Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2205,4,Mod(1,2205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2205.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2205.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: | \(130.099211563\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 34x^{4} + 241x^{2} - 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 34x^{4} + 241x^{2} - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 20\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 25\nu^{2} + 59 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 32\nu^{3} + 195\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 20\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 25\beta_{2} + 216 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 32\beta_{3} + 445\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.91092 | 0 | 16.1171 | −5.00000 | 0 | 0 | −39.8626 | 0 | 24.5546 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.09945 | 0 | 1.60662 | −5.00000 | 0 | 0 | 19.8160 | 0 | 15.4973 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.525584 | 0 | −7.72376 | −5.00000 | 0 | 0 | 8.26415 | 0 | 2.62792 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.525584 | 0 | −7.72376 | −5.00000 | 0 | 0 | −8.26415 | 0 | −2.62792 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.09945 | 0 | 1.60662 | −5.00000 | 0 | 0 | −19.8160 | 0 | −15.4973 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.91092 | 0 | 16.1171 | −5.00000 | 0 | 0 | 39.8626 | 0 | −24.5546 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2205.4.a.bx | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2205.4.a.by | yes | 6 | |
| 7.b | odd | 2 | 1 | 2205.4.a.by | yes | 6 | |
| 21.c | even | 2 | 1 | inner | 2205.4.a.bx | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2205.4.a.bx | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2205.4.a.bx | ✓ | 6 | 21.c | even | 2 | 1 | inner |
| 2205.4.a.by | yes | 6 | 3.b | odd | 2 | 1 | |
| 2205.4.a.by | yes | 6 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):
|
\( T_{2}^{6} - 34T_{2}^{4} + 241T_{2}^{2} - 64 \)
|
|
\( T_{11}^{6} - 3512T_{11}^{4} + 2814992T_{11}^{2} - 335622400 \)
|
|
\( T_{13}^{6} - 6744T_{13}^{4} + 14726800T_{13}^{2} - 10404000000 \)
|
|
\( T_{17}^{3} + 22T_{17}^{2} - 5860T_{17} - 216600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 34 T^{4} + \cdots - 64 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 3512 T^{4} + \cdots - 335622400 \)
|
| $13$ |
\( T^{6} + \cdots - 10404000000 \)
|
| $17$ |
\( (T^{3} + 22 T^{2} + \cdots - 216600)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 23592960000 \)
|
| $23$ |
\( T^{6} + \cdots - 183759683584 \)
|
| $29$ |
\( T^{6} + \cdots - 53084160000 \)
|
| $31$ |
\( T^{6} + \cdots - 19948728960000 \)
|
| $37$ |
\( (T^{3} - 190 T^{2} + \cdots - 193800)^{2} \)
|
| $41$ |
\( (T^{3} + 306 T^{2} + \cdots - 15501000)^{2} \)
|
| $43$ |
\( (T^{3} + 164 T^{2} + \cdots - 369600)^{2} \)
|
| $47$ |
\( (T^{3} + 60 T^{2} + \cdots - 17640000)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 379858852641024 \)
|
| $59$ |
\( (T^{3} + 68 T^{2} + \cdots - 68808000)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 82301184000000 \)
|
| $67$ |
\( (T^{3} - 556 T^{2} + \cdots + 377458560)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 16881580038400 \)
|
| $73$ |
\( T^{6} + \cdots - 87\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} - 700 T^{2} + \cdots - 6055424)^{2} \)
|
| $83$ |
\( (T^{3} + 1456 T^{2} + \cdots - 185865600)^{2} \)
|
| $89$ |
\( (T^{3} + 186 T^{2} + \cdots - 95685000)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 34\!\cdots\!00 \)
|
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