Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [936,2,Mod(313,936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(936, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("936.313");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.q (of order \(3\), 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: | \(7.47399762919\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 217 x^{8} - 394 x^{7} + 555 x^{6} - 598 x^{5} + 483 x^{4} + \cdots + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 217 x^{8} - 394 x^{7} + 555 x^{6} - 598 x^{5} + 483 x^{4} + \cdots + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( - 3 \nu^{10} + 15 \nu^{9} - 71 \nu^{8} + 194 \nu^{7} - 433 \nu^{6} + 683 \nu^{5} - 832 \nu^{4} + \cdots - 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( 5 \nu^{10} - 25 \nu^{9} + 119 \nu^{8} - 326 \nu^{7} + 735 \nu^{6} - 1169 \nu^{5} + 1455 \nu^{4} + \cdots + 58 \)
|
| \(\beta_{3}\) | \(=\) |
\( - 6 \nu^{10} + 30 \nu^{9} - 142 \nu^{8} + 388 \nu^{7} - 867 \nu^{6} + 1369 \nu^{5} - 1678 \nu^{4} + \cdots - 59 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 50 \nu^{11} + 314 \nu^{10} - 1514 \nu^{9} + 4796 \nu^{8} - 11585 \nu^{7} + 21207 \nu^{6} + \cdots + 742 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 118 \nu^{11} - 610 \nu^{10} + 2896 \nu^{9} - 8119 \nu^{8} + 18407 \nu^{7} - 29963 \nu^{6} + 37814 \nu^{5} + \cdots - 149 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 131 \nu^{11} - 688 \nu^{10} + 3273 \nu^{9} - 9289 \nu^{8} + 21228 \nu^{7} - 35085 \nu^{6} + 45029 \nu^{5} + \cdots - 461 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 118 \nu^{11} + 688 \nu^{10} - 3286 \nu^{9} + 9965 \nu^{8} - 23451 \nu^{7} + 41234 \nu^{6} + \cdots + 942 ) / 13 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 135 \nu^{11} + 762 \nu^{10} - 3638 \nu^{9} + 10825 \nu^{8} - 25293 \nu^{7} + 43761 \nu^{6} + \cdots + 836 ) / 13 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 131 \nu^{11} + 753 \nu^{10} - 3598 \nu^{9} + 10823 \nu^{8} - 25414 \nu^{7} + 44393 \nu^{6} + \cdots + 981 ) / 13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 196 \nu^{11} + 1078 \nu^{10} - 5132 \nu^{9} + 15009 \nu^{8} - 34722 \nu^{7} + 59031 \nu^{6} + \cdots + 968 ) / 13 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 193 \nu^{11} + 1094 \nu^{10} - 5232 \nu^{9} + 15625 \nu^{8} - 36649 \nu^{7} + 63756 \nu^{6} + \cdots + 1496 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{10} - \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} - \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{10} - \beta_{9} - 2\beta_{8} + 4\beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} + 4\beta_{3} - \beta _1 - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{11} - 9 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + \cdots - 6 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4 \beta_{11} - 19 \beta_{10} + 7 \beta_{9} + 14 \beta_{8} - 22 \beta_{7} - 19 \beta_{6} - 14 \beta_{5} + \cdots + 28 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11 \beta_{11} + 54 \beta_{10} - 51 \beta_{9} - 15 \beta_{8} - 54 \beta_{7} + 21 \beta_{6} - 51 \beta_{5} + \cdots + 57 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 43 \beta_{11} + 210 \beta_{10} - 102 \beta_{9} - 81 \beta_{8} + 90 \beta_{7} + 180 \beta_{6} + \cdots - 186 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 35 \beta_{11} - 205 \beta_{10} + 325 \beta_{9} + 29 \beta_{8} + 506 \beta_{7} + 53 \beta_{6} + \cdots - 614 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 350 \beta_{11} - 1845 \beta_{10} + 1134 \beta_{9} + 528 \beta_{8} - 102 \beta_{7} - 1332 \beta_{6} + \cdots + 891 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 84 \beta_{11} - 319 \beta_{10} - 1424 \beta_{9} + 209 \beta_{8} - 3892 \beta_{7} - 1861 \beta_{6} + \cdots + 5650 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 842 \beta_{11} + 4603 \beta_{10} - 3412 \beta_{9} - 1185 \beta_{8} - 1102 \beta_{7} + 2727 \beta_{6} + \cdots - 426 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 3098 \beta_{11} + 16465 \beta_{10} + 650 \beta_{9} - 4619 \beta_{8} + 25891 \beta_{7} + 22648 \beta_{6} + \cdots - 44647 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{10}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 313.1 |
|
0 | −1.71033 | − | 0.273454i | 0 | −0.558529 | − | 0.967401i | 0 | 0.697927 | − | 1.20884i | 0 | 2.85045 | + | 0.935391i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 313.2 | 0 | −1.69504 | + | 0.356160i | 0 | 0.348249 | + | 0.603184i | 0 | −0.902727 | + | 1.56357i | 0 | 2.74630 | − | 1.20741i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 313.3 | 0 | −0.793714 | − | 1.53949i | 0 | 0.671409 | + | 1.16292i | 0 | 2.29976 | − | 3.98331i | 0 | −1.74004 | + | 2.44382i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 313.4 | 0 | 0.0876778 | + | 1.72983i | 0 | −0.391618 | − | 0.678302i | 0 | −0.823587 | + | 1.42649i | 0 | −2.98463 | + | 0.303335i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 313.5 | 0 | 0.409577 | − | 1.68293i | 0 | −1.87385 | − | 3.24560i | 0 | 0.175717 | − | 0.304350i | 0 | −2.66449 | − | 1.37858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 313.6 | 0 | 1.70182 | − | 0.322173i | 0 | 1.30434 | + | 2.25918i | 0 | −0.447094 | + | 0.774390i | 0 | 2.79241 | − | 1.09656i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.1 | 0 | −1.71033 | + | 0.273454i | 0 | −0.558529 | + | 0.967401i | 0 | 0.697927 | + | 1.20884i | 0 | 2.85045 | − | 0.935391i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.2 | 0 | −1.69504 | − | 0.356160i | 0 | 0.348249 | − | 0.603184i | 0 | −0.902727 | − | 1.56357i | 0 | 2.74630 | + | 1.20741i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.3 | 0 | −0.793714 | + | 1.53949i | 0 | 0.671409 | − | 1.16292i | 0 | 2.29976 | + | 3.98331i | 0 | −1.74004 | − | 2.44382i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.4 | 0 | 0.0876778 | − | 1.72983i | 0 | −0.391618 | + | 0.678302i | 0 | −0.823587 | − | 1.42649i | 0 | −2.98463 | − | 0.303335i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.5 | 0 | 0.409577 | + | 1.68293i | 0 | −1.87385 | + | 3.24560i | 0 | 0.175717 | + | 0.304350i | 0 | −2.66449 | + | 1.37858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 625.6 | 0 | 1.70182 | + | 0.322173i | 0 | 1.30434 | − | 2.25918i | 0 | −0.447094 | − | 0.774390i | 0 | 2.79241 | + | 1.09656i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 936.2.q.d | ✓ | 12 |
| 3.b | odd | 2 | 1 | 2808.2.q.d | 12 | ||
| 9.c | even | 3 | 1 | inner | 936.2.q.d | ✓ | 12 |
| 9.c | even | 3 | 1 | 8424.2.a.v | 6 | ||
| 9.d | odd | 6 | 1 | 2808.2.q.d | 12 | ||
| 9.d | odd | 6 | 1 | 8424.2.a.u | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 936.2.q.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 936.2.q.d | ✓ | 12 | 9.c | even | 3 | 1 | inner |
| 2808.2.q.d | 12 | 3.b | odd | 2 | 1 | ||
| 2808.2.q.d | 12 | 9.d | odd | 6 | 1 | ||
| 8424.2.a.u | 6 | 9.d | odd | 6 | 1 | ||
| 8424.2.a.v | 6 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):
|
\( T_{5}^{12} + T_{5}^{11} + 13 T_{5}^{10} - 10 T_{5}^{9} + 124 T_{5}^{8} - 29 T_{5}^{7} + 236 T_{5}^{6} + \cdots + 64 \)
|
|
\( T_{7}^{12} - 2 T_{7}^{11} + 17 T_{7}^{10} + 28 T_{7}^{9} + 142 T_{7}^{8} + 122 T_{7}^{7} + 332 T_{7}^{6} + \cdots + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 4 T^{11} + \cdots + 729 \)
|
| $5$ |
\( T^{12} + T^{11} + \cdots + 64 \)
|
| $7$ |
\( T^{12} - 2 T^{11} + \cdots + 36 \)
|
| $11$ |
\( T^{12} + T^{11} + \cdots + 66564 \)
|
| $13$ |
\( (T^{2} + T + 1)^{6} \)
|
| $17$ |
\( (T^{6} - 6 T^{5} + \cdots + 337)^{2} \)
|
| $19$ |
\( (T^{6} + 7 T^{5} + \cdots + 262)^{2} \)
|
| $23$ |
\( T^{12} - 19 T^{11} + \cdots + 729 \)
|
| $29$ |
\( T^{12} + \cdots + 170929476 \)
|
| $31$ |
\( T^{12} - 8 T^{11} + \cdots + 186624 \)
|
| $37$ |
\( (T^{6} + 17 T^{5} + \cdots + 149902)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 23920334244 \)
|
| $43$ |
\( T^{12} - 15 T^{11} + \cdots + 221841 \)
|
| $47$ |
\( T^{12} + 9 T^{11} + \cdots + 46594276 \)
|
| $53$ |
\( (T^{6} - 16 T^{5} + \cdots + 18057)^{2} \)
|
| $59$ |
\( T^{12} - 5 T^{11} + \cdots + 5776 \)
|
| $61$ |
\( T^{12} - 6 T^{11} + \cdots + 2640625 \)
|
| $67$ |
\( T^{12} + \cdots + 5245815184 \)
|
| $71$ |
\( (T^{6} + 21 T^{5} + \cdots - 6912)^{2} \)
|
| $73$ |
\( (T^{6} + T^{5} - 257 T^{4} + \cdots - 4464)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 145660761 \)
|
| $83$ |
\( T^{12} + \cdots + 2338883044 \)
|
| $89$ |
\( (T^{6} - 11 T^{5} + \cdots - 511458)^{2} \)
|
| $97$ |
\( T^{12} - 12 T^{11} + \cdots + 62916624 \)
|
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