Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(3,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.c (of order \(5\), degree \(4\), 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: | \(7.13923111069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.29283765625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 11) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 280\nu ) / 981 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 171 ) / 109 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 1261\nu^{2} ) / 8829 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 10\nu^{6} + 19\nu^{5} - 109\nu^{4} + 1090\nu^{3} - 810\nu^{2} - 729\nu - 6561 ) / 8829 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -19\nu^{7} + 19\nu^{6} - 190\nu^{5} + 1090\nu^{4} - 2071\nu^{3} - 3249\nu^{2} - 15390\nu - 13851 ) / 79461 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\nu^{7} + 3781\nu^{2} ) / 8829 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + 10\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 9\beta_{2} + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 19\beta_{7} + 90\beta_{6} + 19\beta_{5} - 19\beta_{4} + 19\beta_{3} + 19\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 109\beta_{3} - 171 \)
|
| \(\nu^{6}\) | \(=\) |
\( 981\beta_{2} - 280\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1261\beta_{7} - 3781\beta_{4} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−0.747004 | + | 0.542730i | −0.476313 | − | 1.46594i | −2.20868 | + | 6.79761i | −7.05908 | − | 5.12872i | 1.15142 | + | 0.836554i | −0.239524 | + | 0.737179i | −4.32202 | − | 13.3018i | 19.9214 | − | 14.4737i | 8.05666 | ||||||||||||||||||||||||||
| 3.2 | 4.17405 | − | 3.03263i | 1.40336 | + | 4.31911i | 5.75376 | − | 17.7083i | 6.98613 | + | 5.07572i | 18.9560 | + | 13.7723i | −0.513765 | + | 1.58121i | −16.9313 | − | 52.1091i | 5.15818 | − | 3.74763i | 44.5532 | |||||||||||||||||||||||||||
| 9.1 | −0.903364 | + | 2.78027i | −3.67405 | + | 2.66936i | −0.441690 | − | 0.320907i | 1.90871 | + | 5.87440i | −4.10252 | − | 12.6263i | 25.9914 | + | 18.8838i | −17.6291 | + | 12.8083i | −1.97025 | + | 6.06380i | −18.0567 | |||||||||||||||||||||||||||
| 9.2 | 0.976313 | − | 3.00478i | 1.24700 | − | 0.906001i | −1.60339 | − | 1.16493i | −5.33576 | − | 16.4218i | −1.50487 | − | 4.63152i | −7.73807 | − | 5.62204i | 15.3824 | − | 11.1760i | −7.60928 | + | 23.4190i | −54.5532 | |||||||||||||||||||||||||||
| 27.1 | −0.903364 | − | 2.78027i | −3.67405 | − | 2.66936i | −0.441690 | + | 0.320907i | 1.90871 | − | 5.87440i | −4.10252 | + | 12.6263i | 25.9914 | − | 18.8838i | −17.6291 | − | 12.8083i | −1.97025 | − | 6.06380i | −18.0567 | |||||||||||||||||||||||||||
| 27.2 | 0.976313 | + | 3.00478i | 1.24700 | + | 0.906001i | −1.60339 | + | 1.16493i | −5.33576 | + | 16.4218i | −1.50487 | + | 4.63152i | −7.73807 | + | 5.62204i | 15.3824 | + | 11.1760i | −7.60928 | − | 23.4190i | −54.5532 | |||||||||||||||||||||||||||
| 81.1 | −0.747004 | − | 0.542730i | −0.476313 | + | 1.46594i | −2.20868 | − | 6.79761i | −7.05908 | + | 5.12872i | 1.15142 | − | 0.836554i | −0.239524 | − | 0.737179i | −4.32202 | + | 13.3018i | 19.9214 | + | 14.4737i | 8.05666 | |||||||||||||||||||||||||||
| 81.2 | 4.17405 | + | 3.03263i | 1.40336 | − | 4.31911i | 5.75376 | + | 17.7083i | 6.98613 | − | 5.07572i | 18.9560 | − | 13.7723i | −0.513765 | − | 1.58121i | −16.9313 | + | 52.1091i | 5.15818 | + | 3.74763i | 44.5532 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.4.c.i | 8 | |
| 11.b | odd | 2 | 1 | 11.4.c.a | ✓ | 8 | |
| 11.c | even | 5 | 1 | 121.4.a.f | 4 | ||
| 11.c | even | 5 | 2 | 121.4.c.b | 8 | ||
| 11.c | even | 5 | 1 | inner | 121.4.c.i | 8 | |
| 11.d | odd | 10 | 1 | 11.4.c.a | ✓ | 8 | |
| 11.d | odd | 10 | 1 | 121.4.a.g | 4 | ||
| 11.d | odd | 10 | 2 | 121.4.c.h | 8 | ||
| 33.d | even | 2 | 1 | 99.4.f.c | 8 | ||
| 33.f | even | 10 | 1 | 99.4.f.c | 8 | ||
| 33.f | even | 10 | 1 | 1089.4.a.y | 4 | ||
| 33.h | odd | 10 | 1 | 1089.4.a.bh | 4 | ||
| 44.c | even | 2 | 1 | 176.4.m.c | 8 | ||
| 44.g | even | 10 | 1 | 176.4.m.c | 8 | ||
| 44.g | even | 10 | 1 | 1936.4.a.bk | 4 | ||
| 44.h | odd | 10 | 1 | 1936.4.a.bl | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.4.c.a | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 11.4.c.a | ✓ | 8 | 11.d | odd | 10 | 1 | |
| 99.4.f.c | 8 | 33.d | even | 2 | 1 | ||
| 99.4.f.c | 8 | 33.f | even | 10 | 1 | ||
| 121.4.a.f | 4 | 11.c | even | 5 | 1 | ||
| 121.4.a.g | 4 | 11.d | odd | 10 | 1 | ||
| 121.4.c.b | 8 | 11.c | even | 5 | 2 | ||
| 121.4.c.h | 8 | 11.d | odd | 10 | 2 | ||
| 121.4.c.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 121.4.c.i | 8 | 11.c | even | 5 | 1 | inner | |
| 176.4.m.c | 8 | 44.c | even | 2 | 1 | ||
| 176.4.m.c | 8 | 44.g | even | 10 | 1 | ||
| 1089.4.a.y | 4 | 33.f | even | 10 | 1 | ||
| 1089.4.a.bh | 4 | 33.h | odd | 10 | 1 | ||
| 1936.4.a.bk | 4 | 44.g | even | 10 | 1 | ||
| 1936.4.a.bl | 4 | 44.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 7T_{2}^{7} + 31T_{2}^{6} - 71T_{2}^{5} + 319T_{2}^{4} - 78T_{2}^{3} + 1664T_{2}^{2} + 2816T_{2} + 1936 \)
acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 7 T^{7} + \cdots + 1936 \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 2401 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots + 64577296 \)
|
| $7$ |
\( T^{8} - 35 T^{7} + \cdots + 156816 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 2905210000 \)
|
| $17$ |
\( T^{8} + \cdots + 9608691844521 \)
|
| $19$ |
\( T^{8} + \cdots + 90104309844241 \)
|
| $23$ |
\( (T^{4} + 6 T^{3} + \cdots + 49883584)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 97\!\cdots\!96 \)
|
| $31$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $37$ |
\( T^{8} + \cdots + 26\!\cdots\!96 \)
|
| $41$ |
\( T^{8} + \cdots + 27\!\cdots\!61 \)
|
| $43$ |
\( (T^{4} - 325 T^{3} + \cdots - 1288748736)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 70\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 71\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 66\!\cdots\!61 \)
|
| $61$ |
\( T^{8} + \cdots + 22\!\cdots\!36 \)
|
| $67$ |
\( (T^{4} - 43 T^{3} + \cdots - 8869996224)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 55\!\cdots\!16 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!81 \)
|
| $79$ |
\( T^{8} + \cdots + 44\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 32\!\cdots\!81 \)
|
| $89$ |
\( (T^{4} - 1891 T^{3} + \cdots - 2046678844)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 12\!\cdots\!01 \)
|
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