Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,10,Mod(25,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.25");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.b (of order \(2\), degree \(1\), 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: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 121651 x^{8} + 5133263811 x^{6} + 87415793974705 x^{4} + \cdots + 59\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 121651 x^{8} + 5133263811 x^{6} + 87415793974705 x^{4} + \cdots + 59\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 51991 \nu^{8} + 9285130410 \nu^{6} + 409666009386159 \nu^{4} + \cdots - 37\!\cdots\!92 ) / 16\!\cdots\!96 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5959605775 \nu^{8} + 531914418537879 \nu^{6} + \cdots - 26\!\cdots\!88 ) / 56\!\cdots\!08 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 198019383691 \nu^{8} + \cdots + 73\!\cdots\!36 ) / 17\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 970570426 \nu^{9} + \cdots + 30\!\cdots\!64 ) / 30\!\cdots\!44 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6835003 \nu^{9} - 1390826796029 \nu^{7} + \cdots - 50\!\cdots\!28 \nu ) / 10\!\cdots\!16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 22\!\cdots\!21 \nu^{9} + \cdots - 46\!\cdots\!04 \nu ) / 36\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 22\!\cdots\!21 \nu^{9} + \cdots + 10\!\cdots\!84 \nu ) / 36\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 21\!\cdots\!83 \nu^{9} + \cdots + 19\!\cdots\!88 \nu ) / 18\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11\!\cdots\!71 \nu^{9} + \cdots + 17\!\cdots\!56 \nu ) / 73\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{2} + 10\beta _1 - 24328 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5376\beta_{9} - 6144\beta_{8} - 36493\beta_{7} - 38797\beta_{6} + 240924\beta_{5} ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( 50584 \beta_{7} + 50584 \beta_{6} + 50584 \beta_{5} - 101168 \beta_{4} + 6705 \beta_{3} + \cdots + 906077158 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 352513536 \beta_{9} + 393284928 \beta_{8} + 1504352965 \beta_{7} + 1670045125 \beta_{6} - 48600782268 \beta_{5} ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2403703231 \beta_{7} - 2403703231 \beta_{6} - 2403703231 \beta_{5} + 4807406462 \beta_{4} + \cdots - 37800037144084 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 20265392348928 \beta_{9} - 21707555117184 \beta_{8} - 66000334528765 \beta_{7} + \cdots + 37\!\cdots\!72 \beta_{5} ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( 115285693212838 \beta_{7} + 115285693212838 \beta_{6} + 115285693212838 \beta_{5} + \cdots + 16\!\cdots\!42 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 10\!\cdots\!76 \beta_{9} + \cdots - 22\!\cdots\!52 \beta_{5} ) / 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
− | 16.0000i | −162.362 | −256.000 | 767.912i | 2597.79i | 2969.95i | 4096.00i | 6678.30 | 12286.6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | − | 16.0000i | −157.164 | −256.000 | − | 2410.25i | 2514.62i | − | 9056.75i | 4096.00i | 5017.43 | −38563.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | − | 16.0000i | 56.1848 | −256.000 | − | 839.868i | − | 898.956i | 627.345i | 4096.00i | −16526.3 | −13437.9 | ||||||||||||||||||||||||||||||||||||||||||||||
| 25.4 | − | 16.0000i | 103.398 | −256.000 | 646.596i | − | 1654.36i | 6855.66i | 4096.00i | −8991.91 | 10345.5 | |||||||||||||||||||||||||||||||||||||||||||||||
| 25.5 | − | 16.0000i | 240.943 | −256.000 | 622.606i | − | 3855.09i | − | 10111.2i | 4096.00i | 38370.4 | 9961.70 | ||||||||||||||||||||||||||||||||||||||||||||||
| 25.6 | 16.0000i | −162.362 | −256.000 | − | 767.912i | − | 2597.79i | − | 2969.95i | − | 4096.00i | 6678.30 | 12286.6 | |||||||||||||||||||||||||||||||||||||||||||||
| 25.7 | 16.0000i | −157.164 | −256.000 | 2410.25i | − | 2514.62i | 9056.75i | − | 4096.00i | 5017.43 | −38563.9 | |||||||||||||||||||||||||||||||||||||||||||||||
| 25.8 | 16.0000i | 56.1848 | −256.000 | 839.868i | 898.956i | − | 627.345i | − | 4096.00i | −16526.3 | −13437.9 | |||||||||||||||||||||||||||||||||||||||||||||||
| 25.9 | 16.0000i | 103.398 | −256.000 | − | 646.596i | 1654.36i | − | 6855.66i | − | 4096.00i | −8991.91 | 10345.5 | ||||||||||||||||||||||||||||||||||||||||||||||
| 25.10 | 16.0000i | 240.943 | −256.000 | − | 622.606i | 3855.09i | 10111.2i | − | 4096.00i | 38370.4 | 9961.70 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 26.10.b.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 234.10.b.a | 10 | ||
| 4.b | odd | 2 | 1 | 208.10.f.a | 10 | ||
| 13.b | even | 2 | 1 | inner | 26.10.b.a | ✓ | 10 |
| 13.d | odd | 4 | 1 | 338.10.a.g | 5 | ||
| 13.d | odd | 4 | 1 | 338.10.a.j | 5 | ||
| 39.d | odd | 2 | 1 | 234.10.b.a | 10 | ||
| 52.b | odd | 2 | 1 | 208.10.f.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.b.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 26.10.b.a | ✓ | 10 | 13.b | even | 2 | 1 | inner |
| 208.10.f.a | 10 | 4.b | odd | 2 | 1 | ||
| 208.10.f.a | 10 | 52.b | odd | 2 | 1 | ||
| 234.10.b.a | 10 | 3.b | odd | 2 | 1 | ||
| 234.10.b.a | 10 | 39.d | odd | 2 | 1 | ||
| 338.10.a.g | 5 | 13.d | odd | 4 | 1 | ||
| 338.10.a.j | 5 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 256)^{5} \)
|
| $3$ |
\( (T^{5} - 81 T^{4} + \cdots - 35717361264)^{2} \)
|
| $5$ |
\( T^{10} + \cdots + 39\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $11$ |
\( T^{10} + \cdots + 54\!\cdots\!00 \)
|
| $13$ |
\( T^{10} + \cdots + 13\!\cdots\!93 \)
|
| $17$ |
\( (T^{5} + \cdots + 44\!\cdots\!76)^{2} \)
|
| $19$ |
\( T^{10} + \cdots + 94\!\cdots\!00 \)
|
| $23$ |
\( (T^{5} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{5} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots + 48\!\cdots\!76 \)
|
| $41$ |
\( T^{10} + \cdots + 31\!\cdots\!00 \)
|
| $43$ |
\( (T^{5} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 62\!\cdots\!16 \)
|
| $53$ |
\( (T^{5} + \cdots + 84\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 38\!\cdots\!04 \)
|
| $61$ |
\( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 39\!\cdots\!64 \)
|
| $71$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $79$ |
\( (T^{5} + \cdots + 87\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 52\!\cdots\!64 \)
|
| $89$ |
\( T^{10} + \cdots + 76\!\cdots\!64 \)
|
| $97$ |
\( T^{10} + \cdots + 35\!\cdots\!76 \)
|
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