Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,4,Mod(63,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.63");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.n (of order \(4\), degree \(2\), 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: | \(9.44030560092\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 22x^{5} + 532x^{4} - 636x^{3} + 450x^{2} + 2160x + 5184 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 2x^{7} + 2x^{6} + 22x^{5} + 532x^{4} - 636x^{3} + 450x^{2} + 2160x + 5184 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 12827 \nu^{7} + 76486 \nu^{6} - 98830 \nu^{5} - 215714 \nu^{4} - 5831540 \nu^{3} + \cdots - 13371048 ) / 111061800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11087 \nu^{7} + 110714 \nu^{6} - 138830 \nu^{5} - 430606 \nu^{4} + 8851640 \nu^{3} + \cdots - 172204272 ) / 55530900 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4549 \nu^{7} + 31262 \nu^{6} - 19214 \nu^{5} + 59186 \nu^{4} - 2042464 \nu^{3} + \cdots + 33215616 ) / 11106180 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4361 \nu^{7} + 8822 \nu^{6} - 108590 \nu^{5} + 228422 \nu^{4} + 3185780 \nu^{3} + 4136484 \nu^{2} + \cdots + 33234264 ) / 7404120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7537 \nu^{7} + 25076 \nu^{6} - 99680 \nu^{5} - 37954 \nu^{4} - 4290130 \nu^{3} + 8815752 \nu^{2} + \cdots + 5405022 ) / 5553090 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 386857 \nu^{7} + 729026 \nu^{6} - 697730 \nu^{5} - 8895574 \nu^{4} - 197999140 \nu^{3} + \cdots - 839823768 ) / 111061800 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 33307 \nu^{7} - 67126 \nu^{6} + 66380 \nu^{5} + 759274 \nu^{4} + 16985890 \nu^{3} + \cdots + 61844418 ) / 9255150 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} - 5\beta_{3} - 5\beta_{2} + 52\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{7} + 5\beta_{6} - 21\beta_{5} + 21\beta_{4} - 10\beta_{2} + 30\beta _1 - 30 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11\beta_{7} - 27\beta_{6} - 11\beta_{5} + 27\beta_{4} + 140\beta_{3} - 140\beta_{2} - 1094 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -497\beta_{7} - 497\beta_{6} - 129\beta_{5} - 129\beta_{4} + 390\beta_{3} - 1022\beta _1 - 1022 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -379\beta_{7} - 261\beta_{6} + 379\beta_{5} - 261\beta_{4} + 1760\beta_{3} + 1760\beta_{2} - 12922\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1499\beta_{7} - 1499\beta_{6} + 6060\beta_{5} - 6060\beta_{4} + 6130\beta_{2} - 16923\beta _1 + 16923 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | −4.50577 | + | 4.50577i | 0 | −6.30193 | + | 9.23502i | 0 | −19.0427 | − | 19.0427i | 0 | − | 13.6039i | 0 | |||||||||||||||||||||||||||||||||||
| 63.2 | 0 | −3.64336 | + | 3.64336i | 0 | 0.725933 | − | 11.1567i | 0 | 9.23932 | + | 9.23932i | 0 | 0.451865i | 0 | |||||||||||||||||||||||||||||||||||||
| 63.3 | 0 | 2.18370 | − | 2.18370i | 0 | 9.23146 | + | 6.30715i | 0 | 6.10801 | + | 6.10801i | 0 | 17.4629i | 0 | |||||||||||||||||||||||||||||||||||||
| 63.4 | 0 | 4.96543 | − | 4.96543i | 0 | −10.6555 | − | 3.38543i | 0 | −1.30461 | − | 1.30461i | 0 | − | 22.3109i | 0 | ||||||||||||||||||||||||||||||||||||
| 127.1 | 0 | −4.50577 | − | 4.50577i | 0 | −6.30193 | − | 9.23502i | 0 | −19.0427 | + | 19.0427i | 0 | 13.6039i | 0 | |||||||||||||||||||||||||||||||||||||
| 127.2 | 0 | −3.64336 | − | 3.64336i | 0 | 0.725933 | + | 11.1567i | 0 | 9.23932 | − | 9.23932i | 0 | − | 0.451865i | 0 | ||||||||||||||||||||||||||||||||||||
| 127.3 | 0 | 2.18370 | + | 2.18370i | 0 | 9.23146 | − | 6.30715i | 0 | 6.10801 | − | 6.10801i | 0 | − | 17.4629i | 0 | ||||||||||||||||||||||||||||||||||||
| 127.4 | 0 | 4.96543 | + | 4.96543i | 0 | −10.6555 | + | 3.38543i | 0 | −1.30461 | + | 1.30461i | 0 | 22.3109i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.4.n.d | ✓ | 8 |
| 4.b | odd | 2 | 1 | 160.4.n.e | yes | 8 | |
| 5.c | odd | 4 | 1 | 160.4.n.e | yes | 8 | |
| 8.b | even | 2 | 1 | 320.4.n.h | 8 | ||
| 8.d | odd | 2 | 1 | 320.4.n.g | 8 | ||
| 20.e | even | 4 | 1 | inner | 160.4.n.d | ✓ | 8 |
| 40.i | odd | 4 | 1 | 320.4.n.g | 8 | ||
| 40.k | even | 4 | 1 | 320.4.n.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.4.n.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 160.4.n.d | ✓ | 8 | 20.e | even | 4 | 1 | inner |
| 160.4.n.e | yes | 8 | 4.b | odd | 2 | 1 | |
| 160.4.n.e | yes | 8 | 5.c | odd | 4 | 1 | |
| 320.4.n.g | 8 | 8.d | odd | 2 | 1 | ||
| 320.4.n.g | 8 | 40.i | odd | 4 | 1 | ||
| 320.4.n.h | 8 | 8.b | even | 2 | 1 | ||
| 320.4.n.h | 8 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} - 8T_{3}^{5} + 2420T_{3}^{4} + 5768T_{3}^{3} + 6728T_{3}^{2} - 82592T_{3} + 506944 \)
acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 2 T^{7} + \cdots + 506944 \)
|
| $5$ |
\( T^{8} + 14 T^{7} + \cdots + 244140625 \)
|
| $7$ |
\( T^{8} + 10 T^{7} + \cdots + 31449664 \)
|
| $11$ |
\( T^{8} + 2124 T^{6} + \cdots + 92160000 \)
|
| $13$ |
\( T^{8} + \cdots + 4682705569936 \)
|
| $17$ |
\( T^{8} + \cdots + 598290621600400 \)
|
| $19$ |
\( (T^{4} + 40 T^{3} + \cdots + 8404736)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 71\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} - 466 T^{3} + \cdots - 3094032000)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 47\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 41\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} + 8 T^{3} + \cdots + 16035860224)^{2} \)
|
| $61$ |
\( (T^{4} - 1282 T^{3} + \cdots - 61321952)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 39\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
|
| $73$ |
\( T^{8} + \cdots + 16\!\cdots\!04 \)
|
| $79$ |
\( (T^{4} + 1608 T^{3} + \cdots + 217899008)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 49\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 41\!\cdots\!76 \)
|
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