Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2736,3,Mod(1711,2736)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2736.1711");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2736.m (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: | \(74.5506003290\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 111 x^{10} - 500 x^{9} + 4096 x^{8} - 13450 x^{7} + 58525 x^{6} - 130534 x^{5} + \cdots + 52400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 19 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} + 111 x^{10} - 500 x^{9} + 4096 x^{8} - 13450 x^{7} + 58525 x^{6} - 130534 x^{5} + \cdots + 52400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 157 \nu^{10} + 785 \nu^{9} - 20966 \nu^{8} + 79154 \nu^{7} - 989458 \nu^{6} + 2694632 \nu^{5} + \cdots - 147498072 ) / 7949068 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 978 \nu^{10} + 4890 \nu^{9} - 101957 \nu^{8} + 378488 \nu^{7} - 3467529 \nu^{6} + \cdots - 66471296 ) / 7949068 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2777 \nu^{10} - 13885 \nu^{9} + 284905 \nu^{8} - 1056310 \nu^{7} + 9413129 \nu^{6} + \cdots + 306285992 ) / 7949068 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 19501 \nu^{10} + 97505 \nu^{9} - 1993954 \nu^{8} + 7390786 \nu^{7} - 65282682 \nu^{6} + \cdots + 134392672 ) / 55643476 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2633 \nu^{10} - 13165 \nu^{9} + 275668 \nu^{8} - 1023682 \nu^{7} + 9353670 \nu^{6} + \cdots + 128367164 ) / 3974534 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 876762 \nu^{11} + 4822191 \nu^{10} - 101407183 \nu^{9} + 420165891 \nu^{8} + \cdots + 348116921592 ) / 15811532996 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7753002 \nu^{11} - 42641511 \nu^{10} + 838838645 \nu^{9} - 3454962570 \nu^{8} + \cdots - 574093754588 ) / 42917018132 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1987686 \nu^{11} + 10932273 \nu^{10} - 216681429 \nu^{9} + 893074383 \nu^{8} + \cdots + 245743869624 ) / 10729254533 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 62801406 \nu^{11} - 345407733 \nu^{10} + 6745431083 \nu^{9} - 27763881876 \nu^{8} + \cdots - 7346616448904 ) / 300419126924 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 40805158 \nu^{11} + 224428369 \nu^{10} - 4358862651 \nu^{9} + 17931669162 \nu^{8} + \cdots + 1809319947696 ) / 42917018132 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 93702810 \nu^{11} - 515365455 \nu^{10} + 10102801589 \nu^{9} - 41597366238 \nu^{8} + \cdots - 6725081798048 ) / 42917018132 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{11} + 5\beta_{9} + 19\beta_{7} + 2\beta_{6} + 19 ) / 38 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{11} + 5\beta_{9} + 19\beta_{7} + 2\beta_{6} + 19\beta_{3} + 57\beta_{2} - 19\beta _1 - 589 ) / 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 84 \beta_{11} - 57 \beta_{10} - 305 \beta_{9} - 285 \beta_{8} - 1292 \beta_{7} - 122 \beta_{6} + \cdots - 1786 ) / 76 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 86 \beta_{11} - 57 \beta_{10} - 310 \beta_{9} - 285 \beta_{8} - 1311 \beta_{7} - 124 \beta_{6} + \cdots + 17651 ) / 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1942 \beta_{11} + 3135 \beta_{10} + 11201 \beta_{9} + 12065 \beta_{8} + 40888 \beta_{7} + \cdots + 91238 ) / 76 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3129 \beta_{11} + 4845 \beta_{10} + 17579 \beta_{9} + 18810 \beta_{8} + 64619 \beta_{7} + \cdots - 575985 ) / 38 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 46612 \beta_{11} - 138453 \beta_{10} - 402005 \beta_{9} - 424403 \beta_{8} - 1328100 \beta_{7} + \cdots - 4353318 ) / 76 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 108028 \beta_{11} - 299649 \beta_{10} - 886772 \beta_{9} - 937251 \beta_{8} - 2960827 \beta_{7} + \cdots + 19049343 ) / 38 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 1119450 \beta_{11} + 5546271 \beta_{10} + 13931181 \beta_{9} + 13708671 \beta_{8} + 43422752 \beta_{7} + \cdots + 197950778 ) / 76 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 3631171 \beta_{11} + 16147245 \beta_{10} + 41603353 \beta_{9} + 41434155 \beta_{8} + \cdots - 621763543 ) / 38 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 25666696 \beta_{11} - 206080061 \beta_{10} - 463375873 \beta_{9} - 416205735 \beta_{8} + \cdots - 8702847742 ) / 76 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).
| \(n\) | \(1009\) | \(1217\) | \(1711\) | \(2053\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1711.1 |
|
0 | 0 | 0 | −7.69862 | 0 | − | 8.92404i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.2 | 0 | 0 | 0 | −7.69862 | 0 | 8.92404i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.3 | 0 | 0 | 0 | −4.06059 | 0 | − | 8.32166i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.4 | 0 | 0 | 0 | −4.06059 | 0 | 8.32166i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.5 | 0 | 0 | 0 | −3.90421 | 0 | − | 3.75651i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.6 | 0 | 0 | 0 | −3.90421 | 0 | 3.75651i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.7 | 0 | 0 | 0 | 3.90421 | 0 | − | 3.75651i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.8 | 0 | 0 | 0 | 3.90421 | 0 | 3.75651i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.9 | 0 | 0 | 0 | 4.06059 | 0 | − | 8.32166i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.10 | 0 | 0 | 0 | 4.06059 | 0 | 8.32166i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.11 | 0 | 0 | 0 | 7.69862 | 0 | − | 8.92404i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1711.12 | 0 | 0 | 0 | 7.69862 | 0 | 8.92404i | 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 | 2736.3.m.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 2736.3.m.b | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 2736.3.m.b | ✓ | 12 |
| 12.b | even | 2 | 1 | inner | 2736.3.m.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2736.3.m.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 2736.3.m.b | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
| 2736.3.m.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 2736.3.m.b | ✓ | 12 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 91T_{5}^{4} + 2132T_{5}^{2} - 14896 \)
acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{6} - 91 T^{4} + \cdots - 14896)^{2} \)
|
| $7$ |
\( (T^{6} + 163 T^{4} + \cdots + 77824)^{2} \)
|
| $11$ |
\( (T^{6} + 429 T^{4} + \cdots + 2085136)^{2} \)
|
| $13$ |
\( (T^{3} - 184 T + 328)^{4} \)
|
| $17$ |
\( (T^{6} - 863 T^{4} + \cdots - 3040000)^{2} \)
|
| $19$ |
\( (T^{2} + 19)^{6} \)
|
| $23$ |
\( (T^{6} + 1208 T^{4} + \cdots + 22202944)^{2} \)
|
| $29$ |
\( (T^{6} - 1940 T^{4} + \cdots - 20867776)^{2} \)
|
| $31$ |
\( (T^{6} + 3056 T^{4} + \cdots + 37240000)^{2} \)
|
| $37$ |
\( (T^{3} - 26 T^{2} + \cdots - 20432)^{4} \)
|
| $41$ |
\( (T^{6} - 2792 T^{4} + \cdots - 255073024)^{2} \)
|
| $43$ |
\( (T^{6} + 923 T^{4} + \cdots + 8993536)^{2} \)
|
| $47$ |
\( (T^{6} + 8673 T^{4} + \cdots + 3855416464)^{2} \)
|
| $53$ |
\( (T^{6} - 12820 T^{4} + \cdots - 15385134784)^{2} \)
|
| $59$ |
\( (T^{6} + 15568 T^{4} + \cdots + 1159266304)^{2} \)
|
| $61$ |
\( (T^{3} + 51 T^{2} + \cdots - 250124)^{4} \)
|
| $67$ |
\( (T^{6} + 18944 T^{4} + \cdots + 142886809600)^{2} \)
|
| $71$ |
\( (T^{6} + 4832 T^{4} + \cdots + 1420988416)^{2} \)
|
| $73$ |
\( (T^{3} - 25 T^{2} + \cdots + 88996)^{4} \)
|
| $79$ |
\( (T^{6} + 30468 T^{4} + \cdots + 105126609664)^{2} \)
|
| $83$ |
\( (T^{6} + 36860 T^{4} + \cdots + 143132075584)^{2} \)
|
| $89$ |
\( (T^{6} - 17460 T^{4} + \cdots - 15212608704)^{2} \)
|
| $97$ |
\( (T^{3} + 46 T^{2} + \cdots - 544600)^{4} \)
|
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