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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 130x^{5} + 519x^{4} + 2830x^{3} - 18764x^{2} + 30728x - 11136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 130x^{5} + 519x^{4} + 2830x^{3} - 18764x^{2} + 30728x - 11136 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 29\nu^{6} - 352\nu^{5} - 5754\nu^{4} + 43671\nu^{3} + 212900\nu^{2} - 1036756\nu + 365376 ) / 6528 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 130\nu^{4} + 259\nu^{3} + 3348\nu^{2} - 11812\nu + 7104 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{6} + 40\nu^{5} - 1254\nu^{4} - 1359\nu^{3} + 25844\nu^{2} - 59996\nu + 114336 ) / 1632 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -33\nu^{6} + 16\nu^{5} + 4578\nu^{4} - 9795\nu^{3} - 137508\nu^{2} + 417988\nu + 47040 ) / 3264 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 390\nu^{4} - 777\nu^{3} - 10044\nu^{2} + 36460\nu - 21568 ) / 128 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -233\nu^{6} - 260\nu^{5} + 29418\nu^{4} - 29187\nu^{3} - 743504\nu^{2} + 2040028\nu - 874560 ) / 1632 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 3\beta_{2} + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} - 2\beta_{4} - 4\beta_{3} - 7\beta_{2} - 2\beta _1 + 304 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{6} + 73\beta_{5} - 14\beta_{4} + 20\beta_{3} + 259\beta_{2} - 4\beta _1 - 906 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -8\beta_{6} - 459\beta_{5} - 60\beta_{4} - 480\beta_{3} - 1073\beta_{2} - 200\beta _1 + 23610 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 456\beta_{6} + 7127\beta_{5} - 1758\beta_{4} + 3444\beta_{3} + 24519\beta_{2} - 90\beta _1 - 126452 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -2076\beta_{6} - 56721\beta_{5} + 2522\beta_{4} - 54188\beta_{3} - 146675\beta_{2} - 18268\beta _1 + 2252954 ) / 8 \)
|
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 | −20.1582 | 0 | 7.00000 | −8.00000 | 0 | 40.3164 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −18.8618 | 0 | 7.00000 | −8.00000 | 0 | 37.7236 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −11.0503 | 0 | 7.00000 | −8.00000 | 0 | 22.1005 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | −5.52534 | 0 | 7.00000 | −8.00000 | 0 | 11.0507 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 0 | 4.00000 | 1.54016 | 0 | 7.00000 | −8.00000 | 0 | −3.08032 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 0 | 4.00000 | 11.7186 | 0 | 7.00000 | −8.00000 | 0 | −23.4373 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | 0 | 4.00000 | 14.3368 | 0 | 7.00000 | −8.00000 | 0 | −28.6736 | ||||||||||||||||||||||||||||||||||||||||
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.bg | ✓ | 7 |
| 3.b | odd | 2 | 1 | 2394.4.a.bj | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2394.4.a.bg | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2394.4.a.bj | yes | 7 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\):
|
\( T_{5}^{7} + 28T_{5}^{6} - 238T_{5}^{5} - 10028T_{5}^{4} - 4368T_{5}^{3} + 885456T_{5}^{2} + 2584608T_{5} - 6007040 \)
|
|
\( T_{11}^{7} + 82 T_{11}^{6} - 2084 T_{11}^{5} - 203208 T_{11}^{4} + 2660720 T_{11}^{3} + \cdots + 12660279424 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + 28 T^{6} + \cdots - 6007040 \)
|
| $7$ |
\( (T - 7)^{7} \)
|
| $11$ |
\( T^{7} + \cdots + 12660279424 \)
|
| $13$ |
\( T^{7} + \cdots + 450367873024 \)
|
| $17$ |
\( T^{7} + \cdots + 127705604096 \)
|
| $19$ |
\( (T + 19)^{7} \)
|
| $23$ |
\( T^{7} + \cdots + 8778950096384 \)
|
| $29$ |
\( T^{7} + \cdots + 18490155451456 \)
|
| $31$ |
\( T^{7} + \cdots - 57\!\cdots\!40 \)
|
| $37$ |
\( T^{7} + \cdots + 770419852982784 \)
|
| $41$ |
\( T^{7} + \cdots + 341473701795200 \)
|
| $43$ |
\( T^{7} + \cdots + 77\!\cdots\!60 \)
|
| $47$ |
\( T^{7} + \cdots + 17\!\cdots\!32 \)
|
| $53$ |
\( T^{7} + \cdots - 28\!\cdots\!68 \)
|
| $59$ |
\( T^{7} + \cdots + 23\!\cdots\!12 \)
|
| $61$ |
\( T^{7} + \cdots + 62\!\cdots\!80 \)
|
| $67$ |
\( T^{7} + \cdots - 13\!\cdots\!96 \)
|
| $71$ |
\( T^{7} + \cdots + 25\!\cdots\!12 \)
|
| $73$ |
\( T^{7} + \cdots + 91\!\cdots\!28 \)
|
| $79$ |
\( T^{7} + \cdots + 36\!\cdots\!36 \)
|
| $83$ |
\( T^{7} + \cdots - 10\!\cdots\!92 \)
|
| $89$ |
\( T^{7} + \cdots + 39\!\cdots\!44 \)
|
| $97$ |
\( T^{7} + \cdots + 11\!\cdots\!72 \)
|
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