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: | \(1\) |
| 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: |
\( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 18\nu^{4} + 851\nu^{3} - 1814\nu^{2} - 30420\nu - 12312 ) / 2016 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 18\nu^{4} - 851\nu^{3} + 3830\nu^{2} + 30420\nu - 96552 ) / 2016 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 30\nu^{4} - 103\nu^{3} - 3098\nu^{2} + 3060\nu + 29880 ) / 336 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} - 18\nu^{4} + 2705\nu^{3} - 1994\nu^{2} - 41148\nu + 80856 ) / 2016 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} - \beta_{4} - 4\beta_{3} + 4\beta_{2} + 89\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} + 12\beta_{4} + 111\beta_{3} + 107\beta_{2} - 88\beta _1 + 4742 \)
|
| \(\nu^{5}\) | \(=\) |
\( -326\beta_{5} - 127\beta_{4} - 644\beta_{3} + 300\beta_{2} + 8747\beta _1 - 4642 \)
|
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 |
|
−9.61672 | −28.9791 | 60.4813 | 0 | 278.684 | 189.995 | −273.896 | 596.790 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.51599 | −11.2297 | 40.5221 | 0 | 95.6316 | −229.647 | −72.5738 | −116.895 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.34530 | 19.6439 | −30.1902 | 0 | −26.4269 | −48.6530 | 83.6645 | 142.882 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.75663 | −23.9621 | −24.4010 | 0 | −66.0547 | −85.7300 | −155.477 | 331.182 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.93318 | 7.05430 | 3.20258 | 0 | 41.8544 | 185.746 | −170.860 | −193.237 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.7882 | −0.527294 | 84.3852 | 0 | −5.68855 | −231.710 | 565.142 | −242.722 | 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.g | 6 | |
| 5.b | even | 2 | 1 | 65.6.a.d | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 325.6.b.g | 12 | ||
| 15.d | odd | 2 | 1 | 585.6.a.m | 6 | ||
| 20.d | odd | 2 | 1 | 1040.6.a.q | 6 | ||
| 65.d | even | 2 | 1 | 845.6.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.6.a.d | ✓ | 6 | 5.b | even | 2 | 1 | |
| 325.6.a.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 325.6.b.g | 12 | 5.c | odd | 4 | 2 | ||
| 585.6.a.m | 6 | 15.d | odd | 2 | 1 | ||
| 845.6.a.h | 6 | 65.d | even | 2 | 1 | ||
| 1040.6.a.q | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 163T_{2}^{4} + 8T_{2}^{3} + 6120T_{2}^{2} - 6624T_{2} - 19440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 163 T^{4} + \cdots - 19440 \)
|
| $3$ |
\( T^{6} + 38 T^{5} + \cdots + 569784 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 7832693511488 \)
|
| $11$ |
\( T^{6} + \cdots - 57\!\cdots\!56 \)
|
| $13$ |
\( (T + 169)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 60\!\cdots\!68 \)
|
| $19$ |
\( T^{6} + \cdots - 60\!\cdots\!80 \)
|
| $23$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
|
| $29$ |
\( T^{6} + \cdots + 38\!\cdots\!40 \)
|
| $31$ |
\( T^{6} + \cdots - 20\!\cdots\!64 \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!76 \)
|
| $41$ |
\( T^{6} + \cdots - 11\!\cdots\!12 \)
|
| $43$ |
\( T^{6} + \cdots - 18\!\cdots\!04 \)
|
| $47$ |
\( T^{6} + \cdots - 25\!\cdots\!80 \)
|
| $53$ |
\( T^{6} + \cdots + 86\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots + 41\!\cdots\!80 \)
|
| $61$ |
\( T^{6} + \cdots + 91\!\cdots\!28 \)
|
| $67$ |
\( T^{6} + \cdots - 13\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots - 10\!\cdots\!12 \)
|
| $73$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 66\!\cdots\!20 \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!16 \)
|
| $89$ |
\( T^{6} + \cdots - 51\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!20 \)
|
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