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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{514}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 514 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \sqrt{514}\). We also show the integral \(q\)-expansion of the trace form.
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 | −88.0147 | 0 | 63.3578 | −64.0000 | 0 | 352.059 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | 48.0147 | 0 | −163.358 | −64.0000 | 0 | −192.059 | |||||||||||||||||||||||||
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.e | 2 | |
| 3.b | odd | 2 | 1 | 138.6.a.f | ✓ | 2 | |
| 12.b | even | 2 | 1 | 1104.6.a.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.6.a.f | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 414.6.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.g | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 40T_{5} - 4226 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(414))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 40T - 4226 \)
|
| $7$ |
\( T^{2} + 100T - 10350 \)
|
| $11$ |
\( T^{2} + 160T - 26496 \)
|
| $13$ |
\( T^{2} - 632T - 196208 \)
|
| $17$ |
\( T^{2} + 1216 T + 365038 \)
|
| $19$ |
\( T^{2} - 2924 T + 1910770 \)
|
| $23$ |
\( (T - 529)^{2} \)
|
| $29$ |
\( T^{2} + 5300 T + 1674844 \)
|
| $31$ |
\( T^{2} - 2512 T - 15448200 \)
|
| $37$ |
\( T^{2} - 21868 T + 103990492 \)
|
| $41$ |
\( T^{2} + 9532 T + 14293380 \)
|
| $43$ |
\( T^{2} - 6612 T - 23549998 \)
|
| $47$ |
\( T^{2} + 3860 T - 18087204 \)
|
| $53$ |
\( T^{2} + 27744 T + 21306878 \)
|
| $59$ |
\( T^{2} - 59140 T + 873099900 \)
|
| $61$ |
\( T^{2} + 5724 T + 7103420 \)
|
| $67$ |
\( T^{2} + \cdots - 1139059982 \)
|
| $71$ |
\( T^{2} - 16320 T - 275664384 \)
|
| $73$ |
\( T^{2} + \cdots - 2090176092 \)
|
| $79$ |
\( T^{2} + \cdots - 1683484398 \)
|
| $83$ |
\( T^{2} + 61560 T - 106612336 \)
|
| $89$ |
\( T^{2} + \cdots - 10138998258 \)
|
| $97$ |
\( T^{2} + \cdots - 11623465860 \)
|
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