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: | 8.0.324000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
|
| 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
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 27\beta_{7} \)
|
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 |
|
−2.21028 | + | 1.60586i | −2.44995 | − | 7.54017i | −0.165602 | + | 0.509670i | −12.0191 | − | 8.73238i | 17.5235 | + | 12.7316i | 0.949237 | − | 2.92145i | −7.20643 | − | 22.1791i | −29.0084 | + | 21.0759i | 40.5885 | ||||||||||||||||||||||||||
| 3.2 | 0.592242 | − | 0.430289i | 1.83192 | + | 5.63806i | −2.30653 | + | 7.09878i | 10.4011 | + | 7.55681i | 3.51093 | + | 2.55084i | 5.23110 | − | 16.0997i | 3.49823 | + | 10.7664i | −6.58831 | + | 4.78668i | 9.41154 | |||||||||||||||||||||||||||
| 9.1 | −0.226216 | + | 0.696222i | −4.79602 | + | 3.48451i | 6.03859 | + | 4.38729i | −3.97285 | − | 12.2272i | −1.34106 | − | 4.12734i | −13.6952 | − | 9.95015i | −9.15848 | + | 6.65403i | 2.51651 | − | 7.74502i | 9.41154 | |||||||||||||||||||||||||||
| 9.2 | 0.844250 | − | 2.59833i | 6.41405 | − | 4.66008i | 0.433551 | + | 0.314993i | 4.59088 | + | 14.1293i | −6.69339 | − | 20.6001i | −2.48514 | − | 1.80556i | 18.8667 | − | 13.7075i | 11.0802 | − | 34.1015i | 40.5885 | |||||||||||||||||||||||||||
| 27.1 | −0.226216 | − | 0.696222i | −4.79602 | − | 3.48451i | 6.03859 | − | 4.38729i | −3.97285 | + | 12.2272i | −1.34106 | + | 4.12734i | −13.6952 | + | 9.95015i | −9.15848 | − | 6.65403i | 2.51651 | + | 7.74502i | 9.41154 | |||||||||||||||||||||||||||
| 27.2 | 0.844250 | + | 2.59833i | 6.41405 | + | 4.66008i | 0.433551 | − | 0.314993i | 4.59088 | − | 14.1293i | −6.69339 | + | 20.6001i | −2.48514 | + | 1.80556i | 18.8667 | + | 13.7075i | 11.0802 | + | 34.1015i | 40.5885 | |||||||||||||||||||||||||||
| 81.1 | −2.21028 | − | 1.60586i | −2.44995 | + | 7.54017i | −0.165602 | − | 0.509670i | −12.0191 | + | 8.73238i | 17.5235 | − | 12.7316i | 0.949237 | + | 2.92145i | −7.20643 | + | 22.1791i | −29.0084 | − | 21.0759i | 40.5885 | |||||||||||||||||||||||||||
| 81.2 | 0.592242 | + | 0.430289i | 1.83192 | − | 5.63806i | −2.30653 | − | 7.09878i | 10.4011 | − | 7.55681i | 3.51093 | − | 2.55084i | 5.23110 | + | 16.0997i | 3.49823 | − | 10.7664i | −6.58831 | − | 4.78668i | 9.41154 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.4.c.c | 8 | |
| 11.b | odd | 2 | 1 | 121.4.c.f | 8 | ||
| 11.c | even | 5 | 1 | 11.4.a.a | ✓ | 2 | |
| 11.c | even | 5 | 3 | inner | 121.4.c.c | 8 | |
| 11.d | odd | 10 | 1 | 121.4.a.c | 2 | ||
| 11.d | odd | 10 | 3 | 121.4.c.f | 8 | ||
| 33.f | even | 10 | 1 | 1089.4.a.v | 2 | ||
| 33.h | odd | 10 | 1 | 99.4.a.c | 2 | ||
| 44.g | even | 10 | 1 | 1936.4.a.w | 2 | ||
| 44.h | odd | 10 | 1 | 176.4.a.i | 2 | ||
| 55.j | even | 10 | 1 | 275.4.a.b | 2 | ||
| 55.k | odd | 20 | 2 | 275.4.b.c | 4 | ||
| 77.j | odd | 10 | 1 | 539.4.a.e | 2 | ||
| 88.l | odd | 10 | 1 | 704.4.a.n | 2 | ||
| 88.o | even | 10 | 1 | 704.4.a.p | 2 | ||
| 132.o | even | 10 | 1 | 1584.4.a.bc | 2 | ||
| 143.n | even | 10 | 1 | 1859.4.a.a | 2 | ||
| 165.o | odd | 10 | 1 | 2475.4.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.4.a.a | ✓ | 2 | 11.c | even | 5 | 1 | |
| 99.4.a.c | 2 | 33.h | odd | 10 | 1 | ||
| 121.4.a.c | 2 | 11.d | odd | 10 | 1 | ||
| 121.4.c.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 121.4.c.c | 8 | 11.c | even | 5 | 3 | inner | |
| 121.4.c.f | 8 | 11.b | odd | 2 | 1 | ||
| 121.4.c.f | 8 | 11.d | odd | 10 | 3 | ||
| 176.4.a.i | 2 | 44.h | odd | 10 | 1 | ||
| 275.4.a.b | 2 | 55.j | even | 10 | 1 | ||
| 275.4.b.c | 4 | 55.k | odd | 20 | 2 | ||
| 539.4.a.e | 2 | 77.j | odd | 10 | 1 | ||
| 704.4.a.n | 2 | 88.l | odd | 10 | 1 | ||
| 704.4.a.p | 2 | 88.o | even | 10 | 1 | ||
| 1089.4.a.v | 2 | 33.f | even | 10 | 1 | ||
| 1584.4.a.bc | 2 | 132.o | even | 10 | 1 | ||
| 1859.4.a.a | 2 | 143.n | even | 10 | 1 | ||
| 1936.4.a.w | 2 | 44.g | even | 10 | 1 | ||
| 2475.4.a.q | 2 | 165.o | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 6T_{2}^{6} + 16T_{2}^{5} + 44T_{2}^{4} - 32T_{2}^{3} + 24T_{2}^{2} - 16T_{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 4879681 \)
|
| $5$ |
\( T^{8} + \cdots + 1330863361 \)
|
| $7$ |
\( T^{8} + 20 T^{7} + \cdots + 7311616 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 25600000000 \)
|
| $17$ |
\( T^{8} + \cdots + 135530203361536 \)
|
| $19$ |
\( T^{8} + \cdots + 81\!\cdots\!56 \)
|
| $23$ |
\( (T^{2} + 98 T - 1487)^{4} \)
|
| $29$ |
\( T^{8} + \cdots + 318343244414976 \)
|
| $31$ |
\( T^{8} + \cdots + 18113272128961 \)
|
| $37$ |
\( T^{8} + \cdots + 83156680161 \)
|
| $41$ |
\( T^{8} + \cdots + 26\!\cdots\!76 \)
|
| $43$ |
\( (T^{2} + 60 T + 132)^{4} \)
|
| $47$ |
\( T^{8} + \cdots + 37\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots + 68\!\cdots\!96 \)
|
| $59$ |
\( T^{8} + \cdots + 54\!\cdots\!21 \)
|
| $61$ |
\( T^{8} + \cdots + 31\!\cdots\!76 \)
|
| $67$ |
\( (T^{2} - 754 T + 140929)^{4} \)
|
| $71$ |
\( T^{8} + \cdots + 90\!\cdots\!01 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + \cdots + 25\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 17\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + 1842 T + 525489)^{4} \)
|
| $97$ |
\( T^{8} + \cdots + 16\!\cdots\!01 \)
|
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