Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,6,Mod(1,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("768.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.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: | \(123.174773616\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 167x^{4} + 62x^{3} + 7709x^{2} + 6040x - 68866 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{27} \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 167x^{4} + 62x^{3} + 7709x^{2} + 6040x - 68866 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1712\nu^{5} - 5758\nu^{4} - 369348\nu^{3} + 2104280\nu^{2} + 14861998\nu - 82484556 ) / 701925 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 304\nu^{5} - 2222\nu^{4} - 15204\nu^{3} + 205720\nu^{2} - 718498\nu - 5561388 ) / 28077 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -24736\nu^{5} + 139424\nu^{4} + 2974944\nu^{3} - 11301040\nu^{2} - 68580944\nu + 136983168 ) / 701925 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1632\nu^{5} - 24032\nu^{4} - 172256\nu^{3} + 2404800\nu^{2} + 4792768\nu - 38822287 ) / 28077 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2560\nu^{5} - 14208\nu^{4} - 317184\nu^{3} + 1164928\nu^{2} + 10148864\nu - 14653051 ) / 9359 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 8\beta_{3} + \beta_{2} - \beta _1 + 85 ) / 256 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 24\beta_{3} + 31\beta_{2} + 97\beta _1 + 14421 ) / 256 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 152\beta_{5} - 24\beta_{4} + 1376\beta_{3} + 823\beta_{2} - 247\beta _1 + 70256 ) / 512 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 253\beta_{5} - 504\beta_{4} + 3144\beta_{3} + 4152\beta_{2} + 10632\beta _1 + 1157465 ) / 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14674\beta_{5} - 8568\beta_{4} + 120112\beta_{3} + 111915\beta_{2} + 7061\beta _1 + 10684690 ) / 512 \)
|
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 |
|
0 | 9.00000 | 0 | −107.066 | 0 | −150.154 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −46.1613 | 0 | 59.9278 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | −21.1198 | 0 | 241.194 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 21.1198 | 0 | −241.194 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.00000 | 0 | 46.1613 | 0 | −59.9278 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 9.00000 | 0 | 107.066 | 0 | 150.154 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 768.6.a.bf | 6 | |
| 4.b | odd | 2 | 1 | 768.6.a.be | 6 | ||
| 8.b | even | 2 | 1 | 768.6.a.be | 6 | ||
| 8.d | odd | 2 | 1 | inner | 768.6.a.bf | 6 | |
| 16.e | even | 4 | 2 | 384.6.d.j | ✓ | 12 | |
| 16.f | odd | 4 | 2 | 384.6.d.j | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.6.d.j | ✓ | 12 | 16.e | even | 4 | 2 | |
| 384.6.d.j | ✓ | 12 | 16.f | odd | 4 | 2 | |
| 768.6.a.be | 6 | 4.b | odd | 2 | 1 | ||
| 768.6.a.be | 6 | 8.b | even | 2 | 1 | ||
| 768.6.a.bf | 6 | 1.a | even | 1 | 1 | trivial | |
| 768.6.a.bf | 6 | 8.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(768))\):
|
\( T_{5}^{6} - 14040T_{5}^{4} + 30489792T_{5}^{2} - 10895241728 \)
|
|
\( T_{7}^{6} - 84312T_{7}^{4} + 1601504448T_{7}^{2} - 4710436295168 \)
|
|
\( T_{11}^{3} + 20T_{11}^{2} - 310480T_{11} + 56979904 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T - 9)^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 10895241728 \)
|
| $7$ |
\( T^{6} + \cdots - 4710436295168 \)
|
| $11$ |
\( (T^{3} + 20 T^{2} + \cdots + 56979904)^{2} \)
|
| $13$ |
\( T^{6} + \cdots - 31\!\cdots\!32 \)
|
| $17$ |
\( (T^{3} + 1222 T^{2} + \cdots - 2972835064)^{2} \)
|
| $19$ |
\( (T^{3} - 364 T^{2} + \cdots + 2897271232)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 30\!\cdots\!28 \)
|
| $29$ |
\( T^{6} + \cdots - 14\!\cdots\!88 \)
|
| $31$ |
\( T^{6} + \cdots - 33\!\cdots\!12 \)
|
| $37$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} - 6274 T^{2} + \cdots + 560226749800)^{2} \)
|
| $43$ |
\( (T^{3} - 12036 T^{2} + \cdots + 745289149248)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 13\!\cdots\!52 \)
|
| $53$ |
\( T^{6} + \cdots - 22\!\cdots\!48 \)
|
| $59$ |
\( (T^{3} + \cdots - 33069211542976)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 22\!\cdots\!48 \)
|
| $67$ |
\( (T^{3} + \cdots - 14430431514176)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 14\!\cdots\!48 \)
|
| $73$ |
\( (T^{3} + \cdots + 12280634576248)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 20\!\cdots\!92 \)
|
| $83$ |
\( (T^{3} + \cdots + 11704705520192)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots - 28858879919016)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 592920131000504)^{2} \)
|
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