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: |
\( x^{12} - x^{11} - 1763 x^{10} + 5125 x^{9} + 1130618 x^{8} - 5262500 x^{7} - 310237136 x^{6} + \cdots - 9785332666368 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| 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
\( x^{12} - x^{11} - 1763 x^{10} + 5125 x^{9} + 1130618 x^{8} - 5262500 x^{7} - 310237136 x^{6} + \cdots - 9785332666368 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3\nu - 294 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47\!\cdots\!53 \nu^{11} + \cdots - 53\!\cdots\!32 ) / 75\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 90\!\cdots\!77 \nu^{11} + \cdots + 27\!\cdots\!88 ) / 93\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\!\cdots\!87 \nu^{11} + \cdots + 46\!\cdots\!08 ) / 15\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 71\!\cdots\!23 \nu^{11} + \cdots - 13\!\cdots\!52 ) / 28\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24\!\cdots\!63 \nu^{11} + \cdots + 22\!\cdots\!72 ) / 37\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29\!\cdots\!47 \nu^{11} + \cdots - 34\!\cdots\!32 ) / 40\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 33\!\cdots\!33 \nu^{11} + \cdots + 50\!\cdots\!52 ) / 44\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 62\!\cdots\!51 \nu^{11} + \cdots + 10\!\cdots\!44 ) / 75\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 31\!\cdots\!61 \nu^{11} + \cdots - 52\!\cdots\!84 ) / 18\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 + 294 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{11} + \beta_{10} + 4\beta_{9} - 2\beta_{8} + 9\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + 482\beta _1 - 880 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 21 \beta_{11} - 18 \beta_{10} - 3 \beta_{9} - 24 \beta_{7} - 12 \beta_{6} - 66 \beta_{5} + \cdots + 140160 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1364 \beta_{11} + 460 \beta_{10} + 3244 \beta_{9} - 1259 \beta_{8} - 249 \beta_{7} + 84 \beta_{6} + \cdots - 770707 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 18948 \beta_{11} - 18075 \beta_{10} - 8910 \beta_{9} + 24 \beta_{8} - 21444 \beta_{7} + \cdots + 73814769 \)
|
| \(\nu^{7}\) | \(=\) |
\( 837329 \beta_{11} + 214033 \beta_{10} + 2242303 \beta_{9} - 719624 \beta_{8} - 300612 \beta_{7} + \cdots - 555490357 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 13935837 \beta_{11} - 13632234 \beta_{10} - 10562748 \beta_{9} - 30642 \beta_{8} - 14994699 \beta_{7} + \cdots + 40088620104 \)
|
| \(\nu^{9}\) | \(=\) |
\( 512319299 \beta_{11} + 115491085 \beta_{10} + 1477066429 \beta_{9} - 403628828 \beta_{8} + \cdots - 370343383825 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 9545609001 \beta_{11} - 9315814284 \beta_{10} - 9633602442 \beta_{9} - 44334486 \beta_{8} + \cdots + 22121222434230 \)
|
| \(\nu^{11}\) | \(=\) |
\( 315235367735 \beta_{11} + 72106652107 \beta_{10} + 949495053091 \beta_{9} - 224086521752 \beta_{8} + \cdots - 237123632948023 \)
|
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 | −25.4038 | 0 | 15.1006 | 0 | −49.0000 | 0 | 402.354 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −22.8722 | 0 | −107.801 | 0 | −49.0000 | 0 | 280.136 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −22.2846 | 0 | 26.2708 | 0 | −49.0000 | 0 | 253.604 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −12.9267 | 0 | −15.1197 | 0 | −49.0000 | 0 | −75.9002 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −11.7979 | 0 | 84.7081 | 0 | −49.0000 | 0 | −103.810 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −6.62582 | 0 | −52.3131 | 0 | −49.0000 | 0 | −199.098 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −3.65045 | 0 | −86.6053 | 0 | −49.0000 | 0 | −229.674 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 6.66525 | 0 | 76.4329 | 0 | −49.0000 | 0 | −198.574 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 8.24722 | 0 | 23.4887 | 0 | −49.0000 | 0 | −174.983 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 21.1340 | 0 | −16.7834 | 0 | −49.0000 | 0 | 203.645 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 21.7526 | 0 | −70.4012 | 0 | −49.0000 | 0 | 230.177 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 22.7624 | 0 | 59.0231 | 0 | −49.0000 | 0 | 275.125 | 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.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.6.a.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 25 T_{3}^{11} - 1477 T_{3}^{10} - 38405 T_{3}^{9} + 730268 T_{3}^{8} + \cdots - 27474145906176 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(644))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots - 27474145906176 \)
|
| $5$ |
\( T^{12} + \cdots + 31\!\cdots\!00 \)
|
| $7$ |
\( (T + 49)^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 35\!\cdots\!72 \)
|
| $13$ |
\( T^{12} + \cdots - 40\!\cdots\!92 \)
|
| $17$ |
\( T^{12} + \cdots + 14\!\cdots\!00 \)
|
| $19$ |
\( T^{12} + \cdots - 14\!\cdots\!00 \)
|
| $23$ |
\( (T - 529)^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 63\!\cdots\!44 \)
|
| $31$ |
\( T^{12} + \cdots - 79\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots - 18\!\cdots\!72 \)
|
| $41$ |
\( T^{12} + \cdots - 37\!\cdots\!92 \)
|
| $43$ |
\( T^{12} + \cdots - 36\!\cdots\!04 \)
|
| $47$ |
\( T^{12} + \cdots - 11\!\cdots\!64 \)
|
| $53$ |
\( T^{12} + \cdots - 12\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 36\!\cdots\!28 \)
|
| $61$ |
\( T^{12} + \cdots - 91\!\cdots\!80 \)
|
| $67$ |
\( T^{12} + \cdots - 75\!\cdots\!24 \)
|
| $71$ |
\( T^{12} + \cdots - 84\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 41\!\cdots\!80 \)
|
| $79$ |
\( T^{12} + \cdots - 71\!\cdots\!88 \)
|
| $83$ |
\( T^{12} + \cdots + 63\!\cdots\!84 \)
|
| $89$ |
\( T^{12} + \cdots - 32\!\cdots\!96 \)
|
| $97$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| show more | |
| show less |