Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,6,Mod(1,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("644.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 644.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: | \(103.287179960\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 1776 x^{10} + 4180 x^{9} + 1126170 x^{8} - 1790022 x^{7} - 313399552 x^{6} + \cdots + 7245024277248 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 1776 x^{10} + 4180 x^{9} + 1126170 x^{8} - 1790022 x^{7} - 313399552 x^{6} + \cdots + 7245024277248 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 297 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13\!\cdots\!29 \nu^{11} + \cdots + 62\!\cdots\!76 ) / 66\!\cdots\!04 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!01 \nu^{11} + \cdots + 14\!\cdots\!20 ) / 66\!\cdots\!04 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\!\cdots\!41 \nu^{11} + \cdots + 43\!\cdots\!68 ) / 82\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 85\!\cdots\!99 \nu^{11} + \cdots + 19\!\cdots\!40 ) / 22\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 91\!\cdots\!07 \nu^{11} + \cdots + 15\!\cdots\!52 ) / 18\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17\!\cdots\!25 \nu^{11} + \cdots - 29\!\cdots\!12 ) / 33\!\cdots\!52 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13\!\cdots\!29 \nu^{11} + \cdots - 25\!\cdots\!56 ) / 18\!\cdots\!64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 31\!\cdots\!05 \nu^{11} + \cdots + 17\!\cdots\!36 ) / 33\!\cdots\!52 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 54\!\cdots\!34 \nu^{11} + \cdots - 36\!\cdots\!76 ) / 55\!\cdots\!92 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 297 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 4 \beta_{11} - 4 \beta_{10} - 8 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 5 \beta_{5} + \cdots + 172 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 33 \beta_{11} + 3 \beta_{10} - 117 \beta_{9} - 96 \beta_{8} - 39 \beta_{7} - 21 \beta_{6} + \cdots + 151653 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3451 \beta_{11} - 3205 \beta_{10} - 5726 \beta_{9} + 2915 \beta_{8} - 193 \beta_{7} + 1639 \beta_{6} + \cdots + 124273 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 28695 \beta_{11} + 3018 \beta_{10} - 122451 \beta_{9} - 108504 \beta_{8} - 43698 \beta_{7} + \cdots + 91449936 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2481667 \beta_{11} - 2300680 \beta_{10} - 3544184 \beta_{9} + 2829413 \beta_{8} + 120149 \beta_{7} + \cdots + 67215496 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 20872989 \beta_{11} + 5262627 \beta_{10} - 98548716 \beta_{9} - 93012618 \beta_{8} + \cdots + 59129098890 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1736628283 \beta_{11} - 1620051607 \beta_{10} - 2155141964 \beta_{9} + 2391244682 \beta_{8} + \cdots + 10811347585 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 14297771418 \beta_{11} + 6592149336 \beta_{10} - 72875768418 \beta_{9} - 72862377072 \beta_{8} + \cdots + 39527843893815 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1212914488990 \beta_{11} - 1137559272328 \beta_{10} - 1319808865592 \beta_{9} + 1898967039740 \beta_{8} + \cdots - 29729675036816 \)
|
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 | −26.8017 | 0 | −27.9604 | 0 | 49.0000 | 0 | 475.333 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −26.2957 | 0 | 77.1544 | 0 | 49.0000 | 0 | 448.464 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −15.6073 | 0 | 41.3465 | 0 | 49.0000 | 0 | 0.587120 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −14.9941 | 0 | −74.7854 | 0 | 49.0000 | 0 | −18.1771 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −12.4133 | 0 | −20.6894 | 0 | 49.0000 | 0 | −88.9100 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −2.30621 | 0 | 29.8669 | 0 | 49.0000 | 0 | −237.681 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.25864 | 0 | −93.7698 | 0 | 49.0000 | 0 | −232.381 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 5.86870 | 0 | 56.2346 | 0 | 49.0000 | 0 | −208.558 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 11.0359 | 0 | −11.8262 | 0 | 49.0000 | 0 | −121.210 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 17.3857 | 0 | 71.3605 | 0 | 49.0000 | 0 | 59.2637 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 19.7511 | 0 | −16.6940 | 0 | 49.0000 | 0 | 147.105 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 26.1183 | 0 | −44.2378 | 0 | 49.0000 | 0 | 439.166 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 644.6.a.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.6.a.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 15 T_{3}^{11} - 1677 T_{3}^{10} - 21555 T_{3}^{9} + 1009620 T_{3}^{8} + \cdots + 8937044140032 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(644))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 8937044140032 \)
|
| $5$ |
\( T^{12} + \cdots - 13\!\cdots\!52 \)
|
| $7$ |
\( (T - 49)^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 25\!\cdots\!52 \)
|
| $13$ |
\( T^{12} + \cdots + 26\!\cdots\!12 \)
|
| $17$ |
\( T^{12} + \cdots - 81\!\cdots\!40 \)
|
| $19$ |
\( T^{12} + \cdots + 35\!\cdots\!44 \)
|
| $23$ |
\( (T + 529)^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 67\!\cdots\!48 \)
|
| $31$ |
\( T^{12} + \cdots + 11\!\cdots\!32 \)
|
| $37$ |
\( T^{12} + \cdots - 34\!\cdots\!36 \)
|
| $41$ |
\( T^{12} + \cdots - 32\!\cdots\!96 \)
|
| $43$ |
\( T^{12} + \cdots + 50\!\cdots\!20 \)
|
| $47$ |
\( T^{12} + \cdots - 48\!\cdots\!12 \)
|
| $53$ |
\( T^{12} + \cdots + 18\!\cdots\!56 \)
|
| $59$ |
\( T^{12} + \cdots - 96\!\cdots\!12 \)
|
| $61$ |
\( T^{12} + \cdots - 17\!\cdots\!60 \)
|
| $67$ |
\( T^{12} + \cdots + 27\!\cdots\!68 \)
|
| $71$ |
\( T^{12} + \cdots - 78\!\cdots\!20 \)
|
| $73$ |
\( T^{12} + \cdots + 11\!\cdots\!44 \)
|
| $79$ |
\( T^{12} + \cdots - 57\!\cdots\!96 \)
|
| $83$ |
\( T^{12} + \cdots - 15\!\cdots\!48 \)
|
| $89$ |
\( T^{12} + \cdots - 59\!\cdots\!96 \)
|
| $97$ |
\( T^{12} + \cdots - 21\!\cdots\!28 \)
|
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