Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2394,4,Mod(1,2394)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2394, 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("2394.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2394.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: | \(141.250572554\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - 2x^{5} - 113x^{4} + 312x^{3} + 2070x^{2} - 3564x - 11664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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
\( x^{6} - 2x^{5} - 113x^{4} + 312x^{3} + 2070x^{2} - 3564x - 11664 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{5} - \nu^{4} - 502\nu^{3} + 624\nu^{2} + 5706\nu + 324 ) / 486 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} - 5\nu^{4} + 163\nu^{3} + 312\nu^{2} + 504\nu - 9072 ) / 486 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{5} + 7\nu^{4} + 841\nu^{3} - 1560\nu^{2} - 9972\nu + 7938 ) / 486 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 92\nu^{2} + 54\nu + 792 ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 11\nu^{4} - 95\nu^{3} + 1329\nu^{2} - 252\nu - 15633 ) / 243 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{5} + 4\beta_{4} - 3\beta_{3} + \beta_{2} - 6\beta _1 + 153 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{5} - 4\beta_{4} + 89\beta_{3} + 37\beta_{2} + 154\beta _1 - 175 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 360\beta_{5} + 444\beta_{4} - 419\beta_{3} + \beta_{2} - 814\beta _1 + 11029 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 376\beta_{5} - 812\beta_{4} + 8085\beta_{3} + 2449\beta_{2} + 14154\beta _1 - 35859 ) / 4 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | −11.5234 | 0 | 7.00000 | −8.00000 | 0 | 23.0468 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −7.17904 | 0 | 7.00000 | −8.00000 | 0 | 14.3581 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −7.17505 | 0 | 7.00000 | −8.00000 | 0 | 14.3501 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 8.88203 | 0 | 7.00000 | −8.00000 | 0 | −17.7641 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 0 | 4.00000 | 11.2625 | 0 | 7.00000 | −8.00000 | 0 | −22.5250 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 0 | 4.00000 | 17.7330 | 0 | 7.00000 | −8.00000 | 0 | −35.4659 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 2394.4.a.be | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2394.4.a.bf | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2394.4.a.be | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2394.4.a.bf | yes | 6 | 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\):
|
\( T_{5}^{6} - 12T_{5}^{5} - 306T_{5}^{4} + 2436T_{5}^{3} + 30800T_{5}^{2} - 113376T_{5} - 1052928 \)
|
|
\( T_{11}^{6} - 82T_{11}^{5} - 1800T_{11}^{4} + 256576T_{11}^{3} - 4379536T_{11}^{2} - 18667488T_{11} + 181671552 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 12 T^{5} + \cdots - 1052928 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( T^{6} - 82 T^{5} + \cdots + 181671552 \)
|
| $13$ |
\( T^{6} + 64 T^{5} + \cdots - 856909824 \)
|
| $17$ |
\( T^{6} + \cdots - 50453705088 \)
|
| $19$ |
\( (T - 19)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 623173493952 \)
|
| $29$ |
\( T^{6} + \cdots + 3449391842880 \)
|
| $31$ |
\( T^{6} + \cdots + 7542490880 \)
|
| $37$ |
\( T^{6} + \cdots - 19453533208448 \)
|
| $41$ |
\( T^{6} + \cdots + 150379505525952 \)
|
| $43$ |
\( T^{6} + \cdots + 54465549847552 \)
|
| $47$ |
\( T^{6} + \cdots + 48870977453568 \)
|
| $53$ |
\( T^{6} + \cdots - 410953363520256 \)
|
| $59$ |
\( T^{6} + \cdots + 783873068433408 \)
|
| $61$ |
\( T^{6} + \cdots + 528069564559296 \)
|
| $67$ |
\( T^{6} + \cdots - 27\!\cdots\!08 \)
|
| $71$ |
\( T^{6} + \cdots + 23\!\cdots\!32 \)
|
| $73$ |
\( T^{6} + \cdots + 510095489476800 \)
|
| $79$ |
\( T^{6} + \cdots - 13\!\cdots\!76 \)
|
| $83$ |
\( T^{6} + \cdots + 39\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots - 36\!\cdots\!40 \)
|
| $97$ |
\( T^{6} + \cdots + 49\!\cdots\!80 \)
|
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