Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,6,Mod(47,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.47");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.n (of order \(4\), degree \(2\), 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: | \(12.8307055850\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{65}\cdot 3^{4}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 12\!\cdots\!45 \nu^{18} + \cdots - 59\!\cdots\!24 ) / 11\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!57 \nu^{19} + \cdots + 27\!\cdots\!16 \nu ) / 16\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 31\!\cdots\!07 \nu^{19} + \cdots - 10\!\cdots\!40 ) / 43\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 43\!\cdots\!47 \nu^{19} + \cdots - 13\!\cdots\!08 ) / 57\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43\!\cdots\!47 \nu^{19} + \cdots - 23\!\cdots\!80 ) / 57\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 46\!\cdots\!17 \nu^{19} + \cdots + 33\!\cdots\!40 ) / 57\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 42\!\cdots\!31 \nu^{19} + \cdots - 25\!\cdots\!12 ) / 26\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 42\!\cdots\!31 \nu^{19} + \cdots - 25\!\cdots\!12 ) / 26\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 12\!\cdots\!29 \nu^{19} + \cdots - 52\!\cdots\!60 ) / 57\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15\!\cdots\!75 \nu^{19} + \cdots + 53\!\cdots\!76 ) / 57\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 23\!\cdots\!39 \nu^{19} + \cdots + 98\!\cdots\!56 ) / 67\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 24\!\cdots\!13 \nu^{19} + \cdots - 43\!\cdots\!16 ) / 57\!\cdots\!40 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 22\!\cdots\!60 \nu^{19} + \cdots - 12\!\cdots\!64 ) / 20\!\cdots\!05 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 22\!\cdots\!60 \nu^{19} + \cdots - 12\!\cdots\!64 ) / 20\!\cdots\!05 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 25\!\cdots\!11 \nu^{19} + \cdots - 17\!\cdots\!64 ) / 22\!\cdots\!40 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 25\!\cdots\!11 \nu^{19} + \cdots - 17\!\cdots\!64 ) / 22\!\cdots\!40 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 39\!\cdots\!47 \nu^{19} + \cdots + 14\!\cdots\!56 \nu ) / 30\!\cdots\!40 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 10\!\cdots\!21 \nu^{19} + \cdots + 14\!\cdots\!20 ) / 57\!\cdots\!40 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 24\!\cdots\!10 \nu^{19} + \cdots + 26\!\cdots\!24 ) / 67\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( - 36 \beta_{19} + 80 \beta_{18} + 4 \beta_{17} + 72 \beta_{16} - 96 \beta_{15} - 19 \beta_{14} + \cdots + 132 ) / 3840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 24 \beta_{19} + 24 \beta_{17} - 1968 \beta_{16} - 2016 \beta_{15} + 101 \beta_{14} + 77 \beta_{13} + \cdots - 104400 ) / 3840 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1017 \beta_{19} + 120 \beta_{18} - 133 \beta_{17} - 354 \beta_{16} + 1032 \beta_{15} + 1588 \beta_{14} + \cdots + 2382 ) / 480 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 38844 \beta_{19} - 38844 \beta_{17} - 54912 \beta_{16} + 22776 \beta_{15} - 2211 \beta_{14} + \cdots - 55641012 ) / 3840 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 151344 \beta_{19} + 648560 \beta_{18} - 1533904 \beta_{17} - 4981392 \beta_{16} + 4880496 \beta_{15} + \cdots - 1520808 ) / 3840 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 726150 \beta_{19} + 726150 \beta_{17} + 6352440 \beta_{16} + 4900140 \beta_{15} - 4553905 \beta_{14} + \cdots - 420549309 ) / 480 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 50802300 \beta_{19} - 97402960 \beta_{18} + 374650340 \beta_{17} + 356054760 \beta_{16} + \cdots - 406138668 ) / 3840 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 728816424 \beta_{19} + 728816424 \beta_{17} + 7550452272 \beta_{16} + 6092819424 \beta_{15} + \cdots + 1263465976176 ) / 3840 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 1277012079 \beta_{19} - 7813909080 \beta_{18} + 12278449931 \beta_{17} + 20408007798 \beta_{16} + \cdots - 16910733408 ) / 480 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 172680569364 \beta_{19} - 172680569364 \beta_{17} - 2132499788352 \beta_{16} + \cdots + 360137469933540 ) / 3840 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 6921312769440 \beta_{19} - 1703697317680 \beta_{18} - 23877482320160 \beta_{17} + \cdots + 18855140337432 ) / 3840 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 13367968300 \beta_{19} - 13367968300 \beta_{17} - 12410747120880 \beta_{16} - 12384011184280 \beta_{15} + \cdots - 23\!\cdots\!67 ) / 160 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 13\!\cdots\!28 \beta_{19} + \cdots + 56\!\cdots\!92 ) / 3840 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 133874082937224 \beta_{19} + 133874082937224 \beta_{17} + \cdots - 14\!\cdots\!76 ) / 3840 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 10\!\cdots\!53 \beta_{19} + \cdots - 69\!\cdots\!98 ) / 480 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 11\!\cdots\!56 \beta_{19} + \cdots + 33\!\cdots\!16 ) / 3840 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 45\!\cdots\!08 \beta_{19} + \cdots - 45\!\cdots\!08 ) / 3840 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 61\!\cdots\!70 \beta_{19} + \cdots + 52\!\cdots\!03 ) / 480 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 54\!\cdots\!32 \beta_{19} + \cdots + 64\!\cdots\!32 ) / 3840 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −20.3843 | − | 20.3843i | 0 | −46.4503 | + | 31.1026i | 0 | −76.9082 | + | 76.9082i | 0 | 588.037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −17.2921 | − | 17.2921i | 0 | 46.1930 | + | 31.4834i | 0 | 154.079 | − | 154.079i | 0 | 355.037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.3 | 0 | −9.68301 | − | 9.68301i | 0 | −49.1893 | − | 26.5597i | 0 | −48.6629 | + | 48.6629i | 0 | − | 55.4787i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.4 | 0 | −7.48311 | − | 7.48311i | 0 | 34.7301 | − | 43.8043i | 0 | −19.2260 | + | 19.2260i | 0 | − | 131.006i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.5 | 0 | −0.839817 | − | 0.839817i | 0 | 3.71634 | + | 55.7780i | 0 | 99.3589 | − | 99.3589i | 0 | − | 241.589i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.6 | 0 | 0.839817 | + | 0.839817i | 0 | 3.71634 | + | 55.7780i | 0 | −99.3589 | + | 99.3589i | 0 | − | 241.589i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.7 | 0 | 7.48311 | + | 7.48311i | 0 | 34.7301 | − | 43.8043i | 0 | 19.2260 | − | 19.2260i | 0 | − | 131.006i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.8 | 0 | 9.68301 | + | 9.68301i | 0 | −49.1893 | − | 26.5597i | 0 | 48.6629 | − | 48.6629i | 0 | − | 55.4787i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.9 | 0 | 17.2921 | + | 17.2921i | 0 | 46.1930 | + | 31.4834i | 0 | −154.079 | + | 154.079i | 0 | 355.037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.10 | 0 | 20.3843 | + | 20.3843i | 0 | −46.4503 | + | 31.1026i | 0 | 76.9082 | − | 76.9082i | 0 | 588.037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.1 | 0 | −20.3843 | + | 20.3843i | 0 | −46.4503 | − | 31.1026i | 0 | −76.9082 | − | 76.9082i | 0 | − | 588.037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.2 | 0 | −17.2921 | + | 17.2921i | 0 | 46.1930 | − | 31.4834i | 0 | 154.079 | + | 154.079i | 0 | − | 355.037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.3 | 0 | −9.68301 | + | 9.68301i | 0 | −49.1893 | + | 26.5597i | 0 | −48.6629 | − | 48.6629i | 0 | 55.4787i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.4 | 0 | −7.48311 | + | 7.48311i | 0 | 34.7301 | + | 43.8043i | 0 | −19.2260 | − | 19.2260i | 0 | 131.006i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.5 | 0 | −0.839817 | + | 0.839817i | 0 | 3.71634 | − | 55.7780i | 0 | 99.3589 | + | 99.3589i | 0 | 241.589i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.6 | 0 | 0.839817 | − | 0.839817i | 0 | 3.71634 | − | 55.7780i | 0 | −99.3589 | − | 99.3589i | 0 | 241.589i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.7 | 0 | 7.48311 | − | 7.48311i | 0 | 34.7301 | + | 43.8043i | 0 | 19.2260 | + | 19.2260i | 0 | 131.006i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.8 | 0 | 9.68301 | − | 9.68301i | 0 | −49.1893 | + | 26.5597i | 0 | 48.6629 | + | 48.6629i | 0 | 55.4787i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.9 | 0 | 17.2921 | − | 17.2921i | 0 | 46.1930 | − | 31.4834i | 0 | −154.079 | − | 154.079i | 0 | − | 355.037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.10 | 0 | 20.3843 | − | 20.3843i | 0 | −46.4503 | − | 31.1026i | 0 | 76.9082 | + | 76.9082i | 0 | − | 588.037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 80.6.n.d | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 80.6.n.d | ✓ | 20 |
| 5.b | even | 2 | 1 | 400.6.n.g | 20 | ||
| 5.c | odd | 4 | 1 | inner | 80.6.n.d | ✓ | 20 |
| 5.c | odd | 4 | 1 | 400.6.n.g | 20 | ||
| 20.d | odd | 2 | 1 | 400.6.n.g | 20 | ||
| 20.e | even | 4 | 1 | inner | 80.6.n.d | ✓ | 20 |
| 20.e | even | 4 | 1 | 400.6.n.g | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.6.n.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 80.6.n.d | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 80.6.n.d | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 80.6.n.d | ✓ | 20 | 20.e | even | 4 | 1 | inner |
| 400.6.n.g | 20 | 5.b | even | 2 | 1 | ||
| 400.6.n.g | 20 | 5.c | odd | 4 | 1 | ||
| 400.6.n.g | 20 | 20.d | odd | 2 | 1 | ||
| 400.6.n.g | 20 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 1095980 T_{3}^{16} + 297452922160 T_{3}^{12} + \cdots + 21\!\cdots\!76 \)
acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + \cdots + 21\!\cdots\!76 \)
|
| $5$ |
\( (T^{10} + \cdots + 29\!\cdots\!25)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 15\!\cdots\!76 \)
|
| $11$ |
\( (T^{10} + \cdots + 14\!\cdots\!32)^{2} \)
|
| $13$ |
\( (T^{10} + \cdots + 59\!\cdots\!32)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots - 45\!\cdots\!68)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 21\!\cdots\!76 \)
|
| $29$ |
\( (T^{10} + \cdots + 10\!\cdots\!76)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 43\!\cdots\!32)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 57\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{5} + \cdots + 19\!\cdots\!68)^{4} \)
|
| $43$ |
\( T^{20} + \cdots + 52\!\cdots\!76 \)
|
| $47$ |
\( T^{20} + \cdots + 94\!\cdots\!76 \)
|
| $53$ |
\( (T^{10} + \cdots + 11\!\cdots\!68)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots - 43\!\cdots\!68)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots + 24\!\cdots\!24)^{4} \)
|
| $67$ |
\( T^{20} + \cdots + 10\!\cdots\!76 \)
|
| $71$ |
\( (T^{10} + \cdots + 26\!\cdots\!68)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 32\!\cdots\!68)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots - 82\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 33\!\cdots\!76 \)
|
| $89$ |
\( (T^{10} + \cdots + 41\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 34\!\cdots\!68)^{2} \)
|
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