Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(1,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.a (trivial) |
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: | yes |
| Analytic conductor: | \(7.13923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{5}, \sqrt{37})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 21x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 11) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 21x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 13\nu + 8 ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 29\nu - 8 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} + \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - \beta_{2} - \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 13\beta_{2} + 29\beta _1 - 8 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.15942 | 4.54138 | 18.6196 | −8.63533 | −23.4309 | −1.66258 | −54.7907 | −6.37586 | 44.5532 | ||||||||||||||||||||||||||||||
| 1.2 | −2.92335 | 4.54138 | 0.545959 | 6.17671 | −13.2760 | −32.1271 | 21.7907 | −6.37586 | −18.0567 | |||||||||||||||||||||||||||||||
| 1.3 | 0.923347 | −1.54138 | −7.14743 | 8.72550 | −1.42323 | −0.775116 | −13.9863 | −24.6241 | 8.05666 | |||||||||||||||||||||||||||||||
| 1.4 | 3.15942 | −1.54138 | 1.98190 | −17.2669 | −4.86986 | 9.56478 | −19.0137 | −24.6241 | −54.5532 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.4.a.f | 4 | |
| 3.b | odd | 2 | 1 | 1089.4.a.bh | 4 | ||
| 4.b | odd | 2 | 1 | 1936.4.a.bl | 4 | ||
| 11.b | odd | 2 | 1 | 121.4.a.g | 4 | ||
| 11.c | even | 5 | 2 | 121.4.c.b | 8 | ||
| 11.c | even | 5 | 2 | 121.4.c.i | 8 | ||
| 11.d | odd | 10 | 2 | 11.4.c.a | ✓ | 8 | |
| 11.d | odd | 10 | 2 | 121.4.c.h | 8 | ||
| 33.d | even | 2 | 1 | 1089.4.a.y | 4 | ||
| 33.f | even | 10 | 2 | 99.4.f.c | 8 | ||
| 44.c | even | 2 | 1 | 1936.4.a.bk | 4 | ||
| 44.g | even | 10 | 2 | 176.4.m.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.4.c.a | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 99.4.f.c | 8 | 33.f | even | 10 | 2 | ||
| 121.4.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 121.4.a.g | 4 | 11.b | odd | 2 | 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 | 11.c | even | 5 | 2 | ||
| 176.4.m.c | 8 | 44.g | even | 10 | 2 | ||
| 1089.4.a.y | 4 | 33.d | even | 2 | 1 | ||
| 1089.4.a.bh | 4 | 3.b | odd | 2 | 1 | ||
| 1936.4.a.bk | 4 | 44.c | even | 2 | 1 | ||
| 1936.4.a.bl | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 15T_{2}^{2} - 38T_{2} + 44 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 44 \)
|
| $3$ |
\( (T^{2} - 3 T - 7)^{2} \)
|
| $5$ |
\( T^{4} + 11 T^{3} + \cdots + 8036 \)
|
| $7$ |
\( T^{4} + 25 T^{3} + \cdots - 396 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 25 T^{3} + \cdots - 53900 \)
|
| $17$ |
\( T^{4} + 232 T^{3} + \cdots + 3099789 \)
|
| $19$ |
\( T^{4} + 154 T^{3} + \cdots - 9492329 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 49883584 \)
|
| $29$ |
\( T^{4} + 363 T^{3} + \cdots - 98528364 \)
|
| $31$ |
\( T^{4} - 37 T^{3} + \cdots - 33222196 \)
|
| $37$ |
\( T^{4} - 93 T^{3} + \cdots + 163244164 \)
|
| $41$ |
\( T^{4} + \cdots + 1659084581 \)
|
| $43$ |
\( T^{4} + \cdots - 1288748736 \)
|
| $47$ |
\( T^{4} + 869 T^{3} + \cdots + 837687536 \)
|
| $53$ |
\( T^{4} - 811 T^{3} + \cdots - 847714576 \)
|
| $59$ |
\( T^{4} - 178 T^{3} + \cdots + 258148219 \)
|
| $61$ |
\( T^{4} + \cdots + 47885889744 \)
|
| $67$ |
\( T^{4} + \cdots - 8869996224 \)
|
| $71$ |
\( T^{4} - 629 T^{3} + \cdots - 744521796 \)
|
| $73$ |
\( T^{4} + \cdots + 38050128809 \)
|
| $79$ |
\( T^{4} + 977 T^{3} + \cdots - 210591964 \)
|
| $83$ |
\( T^{4} + \cdots - 181259356509 \)
|
| $89$ |
\( T^{4} + \cdots - 2046678844 \)
|
| $97$ |
\( T^{4} + \cdots - 357121332099 \)
|
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