Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,8,Mod(1,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.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: | \(114.957689378\) |
| 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: |
\( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 8769333 \nu^{7} - 68795078 \nu^{6} + 5472097628 \nu^{5} + 25057870375 \nu^{4} + \cdots + 21\!\cdots\!08 ) / 3315013604640 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 70454057 \nu^{7} + 418096898 \nu^{6} + 53704469932 \nu^{5} - 276596270045 \nu^{4} + \cdots + 209986539079392 ) / 3315013604640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 70454057 \nu^{7} + 418096898 \nu^{6} + 53704469932 \nu^{5} - 276596270045 \nu^{4} + \cdots + 209986539079392 ) / 3315013604640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 34071679 \nu^{7} - 199982746 \nu^{6} - 28011140084 \nu^{5} + 111949525435 \nu^{4} + \cdots - 640690545729384 ) / 828753401160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2454119 \nu^{7} - 23811754 \nu^{6} - 1978218156 \nu^{5} + 14359538067 \nu^{4} + \cdots - 17165478315440 ) / 36833484496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 134637239 \nu^{7} + 1818970006 \nu^{6} + 119496719684 \nu^{5} - 1108150909635 \nu^{4} + \cdots - 116254552795536 ) / 1657506802320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 531963853 \nu^{7} - 2557335362 \nu^{6} - 406400960428 \nu^{5} + 1694921846865 \nu^{4} + \cdots - 41\!\cdots\!08 ) / 3315013604640 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{5} - 2\beta_{4} + 5\beta_{3} + 13\beta_{2} - 4\beta _1 + 3328 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -40\beta_{7} + 2\beta_{6} - 20\beta_{5} + 40\beta_{4} + 311\beta_{3} - 603\beta_{2} - 24\beta _1 + 6338 ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 537\beta_{7} - 125\beta_{6} + 207\beta_{5} - 685\beta_{4} + 1150\beta_{3} + 2821\beta_{2} - 922\beta _1 + 572187 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 22654 \beta_{7} + 3532 \beta_{6} - 5830 \beta_{5} + 19198 \beta_{4} + 114901 \beta_{3} + \cdots + 2023612 ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 521640 \beta_{7} - 158274 \beta_{6} + 30292 \beta_{5} - 632680 \beta_{4} + 900637 \beta_{3} + \cdots + 444146654 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 5437557 \beta_{7} + 1204581 \beta_{6} - 814651 \beta_{5} + 4371929 \beta_{4} + 22455552 \beta_{3} + \cdots + 188434669 ) / 8 \)
|
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 | −86.9475 | 0 | −188.239 | 0 | −639.155 | 0 | 5372.88 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −65.3727 | 0 | 376.733 | 0 | 902.074 | 0 | 2086.59 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −60.8046 | 0 | 165.526 | 0 | −952.148 | 0 | 1510.20 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −36.3647 | 0 | −31.9528 | 0 | 461.175 | 0 | −864.611 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 10.7680 | 0 | −404.860 | 0 | 387.911 | 0 | −2071.05 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 37.8447 | 0 | 128.909 | 0 | −1733.23 | 0 | −754.777 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 76.4323 | 0 | 522.229 | 0 | −653.513 | 0 | 3654.90 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 84.4445 | 0 | −124.345 | 0 | 780.885 | 0 | 4943.87 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 368.8.a.h | 8 | |
| 4.b | odd | 2 | 1 | 23.8.a.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 207.8.a.f | 8 | ||
| 20.d | odd | 2 | 1 | 575.8.a.b | 8 | ||
| 92.b | even | 2 | 1 | 529.8.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.8.a.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 207.8.a.f | 8 | 12.b | even | 2 | 1 | ||
| 368.8.a.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 529.8.a.c | 8 | 92.b | even | 2 | 1 | ||
| 575.8.a.b | 8 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 40 T_{3}^{7} - 14887 T_{3}^{6} - 581660 T_{3}^{5} + 67535395 T_{3}^{4} + \cdots + 33056528652000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 33056528652000 \)
|
| $5$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 86\!\cdots\!92 \)
|
| $11$ |
\( T^{8} + \cdots + 23\!\cdots\!20 \)
|
| $13$ |
\( T^{8} + \cdots + 39\!\cdots\!68 \)
|
| $17$ |
\( T^{8} + \cdots - 38\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $23$ |
\( (T - 12167)^{8} \)
|
| $29$ |
\( T^{8} + \cdots - 13\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 80\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $41$ |
\( T^{8} + \cdots - 38\!\cdots\!40 \)
|
| $43$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 82\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 18\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots - 15\!\cdots\!60 \)
|
| $61$ |
\( T^{8} + \cdots + 59\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 90\!\cdots\!68 \)
|
| $79$ |
\( T^{8} + \cdots + 16\!\cdots\!60 \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots - 38\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 17\!\cdots\!92 \)
|
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