Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [87,18,Mod(1,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 87.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: | \(159.403215990\) |
| Analytic rank: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 3 x^{17} - 1690980 x^{16} + 53199640 x^{15} + 1179531333560 x^{14} - 63426587773856 x^{13} + \cdots - 46\!\cdots\!28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{38}\cdot 3^{18} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 3 x^{17} - 1690980 x^{16} + 53199640 x^{15} + 1179531333560 x^{14} - 63426587773856 x^{13} + \cdots - 46\!\cdots\!28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 43\nu - 187894 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 44\!\cdots\!69 \nu^{17} + \cdots + 57\!\cdots\!04 ) / 30\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 54\!\cdots\!21 \nu^{17} + \cdots + 35\!\cdots\!64 ) / 75\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!73 \nu^{17} + \cdots - 27\!\cdots\!28 ) / 15\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17\!\cdots\!99 \nu^{17} + \cdots - 33\!\cdots\!84 ) / 15\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 80\!\cdots\!83 \nu^{17} + \cdots + 63\!\cdots\!12 ) / 30\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21\!\cdots\!29 \nu^{17} + \cdots + 16\!\cdots\!36 ) / 75\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!37 \nu^{17} + \cdots + 41\!\cdots\!08 ) / 30\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11\!\cdots\!13 \nu^{17} + \cdots - 17\!\cdots\!92 ) / 16\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 20\!\cdots\!91 \nu^{17} + \cdots - 16\!\cdots\!56 ) / 30\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 12\!\cdots\!03 \nu^{17} + \cdots + 36\!\cdots\!48 ) / 15\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 43\!\cdots\!41 \nu^{17} + \cdots + 43\!\cdots\!44 ) / 30\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 56\!\cdots\!57 \nu^{17} + \cdots + 54\!\cdots\!12 ) / 30\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 43\!\cdots\!33 \nu^{17} + \cdots + 51\!\cdots\!28 ) / 15\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 15\!\cdots\!57 \nu^{17} + \cdots - 21\!\cdots\!12 ) / 30\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 21\!\cdots\!39 \nu^{17} + \cdots - 23\!\cdots\!24 ) / 37\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 43\beta _1 + 187894 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{3} - 104\beta_{2} + 295744\beta _1 - 8070370 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 9 \beta_{17} + 9 \beta_{16} - \beta_{15} + 26 \beta_{14} - 10 \beta_{13} - 18 \beta_{11} + \cdots + 55567606262 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2075 \beta_{17} + 6505 \beta_{16} + 2603 \beta_{15} + 4334 \beta_{14} - 4866 \beta_{13} + \cdots - 5987634901226 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 5584579 \beta_{17} + 4710175 \beta_{16} - 2275795 \beta_{15} + 13244898 \beta_{14} + \cdots + 19\!\cdots\!82 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1193914811 \beta_{17} + 4304180745 \beta_{16} + 2956921419 \beta_{15} + 2181201422 \beta_{14} + \cdots - 33\!\cdots\!38 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2637423975179 \beta_{17} + 1710186043879 \beta_{16} - 2264518130907 \beta_{15} + \cdots + 73\!\cdots\!38 \)
|
| \(\nu^{9}\) | \(=\) |
\( 524207029872939 \beta_{17} + \cdots - 16\!\cdots\!22 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 11\!\cdots\!23 \beta_{17} + \cdots + 29\!\cdots\!94 \)
|
| \(\nu^{11}\) | \(=\) |
\( 21\!\cdots\!15 \beta_{17} + \cdots - 83\!\cdots\!50 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 46\!\cdots\!43 \beta_{17} + \cdots + 11\!\cdots\!86 \)
|
| \(\nu^{13}\) | \(=\) |
\( 81\!\cdots\!11 \beta_{17} + \cdots - 40\!\cdots\!30 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 18\!\cdots\!11 \beta_{17} + \cdots + 49\!\cdots\!46 \)
|
| \(\nu^{15}\) | \(=\) |
\( 31\!\cdots\!43 \beta_{17} + \cdots - 19\!\cdots\!74 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 74\!\cdots\!07 \beta_{17} + \cdots + 21\!\cdots\!58 \)
|
| \(\nu^{17}\) | \(=\) |
\( 11\!\cdots\!63 \beta_{17} + \cdots - 91\!\cdots\!58 \)
|
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 |
|
−642.906 | 6561.00 | 282256. | −1.12165e6 | −4.21810e6 | 1.41273e7 | −9.71970e7 | 4.30467e7 | 7.21115e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −636.850 | 6561.00 | 274505. | 1.29793e6 | −4.17837e6 | −3.41565e6 | −9.13455e7 | 4.30467e7 | −8.26589e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −547.760 | 6561.00 | 168969. | −200504. | −3.59385e6 | 800365. | −2.07586e7 | 4.30467e7 | 1.09828e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −466.847 | 6561.00 | 86874.6 | −649294. | −3.06299e6 | −1.85104e7 | 2.06335e7 | 4.30467e7 | 3.03121e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −411.655 | 6561.00 | 38387.5 | −231428. | −2.70087e6 | 2.10324e7 | 3.81540e7 | 4.30467e7 | 9.52685e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −389.279 | 6561.00 | 20466.1 | −1.64566e6 | −2.55406e6 | −1.35131e7 | 4.30566e7 | 4.30467e7 | 6.40620e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −384.829 | 6561.00 | 17021.3 | 1.31181e6 | −2.52486e6 | −8.52335e6 | 4.38900e7 | 4.30467e7 | −5.04824e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −214.150 | 6561.00 | −85212.0 | 221270. | −1.40504e6 | −6.08201e6 | 4.63171e7 | 4.30467e7 | −4.73848e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −144.226 | 6561.00 | −110271. | 351878. | −946266. | −26516.8 | 3.48079e7 | 4.30467e7 | −5.07500e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 101.622 | 6561.00 | −120745. | −705531. | 666741. | 4.96625e6 | −2.55901e7 | 4.30467e7 | −7.16974e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 110.308 | 6561.00 | −118904. | −903271. | 723729. | −2.45287e7 | −2.75743e7 | 4.30467e7 | −9.96378e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 172.456 | 6561.00 | −101331. | −1.11025e6 | 1.13148e6 | 2.89987e7 | −4.00793e7 | 4.30467e7 | −1.91468e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 305.671 | 6561.00 | −37637.1 | 1.21061e6 | 2.00551e6 | −9.33111e6 | −5.15695e7 | 4.30467e7 | 3.70048e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 338.905 | 6561.00 | −16215.6 | 238866. | 2.22355e6 | 6.65784e6 | −4.99165e7 | 4.30467e7 | 8.09529e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 451.830 | 6561.00 | 73078.7 | 352836. | 2.96446e6 | −6.13826e6 | −2.62031e7 | 4.30467e7 | 1.59422e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 521.962 | 6561.00 | 141372. | 140839. | 3.42459e6 | 2.41660e7 | 5.37620e6 | 4.30467e7 | 7.35126e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 604.718 | 6561.00 | 234611. | −887018. | 3.96755e6 | −6.01369e6 | 6.26121e7 | 4.30467e7 | −5.36396e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 652.030 | 6561.00 | 294071. | 683978. | 4.27797e6 | −2.89649e7 | 1.06280e8 | 4.30467e7 | 4.45974e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 87.18.a.a | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.18.a.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} + 579 T_{2}^{17} - 1532676 T_{2}^{16} - 891824920 T_{2}^{15} + 949483373240 T_{2}^{14} + \cdots - 37\!\cdots\!00 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(87))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots - 37\!\cdots\!00 \)
|
| $3$ |
\( (T - 6561)^{18} \)
|
| $5$ |
\( T^{18} + \cdots - 45\!\cdots\!00 \)
|
| $7$ |
\( T^{18} + \cdots - 15\!\cdots\!64 \)
|
| $11$ |
\( T^{18} + \cdots + 20\!\cdots\!00 \)
|
| $13$ |
\( T^{18} + \cdots - 28\!\cdots\!76 \)
|
| $17$ |
\( T^{18} + \cdots + 84\!\cdots\!44 \)
|
| $19$ |
\( T^{18} + \cdots - 99\!\cdots\!00 \)
|
| $23$ |
\( T^{18} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( (T - 500246412961)^{18} \)
|
| $31$ |
\( T^{18} + \cdots + 13\!\cdots\!56 \)
|
| $37$ |
\( T^{18} + \cdots + 11\!\cdots\!72 \)
|
| $41$ |
\( T^{18} + \cdots - 32\!\cdots\!00 \)
|
| $43$ |
\( T^{18} + \cdots - 42\!\cdots\!00 \)
|
| $47$ |
\( T^{18} + \cdots + 32\!\cdots\!00 \)
|
| $53$ |
\( T^{18} + \cdots - 31\!\cdots\!00 \)
|
| $59$ |
\( T^{18} + \cdots - 13\!\cdots\!00 \)
|
| $61$ |
\( T^{18} + \cdots + 28\!\cdots\!00 \)
|
| $67$ |
\( T^{18} + \cdots - 22\!\cdots\!00 \)
|
| $71$ |
\( T^{18} + \cdots + 17\!\cdots\!40 \)
|
| $73$ |
\( T^{18} + \cdots - 26\!\cdots\!88 \)
|
| $79$ |
\( T^{18} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots + 13\!\cdots\!24 \)
|
| $89$ |
\( T^{18} + \cdots - 22\!\cdots\!00 \)
|
| $97$ |
\( T^{18} + \cdots - 15\!\cdots\!00 \)
|
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