Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,9,Mod(73,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.73");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.z (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(102.659409735\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} + \cdots + 51\!\cdots\!63 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{31}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{26}\cdot 7^{16} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} + \cdots + 51\!\cdots\!63 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 12\!\cdots\!75 \nu^{18} + \cdots + 51\!\cdots\!39 ) / 27\!\cdots\!72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!49 \nu^{18} + \cdots - 34\!\cdots\!13 ) / 10\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41\!\cdots\!46 \nu^{19} + \cdots - 63\!\cdots\!61 ) / 21\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 20\!\cdots\!23 \nu^{19} + \cdots - 74\!\cdots\!32 ) / 72\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 43\!\cdots\!25 \nu^{19} + \cdots - 74\!\cdots\!27 ) / 31\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 18\!\cdots\!80 \nu^{19} + \cdots + 10\!\cdots\!73 ) / 51\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 18\!\cdots\!80 \nu^{19} + \cdots - 66\!\cdots\!89 ) / 51\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 19\!\cdots\!15 \nu^{19} + \cdots - 17\!\cdots\!70 ) / 51\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!15 \nu^{19} + \cdots + 13\!\cdots\!23 ) / 51\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13\!\cdots\!09 \nu^{19} + \cdots + 13\!\cdots\!41 ) / 28\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\!\cdots\!09 \nu^{19} + \cdots - 12\!\cdots\!08 ) / 28\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15\!\cdots\!05 \nu^{19} + \cdots + 45\!\cdots\!84 ) / 31\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 19\!\cdots\!79 \nu^{19} + \cdots - 19\!\cdots\!21 ) / 18\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 69\!\cdots\!55 \nu^{19} + \cdots + 47\!\cdots\!16 ) / 49\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 69\!\cdots\!55 \nu^{19} + \cdots - 62\!\cdots\!15 ) / 49\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 38\!\cdots\!58 \nu^{19} + \cdots + 14\!\cdots\!23 ) / 18\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 27\!\cdots\!01 \nu^{19} + \cdots + 10\!\cdots\!12 ) / 56\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 33\!\cdots\!09 \nu^{19} + \cdots + 39\!\cdots\!20 ) / 18\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 33\!\cdots\!09 \nu^{19} + \cdots + 37\!\cdots\!87 ) / 18\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} + 3\beta_{3} + \beta _1 + 3 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 24 \beta_{15} - 24 \beta_{14} + 31 \beta_{12} - \beta_{9} + \beta_{8} + 37 \beta_{7} - 6 \beta_{6} + \cdots + 525513 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 312 \beta_{19} - 312 \beta_{18} - 82 \beta_{17} + 72 \beta_{16} + 432 \beta_{15} + 216 \beta_{14} + \cdots + 4729401 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1248 \beta_{18} - 164 \beta_{17} + 25204541 \beta_{15} - 25203245 \beta_{14} - 288 \beta_{13} + \cdots + 442327961807 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 140042625 \beta_{19} - 140045745 \beta_{18} + 1780888 \beta_{17} - 11773292 \beta_{16} + \cdots + 2211515128385 ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 6240 \beta_{19} - 840268230 \beta_{18} + 5343074 \beta_{17} - 1440 \beta_{16} + 8522005327209 \beta_{15} + \cdots + 13\!\cdots\!11 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 52072729741221 \beta_{19} - 52075670690946 \beta_{18} + 28432828051446 \beta_{17} + \cdots + 97\!\cdots\!28 ) / 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 7842544248 \beta_{19} - 416597523053208 \beta_{18} + 113731287271056 \beta_{17} + \cdots + 45\!\cdots\!30 ) / 9 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 18\!\cdots\!59 \beta_{19} + \cdots + 41\!\cdots\!58 ) / 9 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 62\!\cdots\!80 \beta_{19} + \cdots + 15\!\cdots\!10 ) / 9 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 64\!\cdots\!37 \beta_{19} + \cdots + 16\!\cdots\!49 ) / 9 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 40\!\cdots\!08 \beta_{19} + \cdots + 51\!\cdots\!73 ) / 9 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 22\!\cdots\!10 \beta_{19} + \cdots + 66\!\cdots\!95 ) / 9 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 23\!\cdots\!36 \beta_{19} + \cdots + 17\!\cdots\!37 ) / 9 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 78\!\cdots\!86 \beta_{19} + \cdots + 25\!\cdots\!43 ) / 9 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 12\!\cdots\!64 \beta_{19} + \cdots + 58\!\cdots\!57 ) / 9 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 27\!\cdots\!33 \beta_{19} + \cdots + 98\!\cdots\!93 ) / 9 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 64\!\cdots\!24 \beta_{19} + \cdots + 19\!\cdots\!67 ) / 9 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 95\!\cdots\!99 \beta_{19} + \cdots + 37\!\cdots\!74 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | 0 | 0 | −885.405 | + | 511.189i | 0 | 765.093 | + | 2275.84i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0 | 0 | 0 | −846.691 | + | 488.837i | 0 | −1709.42 | − | 1686.02i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | 0 | 0 | 0 | −534.755 | + | 308.741i | 0 | −750.736 | − | 2280.61i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 0 | 0 | 0 | −428.745 | + | 247.536i | 0 | 1933.79 | − | 1423.12i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 0 | 0 | 0 | −6.56919 | + | 3.79272i | 0 | −2048.22 | + | 1252.83i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 0 | 0 | 0 | 6.56919 | − | 3.79272i | 0 | −2048.22 | + | 1252.83i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 0 | 0 | 0 | 428.745 | − | 247.536i | 0 | 1933.79 | − | 1423.12i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 0 | 0 | 0 | 534.755 | − | 308.741i | 0 | −750.736 | − | 2280.61i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.9 | 0 | 0 | 0 | 846.691 | − | 488.837i | 0 | −1709.42 | − | 1686.02i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.10 | 0 | 0 | 0 | 885.405 | − | 511.189i | 0 | 765.093 | + | 2275.84i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.1 | 0 | 0 | 0 | −885.405 | − | 511.189i | 0 | 765.093 | − | 2275.84i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.2 | 0 | 0 | 0 | −846.691 | − | 488.837i | 0 | −1709.42 | + | 1686.02i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.3 | 0 | 0 | 0 | −534.755 | − | 308.741i | 0 | −750.736 | + | 2280.61i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.4 | 0 | 0 | 0 | −428.745 | − | 247.536i | 0 | 1933.79 | + | 1423.12i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.5 | 0 | 0 | 0 | −6.56919 | − | 3.79272i | 0 | −2048.22 | − | 1252.83i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.6 | 0 | 0 | 0 | 6.56919 | + | 3.79272i | 0 | −2048.22 | − | 1252.83i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.7 | 0 | 0 | 0 | 428.745 | + | 247.536i | 0 | 1933.79 | + | 1423.12i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.8 | 0 | 0 | 0 | 534.755 | + | 308.741i | 0 | −750.736 | + | 2280.61i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.9 | 0 | 0 | 0 | 846.691 | + | 488.837i | 0 | −1709.42 | + | 1686.02i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.10 | 0 | 0 | 0 | 885.405 | + | 511.189i | 0 | 765.093 | − | 2275.84i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 252.9.z.e | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 252.9.z.e | ✓ | 20 |
| 7.d | odd | 6 | 1 | inner | 252.9.z.e | ✓ | 20 |
| 21.g | even | 6 | 1 | inner | 252.9.z.e | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 252.9.z.e | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 252.9.z.e | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 252.9.z.e | ✓ | 20 | 7.d | odd | 6 | 1 | inner |
| 252.9.z.e | ✓ | 20 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{20} - 2627541 T_{5}^{18} + 4557812756238 T_{5}^{16} + \cdots + 28\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 28\!\cdots\!00 \)
|
| $7$ |
\( (T^{10} + \cdots + 63\!\cdots\!01)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 76\!\cdots\!00 \)
|
| $13$ |
\( (T^{10} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 52\!\cdots\!00 \)
|
| $19$ |
\( (T^{10} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 87\!\cdots\!00 \)
|
| $29$ |
\( (T^{10} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 27\!\cdots\!07)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 85\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{5} + \cdots + 18\!\cdots\!00)^{4} \)
|
| $47$ |
\( T^{20} + \cdots + 38\!\cdots\!00 \)
|
| $53$ |
\( T^{20} + \cdots + 76\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 34\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 49\!\cdots\!52)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 24\!\cdots\!25)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 34\!\cdots\!00 \)
|
| $97$ |
\( (T^{10} + \cdots + 45\!\cdots\!00)^{2} \)
|
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