Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(9.49930751092\) |
| 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: |
\( x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 25\nu^{2} - 50\nu + 40 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 25\nu^{2} + 26\nu + 72 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 35\nu^{4} + 83\nu^{3} + 306\nu^{2} - 480\nu - 288 ) / 32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 40\nu^{6} + 142\nu^{5} + 455\nu^{4} - 1354\nu^{3} - 1024\nu^{2} + 2592\nu - 768 ) / 128 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} + 44\nu^{6} - 138\nu^{5} - 611\nu^{4} + 1190\nu^{3} + 2520\nu^{2} - 1472\nu - 1536 ) / 128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} - 56\nu^{6} - 18\nu^{5} + 1007\nu^{4} + 498\nu^{3} - 5912\nu^{2} - 2304\nu + 5120 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + 2\beta_{3} + 19\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 4\beta_{3} + 25\beta_{2} + 37\beta _1 + 269 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{7} + 8\beta_{6} - 8\beta_{5} - 58\beta_{4} + 66\beta_{3} + 8\beta_{2} + 419\beta _1 + 368 \)
|
| \(\nu^{6}\) | \(=\) |
\( 24\beta_{7} + 24\beta_{6} + 8\beta_{5} + 132\beta_{4} + 172\beta_{3} + 593\beta_{2} + 1149\beta _1 + 6165 \)
|
| \(\nu^{7}\) | \(=\) |
\( 32 \beta_{8} + 416 \beta_{7} + 384 \beta_{6} - 288 \beta_{5} - 1418 \beta_{4} + 1850 \beta_{3} + \cdots + 12968 \)
|
| \(\nu^{8}\) | \(=\) |
\( 128 \beta_{8} + 1488 \beta_{7} + 1488 \beta_{6} + 304 \beta_{5} + 3316 \beta_{4} + 5796 \beta_{3} + \cdots + 148733 \)
|
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 |
|
−4.55260 | −5.72089 | 12.7261 | 17.5795 | 26.0449 | −7.00000 | −21.5162 | 5.72860 | −80.0322 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.38565 | −0.971823 | 11.2339 | −20.4381 | 4.26207 | −7.00000 | −14.1827 | −26.0556 | 89.6341 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.73922 | 10.3390 | 5.98179 | 7.23273 | −38.6599 | −7.00000 | 7.54655 | 79.8951 | −27.0448 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.58693 | 3.23551 | −5.48164 | −8.51310 | −5.13454 | −7.00000 | 21.3945 | −16.5315 | 13.5097 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.264899 | −6.92505 | −7.92983 | −11.7130 | 1.83444 | −7.00000 | 4.21980 | 20.9564 | 3.10276 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.22159 | −7.12940 | −6.50771 | −2.67729 | −8.70922 | −7.00000 | −17.7225 | 23.8284 | −3.27056 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.04896 | 7.42499 | 1.29618 | 11.7973 | 22.6385 | −7.00000 | −20.4397 | 28.1305 | 35.9694 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.95572 | 7.67438 | 16.5591 | −10.4913 | 38.0320 | −7.00000 | 42.4166 | 31.8961 | −51.9921 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.30303 | 1.07328 | 20.1221 | 13.2233 | 5.69164 | −7.00000 | 64.2837 | −25.8481 | 70.1237 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 161.4.a.c | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1449.4.a.n | 9 | ||
| 7.b | odd | 2 | 1 | 1127.4.a.f | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.4.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1127.4.a.f | 9 | 7.b | odd | 2 | 1 | ||
| 1449.4.a.n | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 60T_{2}^{7} - 22T_{2}^{6} + 1179T_{2}^{5} + 694T_{2}^{4} - 7936T_{2}^{3} - 4352T_{2}^{2} + 11008T_{2} + 3072 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 60 T^{7} + \cdots + 3072 \)
|
| $3$ |
\( T^{9} - 9 T^{8} + \cdots - 561568 \)
|
| $5$ |
\( T^{9} + \cdots + 1135409856 \)
|
| $7$ |
\( (T + 7)^{9} \)
|
| $11$ |
\( T^{9} + \cdots + 43666792951552 \)
|
| $13$ |
\( T^{9} + \cdots - 6636351117312 \)
|
| $17$ |
\( T^{9} + \cdots + 80\!\cdots\!88 \)
|
| $19$ |
\( T^{9} + \cdots + 95\!\cdots\!20 \)
|
| $23$ |
\( (T + 23)^{9} \)
|
| $29$ |
\( T^{9} + \cdots + 87\!\cdots\!60 \)
|
| $31$ |
\( T^{9} + \cdots - 42\!\cdots\!44 \)
|
| $37$ |
\( T^{9} + \cdots + 78\!\cdots\!00 \)
|
| $41$ |
\( T^{9} + \cdots + 51\!\cdots\!48 \)
|
| $43$ |
\( T^{9} + \cdots + 13\!\cdots\!28 \)
|
| $47$ |
\( T^{9} + \cdots + 15\!\cdots\!76 \)
|
| $53$ |
\( T^{9} + \cdots + 27\!\cdots\!12 \)
|
| $59$ |
\( T^{9} + \cdots + 12\!\cdots\!40 \)
|
| $61$ |
\( T^{9} + \cdots + 12\!\cdots\!56 \)
|
| $67$ |
\( T^{9} + \cdots - 11\!\cdots\!56 \)
|
| $71$ |
\( T^{9} + \cdots - 36\!\cdots\!72 \)
|
| $73$ |
\( T^{9} + \cdots + 17\!\cdots\!12 \)
|
| $79$ |
\( T^{9} + \cdots - 92\!\cdots\!20 \)
|
| $83$ |
\( T^{9} + \cdots + 20\!\cdots\!56 \)
|
| $89$ |
\( T^{9} + \cdots + 79\!\cdots\!60 \)
|
| $97$ |
\( T^{9} + \cdots - 44\!\cdots\!92 \)
|
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