Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [285,10,Mod(1,285)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(285, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("285.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 285 = 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 285.a (trivial) |
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: | yes |
| Analytic conductor: | \(146.785213307\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 3 x^{11} - 4398 x^{10} + 4080 x^{9} + 7026370 x^{8} + 7294322 x^{7} - 5023445596 x^{6} + \cdots + 49\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{6}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{12} - 3 x^{11} - 4398 x^{10} + 4080 x^{9} + 7026370 x^{8} + 7294322 x^{7} - 5023445596 x^{6} + \cdots + 49\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 733 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 1208\nu - 511 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!87 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37\!\cdots\!57 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 64\!\cdots\!53 \nu^{11} + \cdots + 23\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14\!\cdots\!77 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 89\!\cdots\!61 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!87 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 41\!\cdots\!21 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 53\!\cdots\!09 \nu^{11} + \cdots - 28\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 733 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 1214\beta _1 + 1977 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 8 \beta_{11} - 10 \beta_{10} - 4 \beta_{9} + 9 \beta_{8} + 6 \beta_{7} + 13 \beta_{6} - 4 \beta_{5} + \cdots + 889061 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 16 \beta_{11} - 174 \beta_{10} - 108 \beta_{9} - 181 \beta_{8} + 2 \beta_{7} + 39 \beta_{6} + \cdots + 4477443 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 19040 \beta_{11} - 23520 \beta_{10} - 12624 \beta_{9} + 23604 \beta_{8} + 19240 \beta_{7} + \cdots + 1273776211 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 78656 \beta_{11} - 430772 \beta_{10} - 281592 \beta_{9} - 288086 \beta_{8} + 32652 \beta_{7} + \cdots + 9236613561 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 35476472 \beta_{11} - 44749286 \beta_{10} - 30683692 \beta_{9} + 48869527 \beta_{8} + \cdots + 1955414213881 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 214410992 \beta_{11} - 833125658 \beta_{10} - 598042548 \beta_{9} - 240067127 \beta_{8} + \cdots + 18668768175903 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 61607399824 \beta_{11} - 79944882156 \beta_{10} - 65393752120 \beta_{9} + 93786784746 \beta_{8} + \cdots + 31\!\cdots\!71 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 482806042272 \beta_{11} - 1514333547008 \beta_{10} - 1211990053344 \beta_{9} + 99195125208 \beta_{8} + \cdots + 37\!\cdots\!01 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−42.7851 | −81.0000 | 1318.56 | 625.000 | 3465.59 | −1551.27 | −34508.7 | 6561.00 | −26740.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −38.8200 | −81.0000 | 994.993 | 625.000 | 3144.42 | 2821.42 | −18749.8 | 6561.00 | −24262.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −26.3522 | −81.0000 | 182.438 | 625.000 | 2134.53 | −1882.26 | 8684.68 | 6561.00 | −16470.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −24.0754 | −81.0000 | 67.6236 | 625.000 | 1950.11 | −10502.2 | 10698.5 | 6561.00 | −15047.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −17.0719 | −81.0000 | −220.549 | 625.000 | 1382.83 | 11804.8 | 12506.0 | 6561.00 | −10670.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −13.1882 | −81.0000 | −338.072 | 625.000 | 1068.24 | 296.534 | 11210.9 | 6561.00 | −8242.61 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.549710 | −81.0000 | −511.698 | 625.000 | −44.5265 | 3867.71 | −562.737 | 6561.00 | 343.569 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.8734 | −81.0000 | −371.022 | 625.000 | −961.747 | −8400.74 | −10484.5 | 6561.00 | 7420.89 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 18.3302 | −81.0000 | −176.005 | 625.000 | −1484.74 | 8938.51 | −12611.2 | 6561.00 | 11456.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 25.1327 | −81.0000 | 119.655 | 625.000 | −2035.75 | 116.881 | −9860.71 | 6561.00 | 15708.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 33.9825 | −81.0000 | 642.811 | 625.000 | −2752.58 | −5887.61 | 4445.28 | 6561.00 | 21239.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 39.4242 | −81.0000 | 1042.27 | 625.000 | −3193.36 | −3071.79 | 20905.3 | 6561.00 | 24640.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(19\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 285.10.a.c | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.10.a.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 33 T_{2}^{11} - 3903 T_{2}^{10} - 123405 T_{2}^{9} + 5382070 T_{2}^{8} + \cdots + 955735809196032 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 955735809196032 \)
|
| $3$ |
\( (T + 81)^{12} \)
|
| $5$ |
\( (T - 625)^{12} \)
|
| $7$ |
\( T^{12} + \cdots + 18\!\cdots\!08 \)
|
| $11$ |
\( T^{12} + \cdots + 16\!\cdots\!08 \)
|
| $13$ |
\( T^{12} + \cdots - 95\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $19$ |
\( (T - 130321)^{12} \)
|
| $23$ |
\( T^{12} + \cdots - 16\!\cdots\!08 \)
|
| $29$ |
\( T^{12} + \cdots + 53\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 39\!\cdots\!80 \)
|
| $37$ |
\( T^{12} + \cdots - 51\!\cdots\!36 \)
|
| $41$ |
\( T^{12} + \cdots + 72\!\cdots\!20 \)
|
| $43$ |
\( T^{12} + \cdots - 46\!\cdots\!88 \)
|
| $47$ |
\( T^{12} + \cdots - 20\!\cdots\!36 \)
|
| $53$ |
\( T^{12} + \cdots - 19\!\cdots\!12 \)
|
| $59$ |
\( T^{12} + \cdots + 87\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots - 55\!\cdots\!20 \)
|
| $67$ |
\( T^{12} + \cdots - 17\!\cdots\!80 \)
|
| $71$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 25\!\cdots\!16 \)
|
| $79$ |
\( T^{12} + \cdots + 30\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots - 18\!\cdots\!40 \)
|
| $89$ |
\( T^{12} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 12\!\cdots\!44 \)
|
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