Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,3,Mod(97,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 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("960.97");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 960.x (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: | \(26.1581053786\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\zeta_{24}^{4} - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} - \zeta_{24}^{3} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{4} + 1 ) / 2 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
| \(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | −3.53553 | − | 3.53553i | 0 | 3.92480 | + | 3.92480i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −1.22474 | − | 1.22474i | 0 | 3.53553 | + | 3.53553i | 0 | −3.92480 | − | 3.92480i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | 1.22474 | + | 1.22474i | 0 | −3.53553 | − | 3.53553i | 0 | 7.38891 | + | 7.38891i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | 1.22474 | + | 1.22474i | 0 | 3.53553 | + | 3.53553i | 0 | −7.38891 | − | 7.38891i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 673.1 | 0 | −1.22474 | + | 1.22474i | 0 | −3.53553 | + | 3.53553i | 0 | 3.92480 | − | 3.92480i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 673.2 | 0 | −1.22474 | + | 1.22474i | 0 | 3.53553 | − | 3.53553i | 0 | −3.92480 | + | 3.92480i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 673.3 | 0 | 1.22474 | − | 1.22474i | 0 | −3.53553 | + | 3.53553i | 0 | 7.38891 | − | 7.38891i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 673.4 | 0 | 1.22474 | − | 1.22474i | 0 | 3.53553 | − | 3.53553i | 0 | −7.38891 | + | 7.38891i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
| 40.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 960.3.x.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 960.3.x.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 960.3.x.b | yes | 8 | |
| 8.b | even | 2 | 1 | 960.3.x.b | yes | 8 | |
| 8.d | odd | 2 | 1 | inner | 960.3.x.a | ✓ | 8 |
| 20.e | even | 4 | 1 | inner | 960.3.x.a | ✓ | 8 |
| 40.i | odd | 4 | 1 | inner | 960.3.x.a | ✓ | 8 |
| 40.k | even | 4 | 1 | 960.3.x.b | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 960.3.x.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 960.3.x.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 960.3.x.a | ✓ | 8 | 20.e | even | 4 | 1 | inner |
| 960.3.x.a | ✓ | 8 | 40.i | odd | 4 | 1 | inner |
| 960.3.x.b | yes | 8 | 4.b | odd | 2 | 1 | |
| 960.3.x.b | yes | 8 | 5.c | odd | 4 | 1 | |
| 960.3.x.b | yes | 8 | 8.b | even | 2 | 1 | |
| 960.3.x.b | yes | 8 | 40.k | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):
|
\( T_{7}^{8} + 12872T_{7}^{4} + 11316496 \)
|
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\( T_{19}^{2} + 40T_{19} + 376 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
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| $3$ |
\( (T^{4} + 9)^{2} \)
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| $5$ |
\( (T^{4} + 625)^{2} \)
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| $7$ |
\( T^{8} + 12872 T^{4} + 11316496 \)
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| $11$ |
\( (T^{2} + 150)^{4} \)
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| $13$ |
\( T^{8} + 321056 T^{4} + 100000000 \)
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| $17$ |
\( (T^{4} + 32 T^{3} + \cdots + 6400)^{2} \)
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| $19$ |
\( (T^{2} + 40 T + 376)^{4} \)
|
| $23$ |
\( T^{8} + 37376 T^{4} + 40960000 \)
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| $29$ |
\( (T^{4} - 708 T^{2} + 900)^{2} \)
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| $31$ |
\( (T^{4} - 1120 T^{2} + 92416)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 85030560000 \)
|
| $41$ |
\( (T^{2} + 20 T + 4)^{4} \)
|
| $43$ |
\( (T^{4} + 16 T^{3} + \cdots + 5382400)^{2} \)
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| $47$ |
\( T^{8} + \cdots + 1146228736 \)
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| $53$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
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| $59$ |
\( (T^{2} + 176 T + 7738)^{4} \)
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| $61$ |
\( (T^{4} + 2800 T^{2} + 1700416)^{2} \)
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| $67$ |
\( (T^{4} + 144 T^{3} + \cdots + 3326976)^{2} \)
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| $71$ |
\( (T^{2} - 2048)^{4} \)
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| $73$ |
\( (T^{4} + 300 T^{3} + \cdots + 122279364)^{2} \)
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| $79$ |
\( (T^{4} + 11104 T^{2} + 719104)^{2} \)
|
| $83$ |
\( (T^{4} + 256 T^{3} + \cdots + 14899600)^{2} \)
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| $89$ |
\( (T^{4} + 49800 T^{2} + 588159504)^{2} \)
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| $97$ |
\( (T^{4} + 36 T^{3} + \cdots + 45562500)^{2} \)
|
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