Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,6,Mod(199,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.199");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.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: | \(44.1055504486\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.11877512256.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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
\( x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -137\nu^{5} + 578\nu^{4} - 4930\nu^{3} + 27490\nu^{2} + 2527\nu + 713224 ) / 138448 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 167\nu^{5} + 306\nu^{4} + 1462\nu^{3} - 30478\nu^{2} - 37945\nu - 804728 ) / 138448 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{5} - 58\nu^{4} - 1182\nu^{3} + 454\nu^{2} - 15277\nu + 127832 ) / 20360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1761\nu^{5} - 3434\nu^{4} - 66402\nu^{3} - 25354\nu^{2} - 683001\nu + 7434160 ) / 346120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4207\nu^{5} + 3978\nu^{4} + 157454\nu^{3} - 86742\nu^{2} + 2737847\nu - 19126680 ) / 692240 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{4} + 15\beta_{3} - 5\beta_{2} + \beta _1 - 21 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -21\beta_{5} - \beta_{4} - 100\beta_{3} - 24\beta_{2} - 29\beta _1 + 79 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -60\beta_{5} + 45\beta_{4} - 495\beta_{3} + 181\beta_{2} + 235\beta _1 + 325 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 521\beta_{5} - 759\beta_{4} + 4520\beta_{3} + 624\beta_{2} + 233\beta _1 + 4745 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 10.3963i | − | 20.6466i | −76.0833 | 0 | −214.649 | 164.454i | 458.304i | −183.283 | 0 | ||||||||||||||||||||||||||||||||||
| 199.2 | − | 8.18772i | − | 3.48600i | −35.0388 | 0 | −28.5424 | − | 145.071i | 24.8808i | 230.848 | 0 | ||||||||||||||||||||||||||||||||||
| 199.3 | − | 2.20859i | 16.8394i | 27.1221 | 0 | 37.1913 | 225.525i | − | 130.577i | −40.5643 | 0 | |||||||||||||||||||||||||||||||||||
| 199.4 | 2.20859i | − | 16.8394i | 27.1221 | 0 | 37.1913 | − | 225.525i | 130.577i | −40.5643 | 0 | |||||||||||||||||||||||||||||||||||
| 199.5 | 8.18772i | 3.48600i | −35.0388 | 0 | −28.5424 | 145.071i | − | 24.8808i | 230.848 | 0 | ||||||||||||||||||||||||||||||||||||
| 199.6 | 10.3963i | 20.6466i | −76.0833 | 0 | −214.649 | − | 164.454i | − | 458.304i | −183.283 | 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 | 275.6.b.b | 6 | |
| 5.b | even | 2 | 1 | inner | 275.6.b.b | 6 | |
| 5.c | odd | 4 | 1 | 11.6.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 275.6.a.b | 3 | ||
| 15.e | even | 4 | 1 | 99.6.a.g | 3 | ||
| 20.e | even | 4 | 1 | 176.6.a.i | 3 | ||
| 35.f | even | 4 | 1 | 539.6.a.e | 3 | ||
| 40.i | odd | 4 | 1 | 704.6.a.q | 3 | ||
| 40.k | even | 4 | 1 | 704.6.a.t | 3 | ||
| 55.e | even | 4 | 1 | 121.6.a.d | 3 | ||
| 165.l | odd | 4 | 1 | 1089.6.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.6.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 99.6.a.g | 3 | 15.e | even | 4 | 1 | ||
| 121.6.a.d | 3 | 55.e | even | 4 | 1 | ||
| 176.6.a.i | 3 | 20.e | even | 4 | 1 | ||
| 275.6.a.b | 3 | 5.c | odd | 4 | 1 | ||
| 275.6.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 275.6.b.b | 6 | 5.b | even | 2 | 1 | inner | |
| 539.6.a.e | 3 | 35.f | even | 4 | 1 | ||
| 704.6.a.q | 3 | 40.i | odd | 4 | 1 | ||
| 704.6.a.t | 3 | 40.k | even | 4 | 1 | ||
| 1089.6.a.r | 3 | 165.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 180T_{2}^{4} + 8100T_{2}^{2} + 35344 \)
acting on \(S_{6}^{\mathrm{new}}(275, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 180 T^{4} + \cdots + 35344 \)
|
| $3$ |
\( T^{6} + 722 T^{4} + \cdots + 1468944 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 28949220680704 \)
|
| $11$ |
\( (T - 121)^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 26\!\cdots\!64 \)
|
| $17$ |
\( T^{6} + \cdots + 11\!\cdots\!36 \)
|
| $19$ |
\( (T^{3} + 1380 T^{2} + \cdots - 57024000)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 28\!\cdots\!84 \)
|
| $29$ |
\( (T^{3} - 3426 T^{2} + \cdots + 4029189120)^{2} \)
|
| $31$ |
\( (T^{3} + 4098 T^{2} + \cdots + 1094344400)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 29\!\cdots\!56 \)
|
| $41$ |
\( (T^{3} - 5994 T^{2} + \cdots + 201929821568)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 59\!\cdots\!36 \)
|
| $47$ |
\( T^{6} + \cdots + 49\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!36 \)
|
| $59$ |
\( (T^{3} + \cdots - 7759637437060)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 15233874751008)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} + \cdots - 1290398551704)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $79$ |
\( (T^{3} + \cdots - 1279883216320)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + \cdots + 90320980174650)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
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