Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,3,Mod(51,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.51");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.x (of order \(10\), degree \(4\), 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: | \(7.49320726991\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(\zeta_{10}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 51.1 |
|
0.690983 | − | 0.951057i | −1.38197 | + | 4.25325i | 0.809017 | + | 2.48990i | 0 | 3.09017 | + | 4.25325i | −7.39919 | + | 2.40414i | 7.39919 | + | 2.40414i | −8.89919 | − | 6.46564i | 0 | ||||||||||||||||
| 101.1 | 1.80902 | + | 0.587785i | −3.61803 | + | 2.62866i | −0.309017 | − | 0.224514i | 0 | −8.09017 | + | 2.62866i | 4.89919 | − | 6.74315i | −4.89919 | − | 6.74315i | 3.39919 | − | 10.4616i | 0 | |||||||||||||||||
| 151.1 | 0.690983 | + | 0.951057i | −1.38197 | − | 4.25325i | 0.809017 | − | 2.48990i | 0 | 3.09017 | − | 4.25325i | −7.39919 | − | 2.40414i | 7.39919 | − | 2.40414i | −8.89919 | + | 6.46564i | 0 | |||||||||||||||||
| 226.1 | 1.80902 | − | 0.587785i | −3.61803 | − | 2.62866i | −0.309017 | + | 0.224514i | 0 | −8.09017 | − | 2.62866i | 4.89919 | + | 6.74315i | −4.89919 | + | 6.74315i | 3.39919 | + | 10.4616i | 0 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.d | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.3.x.c | yes | 4 |
| 5.b | even | 2 | 1 | 275.3.x.b | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 275.3.q.b | 8 | ||
| 11.d | odd | 10 | 1 | inner | 275.3.x.c | yes | 4 |
| 55.h | odd | 10 | 1 | 275.3.x.b | ✓ | 4 | |
| 55.l | even | 20 | 2 | 275.3.q.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 275.3.q.b | 8 | 5.c | odd | 4 | 2 | ||
| 275.3.q.b | 8 | 55.l | even | 20 | 2 | ||
| 275.3.x.b | ✓ | 4 | 5.b | even | 2 | 1 | |
| 275.3.x.b | ✓ | 4 | 55.h | odd | 10 | 1 | |
| 275.3.x.c | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 275.3.x.c | yes | 4 | 11.d | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(275, [\chi])\):
|
\( T_{2}^{4} - 5T_{2}^{3} + 10T_{2}^{2} - 10T_{2} + 5 \)
|
|
\( T_{3}^{4} + 10T_{3}^{3} + 60T_{3}^{2} + 200T_{3} + 400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 5 \)
|
| $3$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 5 T^{3} + \cdots + 4205 \)
|
| $11$ |
\( T^{4} - T^{3} + \cdots + 14641 \)
|
| $13$ |
\( T^{4} - 30 T^{3} + \cdots + 15125 \)
|
| $17$ |
\( T^{4} - 35 T^{3} + \cdots + 12005 \)
|
| $19$ |
\( T^{4} + 50 T^{3} + \cdots + 67280 \)
|
| $23$ |
\( (T^{2} + 25 T - 205)^{2} \)
|
| $29$ |
\( T^{4} - 60 T^{3} + \cdots + 17405 \)
|
| $31$ |
\( T^{4} + 43 T^{3} + \cdots + 1560001 \)
|
| $37$ |
\( T^{4} + 30 T^{3} + \cdots + 9954025 \)
|
| $41$ |
\( T^{4} - 95 T^{3} + \cdots + 3880805 \)
|
| $43$ |
\( T^{4} + 6130 T^{2} + 7925405 \)
|
| $47$ |
\( T^{4} - 35 T^{3} + \cdots + 42025 \)
|
| $53$ |
\( T^{4} - 10 T^{3} + \cdots + 9610000 \)
|
| $59$ |
\( T^{4} - 33 T^{3} + \cdots + 7921 \)
|
| $61$ |
\( T^{4} - 85 T^{3} + \cdots + 8179205 \)
|
| $67$ |
\( (T^{2} - 170 T + 7145)^{2} \)
|
| $71$ |
\( T^{4} + 137 T^{3} + \cdots + 22005481 \)
|
| $73$ |
\( T^{4} + 110 T^{3} + \cdots + 3519605 \)
|
| $79$ |
\( T^{4} - 70 T^{3} + \cdots + 41789405 \)
|
| $83$ |
\( T^{4} + 115 T^{3} + \cdots + 4041005 \)
|
| $89$ |
\( (T^{2} + 159 T + 5659)^{2} \)
|
| $97$ |
\( T^{4} - 155 T^{3} + \cdots + 29757025 \)
|
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