Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,6,Mod(143,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("144.143");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.c (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: | \(23.0952700531\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.9421854806016.47 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 70x^{6} + 224x^{5} + 1799x^{4} - 3976x^{3} - 19214x^{2} + 21240x + 84900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{12} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 70x^{6} + 224x^{5} + 1799x^{4} - 3976x^{3} - 19214x^{2} + 21240x + 84900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 256\nu^{6} - 768\nu^{5} - 23168\nu^{4} + 47616\nu^{3} + 592000\nu^{2} - 615936\nu - 3995520 ) / 5025 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11926 \nu^{7} - 186011 \nu^{6} - 261779 \nu^{5} + 10064215 \nu^{4} - 3331223 \nu^{3} + \cdots + 1072442100 ) / 3222030 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11552 \nu^{7} + 40432 \nu^{6} + 867328 \nu^{5} - 2269400 \nu^{4} - 25398464 \nu^{3} + \cdots - 220698420 ) / 2685025 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4100 \nu^{7} - 45036 \nu^{6} - 81892 \nu^{5} + 2097108 \nu^{4} - 2647696 \nu^{3} - 23095800 \nu^{2} + \cdots + 547920 ) / 383575 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 32160 \nu^{7} + 125384 \nu^{6} + 2269008 \nu^{5} - 7210672 \nu^{4} - 48829536 \nu^{3} + \cdots - 365046960 ) / 1611015 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1602 \nu^{7} - 5607 \nu^{6} - 93303 \nu^{5} + 247275 \nu^{4} + 1858689 \nu^{3} - 3038112 \nu^{2} + \cdots + 5844420 ) / 80150 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 360800 \nu^{7} + 455616 \nu^{6} - 20462848 \nu^{5} - 49650048 \nu^{4} + 257487776 \nu^{3} + \cdots - 1807672320 ) / 2685025 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{7} + 64\beta_{6} - 24\beta_{4} + 144\beta_{3} + 5184 ) / 10368 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -24\beta_{7} + 224\beta_{6} + 240\beta_{4} + 144\beta_{3} - 864\beta_{2} - 81\beta _1 + 202176 ) / 10368 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 99 \beta_{7} + 2024 \beta_{6} + 648 \beta_{5} - 468 \beta_{4} + 1584 \beta_{3} - 648 \beta_{2} + \cdots + 150336 ) / 5184 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 564 \beta_{7} + 7456 \beta_{6} + 1296 \beta_{5} + 3696 \beta_{4} + 3096 \beta_{3} - 19872 \beta_{2} + \cdots + 2433888 ) / 5184 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 3669 \beta_{7} + 90488 \beta_{6} + 43200 \beta_{5} - 9372 \beta_{4} + 40248 \beta_{3} + \cdots + 5834592 ) / 5184 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 19494 \beta_{7} + 387760 \beta_{6} + 126360 \beta_{5} + 87048 \beta_{4} + 113040 \beta_{3} + \cdots + 63187776 ) / 5184 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 112737 \beta_{7} + 3812840 \beta_{6} + 2006424 \beta_{5} - 120948 \beta_{4} + 1039680 \beta_{3} + \cdots + 200911104 ) / 5184 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 143.1 |
|
0 | 0 | 0 | − | 93.7115i | 0 | − | 212.225i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 143.2 | 0 | 0 | 0 | − | 93.7115i | 0 | 212.225i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 143.3 | 0 | 0 | 0 | − | 51.2851i | 0 | − | 142.943i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 143.4 | 0 | 0 | 0 | − | 51.2851i | 0 | 142.943i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 143.5 | 0 | 0 | 0 | 51.2851i | 0 | − | 142.943i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 143.6 | 0 | 0 | 0 | 51.2851i | 0 | 142.943i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 143.7 | 0 | 0 | 0 | 93.7115i | 0 | − | 212.225i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 143.8 | 0 | 0 | 0 | 93.7115i | 0 | 212.225i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.6.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 144.6.c.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 144.6.c.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 576.6.c.d | 8 | ||
| 8.d | odd | 2 | 1 | 576.6.c.d | 8 | ||
| 12.b | even | 2 | 1 | inner | 144.6.c.b | ✓ | 8 |
| 24.f | even | 2 | 1 | 576.6.c.d | 8 | ||
| 24.h | odd | 2 | 1 | 576.6.c.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 144.6.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 144.6.c.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 144.6.c.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 144.6.c.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 576.6.c.d | 8 | 8.b | even | 2 | 1 | ||
| 576.6.c.d | 8 | 8.d | odd | 2 | 1 | ||
| 576.6.c.d | 8 | 24.f | even | 2 | 1 | ||
| 576.6.c.d | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 11412T_{5}^{2} + 23097636 \)
acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 11412 T^{2} + 23097636)^{2} \)
|
| $7$ |
\( (T^{4} + 65472 T^{2} + 920272896)^{2} \)
|
| $11$ |
\( (T^{4} - 210816 T^{2} + 429981696)^{2} \)
|
| $13$ |
\( (T^{2} + 64 T - 93584)^{4} \)
|
| $17$ |
\( (T^{4} + 1311300 T^{2} + 156459802500)^{2} \)
|
| $19$ |
\( (T^{4} + 7876608 T^{2} + 9663676416)^{2} \)
|
| $23$ |
\( (T^{4} - 21555072 T^{2} + 313456656384)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 94259127533796)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 312247559798784)^{2} \)
|
| $37$ |
\( (T^{2} + 3836 T - 118933244)^{4} \)
|
| $41$ |
\( (T^{4} + \cdots + 41\!\cdots\!84)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 17\!\cdots\!36)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 25\!\cdots\!76)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 14\!\cdots\!24)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 19\!\cdots\!96)^{2} \)
|
| $61$ |
\( (T^{2} + 8276 T - 420344348)^{4} \)
|
| $67$ |
\( (T^{4} + \cdots + 526110005919744)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 50\!\cdots\!64)^{2} \)
|
| $73$ |
\( (T^{2} - 1856 T - 3342963968)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 34\!\cdots\!64)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 35\!\cdots\!56)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 30\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{2} - 12416 T - 597604928)^{4} \)
|
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