Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,3,Mod(193,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.p (of order \(4\), 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: | \(21.7984211488\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.151613669376.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 7x^{4} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 7x^{4} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 29\nu ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 9\nu^{2} ) / 80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 11\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 16\nu^{5} - 64\nu^{4} - 71\nu^{3} + 176\nu + 224 ) / 160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 10\nu^{6} - 16\nu^{5} + 71\nu^{3} + 230\nu^{2} + 176\nu ) / 160 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 71\nu^{3} ) / 64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9\nu^{7} + \nu^{3} ) / 320 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} + 5\beta_{5} - 2\beta_{3} + 25\beta_{2} ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} + 9\beta_{6} ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{6} - 5\beta_{4} + 2\beta_{3} + 7 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -29\beta_{3} + 55\beta_1 ) / 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 18\beta_{6} - 45\beta_{5} + 18\beta_{3} + 575\beta_{2} ) / 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 355\beta_{7} - \beta_{6} ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −1.54951 | − | 1.54951i | 0 | 0 | 0 | −2.00000 | + | 2.00000i | 0 | − | 4.19804i | 0 | |||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −1.54951 | − | 1.54951i | 0 | 0 | 0 | −2.00000 | + | 2.00000i | 0 | − | 4.19804i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 3.54951 | + | 3.54951i | 0 | 0 | 0 | −2.00000 | + | 2.00000i | 0 | 16.1980i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 3.54951 | + | 3.54951i | 0 | 0 | 0 | −2.00000 | + | 2.00000i | 0 | 16.1980i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | −1.54951 | + | 1.54951i | 0 | 0 | 0 | −2.00000 | − | 2.00000i | 0 | 4.19804i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −1.54951 | + | 1.54951i | 0 | 0 | 0 | −2.00000 | − | 2.00000i | 0 | 4.19804i | 0 | |||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 3.54951 | − | 3.54951i | 0 | 0 | 0 | −2.00000 | − | 2.00000i | 0 | − | 16.1980i | 0 | ||||||||||||||||||||||||||||||||||||||
| 257.4 | 0 | 3.54951 | − | 3.54951i | 0 | 0 | 0 | −2.00000 | − | 2.00000i | 0 | − | 16.1980i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 20.d | odd | 2 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 800.3.p.n | yes | 8 |
| 4.b | odd | 2 | 1 | 800.3.p.k | ✓ | 8 | |
| 5.b | even | 2 | 1 | 800.3.p.k | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 800.3.p.k | ✓ | 8 | |
| 5.c | odd | 4 | 1 | inner | 800.3.p.n | yes | 8 |
| 20.d | odd | 2 | 1 | inner | 800.3.p.n | yes | 8 |
| 20.e | even | 4 | 1 | 800.3.p.k | ✓ | 8 | |
| 20.e | even | 4 | 1 | inner | 800.3.p.n | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.3.p.k | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 800.3.p.k | ✓ | 8 | 5.b | even | 2 | 1 | |
| 800.3.p.k | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 800.3.p.k | ✓ | 8 | 20.e | even | 4 | 1 | |
| 800.3.p.n | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 800.3.p.n | yes | 8 | 5.c | odd | 4 | 1 | inner |
| 800.3.p.n | yes | 8 | 20.d | odd | 2 | 1 | inner |
| 800.3.p.n | yes | 8 | 20.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(800, [\chi])\):
|
\( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 44T_{3} + 121 \)
|
|
\( T_{13}^{8} + 239904T_{13}^{4} + 8100000000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 4 T^{3} + \cdots + 121)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{2} + 4 T + 8)^{4} \)
|
| $11$ |
\( (T^{4} - 186 T^{2} + 225)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 8100000000 \)
|
| $17$ |
\( T^{8} + 788274 T^{4} + 741200625 \)
|
| $19$ |
\( (T^{4} + 666 T^{2} + 65025)^{2} \)
|
| $23$ |
\( (T^{4} + 40 T^{3} + \cdots + 71824)^{2} \)
|
| $29$ |
\( (T^{4} + 1880 T^{2} + 868624)^{2} \)
|
| $31$ |
\( (T^{4} - 2856 T^{2} + 1904400)^{2} \)
|
| $37$ |
\( T^{8} + 987264 T^{4} + 207360000 \)
|
| $41$ |
\( (T + 15)^{8} \)
|
| $43$ |
\( (T^{4} + 8 T^{3} + \cdots + 3474496)^{2} \)
|
| $47$ |
\( (T^{4} + 160 T^{3} + \cdots + 7463824)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 24289126560000 \)
|
| $59$ |
\( (T^{4} + 12600 T^{2} + 2250000)^{2} \)
|
| $61$ |
\( (T^{2} - 72 T + 360)^{4} \)
|
| $67$ |
\( (T^{4} + 236 T^{3} + \cdots + 34916281)^{2} \)
|
| $71$ |
\( (T^{4} - 7224 T^{2} + 10890000)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $79$ |
\( (T^{4} + 18576 T^{2} + 64641600)^{2} \)
|
| $83$ |
\( (T^{4} - 4 T^{3} + \cdots + 13225)^{2} \)
|
| $89$ |
\( (T^{4} + 19298 T^{2} + 51825601)^{2} \)
|
| $97$ |
\( (T^{4} + 116640000)^{2} \)
|
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