Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(249,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.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: | \(6.18840615665\) |
| 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: |
\( x^{10} + 12x^{8} + 48x^{6} + 73x^{4} + 34x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{10} + 12x^{8} + 48x^{6} + 73x^{4} + 34x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 48\nu^{4} + 43\nu^{2} - 1 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} + 13\nu^{7} + 61\nu^{5} + 121\nu^{3} + 77\nu ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{8} + 26\nu^{6} + 109\nu^{4} + 151\nu^{2} + 24 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{8} + 52\nu^{6} + 162\nu^{4} + 150\nu^{2} + 21 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{9} - 39\nu^{7} - 170\nu^{5} - 272\nu^{3} - 101\nu ) / 13 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} + 39\nu^{7} + 170\nu^{5} + 285\nu^{3} + 153\nu ) / 13 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8\nu^{9} + 91\nu^{7} + 332\nu^{5} + 422\nu^{3} + 122\nu ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 2\beta_{3} - 5\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{8} - 6\beta_{7} + 3\beta_{4} + 18\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{6} - 6\beta_{5} + 17\beta_{3} + 25\beta_{2} - 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{9} + 42\beta_{8} + 30\beta_{7} - 28\beta_{4} - 86\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13\beta_{6} + 30\beta_{5} - 112\beta_{3} - 128\beta_{2} + 171 \)
|
| \(\nu^{9}\) | \(=\) |
\( 13\beta_{9} - 240\beta_{8} - 145\beta_{7} + 194\beta_{4} + 427\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(652\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
− | 2.58398i | 2.24029i | −4.67698 | 0 | 5.78888 | 2.11190i | 6.91727i | −2.01891 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 249.2 | − | 2.23959i | − | 0.740841i | −3.01578 | 0 | −1.65918 | − | 3.67497i | 2.27494i | 2.45115 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 249.3 | − | 1.70816i | − | 0.930775i | −0.917797 | 0 | −1.58991 | 1.50771i | − | 1.84857i | 2.13366 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 249.4 | − | 1.14876i | − | 2.39873i | 0.680351 | 0 | −2.75556 | − | 1.07521i | − | 3.07908i | −2.75390 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 249.5 | − | 0.264183i | − | 2.96850i | 1.93021 | 0 | −0.784228 | − | 2.14598i | − | 1.03829i | −5.81200 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 249.6 | 0.264183i | 2.96850i | 1.93021 | 0 | −0.784228 | 2.14598i | 1.03829i | −5.81200 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 249.7 | 1.14876i | 2.39873i | 0.680351 | 0 | −2.75556 | 1.07521i | 3.07908i | −2.75390 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 249.8 | 1.70816i | 0.930775i | −0.917797 | 0 | −1.58991 | − | 1.50771i | 1.84857i | 2.13366 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 249.9 | 2.23959i | 0.740841i | −3.01578 | 0 | −1.65918 | 3.67497i | − | 2.27494i | 2.45115 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 249.10 | 2.58398i | − | 2.24029i | −4.67698 | 0 | 5.78888 | − | 2.11190i | − | 6.91727i | −2.01891 | 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 | 775.2.b.g | 10 | |
| 5.b | even | 2 | 1 | inner | 775.2.b.g | 10 | |
| 5.c | odd | 4 | 1 | 775.2.a.h | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 775.2.a.k | yes | 5 | |
| 15.e | even | 4 | 1 | 6975.2.a.bp | 5 | ||
| 15.e | even | 4 | 1 | 6975.2.a.by | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 775.2.a.h | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 775.2.a.k | yes | 5 | 5.c | odd | 4 | 1 | |
| 775.2.b.g | 10 | 1.a | even | 1 | 1 | trivial | |
| 775.2.b.g | 10 | 5.b | even | 2 | 1 | inner | |
| 6975.2.a.bp | 5 | 15.e | even | 4 | 1 | ||
| 6975.2.a.by | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 16T_{2}^{8} + 88T_{2}^{6} + 193T_{2}^{4} + 142T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 16 T^{8} + \cdots + 9 \)
|
| $3$ |
\( T^{10} + 21 T^{8} + \cdots + 121 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 26 T^{8} + \cdots + 729 \)
|
| $11$ |
\( (T^{5} + 2 T^{4} + \cdots - 233)^{2} \)
|
| $13$ |
\( T^{10} + 66 T^{8} + \cdots + 23409 \)
|
| $17$ |
\( T^{10} + 103 T^{8} + \cdots + 3481 \)
|
| $19$ |
\( (T^{5} + 8 T^{4} - 3 T^{3} + \cdots - 59)^{2} \)
|
| $23$ |
\( T^{10} + 138 T^{8} + \cdots + 253009 \)
|
| $29$ |
\( (T^{5} + 6 T^{4} + \cdots + 8711)^{2} \)
|
| $31$ |
\( (T + 1)^{10} \)
|
| $37$ |
\( T^{10} + 256 T^{8} + \cdots + 19088161 \)
|
| $41$ |
\( (T^{5} + 2 T^{4} + \cdots - 507)^{2} \)
|
| $43$ |
\( T^{10} + 81 T^{8} + \cdots + 289 \)
|
| $47$ |
\( T^{10} + 246 T^{8} + \cdots + 84681 \)
|
| $53$ |
\( T^{10} + 213 T^{8} + \cdots + 151321 \)
|
| $59$ |
\( (T^{5} - 4 T^{4} + \cdots - 15079)^{2} \)
|
| $61$ |
\( (T^{5} + 5 T^{4} - 29 T^{3} + \cdots - 3)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 15539866281 \)
|
| $71$ |
\( (T^{5} - 6 T^{4} + \cdots + 57279)^{2} \)
|
| $73$ |
\( T^{10} + 383 T^{8} + \cdots + 10830681 \)
|
| $79$ |
\( (T^{5} - 6 T^{4} + \cdots - 1873)^{2} \)
|
| $83$ |
\( T^{10} + 390 T^{8} + \cdots + 3932289 \)
|
| $89$ |
\( (T^{5} - 31 T^{4} + \cdots - 729)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 673973521 \)
|
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