Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,2,Mod(961,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1110.961");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.h (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: | \(8.86339462436\) |
| 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: |
\( x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 10\nu^{4} + 18\nu^{2} - 9 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} + 44\nu^{5} + 180\nu^{3} + 155\nu ) / 56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 44\nu^{5} - 124\nu^{3} + 181\nu ) / 56 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 527\nu ) / 56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} - 27\nu^{4} - 85\nu^{2} + 11 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 639\nu ) / 56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{6} + 27\nu^{4} + 99\nu^{2} + 59 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{6} + 3\beta_{4} + \beta_{3} + \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{7} - 9\beta_{5} - 4\beta _1 + 68 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41\beta_{6} - 35\beta_{4} - 24\beta_{3} - 22\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 26\beta_{7} + 36\beta_{5} + 27\beta _1 - 241 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -293\beta_{6} + 205\beta_{4} + 232\beta_{3} + 240\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) | \(667\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 961.1 |
|
− | 1.00000i | 1.00000 | −1.00000 | 1.00000i | − | 1.00000i | −4.26968 | 1.00000i | 1.00000 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 961.2 | − | 1.00000i | 1.00000 | −1.00000 | 1.00000i | − | 1.00000i | −2.35622 | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.3 | − | 1.00000i | 1.00000 | −1.00000 | 1.00000i | − | 1.00000i | 0.374052 | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.4 | − | 1.00000i | 1.00000 | −1.00000 | 1.00000i | − | 1.00000i | 4.25184 | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.5 | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | 1.00000i | −4.26968 | − | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.6 | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | 1.00000i | −2.35622 | − | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.7 | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | 1.00000i | 0.374052 | − | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
| 961.8 | 1.00000i | 1.00000 | −1.00000 | − | 1.00000i | 1.00000i | 4.25184 | − | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1110.2.h.g | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3330.2.h.o | 8 | ||
| 37.b | even | 2 | 1 | inner | 1110.2.h.g | ✓ | 8 |
| 111.d | odd | 2 | 1 | 3330.2.h.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.2.h.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1110.2.h.g | ✓ | 8 | 37.b | even | 2 | 1 | inner |
| 3330.2.h.o | 8 | 3.b | odd | 2 | 1 | ||
| 3330.2.h.o | 8 | 111.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):
|
\( T_{7}^{4} + 2T_{7}^{3} - 19T_{7}^{2} - 36T_{7} + 16 \)
|
|
\( T_{13}^{8} + 53T_{13}^{6} + 852T_{13}^{4} + 5056T_{13}^{2} + 9216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T^{4} + 2 T^{3} - 19 T^{2} + \cdots + 16)^{2} \)
|
| $11$ |
\( (T^{4} + 2 T^{3} - 33 T^{2} + \cdots + 60)^{2} \)
|
| $13$ |
\( T^{8} + 53 T^{6} + \cdots + 9216 \)
|
| $17$ |
\( T^{8} + 50 T^{6} + \cdots + 144 \)
|
| $19$ |
\( T^{8} + 81 T^{6} + \cdots + 46656 \)
|
| $23$ |
\( T^{8} + 53 T^{6} + \cdots + 9216 \)
|
| $29$ |
\( T^{8} + 213 T^{6} + \cdots + 3594816 \)
|
| $31$ |
\( T^{8} + 117 T^{6} + \cdots + 129600 \)
|
| $37$ |
\( T^{8} - 18 T^{7} + \cdots + 1874161 \)
|
| $41$ |
\( (T^{4} + 5 T^{3} + \cdots + 360)^{2} \)
|
| $43$ |
\( T^{8} + 89 T^{6} + \cdots + 2304 \)
|
| $47$ |
\( (T^{4} - 14 T^{3} + \cdots - 9216)^{2} \)
|
| $53$ |
\( (T^{4} - 6 T^{3} - 7 T^{2} + \cdots + 12)^{2} \)
|
| $59$ |
\( T^{8} + 212 T^{6} + \cdots + 2359296 \)
|
| $61$ |
\( T^{8} + 177 T^{6} + \cdots + 484416 \)
|
| $67$ |
\( (T^{4} - 6 T^{3} + \cdots + 4096)^{2} \)
|
| $71$ |
\( (T^{4} - 12 T^{3} + \cdots + 192)^{2} \)
|
| $73$ |
\( (T^{4} - 5 T^{3} + \cdots + 1880)^{2} \)
|
| $79$ |
\( T^{8} + 492 T^{6} + \cdots + 589824 \)
|
| $83$ |
\( (T^{4} - 3 T^{3} + \cdots - 576)^{2} \)
|
| $89$ |
\( T^{8} + 185 T^{6} + \cdots + 36864 \)
|
| $97$ |
\( T^{8} + 501 T^{6} + \cdots + 191102976 \)
|
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