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: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} - 5365 x^{12} + 7107 x^{11} + 10970098 x^{10} - 19024208 x^{9} - 10608934432 x^{8} + \cdots - 480881506516992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{8}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} - 5365 x^{12} + 7107 x^{11} + 10970098 x^{10} - 19024208 x^{9} - 10608934432 x^{8} + \cdots - 480881506516992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 767 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 1275\nu + 1231 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\!\cdots\!11 \nu^{13} + \cdots - 31\!\cdots\!92 ) / 68\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15\!\cdots\!57 \nu^{13} + \cdots + 38\!\cdots\!04 ) / 68\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10\!\cdots\!77 \nu^{13} + \cdots - 35\!\cdots\!44 ) / 68\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 40\!\cdots\!97 \nu^{13} + \cdots + 42\!\cdots\!16 ) / 21\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 66\!\cdots\!49 \nu^{13} + \cdots - 13\!\cdots\!28 ) / 34\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 35\!\cdots\!01 \nu^{13} + \cdots - 22\!\cdots\!28 ) / 13\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 90\!\cdots\!71 \nu^{13} + \cdots - 30\!\cdots\!12 ) / 31\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 59\!\cdots\!87 \nu^{13} + \cdots + 13\!\cdots\!64 ) / 17\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 63\!\cdots\!51 \nu^{13} + \cdots + 14\!\cdots\!28 ) / 17\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 57\!\cdots\!87 \nu^{13} + \cdots - 16\!\cdots\!64 ) / 68\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 767 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 1274\beta _1 - 464 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + \cdots + 977878 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{13} - 9 \beta_{12} - 24 \beta_{11} + 30 \beta_{10} - 13 \beta_{9} - 26 \beta_{8} + \cdots - 591005 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2893 \beta_{13} + 2508 \beta_{12} + 3898 \beta_{11} + 3269 \beta_{10} - 5125 \beta_{9} + \cdots + 1384970143 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2972 \beta_{13} - 13756 \beta_{12} - 88220 \beta_{11} + 96868 \beta_{10} - 43060 \beta_{9} + \cdots - 1075101020 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 6541337 \beta_{13} + 4959033 \beta_{12} + 5964170 \beta_{11} + 7716633 \beta_{10} + \cdots + 2077546046810 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 21173394 \beta_{13} - 8087749 \beta_{12} - 224900256 \beta_{11} + 219745618 \beta_{10} + \cdots - 2140755470997 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 13343039329 \beta_{13} + 8969110260 \beta_{12} + 8366862002 \beta_{11} + 16090591321 \beta_{10} + \cdots + 32\!\cdots\!19 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 62448351928 \beta_{13} + 18820392872 \beta_{12} - 490975237460 \beta_{11} + 436660818656 \beta_{10} + \cdots - 39\!\cdots\!72 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 25707542042197 \beta_{13} + 15550593556265 \beta_{12} + 11224301212978 \beta_{11} + \cdots + 51\!\cdots\!46 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 143949912659366 \beta_{13} + 87449312953599 \beta_{12} - 986142552578248 \beta_{11} + \cdots - 66\!\cdots\!41 \)
|
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 |
|
−41.5454 | −81.0000 | 1214.02 | −625.000 | 3365.17 | 6093.78 | −29165.5 | 6561.00 | 25965.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −36.7045 | −81.0000 | 835.221 | −625.000 | 2973.07 | −10828.2 | −11863.7 | 6561.00 | 22940.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −33.9313 | −81.0000 | 639.331 | −625.000 | 2748.43 | 9880.04 | −4320.49 | 6561.00 | 21207.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −27.3479 | −81.0000 | 235.909 | −625.000 | 2215.18 | −4985.88 | 7550.51 | 6561.00 | 17092.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −16.1367 | −81.0000 | −251.607 | −625.000 | 1307.07 | −1176.10 | 12322.1 | 6561.00 | 10085.4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −8.32294 | −81.0000 | −442.729 | −625.000 | 674.158 | −7295.26 | 7946.15 | 6561.00 | 5201.84 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.187735 | −81.0000 | −511.965 | −625.000 | 15.2065 | 905.223 | 192.234 | 6561.00 | 117.334 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.198399 | −81.0000 | −511.961 | −625.000 | −16.0704 | 11709.0 | −203.153 | 6561.00 | −124.000 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.26189 | −81.0000 | −506.884 | −625.000 | −183.213 | −10.0609 | −2304.60 | 6561.00 | −1413.68 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 21.3757 | −81.0000 | −55.0809 | −625.000 | −1731.43 | 6106.61 | −12121.7 | 6561.00 | −13359.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 28.9110 | −81.0000 | 323.844 | −625.000 | −2341.79 | −5747.14 | −5439.76 | 6561.00 | −18069.4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 30.3072 | −81.0000 | 406.529 | −625.000 | −2454.89 | 9233.36 | −3196.55 | 6561.00 | −18942.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 38.9237 | −81.0000 | 1003.06 | −625.000 | −3152.82 | −5904.55 | 19113.8 | 6561.00 | −24327.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 41.1985 | −81.0000 | 1185.32 | −625.000 | −3337.08 | 5073.19 | 27739.7 | 6561.00 | −25749.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.f | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.10.a.f | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} + T_{2}^{13} - 5365 T_{2}^{12} - 7107 T_{2}^{11} + 10970098 T_{2}^{10} + \cdots - 480881506516992 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + \cdots - 480881506516992 \)
|
| $3$ |
\( (T + 81)^{14} \)
|
| $5$ |
\( (T + 625)^{14} \)
|
| $7$ |
\( T^{14} + \cdots - 28\!\cdots\!84 \)
|
| $11$ |
\( T^{14} + \cdots - 10\!\cdots\!96 \)
|
| $13$ |
\( T^{14} + \cdots - 87\!\cdots\!00 \)
|
| $17$ |
\( T^{14} + \cdots + 41\!\cdots\!00 \)
|
| $19$ |
\( (T - 130321)^{14} \)
|
| $23$ |
\( T^{14} + \cdots + 55\!\cdots\!44 \)
|
| $29$ |
\( T^{14} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{14} + \cdots - 17\!\cdots\!60 \)
|
| $37$ |
\( T^{14} + \cdots - 53\!\cdots\!96 \)
|
| $41$ |
\( T^{14} + \cdots + 18\!\cdots\!80 \)
|
| $43$ |
\( T^{14} + \cdots - 13\!\cdots\!76 \)
|
| $47$ |
\( T^{14} + \cdots - 22\!\cdots\!84 \)
|
| $53$ |
\( T^{14} + \cdots + 22\!\cdots\!88 \)
|
| $59$ |
\( T^{14} + \cdots - 37\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots + 40\!\cdots\!40 \)
|
| $67$ |
\( T^{14} + \cdots + 22\!\cdots\!80 \)
|
| $71$ |
\( T^{14} + \cdots - 15\!\cdots\!00 \)
|
| $73$ |
\( T^{14} + \cdots + 20\!\cdots\!84 \)
|
| $79$ |
\( T^{14} + \cdots + 11\!\cdots\!00 \)
|
| $83$ |
\( T^{14} + \cdots + 59\!\cdots\!20 \)
|
| $89$ |
\( T^{14} + \cdots + 59\!\cdots\!00 \)
|
| $97$ |
\( T^{14} + \cdots + 13\!\cdots\!88 \)
|
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