Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,7,Mod(7,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.7");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.c (of order \(4\), degree \(2\), 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: | \(11.5027041810\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{129})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 65x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 10) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 65x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 33\nu ) / 32 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 160\nu^{2} + 259\nu + 5216 ) / 32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 80\nu^{2} + 113\nu - 2608 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta _1 - 326 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -33\beta_{3} - 33\beta_{2} + 485\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−4.00000 | − | 4.00000i | −23.8945 | + | 23.8945i | 32.0000i | 0 | 191.156 | 362.840 | + | 362.840i | 128.000 | − | 128.000i | − | 412.898i | 0 | |||||||||||||||||||||
| 7.2 | −4.00000 | − | 4.00000i | 32.8945 | − | 32.8945i | 32.0000i | 0 | −263.156 | −261.840 | − | 261.840i | 128.000 | − | 128.000i | − | 1435.10i | 0 | ||||||||||||||||||||||
| 43.1 | −4.00000 | + | 4.00000i | −23.8945 | − | 23.8945i | − | 32.0000i | 0 | 191.156 | 362.840 | − | 362.840i | 128.000 | + | 128.000i | 412.898i | 0 | ||||||||||||||||||||||
| 43.2 | −4.00000 | + | 4.00000i | 32.8945 | + | 32.8945i | − | 32.0000i | 0 | −263.156 | −261.840 | + | 261.840i | 128.000 | + | 128.000i | 1435.10i | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.7.c.d | 4 | |
| 3.b | odd | 2 | 1 | 450.7.g.m | 4 | ||
| 5.b | even | 2 | 1 | 10.7.c.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 10.7.c.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 50.7.c.d | 4 | |
| 15.d | odd | 2 | 1 | 90.7.g.b | 4 | ||
| 15.e | even | 4 | 1 | 90.7.g.b | 4 | ||
| 15.e | even | 4 | 1 | 450.7.g.m | 4 | ||
| 20.d | odd | 2 | 1 | 80.7.p.c | 4 | ||
| 20.e | even | 4 | 1 | 80.7.p.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.7.c.b | ✓ | 4 | 5.b | even | 2 | 1 | |
| 10.7.c.b | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 50.7.c.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 50.7.c.d | 4 | 5.c | odd | 4 | 1 | inner | |
| 80.7.p.c | 4 | 20.d | odd | 2 | 1 | ||
| 80.7.p.c | 4 | 20.e | even | 4 | 1 | ||
| 90.7.g.b | 4 | 15.d | odd | 2 | 1 | ||
| 90.7.g.b | 4 | 15.e | even | 4 | 1 | ||
| 450.7.g.m | 4 | 3.b | odd | 2 | 1 | ||
| 450.7.g.m | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 18T_{3}^{3} + 162T_{3}^{2} + 28296T_{3} + 2471184 \)
acting on \(S_{7}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 32)^{2} \)
|
| $3$ |
\( T^{4} - 18 T^{3} + \cdots + 2471184 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 36104560144 \)
|
| $11$ |
\( (T^{2} + 1166 T - 205136)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 5177953764 \)
|
| $17$ |
\( T^{4} + \cdots + 246768848018884 \)
|
| $19$ |
\( T^{4} + \cdots + 14702623360000 \)
|
| $23$ |
\( T^{4} + \cdots + 21\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 990990400000000 \)
|
| $31$ |
\( (T^{2} + 9466 T - 370406936)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 57\!\cdots\!24 \)
|
| $41$ |
\( (T^{2} - 55114 T - 4599016976)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + 121686 T - 63858425376)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 60\!\cdots\!44 \)
|
| $71$ |
\( (T^{2} + 523066 T + 68302989064)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 77\!\cdots\!04 \)
|
| $79$ |
\( T^{4} + \cdots + 73\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
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