Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,4,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, 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("242.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(14.2784622214\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.978025.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 11 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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
\( x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 7\nu - 54 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 2\nu^{2} + 57\nu + 146 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 9\nu^{2} - 46\nu + 400 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 11\beta _1 + 6 ) / 11 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{3} - 7\beta_{2} + 11\beta _1 + 552 ) / 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 71\beta_{3} - 17\beta_{2} + 605\beta _1 + 844 ) / 11 \)
|
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 |
|
2.00000 | −7.81182 | 4.00000 | 14.9181 | −15.6236 | −21.7679 | 8.00000 | 34.0245 | 29.8363 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −4.30892 | 4.00000 | −8.06215 | −8.61784 | 26.0792 | 8.00000 | −8.43321 | −16.1243 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 7.57575 | 4.00000 | 5.40810 | 15.1515 | 22.1498 | 8.00000 | 30.3919 | 10.8162 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 8.54499 | 4.00000 | 12.7359 | 17.0900 | −23.4611 | 8.00000 | 46.0168 | 25.4718 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 242.4.a.o | 4 | |
| 3.b | odd | 2 | 1 | 2178.4.a.bt | 4 | ||
| 4.b | odd | 2 | 1 | 1936.4.a.bm | 4 | ||
| 11.b | odd | 2 | 1 | 242.4.a.n | 4 | ||
| 11.c | even | 5 | 2 | 242.4.c.n | 8 | ||
| 11.c | even | 5 | 2 | 242.4.c.q | 8 | ||
| 11.d | odd | 10 | 2 | 22.4.c.b | ✓ | 8 | |
| 11.d | odd | 10 | 2 | 242.4.c.r | 8 | ||
| 33.d | even | 2 | 1 | 2178.4.a.by | 4 | ||
| 33.f | even | 10 | 2 | 198.4.f.d | 8 | ||
| 44.c | even | 2 | 1 | 1936.4.a.bn | 4 | ||
| 44.g | even | 10 | 2 | 176.4.m.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.4.c.b | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 176.4.m.b | 8 | 44.g | even | 10 | 2 | ||
| 198.4.f.d | 8 | 33.f | even | 10 | 2 | ||
| 242.4.a.n | 4 | 11.b | odd | 2 | 1 | ||
| 242.4.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 242.4.c.n | 8 | 11.c | even | 5 | 2 | ||
| 242.4.c.q | 8 | 11.c | even | 5 | 2 | ||
| 242.4.c.r | 8 | 11.d | odd | 10 | 2 | ||
| 1936.4.a.bm | 4 | 4.b | odd | 2 | 1 | ||
| 1936.4.a.bn | 4 | 44.c | even | 2 | 1 | ||
| 2178.4.a.bt | 4 | 3.b | odd | 2 | 1 | ||
| 2178.4.a.by | 4 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{4} - 4T_{3}^{3} - 97T_{3}^{2} + 242T_{3} + 2179 \)
|
|
\( T_{5}^{4} - 25T_{5}^{3} + 73T_{5}^{2} + 1710T_{5} - 8284 \)
|
|
\( T_{7}^{4} - 3T_{7}^{3} - 1093T_{7}^{2} + 1496T_{7} + 295004 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 2179 \)
|
| $5$ |
\( T^{4} - 25 T^{3} + \cdots - 8284 \)
|
| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 295004 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 41 T^{3} + \cdots + 126580 \)
|
| $17$ |
\( T^{4} - 52 T^{3} + \cdots - 2019455 \)
|
| $19$ |
\( T^{4} - 16 T^{3} + \cdots + 1910345 \)
|
| $23$ |
\( T^{4} - 314 T^{3} + \cdots - 257882816 \)
|
| $29$ |
\( T^{4} - 561 T^{3} + \cdots + 102492820 \)
|
| $31$ |
\( T^{4} - 199 T^{3} + \cdots - 922892780 \)
|
| $37$ |
\( T^{4} + \cdots + 1010724084 \)
|
| $41$ |
\( T^{4} - 32 T^{3} + \cdots + 777346421 \)
|
| $43$ |
\( T^{4} + \cdots - 2713875120 \)
|
| $47$ |
\( T^{4} - 403 T^{3} + \cdots + 314788720 \)
|
| $53$ |
\( T^{4} + 133 T^{3} + \cdots + 628715536 \)
|
| $59$ |
\( T^{4} - 1016 T^{3} + \cdots + 184247305 \)
|
| $61$ |
\( T^{4} + \cdots - 4825318480 \)
|
| $67$ |
\( T^{4} + \cdots + 35027256944 \)
|
| $71$ |
\( T^{4} + \cdots + 4828358036 \)
|
| $73$ |
\( T^{4} + \cdots + 17904806629 \)
|
| $79$ |
\( T^{4} + \cdots + 26212027500 \)
|
| $83$ |
\( T^{4} + \cdots + 40525024445 \)
|
| $89$ |
\( T^{4} + \cdots - 64943655580 \)
|
| $97$ |
\( T^{4} + \cdots + 216660545541 \)
|
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