Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,4,Mod(199,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.199");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.c (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: | \(48.6765757547\) |
| 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: |
\( x^{10} + 43x^{8} + 631x^{6} + 3625x^{4} + 7104x^{2} + 900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 43x^{8} + 631x^{6} + 3625x^{4} + 7104x^{2} + 900 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{9} + 37\nu^{7} + 445\nu^{5} + 1891\nu^{3} + 2202\nu ) / 720 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} + 31\nu^{6} + 289\nu^{4} + 817\nu^{2} + 150 ) / 120 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{3} + 14\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{4} + 19\nu^{2} + 46 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} + 49\nu^{7} + 757\nu^{5} + 4039\nu^{3} + 6306\nu ) / 240 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} - 46\nu^{6} - 619\nu^{4} - 2242\nu^{2} - 600 ) / 60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} + 55\nu^{7} + 1093\nu^{5} + 8893\nu^{3} + 22398\nu ) / 360 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 14\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} - 19\beta_{3} + 125 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 2\beta_{7} - 21\beta_{5} + 2\beta_{2} + 216\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{8} - 44\beta_{6} - 8\beta_{4} + 323\beta_{3} - 1925 \)
|
| \(\nu^{7}\) | \(=\) |
\( -52\beta_{9} + 72\beta_{7} + 367\beta_{5} - 112\beta_{2} - 3452\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 124\beta_{8} + 786\beta_{6} + 368\beta_{4} - 5339\beta_{3} + 30753 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1034\beta_{9} - 1774\beta_{7} - 6125\beta_{5} + 3974\beta_{2} + 55876\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 4.08549i | − | 3.00000i | −8.69126 | 0 | −12.2565 | − | 7.95915i | 2.82414i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 199.2 | − | 3.98707i | 3.00000i | −7.89672 | 0 | 11.9612 | 12.5627i | − | 0.411800i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 199.3 | − | 2.51748i | − | 3.00000i | 1.66228 | 0 | −7.55245 | − | 18.4627i | − | 24.3246i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 199.4 | − | 1.98454i | 3.00000i | 4.06159 | 0 | 5.95363 | 5.59777i | − | 23.9367i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 199.5 | − | 0.368634i | − | 3.00000i | 7.86411 | 0 | −1.10590 | 26.5824i | − | 5.84806i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 199.6 | 0.368634i | 3.00000i | 7.86411 | 0 | −1.10590 | − | 26.5824i | 5.84806i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 199.7 | 1.98454i | − | 3.00000i | 4.06159 | 0 | 5.95363 | − | 5.59777i | 23.9367i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 199.8 | 2.51748i | 3.00000i | 1.66228 | 0 | −7.55245 | 18.4627i | 24.3246i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 199.9 | 3.98707i | − | 3.00000i | −7.89672 | 0 | 11.9612 | − | 12.5627i | 0.411800i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 199.10 | 4.08549i | 3.00000i | −8.69126 | 0 | −12.2565 | 7.95915i | − | 2.82414i | −9.00000 | 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 | 825.4.c.s | 10 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.s | 10 | |
| 5.c | odd | 4 | 1 | 825.4.a.x | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 825.4.a.y | yes | 5 | |
| 15.e | even | 4 | 1 | 2475.4.a.bi | 5 | ||
| 15.e | even | 4 | 1 | 2475.4.a.bj | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.4.a.x | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 825.4.a.y | yes | 5 | 5.c | odd | 4 | 1 | |
| 825.4.c.s | 10 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.s | 10 | 5.b | even | 2 | 1 | inner | |
| 2475.4.a.bi | 5 | 15.e | even | 4 | 1 | ||
| 2475.4.a.bj | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{10} + 43T_{2}^{8} + 631T_{2}^{6} + 3625T_{2}^{4} + 7104T_{2}^{2} + 900 \)
|
|
\( T_{7}^{10} + 1300T_{7}^{8} + 522294T_{7}^{6} + 78865816T_{7}^{4} + 4405598425T_{7}^{2} + 75458991204 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 43 T^{8} + 631 T^{6} + \cdots + 900 \)
|
| $3$ |
\( (T^{2} + 9)^{5} \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 1300 T^{8} + \cdots + 75458991204 \)
|
| $11$ |
\( (T + 11)^{10} \)
|
| $13$ |
\( T^{10} + 9891 T^{8} + \cdots + 25\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + 40518 T^{8} + \cdots + 89\!\cdots\!00 \)
|
| $19$ |
\( (T^{5} - 57 T^{4} - 11988 T^{3} + \cdots + 92396817)^{2} \)
|
| $23$ |
\( T^{10} + 64211 T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $29$ |
\( (T^{5} - 107 T^{4} + \cdots - 16041054816)^{2} \)
|
| $31$ |
\( (T^{5} + 295 T^{4} - 33433 T^{3} + \cdots + 9419016168)^{2} \)
|
| $37$ |
\( T^{10} + 238398 T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $41$ |
\( (T^{5} + 128 T^{4} + \cdots + 144243119304)^{2} \)
|
| $43$ |
\( T^{10} + 545979 T^{8} + \cdots + 11\!\cdots\!04 \)
|
| $47$ |
\( T^{10} + 928978 T^{8} + \cdots + 98\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + 1187744 T^{8} + \cdots + 42\!\cdots\!24 \)
|
| $59$ |
\( (T^{5} - 1152 T^{4} + \cdots + 849531990720)^{2} \)
|
| $61$ |
\( (T^{5} + 344 T^{4} + \cdots + 105472451720)^{2} \)
|
| $67$ |
\( T^{10} + 1790126 T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $71$ |
\( (T^{5} + 707 T^{4} + \cdots + 13467667404912)^{2} \)
|
| $73$ |
\( T^{10} + 2445732 T^{8} + \cdots + 79\!\cdots\!56 \)
|
| $79$ |
\( (T^{5} - 2494 T^{4} + \cdots - 57405986771584)^{2} \)
|
| $83$ |
\( T^{10} + 1803841 T^{8} + \cdots + 49\!\cdots\!56 \)
|
| $89$ |
\( (T^{5} - 2435 T^{4} + \cdots + 205516733649744)^{2} \)
|
| $97$ |
\( T^{10} + 6962517 T^{8} + \cdots + 21\!\cdots\!25 \)
|
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