Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,7,Mod(37,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.37");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 76.c (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: | \(17.4841103551\) |
| 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: |
\( x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 5 \) |
| 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
\( x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{6} - 46859\nu^{4} - 67416687\nu^{2} - 4315657896 ) / 21128796 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -55\nu^{6} - 204128\nu^{4} - 211681665\nu^{2} - 40102527312 ) / 42257592 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 55\nu^{7} + 204128\nu^{5} + 211681665\nu^{3} + 41750573400\nu ) / 42257592 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 157\nu^{7} + 518666\nu^{5} + 500211621\nu^{3} + 121944861000\nu ) / 84515184 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 106\nu^{7} + 361397\nu^{5} + 334817847\nu^{3} + 48252931560\nu ) / 21128796 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -89\nu^{6} - 308974\nu^{4} - 293772453\nu^{2} - 46015049520 ) / 14085864 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - 5\beta_{3} + \beta_{2} - 1274 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} + \beta_{4} - 1590\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -1625\beta_{7} + 8221\beta_{3} - 2285\beta_{2} + 2026546 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1625\beta_{6} - 4570\beta_{5} + 259\beta_{4} + 2626950\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2182297\beta_{7} - 12036101\beta_{3} + 4631833\beta_{2} - 3347179418 \)
|
| \(\nu^{7}\) | \(=\) |
\( -2182297\beta_{6} + 9263666\beta_{5} - 4041695\beta_{4} - 4389286830\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(39\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | − | 42.5965i | 0 | 103.439 | 0 | −619.422 | 0 | −1085.46 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | − | 40.3415i | 0 | −186.796 | 0 | 202.293 | 0 | −898.437 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 37.3 | 0 | − | 38.1507i | 0 | 102.472 | 0 | 512.609 | 0 | −726.473 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 37.4 | 0 | − | 13.8790i | 0 | −18.1155 | 0 | 85.5200 | 0 | 536.374 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 37.5 | 0 | 13.8790i | 0 | −18.1155 | 0 | 85.5200 | 0 | 536.374 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 0 | 38.1507i | 0 | 102.472 | 0 | 512.609 | 0 | −726.473 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 37.7 | 0 | 40.3415i | 0 | −186.796 | 0 | 202.293 | 0 | −898.437 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 37.8 | 0 | 42.5965i | 0 | 103.439 | 0 | −619.422 | 0 | −1085.46 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 76.7.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 684.7.h.c | 8 | ||
| 4.b | odd | 2 | 1 | 304.7.e.c | 8 | ||
| 19.b | odd | 2 | 1 | inner | 76.7.c.b | ✓ | 8 |
| 57.d | even | 2 | 1 | 684.7.h.c | 8 | ||
| 76.d | even | 2 | 1 | 304.7.e.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.7.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 76.7.c.b | ✓ | 8 | 19.b | odd | 2 | 1 | inner |
| 304.7.e.c | 8 | 4.b | odd | 2 | 1 | ||
| 304.7.e.c | 8 | 76.d | even | 2 | 1 | ||
| 684.7.h.c | 8 | 3.b | odd | 2 | 1 | ||
| 684.7.h.c | 8 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 5090T_{3}^{6} + 8905881T_{3}^{4} + 5831691048T_{3}^{2} + 827887219200 \)
acting on \(S_{7}^{\mathrm{new}}(76, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 827887219200 \)
|
| $5$ |
\( (T^{4} - T^{3} + \cdots + 35868000)^{2} \)
|
| $7$ |
\( (T^{4} - 181 T^{3} + \cdots - 5493140806)^{2} \)
|
| $11$ |
\( (T^{4} - 451 T^{3} + \cdots - 70045150776)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $17$ |
\( (T^{4} + \cdots + 56395405380078)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 48\!\cdots\!21 \)
|
| $23$ |
\( (T^{4} + \cdots - 329689946145300)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 85\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $43$ |
\( (T^{4} + \cdots - 43\!\cdots\!56)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots - 15\!\cdots\!12)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots + 72\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots - 72\!\cdots\!80)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 77\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + \cdots + 16\!\cdots\!62)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
|
| $83$ |
\( (T^{4} + \cdots + 46\!\cdots\!88)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
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