Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,6,Mod(1,688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(688, 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("688.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 688.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: | \(110.344068031\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 43) |
| 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
\( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 12949 \nu^{7} + 681148 \nu^{6} - 2601727 \nu^{5} - 80176782 \nu^{4} + 378607946 \nu^{3} + \cdots - 1162574592 ) / 547265856 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36205 \nu^{7} - 1830508 \nu^{6} - 539057 \nu^{5} + 265985910 \nu^{4} - 188725898 \nu^{3} + \cdots + 58885512576 ) / 1094531712 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 67489 \nu^{7} + 623356 \nu^{6} + 8964549 \nu^{5} - 78907518 \nu^{4} - 250466958 \nu^{3} + \cdots - 8596825856 ) / 364843904 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 204587 \nu^{7} + 1315940 \nu^{6} + 28380103 \nu^{5} - 147318570 \nu^{4} - 993160922 \nu^{3} + \cdots - 37440050688 ) / 1094531712 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 309497 \nu^{7} + 2935676 \nu^{6} + 38047933 \nu^{5} - 372264846 \nu^{4} - 955734974 \nu^{3} + \cdots - 76606929024 ) / 1094531712 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73259 \nu^{7} - 190788 \nu^{6} - 10643911 \nu^{5} + 12835930 \nu^{4} + 347431274 \nu^{3} + \cdots - 3291957568 ) / 182421952 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 34066 \nu^{7} + 259912 \nu^{6} + 4750721 \nu^{5} - 30187209 \nu^{4} - 176761426 \nu^{3} + \cdots - 2232499728 ) / 68408232 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 4 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{4} + \beta_{3} + 180 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -14\beta_{7} + 33\beta_{6} + 15\beta_{5} + 32\beta_{4} + 65\beta_{3} + 64\beta_{2} + 4\beta _1 + 380 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -99\beta_{7} + 84\beta_{6} - 117\beta_{5} + 488\beta_{4} + 140\beta_{3} + 63\beta_{2} + 46\beta _1 + 6796 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 869 \beta_{7} + 1527 \beta_{6} - 276 \beta_{5} + 2768 \beta_{4} + 3711 \beta_{3} + 2515 \beta_{2} + \cdots + 27000 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 10339 \beta_{7} + 11590 \beta_{6} - 16231 \beta_{5} + 55680 \beta_{4} + 22478 \beta_{3} + \cdots + 639708 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 102849 \beta_{7} + 166838 \beta_{6} - 94865 \beta_{5} + 386624 \beta_{4} + 421838 \beta_{3} + \cdots + 3616388 ) / 2 \)
|
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 | −25.6605 | 0 | −61.4284 | 0 | 184.774 | 0 | 415.460 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −11.1803 | 0 | −63.3756 | 0 | −223.489 | 0 | −118.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −7.84314 | 0 | −107.102 | 0 | 25.5214 | 0 | −181.485 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.05838 | 0 | 27.7074 | 0 | 103.690 | 0 | −233.646 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.19190 | 0 | 73.4416 | 0 | 4.24720 | 0 | −158.509 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 11.2683 | 0 | −9.80186 | 0 | 11.1041 | 0 | −116.025 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 25.1057 | 0 | −61.2927 | 0 | 166.001 | 0 | 387.294 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 28.1764 | 0 | −10.1483 | 0 | −135.849 | 0 | 550.912 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 688.6.a.e | 8 | |
| 4.b | odd | 2 | 1 | 43.6.a.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 387.6.a.c | 8 | ||
| 20.d | odd | 2 | 1 | 1075.6.a.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.6.a.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 387.6.a.c | 8 | 12.b | even | 2 | 1 | ||
| 688.6.a.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1075.6.a.a | 8 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 26 T_{3}^{7} - 907 T_{3}^{6} + 22242 T_{3}^{5} + 184435 T_{3}^{4} - 3627880 T_{3}^{3} + \cdots + 504223128 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 26 T^{7} + \cdots + 504223128 \)
|
| $5$ |
\( T^{8} + \cdots + 5172924974752 \)
|
| $7$ |
\( T^{8} + \cdots + 116222354316288 \)
|
| $11$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $13$ |
\( T^{8} + \cdots - 20\!\cdots\!44 \)
|
| $17$ |
\( T^{8} + \cdots + 37\!\cdots\!33 \)
|
| $19$ |
\( T^{8} + \cdots + 20\!\cdots\!68 \)
|
| $23$ |
\( T^{8} + \cdots - 18\!\cdots\!83 \)
|
| $29$ |
\( T^{8} + \cdots - 39\!\cdots\!36 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!13 \)
|
| $37$ |
\( T^{8} + \cdots - 10\!\cdots\!04 \)
|
| $41$ |
\( T^{8} + \cdots + 17\!\cdots\!57 \)
|
| $43$ |
\( (T - 1849)^{8} \)
|
| $47$ |
\( T^{8} + \cdots - 19\!\cdots\!04 \)
|
| $53$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots - 68\!\cdots\!68 \)
|
| $61$ |
\( T^{8} + \cdots + 23\!\cdots\!84 \)
|
| $67$ |
\( T^{8} + \cdots + 69\!\cdots\!48 \)
|
| $71$ |
\( T^{8} + \cdots - 15\!\cdots\!64 \)
|
| $73$ |
\( T^{8} + \cdots + 80\!\cdots\!92 \)
|
| $79$ |
\( T^{8} + \cdots + 19\!\cdots\!12 \)
|
| $83$ |
\( T^{8} + \cdots - 19\!\cdots\!08 \)
|
| $89$ |
\( T^{8} + \cdots - 13\!\cdots\!64 \)
|
| $97$ |
\( T^{8} + \cdots + 38\!\cdots\!17 \)
|
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