Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,6,Mod(1,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.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: | \(52.1247414392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1878612.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 16x^{2} + 16x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{4} - 2x^{3} - 16x^{2} + 16x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 13\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 9\nu^{2} + \nu - 67 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 3\beta _1 + 36 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 9\beta_{2} + 29\beta _1 + 58 ) / 4 \)
|
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 |
|
−7.55727 | 14.4912 | 25.1123 | 0 | −109.514 | −26.5515 | 52.0524 | −33.0063 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.42789 | −13.6012 | −29.9611 | 0 | 19.4210 | 9.51473 | 88.4737 | −58.0072 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 8.75180 | 26.2145 | 44.5940 | 0 | 229.424 | −8.06989 | 110.220 | 444.199 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 9.23335 | −23.1044 | 53.2548 | 0 | −213.331 | 161.107 | 196.253 | 290.815 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 325.6.a.e | 4 | |
| 5.b | even | 2 | 1 | 65.6.a.c | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 325.6.b.e | 8 | ||
| 15.d | odd | 2 | 1 | 585.6.a.g | 4 | ||
| 20.d | odd | 2 | 1 | 1040.6.a.o | 4 | ||
| 65.d | even | 2 | 1 | 845.6.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.6.a.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 325.6.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 325.6.b.e | 8 | 5.c | odd | 4 | 2 | ||
| 585.6.a.g | 4 | 15.d | odd | 2 | 1 | ||
| 845.6.a.f | 4 | 65.d | even | 2 | 1 | ||
| 1040.6.a.o | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 9T_{2}^{3} - 70T_{2}^{2} + 532T_{2} + 872 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9 T^{3} + \cdots + 872 \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 119376 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 136 T^{3} + \cdots + 328448 \)
|
| $11$ |
\( T^{4} + \cdots - 10689924208 \)
|
| $13$ |
\( (T - 169)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 100880603792 \)
|
| $19$ |
\( T^{4} + \cdots + 161727486480 \)
|
| $23$ |
\( T^{4} + \cdots - 22398353692848 \)
|
| $29$ |
\( T^{4} + \cdots - 170255904416240 \)
|
| $31$ |
\( T^{4} + \cdots - 7222593222000 \)
|
| $37$ |
\( T^{4} + \cdots + 18\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots + 6842070199824 \)
|
| $43$ |
\( T^{4} + \cdots - 44\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 25\!\cdots\!08 \)
|
| $59$ |
\( T^{4} + \cdots + 34\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 76\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots - 48\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots + 54\!\cdots\!52 \)
|
| $73$ |
\( T^{4} + \cdots - 96\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 11\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 68\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots - 49\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 36\!\cdots\!56 \)
|
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