Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1682,2,Mod(1,1682)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1682.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.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: | \(13.4308376200\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.32836640625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18\nu^{7} - 151\nu^{6} - 168\nu^{5} + 2008\nu^{4} + 386\nu^{3} - 6004\nu^{2} - 1858\nu + 1364 ) / 2073 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -28\nu^{7} - 149\nu^{6} + 31\nu^{5} + 1867\nu^{4} + 2010\nu^{3} - 5325\nu^{2} - 3252\nu + 28 ) / 2073 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 44\nu^{7} - 62\nu^{6} - 641\nu^{5} + 916\nu^{4} + 2172\nu^{3} - 3774\nu^{2} + 372\nu + 4102 ) / 2073 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 45\nu^{7} - 32\nu^{6} - 420\nu^{5} + 1565\nu^{4} + 274\nu^{3} - 11555\nu^{2} + 883\nu + 8938 ) / 2073 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 54\nu^{7} + 238\nu^{6} - 504\nu^{5} - 2959\nu^{4} - 224\nu^{3} + 9628\nu^{2} + 5482\nu - 7655 ) / 2073 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 140\nu^{7} + 54\nu^{6} - 2228\nu^{5} - 352\nu^{4} + 9989\nu^{3} - 1015\nu^{2} - 9307\nu + 1242 ) / 2073 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - 3\beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{6} + 2\beta_{5} - 10\beta_{4} + 10\beta_{3} + 8\beta_{2} + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{7} - 10\beta_{6} - 27\beta_{4} - 14\beta_{3} + 12\beta_{2} + 56\beta _1 + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 78\beta_{6} + 26\beta_{5} - 96\beta_{4} + 88\beta_{3} + 57\beta_{2} - 2\beta _1 + 326 \)
|
| \(\nu^{7}\) | \(=\) |
\( 70\beta_{7} - 88\beta_{6} - 5\beta_{5} - 211\beta_{4} - 153\beta_{3} + 125\beta_{2} + 459\beta _1 - 8 \)
|
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 |
|
−1.00000 | −2.88972 | 1.00000 | −2.81297 | 2.88972 | 3.85101 | −1.00000 | 5.35050 | 2.81297 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.79588 | 1.00000 | 0.869361 | 2.79588 | 2.55676 | −1.00000 | 4.81692 | −0.869361 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.70955 | 1.00000 | 4.23180 | 1.70955 | 1.69226 | −1.00000 | −0.0774536 | −4.23180 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.710020 | 1.00000 | −3.33469 | 0.710020 | 4.40617 | −1.00000 | −2.49587 | 3.33469 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.371284 | 1.00000 | −3.64078 | −0.371284 | −3.75542 | −1.00000 | −2.86215 | 3.64078 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.06263 | 1.00000 | 4.09274 | −1.06263 | 2.34135 | −1.00000 | −1.87081 | −4.09274 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.66631 | 1.00000 | 1.88957 | −2.66631 | −4.43319 | −1.00000 | 4.10924 | −1.88957 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.00493 | 1.00000 | 3.70497 | −3.00493 | 0.341047 | −1.00000 | 6.02962 | −3.70497 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(29\) | \(-1\) |
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 | 1682.2.a.u | ✓ | 8 |
| 29.b | even | 2 | 1 | 1682.2.a.v | yes | 8 | |
| 29.c | odd | 4 | 2 | 1682.2.b.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1682.2.a.u | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1682.2.a.v | yes | 8 | 29.b | even | 2 | 1 | |
| 1682.2.b.k | 16 | 29.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1682))\):
|
\( T_{3}^{8} + T_{3}^{7} - 18T_{3}^{6} - 17T_{3}^{5} + 95T_{3}^{4} + 77T_{3}^{3} - 128T_{3}^{2} - 51T_{3} + 31 \)
|
|
\( T_{5}^{8} - 5T_{5}^{7} - 30T_{5}^{6} + 160T_{5}^{5} + 265T_{5}^{4} - 1650T_{5}^{3} - 300T_{5}^{2} + 5400T_{5} - 3600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 31 \)
|
| $5$ |
\( T^{8} - 5 T^{7} + \cdots - 3600 \)
|
| $7$ |
\( T^{8} - 7 T^{7} + \cdots + 976 \)
|
| $11$ |
\( T^{8} - 7 T^{7} + \cdots - 1629 \)
|
| $13$ |
\( T^{8} - 13 T^{7} + \cdots - 464 \)
|
| $17$ |
\( T^{8} + 9 T^{7} + \cdots + 2511 \)
|
| $19$ |
\( T^{8} - 13 T^{7} + \cdots + 1611 \)
|
| $23$ |
\( T^{8} - 12 T^{7} + \cdots - 144 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} + 15 T^{7} + \cdots - 26480 \)
|
| $37$ |
\( T^{8} + 4 T^{7} + \cdots + 50896 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots - 279 \)
|
| $43$ |
\( T^{8} + 12 T^{7} + \cdots - 3689264 \)
|
| $47$ |
\( T^{8} - 13 T^{7} + \cdots - 2471184 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots - 924624 \)
|
| $59$ |
\( T^{8} - 8 T^{7} + \cdots + 5024961 \)
|
| $61$ |
\( T^{8} + 9 T^{7} + \cdots + 416656 \)
|
| $67$ |
\( T^{8} - 34 T^{7} + \cdots + 967471 \)
|
| $71$ |
\( T^{8} + 11 T^{7} + \cdots + 6717456 \)
|
| $73$ |
\( T^{8} - 13 T^{7} + \cdots + 1321 \)
|
| $79$ |
\( T^{8} - 8 T^{7} + \cdots - 3550544 \)
|
| $83$ |
\( T^{8} - 10 T^{7} + \cdots + 110386845 \)
|
| $89$ |
\( T^{8} + 6 T^{7} + \cdots - 2505339 \)
|
| $97$ |
\( T^{8} - 11 T^{7} + \cdots + 49865776 \)
|
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