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: |
\( x^{20} + 1948739 x^{18} + 1581500033760 x^{16} + \cdots + 11\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{59}\cdot 3^{31}\cdot 13^{11} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
\( x^{20} + 1948739 x^{18} + 1581500033760 x^{16} + \cdots + 11\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 194874 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 41\!\cdots\!13 \nu^{18} + \cdots - 67\!\cdots\!80 ) / 53\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 26\!\cdots\!83 \nu^{18} + \cdots - 57\!\cdots\!00 ) / 24\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!73 \nu^{18} + \cdots + 17\!\cdots\!40 ) / 12\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 46\!\cdots\!01 \nu^{18} + \cdots + 29\!\cdots\!00 ) / 48\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 78\!\cdots\!23 \nu^{19} + \cdots + 32\!\cdots\!76 \nu ) / 74\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 78\!\cdots\!23 \nu^{19} + \cdots + 56\!\cdots\!32 \nu ) / 74\!\cdots\!16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!79 \nu^{19} + \cdots - 48\!\cdots\!76 \nu ) / 67\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 36\!\cdots\!13 \nu^{19} + \cdots - 55\!\cdots\!16 \nu ) / 67\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 23\!\cdots\!49 \nu^{19} + \cdots - 74\!\cdots\!40 ) / 83\!\cdots\!80 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 23\!\cdots\!49 \nu^{19} + \cdots + 17\!\cdots\!20 ) / 83\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 34\!\cdots\!43 \nu^{19} + \cdots + 39\!\cdots\!20 ) / 67\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 44\!\cdots\!69 \nu^{19} + \cdots - 12\!\cdots\!20 ) / 55\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 21\!\cdots\!77 \nu^{19} + \cdots + 50\!\cdots\!20 ) / 67\!\cdots\!40 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 27\!\cdots\!67 \nu^{19} + \cdots + 22\!\cdots\!80 ) / 74\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 13\!\cdots\!01 \nu^{19} + \cdots + 65\!\cdots\!20 ) / 33\!\cdots\!20 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 13\!\cdots\!77 \nu^{19} + \cdots - 64\!\cdots\!60 ) / 33\!\cdots\!20 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 18\!\cdots\!05 \nu^{19} + \cdots - 41\!\cdots\!80 ) / 22\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 194874 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - 10\beta_{7} - 325641\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{4} + 6 \beta_{3} + \cdots + 63458408098 \)
|
| \(\nu^{5}\) | \(=\) |
\( 76 \beta_{19} - 25 \beta_{18} - 33 \beta_{17} + 33 \beta_{16} + 27 \beta_{15} - 27 \beta_{13} + \cdots - 182 \)
|
| \(\nu^{6}\) | \(=\) |
\( 18896 \beta_{15} - 51192 \beta_{14} - 727135 \beta_{13} - 21596 \beta_{12} - 1549042 \beta_{11} + \cdots - 24\!\cdots\!02 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 75337700 \beta_{19} + 16568391 \beta_{18} + 18034431 \beta_{17} - 25488991 \beta_{16} + \cdots + 149879562 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 21241054384 \beta_{15} + 47417563432 \beta_{14} + 415880935773 \beta_{13} + 23489624148 \beta_{12} + \cdots + 98\!\cdots\!38 \)
|
| \(\nu^{9}\) | \(=\) |
\( 52327426431724 \beta_{19} - 7741148008053 \beta_{18} - 6891228340957 \beta_{17} + 14939170346045 \beta_{16} + \cdots - 91183454271134 \)
|
| \(\nu^{10}\) | \(=\) |
\( 15\!\cdots\!36 \beta_{15} + \cdots - 42\!\cdots\!54 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 31\!\cdots\!84 \beta_{19} + \cdots + 49\!\cdots\!94 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 10\!\cdots\!28 \beta_{15} + \cdots + 18\!\cdots\!94 \)
|
| \(\nu^{13}\) | \(=\) |
\( 17\!\cdots\!20 \beta_{19} + \cdots - 25\!\cdots\!82 \)
|
| \(\nu^{14}\) | \(=\) |
\( 58\!\cdots\!16 \beta_{15} + \cdots - 86\!\cdots\!54 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 94\!\cdots\!68 \beta_{19} + \cdots + 13\!\cdots\!02 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 32\!\cdots\!04 \beta_{15} + \cdots + 40\!\cdots\!18 \)
|
| \(\nu^{17}\) | \(=\) |
\( 49\!\cdots\!08 \beta_{19} + \cdots - 66\!\cdots\!78 \)
|
| \(\nu^{18}\) | \(=\) |
\( 17\!\cdots\!80 \beta_{15} + \cdots - 19\!\cdots\!94 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 25\!\cdots\!76 \beta_{19} + \cdots + 33\!\cdots\!86 \)
|
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 |
|
− | 702.468i | 0 | −362389. | 46101.0i | 0 | 1.54285e7i | 1.62492e8i | 0 | 3.23845e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.2 | − | 599.814i | 0 | −228705. | − | 503961.i | 0 | 8.24822e6i | 5.85614e7i | 0 | −3.02283e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.3 | − | 593.865i | 0 | −221603. | 645057.i | 0 | − | 2.60356e7i | 5.37634e7i | 0 | 3.83076e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.4 | − | 493.882i | 0 | −112847. | 1.61969e6i | 0 | 1.46269e7i | − | 9.00098e6i | 0 | 7.99936e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.5 | − | 487.274i | 0 | −106364. | − | 479196.i | 0 | 4.45969e6i | − | 1.20396e7i | 0 | −2.33500e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.6 | − | 335.072i | 0 | 18799.0 | − | 1.41968e6i | 0 | − | 1.14676e7i | − | 5.02175e7i | 0 | −4.75693e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.7 | − | 259.347i | 0 | 63811.1 | − | 1.10108e6i | 0 | 1.56033e7i | − | 5.05424e7i | 0 | −2.85562e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.8 | − | 252.345i | 0 | 67393.9 | 293664.i | 0 | − | 1.44137e7i | − | 5.00819e7i | 0 | 7.41048e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.9 | − | 116.120i | 0 | 117588. | 344995.i | 0 | 2.96717e7i | − | 2.88745e7i | 0 | 4.00610e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.10 | − | 69.1047i | 0 | 126297. | 1.07168e6i | 0 | − | 500846.i | − | 1.77854e7i | 0 | 7.40578e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.11 | 69.1047i | 0 | 126297. | − | 1.07168e6i | 0 | 500846.i | 1.77854e7i | 0 | 7.40578e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.12 | 116.120i | 0 | 117588. | − | 344995.i | 0 | − | 2.96717e7i | 2.88745e7i | 0 | 4.00610e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.13 | 252.345i | 0 | 67393.9 | − | 293664.i | 0 | 1.44137e7i | 5.00819e7i | 0 | 7.41048e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.14 | 259.347i | 0 | 63811.1 | 1.10108e6i | 0 | − | 1.56033e7i | 5.05424e7i | 0 | −2.85562e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.15 | 335.072i | 0 | 18799.0 | 1.41968e6i | 0 | 1.14676e7i | 5.02175e7i | 0 | −4.75693e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.16 | 487.274i | 0 | −106364. | 479196.i | 0 | − | 4.45969e6i | 1.20396e7i | 0 | −2.33500e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.17 | 493.882i | 0 | −112847. | − | 1.61969e6i | 0 | − | 1.46269e7i | 9.00098e6i | 0 | 7.99936e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.18 | 593.865i | 0 | −221603. | − | 645057.i | 0 | 2.60356e7i | − | 5.37634e7i | 0 | 3.83076e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.19 | 599.814i | 0 | −228705. | 503961.i | 0 | − | 8.24822e6i | − | 5.85614e7i | 0 | −3.02283e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.20 | 702.468i | 0 | −362389. | − | 46101.0i | 0 | − | 1.54285e7i | − | 1.62492e8i | 0 | 3.23845e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.d | 20 | |
| 3.b | odd | 2 | 1 | 39.18.b.b | ✓ | 20 | |
| 13.b | even | 2 | 1 | inner | 117.18.b.d | 20 | |
| 39.d | odd | 2 | 1 | 39.18.b.b | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.18.b.b | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 39.18.b.b | ✓ | 20 | 39.d | odd | 2 | 1 | |
| 117.18.b.d | 20 | 1.a | even | 1 | 1 | trivial | |
| 117.18.b.d | 20 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 1948739 T_{2}^{18} + 1581500033760 T_{2}^{16} + \cdots + 11\!\cdots\!00 \)
acting on \(S_{18}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 38\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 68\!\cdots\!44 \)
|
| $11$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( T^{20} + \cdots + 23\!\cdots\!49 \)
|
| $17$ |
\( (T^{10} + \cdots - 21\!\cdots\!08)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 27\!\cdots\!00 \)
|
| $23$ |
\( (T^{10} + \cdots - 24\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots + 82\!\cdots\!60)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 24\!\cdots\!36 \)
|
| $41$ |
\( T^{20} + \cdots + 63\!\cdots\!00 \)
|
| $43$ |
\( (T^{10} + \cdots + 50\!\cdots\!40)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 89\!\cdots\!56 \)
|
| $53$ |
\( (T^{10} + \cdots + 52\!\cdots\!20)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 15\!\cdots\!64 \)
|
| $61$ |
\( (T^{10} + \cdots - 77\!\cdots\!20)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 68\!\cdots\!00 \)
|
| $71$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $73$ |
\( T^{20} + \cdots + 22\!\cdots\!00 \)
|
| $79$ |
\( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 24\!\cdots\!56 \)
|
| $89$ |
\( T^{20} + \cdots + 70\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 44\!\cdots\!00 \)
|
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