Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,16,Mod(19,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.19");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(128.424154590\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 1960198978 x^{14} + \cdots + 25\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{94}\cdot 3^{32}\cdot 5^{18} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 1960198978 x^{14} + \cdots + 25\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10\!\cdots\!99 \nu^{15} + \cdots + 11\!\cdots\!00 \nu ) / 70\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13\!\cdots\!09 \nu^{15} + \cdots + 31\!\cdots\!00 \nu ) / 32\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 38\!\cdots\!21 \nu^{15} + \cdots - 59\!\cdots\!00 ) / 92\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!54 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!89 \nu^{15} + \cdots - 34\!\cdots\!00 \nu ) / 34\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 53\!\cdots\!87 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 76\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 34\!\cdots\!10 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25\!\cdots\!95 \nu^{15} + \cdots - 29\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!55 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 69\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 95\!\cdots\!83 \nu^{15} + \cdots + 63\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 52\!\cdots\!83 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 92\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 23\!\cdots\!97 \nu^{15} + \cdots + 92\!\cdots\!00 ) / 38\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 51\!\cdots\!29 \nu^{15} + \cdots - 68\!\cdots\!00 ) / 57\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 80\!\cdots\!54 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 69\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 39\!\cdots\!37 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{13} + 40\beta_{12} + 680\beta_{5} + 1258\beta_{4} - 8520\beta_{2} - 81000\beta_1 ) / 10368000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 348939 \beta_{13} - 20070 \beta_{12} + 1413 \beta_{11} + 228060 \beta_{10} + \cdots - 12\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 271620 \beta_{15} + 41040 \beta_{14} + 2766996878 \beta_{13} - 5170730915 \beta_{12} + \cdots + 29767803508134 \beta_1 ) / 5184000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 267089079320649 \beta_{13} + 6085761013170 \beta_{12} - 2028868411824 \beta_{11} + \cdots + 65\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 265181323032900 \beta_{15} - 191288139166800 \beta_{14} + \cdots - 25\!\cdots\!30 \beta_1 ) / 5184000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 18\!\cdots\!29 \beta_{13} + \cdots - 39\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 22\!\cdots\!20 \beta_{15} + \cdots + 21\!\cdots\!54 \beta_1 ) / 5184000 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 12\!\cdots\!29 \beta_{13} + \cdots + 25\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 19\!\cdots\!00 \beta_{15} + \cdots - 18\!\cdots\!70 \beta_1 ) / 5184000 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 81\!\cdots\!69 \beta_{13} + \cdots - 16\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 15\!\cdots\!20 \beta_{15} + \cdots + 14\!\cdots\!74 \beta_1 ) / 5184000 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 54\!\cdots\!09 \beta_{13} + \cdots + 11\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 11\!\cdots\!00 \beta_{15} + \cdots - 11\!\cdots\!10 \beta_1 ) / 5184000 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 36\!\cdots\!09 \beta_{13} + \cdots - 73\!\cdots\!00 ) / 5184000 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 91\!\cdots\!20 \beta_{15} + \cdots + 86\!\cdots\!94 \beta_1 ) / 5184000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 128.000i | 0 | −16384.0 | −173529. | + | 20129.2i | 0 | 2.89271e6i | 2.09715e6i | 0 | 2.57653e6 | + | 2.22117e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | − | 128.000i | 0 | −16384.0 | −142270. | − | 101375.i | 0 | − | 486402.i | 2.09715e6i | 0 | −1.29760e7 | + | 1.82105e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | − | 128.000i | 0 | −16384.0 | −101852. | + | 141929.i | 0 | − | 859408.i | 2.09715e6i | 0 | 1.81669e7 | + | 1.30370e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | − | 128.000i | 0 | −16384.0 | −74905.5 | − | 157819.i | 0 | − | 3.73633e6i | 2.09715e6i | 0 | −2.02008e7 | + | 9.58790e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | − | 128.000i | 0 | −16384.0 | 74905.5 | − | 157819.i | 0 | 3.73633e6i | 2.09715e6i | 0 | −2.02008e7 | − | 9.58790e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.6 | − | 128.000i | 0 | −16384.0 | 101852. | + | 141929.i | 0 | 859408.i | 2.09715e6i | 0 | 1.81669e7 | − | 1.30370e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.7 | − | 128.000i | 0 | −16384.0 | 142270. | − | 101375.i | 0 | 486402.i | 2.09715e6i | 0 | −1.29760e7 | − | 1.82105e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.8 | − | 128.000i | 0 | −16384.0 | 173529. | + | 20129.2i | 0 | − | 2.89271e6i | 2.09715e6i | 0 | 2.57653e6 | − | 2.22117e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.9 | 128.000i | 0 | −16384.0 | −173529. | − | 20129.2i | 0 | − | 2.89271e6i | − | 2.09715e6i | 0 | 2.57653e6 | − | 2.22117e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.10 | 128.000i | 0 | −16384.0 | −142270. | + | 101375.i | 0 | 486402.i | − | 2.09715e6i | 0 | −1.29760e7 | − | 1.82105e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.11 | 128.000i | 0 | −16384.0 | −101852. | − | 141929.i | 0 | 859408.i | − | 2.09715e6i | 0 | 1.81669e7 | − | 1.30370e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.12 | 128.000i | 0 | −16384.0 | −74905.5 | + | 157819.i | 0 | 3.73633e6i | − | 2.09715e6i | 0 | −2.02008e7 | − | 9.58790e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.13 | 128.000i | 0 | −16384.0 | 74905.5 | + | 157819.i | 0 | − | 3.73633e6i | − | 2.09715e6i | 0 | −2.02008e7 | + | 9.58790e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.14 | 128.000i | 0 | −16384.0 | 101852. | − | 141929.i | 0 | − | 859408.i | − | 2.09715e6i | 0 | 1.81669e7 | + | 1.30370e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.15 | 128.000i | 0 | −16384.0 | 142270. | + | 101375.i | 0 | − | 486402.i | − | 2.09715e6i | 0 | −1.29760e7 | + | 1.82105e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.16 | 128.000i | 0 | −16384.0 | 173529. | − | 20129.2i | 0 | 2.89271e6i | − | 2.09715e6i | 0 | 2.57653e6 | + | 2.22117e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 90.16.c.e | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 90.16.c.e | ✓ | 16 |
| 5.b | even | 2 | 1 | inner | 90.16.c.e | ✓ | 16 |
| 15.d | odd | 2 | 1 | inner | 90.16.c.e | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.16.c.e | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 90.16.c.e | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 90.16.c.e | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 90.16.c.e | ✓ | 16 | 15.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 23303094522784 T_{7}^{6} + \cdots + 20\!\cdots\!56 \)
acting on \(S_{16}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16384)^{8} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + \cdots + 75\!\cdots\!25 \)
|
| $7$ |
\( (T^{8} + \cdots + 20\!\cdots\!56)^{2} \)
|
| $11$ |
\( (T^{8} + \cdots + 32\!\cdots\!76)^{2} \)
|
| $13$ |
\( (T^{8} + \cdots + 25\!\cdots\!76)^{2} \)
|
| $17$ |
\( (T^{8} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots - 12\!\cdots\!04)^{4} \)
|
| $23$ |
\( (T^{8} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{8} + \cdots + 53\!\cdots\!36)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 12\!\cdots\!04)^{4} \)
|
| $37$ |
\( (T^{8} + \cdots + 84\!\cdots\!56)^{2} \)
|
| $41$ |
\( (T^{8} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \)
|
| $47$ |
\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{8} + \cdots + 14\!\cdots\!36)^{2} \)
|
| $59$ |
\( (T^{8} + \cdots + 99\!\cdots\!56)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 16\!\cdots\!96)^{4} \)
|
| $67$ |
\( (T^{8} + \cdots + 19\!\cdots\!16)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 44\!\cdots\!16)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots - 11\!\cdots\!84)^{4} \)
|
| $83$ |
\( (T^{8} + \cdots + 72\!\cdots\!76)^{2} \)
|
| $89$ |
\( (T^{8} + \cdots + 36\!\cdots\!56)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 93\!\cdots\!76)^{2} \)
|
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