Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,3,Mod(145,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.145");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.be (of order \(6\), degree \(2\), 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: | \(24.8502001097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 92x^{6} + 2680x^{4} - 23592x^{2} + 79524 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 114) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 92x^{6} + 2680x^{4} - 23592x^{2} + 79524 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 68\nu^{4} - 706\nu^{2} - 9084 ) / 4788 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 71 \nu^{7} - 6204 \nu^{6} - 11326 \nu^{5} + 518316 \nu^{4} + 564494 \nu^{3} - 12867096 \nu^{2} + \cdots + 64004976 ) / 2700432 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -71\nu^{7} + 11326\nu^{5} - 564494\nu^{3} + 8628024\nu ) / 1350216 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 92\nu^{5} + 2398\nu^{3} - 10620\nu - 7896 ) / 15792 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 91 \nu^{7} + 141 \nu^{6} + 7385 \nu^{5} - 33699 \nu^{4} - 176764 \nu^{3} + 1208652 \nu^{2} + \cdots - 5518458 ) / 1350216 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 91 \nu^{7} + 141 \nu^{6} - 7385 \nu^{5} - 33699 \nu^{4} + 176764 \nu^{3} + 1208652 \nu^{2} + \cdots - 5518458 ) / 1350216 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 91\nu^{7} - 47\nu^{6} - 7385\nu^{5} + 3196\nu^{4} + 176764\nu^{3} - 33182\nu^{2} - 774942\nu - 426948 ) / 450072 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{7} + 3\beta_{6} - 3\beta_{5} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{5} - \beta_{3} - 2\beta_{2} + 20\beta _1 + 69 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -38\beta_{7} + 108\beta_{6} - 108\beta_{5} - 126\beta_{4} - 21\beta_{3} + 19\beta _1 - 63 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -176\beta_{6} - 176\beta_{5} - 46\beta_{3} - 92\beta_{2} + 836\beta _1 + 2328 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 160\beta_{7} + 3954\beta_{6} - 3954\beta_{5} - 9660\beta_{4} - 882\beta_{3} - 80\beta _1 - 4830 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -10556\beta_{6} - 10556\beta_{5} - 2422\beta_{3} - 4844\beta_{2} + 28364\beta _1 + 82338 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 84604\beta_{7} + 136644\beta_{6} - 136644\beta_{5} - 539196\beta_{4} - 30786\beta_{3} - 42302\beta _1 - 269598 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(1 + \beta_{4}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 145.1 |
|
0 | 1.50000 | − | 0.866025i | 0 | −4.80010 | − | 8.31402i | 0 | 8.01641 | 0 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||||||||||
| 145.2 | 0 | 1.50000 | − | 0.866025i | 0 | −1.64794 | − | 2.85432i | 0 | −7.47014 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.3 | 0 | 1.50000 | − | 0.866025i | 0 | 0.872687 | + | 1.51154i | 0 | 9.91963 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.4 | 0 | 1.50000 | − | 0.866025i | 0 | 1.57536 | + | 2.72860i | 0 | −10.4659 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 673.1 | 0 | 1.50000 | + | 0.866025i | 0 | −4.80010 | + | 8.31402i | 0 | 8.01641 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 673.2 | 0 | 1.50000 | + | 0.866025i | 0 | −1.64794 | + | 2.85432i | 0 | −7.47014 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 673.3 | 0 | 1.50000 | + | 0.866025i | 0 | 0.872687 | − | 1.51154i | 0 | 9.91963 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
| 673.4 | 0 | 1.50000 | + | 0.866025i | 0 | 1.57536 | − | 2.72860i | 0 | −10.4659 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.3.be.h | 8 | |
| 4.b | odd | 2 | 1 | 114.3.f.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 342.3.m.c | 8 | ||
| 19.d | odd | 6 | 1 | inner | 912.3.be.h | 8 | |
| 76.f | even | 6 | 1 | 114.3.f.a | ✓ | 8 | |
| 228.n | odd | 6 | 1 | 342.3.m.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.3.f.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 114.3.f.a | ✓ | 8 | 76.f | even | 6 | 1 | |
| 342.3.m.c | 8 | 12.b | even | 2 | 1 | ||
| 342.3.m.c | 8 | 228.n | odd | 6 | 1 | ||
| 912.3.be.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 912.3.be.h | 8 | 19.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):
|
\( T_{5}^{8} + 8T_{5}^{7} + 90T_{5}^{6} - 40T_{5}^{5} + 1174T_{5}^{4} - 600T_{5}^{3} + 11580T_{5}^{2} - 14616T_{5} + 30276 \)
|
|
\( T_{7}^{4} - 164T_{7}^{2} + 24T_{7} + 6217 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{4} \)
|
| $5$ |
\( T^{8} + 8 T^{7} + \cdots + 30276 \)
|
| $7$ |
\( (T^{4} - 164 T^{2} + \cdots + 6217)^{2} \)
|
| $11$ |
\( (T^{4} + 32 T^{3} + \cdots + 1182)^{2} \)
|
| $13$ |
\( T^{8} + 24 T^{7} + \cdots + 21316689 \)
|
| $17$ |
\( T^{8} + 20 T^{7} + \cdots + 14400 \)
|
| $19$ |
\( T^{8} + \cdots + 16983563041 \)
|
| $23$ |
\( T^{8} + \cdots + 27238861764 \)
|
| $29$ |
\( T^{8} + \cdots + 366828880896 \)
|
| $31$ |
\( T^{8} + \cdots + 586340901441 \)
|
| $37$ |
\( T^{8} + \cdots + 129865816161 \)
|
| $41$ |
\( T^{8} + \cdots + 50792988686400 \)
|
| $43$ |
\( T^{8} + \cdots + 7809080161 \)
|
| $47$ |
\( T^{8} + \cdots + 101750792256 \)
|
| $53$ |
\( T^{8} + \cdots + 5979932252100 \)
|
| $59$ |
\( T^{8} + 72 T^{7} + \cdots + 198640836 \)
|
| $61$ |
\( T^{8} + \cdots + 49371772515025 \)
|
| $67$ |
\( T^{8} + \cdots + 284883725025 \)
|
| $71$ |
\( T^{8} + \cdots + 3674889000000 \)
|
| $73$ |
\( T^{8} + \cdots + 8048063521 \)
|
| $79$ |
\( T^{8} + \cdots + 195890003437041 \)
|
| $83$ |
\( (T^{4} + 116 T^{3} + \cdots - 30693384)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 232835310959364 \)
|
| $97$ |
\( T^{8} + \cdots + 34707690000 \)
|
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