Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [575,2,Mod(24,575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("575.24");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 575.b (of order \(2\), degree \(1\), 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: | \(4.59139811622\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.60060285184.11 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 12x^{6} + 40x^{4} + 41x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 115) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 12x^{6} + 40x^{4} + 41x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{6} + 19\nu^{4} + 36\nu^{2} + 6 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{6} - 19\nu^{4} - 29\nu^{2} + 15 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 6\nu^{5} + 24\nu^{3} + 95\nu ) / 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 13\nu^{4} - 46\nu^{2} - 31 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} + 32\nu^{5} + 82\nu^{3} + 51\nu ) / 14 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 13\nu^{5} + 46\nu^{3} + 38\nu ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} - 8\beta_{3} - 9\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{7} - 10\beta_{6} - 8\beta_{4} + 31\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 19\beta_{5} + 58\beta_{3} + 71\beta_{2} - 101 \)
|
| \(\nu^{7}\) | \(=\) |
\( -90\beta_{7} + 84\beta_{6} + 58\beta_{4} - 211\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 2.69353i | − | 2.56155i | −5.25508 | 0 | −6.89961 | 2.74252i | 8.76763i | −3.56155 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 24.2 | − | 1.69353i | 2.56155i | −0.868028 | 0 | 4.33805 | − | 0.819031i | − | 1.91702i | −3.56155 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 24.3 | − | 1.32973i | 1.56155i | 0.231826 | 0 | 2.07644 | 3.50407i | − | 2.96772i | 0.561553 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.4 | − | 0.329727i | − | 1.56155i | 1.89128 | 0 | −0.514886 | 4.06562i | − | 1.28306i | 0.561553 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 24.5 | 0.329727i | 1.56155i | 1.89128 | 0 | −0.514886 | − | 4.06562i | 1.28306i | 0.561553 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 24.6 | 1.32973i | − | 1.56155i | 0.231826 | 0 | 2.07644 | − | 3.50407i | 2.96772i | 0.561553 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.7 | 1.69353i | − | 2.56155i | −0.868028 | 0 | 4.33805 | 0.819031i | 1.91702i | −3.56155 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 24.8 | 2.69353i | 2.56155i | −5.25508 | 0 | −6.89961 | − | 2.74252i | − | 8.76763i | −3.56155 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 575.2.b.e | 8 | |
| 5.b | even | 2 | 1 | inner | 575.2.b.e | 8 | |
| 5.c | odd | 4 | 1 | 115.2.a.c | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 575.2.a.h | 4 | ||
| 15.e | even | 4 | 1 | 1035.2.a.o | 4 | ||
| 15.e | even | 4 | 1 | 5175.2.a.bx | 4 | ||
| 20.e | even | 4 | 1 | 1840.2.a.u | 4 | ||
| 20.e | even | 4 | 1 | 9200.2.a.cl | 4 | ||
| 35.f | even | 4 | 1 | 5635.2.a.v | 4 | ||
| 40.i | odd | 4 | 1 | 7360.2.a.cj | 4 | ||
| 40.k | even | 4 | 1 | 7360.2.a.cg | 4 | ||
| 115.e | even | 4 | 1 | 2645.2.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.2.a.c | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 575.2.a.h | 4 | 5.c | odd | 4 | 1 | ||
| 575.2.b.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 575.2.b.e | 8 | 5.b | even | 2 | 1 | inner | |
| 1035.2.a.o | 4 | 15.e | even | 4 | 1 | ||
| 1840.2.a.u | 4 | 20.e | even | 4 | 1 | ||
| 2645.2.a.m | 4 | 115.e | even | 4 | 1 | ||
| 5175.2.a.bx | 4 | 15.e | even | 4 | 1 | ||
| 5635.2.a.v | 4 | 35.f | even | 4 | 1 | ||
| 7360.2.a.cg | 4 | 40.k | even | 4 | 1 | ||
| 7360.2.a.cj | 4 | 40.i | odd | 4 | 1 | ||
| 9200.2.a.cl | 4 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 12T_{2}^{6} + 40T_{2}^{4} + 41T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 12 T^{6} + \cdots + 4 \)
|
| $3$ |
\( (T^{4} + 9 T^{2} + 16)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 37 T^{6} + \cdots + 1024 \)
|
| $11$ |
\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots + 32)^{2} \)
|
| $13$ |
\( (T^{4} + 41 T^{2} + 212)^{2} \)
|
| $17$ |
\( T^{8} + 37 T^{6} + \cdots + 1024 \)
|
| $19$ |
\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots + 32)^{2} \)
|
| $23$ |
\( (T^{2} + 1)^{4} \)
|
| $29$ |
\( (T^{4} + 19 T^{3} + \cdots + 202)^{2} \)
|
| $31$ |
\( (T^{4} + T^{3} - 101 T^{2} + \cdots + 2144)^{2} \)
|
| $37$ |
\( T^{8} + 241 T^{6} + \cdots + 4032064 \)
|
| $41$ |
\( (T^{4} - 13 T^{3} + \cdots - 94)^{2} \)
|
| $43$ |
\( T^{8} + 108 T^{6} + \cdots + 16384 \)
|
| $47$ |
\( T^{8} + 202 T^{6} + \cdots + 16384 \)
|
| $53$ |
\( T^{8} + 429 T^{6} + \cdots + 77018176 \)
|
| $59$ |
\( (T^{4} + 23 T^{3} + \cdots - 3136)^{2} \)
|
| $61$ |
\( (T^{4} - 56 T^{2} + \cdots - 32)^{2} \)
|
| $67$ |
\( T^{8} + 205 T^{6} + \cdots + 4129024 \)
|
| $71$ |
\( (T^{4} + 3 T^{3} - 149 T^{2} + \cdots - 8)^{2} \)
|
| $73$ |
\( T^{8} + 338 T^{6} + \cdots + 2835856 \)
|
| $79$ |
\( (T^{4} + 2 T^{3} + \cdots + 512)^{2} \)
|
| $83$ |
\( T^{8} + 249 T^{6} + \cdots + 1478656 \)
|
| $89$ |
\( (T^{4} - 216 T^{2} + \cdots - 2752)^{2} \)
|
| $97$ |
\( T^{8} + 180 T^{6} + \cdots + 1149184 \)
|
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