Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,8,Mod(3,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.3");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.c (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: | \(8.12201066259\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 4654\nu^{5} + 6213649\nu^{3} + 1905097728\nu - 3143761920 ) / 6287523840 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 2327\nu^{2} + 11808\nu + 798720 ) / 7872 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 2327\nu^{2} + 11808\nu - 798720 ) / 7872 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 3766\nu^{4} - 3863881\nu^{2} - 854819328 ) / 661248 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 4018\nu^{4} + 4780909\nu^{2} + 1440612480 ) / 330624 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 381 \nu^{7} - 13312 \nu^{6} - 1613430 \nu^{5} - 53487616 \nu^{4} - 2152864077 \nu^{3} + \cdots - 19177433333760 ) / 8802533376 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 625 \nu^{7} - 6656 \nu^{6} - 2429518 \nu^{5} - 25066496 \nu^{4} - 2689763713 \nu^{3} + \cdots - 5685276180480 ) / 8802533376 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta_{2} - 1163 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 48\beta_{7} - 144\beta_{6} - 72\beta_{5} - 24\beta_{4} - 1655\beta_{3} - 1655\beta_{2} - 17760\beta _1 - 8904 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2327\beta_{5} - 4654\beta_{4} - 10917\beta_{3} + 10917\beta_{2} + 1907581 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64464 \beta_{7} + 358704 \beta_{6} + 179352 \beta_{5} + 32232 \beta_{4} + 3087889 \beta_{3} + \cdots + 34452312 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4899601\beta_{5} + 9137954\beta_{4} + 29521779\beta_{3} - 29521779\beta_{2} - 3545075771 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1760304 \beta_{7} - 774642960 \beta_{6} - 387321480 \beta_{5} - 880152 \beta_{4} - 5992544039 \beta_{3} + \cdots - 95583443592 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
4.00000 | + | 6.92820i | −39.6356 | − | 68.6509i | −32.0000 | + | 55.4256i | 132.638 | 317.085 | − | 549.207i | −754.459 | + | 1306.76i | −512.000 | −2048.46 | + | 3548.04i | 530.551 | + | 918.941i | ||||||||||||||||||||||||||||
| 3.2 | 4.00000 | + | 6.92820i | −15.9358 | − | 27.6017i | −32.0000 | + | 55.4256i | −54.4265 | 127.487 | − | 220.813i | 556.354 | − | 963.633i | −512.000 | 585.599 | − | 1014.29i | −217.706 | − | 377.078i | |||||||||||||||||||||||||||||
| 3.3 | 4.00000 | + | 6.92820i | 19.9922 | + | 34.6275i | −32.0000 | + | 55.4256i | −323.700 | −159.938 | + | 277.020i | −284.296 | + | 492.415i | −512.000 | 294.123 | − | 509.436i | −1294.80 | − | 2242.66i | |||||||||||||||||||||||||||||
| 3.4 | 4.00000 | + | 6.92820i | 35.5792 | + | 61.6250i | −32.0000 | + | 55.4256i | 523.489 | −284.634 | + | 493.000i | 208.401 | − | 360.960i | −512.000 | −1438.26 | + | 2491.14i | 2093.96 | + | 3626.84i | |||||||||||||||||||||||||||||
| 9.1 | 4.00000 | − | 6.92820i | −39.6356 | + | 68.6509i | −32.0000 | − | 55.4256i | 132.638 | 317.085 | + | 549.207i | −754.459 | − | 1306.76i | −512.000 | −2048.46 | − | 3548.04i | 530.551 | − | 918.941i | |||||||||||||||||||||||||||||
| 9.2 | 4.00000 | − | 6.92820i | −15.9358 | + | 27.6017i | −32.0000 | − | 55.4256i | −54.4265 | 127.487 | + | 220.813i | 556.354 | + | 963.633i | −512.000 | 585.599 | + | 1014.29i | −217.706 | + | 377.078i | |||||||||||||||||||||||||||||
| 9.3 | 4.00000 | − | 6.92820i | 19.9922 | − | 34.6275i | −32.0000 | − | 55.4256i | −323.700 | −159.938 | − | 277.020i | −284.296 | − | 492.415i | −512.000 | 294.123 | + | 509.436i | −1294.80 | + | 2242.66i | |||||||||||||||||||||||||||||
| 9.4 | 4.00000 | − | 6.92820i | 35.5792 | − | 61.6250i | −32.0000 | − | 55.4256i | 523.489 | −284.634 | − | 493.000i | 208.401 | + | 360.960i | −512.000 | −1438.26 | − | 2491.14i | 2093.96 | − | 3626.84i | |||||||||||||||||||||||||||||
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 | 26.8.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 234.8.h.b | 8 | ||
| 4.b | odd | 2 | 1 | 208.8.i.b | 8 | ||
| 13.c | even | 3 | 1 | inner | 26.8.c.b | ✓ | 8 |
| 13.c | even | 3 | 1 | 338.8.a.i | 4 | ||
| 13.e | even | 6 | 1 | 338.8.a.j | 4 | ||
| 13.f | odd | 12 | 2 | 338.8.b.h | 8 | ||
| 39.i | odd | 6 | 1 | 234.8.h.b | 8 | ||
| 52.j | odd | 6 | 1 | 208.8.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.8.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 26.8.c.b | ✓ | 8 | 13.c | even | 3 | 1 | inner |
| 208.8.i.b | 8 | 4.b | odd | 2 | 1 | ||
| 208.8.i.b | 8 | 52.j | odd | 6 | 1 | ||
| 234.8.h.b | 8 | 3.b | odd | 2 | 1 | ||
| 234.8.h.b | 8 | 39.i | odd | 6 | 1 | ||
| 338.8.a.i | 4 | 13.c | even | 3 | 1 | ||
| 338.8.a.j | 4 | 13.e | even | 6 | 1 | ||
| 338.8.b.h | 8 | 13.f | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 6981 T_{3}^{6} - 70848 T_{3}^{5} + 41545881 T_{3}^{4} - 247294944 T_{3}^{3} + \cdots + 51674244710400 \)
acting on \(S_{8}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 51674244710400 \)
|
| $5$ |
\( (T^{4} - 278 T^{3} + \cdots + 1223288100)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots + 28\!\cdots\!36 \)
|
| $13$ |
\( T^{8} + \cdots + 15\!\cdots\!21 \)
|
| $17$ |
\( T^{8} + \cdots + 35\!\cdots\!41 \)
|
| $19$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 64\!\cdots\!36 \)
|
| $29$ |
\( T^{8} + \cdots + 65\!\cdots\!69 \)
|
| $31$ |
\( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 27\!\cdots\!41 \)
|
| $41$ |
\( T^{8} + \cdots + 75\!\cdots\!25 \)
|
| $43$ |
\( T^{8} + \cdots + 19\!\cdots\!96 \)
|
| $47$ |
\( (T^{4} + \cdots + 87\!\cdots\!28)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 10\!\cdots\!32)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 87\!\cdots\!64 \)
|
| $61$ |
\( T^{8} + \cdots + 36\!\cdots\!25 \)
|
| $67$ |
\( T^{8} + \cdots + 16\!\cdots\!96 \)
|
| $71$ |
\( T^{8} + \cdots + 58\!\cdots\!44 \)
|
| $73$ |
\( (T^{4} + \cdots - 21\!\cdots\!40)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 46\!\cdots\!08)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots - 67\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 27\!\cdots\!36 \)
|
| $97$ |
\( T^{8} + \cdots + 23\!\cdots\!84 \)
|
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