Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,2,Mod(324,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.324");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.d (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: | \(2.59513806569\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5089536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 24\nu^{4} + 6\nu^{3} + \nu^{2} - 6\nu - 285 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 46\nu^{5} - 56\nu^{4} + 14\nu^{3} + 308\nu^{2} + 772\nu - 534 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -92\nu^{5} + 112\nu^{4} - 28\nu^{3} - 223\nu^{2} - 1544\nu + 1068 ) / 393 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} + 4\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 6\beta _1 - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 5\beta_{4} + 9\beta_{3} - 9\beta_{2} - 7\beta _1 - 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
−2.51414 | − | 1.51414i | 4.32088 | 0 | 3.80675i | 3.32088 | −5.83502 | 0.707389 | 0 | |||||||||||||||||||||||||||||||||||
| 324.2 | −2.51414 | 1.51414i | 4.32088 | 0 | − | 3.80675i | 3.32088 | −5.83502 | 0.707389 | 0 | ||||||||||||||||||||||||||||||||||||
| 324.3 | −0.571993 | − | 0.428007i | −1.67282 | 0 | 0.244817i | −2.67282 | 2.10083 | 2.81681 | 0 | ||||||||||||||||||||||||||||||||||||
| 324.4 | −0.571993 | 0.428007i | −1.67282 | 0 | − | 0.244817i | −2.67282 | 2.10083 | 2.81681 | 0 | ||||||||||||||||||||||||||||||||||||
| 324.5 | 2.08613 | − | 3.08613i | 2.35194 | 0 | − | 6.43807i | 1.35194 | 0.734191 | −6.52420 | 0 | |||||||||||||||||||||||||||||||||||
| 324.6 | 2.08613 | 3.08613i | 2.35194 | 0 | 6.43807i | 1.35194 | 0.734191 | −6.52420 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.2.d.e | 6 | |
| 5.b | even | 2 | 1 | 325.2.d.f | 6 | ||
| 5.c | odd | 4 | 1 | 65.2.c.a | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 325.2.c.g | 6 | ||
| 13.b | even | 2 | 1 | 325.2.d.f | 6 | ||
| 15.e | even | 4 | 1 | 585.2.b.g | 6 | ||
| 20.e | even | 4 | 1 | 1040.2.k.d | 6 | ||
| 65.d | even | 2 | 1 | inner | 325.2.d.e | 6 | |
| 65.f | even | 4 | 1 | 845.2.a.k | 3 | ||
| 65.f | even | 4 | 1 | 4225.2.a.be | 3 | ||
| 65.h | odd | 4 | 1 | 65.2.c.a | ✓ | 6 | |
| 65.h | odd | 4 | 1 | 325.2.c.g | 6 | ||
| 65.k | even | 4 | 1 | 845.2.a.i | 3 | ||
| 65.k | even | 4 | 1 | 4225.2.a.bc | 3 | ||
| 65.o | even | 12 | 2 | 845.2.e.k | 6 | ||
| 65.q | odd | 12 | 2 | 845.2.m.h | 12 | ||
| 65.r | odd | 12 | 2 | 845.2.m.h | 12 | ||
| 65.t | even | 12 | 2 | 845.2.e.i | 6 | ||
| 195.j | odd | 4 | 1 | 7605.2.a.cc | 3 | ||
| 195.s | even | 4 | 1 | 585.2.b.g | 6 | ||
| 195.u | odd | 4 | 1 | 7605.2.a.bs | 3 | ||
| 260.p | even | 4 | 1 | 1040.2.k.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.c.a | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 65.2.c.a | ✓ | 6 | 65.h | odd | 4 | 1 | |
| 325.2.c.g | 6 | 5.c | odd | 4 | 1 | ||
| 325.2.c.g | 6 | 65.h | odd | 4 | 1 | ||
| 325.2.d.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 325.2.d.e | 6 | 65.d | even | 2 | 1 | inner | |
| 325.2.d.f | 6 | 5.b | even | 2 | 1 | ||
| 325.2.d.f | 6 | 13.b | even | 2 | 1 | ||
| 585.2.b.g | 6 | 15.e | even | 4 | 1 | ||
| 585.2.b.g | 6 | 195.s | even | 4 | 1 | ||
| 845.2.a.i | 3 | 65.k | even | 4 | 1 | ||
| 845.2.a.k | 3 | 65.f | even | 4 | 1 | ||
| 845.2.e.i | 6 | 65.t | even | 12 | 2 | ||
| 845.2.e.k | 6 | 65.o | even | 12 | 2 | ||
| 845.2.m.h | 12 | 65.q | odd | 12 | 2 | ||
| 845.2.m.h | 12 | 65.r | odd | 12 | 2 | ||
| 1040.2.k.d | 6 | 20.e | even | 4 | 1 | ||
| 1040.2.k.d | 6 | 260.p | even | 4 | 1 | ||
| 4225.2.a.bc | 3 | 65.k | even | 4 | 1 | ||
| 4225.2.a.be | 3 | 65.f | even | 4 | 1 | ||
| 7605.2.a.bs | 3 | 195.u | odd | 4 | 1 | ||
| 7605.2.a.cc | 3 | 195.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 5 T - 3)^{2} \)
|
| $3$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \)
|
| $11$ |
\( T^{6} + 48 T^{4} + \cdots + 2916 \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( T^{6} + 80 T^{4} + \cdots + 9216 \)
|
| $19$ |
\( T^{6} + 44 T^{4} + \cdots + 36 \)
|
| $23$ |
\( T^{6} + 32 T^{4} + \cdots + 36 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \)
|
| $31$ |
\( T^{6} + 56 T^{4} + \cdots + 36 \)
|
| $37$ |
\( (T^{3} - 20 T^{2} + \cdots - 228)^{2} \)
|
| $41$ |
\( T^{6} + 92 T^{4} + \cdots + 5184 \)
|
| $43$ |
\( T^{6} + 120 T^{4} + \cdots + 4 \)
|
| $47$ |
\( (T^{3} + 2 T^{2} - 8 T - 12)^{2} \)
|
| $53$ |
\( T^{6} + 156 T^{4} + \cdots + 5184 \)
|
| $59$ |
\( T^{6} + 84 T^{4} + \cdots + 324 \)
|
| $61$ |
\( (T^{3} + 6 T^{2} - 12 T - 76)^{2} \)
|
| $67$ |
\( (T^{3} + 18 T^{2} + \cdots + 108)^{2} \)
|
| $71$ |
\( T^{6} + 44 T^{4} + \cdots + 36 \)
|
| $73$ |
\( (T^{3} + 20 T^{2} + \cdots - 516)^{2} \)
|
| $79$ |
\( (T^{3} + 12 T^{2} + \cdots - 32)^{2} \)
|
| $83$ |
\( (T^{3} - 18 T^{2} + \cdots - 36)^{2} \)
|
| $89$ |
\( T^{6} + 192 T^{4} + \cdots + 82944 \)
|
| $97$ |
\( (T^{3} - 14 T^{2} + \cdots + 24)^{2} \)
|
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