Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,6,Mod(1,275)]
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([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("275.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.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: | \(44.1055504486\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 166x^{6} + 465x^{5} + 8430x^{4} - 12980x^{3} - 117384x^{2} + 34576x + 83744 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 166x^{6} + 465x^{5} + 8430x^{4} - 12980x^{3} - 117384x^{2} + 34576x + 83744 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19 \nu^{7} + 172 \nu^{6} + 2982 \nu^{5} - 11355 \nu^{4} - 190626 \nu^{3} - 113920 \nu^{2} + \cdots + 2862544 ) / 317856 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 19 \nu^{7} + 172 \nu^{6} + 2982 \nu^{5} - 11355 \nu^{4} - 31698 \nu^{3} - 431776 \nu^{2} + \cdots + 8901808 ) / 158928 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 67 \nu^{7} - 258 \nu^{6} - 4242 \nu^{5} - 3873 \nu^{4} - 148572 \nu^{3} + 1568584 \nu^{2} + \cdots - 13986576 ) / 158928 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 163 \nu^{7} - 430 \nu^{6} - 26628 \nu^{5} + 45135 \nu^{4} + 1278828 \nu^{3} - 1024970 \nu^{2} + \cdots + 80408 ) / 79464 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\nu^{7} - 129\nu^{6} - 2226\nu^{5} + 14790\nu^{4} + 94323\nu^{3} - 414500\nu^{2} - 1057280\nu + 1241232 ) / 7224 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + 2\beta_{2} + 70\beta _1 + 48 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{5} + 4\beta_{4} + 22\beta_{3} + 86\beta_{2} + 158\beta _1 + 3016 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 4\beta_{6} + 20\beta_{5} + 122\beta_{4} - 92\beta_{3} + 279\beta_{2} + 5433\beta _1 + 7069 \)
|
| \(\nu^{6}\) | \(=\) |
\( 54\beta_{7} + 72\beta_{6} - 30\beta_{5} + 663\beta_{4} + 2934\beta_{3} + 7176\beta_{2} + 19040\beta _1 + 233018 \)
|
| \(\nu^{7}\) | \(=\) |
\( 519 \beta_{7} + 24 \beta_{6} + 3465 \beta_{5} + 12726 \beta_{4} + 2310 \beta_{3} + 31292 \beta_{2} + \cdots + 827690 \)
|
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 |
|
−9.78761 | −20.4642 | 63.7972 | 0 | 200.296 | −32.0165 | −311.219 | 175.785 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.87340 | 18.5105 | 46.7372 | 0 | −164.251 | −198.142 | −130.769 | 99.6396 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.14470 | −1.72709 | −5.53211 | 0 | 8.88534 | 109.947 | 193.091 | −240.017 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.983607 | −24.0177 | −31.0325 | 0 | 23.6240 | −206.531 | 61.9992 | 333.852 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.748788 | −2.18200 | −31.4393 | 0 | −1.63385 | 85.2879 | −47.5026 | −238.239 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.92769 | 23.3337 | −16.5733 | 0 | 91.6475 | −95.2818 | −190.781 | 301.461 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.71649 | 7.11345 | 27.5442 | 0 | 54.8909 | −45.9738 | −34.3829 | −192.399 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.39634 | −27.5666 | 38.4985 | 0 | −231.459 | 23.7098 | 54.5638 | 516.917 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 275.6.a.g | ✓ | 8 |
| 5.b | even | 2 | 1 | 275.6.a.j | yes | 8 | |
| 5.c | odd | 4 | 2 | 275.6.b.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 275.6.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 275.6.a.j | yes | 8 | 5.b | even | 2 | 1 | |
| 275.6.b.h | 16 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{7} - 166T_{2}^{6} - 465T_{2}^{5} + 8430T_{2}^{4} + 12980T_{2}^{3} - 117384T_{2}^{2} - 34576T_{2} + 83744 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(275))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 4 T^{7} + \cdots + 83744 \)
|
| $3$ |
\( T^{8} + 27 T^{7} + \cdots - 156877884 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 12\!\cdots\!56 \)
|
| $11$ |
\( (T - 121)^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 24\!\cdots\!89 \)
|
| $17$ |
\( T^{8} + \cdots - 11\!\cdots\!44 \)
|
| $19$ |
\( T^{8} + \cdots + 59\!\cdots\!25 \)
|
| $23$ |
\( T^{8} + \cdots + 26\!\cdots\!01 \)
|
| $29$ |
\( T^{8} + \cdots - 14\!\cdots\!75 \)
|
| $31$ |
\( T^{8} + \cdots - 27\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots - 14\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots - 90\!\cdots\!24 \)
|
| $43$ |
\( T^{8} + \cdots + 41\!\cdots\!44 \)
|
| $47$ |
\( T^{8} + \cdots - 77\!\cdots\!64 \)
|
| $53$ |
\( T^{8} + \cdots + 79\!\cdots\!44 \)
|
| $59$ |
\( T^{8} + \cdots - 67\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 91\!\cdots\!76 \)
|
| $67$ |
\( T^{8} + \cdots + 22\!\cdots\!04 \)
|
| $71$ |
\( T^{8} + \cdots + 40\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots + 15\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 37\!\cdots\!11 \)
|
| $89$ |
\( T^{8} + \cdots + 85\!\cdots\!25 \)
|
| $97$ |
\( T^{8} + \cdots + 21\!\cdots\!11 \)
|
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