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: |
\( x^{12} - 3 x^{11} - 4398 x^{10} + 11376 x^{9} + 7070146 x^{8} - 15274638 x^{7} - 5114407260 x^{6} + \cdots + 43\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{3}\cdot 5^{2} \) |
| 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
\( x^{12} - 3 x^{11} - 4398 x^{10} + 11376 x^{9} + 7070146 x^{8} - 15274638 x^{7} - 5114407260 x^{6} + \cdots + 43\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 734 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 60\!\cdots\!33 \nu^{11} + \cdots + 98\!\cdots\!80 ) / 66\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25\!\cdots\!72 \nu^{11} + \cdots - 10\!\cdots\!20 ) / 33\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25\!\cdots\!72 \nu^{11} + \cdots - 12\!\cdots\!60 ) / 33\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 81\!\cdots\!91 \nu^{11} + \cdots + 17\!\cdots\!88 ) / 66\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!08 \nu^{11} + \cdots - 78\!\cdots\!60 ) / 11\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16\!\cdots\!11 \nu^{11} + \cdots - 24\!\cdots\!60 ) / 33\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 48\!\cdots\!39 \nu^{11} + \cdots + 97\!\cdots\!40 ) / 83\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 20\!\cdots\!57 \nu^{11} + \cdots + 40\!\cdots\!60 ) / 22\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 38\!\cdots\!33 \nu^{11} + \cdots - 82\!\cdots\!40 ) / 16\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 734 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{2} + 1184\beta _1 + 161 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 10 \beta_{10} - 4 \beta_{9} + 10 \beta_{8} - 6 \beta_{7} + 30 \beta_{6} - 4 \beta_{5} + 14 \beta_{4} + \cdots + 869182 \)
|
| \(\nu^{5}\) | \(=\) |
\( 272 \beta_{11} + 314 \beta_{10} - 196 \beta_{9} + 342 \beta_{8} - 106 \beta_{7} + 274 \beta_{6} + \cdots + 459699 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1800 \beta_{11} - 20936 \beta_{10} - 6312 \beta_{9} + 29496 \beta_{8} - 11576 \beta_{7} + \cdots + 1192208620 \)
|
| \(\nu^{7}\) | \(=\) |
\( 814672 \beta_{11} + 979580 \beta_{10} - 602248 \beta_{9} + 1029620 \beta_{8} - 476428 \beta_{7} + \cdots + 1318783089 \)
|
| \(\nu^{8}\) | \(=\) |
\( 6298704 \beta_{11} - 33653574 \beta_{10} - 4914508 \beta_{9} + 64963350 \beta_{8} - 18138650 \beta_{7} + \cdots + 1771384672930 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1796504560 \beta_{11} + 2222145966 \beta_{10} - 1349736652 \beta_{9} + 2260779634 \beta_{8} + \cdots + 3425843328143 \)
|
| \(\nu^{10}\) | \(=\) |
\( 15881066552 \beta_{11} - 49671714164 \beta_{10} + 1717261456 \beta_{9} + 128958154260 \beta_{8} + \cdots + 27\!\cdots\!56 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3526759027120 \beta_{11} + 4435915245424 \beta_{10} - 2666403632272 \beta_{9} + 4419288994480 \beta_{8} + \cdots + 79\!\cdots\!01 \)
|
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 |
|
−43.7797 | 81.0000 | 1404.66 | 625.000 | −3546.16 | 6872.55 | −39080.6 | 6561.00 | −27362.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −37.2245 | 81.0000 | 873.663 | 625.000 | −3015.18 | −2357.55 | −13462.7 | 6561.00 | −23265.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −31.9288 | 81.0000 | 507.448 | 625.000 | −2586.23 | −632.496 | 145.335 | 6561.00 | −19955.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −25.0115 | 81.0000 | 113.577 | 625.000 | −2025.93 | −7116.10 | 9965.17 | 6561.00 | −15632.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −15.3423 | 81.0000 | −276.614 | 625.000 | −1242.73 | 8224.35 | 12099.1 | 6561.00 | −9588.93 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −8.36255 | 81.0000 | −442.068 | 625.000 | −677.366 | 1660.91 | 7978.44 | 6561.00 | −5226.59 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.42527 | 81.0000 | −500.268 | 625.000 | 277.446 | −1060.52 | −3467.29 | 6561.00 | 2140.79 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.50131 | 81.0000 | −439.728 | 625.000 | 688.606 | −8602.42 | −8090.93 | 6561.00 | 5313.32 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 23.4167 | 81.0000 | 36.3418 | 625.000 | 1896.75 | 2524.82 | −11138.3 | 6561.00 | 14635.4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 23.9112 | 81.0000 | 59.7445 | 625.000 | 1936.81 | 7881.01 | −10814.0 | 6561.00 | 14944.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 30.7980 | 81.0000 | 436.517 | 625.000 | 2494.64 | −2782.02 | −2324.71 | 6561.00 | 19248.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 38.5969 | 81.0000 | 977.722 | 625.000 | 3126.35 | −9712.55 | 17975.4 | 6561.00 | 24123.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.10.a.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 33 T_{2}^{11} - 3903 T_{2}^{10} - 116109 T_{2}^{9} + 5622838 T_{2}^{8} + \cdots + 32\!\cdots\!12 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 32\!\cdots\!12 \)
|
| $3$ |
\( (T - 81)^{12} \)
|
| $5$ |
\( (T - 625)^{12} \)
|
| $7$ |
\( T^{12} + \cdots - 48\!\cdots\!00 \)
|
| $11$ |
\( T^{12} + \cdots - 32\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots - 12\!\cdots\!64 \)
|
| $17$ |
\( T^{12} + \cdots + 45\!\cdots\!88 \)
|
| $19$ |
\( (T + 130321)^{12} \)
|
| $23$ |
\( T^{12} + \cdots + 55\!\cdots\!44 \)
|
| $29$ |
\( T^{12} + \cdots + 40\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 27\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 31\!\cdots\!00 \)
|
| $41$ |
\( T^{12} + \cdots + 66\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots - 52\!\cdots\!20 \)
|
| $47$ |
\( T^{12} + \cdots + 19\!\cdots\!08 \)
|
| $53$ |
\( T^{12} + \cdots + 85\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots + 36\!\cdots\!24 \)
|
| $67$ |
\( T^{12} + \cdots - 80\!\cdots\!88 \)
|
| $71$ |
\( T^{12} + \cdots + 17\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 99\!\cdots\!00 \)
|
| $79$ |
\( T^{12} + \cdots - 88\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots - 28\!\cdots\!60 \)
|
| $89$ |
\( T^{12} + \cdots - 33\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 17\!\cdots\!00 \)
|
| show more | |
| show less |