Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,18,Mod(64,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.64");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.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: | \(214.369842193\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 2031617 x^{18} + 1715857968816 x^{16} + \cdots + 62\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{43}\cdot 3^{33}\cdot 13^{11} \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 2031617 x^{18} + 1715857968816 x^{16} + \cdots + 62\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 203162 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!83 \nu^{18} + \cdots - 13\!\cdots\!84 ) / 35\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 42\!\cdots\!63 \nu^{18} + \cdots + 55\!\cdots\!60 ) / 39\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 63\!\cdots\!07 \nu^{18} + \cdots - 67\!\cdots\!40 ) / 39\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10\!\cdots\!91 \nu^{18} + \cdots - 75\!\cdots\!00 ) / 19\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 62\!\cdots\!73 \nu^{18} + \cdots - 40\!\cdots\!08 ) / 14\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25\!\cdots\!77 \nu^{19} + \cdots - 47\!\cdots\!56 \nu ) / 73\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\!\cdots\!77 \nu^{19} + \cdots + 97\!\cdots\!12 \nu ) / 14\!\cdots\!88 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 36\!\cdots\!71 \nu^{19} + \cdots - 29\!\cdots\!08 \nu ) / 23\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 40\!\cdots\!67 \nu^{19} + \cdots - 25\!\cdots\!24 \nu ) / 23\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14\!\cdots\!41 \nu^{19} + \cdots + 12\!\cdots\!36 \nu ) / 23\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 19\!\cdots\!19 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 11\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 19\!\cdots\!19 \nu^{19} + \cdots + 24\!\cdots\!76 ) / 11\!\cdots\!80 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 79\!\cdots\!59 \nu^{19} + \cdots + 24\!\cdots\!20 ) / 23\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 98\!\cdots\!45 \nu^{19} + \cdots + 95\!\cdots\!36 \nu ) / 10\!\cdots\!80 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 18\!\cdots\!11 \nu^{19} + \cdots - 29\!\cdots\!80 ) / 98\!\cdots\!40 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 12\!\cdots\!33 \nu^{19} + \cdots + 14\!\cdots\!04 ) / 23\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 86\!\cdots\!39 \nu^{19} + \cdots - 10\!\cdots\!32 ) / 11\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 203162 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 5\beta_{8} - 342462\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + \beta_{4} - 29285\beta_{3} - 461416\beta_{2} + 69575308128 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{19} - 3 \beta_{18} - 77 \beta_{17} + 35 \beta_{16} - 77 \beta_{15} + 977 \beta_{13} + \cdots - 375 \)
|
| \(\nu^{6}\) | \(=\) |
\( 19120 \beta_{17} + 20080 \beta_{15} + 32160 \beta_{14} + 218320 \beta_{13} + 57360 \beta_{11} + \cdots - 27\!\cdots\!48 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3993642 \beta_{19} + 4757943 \beta_{18} + 60371913 \beta_{17} - 25010055 \beta_{16} + \cdots + 284356395 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 16064571760 \beta_{17} - 18771196240 \beta_{15} - 26788390560 \beta_{14} - 187464291280 \beta_{13} + \cdots + 11\!\cdots\!00 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3842331412810 \beta_{19} - 3263078487015 \beta_{18} - 34759834780825 \beta_{17} + 14280768762295 \beta_{16} + \cdots - 159588354104475 \)
|
| \(\nu^{10}\) | \(=\) |
\( 94\!\cdots\!20 \beta_{17} + \cdots - 48\!\cdots\!72 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 27\!\cdots\!66 \beta_{19} + \cdots + 80\!\cdots\!15 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 48\!\cdots\!60 \beta_{17} + \cdots + 21\!\cdots\!76 \)
|
| \(\nu^{13}\) | \(=\) |
\( 17\!\cdots\!94 \beta_{19} + \cdots - 39\!\cdots\!75 \)
|
| \(\nu^{14}\) | \(=\) |
\( 23\!\cdots\!00 \beta_{17} + \cdots - 93\!\cdots\!72 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 10\!\cdots\!38 \beta_{19} + \cdots + 18\!\cdots\!95 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 11\!\cdots\!80 \beta_{17} + \cdots + 41\!\cdots\!36 \)
|
| \(\nu^{17}\) | \(=\) |
\( 56\!\cdots\!98 \beta_{19} + \cdots - 88\!\cdots\!75 \)
|
| \(\nu^{18}\) | \(=\) |
\( 53\!\cdots\!40 \beta_{17} + \cdots - 18\!\cdots\!36 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 30\!\cdots\!26 \beta_{19} + \cdots + 41\!\cdots\!55 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
− | 687.035i | 0 | −340946. | 1.30332e6i | 0 | − | 225885.i | 1.44191e8i | 0 | 8.95425e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.2 | − | 653.708i | 0 | −296262. | − | 1.18636e6i | 0 | 6.26004e6i | 1.07986e8i | 0 | −7.75534e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.3 | − | 626.928i | 0 | −261967. | − | 1.55925e6i | 0 | − | 2.20400e7i | 8.20619e7i | 0 | −9.77538e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.4 | − | 483.872i | 0 | −103060. | − | 411183.i | 0 | 2.45337e7i | − | 1.35541e7i | 0 | −1.98960e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.5 | − | 476.204i | 0 | −95698.4 | 450866.i | 0 | − | 9.05916e6i | − | 1.68451e7i | 0 | 2.14704e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.6 | − | 354.573i | 0 | 5349.95 | 907528.i | 0 | − | 1.26328e6i | − | 4.83715e7i | 0 | 3.21785e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.7 | − | 338.300i | 0 | 16625.1 | − | 129285.i | 0 | − | 1.08835e7i | − | 4.99659e7i | 0 | −4.37372e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.8 | − | 157.503i | 0 | 106265. | 861342.i | 0 | 2.37605e7i | − | 3.73812e7i | 0 | 1.35664e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.9 | − | 91.7275i | 0 | 122658. | − | 430609.i | 0 | − | 1.71984e7i | − | 2.32740e7i | 0 | −3.94986e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.10 | − | 70.2433i | 0 | 126138. | − | 1.58264e6i | 0 | 1.05039e7i | − | 1.80673e7i | 0 | −1.11170e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.11 | 70.2433i | 0 | 126138. | 1.58264e6i | 0 | − | 1.05039e7i | 1.80673e7i | 0 | −1.11170e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.12 | 91.7275i | 0 | 122658. | 430609.i | 0 | 1.71984e7i | 2.32740e7i | 0 | −3.94986e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.13 | 157.503i | 0 | 106265. | − | 861342.i | 0 | − | 2.37605e7i | 3.73812e7i | 0 | 1.35664e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.14 | 338.300i | 0 | 16625.1 | 129285.i | 0 | 1.08835e7i | 4.99659e7i | 0 | −4.37372e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.15 | 354.573i | 0 | 5349.95 | − | 907528.i | 0 | 1.26328e6i | 4.83715e7i | 0 | 3.21785e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.16 | 476.204i | 0 | −95698.4 | − | 450866.i | 0 | 9.05916e6i | 1.68451e7i | 0 | 2.14704e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.17 | 483.872i | 0 | −103060. | 411183.i | 0 | − | 2.45337e7i | 1.35541e7i | 0 | −1.98960e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.18 | 626.928i | 0 | −261967. | 1.55925e6i | 0 | 2.20400e7i | − | 8.20619e7i | 0 | −9.77538e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.19 | 653.708i | 0 | −296262. | 1.18636e6i | 0 | − | 6.26004e6i | − | 1.07986e8i | 0 | −7.75534e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.20 | 687.035i | 0 | −340946. | − | 1.30332e6i | 0 | 225885.i | − | 1.44191e8i | 0 | 8.95425e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.18.b.c | 20 | |
| 3.b | odd | 2 | 1 | 13.18.b.a | ✓ | 20 | |
| 13.b | even | 2 | 1 | inner | 117.18.b.c | 20 | |
| 39.d | odd | 2 | 1 | 13.18.b.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.18.b.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 13.18.b.a | ✓ | 20 | 39.d | odd | 2 | 1 | |
| 117.18.b.c | 20 | 1.a | even | 1 | 1 | trivial | |
| 117.18.b.c | 20 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 2031617 T_{2}^{18} + 1715857968816 T_{2}^{16} + \cdots + 62\!\cdots\!36 \)
acting on \(S_{18}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 62\!\cdots\!36 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 94\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 16\!\cdots\!00 \)
|
| $11$ |
\( T^{20} + \cdots + 37\!\cdots\!00 \)
|
| $13$ |
\( T^{20} + \cdots + 23\!\cdots\!49 \)
|
| $17$ |
\( (T^{10} + \cdots + 71\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 92\!\cdots\!56 \)
|
| $23$ |
\( (T^{10} + \cdots - 94\!\cdots\!96)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots + 53\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 42\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 23\!\cdots\!56 \)
|
| $41$ |
\( T^{20} + \cdots + 31\!\cdots\!00 \)
|
| $43$ |
\( (T^{10} + \cdots + 41\!\cdots\!16)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 63\!\cdots\!96 \)
|
| $53$ |
\( (T^{10} + \cdots - 78\!\cdots\!56)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 12\!\cdots\!36 \)
|
| $61$ |
\( (T^{10} + \cdots - 90\!\cdots\!96)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 88\!\cdots\!76 \)
|
| $71$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $73$ |
\( T^{20} + \cdots + 23\!\cdots\!36 \)
|
| $79$ |
\( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 21\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 49\!\cdots\!16 \)
|
| $97$ |
\( T^{20} + \cdots + 41\!\cdots\!96 \)
|
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