Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,10,Mod(1,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 19.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: | \(9.78568088711\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2442607 \nu^{7} + 89863001 \nu^{6} + 5717983110 \nu^{5} - 218132022350 \nu^{4} + \cdots + 25\!\cdots\!72 ) / 268007397120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2442607 \nu^{7} + 89863001 \nu^{6} + 5717983110 \nu^{5} - 218132022350 \nu^{4} + \cdots + 27\!\cdots\!52 ) / 268007397120 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4307827 \nu^{7} - 91728929 \nu^{6} - 11718846762 \nu^{5} + 209079010526 \nu^{4} + \cdots + 19\!\cdots\!04 ) / 268007397120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2957621 \nu^{7} - 103632415 \nu^{6} - 7003343598 \nu^{5} + 249515611234 \nu^{4} + \cdots - 34\!\cdots\!12 ) / 89335799040 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10018537 \nu^{7} - 278554463 \nu^{6} - 25382046138 \nu^{5} + 653708092034 \nu^{4} + \cdots - 52\!\cdots\!88 ) / 268007397120 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3205844 \nu^{7} - 67957249 \nu^{6} - 8652761913 \nu^{5} + 153606755494 \nu^{4} + \cdots + 634795682678720 ) / 67001849280 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2\beta _1 + 839 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{7} - 5\beta_{6} - \beta_{5} - 8\beta_{4} - 16\beta_{3} + 4\beta_{2} + 1457\beta _1 + 1575 \)
|
| \(\nu^{4}\) | \(=\) |
\( -93\beta_{7} + 257\beta_{6} - 301\beta_{5} + 2\beta_{4} - 2404\beta_{3} + 1880\beta_{2} + 10395\beta _1 + 1224153 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11883 \beta_{7} - 8443 \beta_{6} - 7181 \beta_{5} - 22534 \beta_{4} - 46878 \beta_{3} + 8806 \beta_{2} + \cdots + 8671551 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 235743 \beta_{7} + 751431 \beta_{6} - 911307 \beta_{5} - 91222 \beta_{4} - 5100850 \beta_{3} + \cdots + 2014920109 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23484551 \beta_{7} - 11105367 \beta_{6} - 22663853 \beta_{5} - 49941242 \beta_{4} - 115934890 \beta_{3} + \cdots + 23420404735 \)
|
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 |
|
−43.0760 | 210.029 | 1343.54 | 1106.01 | −9047.23 | 1237.92 | −35819.6 | 24429.4 | −47642.6 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −32.1433 | 6.24328 | 521.193 | −1240.15 | −200.680 | −8874.26 | −295.489 | −19644.0 | 39862.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.83232 | −260.789 | −465.319 | −1115.83 | 1781.79 | −8602.80 | 6677.36 | 48328.0 | 7623.74 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.59102 | 196.638 | −490.923 | 2263.73 | −902.769 | −2177.39 | 4604.44 | 18983.5 | −10392.8 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.24681 | −81.1272 | −501.458 | −574.796 | 263.404 | 7689.68 | 3290.50 | −13101.4 | 1866.25 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 30.7385 | −266.878 | 432.858 | 2199.99 | −8203.46 | 7318.17 | −2432.70 | 51541.1 | 67624.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 30.7538 | 175.481 | 433.799 | 18.5839 | 5396.70 | 6967.83 | −2404.99 | 11110.5 | 571.527 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 43.3971 | 27.4035 | 1371.31 | 1236.46 | 1189.23 | −10692.2 | 37291.4 | −18932.0 | 53658.9 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(-1\) |
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 | 19.10.a.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 171.10.a.f | 8 | ||
| 4.b | odd | 2 | 1 | 304.10.a.i | 8 | ||
| 19.b | odd | 2 | 1 | 361.10.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.10.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 171.10.a.f | 8 | 3.b | odd | 2 | 1 | ||
| 304.10.a.i | 8 | 4.b | odd | 2 | 1 | ||
| 361.10.a.c | 8 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 15 T_{2}^{7} - 3258 T_{2}^{6} + 41238 T_{2}^{5} + 2972568 T_{2}^{4} - 23100984 T_{2}^{3} + \cdots - 5784998400 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(19))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots - 5784998400 \)
|
| $3$ |
\( T^{8} + \cdots - 70\!\cdots\!40 \)
|
| $5$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 86\!\cdots\!12 \)
|
| $11$ |
\( T^{8} + \cdots - 14\!\cdots\!96 \)
|
| $13$ |
\( T^{8} + \cdots - 47\!\cdots\!80 \)
|
| $17$ |
\( T^{8} + \cdots + 41\!\cdots\!34 \)
|
| $19$ |
\( (T - 130321)^{8} \)
|
| $23$ |
\( T^{8} + \cdots + 52\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots - 95\!\cdots\!60 \)
|
| $31$ |
\( T^{8} + \cdots + 47\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 13\!\cdots\!40 \)
|
| $41$ |
\( T^{8} + \cdots - 30\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
|
| $47$ |
\( T^{8} + \cdots - 55\!\cdots\!52 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!20 \)
|
| $59$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 16\!\cdots\!24 \)
|
| $67$ |
\( T^{8} + \cdots - 56\!\cdots\!04 \)
|
| $71$ |
\( T^{8} + \cdots + 27\!\cdots\!88 \)
|
| $73$ |
\( T^{8} + \cdots - 39\!\cdots\!66 \)
|
| $79$ |
\( T^{8} + \cdots + 19\!\cdots\!40 \)
|
| $83$ |
\( T^{8} + \cdots - 50\!\cdots\!40 \)
|
| $89$ |
\( T^{8} + \cdots - 47\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 27\!\cdots\!44 \)
|
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