Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,2,Mod(11,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.k (of order \(12\), degree \(4\), 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: | \(0.622833135766\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 16.0.9349208943630483456.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1548 \nu^{15} - 13188 \nu^{14} + 76652 \nu^{13} - 318997 \nu^{12} + 1019601 \nu^{11} - 2665197 \nu^{10} + \cdots - 204955 ) / 17095 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + \cdots + 142555 ) / 17095 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + \cdots + 5945 ) / 65 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1548 \nu^{15} + 26338 \nu^{14} - 168702 \nu^{13} + 849994 \nu^{12} - 3008933 \nu^{11} + \cdots + 979490 ) / 17095 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15207 \nu^{15} - 93144 \nu^{14} + 525827 \nu^{13} - 1795712 \nu^{12} + 5254311 \nu^{11} + \cdots + 281130 ) / 17095 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15977 \nu^{15} - 98393 \nu^{14} + 556086 \nu^{13} - 1902441 \nu^{12} + 5567337 \nu^{11} + \cdots + 610830 ) / 17095 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} + \cdots - 1883725 ) / 17095 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15207 \nu^{15} - 151267 \nu^{14} + 932688 \nu^{13} - 4151403 \nu^{12} + 14099264 \nu^{11} + \cdots - 2984015 ) / 17095 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 49663 \nu^{15} - 358139 \nu^{14} + 2089429 \nu^{13} - 8006091 \nu^{12} + 25109764 \nu^{11} + \cdots - 1090565 ) / 17095 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 46358 \nu^{15} - 350841 \nu^{14} + 2068305 \nu^{13} - 8157610 \nu^{12} + 26035188 \nu^{11} + \cdots - 2064105 ) / 17095 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 46358 \nu^{15} + 373985 \nu^{14} - 2230313 \nu^{13} + 9095994 \nu^{12} - 29559388 \nu^{11} + \cdots + 3439595 ) / 17095 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 54184 \nu^{15} + 399016 \nu^{14} - 2335837 \nu^{13} + 9056462 \nu^{12} - 28575749 \nu^{11} + \cdots + 1312670 ) / 17095 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} + \cdots - 370592 ) / 3419 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + \cdots + 1951760 ) / 17095 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{15} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{15} + 7 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots - 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 10 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 8 \beta_{12} - 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} + \cdots + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 21 \beta_{15} - 46 \beta_{14} + 44 \beta_{13} - 19 \beta_{12} + 44 \beta_{11} + 19 \beta_{10} + \cdots + 36 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 60 \beta_{15} - 43 \beta_{14} - 5 \beta_{13} - 70 \beta_{12} + 55 \beta_{11} + 49 \beta_{10} - 30 \beta_{9} + \cdots - 50 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 321 \beta_{15} + 267 \beta_{14} - 365 \beta_{13} + 17 \beta_{12} - 213 \beta_{11} - 28 \beta_{10} + \cdots - 307 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 256 \beta_{15} + 446 \beta_{14} - 108 \beta_{13} + 544 \beta_{12} - 528 \beta_{11} - 304 \beta_{10} + \cdots + 267 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 3026 \beta_{15} - 1225 \beta_{14} + 2701 \beta_{13} + 906 \beta_{12} + 525 \beta_{11} - 297 \beta_{10} + \cdots + 2725 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( 274 \beta_{15} - 4001 \beta_{14} + 2070 \beta_{13} - 3737 \beta_{12} + 4287 \beta_{11} + 1922 \beta_{10} + \cdots - 932 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 23673 \beta_{15} + 1926 \beta_{14} - 17346 \beta_{13} - 14155 \beta_{12} + 4640 \beta_{11} + \cdots - 22758 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 10280 \beta_{15} + 31947 \beta_{14} - 24436 \beta_{13} + 21968 \beta_{12} - 30552 \beta_{11} + \cdots - 3138 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 160655 \beta_{15} + 46695 \beta_{14} + 88659 \beta_{13} + 152123 \beta_{12} - 96769 \beta_{11} + \cdots + 171575 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 160979 \beta_{15} - 225841 \beta_{14} + 234065 \beta_{13} - 95851 \beta_{12} + 187408 \beta_{11} + \cdots + 106657 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 914760 \beta_{15} - 806121 \beta_{14} - 233687 \beta_{13} - 1368388 \beta_{12} + 1121513 \beta_{11} + \cdots - 1132093 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{9} + \beta_{14}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−0.965926 | − | 0.258819i | −1.73022 | + | 0.0795432i | 0.866025 | + | 0.500000i | 2.76293 | − | 2.76293i | 1.69185 | + | 0.370982i | −0.657464 | − | 2.45369i | −0.707107 | − | 0.707107i | 2.98735 | − | 0.275255i | −3.38389 | + | 1.95369i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −0.965926 | − | 0.258819i | 1.73022 | − | 0.0795432i | 0.866025 | + | 0.500000i | −0.313444 | + | 0.313444i | −1.69185 | − | 0.370982i | −0.0745867 | − | 0.278362i | −0.707107 | − | 0.707107i | 2.98735 | − | 0.275255i | 0.383889 | − | 0.221638i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | 0.965926 | + | 0.258819i | −0.933998 | + | 1.45865i | 0.866025 | + | 0.500000i | 0.313444 | − | 0.313444i | −1.27970 | + | 1.16721i | −0.0745867 | − | 0.278362i | 0.707107 | + | 0.707107i | −1.25529 | − | 2.72474i | 0.383889 | − | 0.221638i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | 0.965926 | + | 0.258819i | 0.933998 | − | 1.45865i | 0.866025 | + | 0.500000i | −2.76293 | + | 2.76293i | 1.27970 | − | 1.16721i | −0.657464 | − | 2.45369i | 0.707107 | + | 0.707107i | −1.25529 | − | 2.72474i | −3.38389 | + | 1.95369i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | −0.258819 | + | 0.965926i | −0.0795432 | − | 1.73022i | −0.866025 | − | 0.500000i | 2.02097 | + | 2.02097i | 1.69185 | + | 0.370982i | 3.46723 | − | 0.929042i | 0.707107 | − | 0.707107i | −2.98735 | + | 0.275255i | −2.47517 | + | 1.42904i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | −0.258819 | + | 0.965926i | 0.0795432 | + | 1.73022i | −0.866025 | − | 0.500000i | 0.428520 | + | 0.428520i | −1.69185 | − | 0.370982i | −0.735180 | + | 0.196991i | 0.707107 | − | 0.707107i | −2.98735 | + | 0.275255i | −0.524827 | + | 0.303009i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | 0.258819 | − | 0.965926i | −1.45865 | − | 0.933998i | −0.866025 | − | 0.500000i | −2.02097 | − | 2.02097i | −1.27970 | + | 1.16721i | 3.46723 | − | 0.929042i | −0.707107 | + | 0.707107i | 1.25529 | + | 2.72474i | −2.47517 | + | 1.42904i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | 0.258819 | − | 0.965926i | 1.45865 | + | 0.933998i | −0.866025 | − | 0.500000i | −0.428520 | − | 0.428520i | 1.27970 | − | 1.16721i | −0.735180 | + | 0.196991i | −0.707107 | + | 0.707107i | 1.25529 | + | 2.72474i | −0.524827 | + | 0.303009i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 59.1 | −0.258819 | − | 0.965926i | −0.0795432 | + | 1.73022i | −0.866025 | + | 0.500000i | 2.02097 | − | 2.02097i | 1.69185 | − | 0.370982i | 3.46723 | + | 0.929042i | 0.707107 | + | 0.707107i | −2.98735 | − | 0.275255i | −2.47517 | − | 1.42904i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 59.2 | −0.258819 | − | 0.965926i | 0.0795432 | − | 1.73022i | −0.866025 | + | 0.500000i | 0.428520 | − | 0.428520i | −1.69185 | + | 0.370982i | −0.735180 | − | 0.196991i | 0.707107 | + | 0.707107i | −2.98735 | − | 0.275255i | −0.524827 | − | 0.303009i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 59.3 | 0.258819 | + | 0.965926i | −1.45865 | + | 0.933998i | −0.866025 | + | 0.500000i | −2.02097 | + | 2.02097i | −1.27970 | − | 1.16721i | 3.46723 | + | 0.929042i | −0.707107 | − | 0.707107i | 1.25529 | − | 2.72474i | −2.47517 | − | 1.42904i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 59.4 | 0.258819 | + | 0.965926i | 1.45865 | − | 0.933998i | −0.866025 | + | 0.500000i | −0.428520 | + | 0.428520i | 1.27970 | + | 1.16721i | −0.735180 | − | 0.196991i | −0.707107 | − | 0.707107i | 1.25529 | − | 2.72474i | −0.524827 | − | 0.303009i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 71.1 | −0.965926 | + | 0.258819i | −1.73022 | − | 0.0795432i | 0.866025 | − | 0.500000i | 2.76293 | + | 2.76293i | 1.69185 | − | 0.370982i | −0.657464 | + | 2.45369i | −0.707107 | + | 0.707107i | 2.98735 | + | 0.275255i | −3.38389 | − | 1.95369i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 71.2 | −0.965926 | + | 0.258819i | 1.73022 | + | 0.0795432i | 0.866025 | − | 0.500000i | −0.313444 | − | 0.313444i | −1.69185 | + | 0.370982i | −0.0745867 | + | 0.278362i | −0.707107 | + | 0.707107i | 2.98735 | + | 0.275255i | 0.383889 | + | 0.221638i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 71.3 | 0.965926 | − | 0.258819i | −0.933998 | − | 1.45865i | 0.866025 | − | 0.500000i | 0.313444 | + | 0.313444i | −1.27970 | − | 1.16721i | −0.0745867 | + | 0.278362i | 0.707107 | − | 0.707107i | −1.25529 | + | 2.72474i | 0.383889 | + | 0.221638i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 71.4 | 0.965926 | − | 0.258819i | 0.933998 | + | 1.45865i | 0.866025 | − | 0.500000i | −2.76293 | − | 2.76293i | 1.27970 | + | 1.16721i | −0.657464 | + | 2.45369i | 0.707107 | − | 0.707107i | −1.25529 | + | 2.72474i | −3.38389 | − | 1.95369i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 39.k | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.2.k.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 78.2.k.a | ✓ | 16 |
| 4.b | odd | 2 | 1 | 624.2.cn.d | 16 | ||
| 12.b | even | 2 | 1 | 624.2.cn.d | 16 | ||
| 13.c | even | 3 | 1 | 1014.2.g.c | 16 | ||
| 13.e | even | 6 | 1 | 1014.2.g.d | 16 | ||
| 13.f | odd | 12 | 1 | inner | 78.2.k.a | ✓ | 16 |
| 13.f | odd | 12 | 1 | 1014.2.g.c | 16 | ||
| 13.f | odd | 12 | 1 | 1014.2.g.d | 16 | ||
| 39.h | odd | 6 | 1 | 1014.2.g.d | 16 | ||
| 39.i | odd | 6 | 1 | 1014.2.g.c | 16 | ||
| 39.k | even | 12 | 1 | inner | 78.2.k.a | ✓ | 16 |
| 39.k | even | 12 | 1 | 1014.2.g.c | 16 | ||
| 39.k | even | 12 | 1 | 1014.2.g.d | 16 | ||
| 52.l | even | 12 | 1 | 624.2.cn.d | 16 | ||
| 156.v | odd | 12 | 1 | 624.2.cn.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.2.k.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 78.2.k.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 78.2.k.a | ✓ | 16 | 13.f | odd | 12 | 1 | inner |
| 78.2.k.a | ✓ | 16 | 39.k | even | 12 | 1 | inner |
| 624.2.cn.d | 16 | 4.b | odd | 2 | 1 | ||
| 624.2.cn.d | 16 | 12.b | even | 2 | 1 | ||
| 624.2.cn.d | 16 | 52.l | even | 12 | 1 | ||
| 624.2.cn.d | 16 | 156.v | odd | 12 | 1 | ||
| 1014.2.g.c | 16 | 13.c | even | 3 | 1 | ||
| 1014.2.g.c | 16 | 13.f | odd | 12 | 1 | ||
| 1014.2.g.c | 16 | 39.i | odd | 6 | 1 | ||
| 1014.2.g.c | 16 | 39.k | even | 12 | 1 | ||
| 1014.2.g.d | 16 | 13.e | even | 6 | 1 | ||
| 1014.2.g.d | 16 | 13.f | odd | 12 | 1 | ||
| 1014.2.g.d | 16 | 39.h | odd | 6 | 1 | ||
| 1014.2.g.d | 16 | 39.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{16} - 6 T^{12} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} + 300 T^{12} + \cdots + 81 \)
|
| $7$ |
\( (T^{8} - 4 T^{7} + 2 T^{6} + \cdots + 4)^{2} \)
|
| $11$ |
\( (T^{8} - 30 T^{6} + \cdots + 36)^{2} \)
|
| $13$ |
\( (T^{8} + 12 T^{7} + \cdots + 28561)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 151807041 \)
|
| $19$ |
\( (T^{8} + 8 T^{7} + \cdots + 5476)^{2} \)
|
| $23$ |
\( T^{16} + 60 T^{14} + \cdots + 18974736 \)
|
| $29$ |
\( T^{16} + \cdots + 33871089681 \)
|
| $31$ |
\( (T^{8} - 8 T^{7} + \cdots + 7744)^{2} \)
|
| $37$ |
\( (T^{8} - 8 T^{7} + \cdots + 375769)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 65697655057281 \)
|
| $43$ |
\( (T^{8} - 18 T^{6} + \cdots + 2916)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 41006250000 \)
|
| $53$ |
\( (T^{8} + 156 T^{6} + \cdots + 522729)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 43489065701376 \)
|
| $61$ |
\( (T^{8} + 12 T^{7} + \cdots + 47961)^{2} \)
|
| $67$ |
\( (T^{8} - 16 T^{7} + \cdots + 676)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 5143987297296 \)
|
| $73$ |
\( (T^{8} - 28 T^{7} + \cdots + 16834609)^{2} \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots + 312)^{4} \)
|
| $83$ |
\( T^{16} + \cdots + 36804120336 \)
|
| $89$ |
\( T^{16} + \cdots + 59\!\cdots\!56 \)
|
| $97$ |
\( (T^{8} - 8 T^{7} + \cdots + 43264)^{2} \)
|
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