Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,3,Mod(604,1089)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1089.604");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.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: | \(29.6731007888\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.41108373504.15 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 363) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 13\nu^{7} + 174\nu^{6} - 806\nu^{5} - 354\nu^{4} + 2179\nu^{3} + 5568\nu^{2} - 1746\nu - 825 ) / 6699 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 26\nu^{7} - 58\nu^{6} - 394\nu^{5} + 713\nu^{4} + 1922\nu^{3} - 1450\nu^{2} - 2274\nu + 786 ) / 1827 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 13\nu^{6} - 8\nu^{5} - 86\nu^{4} + 76\nu^{3} + 229\nu^{2} - 102\nu - 66 ) / 99 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -228\nu^{7} + 899\nu^{6} + 738\nu^{5} - 3754\nu^{4} - 4206\nu^{3} + 4205\nu^{2} - 6480\nu - 990 ) / 6699 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -970\nu^{7} + 3335\nu^{6} + 6548\nu^{5} - 19105\nu^{4} - 33760\nu^{3} + 28565\nu^{2} + 45768\nu - 31713 ) / 20097 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 430\nu^{7} - 2146\nu^{6} + 136\nu^{5} + 8216\nu^{4} + 4054\nu^{3} - 6148\nu^{2} - 10344\nu - 23166 ) / 6699 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 44\nu^{7} - 145\nu^{6} - 292\nu^{5} + 785\nu^{4} + 1004\nu^{3} + 29\nu^{2} + 930\nu + 159 ) / 609 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 2\beta_{5} + 2\beta_{2} + 2\beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} + 2\beta_{6} + 6\beta_{5} - 7\beta_{4} + 9\beta_{2} + 6\beta _1 + 13 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{7} + 3\beta_{6} + 8\beta_{5} - 7\beta_{4} - 3\beta_{3} + 11\beta_{2} + 14\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -40\beta_{7} + 15\beta_{6} + 37\beta_{5} - 74\beta_{4} - 15\beta_{3} + 49\beta_{2} + 100\beta _1 + 91 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -130\beta_{7} + 23\beta_{6} + 72\beta_{5} - 232\beta_{4} - 66\beta_{3} + 102\beta_{2} + 366\beta _1 + 150 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -455\beta_{7} + 22\beta_{6} + 38\beta_{5} - 825\beta_{4} - 210\beta_{3} + 41\beta_{2} + 1232\beta _1 + 127 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).
| \(n\) | \(244\) | \(848\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 604.1 |
|
− | 3.73960i | 0 | −9.98460 | 0.556540 | 0 | − | 12.0391i | 22.3800i | 0 | − | 2.08124i | |||||||||||||||||||||||||||||||||||||||
| 604.2 | − | 3.11226i | 0 | −5.68615 | −5.66935 | 0 | − | 9.60304i | 5.24773i | 0 | 17.6445i | |||||||||||||||||||||||||||||||||||||||||
| 604.3 | − | 2.07698i | 0 | −0.313852 | 1.66935 | 0 | 3.02335i | − | 7.65606i | 0 | − | 3.46720i | ||||||||||||||||||||||||||||||||||||||||
| 604.4 | − | 0.124105i | 0 | 3.98460 | −4.55654 | 0 | 5.06958i | − | 0.990926i | 0 | 0.565488i | |||||||||||||||||||||||||||||||||||||||||
| 604.5 | 0.124105i | 0 | 3.98460 | −4.55654 | 0 | − | 5.06958i | 0.990926i | 0 | − | 0.565488i | |||||||||||||||||||||||||||||||||||||||||
| 604.6 | 2.07698i | 0 | −0.313852 | 1.66935 | 0 | − | 3.02335i | 7.65606i | 0 | 3.46720i | ||||||||||||||||||||||||||||||||||||||||||
| 604.7 | 3.11226i | 0 | −5.68615 | −5.66935 | 0 | 9.60304i | − | 5.24773i | 0 | − | 17.6445i | |||||||||||||||||||||||||||||||||||||||||
| 604.8 | 3.73960i | 0 | −9.98460 | 0.556540 | 0 | 12.0391i | − | 22.3800i | 0 | 2.08124i | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1089.3.c.j | 8 | |
| 3.b | odd | 2 | 1 | 363.3.c.d | ✓ | 8 | |
| 11.b | odd | 2 | 1 | inner | 1089.3.c.j | 8 | |
| 33.d | even | 2 | 1 | 363.3.c.d | ✓ | 8 | |
| 33.f | even | 10 | 4 | 363.3.g.h | 32 | ||
| 33.h | odd | 10 | 4 | 363.3.g.h | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.3.c.d | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 363.3.c.d | ✓ | 8 | 33.d | even | 2 | 1 | |
| 363.3.g.h | 32 | 33.f | even | 10 | 4 | ||
| 363.3.g.h | 32 | 33.h | odd | 10 | 4 | ||
| 1089.3.c.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 1089.3.c.j | 8 | 11.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 28T_{2}^{6} + 238T_{2}^{4} + 588T_{2}^{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 28 T^{6} + \cdots + 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 8 T^{3} + 4 T^{2} + \cdots + 24)^{2} \)
|
| $7$ |
\( T^{8} + 272 T^{6} + \cdots + 3139984 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 688 T^{6} + \cdots + 175191696 \)
|
| $17$ |
\( T^{8} + \cdots + 15767322624 \)
|
| $19$ |
\( T^{8} + \cdots + 45425249424 \)
|
| $23$ |
\( (T^{4} - 2004 T^{2} + \cdots + 773208)^{2} \)
|
| $29$ |
\( T^{8} + 2896 T^{6} + \cdots + 158155776 \)
|
| $31$ |
\( (T^{4} + 64 T^{3} + \cdots - 325152)^{2} \)
|
| $37$ |
\( (T^{4} - 40 T^{3} + \cdots - 262944)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 57002517504 \)
|
| $43$ |
\( T^{8} + \cdots + 105357369744 \)
|
| $47$ |
\( (T^{4} - 16 T^{3} + \cdots + 224664)^{2} \)
|
| $53$ |
\( (T^{4} + 40 T^{3} + \cdots - 9192)^{2} \)
|
| $59$ |
\( (T^{4} - 32 T^{3} + \cdots + 157056)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 1600336321936 \)
|
| $67$ |
\( (T^{4} - 232 T^{3} + \cdots + 10956096)^{2} \)
|
| $71$ |
\( (T^{4} - 64 T^{3} + \cdots + 5084376)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 66074202988816 \)
|
| $79$ |
\( T^{8} + \cdots + 48\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 87005017211904 \)
|
| $89$ |
\( (T^{4} - 360 T^{3} + \cdots - 83517552)^{2} \)
|
| $97$ |
\( (T^{4} - 208 T^{3} + \cdots - 33715184)^{2} \)
|
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