Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,34,Mod(24,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.b (of order \(2\), degree \(1\), not 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: | \(172.457072203\) |
| 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: |
\( x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + \cdots + 34\!\cdots\!68 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{10}\cdot 5^{24}\cdot 11^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + \cdots + 34\!\cdots\!68 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 25\!\cdots\!39 \nu^{9} + \cdots + 12\!\cdots\!56 ) / 19\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 16\!\cdots\!35 \nu^{9} + \cdots + 48\!\cdots\!20 ) / 78\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 18\!\cdots\!05 \nu^{9} + \cdots + 57\!\cdots\!92 ) / 65\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!55 \nu^{9} + \cdots + 76\!\cdots\!60 ) / 26\!\cdots\!88 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 34\!\cdots\!51 \nu^{9} + \cdots - 83\!\cdots\!52 ) / 27\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 69\!\cdots\!07 \nu^{9} + \cdots - 20\!\cdots\!40 ) / 17\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 86\!\cdots\!51 \nu^{9} + \cdots + 18\!\cdots\!52 ) / 22\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15\!\cdots\!17 \nu^{9} + \cdots - 44\!\cdots\!96 ) / 17\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 89\!\cdots\!81 \nu^{9} + \cdots + 26\!\cdots\!84 ) / 17\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 154\beta_{2} - 6250\beta _1 + 10000 ) / 50000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{9} - 20\beta_{8} + 17030\beta_{6} - 1756118124\beta_{2} - 3547610\beta_1 ) / 40000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 166525 \beta_{9} + 1258900 \beta_{8} - 1925000 \beta_{7} - 774215850 \beta_{6} + \cdots - 19\!\cdots\!00 ) / 200000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 480675000 \beta_{7} + 20902314240 \beta_{5} - 528932034229 \beta_{4} + 375334179375 \beta_{3} - 26\!\cdots\!00 ) / 20000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 59152625828125 \beta_{9} - 423686085575000 \beta_{8} - 660296588887500 \beta_{7} + \cdots - 11\!\cdots\!00 ) / 100000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 17\!\cdots\!35 \beta_{9} + \cdots + 34\!\cdots\!70 \beta_1 ) / 40000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 71\!\cdots\!75 \beta_{9} + \cdots + 18\!\cdots\!00 ) / 200000 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 27\!\cdots\!00 \beta_{7} + \cdots + 77\!\cdots\!00 ) / 20000 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 21\!\cdots\!25 \beta_{9} + \cdots + 68\!\cdots\!00 ) / 100000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 138106.i | − | 1.45282e7i | −1.04832e10 | 0 | −2.00642e12 | − | 3.75584e13i | 2.61472e14i | 5.34799e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 24.2 | − | 129081.i | 6.82881e7i | −8.07187e9 | 0 | 8.81467e12 | − | 1.39338e14i | − | 6.68716e13i | 8.95800e14 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 24.3 | − | 71937.8i | 1.37573e8i | 3.41489e9 | 0 | 9.89671e12 | 1.07684e14i | − | 8.63600e14i | −1.33673e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 24.4 | − | 56118.7i | − | 5.48591e7i | 5.44063e9 | 0 | −3.07862e12 | 7.38557e13i | − | 7.87377e14i | 2.54954e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 24.5 | − | 5627.97i | − | 1.24605e8i | 8.55826e9 | 0 | −7.01272e11 | − | 7.01560e13i | − | 9.65096e13i | −9.96727e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 24.6 | 5627.97i | 1.24605e8i | 8.55826e9 | 0 | −7.01272e11 | 7.01560e13i | 9.65096e13i | −9.96727e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 24.7 | 56118.7i | 5.48591e7i | 5.44063e9 | 0 | −3.07862e12 | − | 7.38557e13i | 7.87377e14i | 2.54954e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 24.8 | 71937.8i | − | 1.37573e8i | 3.41489e9 | 0 | 9.89671e12 | − | 1.07684e14i | 8.63600e14i | −1.33673e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 24.9 | 129081.i | − | 6.82881e7i | −8.07187e9 | 0 | 8.81467e12 | 1.39338e14i | 6.68716e13i | 8.95800e14 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 24.10 | 138106.i | 1.45282e7i | −1.04832e10 | 0 | −2.00642e12 | 3.75584e13i | − | 2.61472e14i | 5.34799e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.34.b.b | 10 | |
| 5.b | even | 2 | 1 | inner | 25.34.b.b | 10 | |
| 5.c | odd | 4 | 1 | 5.34.a.a | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 25.34.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.a.a | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 25.34.a.b | 5 | 5.c | odd | 4 | 1 | ||
| 25.34.b.b | 10 | 1.a | even | 1 | 1 | trivial | |
| 25.34.b.b | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 44090984320 T_{2}^{8} + \cdots + 16\!\cdots\!24 \)
acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 16\!\cdots\!24 \)
|
| $3$ |
\( T^{10} + \cdots + 87\!\cdots\!76 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 85\!\cdots\!24 \)
|
| $11$ |
\( (T^{5} + \cdots + 62\!\cdots\!68)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 15\!\cdots\!76 \)
|
| $17$ |
\( T^{10} + \cdots + 60\!\cdots\!24 \)
|
| $19$ |
\( (T^{5} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 52\!\cdots\!76 \)
|
| $29$ |
\( (T^{5} + \cdots + 58\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots - 18\!\cdots\!32)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 19\!\cdots\!24 \)
|
| $41$ |
\( (T^{5} + \cdots + 14\!\cdots\!68)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 54\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots + 27\!\cdots\!24 \)
|
| $53$ |
\( T^{10} + \cdots + 25\!\cdots\!76 \)
|
| $59$ |
\( (T^{5} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 88\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 11\!\cdots\!24 \)
|
| $71$ |
\( (T^{5} + \cdots - 43\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 22\!\cdots\!76 \)
|
| $79$ |
\( (T^{5} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 10\!\cdots\!76 \)
|
| $89$ |
\( (T^{5} + \cdots + 20\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 18\!\cdots\!24 \)
|
| show more | |
| show less |