Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,19,Mod(2,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.2");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.c (of order \(4\), degree \(2\), 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: | \(10.2693068855\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 1365649 x^{14} + 735328563216 x^{12} + \cdots + 22\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{39}\cdot 3^{12}\cdot 5^{25} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 1365649 x^{14} + 735328563216 x^{12} + \cdots + 22\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 22\!\cdots\!27 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\!\cdots\!27 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47\!\cdots\!61 \nu^{14} + \cdots + 50\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\!\cdots\!01 \nu^{15} + \cdots + 50\!\cdots\!00 \nu ) / 16\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\!\cdots\!31 \nu^{15} + \cdots - 46\!\cdots\!00 ) / 64\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12\!\cdots\!31 \nu^{15} + \cdots - 46\!\cdots\!00 ) / 64\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 25\!\cdots\!79 \nu^{15} + \cdots - 36\!\cdots\!00 \nu ) / 71\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 47\!\cdots\!13 \nu^{15} + \cdots + 20\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 12\!\cdots\!47 \nu^{15} + \cdots - 55\!\cdots\!00 ) / 35\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 37\!\cdots\!73 \nu^{15} + \cdots - 35\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 37\!\cdots\!73 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 18\!\cdots\!79 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 64\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 59\!\cdots\!91 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 51\!\cdots\!17 \nu^{15} + \cdots - 23\!\cdots\!00 ) / 64\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 55\!\cdots\!01 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 64\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{5} + \beta_{3} + 8\beta_{2} + 8\beta _1 - 341411 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + 13 \beta_{11} + 13 \beta_{9} + 10 \beta_{8} + \cdots - 6 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 187 \beta_{15} + 187 \beta_{14} + 101 \beta_{13} + 795 \beta_{12} + 9711 \beta_{11} + \cdots + 197165209486 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 332621 \beta_{15} - 332621 \beta_{14} + 248173 \beta_{13} + 512493 \beta_{12} - 5689577 \beta_{11} + \cdots + 2281998 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 89122963 \beta_{15} - 89122963 \beta_{14} - 188206861 \beta_{13} - 460116339 \beta_{12} + \cdots - 61\!\cdots\!18 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 91332043533 \beta_{15} + 91332043533 \beta_{14} - 28750839341 \beta_{13} - 209834854061 \beta_{12} + \cdots - 715757080206 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 33766232882451 \beta_{15} + 33766232882451 \beta_{14} + 117458971654797 \beta_{13} + \cdots + 20\!\cdots\!78 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 23\!\cdots\!37 \beta_{15} + \cdots + 21\!\cdots\!62 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 12\!\cdots\!11 \beta_{15} + \cdots - 67\!\cdots\!42 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 58\!\cdots\!93 \beta_{15} + \cdots - 65\!\cdots\!74 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 44\!\cdots\!03 \beta_{15} + \cdots + 23\!\cdots\!86 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 14\!\cdots\!61 \beta_{15} + \cdots + 20\!\cdots\!10 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 16\!\cdots\!87 \beta_{15} + \cdots - 79\!\cdots\!06 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 32\!\cdots\!01 \beta_{15} + \cdots - 63\!\cdots\!98 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−569.773 | − | 569.773i | −12889.4 | + | 12889.4i | 387139.i | 1.81741e6 | − | 715334.i | 1.46881e7 | 8.52951e6 | + | 8.52951e6i | 7.12189e7 | − | 7.12189e7i | 5.51474e7i | −1.44309e9 | − | 6.27936e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.2 | −489.397 | − | 489.397i | 9614.07 | − | 9614.07i | 216875.i | −1.89157e6 | + | 486466.i | −9.41019e6 | −2.11052e7 | − | 2.11052e7i | −2.21547e7 | + | 2.21547e7i | 2.02560e8i | 1.16380e9 | + | 6.87655e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.3 | −148.361 | − | 148.361i | 24463.5 | − | 24463.5i | − | 218122.i | 1.64948e6 | − | 1.04590e6i | −7.25885e6 | 3.87838e7 | + | 3.87838e7i | −7.12526e7 | + | 7.12526e7i | − | 8.09507e8i | −3.99889e8 | − | 8.95466e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | −23.4817 | − | 23.4817i | −19449.6 | + | 19449.6i | − | 261041.i | −1.65425e6 | − | 1.03834e6i | 913420. | 3.92374e7 | + | 3.92374e7i | −1.22853e7 | + | 1.22853e7i | − | 3.69155e8i | 1.44628e7 | + | 6.32266e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.5 | −17.9217 | − | 17.9217i | −4350.59 | + | 4350.59i | − | 261502.i | 823487. | + | 1.77104e6i | 155940. | −2.28757e7 | − | 2.28757e7i | −9.38460e6 | + | 9.38460e6i | 3.49565e8i | 1.69817e7 | − | 4.64982e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.6 | 367.182 | + | 367.182i | 5232.51 | − | 5232.51i | 7501.19i | −39872.2 | − | 1.95272e6i | 3.84257e6 | −4.50520e7 | − | 4.50520e7i | 9.35002e7 | − | 9.35002e7i | 3.32662e8i | 7.02362e8 | − | 7.31643e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.7 | 514.444 | + | 514.444i | 11708.2 | − | 11708.2i | 267162.i | −1.06274e6 | + | 1.63869e6i | 1.20464e7 | 4.54981e7 | + | 4.54981e7i | −2.58155e6 | + | 2.58155e6i | 1.13259e8i | −1.38973e9 | + | 2.96294e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.8 | 622.308 | + | 622.308i | −24393.6 | + | 24393.6i | 512390.i | 1.93064e6 | + | 295511.i | −3.03607e7 | −3.63215e6 | − | 3.63215e6i | −1.55730e8 | + | 1.55730e8i | − | 8.02680e8i | 1.01755e9 | + | 1.38535e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.1 | −569.773 | + | 569.773i | −12889.4 | − | 12889.4i | − | 387139.i | 1.81741e6 | + | 715334.i | 1.46881e7 | 8.52951e6 | − | 8.52951e6i | 7.12189e7 | + | 7.12189e7i | − | 5.51474e7i | −1.44309e9 | + | 6.27936e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −489.397 | + | 489.397i | 9614.07 | + | 9614.07i | − | 216875.i | −1.89157e6 | − | 486466.i | −9.41019e6 | −2.11052e7 | + | 2.11052e7i | −2.21547e7 | − | 2.21547e7i | − | 2.02560e8i | 1.16380e9 | − | 6.87655e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −148.361 | + | 148.361i | 24463.5 | + | 24463.5i | 218122.i | 1.64948e6 | + | 1.04590e6i | −7.25885e6 | 3.87838e7 | − | 3.87838e7i | −7.12526e7 | − | 7.12526e7i | 8.09507e8i | −3.99889e8 | + | 8.95466e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −23.4817 | + | 23.4817i | −19449.6 | − | 19449.6i | 261041.i | −1.65425e6 | + | 1.03834e6i | 913420. | 3.92374e7 | − | 3.92374e7i | −1.22853e7 | − | 1.22853e7i | 3.69155e8i | 1.44628e7 | − | 6.32266e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −17.9217 | + | 17.9217i | −4350.59 | − | 4350.59i | 261502.i | 823487. | − | 1.77104e6i | 155940. | −2.28757e7 | + | 2.28757e7i | −9.38460e6 | − | 9.38460e6i | − | 3.49565e8i | 1.69817e7 | + | 4.64982e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 367.182 | − | 367.182i | 5232.51 | + | 5232.51i | − | 7501.19i | −39872.2 | + | 1.95272e6i | 3.84257e6 | −4.50520e7 | + | 4.50520e7i | 9.35002e7 | + | 9.35002e7i | − | 3.32662e8i | 7.02362e8 | + | 7.31643e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 514.444 | − | 514.444i | 11708.2 | + | 11708.2i | − | 267162.i | −1.06274e6 | − | 1.63869e6i | 1.20464e7 | 4.54981e7 | − | 4.54981e7i | −2.58155e6 | − | 2.58155e6i | − | 1.13259e8i | −1.38973e9 | − | 2.96294e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 622.308 | − | 622.308i | −24393.6 | − | 24393.6i | − | 512390.i | 1.93064e6 | − | 295511.i | −3.03607e7 | −3.63215e6 | + | 3.63215e6i | −1.55730e8 | − | 1.55730e8i | 8.02680e8i | 1.01755e9 | − | 1.38535e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5.19.c.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 45.19.g.a | 16 | ||
| 5.b | even | 2 | 1 | 25.19.c.b | 16 | ||
| 5.c | odd | 4 | 1 | inner | 5.19.c.a | ✓ | 16 |
| 5.c | odd | 4 | 1 | 25.19.c.b | 16 | ||
| 15.e | even | 4 | 1 | 45.19.g.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.19.c.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 5.19.c.a | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 25.19.c.b | 16 | 5.b | even | 2 | 1 | ||
| 25.19.c.b | 16 | 5.c | odd | 4 | 1 | ||
| 45.19.g.a | 16 | 3.b | odd | 2 | 1 | ||
| 45.19.g.a | 16 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(5, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 10\!\cdots\!96 \)
|
| $3$ |
\( T^{16} + \cdots + 37\!\cdots\!36 \)
|
| $5$ |
\( T^{16} + \cdots + 44\!\cdots\!25 \)
|
| $7$ |
\( T^{16} + \cdots + 55\!\cdots\!56 \)
|
| $11$ |
\( (T^{8} + \cdots + 29\!\cdots\!76)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 17\!\cdots\!16 \)
|
| $17$ |
\( T^{16} + \cdots + 30\!\cdots\!76 \)
|
| $19$ |
\( T^{16} + \cdots + 63\!\cdots\!00 \)
|
| $23$ |
\( T^{16} + \cdots + 21\!\cdots\!96 \)
|
| $29$ |
\( T^{16} + \cdots + 25\!\cdots\!00 \)
|
| $31$ |
\( (T^{8} + \cdots - 13\!\cdots\!84)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 11\!\cdots\!16 \)
|
| $41$ |
\( (T^{8} + \cdots - 88\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 51\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 91\!\cdots\!36 \)
|
| $53$ |
\( T^{16} + \cdots + 93\!\cdots\!36 \)
|
| $59$ |
\( T^{16} + \cdots + 29\!\cdots\!00 \)
|
| $61$ |
\( (T^{8} + \cdots + 13\!\cdots\!76)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 28\!\cdots\!76 \)
|
| $71$ |
\( (T^{8} + \cdots - 11\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 23\!\cdots\!96 \)
|
| $79$ |
\( T^{16} + \cdots + 44\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 17\!\cdots\!76 \)
|
| $89$ |
\( T^{16} + \cdots + 80\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 38\!\cdots\!36 \)
|
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