Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,6,Mod(1,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, 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("414.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 414.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: | \(66.3989014026\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 8369x^{2} - 182616x - 370980 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 138) |
| 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
\( x^{4} - x^{3} - 8369x^{2} - 182616x - 370980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{3} + 307\nu^{2} + 58579\nu - 16554 ) / 12684 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{3} + 307\nu^{2} + 83947\nu - 16554 ) / 12684 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{3} + 38\nu^{2} + 65224\nu + 969000 ) / 3171 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -27\beta_{3} + 56\beta_{2} + 40\beta _1 + 8376 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -921\beta_{3} + 8419\beta_{2} - 7963\beta _1 + 282036 ) / 2 \)
|
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 |
|
−4.00000 | 0 | 16.0000 | −93.5694 | 0 | 209.214 | −64.0000 | 0 | 374.278 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | −38.3508 | 0 | −90.6804 | −64.0000 | 0 | 153.403 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 0 | 16.0000 | −24.6410 | 0 | 103.175 | −64.0000 | 0 | 98.5642 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 0 | 16.0000 | 102.561 | 0 | 126.292 | −64.0000 | 0 | −410.245 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 414.6.a.o | 4 | |
| 3.b | odd | 2 | 1 | 138.6.a.i | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1104.6.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.6.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 414.6.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.m | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 54T_{5}^{3} - 9218T_{5}^{2} - 613004T_{5} - 9068808 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(414))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 54 T^{3} + \cdots - 9068808 \)
|
| $7$ |
\( T^{4} - 348 T^{3} + \cdots - 247202152 \)
|
| $11$ |
\( T^{4} + \cdots + 6955214976 \)
|
| $13$ |
\( T^{4} + \cdots - 368203080832 \)
|
| $17$ |
\( T^{4} + \cdots + 791612460000 \)
|
| $19$ |
\( T^{4} + \cdots + 12533606427240 \)
|
| $23$ |
\( (T + 529)^{4} \)
|
| $29$ |
\( T^{4} + \cdots - 7814693873520 \)
|
| $31$ |
\( T^{4} + \cdots + 270426164868224 \)
|
| $37$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 17\!\cdots\!40 \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!72 \)
|
| $47$ |
\( T^{4} + \cdots + 17\!\cdots\!60 \)
|
| $53$ |
\( T^{4} + \cdots - 75\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 21\!\cdots\!68 \)
|
| $61$ |
\( T^{4} + \cdots - 55\!\cdots\!20 \)
|
| $67$ |
\( T^{4} + \cdots + 38\!\cdots\!80 \)
|
| $71$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 61\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 42\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots - 77\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 16\!\cdots\!20 \)
|
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