Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,7,Mod(319,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.319");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.b (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: | \(88.3407681100\) |
| 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: |
\( x^{8} - 4x^{7} + 106x^{6} - 304x^{5} + 4359x^{4} - 8216x^{3} + 73366x^{2} - 69308x + 604693 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{10} \) |
| 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
\( x^{8} - 4x^{7} + 106x^{6} - 304x^{5} + 4359x^{4} - 8216x^{3} + 73366x^{2} - 69308x + 604693 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{6} - 6\nu^{5} + 418\nu^{4} - 826\nu^{3} + 15758\nu^{2} - 15346\nu + 184860 ) / 959 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 72\nu^{6} - 216\nu^{5} + 5184\nu^{4} - 10008\nu^{3} + 103680\nu^{2} - 98712\nu - 42696 ) / 137 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 146\nu^{6} - 438\nu^{5} + 13252\nu^{4} - 25774\nu^{3} + 477116\nu^{2} - 464302\nu + 5226282 ) / 959 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3078 \nu^{7} + 10773 \nu^{6} - 265689 \nu^{5} + 637290 \nu^{4} - 9124497 \nu^{3} + \cdots + 37507653 ) / 13442440 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6066 \nu^{7} - 21231 \nu^{6} + 1938603 \nu^{5} - 4793430 \nu^{4} + 138256659 \nu^{3} + \cdots - 2117169471 ) / 23524270 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 92502 \nu^{7} - 323757 \nu^{6} + 5437665 \nu^{5} - 12784770 \nu^{4} + 57721113 \nu^{3} + \cdots + 518150979 ) / 4704854 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 55910 \nu^{7} + 195685 \nu^{6} - 5832297 \nu^{5} + 14091530 \nu^{4} - 193915041 \nu^{3} + \cdots + 843807893 ) / 2688488 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 9\beta_{5} + 96\beta_{4} + 432 ) / 864 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} + 18\beta_{5} + 192\beta_{4} + 12\beta_{3} - 3\beta_{2} - 120\beta _1 - 42336 ) / 1728 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 162\beta_{7} - 176\beta_{6} - 576\beta_{5} - 30474\beta_{4} + 36\beta_{3} - 9\beta_{2} - 360\beta _1 - 127872 ) / 3456 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 81\beta_{7} - 89\beta_{6} - 297\beta_{5} - 15333\beta_{4} - 264\beta_{3} + 54\beta_{2} + 5664\beta _1 + 364176 ) / 864 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 12960 \beta_{7} + 2852 \beta_{6} + 2484 \beta_{5} + 1424736 \beta_{4} - 2700 \beta_{3} + \cdots + 3855168 ) / 3456 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 19845 \beta_{7} + 4724 \beta_{6} + 5220 \beta_{5} + 2213865 \beta_{4} + 19188 \beta_{3} + \cdots + 7627392 ) / 1728 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 283311 \beta_{7} + 74398 \beta_{6} + 254286 \beta_{5} - 26602971 \beta_{4} + 71904 \beta_{3} + \cdots + 19874592 ) / 1728 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
0 | −15.5885 | 0 | − | 195.235i | 0 | 277.510i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −15.5885 | 0 | − | 85.3891i | 0 | − | 511.824i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | −15.5885 | 0 | 85.3891i | 0 | 511.824i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | −15.5885 | 0 | 195.235i | 0 | − | 277.510i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | 15.5885 | 0 | − | 195.235i | 0 | − | 277.510i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 15.5885 | 0 | − | 85.3891i | 0 | 511.824i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 15.5885 | 0 | 85.3891i | 0 | − | 511.824i | 0 | 243.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 15.5885 | 0 | 195.235i | 0 | 277.510i | 0 | 243.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 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 | 384.7.b.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 384.7.b.c | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 384.7.b.c | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 384.7.b.c | ✓ | 8 |
| 16.e | even | 4 | 2 | 768.7.g.g | 8 | ||
| 16.f | odd | 4 | 2 | 768.7.g.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.7.b.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.7.b.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 384.7.b.c | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.7.b.c | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 768.7.g.g | 8 | 16.e | even | 4 | 2 | ||
| 768.7.g.g | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 45408T_{5}^{2} + 277920000 \)
acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 243)^{4} \)
|
| $5$ |
\( (T^{4} + 45408 T^{2} + 277920000)^{2} \)
|
| $7$ |
\( (T^{4} + 338976 T^{2} + 20174323968)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 4522189383936)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 11711515004928)^{2} \)
|
| $17$ |
\( (T^{2} + 1636 T - 58718780)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 29\!\cdots\!08)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 64\!\cdots\!08)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 10\!\cdots\!88)^{2} \)
|
| $41$ |
\( (T^{2} + 124892 T + 3840115012)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 80\!\cdots\!52)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 91\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 63\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 24\!\cdots\!84)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 55\!\cdots\!92)^{2} \)
|
| $73$ |
\( (T^{2} + 490660 T + 36431647300)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 27\!\cdots\!44)^{2} \)
|
| $89$ |
\( (T^{2} + 423748 T - 40865541500)^{4} \)
|
| $97$ |
\( (T^{2} - 1908188 T + 904356570436)^{4} \)
|
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