Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,16,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, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| 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: | \(35.6733762750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 134x^{4} - 34436x^{3} + 232498x^{2} - 6420444x + 303625485 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 5^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 134x^{4} - 34436x^{3} + 232498x^{2} - 6420444x + 303625485 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 59879 \nu^{5} + 586373 \nu^{4} + 16900633 \nu^{3} - 1022968807 \nu^{2} + 1148021519 \nu - 349012105485 ) / 7048834119 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 95840092 \nu^{5} - 3962701294 \nu^{4} - 123095433974 \nu^{3} + 1463286732596 \nu^{2} + \cdots + 18\!\cdots\!30 ) / 5110404736275 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 187792 \nu^{5} - 8441956 \nu^{4} + 227228524 \nu^{3} - 6694775096 \nu^{2} + \cdots - 5312807364180 ) / 5024980075 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 39734941 \nu^{5} - 619340933 \nu^{4} + 2528243447 \nu^{3} - 1723536772253 \nu^{2} + \cdots - 248663563097475 ) / 1022080947255 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 143984 \nu^{5} + 7227588 \nu^{4} - 103208452 \nu^{3} - 6506468192 \nu^{2} + \cdots + 2105151920135 ) / 1004996015 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} + 15\beta_{4} - 10\beta_{3} - 10\beta_{2} - 13\beta _1 + 266 ) / 800 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -78\beta_{5} - 375\beta_{4} - 170\beta_{3} + 450\beta_{2} + 4773\beta _1 - 35226 ) / 800 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -424\beta_{5} + 405\beta_{4} + 2530\beta_{3} - 13570\beta_{2} - 35791\beta _1 + 6833992 ) / 400 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2242\beta_{5} + 248325\beta_{4} - 319230\beta_{3} - 684150\beta_{2} - 1204623\beta _1 - 82763414 ) / 800 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1032902 \beta_{5} - 9354435 \beta_{4} - 1014610 \beta_{3} + 22239290 \beta_{2} + 207965417 \beta _1 + 1008731966 ) / 800 \)
|
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 |
|
− | 282.188i | 5750.55i | −46862.2 | 0 | 1.62274e6 | 3.32761e6i | 3.97721e6i | −1.87200e7 | 0 | |||||||||||||||||||||||||||||||||||
| 24.2 | − | 152.575i | 6379.22i | 9488.76 | 0 | 973313. | − | 1.77538e6i | − | 6.44734e6i | −2.63456e7 | 0 | ||||||||||||||||||||||||||||||||||
| 24.3 | − | 125.613i | − | 4146.67i | 16989.4 | 0 | −520875. | 4.19779e6i | − | 6.25017e6i | −2.84597e6 | 0 | ||||||||||||||||||||||||||||||||||
| 24.4 | 125.613i | 4146.67i | 16989.4 | 0 | −520875. | − | 4.19779e6i | 6.25017e6i | −2.84597e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 24.5 | 152.575i | − | 6379.22i | 9488.76 | 0 | 973313. | 1.77538e6i | 6.44734e6i | −2.63456e7 | 0 | ||||||||||||||||||||||||||||||||||||
| 24.6 | 282.188i | − | 5750.55i | −46862.2 | 0 | 1.62274e6 | − | 3.32761e6i | − | 3.97721e6i | −1.87200e7 | 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 | 25.16.b.c | 6 | |
| 5.b | even | 2 | 1 | inner | 25.16.b.c | 6 | |
| 5.c | odd | 4 | 1 | 5.16.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 25.16.a.c | 3 | ||
| 15.e | even | 4 | 1 | 45.16.a.f | 3 | ||
| 20.e | even | 4 | 1 | 80.16.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.16.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 25.16.a.c | 3 | 5.c | odd | 4 | 1 | ||
| 25.16.b.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.16.b.c | 6 | 5.b | even | 2 | 1 | inner | |
| 45.16.a.f | 3 | 15.e | even | 4 | 1 | ||
| 80.16.a.g | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 118688T_{2}^{4} + 3477494848T_{2}^{2} + 29249232961536 \)
acting on \(S_{16}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 29249232961536 \)
|
| $3$ |
\( T^{6} + \cdots + 23\!\cdots\!04 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 61\!\cdots\!56 \)
|
| $11$ |
\( (T^{3} + \cdots + 92\!\cdots\!92)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 91\!\cdots\!64 \)
|
| $17$ |
\( T^{6} + \cdots + 43\!\cdots\!96 \)
|
| $19$ |
\( (T^{3} + \cdots - 46\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 77\!\cdots\!24 \)
|
| $29$ |
\( (T^{3} + \cdots - 75\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots - 90\!\cdots\!08)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 50\!\cdots\!76 \)
|
| $41$ |
\( (T^{3} + \cdots - 22\!\cdots\!08)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 68\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 44\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots + 20\!\cdots\!04 \)
|
| $59$ |
\( (T^{3} + \cdots - 44\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 18\!\cdots\!08)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 95\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 13\!\cdots\!92)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 55\!\cdots\!24 \)
|
| $79$ |
\( (T^{3} + \cdots + 65\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!84 \)
|
| $89$ |
\( (T^{3} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 45\!\cdots\!16 \)
|
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