Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2450,4,Mod(1,2450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2450.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.a (trivial) |
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: | yes |
| Analytic conductor: | \(144.554679514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 33x^{2} - 37x + 70 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 350) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 33x^{2} - 37x + 70 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 17\nu - 10 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 32\nu - 10 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 17\nu^{2} + 34\nu - 225 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta _1 + 49 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 17\beta_{2} + 34\beta _1 + 79 ) / 3 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −9.49884 | 4.00000 | 0 | 18.9977 | 0 | −8.00000 | 63.2280 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.07686 | 4.00000 | 0 | 6.15371 | 0 | −8.00000 | −17.5330 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.11328 | 4.00000 | 0 | −6.22657 | 0 | −8.00000 | −17.3075 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 8.46241 | 4.00000 | 0 | −16.9248 | 0 | −8.00000 | 44.6125 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2450.4.a.ck | 4 | |
| 5.b | even | 2 | 1 | 2450.4.a.cu | 4 | ||
| 7.b | odd | 2 | 1 | 2450.4.a.co | 4 | ||
| 7.d | odd | 6 | 2 | 350.4.e.m | yes | 8 | |
| 35.c | odd | 2 | 1 | 2450.4.a.cq | 4 | ||
| 35.i | odd | 6 | 2 | 350.4.e.l | ✓ | 8 | |
| 35.k | even | 12 | 4 | 350.4.j.j | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.4.e.l | ✓ | 8 | 35.i | odd | 6 | 2 | |
| 350.4.e.m | yes | 8 | 7.d | odd | 6 | 2 | |
| 350.4.j.j | 16 | 35.k | even | 12 | 4 | ||
| 2450.4.a.ck | 4 | 1.a | even | 1 | 1 | trivial | |
| 2450.4.a.co | 4 | 7.b | odd | 2 | 1 | ||
| 2450.4.a.cq | 4 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.cu | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{4} + T_{3}^{3} - 90T_{3}^{2} - 7T_{3} + 770 \)
|
|
\( T_{11}^{4} + 10T_{11}^{3} - 4437T_{11}^{2} + 2349T_{11} + 123660 \)
|
|
\( T_{19}^{4} + 246T_{19}^{3} + 16191T_{19}^{2} + 66731T_{19} - 14253120 \)
|
|
\( T_{23}^{4} - 2T_{23}^{3} - 12333T_{23}^{2} - 328167T_{23} + 12485151 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 770 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 10 T^{3} + \cdots + 123660 \)
|
| $13$ |
\( T^{4} - 94 T^{3} + \cdots - 210070 \)
|
| $17$ |
\( T^{4} - 99 T^{3} + \cdots - 17795295 \)
|
| $19$ |
\( T^{4} + 246 T^{3} + \cdots - 14253120 \)
|
| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 12485151 \)
|
| $29$ |
\( T^{4} + 98 T^{3} + \cdots - 38101482 \)
|
| $31$ |
\( T^{4} + 304 T^{3} + \cdots - 93255295 \)
|
| $37$ |
\( T^{4} - 82 T^{3} + \cdots + 33806896 \)
|
| $41$ |
\( T^{4} + 352 T^{3} + \cdots + 84396627 \)
|
| $43$ |
\( T^{4} + 131 T^{3} + \cdots + 79091830 \)
|
| $47$ |
\( T^{4} + \cdots - 22106007945 \)
|
| $53$ |
\( T^{4} + \cdots - 1883661534 \)
|
| $59$ |
\( T^{4} + 673 T^{3} + \cdots - 81648000 \)
|
| $61$ |
\( T^{4} + \cdots - 15350227200 \)
|
| $67$ |
\( T^{4} + \cdots - 13369619760 \)
|
| $71$ |
\( T^{4} + \cdots - 5901875865 \)
|
| $73$ |
\( T^{4} + \cdots + 372143318400 \)
|
| $79$ |
\( T^{4} + \cdots - 1399066375 \)
|
| $83$ |
\( T^{4} + \cdots + 82998924750 \)
|
| $89$ |
\( T^{4} + \cdots + 94994566695 \)
|
| $97$ |
\( T^{4} + \cdots - 7337974175 \)
|
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