Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1672,2,Mod(1,1672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1672, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1672.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1672 = 2^{3} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1672.a (trivial) |
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: | yes |
| Analytic conductor: | \(13.3509872180\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 28 \nu^{8} + 66 \nu^{7} + 685 \nu^{6} - 1506 \nu^{5} - 5225 \nu^{4} + 9908 \nu^{3} + 12355 \nu^{2} + \cdots - 6181 ) / 555 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{8} - 48\nu^{7} - 404\nu^{6} + 954\nu^{5} + 2986\nu^{4} - 5302\nu^{3} - 6863\nu^{2} + 5868\nu + 3170 ) / 222 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{8} + 48\nu^{7} + 404\nu^{6} - 954\nu^{5} - 2986\nu^{4} + 5524\nu^{3} + 6863\nu^{2} - 7644\nu - 2948 ) / 222 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 107 \nu^{8} + 54 \nu^{7} + 2360 \nu^{6} - 1434 \nu^{5} - 16300 \nu^{4} + 10192 \nu^{3} + \cdots - 10994 ) / 1110 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 109 \nu^{8} + 138 \nu^{7} + 2290 \nu^{6} - 3048 \nu^{5} - 14810 \nu^{4} + 19304 \nu^{3} + \cdots - 13378 ) / 1110 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -8\nu^{8} + 3\nu^{7} + 164\nu^{6} - 92\nu^{5} - 996\nu^{4} + 706\nu^{3} + 1606\nu^{2} - 1005\nu - 360 ) / 74 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{5} + 10\beta_{2} - 2\beta _1 + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 14\beta_{5} + 12\beta_{4} - 3\beta_{3} + 71\beta _1 - 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 30\beta_{8} - 36\beta_{7} - 11\beta_{6} + 17\beta_{5} + \beta_{3} + 99\beta_{2} - 33\beta _1 + 335 \)
|
| \(\nu^{7}\) | \(=\) |
\( -2\beta_{8} - 12\beta_{7} + 12\beta_{6} + 162\beta_{5} + 122\beta_{4} - 56\beta_{3} - 6\beta_{2} + 655\beta _1 - 280 \)
|
| \(\nu^{8}\) | \(=\) |
\( 356 \beta_{8} - 482 \beta_{7} - 108 \beta_{6} + 212 \beta_{5} - 4 \beta_{4} + 34 \beta_{3} + 983 \beta_{2} + \cdots + 3019 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.22348 | 0 | −1.96048 | 0 | 3.83651 | 0 | 7.39085 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.93149 | 0 | 3.77651 | 0 | −2.25533 | 0 | 5.59362 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.48312 | 0 | 0.694979 | 0 | −3.89404 | 0 | −0.800348 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.438092 | 0 | 3.19099 | 0 | 3.98768 | 0 | −2.80808 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.223788 | 0 | −4.30360 | 0 | −0.878402 | 0 | −2.94992 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.21469 | 0 | −1.18050 | 0 | 1.93448 | 0 | −1.52452 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.34973 | 0 | 3.03417 | 0 | −3.94780 | 0 | 2.52125 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.61392 | 0 | 4.05097 | 0 | 3.36136 | 0 | 3.83255 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 3.12163 | 0 | −1.30304 | 0 | 0.855540 | 0 | 6.74459 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1672.2.a.k | ✓ | 9 |
| 4.b | odd | 2 | 1 | 3344.2.a.bb | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1672.2.a.k | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 3344.2.a.bb | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1672))\):
|
\( T_{3}^{9} - T_{3}^{8} - 22T_{3}^{7} + 22T_{3}^{6} + 152T_{3}^{5} - 136T_{3}^{4} - 341T_{3}^{3} + 169T_{3}^{2} + 196T_{3} + 32 \)
|
|
\( T_{5}^{9} - 6T_{5}^{8} - 20T_{5}^{7} + 162T_{5}^{6} + 16T_{5}^{5} - 1080T_{5}^{4} + 201T_{5}^{3} + 2654T_{5}^{2} + 316T_{5} - 1336 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} - T^{8} + \cdots + 32 \)
|
| $5$ |
\( T^{9} - 6 T^{8} + \cdots - 1336 \)
|
| $7$ |
\( T^{9} - 3 T^{8} + \cdots - 2592 \)
|
| $11$ |
\( (T - 1)^{9} \)
|
| $13$ |
\( T^{9} - T^{8} + \cdots - 27592 \)
|
| $17$ |
\( T^{9} - 7 T^{8} + \cdots - 752000 \)
|
| $19$ |
\( (T + 1)^{9} \)
|
| $23$ |
\( T^{9} - 13 T^{8} + \cdots - 983040 \)
|
| $29$ |
\( T^{9} - 9 T^{8} + \cdots + 7577384 \)
|
| $31$ |
\( T^{9} + 4 T^{8} + \cdots - 494096 \)
|
| $37$ |
\( T^{9} - 24 T^{8} + \cdots + 190720 \)
|
| $41$ |
\( T^{9} + 6 T^{8} + \cdots - 4555776 \)
|
| $43$ |
\( T^{9} + 14 T^{8} + \cdots + 1550336 \)
|
| $47$ |
\( T^{9} - 24 T^{8} + \cdots - 786432 \)
|
| $53$ |
\( T^{9} - 19 T^{8} + \cdots + 16384 \)
|
| $59$ |
\( T^{9} + 19 T^{8} + \cdots + 10710016 \)
|
| $61$ |
\( T^{9} - 28 T^{8} + \cdots + 21155840 \)
|
| $67$ |
\( T^{9} - 5 T^{8} + \cdots + 34237440 \)
|
| $71$ |
\( T^{9} - 16 T^{8} + \cdots - 22501488 \)
|
| $73$ |
\( T^{9} - 15 T^{8} + \cdots + 38400 \)
|
| $79$ |
\( T^{9} - 2 T^{8} + \cdots + 1196032 \)
|
| $83$ |
\( T^{9} - 8 T^{8} + \cdots + 14477312 \)
|
| $89$ |
\( T^{9} - 12 T^{8} + \cdots + 307200 \)
|
| $97$ |
\( T^{9} + 4 T^{8} + \cdots - 113837312 \)
|
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