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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 429x^{5} + 1799x^{4} + 59687x^{3} - 308117x^{2} - 2682459x + 15997617 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 429x^{5} + 1799x^{4} + 59687x^{3} - 308117x^{2} - 2682459x + 15997617 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 5\nu^{5} - 392\nu^{4} - 1328\nu^{3} + 49879\nu^{2} + 87267\nu - 2035848 ) / 48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 4\nu^{5} - 405\nu^{4} - 1120\nu^{3} + 52487\nu^{2} + 77884\nu - 2149043 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{6} - 49\nu^{5} + 4390\nu^{4} + 13408\nu^{3} - 564365\nu^{2} - 910503\nu + 23092938 ) / 336 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23\nu^{6} + 126\nu^{5} - 8761\nu^{4} - 32352\nu^{3} + 1090945\nu^{2} + 2049554\nu - 44005903 ) / 224 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 125 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{5} + 4\beta_{4} + 5\beta_{3} + \beta_{2} + 141\beta _1 - 280 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} - 30\beta_{5} - 14\beta_{4} - 36\beta_{3} + 242\beta_{2} - 554\beta _1 + 17937 \)
|
| \(\nu^{5}\) | \(=\) |
\( -26\beta_{6} + 1846\beta_{5} + 982\beta_{4} + 1556\beta_{3} - 330\beta_{2} + 21931\beta _1 - 78616 \)
|
| \(\nu^{6}\) | \(=\) |
\( 914\beta_{6} - 11694\beta_{5} - 5086\beta_{4} - 15204\beta_{3} + 47963\beta_{2} - 127084\beta _1 + 2853517 \)
|
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 | −8.84119 | 0 | −7.00000 | 8.00000 | 0 | −17.6824 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −8.79021 | 0 | −7.00000 | 8.00000 | 0 | −17.5804 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | −5.04472 | 0 | −7.00000 | 8.00000 | 0 | −10.0894 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | −5.02869 | 0 | −7.00000 | 8.00000 | 0 | −10.0574 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 0 | 4.00000 | 13.1491 | 0 | −7.00000 | 8.00000 | 0 | 26.2982 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 0 | 4.00000 | 15.0547 | 0 | −7.00000 | 8.00000 | 0 | 30.1093 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 0 | 4.00000 | 17.5010 | 0 | −7.00000 | 8.00000 | 0 | 35.0021 | ||||||||||||||||||||||||||||||||||||||||
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.bi | yes | 7 |
| 3.b | odd | 2 | 1 | 2394.4.a.bh | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2394.4.a.bh | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 2394.4.a.bi | yes | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\):
|
\( T_{5}^{7} - 18T_{5}^{6} - 294T_{5}^{5} + 4096T_{5}^{4} + 43880T_{5}^{3} - 211840T_{5}^{2} - 2898336T_{5} - 6830208 \)
|
|
\( T_{11}^{7} - 28 T_{11}^{6} - 5216 T_{11}^{5} + 127000 T_{11}^{4} + 3367696 T_{11}^{3} + 12642560 T_{11}^{2} + \cdots - 311359104 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 18 T^{6} + \cdots - 6830208 \)
|
| $7$ |
\( (T + 7)^{7} \)
|
| $11$ |
\( T^{7} - 28 T^{6} + \cdots - 311359104 \)
|
| $13$ |
\( T^{7} + \cdots + 8042803200 \)
|
| $17$ |
\( T^{7} + \cdots - 442964929792 \)
|
| $19$ |
\( (T - 19)^{7} \)
|
| $23$ |
\( T^{7} + \cdots - 1193706375552 \)
|
| $29$ |
\( T^{7} + \cdots - 32\!\cdots\!32 \)
|
| $31$ |
\( T^{7} + \cdots + 472109747788800 \)
|
| $37$ |
\( T^{7} + \cdots + 35\!\cdots\!84 \)
|
| $41$ |
\( T^{7} + \cdots - 12\!\cdots\!32 \)
|
| $43$ |
\( T^{7} + \cdots - 16\!\cdots\!12 \)
|
| $47$ |
\( T^{7} + \cdots - 15033207377664 \)
|
| $53$ |
\( T^{7} + \cdots + 28\!\cdots\!92 \)
|
| $59$ |
\( T^{7} + \cdots + 22\!\cdots\!88 \)
|
| $61$ |
\( T^{7} + \cdots + 19\!\cdots\!24 \)
|
| $67$ |
\( T^{7} + \cdots + 50\!\cdots\!12 \)
|
| $71$ |
\( T^{7} + \cdots + 18\!\cdots\!48 \)
|
| $73$ |
\( T^{7} + \cdots - 66\!\cdots\!88 \)
|
| $79$ |
\( T^{7} + \cdots - 75\!\cdots\!64 \)
|
| $83$ |
\( T^{7} + \cdots + 14\!\cdots\!44 \)
|
| $89$ |
\( T^{7} + \cdots - 28\!\cdots\!32 \)
|
| $97$ |
\( T^{7} + \cdots - 20\!\cdots\!56 \)
|
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