Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,3,Mod(191,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1216.191");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.d (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: | \(33.1336001462\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.210056875.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 12x^{4} - 9x^{3} - 39x^{2} + 58x + 92 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 304) |
| 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
\( x^{6} - 3x^{5} + 12x^{4} - 9x^{3} - 39x^{2} + 58x + 92 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{5} + 106\nu^{4} - 274\nu^{3} + 1281\nu^{2} - 86\nu - 2300 ) / 736 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} + 2\nu^{4} - 26\nu^{3} + 45\nu^{2} + 290\nu + 644 ) / 368 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -45\nu^{5} + 118\nu^{4} - 430\nu^{3} + 79\nu^{2} + 3126\nu - 2116 ) / 736 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 10\nu^{4} + 50\nu^{3} - 65\nu^{2} - 10\nu + 156 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 117\nu^{5} - 454\nu^{4} + 1854\nu^{3} - 2855\nu^{2} - 326\nu + 6532 ) / 736 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{3} + 6\beta_{2} + 3\beta _1 - 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{5} + 9\beta_{4} - \beta_{3} + 6\beta_{2} - \beta _1 - 16 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -35\beta_{5} + 55\beta_{4} + 9\beta_{3} - 48\beta_{2} - 11\beta _1 + 118 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 75\beta_{5} - 99\beta_{4} + 45\beta_{3} - 230\beta_{2} + 65\beta _1 + 552 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(705\) | \(837\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
0 | − | 5.71233i | 0 | 2.13439 | 0 | − | 11.8085i | 0 | −23.6307 | 0 | ||||||||||||||||||||||||||||||||||
| 191.2 | 0 | − | 3.58210i | 0 | −8.22273 | 0 | 11.1452i | 0 | −3.83144 | 0 | ||||||||||||||||||||||||||||||||||||
| 191.3 | 0 | − | 2.13023i | 0 | −0.911656 | 0 | − | 1.15921i | 0 | 4.46213 | 0 | |||||||||||||||||||||||||||||||||||
| 191.4 | 0 | 2.13023i | 0 | −0.911656 | 0 | 1.15921i | 0 | 4.46213 | 0 | |||||||||||||||||||||||||||||||||||||
| 191.5 | 0 | 3.58210i | 0 | −8.22273 | 0 | − | 11.1452i | 0 | −3.83144 | 0 | ||||||||||||||||||||||||||||||||||||
| 191.6 | 0 | 5.71233i | 0 | 2.13439 | 0 | 11.8085i | 0 | −23.6307 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1216.3.d.b | 6 | |
| 4.b | odd | 2 | 1 | inner | 1216.3.d.b | 6 | |
| 8.b | even | 2 | 1 | 304.3.d.a | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 304.3.d.a | ✓ | 6 | |
| 24.f | even | 2 | 1 | 2736.3.m.a | 6 | ||
| 24.h | odd | 2 | 1 | 2736.3.m.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 304.3.d.a | ✓ | 6 | 8.b | even | 2 | 1 | |
| 304.3.d.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 1216.3.d.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 1216.3.d.b | 6 | 4.b | odd | 2 | 1 | inner | |
| 2736.3.m.a | 6 | 24.f | even | 2 | 1 | ||
| 2736.3.m.a | 6 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 50T_{3}^{4} + 625T_{3}^{2} + 1900 \)
acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 50 T^{4} + \cdots + 1900 \)
|
| $5$ |
\( (T^{3} + 7 T^{2} - 12 T - 16)^{2} \)
|
| $7$ |
\( T^{6} + 265 T^{4} + \cdots + 23275 \)
|
| $11$ |
\( T^{6} + 675 T^{4} + \cdots + 9310000 \)
|
| $13$ |
\( (T^{3} - 265 T + 1580)^{2} \)
|
| $17$ |
\( (T^{3} + 5 T^{2} + \cdots - 7805)^{2} \)
|
| $19$ |
\( (T^{2} + 19)^{3} \)
|
| $23$ |
\( T^{6} + 1190 T^{4} + \cdots + 18247600 \)
|
| $29$ |
\( (T^{3} - 54 T^{2} + \cdots + 818)^{2} \)
|
| $31$ |
\( T^{6} + 3200 T^{4} + \cdots + 48640000 \)
|
| $37$ |
\( (T^{3} + 50 T^{2} + \cdots - 1960)^{2} \)
|
| $41$ |
\( (T^{3} + 46 T^{2} + \cdots - 3232)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 1608160000 \)
|
| $47$ |
\( T^{6} + 5835 T^{4} + \cdots + 1945600 \)
|
| $53$ |
\( (T^{3} - 160 T^{2} + \cdots - 26540)^{2} \)
|
| $59$ |
\( T^{6} + 10450 T^{4} + \cdots + 840227500 \)
|
| $61$ |
\( (T^{3} + 159 T^{2} + \cdots + 101252)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 4755175600 \)
|
| $71$ |
\( T^{6} + \cdots + 558181240000 \)
|
| $73$ |
\( (T^{3} - 55 T^{2} + \cdots + 515035)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 136629760000 \)
|
| $83$ |
\( T^{6} + \cdots + 135343840000 \)
|
| $89$ |
\( (T^{3} - 96 T^{2} + \cdots + 205632)^{2} \)
|
| $97$ |
\( (T^{3} - 20 T^{2} + \cdots + 10880)^{2} \)
|
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