Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,2,Mod(113,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 10, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 720.cu (of order \(12\), degree \(4\), not 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: | \(5.74922894553\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 90) |
| 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
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 4341 \nu^{15} + 25062 \nu^{14} - 134821 \nu^{13} + 417198 \nu^{12} - 1075015 \nu^{11} + \cdots - 740270 ) / 17095 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 142555 ) / 17095 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5033 \nu^{15} + 21310 \nu^{14} - 104426 \nu^{13} + 187861 \nu^{12} - 172588 \nu^{11} + \cdots - 887140 ) / 17095 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2793 \nu^{15} + 28180 \nu^{14} - 172311 \nu^{13} + 766484 \nu^{12} - 2581266 \nu^{11} + \cdots + 618310 ) / 17095 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} + \cdots - 1883725 ) / 17095 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 26630 \nu^{15} - 210508 \nu^{14} + 1251886 \nu^{13} - 5050631 \nu^{12} + 16323908 \nu^{11} + \cdots - 1160550 ) / 17095 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 26630 \nu^{15} + 212086 \nu^{14} - 1262932 \nu^{13} + 5122693 \nu^{12} - 16612682 \nu^{11} + \cdots + 2084995 ) / 17095 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 36479 \nu^{15} - 258470 \nu^{14} + 1499462 \nu^{13} - 5670854 \nu^{12} + 17608509 \nu^{11} + \cdots + 18365 ) / 17095 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 38207 \nu^{15} + 272482 \nu^{14} - 1582764 \nu^{13} + 6010234 \nu^{12} - 18706521 \nu^{11} + \cdots + 68035 ) / 17095 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 35414 \nu^{15} + 286908 \nu^{14} - 1708695 \nu^{13} + 6974027 \nu^{12} - 22629771 \nu^{11} + \cdots + 2511045 ) / 17095 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 38207 \nu^{15} + 300623 \nu^{14} - 1779751 \nu^{13} + 7154021 \nu^{12} - 23008412 \nu^{11} + \cdots + 1923500 ) / 17095 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 39935 \nu^{15} - 314635 \nu^{14} + 1863053 \nu^{13} - 7493401 \nu^{12} + 24106424 \nu^{11} + \cdots - 2009900 ) / 17095 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} + \cdots - 1312670 ) / 17095 \)
|
| \(\nu\) | \(=\) |
\( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - 3 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} + 3 \beta_{9} + \beta_{7} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 8 \beta_{15} - 6 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} + 5 \beta_{9} + \cdots + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 17 \beta_{15} + 13 \beta_{14} + 27 \beta_{13} - 20 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} + \cdots - 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 43 \beta_{15} + 40 \beta_{14} + 64 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} + 59 \beta_{10} - 63 \beta_{9} + \cdots - 32 \)
|
| \(\nu^{7}\) | \(=\) |
\( 170 \beta_{15} - 68 \beta_{14} - 154 \beta_{13} + 133 \beta_{12} + 208 \beta_{11} + 16 \beta_{10} + \cdots + 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 160 \beta_{15} - 316 \beta_{14} - 556 \beta_{13} + 108 \beta_{12} + 196 \beta_{11} - 388 \beta_{10} + \cdots + 231 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1442 \beta_{15} + 303 \beta_{14} + 759 \beta_{13} - 888 \beta_{12} - 1474 \beta_{11} - 546 \beta_{10} + \cdots + 116 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 163 \beta_{15} + 2609 \beta_{14} + 4787 \beta_{13} - 1507 \beta_{12} - 3305 \beta_{11} + 2222 \beta_{10} + \cdots - 1611 \)
|
| \(\nu^{11}\) | \(=\) |
\( 10909 \beta_{15} - 253 \beta_{14} - 1879 \beta_{13} + 5489 \beta_{12} + 8068 \beta_{11} + 6380 \beta_{10} + \cdots - 2314 \)
|
| \(\nu^{12}\) | \(=\) |
\( 11866 \beta_{15} - 20444 \beta_{14} - 38548 \beta_{13} + 16364 \beta_{12} + 34826 \beta_{11} + \cdots + 10052 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 72687 \beta_{15} - 16328 \beta_{14} - 20114 \beta_{13} - 27664 \beta_{12} - 26597 \beta_{11} + \cdots + 27130 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 163323 \beta_{15} + 144445 \beta_{14} + 280994 \beta_{13} - 152817 \beta_{12} - 300247 \beta_{11} + \cdots - 51022 \)
|
| \(\nu^{15}\) | \(=\) |
\( 402584 \beta_{15} + 264336 \beta_{14} + 422112 \beta_{13} + 70486 \beta_{12} - 100959 \beta_{11} + \cdots - 258101 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{6} - \beta_{9}\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 113.1 |
|
0 | −1.45865 | − | 0.933998i | 0 | −1.29554 | − | 1.82252i | 0 | −1.94786 | − | 0.521929i | 0 | 1.25529 | + | 2.72474i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −0.0795432 | − | 1.73022i | 0 | −2.00265 | + | 0.994679i | 0 | −2.32238 | − | 0.622279i | 0 | −2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | 0.0795432 | + | 1.73022i | 0 | 0.661570 | − | 2.13596i | 0 | −3.75574 | − | 1.00635i | 0 | −2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | 1.45865 | + | 0.933998i | 0 | 1.36868 | − | 1.76825i | 0 | 2.56188 | + | 0.686453i | 0 | 1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | −1.73022 | + | 0.0795432i | 0 | −0.139908 | − | 2.23169i | 0 | 0.622279 | − | 2.32238i | 0 | 2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −0.933998 | + | 1.45865i | 0 | −2.22612 | − | 0.210717i | 0 | 0.521929 | − | 1.94786i | 0 | −1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 0.933998 | − | 1.45865i | 0 | −0.847015 | + | 2.06944i | 0 | −0.686453 | + | 2.56188i | 0 | −1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.4 | 0 | 1.73022 | − | 0.0795432i | 0 | −1.51901 | + | 1.64092i | 0 | 1.00635 | − | 3.75574i | 0 | 2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.1 | 0 | −1.73022 | − | 0.0795432i | 0 | −0.139908 | + | 2.23169i | 0 | 0.622279 | + | 2.32238i | 0 | 2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.2 | 0 | −0.933998 | − | 1.45865i | 0 | −2.22612 | + | 0.210717i | 0 | 0.521929 | + | 1.94786i | 0 | −1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.3 | 0 | 0.933998 | + | 1.45865i | 0 | −0.847015 | − | 2.06944i | 0 | −0.686453 | − | 2.56188i | 0 | −1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.4 | 0 | 1.73022 | + | 0.0795432i | 0 | −1.51901 | − | 1.64092i | 0 | 1.00635 | + | 3.75574i | 0 | 2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 497.1 | 0 | −1.45865 | + | 0.933998i | 0 | −1.29554 | + | 1.82252i | 0 | −1.94786 | + | 0.521929i | 0 | 1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 497.2 | 0 | −0.0795432 | + | 1.73022i | 0 | −2.00265 | − | 0.994679i | 0 | −2.32238 | + | 0.622279i | 0 | −2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 497.3 | 0 | 0.0795432 | − | 1.73022i | 0 | 0.661570 | + | 2.13596i | 0 | −3.75574 | + | 1.00635i | 0 | −2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 497.4 | 0 | 1.45865 | − | 0.933998i | 0 | 1.36868 | + | 1.76825i | 0 | 2.56188 | − | 0.686453i | 0 | 1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 45.l | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 720.2.cu.b | 16 | |
| 4.b | odd | 2 | 1 | 90.2.l.b | ✓ | 16 | |
| 5.c | odd | 4 | 1 | inner | 720.2.cu.b | 16 | |
| 9.d | odd | 6 | 1 | inner | 720.2.cu.b | 16 | |
| 12.b | even | 2 | 1 | 270.2.m.b | 16 | ||
| 20.d | odd | 2 | 1 | 450.2.p.h | 16 | ||
| 20.e | even | 4 | 1 | 90.2.l.b | ✓ | 16 | |
| 20.e | even | 4 | 1 | 450.2.p.h | 16 | ||
| 36.f | odd | 6 | 1 | 270.2.m.b | 16 | ||
| 36.f | odd | 6 | 1 | 810.2.f.c | 16 | ||
| 36.h | even | 6 | 1 | 90.2.l.b | ✓ | 16 | |
| 36.h | even | 6 | 1 | 810.2.f.c | 16 | ||
| 45.l | even | 12 | 1 | inner | 720.2.cu.b | 16 | |
| 60.h | even | 2 | 1 | 1350.2.q.h | 16 | ||
| 60.l | odd | 4 | 1 | 270.2.m.b | 16 | ||
| 60.l | odd | 4 | 1 | 1350.2.q.h | 16 | ||
| 180.n | even | 6 | 1 | 450.2.p.h | 16 | ||
| 180.p | odd | 6 | 1 | 1350.2.q.h | 16 | ||
| 180.v | odd | 12 | 1 | 90.2.l.b | ✓ | 16 | |
| 180.v | odd | 12 | 1 | 450.2.p.h | 16 | ||
| 180.v | odd | 12 | 1 | 810.2.f.c | 16 | ||
| 180.x | even | 12 | 1 | 270.2.m.b | 16 | ||
| 180.x | even | 12 | 1 | 810.2.f.c | 16 | ||
| 180.x | even | 12 | 1 | 1350.2.q.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.2.l.b | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 90.2.l.b | ✓ | 16 | 20.e | even | 4 | 1 | |
| 90.2.l.b | ✓ | 16 | 36.h | even | 6 | 1 | |
| 90.2.l.b | ✓ | 16 | 180.v | odd | 12 | 1 | |
| 270.2.m.b | 16 | 12.b | even | 2 | 1 | ||
| 270.2.m.b | 16 | 36.f | odd | 6 | 1 | ||
| 270.2.m.b | 16 | 60.l | odd | 4 | 1 | ||
| 270.2.m.b | 16 | 180.x | even | 12 | 1 | ||
| 450.2.p.h | 16 | 20.d | odd | 2 | 1 | ||
| 450.2.p.h | 16 | 20.e | even | 4 | 1 | ||
| 450.2.p.h | 16 | 180.n | even | 6 | 1 | ||
| 450.2.p.h | 16 | 180.v | odd | 12 | 1 | ||
| 720.2.cu.b | 16 | 1.a | even | 1 | 1 | trivial | |
| 720.2.cu.b | 16 | 5.c | odd | 4 | 1 | inner | |
| 720.2.cu.b | 16 | 9.d | odd | 6 | 1 | inner | |
| 720.2.cu.b | 16 | 45.l | even | 12 | 1 | inner | |
| 810.2.f.c | 16 | 36.f | odd | 6 | 1 | ||
| 810.2.f.c | 16 | 36.h | even | 6 | 1 | ||
| 810.2.f.c | 16 | 180.v | odd | 12 | 1 | ||
| 810.2.f.c | 16 | 180.x | even | 12 | 1 | ||
| 1350.2.q.h | 16 | 60.h | even | 2 | 1 | ||
| 1350.2.q.h | 16 | 60.l | odd | 4 | 1 | ||
| 1350.2.q.h | 16 | 180.p | odd | 6 | 1 | ||
| 1350.2.q.h | 16 | 180.x | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 8 T_{7}^{15} + 32 T_{7}^{14} + 144 T_{7}^{13} + 476 T_{7}^{12} + 592 T_{7}^{11} + \cdots + 6250000 \)
acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 6 T^{12} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} + 12 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 8 T^{15} + \cdots + 6250000 \)
|
| $11$ |
\( (T^{8} - 22 T^{6} + \cdots + 1849)^{2} \)
|
| $13$ |
\( T^{16} + 48 T^{13} + \cdots + 1296 \)
|
| $17$ |
\( T^{16} + 4132 T^{12} + \cdots + 390625 \)
|
| $19$ |
\( (T^{8} + 52 T^{6} + \cdots + 10201)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 82538991616 \)
|
| $29$ |
\( T^{16} + 80 T^{14} + \cdots + 40960000 \)
|
| $31$ |
\( (T^{8} - 4 T^{7} + \cdots + 448900)^{2} \)
|
| $37$ |
\( (T^{8} - 576 T^{5} + \cdots + 82944)^{2} \)
|
| $41$ |
\( (T^{8} - 12 T^{7} + \cdots + 555025)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 35806100625 \)
|
| $47$ |
\( T^{16} + \cdots + 2981133747216 \)
|
| $53$ |
\( T^{16} + \cdots + 3906250000 \)
|
| $59$ |
\( T^{16} + 44 T^{14} + \cdots + 625 \)
|
| $61$ |
\( (T^{8} + 12 T^{7} + \cdots + 45887076)^{2} \)
|
| $67$ |
\( T^{16} - 16 T^{15} + \cdots + 3418801 \)
|
| $71$ |
\( (T^{8} + 272 T^{6} + \cdots + 14032516)^{2} \)
|
| $73$ |
\( (T^{8} - 8 T^{7} + \cdots + 966289)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 17\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 21743271936 \)
|
| $89$ |
\( (T^{4} - 28 T^{2} + 100)^{4} \)
|
| $97$ |
\( T^{16} + \cdots + 19559470366881 \)
|
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