Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,6,Mod(47,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 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("144.47");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.s (of order \(6\), 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: | \(23.0952700531\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 8 x^{19} - 12929 x^{18} + 122470 x^{17} + 67551337 x^{16} - 634332392 x^{15} + \cdots + 12\!\cdots\!83 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{26} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{20} - 8 x^{19} - 12929 x^{18} + 122470 x^{17} + 67551337 x^{16} - 634332392 x^{15} + \cdots + 12\!\cdots\!83 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 44\!\cdots\!28 \nu^{19} + \cdots + 21\!\cdots\!06 ) / 35\!\cdots\!88 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 94\!\cdots\!01 \nu^{19} + \cdots + 61\!\cdots\!05 ) / 33\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\!\cdots\!99 \nu^{19} + \cdots + 60\!\cdots\!05 ) / 33\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!01 \nu^{19} + \cdots - 52\!\cdots\!95 ) / 33\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 43\!\cdots\!19 \nu^{19} + \cdots - 21\!\cdots\!08 ) / 33\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 40\!\cdots\!54 \nu^{19} + \cdots + 29\!\cdots\!09 ) / 29\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 19\!\cdots\!89 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 12\!\cdots\!66 \nu^{19} + \cdots + 66\!\cdots\!49 ) / 33\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!48 \nu^{19} + \cdots + 43\!\cdots\!77 ) / 33\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 16\!\cdots\!82 \nu^{19} + \cdots + 10\!\cdots\!07 ) / 33\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 71\!\cdots\!78 \nu^{19} + \cdots - 29\!\cdots\!75 ) / 11\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 46\!\cdots\!07 \nu^{19} + \cdots - 22\!\cdots\!96 ) / 56\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 88\!\cdots\!13 \nu^{19} + \cdots + 47\!\cdots\!16 ) / 84\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 87\!\cdots\!19 \nu^{19} + \cdots - 42\!\cdots\!77 ) / 56\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 59\!\cdots\!95 \nu^{19} + \cdots - 29\!\cdots\!65 ) / 33\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 66\!\cdots\!15 \nu^{19} + \cdots + 33\!\cdots\!82 ) / 33\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 42\!\cdots\!86 \nu^{19} + \cdots - 20\!\cdots\!23 ) / 17\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 15\!\cdots\!85 \nu^{19} + \cdots - 72\!\cdots\!38 ) / 33\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 28\!\cdots\!39 \nu^{19} + \cdots + 13\!\cdots\!43 ) / 33\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} - \beta_{3} + 2\beta_{2} + 3\beta _1 + 3 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{19} + 4 \beta_{17} - 3 \beta_{16} + 6 \beta_{15} + 3 \beta_{14} - 2 \beta_{12} + 2 \beta_{11} + \cdots + 3881 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 263 \beta_{19} + 27 \beta_{18} - 942 \beta_{17} + 150 \beta_{16} - 312 \beta_{15} + 628 \beta_{14} + \cdots - 6798 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 41291 \beta_{19} - 546 \beta_{18} + 54024 \beta_{17} - 38805 \beta_{16} + 66642 \beta_{15} + \cdots + 28410696 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1353783 \beta_{19} + 192795 \beta_{18} - 4471674 \beta_{17} + 916518 \beta_{16} - 1801296 \beta_{15} + \cdots - 270776661 ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 164807974 \beta_{19} - 4924818 \beta_{18} + 224954328 \beta_{17} - 150492648 \beta_{16} + \cdots + 92240762019 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6201965962 \beta_{19} + 1038236304 \beta_{18} - 18896025432 \beta_{17} + 4542376206 \beta_{16} + \cdots - 1689073014825 ) / 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 647544263793 \beta_{19} - 32255023350 \beta_{18} + 947635883472 \beta_{17} - 579227937135 \beta_{16} + \cdots + 330605626982403 ) / 9 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 27426732108281 \beta_{19} + 5086599371721 \beta_{18} - 77934100026102 \beta_{17} + \cdots - 82\!\cdots\!28 ) / 9 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 25\!\cdots\!01 \beta_{19} - 185059048423434 \beta_{18} + \cdots + 12\!\cdots\!74 ) / 9 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 11\!\cdots\!59 \beta_{19} + \cdots - 37\!\cdots\!73 ) / 9 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 10\!\cdots\!12 \beta_{19} + \cdots + 46\!\cdots\!73 ) / 9 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 50\!\cdots\!56 \beta_{19} + \cdots - 16\!\cdots\!39 ) / 9 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 40\!\cdots\!13 \beta_{19} + \cdots + 17\!\cdots\!35 ) / 9 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 21\!\cdots\!79 \beta_{19} + \cdots - 68\!\cdots\!94 ) / 9 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 16\!\cdots\!95 \beta_{19} + \cdots + 66\!\cdots\!84 ) / 9 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 89\!\cdots\!19 \beta_{19} + \cdots - 28\!\cdots\!57 ) / 9 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 65\!\cdots\!18 \beta_{19} + \cdots + 25\!\cdots\!83 ) / 9 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 37\!\cdots\!74 \beta_{19} + \cdots - 11\!\cdots\!45 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(-1\) |
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 | −15.4905 | − | 1.74481i | 0 | −36.4570 | + | 21.0484i | 0 | 69.2990 | + | 40.0098i | 0 | 236.911 | + | 54.0560i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −13.3172 | − | 8.10263i | 0 | 55.4166 | − | 31.9948i | 0 | −165.007 | − | 95.2670i | 0 | 111.695 | + | 215.808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.3 | 0 | −9.55954 | + | 12.3132i | 0 | 88.7047 | − | 51.2137i | 0 | 158.148 | + | 91.3070i | 0 | −60.2302 | − | 235.417i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.4 | 0 | −8.29419 | + | 13.1987i | 0 | −35.5670 | + | 20.5346i | 0 | −82.3079 | − | 47.5205i | 0 | −105.413 | − | 218.945i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.5 | 0 | −3.21536 | − | 15.2532i | 0 | 17.0016 | − | 9.81587i | 0 | 59.3512 | + | 34.2664i | 0 | −222.323 | + | 98.0893i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.6 | 0 | 3.06680 | − | 15.2838i | 0 | −91.7356 | + | 52.9636i | 0 | −66.8335 | − | 38.5864i | 0 | −224.190 | − | 93.7446i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.7 | 0 | 7.50461 | + | 13.6631i | 0 | 54.5790 | − | 31.5112i | 0 | −155.514 | − | 89.7863i | 0 | −130.362 | + | 205.073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.8 | 0 | 9.78531 | + | 12.1346i | 0 | −35.2636 | + | 20.3595i | 0 | 165.669 | + | 95.6488i | 0 | −51.4953 | + | 237.481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.9 | 0 | 12.4461 | − | 9.38582i | 0 | 49.0797 | − | 28.3362i | 0 | 75.4185 | + | 43.5429i | 0 | 66.8129 | − | 233.634i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.10 | 0 | 15.5739 | − | 0.673289i | 0 | −22.2583 | + | 12.8508i | 0 | −101.723 | − | 58.7296i | 0 | 242.093 | − | 20.9715i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.1 | 0 | −15.4905 | + | 1.74481i | 0 | −36.4570 | − | 21.0484i | 0 | 69.2990 | − | 40.0098i | 0 | 236.911 | − | 54.0560i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.2 | 0 | −13.3172 | + | 8.10263i | 0 | 55.4166 | + | 31.9948i | 0 | −165.007 | + | 95.2670i | 0 | 111.695 | − | 215.808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.3 | 0 | −9.55954 | − | 12.3132i | 0 | 88.7047 | + | 51.2137i | 0 | 158.148 | − | 91.3070i | 0 | −60.2302 | + | 235.417i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.4 | 0 | −8.29419 | − | 13.1987i | 0 | −35.5670 | − | 20.5346i | 0 | −82.3079 | + | 47.5205i | 0 | −105.413 | + | 218.945i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.5 | 0 | −3.21536 | + | 15.2532i | 0 | 17.0016 | + | 9.81587i | 0 | 59.3512 | − | 34.2664i | 0 | −222.323 | − | 98.0893i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.6 | 0 | 3.06680 | + | 15.2838i | 0 | −91.7356 | − | 52.9636i | 0 | −66.8335 | + | 38.5864i | 0 | −224.190 | + | 93.7446i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.7 | 0 | 7.50461 | − | 13.6631i | 0 | 54.5790 | + | 31.5112i | 0 | −155.514 | + | 89.7863i | 0 | −130.362 | − | 205.073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.8 | 0 | 9.78531 | − | 12.1346i | 0 | −35.2636 | − | 20.3595i | 0 | 165.669 | − | 95.6488i | 0 | −51.4953 | − | 237.481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.9 | 0 | 12.4461 | + | 9.38582i | 0 | 49.0797 | + | 28.3362i | 0 | 75.4185 | − | 43.5429i | 0 | 66.8129 | + | 233.634i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.10 | 0 | 15.5739 | + | 0.673289i | 0 | −22.2583 | − | 12.8508i | 0 | −101.723 | + | 58.7296i | 0 | 242.093 | + | 20.9715i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 36.h | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.6.s.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 432.6.s.a | 20 | ||
| 4.b | odd | 2 | 1 | 144.6.s.c | yes | 20 | |
| 9.c | even | 3 | 1 | 432.6.s.b | 20 | ||
| 9.d | odd | 6 | 1 | 144.6.s.c | yes | 20 | |
| 12.b | even | 2 | 1 | 432.6.s.b | 20 | ||
| 36.f | odd | 6 | 1 | 432.6.s.a | 20 | ||
| 36.h | even | 6 | 1 | inner | 144.6.s.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 144.6.s.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 144.6.s.a | ✓ | 20 | 36.h | even | 6 | 1 | inner |
| 144.6.s.c | yes | 20 | 4.b | odd | 2 | 1 | |
| 144.6.s.c | yes | 20 | 9.d | odd | 6 | 1 | |
| 432.6.s.a | 20 | 3.b | odd | 2 | 1 | ||
| 432.6.s.a | 20 | 36.f | odd | 6 | 1 | ||
| 432.6.s.b | 20 | 9.c | even | 3 | 1 | ||
| 432.6.s.b | 20 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\):
|
\( T_{5}^{20} - 87 T_{5}^{19} - 15792 T_{5}^{18} + 1593405 T_{5}^{17} + 191955933 T_{5}^{16} + \cdots + 77\!\cdots\!76 \)
|
|
\( T_{7}^{20} + 87 T_{7}^{19} - 89196 T_{7}^{18} - 7979553 T_{7}^{17} + 5282435493 T_{7}^{16} + \cdots + 24\!\cdots\!16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + \cdots + 71\!\cdots\!49 \)
|
| $5$ |
\( T^{20} + \cdots + 77\!\cdots\!76 \)
|
| $7$ |
\( T^{20} + \cdots + 24\!\cdots\!16 \)
|
| $11$ |
\( T^{20} + \cdots + 78\!\cdots\!89 \)
|
| $13$ |
\( T^{20} + \cdots + 42\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots + 71\!\cdots\!84 \)
|
| $19$ |
\( T^{20} + \cdots + 36\!\cdots\!44 \)
|
| $23$ |
\( T^{20} + \cdots + 42\!\cdots\!00 \)
|
| $29$ |
\( T^{20} + \cdots + 11\!\cdots\!04 \)
|
| $31$ |
\( T^{20} + \cdots + 12\!\cdots\!00 \)
|
| $37$ |
\( (T^{10} + \cdots - 38\!\cdots\!16)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 63\!\cdots\!29 \)
|
| $43$ |
\( T^{20} + \cdots + 63\!\cdots\!81 \)
|
| $47$ |
\( T^{20} + \cdots + 26\!\cdots\!44 \)
|
| $53$ |
\( T^{20} + \cdots + 12\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 15\!\cdots\!25 \)
|
| $61$ |
\( T^{20} + \cdots + 61\!\cdots\!36 \)
|
| $67$ |
\( T^{20} + \cdots + 24\!\cdots\!25 \)
|
| $71$ |
\( (T^{10} + \cdots - 26\!\cdots\!32)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots - 71\!\cdots\!56)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $89$ |
\( T^{20} + \cdots + 55\!\cdots\!44 \)
|
| $97$ |
\( T^{20} + \cdots + 22\!\cdots\!25 \)
|
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