Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,18,Mod(49,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.49");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.c (of order \(2\), degree \(1\), 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: | \(146.577669876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{35}\cdot 3^{8}\cdot 5^{12}\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 5) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 \)
:
| \(\beta_{1}\) | \(=\) |
\( 256\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -19\nu^{7} - 3990585\nu^{5} - 233779723524\nu^{3} - 2952850120132352\nu ) / 7999594960896 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19613 \nu^{7} - 2309984 \nu^{6} - 3663672279 \nu^{5} - 423805335072 \nu^{4} + \cdots - 20\!\cdots\!04 ) / 66663291340800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 699511 \nu^{7} + 6929952 \nu^{6} - 125502752613 \nu^{5} + 1271416005216 \nu^{4} + \cdots + 60\!\cdots\!12 ) / 199989874022400 \)
|
| \(\beta_{5}\) | \(=\) |
\( 128\nu^{2} + 6510688 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 58839 \nu^{7} - 5556448 \nu^{6} + 10991016837 \nu^{5} + 1382857125216 \nu^{4} + \cdots + 38\!\cdots\!12 ) / 99994937011200 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 235831 \nu^{7} + 41579712 \nu^{6} - 44063831973 \nu^{5} + 7628496031296 \nu^{4} + \cdots + 36\!\cdots\!72 ) / 39997974804480 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 256 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 6510688 ) / 128 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -32\beta_{7} + 96\beta_{4} - 864\beta_{3} + 45088\beta_{2} - 82279\beta_1 ) / 256 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 384\beta_{7} + 5328\beta_{6} - 117487\beta_{5} + 2976\beta_{3} - 1920\beta_{2} + 2256\beta _1 + 533435721248 ) / 128 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4583264\beta_{7} - 11502624\beta_{4} + 125995296\beta_{3} - 10045481824\beta_{2} + 7869549417\beta_1 ) / 256 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3427200 \beta_{7} - 977511024 \beta_{6} + 12626030433 \beta_{5} - 2023566048 \beta_{3} + \cdots - 50\!\cdots\!08 ) / 128 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 568892285088 \beta_{7} + 1234702386144 \beta_{4} - 15832066166496 \beta_{3} + \cdots - 795883900449479 \beta_1 ) / 256 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 18245.5i | 0 | 346646. | − | 801733.i | 0 | 1.34806e7i | 0 | −2.03757e8 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 12817.1i | 0 | 620811. | + | 614437.i | 0 | 4.49140e6i | 0 | −3.51381e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 11309.3i | 0 | −21187.7 | − | 873207.i | 0 | − | 2.37065e7i | 0 | 1.24087e6 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | − | 2990.60i | 0 | −756669. | + | 436339.i | 0 | − | 2.43909e7i | 0 | 1.20196e8 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 2990.60i | 0 | −756669. | − | 436339.i | 0 | 2.43909e7i | 0 | 1.20196e8 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 11309.3i | 0 | −21187.7 | + | 873207.i | 0 | 2.37065e7i | 0 | 1.24087e6 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 12817.1i | 0 | 620811. | − | 614437.i | 0 | − | 4.49140e6i | 0 | −3.51381e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 18245.5i | 0 | 346646. | + | 801733.i | 0 | − | 1.34806e7i | 0 | −2.03757e8 | 0 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 80.18.c.b | 8 | |
| 4.b | odd | 2 | 1 | 5.18.b.a | ✓ | 8 | |
| 5.b | even | 2 | 1 | inner | 80.18.c.b | 8 | |
| 12.b | even | 2 | 1 | 45.18.b.b | 8 | ||
| 20.d | odd | 2 | 1 | 5.18.b.a | ✓ | 8 | |
| 20.e | even | 4 | 2 | 25.18.a.f | 8 | ||
| 60.h | even | 2 | 1 | 45.18.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.18.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 5.18.b.a | ✓ | 8 | 20.d | odd | 2 | 1 | |
| 25.18.a.f | 8 | 20.e | even | 4 | 2 | ||
| 45.18.b.b | 8 | 12.b | even | 2 | 1 | ||
| 45.18.b.b | 8 | 60.h | even | 2 | 1 | ||
| 80.18.c.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 80.18.c.b | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 634018824 T_{3}^{6} + \cdots + 62\!\cdots\!96 \)
acting on \(S_{18}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 62\!\cdots\!96 \)
|
| $5$ |
\( T^{8} + \cdots + 33\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 12\!\cdots\!16 \)
|
| $11$ |
\( (T^{4} + \cdots + 25\!\cdots\!56)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 25\!\cdots\!76 \)
|
| $17$ |
\( T^{8} + \cdots + 24\!\cdots\!76 \)
|
| $19$ |
\( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $29$ |
\( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 78\!\cdots\!64)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 37\!\cdots\!96 \)
|
| $41$ |
\( (T^{4} + \cdots - 51\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 28\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 44\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots + 88\!\cdots\!96 \)
|
| $59$ |
\( (T^{4} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 56\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} + \cdots - 23\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 36\!\cdots\!56 \)
|
| $79$ |
\( (T^{4} + \cdots + 53\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $89$ |
\( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 76\!\cdots\!56 \)
|
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