Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4489,2,Mod(1,4489)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4489, 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("4489.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4489 = 67^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4489.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: | \(35.8448454674\) |
| 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: |
\( x^{10} - 2x^{9} - 21x^{8} + 32x^{7} + 161x^{6} - 147x^{5} - 535x^{4} + 120x^{3} + 670x^{2} + 207x - 67 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 67) |
| 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
\( x^{10} - 2x^{9} - 21x^{8} + 32x^{7} + 161x^{6} - 147x^{5} - 535x^{4} + 120x^{3} + 670x^{2} + 207x - 67 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2283 \nu^{9} + 10964 \nu^{8} + 43501 \nu^{7} - 188233 \nu^{6} - 313411 \nu^{5} + 1005362 \nu^{4} + \cdots + 202774 ) / 56717 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2641 \nu^{9} - 1479 \nu^{8} - 53403 \nu^{7} + 16396 \nu^{6} + 378457 \nu^{5} - 49592 \nu^{4} + \cdots + 59274 ) / 56717 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3873 \nu^{9} - 10029 \nu^{8} - 70369 \nu^{7} + 167437 \nu^{6} + 435320 \nu^{5} - 882742 \nu^{4} + \cdots - 338034 ) / 56717 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5268 \nu^{9} - 11884 \nu^{8} - 95012 \nu^{7} + 194137 \nu^{6} + 567426 \nu^{5} - 975275 \nu^{4} + \cdots - 266222 ) / 56717 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5579 \nu^{9} - 16160 \nu^{8} - 105832 \nu^{7} + 276322 \nu^{6} + 701245 \nu^{5} - 1473872 \nu^{4} + \cdots - 428942 ) / 56717 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 8128 \nu^{9} + 27681 \nu^{8} + 146551 \nu^{7} - 461718 \nu^{6} - 928323 \nu^{5} + 2394869 \nu^{4} + \cdots + 751013 ) / 56717 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 8875 \nu^{9} + 21356 \nu^{8} + 168163 \nu^{7} - 364411 \nu^{6} - 1089079 \nu^{5} + 1942382 \nu^{4} + \cdots + 768544 ) / 56717 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23620 \nu^{9} - 67797 \nu^{8} - 428027 \nu^{7} + 1131466 \nu^{6} + 2669603 \nu^{5} - 5869120 \nu^{4} + \cdots - 1762847 ) / 56717 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - 2\beta_{6} - \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - 9\beta_{8} - 7\beta_{7} - 10\beta_{6} - 8\beta_{5} - 12\beta_{4} + 7\beta_{2} + 11\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 25 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + \beta_{3} + \cdots + 23 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17 \beta_{9} - 79 \beta_{8} - 41 \beta_{7} - 96 \beta_{6} - 71 \beta_{5} - 108 \beta_{4} - 2 \beta_{3} + \cdots + 280 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55 \beta_{9} - 122 \beta_{8} + 40 \beta_{7} - 268 \beta_{6} - 104 \beta_{5} - 63 \beta_{4} + 12 \beta_{3} + \cdots + 241 \)
|
| \(\nu^{8}\) | \(=\) |
\( 217 \beta_{9} - 695 \beta_{8} - 195 \beta_{7} - 914 \beta_{6} - 663 \beta_{5} - 922 \beta_{4} + \cdots + 2254 \)
|
| \(\nu^{9}\) | \(=\) |
\( 717 \beta_{9} - 1200 \beta_{8} + 553 \beta_{7} - 2727 \beta_{6} - 1307 \beta_{5} - 755 \beta_{4} + \cdots + 2474 \)
|
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 |
|
−1.91899 | −2.54887 | 1.68251 | −1.41933 | 4.89125 | −2.21212 | 0.609264 | 3.49675 | 2.72367 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.91899 | 0.954521 | 1.68251 | 3.16914 | −1.83171 | 3.37352 | 0.609264 | −2.08889 | −6.08154 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.30972 | −0.197302 | −0.284630 | 1.64168 | 0.258410 | 3.13024 | 2.99223 | −2.96107 | −2.15015 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.30972 | 2.71064 | −0.284630 | −3.25095 | −3.55018 | −4.17839 | 2.99223 | 4.34756 | 4.25784 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.284630 | −3.06231 | −1.91899 | 2.82889 | 0.871624 | −4.06801 | 1.11546 | 6.37773 | −0.805185 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.284630 | 2.58342 | −1.91899 | −1.86175 | −0.735317 | −1.36432 | 1.11546 | 3.67404 | 0.529909 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.830830 | −1.65335 | −1.30972 | −4.00359 | −1.37365 | −0.416074 | −2.74982 | −0.266433 | −3.32630 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.830830 | 1.41687 | −1.30972 | 1.88813 | 1.17718 | 0.309968 | −2.74982 | −0.992475 | 1.56871 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.68251 | −3.09991 | 0.830830 | −2.56483 | −5.21562 | −3.69046 | −1.96714 | 6.60945 | −4.31535 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.68251 | 0.896295 | 0.830830 | −1.42739 | 1.50802 | 5.11564 | −1.96714 | −2.19666 | −2.40160 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(67\) | \(1\) |
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 | 4489.2.a.l | 10 | |
| 67.b | odd | 2 | 1 | 4489.2.a.m | 10 | ||
| 67.e | even | 11 | 2 | 67.2.e.c | ✓ | 20 | |
| 201.k | odd | 22 | 2 | 603.2.u.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 67.2.e.c | ✓ | 20 | 67.e | even | 11 | 2 | |
| 603.2.u.c | 20 | 201.k | odd | 22 | 2 | ||
| 4489.2.a.l | 10 | 1.a | even | 1 | 1 | trivial | |
| 4489.2.a.m | 10 | 67.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + T_{2}^{4} - 4T_{2}^{3} - 3T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4489))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{10} + 2 T^{9} + \cdots - 67 \)
|
| $5$ |
\( T^{10} + 5 T^{9} + \cdots + 3499 \)
|
| $7$ |
\( T^{10} + 4 T^{9} + \cdots + 1319 \)
|
| $11$ |
\( T^{10} + 11 T^{9} + \cdots - 2047 \)
|
| $13$ |
\( T^{10} + 5 T^{9} + \cdots + 67 \)
|
| $17$ |
\( T^{10} - 20 T^{9} + \cdots - 29611 \)
|
| $19$ |
\( T^{10} - 2 T^{9} + \cdots - 4829 \)
|
| $23$ |
\( T^{10} - T^{9} + \cdots - 33143 \)
|
| $29$ |
\( T^{10} + 3 T^{9} + \cdots + 737 \)
|
| $31$ |
\( T^{10} + 33 T^{9} + \cdots - 931283 \)
|
| $37$ |
\( T^{10} - 12 T^{9} + \cdots - 279553 \)
|
| $41$ |
\( T^{10} - 8 T^{9} + \cdots - 11497189 \)
|
| $43$ |
\( T^{10} - 11 T^{9} + \cdots - 25277891 \)
|
| $47$ |
\( T^{10} + 3 T^{9} + \cdots - 1603513 \)
|
| $53$ |
\( T^{10} + 17 T^{9} + \cdots + 226843 \)
|
| $59$ |
\( T^{10} - 9 T^{9} + \cdots + 16703917 \)
|
| $61$ |
\( T^{10} + 24 T^{9} + \cdots - 8738203 \)
|
| $67$ |
\( T^{10} \)
|
| $71$ |
\( T^{10} + 34 T^{9} + \cdots - 1261073 \)
|
| $73$ |
\( T^{10} + 11 T^{9} + \cdots - 49193849 \)
|
| $79$ |
\( T^{10} + 28 T^{9} + \cdots - 67524797 \)
|
| $83$ |
\( T^{10} + 38 T^{9} + \cdots - 160777 \)
|
| $89$ |
\( T^{10} + \cdots + 2031482707 \)
|
| $97$ |
\( T^{10} + \cdots - 294045257 \)
|
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