Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,4,Mod(27,28)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.27");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 28.d (of order \(2\), degree \(1\), 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: | \(1.65205348016\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2107780 \nu^{7} - 28020178 \nu^{6} + 12013496 \nu^{5} + 856630098 \nu^{4} - 1820971716 \nu^{3} + \cdots + 26653239984 ) / 11759822513 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2107780 \nu^{7} + 28020178 \nu^{6} - 12013496 \nu^{5} - 856630098 \nu^{4} + \cdots - 38413062497 ) / 11759822513 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5280 \nu^{7} + 27552 \nu^{6} + 648572 \nu^{5} - 3328540 \nu^{4} - 9022980 \nu^{3} + \cdots - 204544903 ) / 7201361 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37410820 \nu^{7} - 178195062 \nu^{6} - 946363840 \nu^{5} + 4456400722 \nu^{4} + \cdots + 158760439399 ) / 11759822513 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 71287048 \nu^{7} + 411792664 \nu^{6} + 1067149048 \nu^{5} - 6385173864 \nu^{4} + \cdots - 65434616690 ) / 11759822513 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 85892144 \nu^{7} - 485124166 \nu^{6} - 2515029448 \nu^{5} + 14797753670 \nu^{4} + \cdots + 390648894103 ) / 11759822513 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2876 \nu^{7} + 28790 \nu^{6} - 17704 \nu^{5} - 509450 \nu^{4} + 374248 \nu^{3} + 4834142 \nu^{2} + \cdots - 15131816 ) / 379019 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 38 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} + 6\beta_{6} + 10\beta_{5} + 3\beta_{4} + 6\beta_{3} + 79\beta_{2} + 17\beta _1 + 86 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -16\beta_{7} + 48\beta_{6} + 83\beta_{5} + 39\beta_{4} + 60\beta_{3} + 305\beta_{2} + 32\beta _1 + 330 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -250\beta_{7} + 330\beta_{6} + 965\beta_{5} + 834\beta_{4} + 560\beta_{3} + 3838\beta_{2} - 68\beta _1 + 1764 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 590 \beta_{7} + 754 \beta_{6} + 2877 \beta_{5} + 3229 \beta_{4} + 1576 \beta_{3} + 8337 \beta_{2} + \cdots - 2086 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6440 \beta_{7} + 4344 \beta_{6} + 28219 \beta_{5} + 39754 \beta_{4} + 15134 \beta_{3} + 73906 \beta_{2} + \cdots - 63556 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 27.1 |
|
−2.41421 | − | 1.47363i | −4.79890 | 3.65685 | + | 7.11529i | 17.0728i | 11.5856 | + | 7.07178i | −18.3722 | − | 2.33686i | 1.65685 | − | 22.5667i | −3.97056 | 25.1589 | − | 41.2174i | ||||||||||||||||||||||||||||||
| 27.2 | −2.41421 | − | 1.47363i | 4.79890 | 3.65685 | + | 7.11529i | − | 17.0728i | −11.5856 | − | 7.07178i | 18.3722 | − | 2.33686i | 1.65685 | − | 22.5667i | −3.97056 | −25.1589 | + | 41.2174i | ||||||||||||||||||||||||||||||
| 27.3 | −2.41421 | + | 1.47363i | −4.79890 | 3.65685 | − | 7.11529i | − | 17.0728i | 11.5856 | − | 7.07178i | −18.3722 | + | 2.33686i | 1.65685 | + | 22.5667i | −3.97056 | 25.1589 | + | 41.2174i | ||||||||||||||||||||||||||||||
| 27.4 | −2.41421 | + | 1.47363i | 4.79890 | 3.65685 | − | 7.11529i | 17.0728i | −11.5856 | + | 7.07178i | 18.3722 | + | 2.33686i | 1.65685 | + | 22.5667i | −3.97056 | −25.1589 | − | 41.2174i | |||||||||||||||||||||||||||||||
| 27.5 | 0.414214 | − | 2.79793i | −7.54788 | −7.65685 | − | 2.31788i | − | 8.74756i | −3.12644 | + | 21.1185i | 13.8008 | − | 12.3507i | −9.65685 | + | 20.4633i | 29.9706 | −24.4751 | − | 3.62336i | ||||||||||||||||||||||||||||||
| 27.6 | 0.414214 | − | 2.79793i | 7.54788 | −7.65685 | − | 2.31788i | 8.74756i | 3.12644 | − | 21.1185i | −13.8008 | − | 12.3507i | −9.65685 | + | 20.4633i | 29.9706 | 24.4751 | + | 3.62336i | |||||||||||||||||||||||||||||||
| 27.7 | 0.414214 | + | 2.79793i | −7.54788 | −7.65685 | + | 2.31788i | 8.74756i | −3.12644 | − | 21.1185i | 13.8008 | + | 12.3507i | −9.65685 | − | 20.4633i | 29.9706 | −24.4751 | + | 3.62336i | |||||||||||||||||||||||||||||||
| 27.8 | 0.414214 | + | 2.79793i | 7.54788 | −7.65685 | + | 2.31788i | − | 8.74756i | 3.12644 | + | 21.1185i | −13.8008 | + | 12.3507i | −9.65685 | − | 20.4633i | 29.9706 | 24.4751 | − | 3.62336i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 28.4.d.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 252.4.b.d | 8 | ||
| 4.b | odd | 2 | 1 | inner | 28.4.d.b | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 28.4.d.b | ✓ | 8 |
| 7.c | even | 3 | 2 | 196.4.f.c | 16 | ||
| 7.d | odd | 6 | 2 | 196.4.f.c | 16 | ||
| 8.b | even | 2 | 1 | 448.4.f.d | 8 | ||
| 8.d | odd | 2 | 1 | 448.4.f.d | 8 | ||
| 12.b | even | 2 | 1 | 252.4.b.d | 8 | ||
| 21.c | even | 2 | 1 | 252.4.b.d | 8 | ||
| 28.d | even | 2 | 1 | inner | 28.4.d.b | ✓ | 8 |
| 28.f | even | 6 | 2 | 196.4.f.c | 16 | ||
| 28.g | odd | 6 | 2 | 196.4.f.c | 16 | ||
| 56.e | even | 2 | 1 | 448.4.f.d | 8 | ||
| 56.h | odd | 2 | 1 | 448.4.f.d | 8 | ||
| 84.h | odd | 2 | 1 | 252.4.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.4.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 28.4.d.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 28.4.d.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 28.4.d.b | ✓ | 8 | 28.d | even | 2 | 1 | inner |
| 196.4.f.c | 16 | 7.c | even | 3 | 2 | ||
| 196.4.f.c | 16 | 7.d | odd | 6 | 2 | ||
| 196.4.f.c | 16 | 28.f | even | 6 | 2 | ||
| 196.4.f.c | 16 | 28.g | odd | 6 | 2 | ||
| 252.4.b.d | 8 | 3.b | odd | 2 | 1 | ||
| 252.4.b.d | 8 | 12.b | even | 2 | 1 | ||
| 252.4.b.d | 8 | 21.c | even | 2 | 1 | ||
| 252.4.b.d | 8 | 84.h | odd | 2 | 1 | ||
| 448.4.f.d | 8 | 8.b | even | 2 | 1 | ||
| 448.4.f.d | 8 | 8.d | odd | 2 | 1 | ||
| 448.4.f.d | 8 | 56.e | even | 2 | 1 | ||
| 448.4.f.d | 8 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 80T_{3}^{2} + 1312 \)
acting on \(S_{4}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 4 T^{3} + 12 T^{2} + \cdots + 64)^{2} \)
|
| $3$ |
\( (T^{4} - 80 T^{2} + 1312)^{2} \)
|
| $5$ |
\( (T^{4} + 368 T^{2} + 22304)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 13841287201 \)
|
| $11$ |
\( (T^{4} + 1720 T^{2} + 272)^{2} \)
|
| $13$ |
\( (T^{4} + 7792 T^{2} + 11798816)^{2} \)
|
| $17$ |
\( (T^{4} + 7936 T^{2} + 5709824)^{2} \)
|
| $19$ |
\( (T^{4} - 2128 T^{2} + 694048)^{2} \)
|
| $23$ |
\( (T^{4} + 23800 T^{2} + 132140048)^{2} \)
|
| $29$ |
\( (T^{2} + 148 T - 4892)^{4} \)
|
| $31$ |
\( (T^{4} - 23040 T^{2} + 108822528)^{2} \)
|
| $37$ |
\( (T^{2} - 348 T + 24004)^{4} \)
|
| $41$ |
\( (T^{4} + 58304 T^{2} + 342946304)^{2} \)
|
| $43$ |
\( (T^{4} + 34360 T^{2} + 141396752)^{2} \)
|
| $47$ |
\( (T^{4} - 103680 T^{2} + 1789861888)^{2} \)
|
| $53$ |
\( (T^{2} + 292 T - 163516)^{4} \)
|
| $59$ |
\( (T^{4} - 570320 T^{2} + 67995188512)^{2} \)
|
| $61$ |
\( (T^{4} + 606832 T^{2} + 6445856)^{2} \)
|
| $67$ |
\( (T^{4} + 343672 T^{2} + 12017669648)^{2} \)
|
| $71$ |
\( (T^{4} + 275576 T^{2} + 9248152592)^{2} \)
|
| $73$ |
\( (T^{4} + 92864 T^{2} + 1798951424)^{2} \)
|
| $79$ |
\( (T^{4} + 1466680 T^{2} + 108115639568)^{2} \)
|
| $83$ |
\( (T^{4} - 1267920 T^{2} + 340971272992)^{2} \)
|
| $89$ |
\( (T^{4} + 1846976 T^{2} + 661026681344)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 1135775350784)^{2} \)
|
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