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: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 5 x^{12} - 2061 x^{11} + 9751 x^{10} + 1539985 x^{9} - 6322821 x^{8} - 543149608 x^{7} + \cdots + 89324732764635 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 344) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 5 x^{12} - 2061 x^{11} + 9751 x^{10} + 1539985 x^{9} - 6322821 x^{8} - 543149608 x^{7} + \cdots + 89324732764635 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 40\!\cdots\!83 \nu^{12} + \cdots - 13\!\cdots\!21 ) / 12\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 79\!\cdots\!85 \nu^{12} + \cdots + 96\!\cdots\!69 ) / 18\!\cdots\!08 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 19\!\cdots\!91 \nu^{12} + \cdots - 36\!\cdots\!65 ) / 36\!\cdots\!16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 55\!\cdots\!08 \nu^{12} + \cdots + 40\!\cdots\!97 ) / 45\!\cdots\!02 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 94\!\cdots\!91 \nu^{12} + \cdots - 63\!\cdots\!83 ) / 72\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 46\!\cdots\!01 \nu^{12} + \cdots + 44\!\cdots\!39 ) / 25\!\cdots\!44 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11\!\cdots\!67 \nu^{12} + \cdots - 70\!\cdots\!47 ) / 36\!\cdots\!16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 24\!\cdots\!93 \nu^{12} + \cdots + 72\!\cdots\!15 ) / 72\!\cdots\!32 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 39\!\cdots\!99 \nu^{12} + \cdots - 22\!\cdots\!97 ) / 10\!\cdots\!76 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 28\!\cdots\!97 \nu^{12} + \cdots - 91\!\cdots\!77 ) / 72\!\cdots\!32 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 72\!\cdots\!43 \nu^{12} + \cdots + 18\!\cdots\!07 ) / 12\!\cdots\!72 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + 319 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 7 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 9 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 13 \beta_{6} + \cdots - 86 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 30 \beta_{12} - 147 \beta_{11} + 72 \beta_{10} + 106 \beta_{9} - 82 \beta_{8} + 199 \beta_{7} + \cdots + 180789 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 6478 \beta_{12} - 3470 \beta_{11} + 5042 \beta_{10} + 8892 \beta_{9} - 932 \beta_{8} + 2929 \beta_{7} + \cdots - 250376 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 36255 \beta_{12} - 187167 \beta_{11} + 85818 \beta_{10} + 119941 \beta_{9} - 112393 \beta_{8} + \cdots + 132627669 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 5451553 \beta_{12} - 2187857 \beta_{11} + 4643402 \beta_{10} + 7876452 \beta_{9} - 131327 \beta_{8} + \cdots - 366227609 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 32601300 \beta_{12} - 187007688 \beta_{11} + 82570791 \beta_{10} + 111354571 \beta_{9} + \cdots + 109062795030 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4635152671 \beta_{12} - 1397241872 \beta_{11} + 4203199205 \beta_{10} + 6914503314 \beta_{9} + \cdots - 441625626017 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 26662686039 \beta_{12} - 173973490368 \beta_{11} + 74669090628 \beta_{10} + 97780104049 \beta_{9} + \cdots + 94242319212471 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 4015075102063 \beta_{12} - 908730949199 \beta_{11} + 3770385303098 \beta_{10} + \cdots - 487682604775811 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 20848461759309 \beta_{12} - 157840882048425 \beta_{11} + 65773435548495 \beta_{10} + \cdots + 83\!\cdots\!62 \)
|
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 | −29.3621 | 0 | −92.8141 | 0 | 40.9767 | 0 | 619.133 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −19.7435 | 0 | −3.08190 | 0 | −58.5451 | 0 | 146.807 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −15.1390 | 0 | 45.6869 | 0 | 158.543 | 0 | −13.8092 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −13.5229 | 0 | 96.1595 | 0 | 176.475 | 0 | −60.1323 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −11.1002 | 0 | −29.4977 | 0 | 78.4325 | 0 | −119.786 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −5.64949 | 0 | −27.6002 | 0 | −66.7952 | 0 | −211.083 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.625007 | 0 | 39.7863 | 0 | −101.718 | 0 | −242.609 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 10.9515 | 0 | 15.2510 | 0 | −228.329 | 0 | −123.066 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 13.1732 | 0 | −95.2045 | 0 | −123.374 | 0 | −69.4668 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 14.8554 | 0 | −60.5751 | 0 | 92.0418 | 0 | −22.3159 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 19.4463 | 0 | 43.9087 | 0 | 209.610 | 0 | 135.157 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 22.9332 | 0 | 91.8694 | 0 | −33.0887 | 0 | 282.932 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 30.5326 | 0 | −73.8883 | 0 | 87.7715 | 0 | 689.240 | 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.j | 13 | |
| 4.b | odd | 2 | 1 | 344.6.a.b | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 344.6.a.b | ✓ | 13 | 4.b | odd | 2 | 1 | |
| 688.6.a.j | 13 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{13} - 18 T_{3}^{12} - 1923 T_{3}^{11} + 31806 T_{3}^{10} + 1330935 T_{3}^{9} + \cdots - 135742423466880 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} \)
|
| $3$ |
\( T^{13} + \cdots - 135742423466880 \)
|
| $5$ |
\( T^{13} + \cdots + 10\!\cdots\!88 \)
|
| $7$ |
\( T^{13} + \cdots - 56\!\cdots\!20 \)
|
| $11$ |
\( T^{13} + \cdots - 23\!\cdots\!40 \)
|
| $13$ |
\( T^{13} + \cdots - 92\!\cdots\!64 \)
|
| $17$ |
\( T^{13} + \cdots + 74\!\cdots\!78 \)
|
| $19$ |
\( T^{13} + \cdots + 40\!\cdots\!20 \)
|
| $23$ |
\( T^{13} + \cdots - 14\!\cdots\!52 \)
|
| $29$ |
\( T^{13} + \cdots - 21\!\cdots\!68 \)
|
| $31$ |
\( T^{13} + \cdots + 49\!\cdots\!12 \)
|
| $37$ |
\( T^{13} + \cdots - 19\!\cdots\!52 \)
|
| $41$ |
\( T^{13} + \cdots + 77\!\cdots\!46 \)
|
| $43$ |
\( (T - 1849)^{13} \)
|
| $47$ |
\( T^{13} + \cdots - 64\!\cdots\!00 \)
|
| $53$ |
\( T^{13} + \cdots + 71\!\cdots\!80 \)
|
| $59$ |
\( T^{13} + \cdots - 26\!\cdots\!20 \)
|
| $61$ |
\( T^{13} + \cdots + 16\!\cdots\!00 \)
|
| $67$ |
\( T^{13} + \cdots + 31\!\cdots\!84 \)
|
| $71$ |
\( T^{13} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( T^{13} + \cdots - 11\!\cdots\!20 \)
|
| $79$ |
\( T^{13} + \cdots - 25\!\cdots\!52 \)
|
| $83$ |
\( T^{13} + \cdots + 39\!\cdots\!96 \)
|
| $89$ |
\( T^{13} + \cdots + 28\!\cdots\!16 \)
|
| $97$ |
\( T^{13} + \cdots - 50\!\cdots\!38 \)
|
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