Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1620,4,Mod(1,1620)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, 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("1620.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1620.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: | \(95.5830942093\) |
| 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: |
\( x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{7} \) |
| Twist minimal: | no (minimal twist has level 180) |
| 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
\( x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -217\nu^{6} - 13749\nu^{5} - 35278\nu^{4} + 1794864\nu^{3} + 1185215\nu^{2} - 19480107\nu - 14899152 ) / 727830 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 557\nu^{6} - 1827\nu^{5} - 96604\nu^{4} + 664140\nu^{3} + 966533\nu^{2} - 12799845\nu + 6615822 ) / 485220 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 791\nu^{6} + 6291\nu^{5} - 114016\nu^{4} - 478884\nu^{3} + 3130175\nu^{2} + 6827613\nu - 14987430 ) / 161740 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 7091 \nu^{6} - 10236 \nu^{5} + 1176262 \nu^{4} - 2608284 \nu^{3} - 21933689 \nu^{2} + \cdots + 63927732 ) / 727830 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13165 \nu^{6} + 1986 \nu^{5} - 2205932 \nu^{4} + 7579068 \nu^{3} + 32668543 \nu^{2} + \cdots + 8324382 ) / 727830 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20879 \nu^{6} + 7086 \nu^{5} - 3426478 \nu^{4} + 11721000 \nu^{3} + 44808701 \nu^{2} + \cdots + 54958434 ) / 727830 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} + 3\beta _1 + 7 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{6} - 16\beta_{5} - 22\beta_{4} - 9\beta_{3} + 3\beta_{2} - 3\beta _1 + 877 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27\beta_{6} + 99\beta_{5} + 480\beta_{4} + 364\beta_{3} + 378\beta_{2} + 345\beta _1 - 9693 ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 633\beta_{6} - 2607\beta_{5} - 4080\beta_{4} - 2152\beta_{3} - 546\beta_{2} - 1335\beta _1 + 121083 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3377\beta_{6} + 38585\beta_{5} + 101174\beta_{4} + 68066\beta_{3} + 59364\beta_{2} + 54921\beta _1 - 2450651 ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 92699 \beta_{6} - 431615 \beta_{5} - 766700 \beta_{4} - 440015 \beta_{3} - 212217 \beta_{2} + \cdots + 21104582 ) / 18 \)
|
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 |
|
0 | 0 | 0 | −5.00000 | 0 | −29.9233 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −5.00000 | 0 | −22.5332 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −5.00000 | 0 | −13.7344 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | −5.00000 | 0 | 8.37861 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | −5.00000 | 0 | 13.6186 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | −5.00000 | 0 | 24.6035 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | −5.00000 | 0 | 27.5902 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 1620.4.a.k | 7 | |
| 3.b | odd | 2 | 1 | 1620.4.a.l | 7 | ||
| 9.c | even | 3 | 2 | 540.4.i.c | 14 | ||
| 9.d | odd | 6 | 2 | 180.4.i.c | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 180.4.i.c | ✓ | 14 | 9.d | odd | 6 | 2 | |
| 540.4.i.c | 14 | 9.c | even | 3 | 2 | ||
| 1620.4.a.k | 7 | 1.a | even | 1 | 1 | trivial | |
| 1620.4.a.l | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1620))\):
|
\( T_{7}^{7} - 8T_{7}^{6} - 1575T_{7}^{5} + 13376T_{7}^{4} + 715043T_{7}^{3} - 6030324T_{7}^{2} - 85402745T_{7} + 717295564 \)
|
|
\( T_{11}^{7} + 27 T_{11}^{6} - 6924 T_{11}^{5} - 279972 T_{11}^{4} + 9313200 T_{11}^{3} + \cdots + 54671232528 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( (T + 5)^{7} \)
|
| $7$ |
\( T^{7} - 8 T^{6} + \cdots + 717295564 \)
|
| $11$ |
\( T^{7} + \cdots + 54671232528 \)
|
| $13$ |
\( T^{7} + \cdots + 347644158016 \)
|
| $17$ |
\( T^{7} + \cdots - 21594831010704 \)
|
| $19$ |
\( T^{7} + \cdots - 5746852185008 \)
|
| $23$ |
\( T^{7} + \cdots + 38691437083074 \)
|
| $29$ |
\( T^{7} + \cdots + 96760250272134 \)
|
| $31$ |
\( T^{7} + \cdots + 96576105164320 \)
|
| $37$ |
\( T^{7} + \cdots - 77\!\cdots\!44 \)
|
| $41$ |
\( T^{7} + \cdots - 12\!\cdots\!99 \)
|
| $43$ |
\( T^{7} + \cdots - 13\!\cdots\!36 \)
|
| $47$ |
\( T^{7} + \cdots + 28\!\cdots\!16 \)
|
| $53$ |
\( T^{7} + \cdots + 55\!\cdots\!56 \)
|
| $59$ |
\( T^{7} + \cdots + 18\!\cdots\!72 \)
|
| $61$ |
\( T^{7} + \cdots + 33\!\cdots\!82 \)
|
| $67$ |
\( T^{7} + \cdots + 19\!\cdots\!01 \)
|
| $71$ |
\( T^{7} + \cdots + 64\!\cdots\!96 \)
|
| $73$ |
\( T^{7} + \cdots + 25\!\cdots\!40 \)
|
| $79$ |
\( T^{7} + \cdots + 51\!\cdots\!84 \)
|
| $83$ |
\( T^{7} + \cdots - 24\!\cdots\!64 \)
|
| $89$ |
\( T^{7} + \cdots + 51\!\cdots\!06 \)
|
| $97$ |
\( T^{7} + \cdots + 57\!\cdots\!32 \)
|
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