Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,25,Mod(2,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 25, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 25);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 25 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(10.9490145677\) |
| 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: |
\( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{24}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 ) / 1041302528 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} + \cdots - 624717182509056 ) / 520651264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + \cdots + 14\!\cdots\!32 ) / 520651264 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + \cdots + 96\!\cdots\!28 ) / 1041302528 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 8\beta_{3} - 17\beta_{2} + 8\beta _1 - 27335016 ) / 324 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -736\beta_{5} + 103\beta_{4} + 3393\beta_{3} - 21256\beta_{2} - 18690797\beta_1 ) / 2916 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -250237\beta_{5} + 18272248\beta_{3} + 36794733\beta_{2} - 18272248\beta _1 + 42588282369096 ) / 4374 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 103238176\beta_{5} - 8788481\beta_{4} - 481592855\beta_{3} + 3026837432\beta_{2} + 1194462231243\beta_1 ) / 1458 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
− | 6843.54i | 162008. | − | 506145.i | −3.00569e7 | 3.79015e8i | −3.46383e9 | − | 1.10871e9i | −1.14171e10 | 9.08798e10i | −2.29937e11 | − | 1.63999e11i | 2.59380e12 | |||||||||||||||||||||||||||||
| 2.2 | − | 5372.56i | −24323.9 | + | 530884.i | −1.20872e7 | − | 5.28531e7i | 2.85221e9 | + | 1.30681e8i | 2.07202e10 | − | 2.51975e10i | −2.81246e11 | − | 2.58263e10i | −2.83957e11 | ||||||||||||||||||||||||||||
| 2.3 | − | 2511.29i | −446105. | − | 288825.i | 1.04706e7 | − | 3.68111e8i | −7.25325e8 | + | 1.12030e9i | −8.30906e9 | − | 6.84273e10i | 1.15589e11 | + | 2.57693e11i | −9.24434e11 | ||||||||||||||||||||||||||||
| 2.4 | 2511.29i | −446105. | + | 288825.i | 1.04706e7 | 3.68111e8i | −7.25325e8 | − | 1.12030e9i | −8.30906e9 | 6.84273e10i | 1.15589e11 | − | 2.57693e11i | −9.24434e11 | |||||||||||||||||||||||||||||||
| 2.5 | 5372.56i | −24323.9 | − | 530884.i | −1.20872e7 | 5.28531e7i | 2.85221e9 | − | 1.30681e8i | 2.07202e10 | 2.51975e10i | −2.81246e11 | + | 2.58263e10i | −2.83957e11 | |||||||||||||||||||||||||||||||
| 2.6 | 6843.54i | 162008. | + | 506145.i | −3.00569e7 | − | 3.79015e8i | −3.46383e9 | + | 1.10871e9i | −1.14171e10 | − | 9.08798e10i | −2.29937e11 | + | 1.63999e11i | 2.59380e12 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3.25.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 3.25.b.b | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.25.e.b | 6 | ||
| 12.b | even | 2 | 1 | 48.25.e.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.25.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3.25.b.b | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.25.e.b | 6 | 4.b | odd | 2 | 1 | ||
| 48.25.e.b | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 82005048T_{2}^{4} + 1829235783453696T_{2}^{2} + 8525473984011546132480 \)
acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 85\!\cdots\!80 \)
|
| $3$ |
\( T^{6} + \cdots + 22\!\cdots\!41 \)
|
| $5$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 73\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 54\!\cdots\!20 \)
|
| $19$ |
\( (T^{3} + \cdots + 24\!\cdots\!72)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!80 \)
|
| $29$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 26\!\cdots\!72)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 85\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 31\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 15\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots + 53\!\cdots\!20 \)
|
| $59$ |
\( T^{6} + \cdots + 77\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 18\!\cdots\!88)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 25\!\cdots\!72)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 29\!\cdots\!20 \)
|
| $89$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 61\!\cdots\!00)^{2} \)
|
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