Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,3,Mod(323,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.323");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 414.c (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: | \(11.2806829445\) |
| 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: |
\( x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 373 \nu^{7} - 477 \nu^{6} - 4990 \nu^{5} + 11918 \nu^{4} - 51735 \nu^{3} + 94493 \nu^{2} + \cdots + 1216458 ) / 565056 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{7} - 27\nu^{6} + 14\nu^{5} - 142\nu^{4} + 951\nu^{3} + 11\nu^{2} - 4962\nu + 8262 ) / 5184 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 77\nu^{7} + 180\nu^{6} + 2282\nu^{5} - 88\nu^{4} + 18591\nu^{3} - 57976\nu^{2} + 134574\nu - 379080 ) / 70632 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + 22\nu^{5} - 110\nu^{4} + 411\nu^{3} - 1421\nu^{2} + 2406\nu - 8586 ) / 864 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -59\nu^{7} - 363\nu^{6} - 1154\nu^{5} - 3950\nu^{4} + 1383\nu^{3} - 23045\nu^{2} + 156990\nu - 59562 ) / 47088 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} + 3\nu^{6} - 62\nu^{5} + 238\nu^{4} - 423\nu^{3} + 2269\nu^{2} - 2238\nu + 6858 ) / 1728 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1901 \nu^{7} - 1251 \nu^{6} + 40286 \nu^{5} - 105598 \nu^{4} + 460815 \nu^{3} - 1151821 \nu^{2} + \cdots - 6900714 ) / 565056 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} - 5\beta_{3} - 17 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} + 12\beta_{6} - 5\beta_{5} + 13\beta_{4} - 5\beta_{3} + 6\beta_{2} - 30\beta _1 + 79 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -32\beta_{7} - 2\beta_{6} + 23\beta_{5} - 15\beta_{4} + 87\beta_{3} + 36\beta_{2} - 28\beta _1 - 33 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 210\beta_{7} - 24\beta_{6} - 59\beta_{5} - 145\beta_{4} - 71\beta_{3} - 78\beta_{2} + 802\beta _1 - 839 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -212\beta_{7} + 206\beta_{6} - 295\beta_{5} + 531\beta_{4} - 395\beta_{3} - 432\beta_{2} - 1776\beta _1 + 3889 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2270 \beta_{7} - 1164 \beta_{6} + 1171 \beta_{5} - 611 \beta_{4} + 2491 \beta_{3} + 1914 \beta_{2} + \cdots + 1807 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(235\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 323.1 |
|
− | 1.41421i | 0 | −2.00000 | − | 6.87676i | 0 | −5.43723 | 2.82843i | 0 | −9.72521 | ||||||||||||||||||||||||||||||||||||||||
| 323.2 | − | 1.41421i | 0 | −2.00000 | − | 4.01397i | 0 | 13.7953 | 2.82843i | 0 | −5.67661 | |||||||||||||||||||||||||||||||||||||||||
| 323.3 | − | 1.41421i | 0 | −2.00000 | 2.59976i | 0 | −3.01296 | 2.82843i | 0 | 3.67661 | ||||||||||||||||||||||||||||||||||||||||||
| 323.4 | − | 1.41421i | 0 | −2.00000 | 5.46255i | 0 | 2.65490 | 2.82843i | 0 | 7.72521 | ||||||||||||||||||||||||||||||||||||||||||
| 323.5 | 1.41421i | 0 | −2.00000 | − | 5.46255i | 0 | 2.65490 | − | 2.82843i | 0 | 7.72521 | |||||||||||||||||||||||||||||||||||||||||
| 323.6 | 1.41421i | 0 | −2.00000 | − | 2.59976i | 0 | −3.01296 | − | 2.82843i | 0 | 3.67661 | |||||||||||||||||||||||||||||||||||||||||
| 323.7 | 1.41421i | 0 | −2.00000 | 4.01397i | 0 | 13.7953 | − | 2.82843i | 0 | −5.67661 | ||||||||||||||||||||||||||||||||||||||||||
| 323.8 | 1.41421i | 0 | −2.00000 | 6.87676i | 0 | −5.43723 | − | 2.82843i | 0 | −9.72521 | ||||||||||||||||||||||||||||||||||||||||||
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 | 414.3.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 414.3.c.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | 3312.3.g.b | 8 | ||
| 12.b | even | 2 | 1 | 3312.3.g.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 414.3.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 414.3.c.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 3312.3.g.b | 8 | 4.b | odd | 2 | 1 | ||
| 3312.3.g.b | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 100T_{5}^{6} + 3284T_{5}^{4} + 40672T_{5}^{2} + 153664 \)
acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 100 T^{6} + \cdots + 153664 \)
|
| $7$ |
\( (T^{4} - 8 T^{3} + \cdots + 600)^{2} \)
|
| $11$ |
\( T^{8} + 472 T^{6} + \cdots + 5760000 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 18496)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 1121982016 \)
|
| $19$ |
\( (T^{4} + 24 T^{3} + \cdots - 195752)^{2} \)
|
| $23$ |
\( (T^{2} + 23)^{4} \)
|
| $29$ |
\( T^{8} + 3048 T^{6} + \cdots + 676624144 \)
|
| $31$ |
\( (T^{4} + 16 T^{3} + \cdots - 8064)^{2} \)
|
| $37$ |
\( (T^{4} - 16 T^{3} + \cdots + 203152)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 124832195856 \)
|
| $43$ |
\( (T^{4} - 16 T^{3} + \cdots + 1054936)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 5017886724096 \)
|
| $53$ |
\( T^{8} + \cdots + 1459979223616 \)
|
| $59$ |
\( T^{8} + \cdots + 196109327241216 \)
|
| $61$ |
\( (T^{4} - 24 T^{3} + \cdots - 421360)^{2} \)
|
| $67$ |
\( (T^{4} + 8 T^{3} + \cdots + 7536600)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 309201557520384 \)
|
| $73$ |
\( (T^{4} + 216 T^{3} + \cdots - 3423744)^{2} \)
|
| $79$ |
\( (T^{4} + 208 T^{3} + \cdots + 17983128)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 141567353611264 \)
|
| $89$ |
\( T^{8} + \cdots + 594285713856576 \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots - 1242528)^{2} \)
|
| show more | |
| show less |