Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1716,2,Mod(133,1716)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1716.133");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1716.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: | \(13.7023289869\) |
| 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: |
\( x^{12} + 5 x^{10} - 2 x^{9} - 239 x^{8} - 479 x^{7} + 274 x^{6} + 3168 x^{5} + 8220 x^{4} + 10431 x^{3} + \cdots + 2541 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{12} + 5 x^{10} - 2 x^{9} - 239 x^{8} - 479 x^{7} + 274 x^{6} + 3168 x^{5} + 8220 x^{4} + 10431 x^{3} + \cdots + 2541 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 94985149349627 \nu^{11} + \cdots + 27\!\cdots\!56 ) / 84\!\cdots\!99 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 433164114403101 \nu^{11} + \cdots + 10\!\cdots\!88 ) / 84\!\cdots\!99 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 44107973907793 \nu^{11} + 659328604382085 \nu^{10} - 709599759826685 \nu^{9} + \cdots + 11\!\cdots\!19 ) / 65\!\cdots\!23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 169633174 \nu^{11} - 244324498 \nu^{10} - 89415009 \nu^{9} - 1343122998 \nu^{8} + \cdots - 567784271681 ) / 158309996647 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!43 \nu^{11} + \cdots + 52\!\cdots\!44 ) / 84\!\cdots\!99 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!63 \nu^{11} + \cdots - 12\!\cdots\!17 ) / 84\!\cdots\!99 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 961526272699 \nu^{11} - 1639765168723 \nu^{10} + 5454742863948 \nu^{9} + \cdots + 209660135947339 ) / 418336085286119 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 21\!\cdots\!71 \nu^{11} - 935821235048964 \nu^{10} + \cdots + 36\!\cdots\!64 ) / 84\!\cdots\!99 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 49\!\cdots\!97 \nu^{11} + \cdots + 22\!\cdots\!01 ) / 84\!\cdots\!99 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 424113418161866 \nu^{11} + 434680725694743 \nu^{10} + \cdots - 48\!\cdots\!55 ) / 65\!\cdots\!23 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 506653379111856 \nu^{11} + 168393011690675 \nu^{10} + \cdots - 50\!\cdots\!94 ) / 76\!\cdots\!09 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{10} - 2\beta_{9} + \beta_{7} - \beta_{6} - 2\beta_{4} - \beta_{3} + \beta_{2} - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{11} - \beta_{10} - \beta_{9} + 5 \beta_{8} - \beta_{7} + 6 \beta_{6} + 2 \beta_{5} + 17 \beta_{4} + \cdots + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 18 \beta_{11} + 18 \beta_{10} - 15 \beta_{9} - 4 \beta_{8} - 28 \beta_{7} - 15 \beta_{6} - 23 \beta_{5} + \cdots + 99 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 65 \beta_{11} + 128 \beta_{10} - 118 \beta_{9} - 112 \beta_{8} + 12 \beta_{7} - 117 \beta_{6} + \cdots + 64 \)
|
| \(\nu^{6}\) | \(=\) |
\( 331 \beta_{11} - 158 \beta_{10} + 116 \beta_{9} - 181 \beta_{8} + 689 \beta_{7} + 315 \beta_{6} + \cdots - 1145 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1588 \beta_{11} - 1798 \beta_{10} + 1383 \beta_{9} + 1723 \beta_{8} + 707 \beta_{7} + 2824 \beta_{6} + \cdots - 427 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4613 \beta_{11} + 2784 \beta_{10} - 2523 \beta_{9} + 3536 \beta_{8} - 11589 \beta_{7} - 2285 \beta_{6} + \cdots + 29523 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 35013 \beta_{11} + 45624 \beta_{10} - 38909 \beta_{9} - 34436 \beta_{8} - 20544 \beta_{7} + \cdots + 58415 \)
|
| \(\nu^{10}\) | \(=\) |
\( 61813 \beta_{11} + 5395 \beta_{10} - 7181 \beta_{9} - 116717 \beta_{8} + 221297 \beta_{7} + \cdots - 443437 \)
|
| \(\nu^{11}\) | \(=\) |
\( 743949 \beta_{11} - 808406 \beta_{10} + 659960 \beta_{9} + 517514 \beta_{8} + 648263 \beta_{7} + \cdots - 1278948 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times\).
| \(n\) | \(859\) | \(925\) | \(937\) | \(1145\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 133.1 |
|
0 | 0.500000 | + | 0.866025i | 0 | −3.69369 | 0 | 0.558303 | − | 0.967009i | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 133.2 | 0 | 0.500000 | + | 0.866025i | 0 | −2.88525 | 0 | −1.35166 | + | 2.34114i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 133.3 | 0 | 0.500000 | + | 0.866025i | 0 | 0.0461116 | 0 | 0.647887 | − | 1.12217i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 133.4 | 0 | 0.500000 | + | 0.866025i | 0 | 0.492819 | 0 | −0.209940 | + | 0.363626i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 133.5 | 0 | 0.500000 | + | 0.866025i | 0 | 3.46475 | 0 | −1.60719 | + | 2.78374i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 133.6 | 0 | 0.500000 | + | 0.866025i | 0 | 3.57526 | 0 | 2.46260 | − | 4.26535i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.1 | 0 | 0.500000 | − | 0.866025i | 0 | −3.69369 | 0 | 0.558303 | + | 0.967009i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.2 | 0 | 0.500000 | − | 0.866025i | 0 | −2.88525 | 0 | −1.35166 | − | 2.34114i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.3 | 0 | 0.500000 | − | 0.866025i | 0 | 0.0461116 | 0 | 0.647887 | + | 1.12217i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.4 | 0 | 0.500000 | − | 0.866025i | 0 | 0.492819 | 0 | −0.209940 | − | 0.363626i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.5 | 0 | 0.500000 | − | 0.866025i | 0 | 3.46475 | 0 | −1.60719 | − | 2.78374i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 529.6 | 0 | 0.500000 | − | 0.866025i | 0 | 3.57526 | 0 | 2.46260 | + | 4.26535i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1716.2.q.d | ✓ | 12 |
| 13.c | even | 3 | 1 | inner | 1716.2.q.d | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1716.2.q.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 1716.2.q.d | ✓ | 12 | 13.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - T_{5}^{5} - 23T_{5}^{4} + 19T_{5}^{3} + 128T_{5}^{2} - 71T_{5} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{2} - T + 1)^{6} \)
|
| $5$ |
\( (T^{6} - T^{5} - 23 T^{4} + \cdots + 3)^{2} \)
|
| $7$ |
\( T^{12} - T^{11} + \cdots + 676 \)
|
| $11$ |
\( (T^{2} - T + 1)^{6} \)
|
| $13$ |
\( T^{12} + 4 T^{11} + \cdots + 4826809 \)
|
| $17$ |
\( T^{12} + 2 T^{11} + \cdots + 3568321 \)
|
| $19$ |
\( T^{12} + 5 T^{11} + \cdots + 7056 \)
|
| $23$ |
\( T^{12} - 11 T^{11} + \cdots + 114244 \)
|
| $29$ |
\( T^{12} - 5 T^{11} + \cdots + 1521 \)
|
| $31$ |
\( (T^{6} - 87 T^{4} + \cdots + 9018)^{2} \)
|
| $37$ |
\( T^{12} + 36 T^{10} + \cdots + 59049 \)
|
| $41$ |
\( T^{12} + \cdots + 663938289 \)
|
| $43$ |
\( T^{12} + \cdots + 756690064 \)
|
| $47$ |
\( (T^{6} + 12 T^{5} + \cdots - 34092)^{2} \)
|
| $53$ |
\( (T^{6} + 12 T^{5} + \cdots + 223)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 53719187076 \)
|
| $61$ |
\( T^{12} + \cdots + 32638396921 \)
|
| $67$ |
\( T^{12} + 5 T^{11} + \cdots + 1016064 \)
|
| $71$ |
\( T^{12} + \cdots + 11652338916 \)
|
| $73$ |
\( (T^{6} - 19 T^{5} + \cdots + 8841)^{2} \)
|
| $79$ |
\( (T^{6} - 10 T^{5} + \cdots + 18174)^{2} \)
|
| $83$ |
\( (T^{6} + 14 T^{5} + \cdots + 65082)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 2053721124 \)
|
| $97$ |
\( T^{12} + \cdots + 170349354756 \)
|
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