Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,4,Mod(3,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, 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("242.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(14.2784622214\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).
| \(n\) | \(123\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) |
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 |
|
−1.61803 | + | 1.17557i | −0.972136 | − | 2.99193i | 1.23607 | − | 3.80423i | −0.500000 | − | 0.363271i | 5.09017 | + | 3.69822i | −3.02786 | + | 9.31881i | 2.47214 | + | 7.60845i | 13.8369 | − | 10.0531i | 1.23607 | ||||||||||||||
| 9.1 | 0.618034 | − | 1.90211i | 7.97214 | − | 5.79210i | −3.23607 | − | 2.35114i | −0.500000 | − | 1.53884i | −6.09017 | − | 18.7436i | −11.9721 | − | 8.69827i | −6.47214 | + | 4.70228i | 21.6631 | − | 66.6722i | −3.23607 | |||||||||||||||
| 27.1 | 0.618034 | + | 1.90211i | 7.97214 | + | 5.79210i | −3.23607 | + | 2.35114i | −0.500000 | + | 1.53884i | −6.09017 | + | 18.7436i | −11.9721 | + | 8.69827i | −6.47214 | − | 4.70228i | 21.6631 | + | 66.6722i | −3.23607 | |||||||||||||||
| 81.1 | −1.61803 | − | 1.17557i | −0.972136 | + | 2.99193i | 1.23607 | + | 3.80423i | −0.500000 | + | 0.363271i | 5.09017 | − | 3.69822i | −3.02786 | − | 9.31881i | 2.47214 | − | 7.60845i | 13.8369 | + | 10.0531i | 1.23607 | |||||||||||||||
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 | 242.4.c.f | 4 | |
| 11.b | odd | 2 | 1 | 242.4.c.m | 4 | ||
| 11.c | even | 5 | 2 | 22.4.c.a | ✓ | 4 | |
| 11.c | even | 5 | 1 | 242.4.a.k | 2 | ||
| 11.c | even | 5 | 1 | inner | 242.4.c.f | 4 | |
| 11.d | odd | 10 | 1 | 242.4.a.h | 2 | ||
| 11.d | odd | 10 | 2 | 242.4.c.j | 4 | ||
| 11.d | odd | 10 | 1 | 242.4.c.m | 4 | ||
| 33.f | even | 10 | 1 | 2178.4.a.bi | 2 | ||
| 33.h | odd | 10 | 2 | 198.4.f.b | 4 | ||
| 33.h | odd | 10 | 1 | 2178.4.a.z | 2 | ||
| 44.g | even | 10 | 1 | 1936.4.a.bc | 2 | ||
| 44.h | odd | 10 | 2 | 176.4.m.a | 4 | ||
| 44.h | odd | 10 | 1 | 1936.4.a.bb | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.4.c.a | ✓ | 4 | 11.c | even | 5 | 2 | |
| 176.4.m.a | 4 | 44.h | odd | 10 | 2 | ||
| 198.4.f.b | 4 | 33.h | odd | 10 | 2 | ||
| 242.4.a.h | 2 | 11.d | odd | 10 | 1 | ||
| 242.4.a.k | 2 | 11.c | even | 5 | 1 | ||
| 242.4.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 242.4.c.f | 4 | 11.c | even | 5 | 1 | inner | |
| 242.4.c.j | 4 | 11.d | odd | 10 | 2 | ||
| 242.4.c.m | 4 | 11.b | odd | 2 | 1 | ||
| 242.4.c.m | 4 | 11.d | odd | 10 | 1 | ||
| 1936.4.a.bb | 2 | 44.h | odd | 10 | 1 | ||
| 1936.4.a.bc | 2 | 44.g | even | 10 | 1 | ||
| 2178.4.a.z | 2 | 33.h | odd | 10 | 1 | ||
| 2178.4.a.bi | 2 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):
|
\( T_{3}^{4} - 14T_{3}^{3} + 76T_{3}^{2} + 31T_{3} + 961 \)
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\( T_{5}^{4} + 2T_{5}^{3} + 4T_{5}^{2} + 3T_{5} + 1 \)
|
|
\( T_{7}^{4} + 30T_{7}^{3} + 460T_{7}^{2} + 3625T_{7} + 21025 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} - 14 T^{3} + \cdots + 961 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} + 30 T^{3} + \cdots + 21025 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 54 T^{3} + \cdots + 502681 \)
|
| $17$ |
\( T^{4} - 158 T^{3} + \cdots + 52983841 \)
|
| $19$ |
\( T^{4} - 14 T^{3} + \cdots + 1515361 \)
|
| $23$ |
\( (T^{2} + 112 T + 2416)^{2} \)
|
| $29$ |
\( T^{4} - 370 T^{3} + \cdots + 951414025 \)
|
| $31$ |
\( T^{4} - 488 T^{3} + \cdots + 72777961 \)
|
| $37$ |
\( T^{4} - 334 T^{3} + \cdots + 117310561 \)
|
| $41$ |
\( T^{4} + 298 T^{3} + \cdots + 408080401 \)
|
| $43$ |
\( (T^{2} - 580 T + 78320)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 1509400201 \)
|
| $53$ |
\( T^{4} + \cdots + 1259895025 \)
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| $59$ |
\( T^{4} + \cdots + 11599505401 \)
|
| $61$ |
\( T^{4} + \cdots + 46442405025 \)
|
| $67$ |
\( (T^{2} + 44 T - 16336)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 5373623025 \)
|
| $73$ |
\( T^{4} - 102 T^{3} + \cdots + 6765201 \)
|
| $79$ |
\( T^{4} + \cdots + 6794540041 \)
|
| $83$ |
\( T^{4} + \cdots + 36760776361 \)
|
| $89$ |
\( (T^{2} - 864 T - 917876)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 181211124721 \)
|
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