Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,4,Mod(649,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.649");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.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: | \(39.8262892539\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} - 38x^{5} + 650x^{4} - 2138x^{3} + 3698x^{2} - 3182x + 1369 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 2x^{6} - 38x^{5} + 650x^{4} - 2138x^{3} + 3698x^{2} - 3182x + 1369 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 350401 \nu^{7} - 394627 \nu^{6} + 600643 \nu^{5} - 12854329 \nu^{4} + 216478129 \nu^{3} + \cdots - 544140241 ) / 266653524 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18797 \nu^{7} + 43181 \nu^{6} - 3887 \nu^{5} + 711992 \nu^{4} - 12394355 \nu^{3} + \cdots + 31068308 ) / 1826394 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 30547 \nu^{7} + 8251 \nu^{6} - 697 \nu^{5} + 1161507 \nu^{4} - 17894791 \nu^{3} + \cdots + 2815963 ) / 2402284 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4400605 \nu^{7} - 9409564 \nu^{6} + 1242295 \nu^{5} - 166422865 \nu^{4} + 2892502405 \nu^{3} + \cdots - 7251437785 ) / 133326762 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 41606 \nu^{7} + 11651 \nu^{6} - 11684 \nu^{5} + 1423369 \nu^{4} - 24646604 \nu^{3} + \cdots - 31881919 ) / 1201142 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 571\nu^{7} - 151\nu^{6} - 71\nu^{5} - 21778\nu^{4} + 332359\nu^{3} - 606799\nu^{2} + 535501\nu + 14484 ) / 16454 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 124903 \nu^{7} + 189718 \nu^{6} - 158749 \nu^{5} + 4625842 \nu^{4} - 78692077 \nu^{3} + \cdots + 197632318 ) / 1826394 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + 3\beta_{3} + 3\beta_{2} + \beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{4} - 5\beta_{2} + 38\beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -25\beta_{6} + 25\beta_{4} - 69\beta_{3} + 69\beta_{2} - 95\beta _1 + 95 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{6} - 8\beta_{5} + 60\beta_{3} - 295 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 38\beta_{7} - 639\beta_{6} + 38\beta_{5} - 639\beta_{4} - 1837\beta_{3} - 1837\beta_{2} + 3741\beta _1 + 3741 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -575\beta_{7} + 1481\beta_{4} + 5635\beta_{2} - 22990\beta_1 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1826 \beta_{7} + 16911 \beta_{6} - 1826 \beta_{5} - 16911 \beta_{4} + 50851 \beta_{3} - 50851 \beta_{2} + \cdots - 123429 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 4.90965i | 0 | −16.1047 | 0 | 0 | − | 2.10469i | 39.7912i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 649.2 | − | 4.90965i | 0 | −16.1047 | 0 | 0 | 2.10469i | 39.7912i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 649.3 | − | 2.21254i | 0 | 3.10469 | 0 | 0 | − | 17.1047i | − | 24.5695i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 649.4 | − | 2.21254i | 0 | 3.10469 | 0 | 0 | 17.1047i | − | 24.5695i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 649.5 | 2.21254i | 0 | 3.10469 | 0 | 0 | − | 17.1047i | 24.5695i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 649.6 | 2.21254i | 0 | 3.10469 | 0 | 0 | 17.1047i | 24.5695i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 649.7 | 4.90965i | 0 | −16.1047 | 0 | 0 | − | 2.10469i | − | 39.7912i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 649.8 | 4.90965i | 0 | −16.1047 | 0 | 0 | 2.10469i | − | 39.7912i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 675.4.b.o | 8 | |
| 3.b | odd | 2 | 1 | inner | 675.4.b.o | 8 | |
| 5.b | even | 2 | 1 | inner | 675.4.b.o | 8 | |
| 5.c | odd | 4 | 1 | 675.4.a.w | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 675.4.a.x | yes | 4 | |
| 15.d | odd | 2 | 1 | inner | 675.4.b.o | 8 | |
| 15.e | even | 4 | 1 | 675.4.a.w | ✓ | 4 | |
| 15.e | even | 4 | 1 | 675.4.a.x | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 675.4.a.w | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 675.4.a.w | ✓ | 4 | 15.e | even | 4 | 1 | |
| 675.4.a.x | yes | 4 | 5.c | odd | 4 | 1 | |
| 675.4.a.x | yes | 4 | 15.e | even | 4 | 1 | |
| 675.4.b.o | 8 | 1.a | even | 1 | 1 | trivial | |
| 675.4.b.o | 8 | 3.b | odd | 2 | 1 | inner | |
| 675.4.b.o | 8 | 5.b | even | 2 | 1 | inner | |
| 675.4.b.o | 8 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(675, [\chi])\):
|
\( T_{2}^{4} + 29T_{2}^{2} + 118 \)
|
|
\( T_{7}^{4} + 297T_{7}^{2} + 1296 \)
|
|
\( T_{11}^{4} - 6092T_{11}^{2} + 7257472 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 29 T^{2} + 118)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 297 T^{2} + 1296)^{2} \)
|
| $11$ |
\( (T^{4} - 6092 T^{2} + 7257472)^{2} \)
|
| $13$ |
\( (T^{4} + 9101 T^{2} + 20160100)^{2} \)
|
| $17$ |
\( (T^{4} + 11468 T^{2} + 15980032)^{2} \)
|
| $19$ |
\( (T^{2} + 173 T + 5176)^{4} \)
|
| $23$ |
\( (T^{4} + 47664 T^{2} + 244684800)^{2} \)
|
| $29$ |
\( (T^{4} - 56492 T^{2} + 26288512)^{2} \)
|
| $31$ |
\( (T^{2} - 372 T + 31275)^{4} \)
|
| $37$ |
\( (T^{4} + 3674 T^{2} + 1243225)^{2} \)
|
| $41$ |
\( (T^{4} - 242348 T^{2} + 12373196800)^{2} \)
|
| $43$ |
\( (T^{4} + 69578 T^{2} + 624450121)^{2} \)
|
| $47$ |
\( (T^{4} + 162476 T^{2} + 6419834368)^{2} \)
|
| $53$ |
\( (T^{4} + 89228 T^{2} + 475783552)^{2} \)
|
| $59$ |
\( (T^{4} - 1856 T^{2} + 483328)^{2} \)
|
| $61$ |
\( (T^{2} - 89 T - 516926)^{4} \)
|
| $67$ |
\( (T^{4} + 644625 T^{2} + 102356484624)^{2} \)
|
| $71$ |
\( (T^{4} - 217376 T^{2} + 8294391808)^{2} \)
|
| $73$ |
\( (T^{4} + 137114 T^{2} + 4304016025)^{2} \)
|
| $79$ |
\( (T^{2} + 1038 T - 108495)^{4} \)
|
| $83$ |
\( (T^{4} + 64332 T^{2} + 611712)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 4761627379200)^{2} \)
|
| $97$ |
\( (T^{4} + 2407397 T^{2} + 460207564996)^{2} \)
|
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