Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,5,Mod(193,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.193");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.p (of order \(4\), degree \(2\), not 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: | \(33.0783881868\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{241})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 121x^{2} + 3600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 121x^{2} + 3600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 61\nu ) / 60 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 61 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 60\nu^{2} + 121\nu - 3660 ) / 60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 - 122 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -61\beta_{3} - 61\beta_{2} + 181\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(257\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −10.2621 | − | 10.2621i | 0 | 24.7863 | + | 3.26209i | 0 | −50.7863 | + | 50.7863i | 0 | 129.621i | 0 | ||||||||||||||||||||||||
| 193.2 | 0 | 5.26209 | + | 5.26209i | 0 | −21.7863 | − | 12.2621i | 0 | −4.21374 | + | 4.21374i | 0 | − | 25.6209i | 0 | ||||||||||||||||||||||||
| 257.1 | 0 | −10.2621 | + | 10.2621i | 0 | 24.7863 | − | 3.26209i | 0 | −50.7863 | − | 50.7863i | 0 | − | 129.621i | 0 | ||||||||||||||||||||||||
| 257.2 | 0 | 5.26209 | − | 5.26209i | 0 | −21.7863 | + | 12.2621i | 0 | −4.21374 | − | 4.21374i | 0 | 25.6209i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.5.p.l | 4 | |
| 4.b | odd | 2 | 1 | 320.5.p.m | 4 | ||
| 5.c | odd | 4 | 1 | inner | 320.5.p.l | 4 | |
| 8.b | even | 2 | 1 | 80.5.p.e | 4 | ||
| 8.d | odd | 2 | 1 | 20.5.f.a | ✓ | 4 | |
| 20.e | even | 4 | 1 | 320.5.p.m | 4 | ||
| 24.f | even | 2 | 1 | 180.5.l.a | 4 | ||
| 40.e | odd | 2 | 1 | 100.5.f.c | 4 | ||
| 40.f | even | 2 | 1 | 400.5.p.h | 4 | ||
| 40.i | odd | 4 | 1 | 80.5.p.e | 4 | ||
| 40.i | odd | 4 | 1 | 400.5.p.h | 4 | ||
| 40.k | even | 4 | 1 | 20.5.f.a | ✓ | 4 | |
| 40.k | even | 4 | 1 | 100.5.f.c | 4 | ||
| 120.m | even | 2 | 1 | 900.5.l.a | 4 | ||
| 120.q | odd | 4 | 1 | 180.5.l.a | 4 | ||
| 120.q | odd | 4 | 1 | 900.5.l.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.5.f.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 20.5.f.a | ✓ | 4 | 40.k | even | 4 | 1 | |
| 80.5.p.e | 4 | 8.b | even | 2 | 1 | ||
| 80.5.p.e | 4 | 40.i | odd | 4 | 1 | ||
| 100.5.f.c | 4 | 40.e | odd | 2 | 1 | ||
| 100.5.f.c | 4 | 40.k | even | 4 | 1 | ||
| 180.5.l.a | 4 | 24.f | even | 2 | 1 | ||
| 180.5.l.a | 4 | 120.q | odd | 4 | 1 | ||
| 320.5.p.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.5.p.l | 4 | 5.c | odd | 4 | 1 | inner | |
| 320.5.p.m | 4 | 4.b | odd | 2 | 1 | ||
| 320.5.p.m | 4 | 20.e | even | 4 | 1 | ||
| 400.5.p.h | 4 | 40.f | even | 2 | 1 | ||
| 400.5.p.h | 4 | 40.i | odd | 4 | 1 | ||
| 900.5.l.a | 4 | 120.m | even | 2 | 1 | ||
| 900.5.l.a | 4 | 120.q | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\):
|
\( T_{3}^{4} + 10T_{3}^{3} + 50T_{3}^{2} - 1080T_{3} + 11664 \)
|
|
\( T_{13}^{4} - 360T_{13}^{3} + 64800T_{13}^{2} - 4270320T_{13} + 140707044 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 10 T^{3} + \cdots + 11664 \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 390625 \)
|
| $7$ |
\( T^{4} + 110 T^{3} + \cdots + 183184 \)
|
| $11$ |
\( (T^{2} + 150 T - 400)^{2} \)
|
| $13$ |
\( T^{4} - 360 T^{3} + \cdots + 140707044 \)
|
| $17$ |
\( T^{4} + \cdots + 8387262724 \)
|
| $19$ |
\( T^{4} + \cdots + 45392859136 \)
|
| $23$ |
\( T^{4} + \cdots + 2226707344 \)
|
| $29$ |
\( T^{4} + \cdots + 6544162816 \)
|
| $31$ |
\( (T^{2} - 418 T - 1311944)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 1825368527844 \)
|
| $41$ |
\( (T^{2} - 1482 T + 543056)^{2} \)
|
| $43$ |
\( T^{4} - 3270 T^{3} + \cdots + 65610000 \)
|
| $47$ |
\( T^{4} + \cdots + 1883558615184 \)
|
| $53$ |
\( T^{4} + \cdots + 156481954084 \)
|
| $59$ |
\( T^{4} + \cdots + 341649975218176 \)
|
| $61$ |
\( (T^{2} + 7914 T + 9096624)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 5826179407504 \)
|
| $71$ |
\( (T^{2} - 2994 T - 946216)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 266084019174724 \)
|
| $79$ |
\( T^{4} + \cdots + 189577209839616 \)
|
| $83$ |
\( T^{4} + \cdots + 498765926292624 \)
|
| $89$ |
\( T^{4} + \cdots + 356802481094656 \)
|
| $97$ |
\( T^{4} + \cdots + 16\!\cdots\!64 \)
|
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