Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1850,2,Mod(149,1850)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1850, 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("1850.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1850 = 2 \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1850.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: | \(14.7723243739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3182656.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 370) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 25\nu^{4} + 5\nu^{3} - \nu^{2} + 362 ) / 124 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - \nu^{4} - 25\nu^{3} + 5\nu^{2} + 50 ) / 124 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} + \nu^{4} + 25\nu^{3} - 5\nu^{2} + 248\nu - 50 ) / 124 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{5} - 5\nu^{4} - \nu^{3} + 25\nu^{2} - 620\nu + 126 ) / 124 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -75\nu^{5} - 15\nu^{4} - 3\nu^{3} + 199\nu^{2} - 1860\nu + 378 ) / 124 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} + 5\beta_{3} - 5\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{2} + 5\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} - 16\beta_{4} - 25\beta_{3} - 25\beta_{2} - 2\beta _1 + 16 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1777\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 149.1 |
|
− | 1.00000i | − | 3.34596i | −1.00000 | 0 | −3.34596 | 2.59774i | 1.00000i | −8.19547 | 0 | ||||||||||||||||||||||||||||||||||
| 149.2 | − | 1.00000i | 0.406728i | −1.00000 | 0 | 0.406728 | − | 2.91729i | 1.00000i | 2.83457 | 0 | |||||||||||||||||||||||||||||||||||
| 149.3 | − | 1.00000i | 2.93923i | −1.00000 | 0 | 2.93923 | 1.31955i | 1.00000i | −5.63910 | 0 | ||||||||||||||||||||||||||||||||||||
| 149.4 | 1.00000i | − | 2.93923i | −1.00000 | 0 | 2.93923 | − | 1.31955i | − | 1.00000i | −5.63910 | 0 | ||||||||||||||||||||||||||||||||||
| 149.5 | 1.00000i | − | 0.406728i | −1.00000 | 0 | 0.406728 | 2.91729i | − | 1.00000i | 2.83457 | 0 | |||||||||||||||||||||||||||||||||||
| 149.6 | 1.00000i | 3.34596i | −1.00000 | 0 | −3.34596 | − | 2.59774i | − | 1.00000i | −8.19547 | 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 | 1850.2.b.o | 6 | |
| 5.b | even | 2 | 1 | inner | 1850.2.b.o | 6 | |
| 5.c | odd | 4 | 1 | 370.2.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1850.2.a.z | 3 | ||
| 15.e | even | 4 | 1 | 3330.2.a.bg | 3 | ||
| 20.e | even | 4 | 1 | 2960.2.a.u | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.a.g | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1850.2.a.z | 3 | 5.c | odd | 4 | 1 | ||
| 1850.2.b.o | 6 | 1.a | even | 1 | 1 | trivial | |
| 1850.2.b.o | 6 | 5.b | even | 2 | 1 | inner | |
| 2960.2.a.u | 3 | 20.e | even | 4 | 1 | ||
| 3330.2.a.bg | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1850, [\chi])\):
|
\( T_{3}^{6} + 20T_{3}^{4} + 100T_{3}^{2} + 16 \)
|
|
\( T_{7}^{6} + 17T_{7}^{4} + 84T_{7}^{2} + 100 \)
|
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\( T_{13}^{6} + 80T_{13}^{4} + 1600T_{13}^{2} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} + 20 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 17 T^{4} + \cdots + 100 \)
|
| $11$ |
\( (T^{3} - 11 T^{2} + 28 T + 8)^{2} \)
|
| $13$ |
\( T^{6} + 80 T^{4} + \cdots + 1024 \)
|
| $17$ |
\( T^{6} + 25 T^{4} + \cdots + 64 \)
|
| $19$ |
\( (T^{3} - 10 T - 4)^{2} \)
|
| $23$ |
\( T^{6} + 100 T^{4} + \cdots + 4096 \)
|
| $29$ |
\( (T^{3} - 5 T^{2} - 16 T + 76)^{2} \)
|
| $31$ |
\( (T^{3} - 3 T^{2} + \cdots + 270)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{3} \)
|
| $41$ |
\( (T^{3} + 9 T^{2} + \cdots - 364)^{2} \)
|
| $43$ |
\( T^{6} + 89 T^{4} + \cdots + 6400 \)
|
| $47$ |
\( T^{6} + 64 T^{4} + \cdots + 3136 \)
|
| $53$ |
\( T^{6} + 281 T^{4} + \cdots + 99856 \)
|
| $59$ |
\( (T^{3} + 14 T^{2} + \cdots - 80)^{2} \)
|
| $61$ |
\( (T^{3} - 7 T^{2} - 8 T + 4)^{2} \)
|
| $67$ |
\( T^{6} + 64 T^{4} + \cdots + 3136 \)
|
| $71$ |
\( (T^{3} - 22 T^{2} + \cdots + 640)^{2} \)
|
| $73$ |
\( T^{6} + 184 T^{4} + \cdots + 43264 \)
|
| $79$ |
\( (T^{3} - 26 T^{2} + \cdots - 224)^{2} \)
|
| $83$ |
\( T^{6} + 320 T^{4} + \cdots + 440896 \)
|
| $89$ |
\( (T + 6)^{6} \)
|
| $97$ |
\( T^{6} + 281 T^{4} + \cdots + 99856 \)
|
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