Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [88,2,Mod(9,88)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(88, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("88.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 88 = 2^{3} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 88.i (of order \(5\), degree \(4\), 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: | \(0.702683537787\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.682515625.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + \cdots + 440220 ) / 1168519 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + \cdots + 528473 ) / 1168519 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + \cdots - 701074 ) / 1168519 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + \cdots + 2809884 ) / 1168519 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} + \cdots - 2305776 ) / 1168519 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} + \cdots + 665808 ) / 1168519 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62 \)
|
| \(\nu^{6}\) | \(=\) |
\( 97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221 \)
|
| \(\nu^{7}\) | \(=\) |
\( -449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).
| \(n\) | \(23\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
0 | −1.55261 | + | 1.12804i | 0 | 1.05261 | + | 3.23960i | 0 | −0.703158 | − | 0.510874i | 0 | 0.211078 | − | 0.649631i | 0 | ||||||||||||||||||||||||||||||||||
| 9.2 | 0 | 0.743592 | − | 0.540251i | 0 | −1.24359 | − | 3.82738i | 0 | 3.01217 | + | 2.18847i | 0 | −0.665993 | + | 2.04972i | 0 | |||||||||||||||||||||||||||||||||||
| 25.1 | 0 | −0.632489 | − | 1.94660i | 0 | 0.132489 | + | 0.0962586i | 0 | 1.08188 | − | 3.32969i | 0 | −0.962157 | + | 0.699048i | 0 | |||||||||||||||||||||||||||||||||||
| 25.2 | 0 | 0.941506 | + | 2.89766i | 0 | −1.44151 | − | 1.04732i | 0 | 0.109101 | − | 0.335777i | 0 | −5.08293 | + | 3.69296i | 0 | |||||||||||||||||||||||||||||||||||
| 49.1 | 0 | −1.55261 | − | 1.12804i | 0 | 1.05261 | − | 3.23960i | 0 | −0.703158 | + | 0.510874i | 0 | 0.211078 | + | 0.649631i | 0 | |||||||||||||||||||||||||||||||||||
| 49.2 | 0 | 0.743592 | + | 0.540251i | 0 | −1.24359 | + | 3.82738i | 0 | 3.01217 | − | 2.18847i | 0 | −0.665993 | − | 2.04972i | 0 | |||||||||||||||||||||||||||||||||||
| 81.1 | 0 | −0.632489 | + | 1.94660i | 0 | 0.132489 | − | 0.0962586i | 0 | 1.08188 | + | 3.32969i | 0 | −0.962157 | − | 0.699048i | 0 | |||||||||||||||||||||||||||||||||||
| 81.2 | 0 | 0.941506 | − | 2.89766i | 0 | −1.44151 | + | 1.04732i | 0 | 0.109101 | + | 0.335777i | 0 | −5.08293 | − | 3.69296i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 88.2.i.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 792.2.r.g | 8 | ||
| 4.b | odd | 2 | 1 | 176.2.m.d | 8 | ||
| 8.b | even | 2 | 1 | 704.2.m.l | 8 | ||
| 8.d | odd | 2 | 1 | 704.2.m.i | 8 | ||
| 11.b | odd | 2 | 1 | 968.2.i.p | 8 | ||
| 11.c | even | 5 | 1 | inner | 88.2.i.b | ✓ | 8 |
| 11.c | even | 5 | 1 | 968.2.a.n | 4 | ||
| 11.c | even | 5 | 2 | 968.2.i.s | 8 | ||
| 11.d | odd | 10 | 1 | 968.2.a.m | 4 | ||
| 11.d | odd | 10 | 1 | 968.2.i.p | 8 | ||
| 11.d | odd | 10 | 2 | 968.2.i.t | 8 | ||
| 33.f | even | 10 | 1 | 8712.2.a.cd | 4 | ||
| 33.h | odd | 10 | 1 | 792.2.r.g | 8 | ||
| 33.h | odd | 10 | 1 | 8712.2.a.ce | 4 | ||
| 44.g | even | 10 | 1 | 1936.2.a.bc | 4 | ||
| 44.h | odd | 10 | 1 | 176.2.m.d | 8 | ||
| 44.h | odd | 10 | 1 | 1936.2.a.bb | 4 | ||
| 88.k | even | 10 | 1 | 7744.2.a.ds | 4 | ||
| 88.l | odd | 10 | 1 | 704.2.m.i | 8 | ||
| 88.l | odd | 10 | 1 | 7744.2.a.dr | 4 | ||
| 88.o | even | 10 | 1 | 704.2.m.l | 8 | ||
| 88.o | even | 10 | 1 | 7744.2.a.di | 4 | ||
| 88.p | odd | 10 | 1 | 7744.2.a.dh | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.2.i.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 88.2.i.b | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 176.2.m.d | 8 | 4.b | odd | 2 | 1 | ||
| 176.2.m.d | 8 | 44.h | odd | 10 | 1 | ||
| 704.2.m.i | 8 | 8.d | odd | 2 | 1 | ||
| 704.2.m.i | 8 | 88.l | odd | 10 | 1 | ||
| 704.2.m.l | 8 | 8.b | even | 2 | 1 | ||
| 704.2.m.l | 8 | 88.o | even | 10 | 1 | ||
| 792.2.r.g | 8 | 3.b | odd | 2 | 1 | ||
| 792.2.r.g | 8 | 33.h | odd | 10 | 1 | ||
| 968.2.a.m | 4 | 11.d | odd | 10 | 1 | ||
| 968.2.a.n | 4 | 11.c | even | 5 | 1 | ||
| 968.2.i.p | 8 | 11.b | odd | 2 | 1 | ||
| 968.2.i.p | 8 | 11.d | odd | 10 | 1 | ||
| 968.2.i.s | 8 | 11.c | even | 5 | 2 | ||
| 968.2.i.t | 8 | 11.d | odd | 10 | 2 | ||
| 1936.2.a.bb | 4 | 44.h | odd | 10 | 1 | ||
| 1936.2.a.bc | 4 | 44.g | even | 10 | 1 | ||
| 7744.2.a.dh | 4 | 88.p | odd | 10 | 1 | ||
| 7744.2.a.di | 4 | 88.o | even | 10 | 1 | ||
| 7744.2.a.dr | 4 | 88.l | odd | 10 | 1 | ||
| 7744.2.a.ds | 4 | 88.k | even | 10 | 1 | ||
| 8712.2.a.cd | 4 | 33.f | even | 10 | 1 | ||
| 8712.2.a.ce | 4 | 33.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} + 10T_{3}^{6} + 19T_{3}^{5} + 49T_{3}^{4} + 29T_{3}^{3} + 20T_{3}^{2} - 99T_{3} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(88, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 121 \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots + 16 \)
|
| $7$ |
\( T^{8} - 7 T^{7} + \cdots + 16 \)
|
| $11$ |
\( T^{8} - 7 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} - 7 T^{7} + \cdots + 1936 \)
|
| $17$ |
\( T^{8} - T^{7} + \cdots + 39601 \)
|
| $19$ |
\( (T^{4} + T^{3} + 16 T^{2} + \cdots + 121)^{2} \)
|
| $23$ |
\( (T^{4} + 2 T^{3} - 52 T^{2} + \cdots - 64)^{2} \)
|
| $29$ |
\( T^{8} - 17 T^{7} + \cdots + 1936 \)
|
| $31$ |
\( T^{8} + 13 T^{7} + \cdots + 55696 \)
|
| $37$ |
\( T^{8} - T^{7} + \cdots + 1936 \)
|
| $41$ |
\( T^{8} - 9 T^{7} + \cdots + 121 \)
|
| $43$ |
\( (T^{4} - 3 T^{3} - 15 T^{2} + \cdots - 16)^{2} \)
|
| $47$ |
\( T^{8} + T^{7} + \cdots + 30976 \)
|
| $53$ |
\( T^{8} + 33 T^{7} + \cdots + 30976 \)
|
| $59$ |
\( (T^{4} + 15 T^{3} + \cdots + 625)^{2} \)
|
| $61$ |
\( T^{8} + 9 T^{7} + \cdots + 3748096 \)
|
| $67$ |
\( (T^{4} + 5 T^{3} + \cdots - 176)^{2} \)
|
| $71$ |
\( T^{8} + 25 T^{7} + \cdots + 512656 \)
|
| $73$ |
\( T^{8} + 7 T^{7} + \cdots + 609961 \)
|
| $79$ |
\( T^{8} + T^{7} + \cdots + 712336 \)
|
| $83$ |
\( T^{8} + 167 T^{6} + \cdots + 121 \)
|
| $89$ |
\( (T^{4} - 11 T^{3} + \cdots - 124)^{2} \)
|
| $97$ |
\( T^{8} - 8 T^{7} + \cdots + 241081 \)
|
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