Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [35,4,Mod(11,35)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 35.e (of order \(3\), 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: | \(2.06506685020\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
0.792893 | + | 1.37333i | 2.62132 | − | 4.54026i | 2.74264 | − | 4.75039i | −2.50000 | − | 4.33013i | 8.31371 | −5.10660 | + | 17.8023i | 21.3848 | −0.242641 | − | 0.420266i | 3.96447 | − | 6.86666i | ||||||||||||||||
| 11.2 | 2.20711 | + | 3.82282i | −1.62132 | + | 2.80821i | −5.74264 | + | 9.94655i | −2.50000 | − | 4.33013i | −14.3137 | 16.1066 | − | 9.14207i | −15.3848 | 8.24264 | + | 14.2767i | 11.0355 | − | 19.1141i | |||||||||||||||||
| 16.1 | 0.792893 | − | 1.37333i | 2.62132 | + | 4.54026i | 2.74264 | + | 4.75039i | −2.50000 | + | 4.33013i | 8.31371 | −5.10660 | − | 17.8023i | 21.3848 | −0.242641 | + | 0.420266i | 3.96447 | + | 6.86666i | |||||||||||||||||
| 16.2 | 2.20711 | − | 3.82282i | −1.62132 | − | 2.80821i | −5.74264 | − | 9.94655i | −2.50000 | + | 4.33013i | −14.3137 | 16.1066 | + | 9.14207i | −15.3848 | 8.24264 | − | 14.2767i | 11.0355 | + | 19.1141i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 35.4.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 315.4.j.c | 4 | ||
| 4.b | odd | 2 | 1 | 560.4.q.i | 4 | ||
| 5.b | even | 2 | 1 | 175.4.e.c | 4 | ||
| 5.c | odd | 4 | 2 | 175.4.k.c | 8 | ||
| 7.b | odd | 2 | 1 | 245.4.e.l | 4 | ||
| 7.c | even | 3 | 1 | inner | 35.4.e.b | ✓ | 4 |
| 7.c | even | 3 | 1 | 245.4.a.g | 2 | ||
| 7.d | odd | 6 | 1 | 245.4.a.h | 2 | ||
| 7.d | odd | 6 | 1 | 245.4.e.l | 4 | ||
| 21.g | even | 6 | 1 | 2205.4.a.bg | 2 | ||
| 21.h | odd | 6 | 1 | 315.4.j.c | 4 | ||
| 21.h | odd | 6 | 1 | 2205.4.a.bf | 2 | ||
| 28.g | odd | 6 | 1 | 560.4.q.i | 4 | ||
| 35.i | odd | 6 | 1 | 1225.4.a.v | 2 | ||
| 35.j | even | 6 | 1 | 175.4.e.c | 4 | ||
| 35.j | even | 6 | 1 | 1225.4.a.x | 2 | ||
| 35.l | odd | 12 | 2 | 175.4.k.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 35.4.e.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 175.4.e.c | 4 | 5.b | even | 2 | 1 | ||
| 175.4.e.c | 4 | 35.j | even | 6 | 1 | ||
| 175.4.k.c | 8 | 5.c | odd | 4 | 2 | ||
| 175.4.k.c | 8 | 35.l | odd | 12 | 2 | ||
| 245.4.a.g | 2 | 7.c | even | 3 | 1 | ||
| 245.4.a.h | 2 | 7.d | odd | 6 | 1 | ||
| 245.4.e.l | 4 | 7.b | odd | 2 | 1 | ||
| 245.4.e.l | 4 | 7.d | odd | 6 | 1 | ||
| 315.4.j.c | 4 | 3.b | odd | 2 | 1 | ||
| 315.4.j.c | 4 | 21.h | odd | 6 | 1 | ||
| 560.4.q.i | 4 | 4.b | odd | 2 | 1 | ||
| 560.4.q.i | 4 | 28.g | odd | 6 | 1 | ||
| 1225.4.a.v | 2 | 35.i | odd | 6 | 1 | ||
| 1225.4.a.x | 2 | 35.j | even | 6 | 1 | ||
| 2205.4.a.bf | 2 | 21.h | odd | 6 | 1 | ||
| 2205.4.a.bg | 2 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 6T_{2}^{3} + 29T_{2}^{2} - 42T_{2} + 49 \)
acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 49 \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 289 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} - 22 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} + 28 T^{3} + \cdots + 16 \)
|
| $13$ |
\( (T^{2} + 36 T + 124)^{2} \)
|
| $17$ |
\( T^{4} - 76 T^{3} + \cdots + 19254544 \)
|
| $19$ |
\( T^{4} + 160 T^{3} + \cdots + 25482304 \)
|
| $23$ |
\( T^{4} - 22 T^{3} + \cdots + 742944049 \)
|
| $29$ |
\( (T^{2} + 250 T + 8425)^{2} \)
|
| $31$ |
\( T^{4} - 132 T^{3} + \cdots + 244734736 \)
|
| $37$ |
\( T^{4} - 416 T^{3} + \cdots + 39337984 \)
|
| $41$ |
\( (T^{2} + 106 T + 1009)^{2} \)
|
| $43$ |
\( (T^{2} + 666 T + 110839)^{2} \)
|
| $47$ |
\( T^{4} - 196 T^{3} + \cdots + 1993744 \)
|
| $53$ |
\( T^{4} + \cdots + 34688317504 \)
|
| $59$ |
\( T^{4} + \cdots + 1217172544 \)
|
| $61$ |
\( T^{4} + \cdots + 419125465201 \)
|
| $67$ |
\( T^{4} + \cdots + 163157021329 \)
|
| $71$ |
\( (T^{2} - 1064 T + 38024)^{2} \)
|
| $73$ |
\( T^{4} - 172 T^{3} + \cdots + 418284304 \)
|
| $79$ |
\( T^{4} + \cdots + 85736524864 \)
|
| $83$ |
\( (T^{2} + 1906 T + 908159)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 10884122929 \)
|
| $97$ |
\( (T^{2} + 628 T - 401404)^{2} \)
|
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