Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,8,Mod(1,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(21.5545667584\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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
\( x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 105 \nu^{7} - 413 \nu^{6} - 99508 \nu^{5} - 294286 \nu^{4} + 29391029 \nu^{3} + 156833675 \nu^{2} + \cdots - 2892408736 ) / 5195392 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 105 \nu^{7} - 413 \nu^{6} - 99508 \nu^{5} - 294286 \nu^{4} + 24195637 \nu^{3} + 177615243 \nu^{2} + \cdots - 4544543392 ) / 5195392 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 331 \nu^{7} + 1404 \nu^{6} - 231736 \nu^{5} - 1304986 \nu^{4} + 44182269 \nu^{3} + \cdots - 6314973632 ) / 2597696 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 528 \nu^{7} - 6041 \nu^{6} + 393692 \nu^{5} + 4513576 \nu^{4} - 78508306 \nu^{3} + \cdots + 18986484320 ) / 2597696 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1641 \nu^{7} - 9781 \nu^{6} + 1156236 \nu^{5} + 8996798 \nu^{4} - 219270937 \nu^{3} + \cdots + 48102725728 ) / 5195392 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 189 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 4\beta_{2} + 305\beta _1 + 438 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{7} - 2\beta_{6} + 12\beta_{5} - 2\beta_{3} + 375\beta_{2} + 1406\beta _1 + 57999 \)
|
| \(\nu^{5}\) | \(=\) |
\( -82\beta_{7} + 46\beta_{6} - 108\beta_{5} - 547\beta_{4} + 409\beta_{3} + 2002\beta_{2} + 103317\beta _1 + 288060 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3320 \beta_{7} - 1672 \beta_{6} + 5904 \beta_{5} - 1346 \beta_{4} - 806 \beta_{3} + 134865 \beta_{2} + \cdots + 19697569 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 47836 \beta_{7} + 31412 \beta_{6} - 45496 \beta_{5} - 243769 \beta_{4} + 148397 \beta_{3} + \cdots + 135668930 \)
|
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 |
|
−16.5036 | 27.0000 | 144.368 | −425.514 | −445.597 | −1501.62 | −270.137 | 729.000 | 7022.50 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.1553 | 27.0000 | 132.994 | 118.315 | −436.193 | 738.380 | −80.6798 | 729.000 | −1911.41 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.38924 | 27.0000 | −122.292 | −147.281 | −64.5095 | −1223.93 | 598.006 | 729.000 | 351.889 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.96449 | 27.0000 | −124.141 | 495.444 | −53.0413 | 824.524 | 495.329 | 729.000 | −973.296 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 7.11469 | 27.0000 | −77.3812 | −244.007 | 192.097 | 549.357 | −1461.22 | 729.000 | −1736.03 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 14.6014 | 27.0000 | 85.2018 | 493.797 | 394.239 | −368.336 | −624.915 | 729.000 | 7210.15 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 17.7586 | 27.0000 | 187.368 | 32.5148 | 479.482 | 1672.82 | 1054.29 | 729.000 | 577.418 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 21.5379 | 27.0000 | 335.881 | 54.7305 | 581.523 | −565.198 | 4477.33 | 729.000 | 1178.78 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 69.8.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 207.8.a.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.8.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 207.8.a.e | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 24 T_{2}^{7} - 505 T_{2}^{6} + 13284 T_{2}^{5} + 56268 T_{2}^{4} - 1967776 T_{2}^{3} + \cdots + 49724160 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 24 T^{7} + \cdots + 49724160 \)
|
| $3$ |
\( (T - 27)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 78\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 21\!\cdots\!72 \)
|
| $11$ |
\( T^{8} + \cdots - 27\!\cdots\!60 \)
|
| $13$ |
\( T^{8} + \cdots - 32\!\cdots\!68 \)
|
| $17$ |
\( T^{8} + \cdots + 47\!\cdots\!40 \)
|
| $19$ |
\( T^{8} + \cdots - 91\!\cdots\!40 \)
|
| $23$ |
\( (T - 12167)^{8} \)
|
| $29$ |
\( T^{8} + \cdots - 21\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 28\!\cdots\!12 \)
|
| $37$ |
\( T^{8} + \cdots + 93\!\cdots\!08 \)
|
| $41$ |
\( T^{8} + \cdots + 42\!\cdots\!40 \)
|
| $43$ |
\( T^{8} + \cdots + 94\!\cdots\!60 \)
|
| $47$ |
\( T^{8} + \cdots - 32\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots - 11\!\cdots\!40 \)
|
| $59$ |
\( T^{8} + \cdots + 16\!\cdots\!60 \)
|
| $61$ |
\( T^{8} + \cdots + 13\!\cdots\!56 \)
|
| $67$ |
\( T^{8} + \cdots + 30\!\cdots\!40 \)
|
| $71$ |
\( T^{8} + \cdots - 28\!\cdots\!80 \)
|
| $73$ |
\( T^{8} + \cdots - 19\!\cdots\!64 \)
|
| $79$ |
\( T^{8} + \cdots - 56\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 68\!\cdots\!40 \)
|
| $89$ |
\( T^{8} + \cdots + 42\!\cdots\!20 \)
|
| $97$ |
\( T^{8} + \cdots + 70\!\cdots\!48 \)
|
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