Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(122,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.122");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.b (of order \(2\), degree \(1\), 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: | \(9.89103359628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 122.1 |
|
− | 2.52434i | −2.68614 | + | 1.33591i | −2.37228 | 0.792287i | 3.37228 | + | 6.78073i | −6.74456 | − | 4.10891i | 5.43070 | − | 7.17687i | 2.00000 | ||||||||||||||||||||||
| 122.2 | − | 0.792287i | 0.186141 | − | 2.99422i | 3.37228 | 2.52434i | −2.37228 | − | 0.147477i | 4.74456 | − | 5.84096i | −8.93070 | − | 1.11469i | 2.00000 | |||||||||||||||||||||||
| 122.3 | 0.792287i | 0.186141 | + | 2.99422i | 3.37228 | − | 2.52434i | −2.37228 | + | 0.147477i | 4.74456 | 5.84096i | −8.93070 | + | 1.11469i | 2.00000 | ||||||||||||||||||||||||
| 122.4 | 2.52434i | −2.68614 | − | 1.33591i | −2.37228 | − | 0.792287i | 3.37228 | − | 6.78073i | −6.74456 | 4.10891i | 5.43070 | + | 7.17687i | 2.00000 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.3.b.h | 4 | |
| 3.b | odd | 2 | 1 | inner | 363.3.b.h | 4 | |
| 11.b | odd | 2 | 1 | 33.3.b.b | ✓ | 4 | |
| 11.c | even | 5 | 4 | 363.3.h.l | 16 | ||
| 11.d | odd | 10 | 4 | 363.3.h.m | 16 | ||
| 33.d | even | 2 | 1 | 33.3.b.b | ✓ | 4 | |
| 33.f | even | 10 | 4 | 363.3.h.m | 16 | ||
| 33.h | odd | 10 | 4 | 363.3.h.l | 16 | ||
| 44.c | even | 2 | 1 | 528.3.i.d | 4 | ||
| 132.d | odd | 2 | 1 | 528.3.i.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.3.b.b | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 33.3.b.b | ✓ | 4 | 33.d | even | 2 | 1 | |
| 363.3.b.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 363.3.b.h | 4 | 3.b | odd | 2 | 1 | inner | |
| 363.3.h.l | 16 | 11.c | even | 5 | 4 | ||
| 363.3.h.l | 16 | 33.h | odd | 10 | 4 | ||
| 363.3.h.m | 16 | 11.d | odd | 10 | 4 | ||
| 363.3.h.m | 16 | 33.f | even | 10 | 4 | ||
| 528.3.i.d | 4 | 44.c | even | 2 | 1 | ||
| 528.3.i.d | 4 | 132.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{4} + 7T_{2}^{2} + 4 \)
|
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\( T_{5}^{4} + 7T_{5}^{2} + 4 \)
|
|
\( T_{7}^{2} + 2T_{7} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $3$ |
\( T^{4} + 5 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $7$ |
\( (T^{2} + 2 T - 32)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} - 4 T - 128)^{2} \)
|
| $17$ |
\( T^{4} + 1372 T^{2} + 440896 \)
|
| $19$ |
\( (T^{2} - 18 T - 216)^{2} \)
|
| $23$ |
\( T^{4} + 1639 T^{2} + 662596 \)
|
| $29$ |
\( T^{4} + 3552 T^{2} + 1937664 \)
|
| $31$ |
\( (T^{2} - 23 T - 74)^{2} \)
|
| $37$ |
\( (T^{2} + 45 T + 102)^{2} \)
|
| $41$ |
\( T^{4} + 1264 T^{2} + 295936 \)
|
| $43$ |
\( (T^{2} + 48 T + 444)^{2} \)
|
| $47$ |
\( T^{4} + 2716 T^{2} + 1700416 \)
|
| $53$ |
\( T^{4} + 8124 T^{2} + 788544 \)
|
| $59$ |
\( T^{4} + 4003 T^{2} + 824464 \)
|
| $61$ |
\( (T^{2} - 12 T - 3264)^{2} \)
|
| $67$ |
\( (T^{2} + 29 T - 2174)^{2} \)
|
| $71$ |
\( T^{4} + 231T^{2} + 4356 \)
|
| $73$ |
\( (T^{2} - 142 T + 4744)^{2} \)
|
| $79$ |
\( (T^{2} - 38 T - 5216)^{2} \)
|
| $83$ |
\( T^{4} + 5436 T^{2} + 4981824 \)
|
| $89$ |
\( T^{4} + 20787 T^{2} + 4112784 \)
|
| $97$ |
\( (T^{2} - 109 T - 6014)^{2} \)
|
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