Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1100,3,Mod(901,1100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1100.901");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1100.f (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: | \(29.9728290796\) |
| 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: |
\( x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 220) |
| 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
\( x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} - 59\nu^{4} - 763\nu^{2} - 420 ) / 605 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{7} - 5\nu^{6} - 129\nu^{5} - 350\nu^{4} - 2692\nu^{3} - 6620\nu^{2} - 25315\nu - 24650 ) / 3025 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} + 608\nu^{5} + 12004\nu^{3} + 60155\nu ) / 3025 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 57\nu^{5} - 826\nu^{3} - 2020\nu ) / 275 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} - 140\nu^{4} - 2648\nu^{2} - 9860 ) / 605 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -23\nu^{7} - 1456\nu^{5} - 25018\nu^{3} - 96835\nu ) / 3025 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6\nu^{6} + 332\nu^{4} + 4666\nu^{2} + 14070 ) / 605 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{6} + 3\beta_{5} + 2\beta_{4} - 4\beta_{3} - 6\beta_{2} ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} + 8\beta _1 - 34 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 92\beta_{6} - 43\beta_{5} - 82\beta_{4} + 154\beta_{3} + 86\beta_{2} ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -51\beta_{7} - 4\beta_{5} - 298\beta _1 + 914 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3542\beta_{6} + 743\beta_{5} + 3582\beta_{4} - 5004\beta_{3} - 1486\beta_{2} ) / 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2246\beta_{7} + 999\beta_{5} + 10268\beta _1 - 28824 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 129942\beta_{6} - 12893\beta_{5} - 145982\beta_{4} + 166104\beta_{3} + 25786\beta_{2} ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(551\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 901.1 |
|
0 | −3.65160 | 0 | 0 | 0 | − | 9.09184i | 0 | 4.33416 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 901.2 | 0 | −3.65160 | 0 | 0 | 0 | 9.09184i | 0 | 4.33416 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 901.3 | 0 | −0.712855 | 0 | 0 | 0 | − | 7.86633i | 0 | −8.49184 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.4 | 0 | −0.712855 | 0 | 0 | 0 | 7.86633i | 0 | −8.49184 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 901.5 | 0 | 3.41553 | 0 | 0 | 0 | − | 0.558647i | 0 | 2.66584 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.6 | 0 | 3.41553 | 0 | 0 | 0 | 0.558647i | 0 | 2.66584 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 901.7 | 0 | 4.94892 | 0 | 0 | 0 | − | 13.1585i | 0 | 15.4918 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 901.8 | 0 | 4.94892 | 0 | 0 | 0 | 13.1585i | 0 | 15.4918 | 0 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1100.3.f.f | 8 | |
| 5.b | even | 2 | 1 | 220.3.f.a | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 1100.3.e.b | 16 | ||
| 11.b | odd | 2 | 1 | inner | 1100.3.f.f | 8 | |
| 15.d | odd | 2 | 1 | 1980.3.b.a | 8 | ||
| 20.d | odd | 2 | 1 | 880.3.j.b | 8 | ||
| 55.d | odd | 2 | 1 | 220.3.f.a | ✓ | 8 | |
| 55.e | even | 4 | 2 | 1100.3.e.b | 16 | ||
| 165.d | even | 2 | 1 | 1980.3.b.a | 8 | ||
| 220.g | even | 2 | 1 | 880.3.j.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.3.f.a | ✓ | 8 | 5.b | even | 2 | 1 | |
| 220.3.f.a | ✓ | 8 | 55.d | odd | 2 | 1 | |
| 880.3.j.b | 8 | 20.d | odd | 2 | 1 | ||
| 880.3.j.b | 8 | 220.g | even | 2 | 1 | ||
| 1100.3.e.b | 16 | 5.c | odd | 4 | 2 | ||
| 1100.3.e.b | 16 | 55.e | even | 4 | 2 | ||
| 1100.3.f.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 1100.3.f.f | 8 | 11.b | odd | 2 | 1 | inner | |
| 1980.3.b.a | 8 | 15.d | odd | 2 | 1 | ||
| 1980.3.b.a | 8 | 165.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} - 17T_{3}^{2} + 52T_{3} + 44 \)
acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 4 T^{3} - 17 T^{2} + \cdots + 44)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 318 T^{6} + \cdots + 276400 \)
|
| $11$ |
\( T^{8} - 24 T^{7} + \cdots + 214358881 \)
|
| $13$ |
\( T^{8} + 1052 T^{6} + \cdots + 110560000 \)
|
| $17$ |
\( T^{8} + \cdots + 12073428400 \)
|
| $19$ |
\( T^{8} + 650 T^{6} + \cdots + 110560000 \)
|
| $23$ |
\( (T^{4} + 28 T^{3} + \cdots - 704)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 100835142400 \)
|
| $31$ |
\( (T^{4} - 10 T^{3} + \cdots + 1062716)^{2} \)
|
| $37$ |
\( (T^{4} + 36 T^{3} + \cdots - 36620)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 7245660160000 \)
|
| $43$ |
\( T^{8} + \cdots + 840680550400 \)
|
| $47$ |
\( (T^{4} - 92 T^{3} + \cdots - 1600)^{2} \)
|
| $53$ |
\( (T^{4} + 68 T^{3} + \cdots + 181444)^{2} \)
|
| $59$ |
\( (T^{4} + 8 T^{3} + \cdots - 735680)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 75981364960000 \)
|
| $67$ |
\( (T^{4} - 132 T^{3} + \cdots + 10804784)^{2} \)
|
| $71$ |
\( (T^{4} + 110 T^{3} + \cdots - 24870124)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 22293318400 \)
|
| $79$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} - 110 T^{3} + \cdots + 44004556)^{2} \)
|
| $97$ |
\( (T^{4} - 4 T^{3} + \cdots + 7804400)^{2} \)
|
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