Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,6,Mod(1,325)]
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([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.a (trivial) |
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: | yes |
| Analytic conductor: | \(52.1247414392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 65) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 2\nu^{4} + 125\nu^{3} + 212\nu^{2} - 2956\nu - 4800 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 125\nu^{3} + 28\nu^{2} + 3196\nu - 8400 ) / 240 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 10\nu^{3} + 125\nu^{2} - 808\nu - 2260 ) / 60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 101\nu^{2} - 116\nu + 1288 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - \beta _1 + 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} - 2\beta_{3} - 2\beta_{2} + 79\beta _1 - 29 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{5} - 10\beta_{4} + 105\beta_{3} + 105\beta_{2} - 143\beta _1 + 4325 \)
|
| \(\nu^{5}\) | \(=\) |
\( 105\beta_{5} + 645\beta_{4} - 248\beta_{3} - 488\beta_{2} + 6993\beta _1 - 5415 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9.62003 | −10.7175 | 60.5450 | 0 | 103.103 | −18.4289 | −274.604 | −128.135 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −7.62628 | 29.7832 | 26.1601 | 0 | −227.135 | −171.199 | 44.5366 | 644.042 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.15349 | −29.2293 | −14.7485 | 0 | 121.404 | −42.5171 | 194.170 | 611.349 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.16876 | 2.56930 | −27.2965 | 0 | 5.57220 | 75.5966 | −128.600 | −236.399 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 7.05260 | −22.8190 | 17.7392 | 0 | −160.933 | −141.347 | −100.576 | 277.708 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.1784 | 10.4132 | 71.6007 | 0 | 105.990 | 125.896 | 403.073 | −134.565 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.6.a.f | 6 | |
| 5.b | even | 2 | 1 | 65.6.a.e | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 325.6.b.f | 12 | ||
| 15.d | odd | 2 | 1 | 585.6.a.k | 6 | ||
| 20.d | odd | 2 | 1 | 1040.6.a.r | 6 | ||
| 65.d | even | 2 | 1 | 845.6.a.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.6.a.e | ✓ | 6 | 5.b | even | 2 | 1 | |
| 325.6.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 325.6.b.f | 12 | 5.c | odd | 4 | 2 | ||
| 585.6.a.k | 6 | 15.d | odd | 2 | 1 | ||
| 845.6.a.g | 6 | 65.d | even | 2 | 1 | ||
| 1040.6.a.r | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} - 161T_{2}^{4} - 328T_{2}^{3} + 6584T_{2}^{2} + 10688T_{2} - 47440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots - 47440 \)
|
| $3$ |
\( T^{6} + 20 T^{5} + \cdots - 5696136 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 180452981952 \)
|
| $11$ |
\( T^{6} + \cdots - 674919089487832 \)
|
| $13$ |
\( (T - 169)^{6} \)
|
| $17$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots + 58\!\cdots\!40 \)
|
| $23$ |
\( T^{6} + \cdots + 84\!\cdots\!48 \)
|
| $29$ |
\( T^{6} + \cdots - 24\!\cdots\!20 \)
|
| $31$ |
\( T^{6} + \cdots - 47\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots - 39\!\cdots\!12 \)
|
| $41$ |
\( T^{6} + \cdots - 78\!\cdots\!32 \)
|
| $43$ |
\( T^{6} + \cdots - 34\!\cdots\!28 \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 51\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots + 29\!\cdots\!80 \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!68 \)
|
| $67$ |
\( T^{6} + \cdots - 31\!\cdots\!72 \)
|
| $71$ |
\( T^{6} + \cdots - 14\!\cdots\!12 \)
|
| $73$ |
\( T^{6} + \cdots - 20\!\cdots\!08 \)
|
| $79$ |
\( T^{6} + \cdots + 22\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots - 22\!\cdots\!12 \)
|
| $89$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!20 \)
|
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