Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,13,Mod(17,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.17");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.p (of order \(4\), degree \(2\), not 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: | \(73.1195053821\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{29}\cdot 3^{2}\cdot 5^{9} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 26465 \nu^{9} - 176059482 \nu^{7} - 167682655437 \nu^{5} + 471206316514012 \nu^{3} + 74\!\cdots\!92 \nu ) / 33\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 38463776647 \nu^{8} + 302018459813961 \nu^{6} + \cdots + 47\!\cdots\!16 ) / 11\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12053958997610 \nu^{9} - 308572739015625 \nu^{8} + \cdots - 15\!\cdots\!60 ) / 36\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12053958997610 \nu^{9} - 308572739015625 \nu^{8} + \cdots - 15\!\cdots\!60 ) / 36\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 801019586547595 \nu^{9} + \cdots - 61\!\cdots\!60 ) / 73\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 801019586547595 \nu^{9} + \cdots + 61\!\cdots\!60 ) / 73\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5822655238552 \nu^{9} - 143756862844860 \nu^{8} + \cdots - 23\!\cdots\!16 ) / 52\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 633929768841205 \nu^{9} + \cdots - 10\!\cdots\!56 \nu ) / 18\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 272554049849920 \nu^{9} + 455936994874375 \nu^{8} + \cdots + 74\!\cdots\!00 ) / 61\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( 192 \beta_{9} - 125 \beta_{8} + 64 \beta_{7} + 288 \beta_{6} + 224 \beta_{5} + 937 \beta_{4} + \cdots - 64 ) / 96000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 32 \beta_{9} - 96 \beta_{7} - 672 \beta_{6} + 704 \beta_{5} + 6757 \beta_{4} + 6789 \beta_{3} + \cdots - 10173684 ) / 6400 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 762528 \beta_{9} + 550625 \beta_{8} - 254176 \beta_{7} - 993792 \beta_{6} - 739616 \beta_{5} + \cdots + 254176 ) / 96000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 29024 \beta_{9} + 87072 \beta_{7} + 564512 \beta_{6} - 593536 \beta_{5} - 5108065 \beta_{4} + \cdots + 7505310628 ) / 1280 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2930139552 \beta_{9} - 2077180625 \beta_{8} + 976713184 \beta_{7} + 3855779328 \beta_{6} + \cdots - 976713184 ) / 96000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 673331168 \beta_{9} - 2019993504 \beta_{7} - 11174674848 \beta_{6} + 11848006016 \beta_{5} + \cdots - 143566031529716 ) / 6400 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 11257924759968 \beta_{9} + 7779326218625 \beta_{8} - 3752641586656 \beta_{7} + \cdots + 3752641586656 ) / 96000 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 3035619636704 \beta_{9} + 9106858910112 \beta_{7} + 44116123468704 \beta_{6} + \cdots + 55\!\cdots\!48 ) / 6400 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 43\!\cdots\!12 \beta_{9} + \cdots - 14\!\cdots\!04 ) / 96000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −877.187 | + | 877.187i | 0 | −13781.9 | − | 7362.02i | 0 | −67595.3 | − | 67595.3i | 0 | − | 1.00747e6i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −539.020 | + | 539.020i | 0 | 12911.0 | + | 8800.45i | 0 | −25919.1 | − | 25919.1i | 0 | − | 49644.2i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | 299.721 | − | 299.721i | 0 | −15614.1 | + | 583.350i | 0 | 139850. | + | 139850.i | 0 | 351775.i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | 451.057 | − | 451.057i | 0 | −1256.84 | − | 15574.4i | 0 | −107575. | − | 107575.i | 0 | 124537.i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 506.429 | − | 506.429i | 0 | 15616.9 | + | 502.591i | 0 | −78559.8 | − | 78559.8i | 0 | 18500.3i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.1 | 0 | −877.187 | − | 877.187i | 0 | −13781.9 | + | 7362.02i | 0 | −67595.3 | + | 67595.3i | 0 | 1.00747e6i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −539.020 | − | 539.020i | 0 | 12911.0 | − | 8800.45i | 0 | −25919.1 | + | 25919.1i | 0 | 49644.2i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | 299.721 | + | 299.721i | 0 | −15614.1 | − | 583.350i | 0 | 139850. | − | 139850.i | 0 | − | 351775.i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | 451.057 | + | 451.057i | 0 | −1256.84 | + | 15574.4i | 0 | −107575. | + | 107575.i | 0 | − | 124537.i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 506.429 | + | 506.429i | 0 | 15616.9 | − | 502.591i | 0 | −78559.8 | + | 78559.8i | 0 | − | 18500.3i | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 80.13.p.c | 10 | |
| 4.b | odd | 2 | 1 | 5.13.c.a | ✓ | 10 | |
| 5.c | odd | 4 | 1 | inner | 80.13.p.c | 10 | |
| 12.b | even | 2 | 1 | 45.13.g.a | 10 | ||
| 20.d | odd | 2 | 1 | 25.13.c.b | 10 | ||
| 20.e | even | 4 | 1 | 5.13.c.a | ✓ | 10 | |
| 20.e | even | 4 | 1 | 25.13.c.b | 10 | ||
| 60.l | odd | 4 | 1 | 45.13.g.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.13.c.a | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 5.13.c.a | ✓ | 10 | 20.e | even | 4 | 1 | |
| 25.13.c.b | 10 | 20.d | odd | 2 | 1 | ||
| 25.13.c.b | 10 | 20.e | even | 4 | 1 | ||
| 45.13.g.a | 10 | 12.b | even | 2 | 1 | ||
| 45.13.g.a | 10 | 60.l | odd | 4 | 1 | ||
| 80.13.p.c | 10 | 1.a | even | 1 | 1 | trivial | |
| 80.13.p.c | 10 | 5.c | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 318 T_{3}^{9} + 50562 T_{3}^{8} - 771958800 T_{3}^{7} + 1336676660388 T_{3}^{6} + \cdots + 33\!\cdots\!32 \)
acting on \(S_{13}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots + 33\!\cdots\!32 \)
|
| $5$ |
\( T^{10} + \cdots + 86\!\cdots\!25 \)
|
| $7$ |
\( T^{10} + \cdots + 13\!\cdots\!32 \)
|
| $11$ |
\( (T^{5} + \cdots + 33\!\cdots\!32)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 12\!\cdots\!32 \)
|
| $17$ |
\( T^{10} + \cdots + 18\!\cdots\!32 \)
|
| $19$ |
\( T^{10} + \cdots + 47\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 29\!\cdots\!32 \)
|
| $29$ |
\( T^{10} + \cdots + 76\!\cdots\!00 \)
|
| $31$ |
\( (T^{5} + \cdots + 82\!\cdots\!32)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 22\!\cdots\!32 \)
|
| $41$ |
\( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 22\!\cdots\!32 \)
|
| $47$ |
\( T^{10} + \cdots + 27\!\cdots\!32 \)
|
| $53$ |
\( T^{10} + \cdots + 91\!\cdots\!32 \)
|
| $59$ |
\( T^{10} + \cdots + 99\!\cdots\!00 \)
|
| $61$ |
\( (T^{5} + \cdots + 13\!\cdots\!68)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 28\!\cdots\!32 \)
|
| $71$ |
\( (T^{5} + \cdots + 61\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 16\!\cdots\!32 \)
|
| $79$ |
\( T^{10} + \cdots + 22\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 48\!\cdots\!32 \)
|
| $89$ |
\( T^{10} + \cdots + 35\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 32\!\cdots\!32 \)
|
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