Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [30,9,Mod(7,30)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.f (of order \(4\), degree \(2\), 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: | \(12.2213583018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 90x^{6} + 2693x^{4} + 30660x^{2} + 111556 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{12}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 90x^{6} + 2693x^{4} + 30660x^{2} + 111556 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 21\nu^{7} + 1556\nu^{5} + 33841\nu^{3} + 201310\nu ) / 85504 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 714 \nu^{7} - 11523 \nu^{6} - 52904 \nu^{5} - 719436 \nu^{4} - 765826 \nu^{3} - 11997447 \nu^{2} + \cdots - 52221234 ) / 427520 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 240 \nu^{7} - 46593 \nu^{6} - 5568 \nu^{5} - 3360708 \nu^{4} + 315600 \nu^{3} - 70008237 \nu^{2} + \cdots - 388579942 ) / 427520 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1728 \nu^{7} + 4509 \nu^{6} + 155520 \nu^{5} + 306612 \nu^{4} + 4076352 \nu^{3} + 4820121 \nu^{2} + \cdots + 6231438 ) / 427520 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1728 \nu^{7} - 4509 \nu^{6} + 155520 \nu^{5} - 306612 \nu^{4} + 4076352 \nu^{3} - 4820121 \nu^{2} + \cdots - 6231438 ) / 427520 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4731 \nu^{7} - 18036 \nu^{6} + 292524 \nu^{5} - 649296 \nu^{4} + 3229599 \nu^{3} + 2843676 \nu^{2} + \cdots + 81056456 ) / 427520 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1941 \nu^{7} - 8517 \nu^{6} + 120916 \nu^{5} - 995988 \nu^{4} + 1474289 \nu^{3} - 30427233 \nu^{2} + \cdots - 208707582 ) / 213760 \)
|
| \(\nu\) | \(=\) |
\( ( 9\beta_{7} + 3\beta_{6} + 11\beta_{5} + 17\beta_{4} - 15\beta_{3} + 45\beta_{2} - 657\beta _1 - 6 ) / 1350 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 3\beta_{6} + 89\beta_{5} - 87\beta_{4} - 15\beta_{3} - 5\beta_{2} - 3\beta _1 - 10134 ) / 450 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 387 \beta_{7} - 279 \beta_{6} - 673 \beta_{5} - 781 \beta_{4} + 495 \beta_{3} - 1035 \beta_{2} + \cdots + 108 ) / 1350 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 111 \beta_{7} + 153 \beta_{6} - 2459 \beta_{5} + 2417 \beta_{4} + 405 \beta_{3} + 195 \beta_{2} + \cdots + 183474 ) / 270 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14769 \beta_{7} + 11973 \beta_{6} + 37951 \beta_{5} + 40747 \beta_{4} - 17565 \beta_{3} + 31545 \beta_{2} + \cdots - 2796 ) / 1350 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 34533 \beta_{7} - 42399 \beta_{6} + 486637 \beta_{5} - 478771 \beta_{4} - 89595 \beta_{3} + \cdots - 31747122 ) / 1350 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 61883 \beta_{7} - 51811 \beta_{6} - 203657 \beta_{5} - 213729 \beta_{4} + 71955 \beta_{3} + \cdots + 10072 ) / 150 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−8.00000 | − | 8.00000i | −33.0681 | + | 33.0681i | 128.000i | −470.455 | + | 411.457i | 529.090 | 630.204 | + | 630.204i | 1024.00 | − | 1024.00i | − | 2187.00i | 7055.30 | + | 471.989i | |||||||||||||||||||||||||||||
| 7.2 | −8.00000 | − | 8.00000i | −33.0681 | + | 33.0681i | 128.000i | 192.221 | − | 594.707i | 529.090 | −332.493 | − | 332.493i | 1024.00 | − | 1024.00i | − | 2187.00i | −6295.42 | + | 3219.88i | ||||||||||||||||||||||||||||||
| 7.3 | −8.00000 | − | 8.00000i | 33.0681 | − | 33.0681i | 128.000i | 243.876 | + | 575.456i | −529.090 | −2647.20 | − | 2647.20i | 1024.00 | − | 1024.00i | − | 2187.00i | 2652.64 | − | 6554.65i | ||||||||||||||||||||||||||||||
| 7.4 | −8.00000 | − | 8.00000i | 33.0681 | − | 33.0681i | 128.000i | 352.358 | − | 516.206i | −529.090 | 1607.49 | + | 1607.49i | 1024.00 | − | 1024.00i | − | 2187.00i | −6948.51 | + | 1310.79i | ||||||||||||||||||||||||||||||
| 13.1 | −8.00000 | + | 8.00000i | −33.0681 | − | 33.0681i | − | 128.000i | −470.455 | − | 411.457i | 529.090 | 630.204 | − | 630.204i | 1024.00 | + | 1024.00i | 2187.00i | 7055.30 | − | 471.989i | ||||||||||||||||||||||||||||||
| 13.2 | −8.00000 | + | 8.00000i | −33.0681 | − | 33.0681i | − | 128.000i | 192.221 | + | 594.707i | 529.090 | −332.493 | + | 332.493i | 1024.00 | + | 1024.00i | 2187.00i | −6295.42 | − | 3219.88i | ||||||||||||||||||||||||||||||
| 13.3 | −8.00000 | + | 8.00000i | 33.0681 | + | 33.0681i | − | 128.000i | 243.876 | − | 575.456i | −529.090 | −2647.20 | + | 2647.20i | 1024.00 | + | 1024.00i | 2187.00i | 2652.64 | + | 6554.65i | ||||||||||||||||||||||||||||||
| 13.4 | −8.00000 | + | 8.00000i | 33.0681 | + | 33.0681i | − | 128.000i | 352.358 | + | 516.206i | −529.090 | 1607.49 | − | 1607.49i | 1024.00 | + | 1024.00i | 2187.00i | −6948.51 | − | 1310.79i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 30.9.f.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 90.9.g.e | 8 | ||
| 4.b | odd | 2 | 1 | 240.9.bg.a | 8 | ||
| 5.b | even | 2 | 1 | 150.9.f.h | 8 | ||
| 5.c | odd | 4 | 1 | inner | 30.9.f.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 150.9.f.h | 8 | ||
| 15.e | even | 4 | 1 | 90.9.g.e | 8 | ||
| 20.e | even | 4 | 1 | 240.9.bg.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.9.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 30.9.f.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 90.9.g.e | 8 | 3.b | odd | 2 | 1 | ||
| 90.9.g.e | 8 | 15.e | even | 4 | 1 | ||
| 150.9.f.h | 8 | 5.b | even | 2 | 1 | ||
| 150.9.f.h | 8 | 5.c | odd | 4 | 1 | ||
| 240.9.bg.a | 8 | 4.b | odd | 2 | 1 | ||
| 240.9.bg.a | 8 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 1484 T_{7}^{7} + 1101128 T_{7}^{6} - 18366461848 T_{7}^{5} + 84046808703844 T_{7}^{4} + \cdots + 12\!\cdots\!56 \)
acting on \(S_{9}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 128)^{4} \)
|
| $3$ |
\( (T^{4} + 4782969)^{2} \)
|
| $5$ |
\( T^{8} + \cdots + 23\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
|
| $11$ |
\( (T^{4} + \cdots - 502886699232800)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 80\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots + 56\!\cdots\!76 \)
|
| $19$ |
\( T^{8} + \cdots + 78\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 87\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 21\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $41$ |
\( (T^{4} + \cdots + 31\!\cdots\!44)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 77\!\cdots\!56 \)
|
| $47$ |
\( T^{8} + \cdots + 41\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 67\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 93\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 77\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} + \cdots + 76\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 20\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 25\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 35\!\cdots\!76 \)
|
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