Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,2,Mod(101,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.d (of order \(2\), degree \(1\), 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: | \(3.39364208590\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.93924352.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 17x^{4} + 73x^{2} + 67 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} + 17x^{4} + 73x^{2} + 67 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} + 21\nu^{2} + 18 ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 19\nu^{2} - 60 ) / 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 19\nu^{3} + 77\nu ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} + 40\nu^{3} + 95\nu ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{3} - \beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21\beta_{3} + 19\beta_{2} + 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{5} - 40\beta_{4} + 75\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
−2.17009 | − | 2.21547i | 2.70928 | 0 | 4.80775i | 1.02091i | −1.53919 | −1.90829 | 0 | |||||||||||||||||||||||||||||||||||
| 101.2 | −2.17009 | 2.21547i | 2.70928 | 0 | − | 4.80775i | − | 1.02091i | −1.53919 | −1.90829 | 0 | |||||||||||||||||||||||||||||||||||
| 101.3 | −0.311108 | − | 1.12261i | −1.90321 | 0 | 0.349253i | 3.60843i | 1.21432 | 1.73975 | 0 | ||||||||||||||||||||||||||||||||||||
| 101.4 | −0.311108 | 1.12261i | −1.90321 | 0 | − | 0.349253i | − | 3.60843i | 1.21432 | 1.73975 | 0 | |||||||||||||||||||||||||||||||||||
| 101.5 | 1.48119 | − | 3.29112i | 0.193937 | 0 | − | 4.87478i | − | 2.22194i | −2.67513 | −7.83146 | 0 | ||||||||||||||||||||||||||||||||||
| 101.6 | 1.48119 | 3.29112i | 0.193937 | 0 | 4.87478i | 2.22194i | −2.67513 | −7.83146 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 425.2.d.a | ✓ | 6 |
| 5.b | even | 2 | 1 | 425.2.d.b | yes | 6 | |
| 5.c | odd | 4 | 2 | 425.2.c.c | 12 | ||
| 17.b | even | 2 | 1 | inner | 425.2.d.a | ✓ | 6 |
| 17.c | even | 4 | 2 | 7225.2.a.bg | 6 | ||
| 85.c | even | 2 | 1 | 425.2.d.b | yes | 6 | |
| 85.g | odd | 4 | 2 | 425.2.c.c | 12 | ||
| 85.j | even | 4 | 2 | 7225.2.a.ba | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 425.2.c.c | 12 | 5.c | odd | 4 | 2 | ||
| 425.2.c.c | 12 | 85.g | odd | 4 | 2 | ||
| 425.2.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 425.2.d.a | ✓ | 6 | 17.b | even | 2 | 1 | inner |
| 425.2.d.b | yes | 6 | 5.b | even | 2 | 1 | |
| 425.2.d.b | yes | 6 | 85.c | even | 2 | 1 | |
| 7225.2.a.ba | 6 | 85.j | even | 4 | 2 | ||
| 7225.2.a.bg | 6 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \)
|
|
\( T_{3}^{6} + 17T_{3}^{4} + 73T_{3}^{2} + 67 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 3 T - 1)^{2} \)
|
| $3$ |
\( T^{6} + 17 T^{4} + \cdots + 67 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 19 T^{4} + \cdots + 67 \)
|
| $11$ |
\( T^{6} + 66 T^{4} + \cdots + 6700 \)
|
| $13$ |
\( (T^{3} - T^{2} - 13 T + 23)^{2} \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} - 6 T^{2} + \cdots + 100)^{2} \)
|
| $23$ |
\( T^{6} + 54 T^{4} + \cdots + 268 \)
|
| $29$ |
\( T^{6} + 116 T^{4} + \cdots + 1072 \)
|
| $31$ |
\( T^{6} + 89 T^{4} + \cdots + 1675 \)
|
| $37$ |
\( T^{6} + 144 T^{4} + \cdots + 1072 \)
|
| $41$ |
\( T^{6} + 100 T^{4} + \cdots + 26800 \)
|
| $43$ |
\( (T^{3} + 4 T^{2} + \cdots - 452)^{2} \)
|
| $47$ |
\( (T^{3} - 20 T^{2} + \cdots - 208)^{2} \)
|
| $53$ |
\( (T^{3} + 5 T^{2} - 69 T + 43)^{2} \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 44 T - 20)^{2} \)
|
| $61$ |
\( T^{6} + 228 T^{4} + \cdots + 107200 \)
|
| $67$ |
\( (T^{3} - 12 T^{2} + \cdots + 16)^{2} \)
|
| $71$ |
\( T^{6} + 91 T^{4} + \cdots + 1675 \)
|
| $73$ |
\( T^{6} + 32 T^{4} + \cdots + 1072 \)
|
| $79$ |
\( T^{6} + 135 T^{4} + \cdots + 67 \)
|
| $83$ |
\( (T^{3} + 10 T^{2} + \cdots + 184)^{2} \)
|
| $89$ |
\( (T^{3} + 4 T^{2} - 4 T - 20)^{2} \)
|
| $97$ |
\( T^{6} + 332 T^{4} + \cdots + 1239232 \)
|
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