( T^{10} + 20 T^{9} + \cdots - 7623452 \)
|
| $83$ |
\( T^{10} + \cdots + 3533457544 \)
|
| $89$ |
\( T^{10} + \cdots + 344547208 \)
|
| $97$ |
\( T^{10} + 4 T^{9} + \cdots - 706936 \)
|
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