Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(116,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.116");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.e (of order \(3\), 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: | \(14.4554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 116.1 |
|
−2.70711 | − | 4.68885i | 2.32843 | − | 4.03295i | −10.6569 | + | 18.4582i | 2.50000 | + | 4.33013i | −25.2132 | 0 | 72.0833 | 2.65685 | + | 4.60181i | 13.5355 | − | 23.4442i | ||||||||||||||||||
| 116.2 | −1.29289 | − | 2.23936i | −3.32843 | + | 5.76500i | 0.656854 | − | 1.13770i | 2.50000 | + | 4.33013i | 17.2132 | 0 | −24.0833 | −8.65685 | − | 14.9941i | 6.46447 | − | 11.1968i | |||||||||||||||||||
| 226.1 | −2.70711 | + | 4.68885i | 2.32843 | + | 4.03295i | −10.6569 | − | 18.4582i | 2.50000 | − | 4.33013i | −25.2132 | 0 | 72.0833 | 2.65685 | − | 4.60181i | 13.5355 | + | 23.4442i | |||||||||||||||||||
| 226.2 | −1.29289 | + | 2.23936i | −3.32843 | − | 5.76500i | 0.656854 | + | 1.13770i | 2.50000 | − | 4.33013i | 17.2132 | 0 | −24.0833 | −8.65685 | + | 14.9941i | 6.46447 | + | 11.1968i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.e.h | 4 | |
| 7.b | odd | 2 | 1 | 245.4.e.i | 4 | ||
| 7.c | even | 3 | 1 | 35.4.a.b | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 245.4.e.h | 4 | |
| 7.d | odd | 6 | 1 | 245.4.a.k | 2 | ||
| 7.d | odd | 6 | 1 | 245.4.e.i | 4 | ||
| 21.g | even | 6 | 1 | 2205.4.a.u | 2 | ||
| 21.h | odd | 6 | 1 | 315.4.a.f | 2 | ||
| 28.g | odd | 6 | 1 | 560.4.a.r | 2 | ||
| 35.i | odd | 6 | 1 | 1225.4.a.m | 2 | ||
| 35.j | even | 6 | 1 | 175.4.a.c | 2 | ||
| 35.l | odd | 12 | 2 | 175.4.b.c | 4 | ||
| 56.k | odd | 6 | 1 | 2240.4.a.bo | 2 | ||
| 56.p | even | 6 | 1 | 2240.4.a.bn | 2 | ||
| 105.o | odd | 6 | 1 | 1575.4.a.z | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.b | ✓ | 2 | 7.c | even | 3 | 1 | |
| 175.4.a.c | 2 | 35.j | even | 6 | 1 | ||
| 175.4.b.c | 4 | 35.l | odd | 12 | 2 | ||
| 245.4.a.k | 2 | 7.d | odd | 6 | 1 | ||
| 245.4.e.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 245.4.e.h | 4 | 7.c | even | 3 | 1 | inner | |
| 245.4.e.i | 4 | 7.b | odd | 2 | 1 | ||
| 245.4.e.i | 4 | 7.d | odd | 6 | 1 | ||
| 315.4.a.f | 2 | 21.h | odd | 6 | 1 | ||
| 560.4.a.r | 2 | 28.g | odd | 6 | 1 | ||
| 1225.4.a.m | 2 | 35.i | odd | 6 | 1 | ||
| 1575.4.a.z | 2 | 105.o | odd | 6 | 1 | ||
| 2205.4.a.u | 2 | 21.g | even | 6 | 1 | ||
| 2240.4.a.bn | 2 | 56.p | even | 6 | 1 | ||
| 2240.4.a.bo | 2 | 56.k | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{4} + 8T_{2}^{3} + 50T_{2}^{2} + 112T_{2} + 196 \)
|
|
\( T_{3}^{4} + 2T_{3}^{3} + 35T_{3}^{2} - 62T_{3} + 961 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8 T^{3} + \cdots + 196 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 961 \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 14 T^{3} + \cdots + 3996001 \)
|
| $13$ |
\( (T^{2} - 50 T + 593)^{2} \)
|
| $17$ |
\( T^{4} - 50 T^{3} + \cdots + 10543009 \)
|
| $19$ |
\( T^{4} + 36 T^{3} + \cdots + 12588304 \)
|
| $23$ |
\( T^{4} + 244 T^{3} + \cdots + 31764496 \)
|
| $29$ |
\( (T^{2} + 26 T - 983)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 3745440000 \)
|
| $37$ |
\( T^{4} + \cdots + 5230760976 \)
|
| $41$ |
\( (T^{2} + 328 T - 3856)^{2} \)
|
| $43$ |
\( (T^{2} + 260 T + 7652)^{2} \)
|
| $47$ |
\( T^{4} - 350 T^{3} + \cdots + 17833729 \)
|
| $53$ |
\( T^{4} + \cdots + 1022976256 \)
|
| $59$ |
\( (T^{2} - 616 T + 379456)^{2} \)
|
| $61$ |
\( T^{4} + 336 T^{3} + \cdots + 23970816 \)
|
| $67$ |
\( T^{4} - 152 T^{3} + \cdots + 5837056 \)
|
| $71$ |
\( (T + 952)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 14988615184 \)
|
| $79$ |
\( T^{4} + \cdots + 17966989681 \)
|
| $83$ |
\( (T^{2} + 376 T - 684656)^{2} \)
|
| $89$ |
\( T^{4} - 216 T^{3} + \cdots + 60715264 \)
|
| $97$ |
\( (T^{2} - 2742 T + 1782841)^{2} \)
|
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