Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,6,Mod(1,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 161.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: | \(25.8217949899\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 2 x^{9} - 187 x^{8} + 322 x^{7} + 11471 x^{6} - 16782 x^{5} - 253209 x^{4} + 251398 x^{3} + \cdots - 6912 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} - 187 x^{8} + 322 x^{7} + 11471 x^{6} - 16782 x^{5} - 253209 x^{4} + 251398 x^{3} + \cdots - 6912 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6451 \nu^{9} + 58176 \nu^{8} + 1154512 \nu^{7} - 9806446 \nu^{6} - 67994593 \nu^{5} + \cdots - 3726983424 ) / 397220992 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6451 \nu^{9} + 58176 \nu^{8} + 1154512 \nu^{7} - 9806446 \nu^{6} - 67994593 \nu^{5} + \cdots + 11367414272 ) / 397220992 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 72137 \nu^{9} - 170929 \nu^{8} - 13773110 \nu^{7} + 29208798 \nu^{6} + 870840937 \nu^{5} + \cdots + 19465604352 ) / 794441984 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 51388 \nu^{9} + 99265 \nu^{8} + 9288605 \nu^{7} - 14972830 \nu^{6} - 541463766 \nu^{5} + \cdots - 19576285952 ) / 397220992 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 69053 \nu^{9} + 229226 \nu^{8} + 12477947 \nu^{7} - 39418964 \nu^{6} - 718081783 \nu^{5} + \cdots + 20002810112 ) / 397220992 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 164671 \nu^{9} - 448351 \nu^{8} - 30252766 \nu^{7} + 74116778 \nu^{6} + 1810117951 \nu^{5} + \cdots + 51831550208 ) / 794441984 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 189801 \nu^{9} - 659583 \nu^{8} - 35680552 \nu^{7} + 111665674 \nu^{6} + 2198985269 \nu^{5} + \cdots + 82638427904 ) / 794441984 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 37087 \nu^{9} + 94890 \nu^{8} + 6763362 \nu^{7} - 15968902 \nu^{6} - 394953521 \nu^{5} + \cdots + 1055465440 ) / 99305248 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + 2\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + 2\beta_{2} + 65\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + 4\beta_{8} + 8\beta_{7} + 4\beta_{5} - 12\beta_{4} - 73\beta_{3} + 89\beta_{2} + 97\beta _1 + 2456 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 69 \beta_{9} - 12 \beta_{8} + 254 \beta_{7} + 174 \beta_{6} + 210 \beta_{5} - 194 \beta_{4} + \cdots + 943 \)
|
| \(\nu^{6}\) | \(=\) |
\( 366 \beta_{9} + 400 \beta_{8} + 1204 \beta_{7} - 252 \beta_{6} + 692 \beta_{5} - 1380 \beta_{4} + \cdots + 178092 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3477 \beta_{9} - 2104 \beta_{8} + 26018 \beta_{7} + 12306 \beta_{6} + 18650 \beta_{5} - 17002 \beta_{4} + \cdots + 171775 \)
|
| \(\nu^{8}\) | \(=\) |
\( 43794 \beta_{9} + 31148 \beta_{8} + 139952 \beta_{7} - 40664 \beta_{6} + 85876 \beta_{5} + \cdots + 13665144 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 107733 \beta_{9} - 252756 \beta_{8} + 2487014 \beta_{7} + 811814 \beta_{6} + 1598338 \beta_{5} + \cdots + 22834463 \)
|
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 |
|
−9.69455 | −4.68259 | 61.9843 | 64.1472 | 45.3956 | 49.0000 | −290.685 | −221.073 | −621.878 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.25569 | 27.4656 | 53.6679 | −15.1756 | −254.213 | 49.0000 | −200.551 | 511.357 | 140.460 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.68327 | 5.69706 | 12.6661 | −77.3301 | −38.0750 | 49.0000 | 129.213 | −210.543 | 516.818 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.43677 | 5.83141 | −26.0621 | 50.4902 | −14.2098 | 49.0000 | 141.484 | −208.995 | −123.033 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.96617 | −16.3242 | −28.1342 | −30.8090 | 32.0962 | 49.0000 | 118.234 | 23.4806 | 60.5757 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.992978 | −28.5410 | −31.0140 | −30.0589 | 28.3406 | 49.0000 | 62.5715 | 571.590 | 29.8478 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.49579 | 14.3986 | −19.7795 | 6.33528 | 50.3346 | 49.0000 | −181.010 | −35.6792 | 22.1468 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.18996 | −14.3126 | −14.4443 | 53.9160 | −59.9691 | 49.0000 | −194.599 | −38.1500 | 225.906 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 6.83480 | 14.1878 | 14.7145 | −89.0298 | 96.9708 | 49.0000 | −118.143 | −41.7064 | −608.501 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.50889 | −5.72003 | 40.4012 | −24.4853 | −48.6711 | 49.0000 | 71.4849 | −210.281 | −208.342 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 161.6.a.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.6.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 8 T_{2}^{9} - 160 T_{2}^{8} - 1126 T_{2}^{7} + 8531 T_{2}^{6} + 48334 T_{2}^{5} + \cdots + 2430272 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(161))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 8 T^{9} + \cdots + 2430272 \)
|
| $3$ |
\( T^{10} + \cdots - 33293161728 \)
|
| $5$ |
\( T^{10} + \cdots + 26\!\cdots\!56 \)
|
| $7$ |
\( (T - 49)^{10} \)
|
| $11$ |
\( T^{10} + \cdots - 21\!\cdots\!40 \)
|
| $13$ |
\( T^{10} + \cdots + 63\!\cdots\!28 \)
|
| $17$ |
\( T^{10} + \cdots + 27\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots - 10\!\cdots\!60 \)
|
| $23$ |
\( (T - 529)^{10} \)
|
| $29$ |
\( T^{10} + \cdots + 34\!\cdots\!72 \)
|
| $31$ |
\( T^{10} + \cdots - 17\!\cdots\!12 \)
|
| $37$ |
\( T^{10} + \cdots + 87\!\cdots\!12 \)
|
| $41$ |
\( T^{10} + \cdots - 29\!\cdots\!60 \)
|
| $43$ |
\( T^{10} + \cdots - 12\!\cdots\!04 \)
|
| $47$ |
\( T^{10} + \cdots - 56\!\cdots\!84 \)
|
| $53$ |
\( T^{10} + \cdots + 13\!\cdots\!76 \)
|
| $59$ |
\( T^{10} + \cdots - 60\!\cdots\!48 \)
|
| $61$ |
\( T^{10} + \cdots + 47\!\cdots\!40 \)
|
| $67$ |
\( T^{10} + \cdots + 45\!\cdots\!04 \)
|
| $71$ |
\( T^{10} + \cdots - 29\!\cdots\!12 \)
|
| $73$ |
\( T^{10} + \cdots - 28\!\cdots\!20 \)
|
| $79$ |
\( T^{10} + \cdots + 49\!\cdots\!80 \)
|
| $83$ |
\( T^{10} + \cdots + 12\!\cdots\!08 \)
|
| $89$ |
\( T^{10} + \cdots - 64\!\cdots\!60 \)
|
| $97$ |
\( T^{10} + \cdots - 94\!\cdots\!12 \)
|
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