Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.3");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(13.3909317403\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 55444 x^{8} + 204985 x^{7} + 2800183688 x^{6} + 11834012485 x^{5} + \cdots + 50\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 55444 x^{8} + 204985 x^{7} + 2800183688 x^{6} + 11834012485 x^{5} + \cdots + 50\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\!\cdots\!21 \nu^{9} + \cdots + 32\!\cdots\!20 ) / 32\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 46\!\cdots\!01 \nu^{9} + \cdots + 93\!\cdots\!76 ) / 35\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\!\cdots\!43 \nu^{9} + \cdots + 77\!\cdots\!04 ) / 16\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 63\!\cdots\!19 \nu^{9} + \cdots + 41\!\cdots\!84 ) / 78\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 44\!\cdots\!96 \nu^{9} + \cdots + 22\!\cdots\!48 ) / 41\!\cdots\!85 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!25 \nu^{9} + \cdots + 19\!\cdots\!36 ) / 79\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 28\!\cdots\!29 \nu^{9} + \cdots - 71\!\cdots\!96 ) / 83\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 18\!\cdots\!17 \nu^{9} + \cdots - 88\!\cdots\!96 ) / 41\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -3\beta_{9} + 6\beta_{8} - 2\beta_{7} - 3\beta_{6} + 2\beta_{5} + 6\beta_{4} + 2\beta_{3} - 22178\beta_{2} + 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -116\beta_{6} - 465\beta_{5} + 428\beta_{4} + 45215\beta_{3} - 69055 \)
|
| \(\nu^{4}\) | \(=\) |
\( 167740\beta_{9} - 316237\beta_{8} + 86651\beta_{7} + 1010806093\beta_{2} + 93122\beta _1 - 1010806093 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6823441 \beta_{9} - 24494467 \beta_{8} - 25544802 \beta_{7} + 6823441 \beta_{6} + \cdots - 2233243875 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8494502327\beta_{6} - 4196542496\beta_{5} - 15947617706\beta_{4} + 7237432338\beta_{3} + 49977237824328 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 346179720534 \beta_{9} + 1237663004082 \beta_{8} + 1292631707223 \beta_{7} + 87393353677683 \beta_{2} + \cdots - 87393353677683 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 425083659439974 \beta_{9} + 797652614869011 \beta_{8} - 209073309820553 \beta_{7} + \cdots - 420270245650738 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 17\!\cdots\!83 \beta_{6} + \cdots + 56\!\cdots\!62 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 |
|
8.00000 | + | 13.8564i | −119.661 | − | 207.259i | −128.000 | + | 221.703i | −974.315 | 1914.57 | − | 3316.14i | 1908.72 | − | 3306.00i | −4096.00 | −18796.0 | + | 32555.6i | −7794.52 | − | 13500.5i | ||||||||||||||||||||||||||||||||||
| 3.2 | 8.00000 | + | 13.8564i | −50.6451 | − | 87.7198i | −128.000 | + | 221.703i | 496.759 | 810.321 | − | 1403.52i | −2351.28 | + | 4072.54i | −4096.00 | 4711.66 | − | 8160.83i | 3974.07 | + | 6883.29i | |||||||||||||||||||||||||||||||||||
| 3.3 | 8.00000 | + | 13.8564i | 8.40820 | + | 14.5634i | −128.000 | + | 221.703i | 1992.94 | −134.531 | + | 233.015i | 422.689 | − | 732.119i | −4096.00 | 9700.10 | − | 16801.1i | 15943.5 | + | 27614.9i | |||||||||||||||||||||||||||||||||||
| 3.4 | 8.00000 | + | 13.8564i | 17.5324 | + | 30.3670i | −128.000 | + | 221.703i | −2051.83 | −280.518 | + | 485.872i | 4269.09 | − | 7394.28i | −4096.00 | 9226.73 | − | 15981.2i | −16414.6 | − | 28431.0i | |||||||||||||||||||||||||||||||||||
| 3.5 | 8.00000 | + | 13.8564i | 103.865 | + | 179.900i | −128.000 | + | 221.703i | −377.553 | −1661.85 | + | 2878.40i | −2587.72 | + | 4482.06i | −4096.00 | −11734.5 | + | 20324.8i | −3020.42 | − | 5231.52i | |||||||||||||||||||||||||||||||||||
| 9.1 | 8.00000 | − | 13.8564i | −119.661 | + | 207.259i | −128.000 | − | 221.703i | −974.315 | 1914.57 | + | 3316.14i | 1908.72 | + | 3306.00i | −4096.00 | −18796.0 | − | 32555.6i | −7794.52 | + | 13500.5i | |||||||||||||||||||||||||||||||||||
| 9.2 | 8.00000 | − | 13.8564i | −50.6451 | + | 87.7198i | −128.000 | − | 221.703i | 496.759 | 810.321 | + | 1403.52i | −2351.28 | − | 4072.54i | −4096.00 | 4711.66 | + | 8160.83i | 3974.07 | − | 6883.29i | |||||||||||||||||||||||||||||||||||
| 9.3 | 8.00000 | − | 13.8564i | 8.40820 | − | 14.5634i | −128.000 | − | 221.703i | 1992.94 | −134.531 | − | 233.015i | 422.689 | + | 732.119i | −4096.00 | 9700.10 | + | 16801.1i | 15943.5 | − | 27614.9i | |||||||||||||||||||||||||||||||||||
| 9.4 | 8.00000 | − | 13.8564i | 17.5324 | − | 30.3670i | −128.000 | − | 221.703i | −2051.83 | −280.518 | − | 485.872i | 4269.09 | + | 7394.28i | −4096.00 | 9226.73 | + | 15981.2i | −16414.6 | + | 28431.0i | |||||||||||||||||||||||||||||||||||
| 9.5 | 8.00000 | − | 13.8564i | 103.865 | − | 179.900i | −128.000 | − | 221.703i | −377.553 | −1661.85 | − | 2878.40i | −2587.72 | − | 4482.06i | −4096.00 | −11734.5 | − | 20324.8i | −3020.42 | + | 5231.52i | |||||||||||||||||||||||||||||||||||
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 | 26.10.c.a | ✓ | 10 |
| 13.c | even | 3 | 1 | inner | 26.10.c.a | ✓ | 10 |
| 13.c | even | 3 | 1 | 338.10.a.h | 5 | ||
| 13.e | even | 6 | 1 | 338.10.a.i | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 26.10.c.a | ✓ | 10 | 13.c | even | 3 | 1 | inner |
| 338.10.a.h | 5 | 13.c | even | 3 | 1 | ||
| 338.10.a.i | 5 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 81 T_{3}^{9} + 59380 T_{3}^{8} + 809655 T_{3}^{7} + 2762086216 T_{3}^{6} + \cdots + 88\!\cdots\!04 \)
acting on \(S_{10}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{5} \)
|
| $3$ |
\( T^{10} + \cdots + 88\!\cdots\!04 \)
|
| $5$ |
\( (T^{5} + \cdots + 747234600028200)^{2} \)
|
| $7$ |
\( T^{10} + \cdots + 44\!\cdots\!44 \)
|
| $11$ |
\( T^{10} + \cdots + 40\!\cdots\!44 \)
|
| $13$ |
\( T^{10} + \cdots + 13\!\cdots\!93 \)
|
| $17$ |
\( T^{10} + \cdots + 44\!\cdots\!29 \)
|
| $19$ |
\( T^{10} + \cdots + 75\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 19\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots + 32\!\cdots\!25 \)
|
| $31$ |
\( (T^{5} + \cdots + 15\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 34\!\cdots\!49 \)
|
| $41$ |
\( T^{10} + \cdots + 19\!\cdots\!29 \)
|
| $43$ |
\( T^{10} + \cdots + 15\!\cdots\!04 \)
|
| $47$ |
\( (T^{5} + \cdots - 28\!\cdots\!96)^{2} \)
|
| $53$ |
\( (T^{5} + \cdots + 11\!\cdots\!04)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 11\!\cdots\!09 \)
|
| $67$ |
\( T^{10} + \cdots + 64\!\cdots\!36 \)
|
| $71$ |
\( T^{10} + \cdots + 26\!\cdots\!84 \)
|
| $73$ |
\( (T^{5} + \cdots - 42\!\cdots\!88)^{2} \)
|
| $79$ |
\( (T^{5} + \cdots - 51\!\cdots\!64)^{2} \)
|
| $83$ |
\( (T^{5} + \cdots + 76\!\cdots\!36)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 18\!\cdots\!96 \)
|
| $97$ |
\( T^{10} + \cdots + 93\!\cdots\!00 \)
|
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