Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [816,2,Mod(239,816)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(816, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("816.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 816.e (of order \(2\), degree \(1\), 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: | \(6.51579280494\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 11 x^{16} - 28 x^{15} + 36 x^{14} - 38 x^{13} + 61 x^{12} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{16} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 11 x^{16} - 28 x^{15} + 36 x^{14} - 38 x^{13} + 61 x^{12} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3 \nu^{19} + 38 \nu^{18} - 170 \nu^{17} + 378 \nu^{16} - 551 \nu^{15} + 880 \nu^{14} - 1980 \nu^{13} + \cdots + 27136 ) / 512 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 207 \nu^{19} + 876 \nu^{18} - 2974 \nu^{17} + 7978 \nu^{16} - 13849 \nu^{15} + 22134 \nu^{14} + \cdots + 581632 ) / 24064 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{19} - 25 \nu^{18} + 130 \nu^{17} - 288 \nu^{16} + 401 \nu^{15} - 621 \nu^{14} + 1470 \nu^{13} + \cdots - 16384 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 159 \nu^{19} + 308 \nu^{18} - 4188 \nu^{17} + 11702 \nu^{16} - 16715 \nu^{15} + 22454 \nu^{14} + \cdots + 846592 ) / 12032 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9 \nu^{19} - 52 \nu^{18} + 238 \nu^{17} - 466 \nu^{16} + 625 \nu^{15} - 1070 \nu^{14} + \cdots - 22016 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11 \nu^{19} + 52 \nu^{18} - 234 \nu^{17} + 422 \nu^{16} - 531 \nu^{15} + 978 \nu^{14} - 2512 \nu^{13} + \cdots + 12800 ) / 512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 827 \nu^{19} - 2602 \nu^{18} + 5862 \nu^{17} - 11198 \nu^{16} + 20273 \nu^{15} - 38496 \nu^{14} + \cdots - 429056 ) / 24064 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 10 \nu^{19} - 5 \nu^{18} - 40 \nu^{17} + 106 \nu^{16} - 108 \nu^{15} + 97 \nu^{14} - 382 \nu^{13} + \cdots + 10240 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 318 \nu^{19} - 982 \nu^{18} + 1447 \nu^{17} - 1224 \nu^{16} + 2572 \nu^{15} - 8296 \nu^{14} + \cdots + 153088 ) / 6016 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 629 \nu^{19} - 2700 \nu^{18} + 6340 \nu^{17} - 12174 \nu^{16} + 21543 \nu^{15} - 40150 \nu^{14} + \cdots - 645376 ) / 12032 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1485 \nu^{19} + 5328 \nu^{18} - 11502 \nu^{17} + 16838 \nu^{16} - 27323 \nu^{15} + 60398 \nu^{14} + \cdots - 124416 ) / 24064 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1579 \nu^{19} - 3824 \nu^{18} + 2666 \nu^{17} + 5910 \nu^{16} - 5859 \nu^{15} - 13586 \nu^{14} + \cdots + 1808896 ) / 24064 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 395 \nu^{19} - 1675 \nu^{18} + 3350 \nu^{17} - 4876 \nu^{16} + 8303 \nu^{15} - 18703 \nu^{14} + \cdots - 52224 ) / 6016 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 931 \nu^{19} + 2175 \nu^{18} - 590 \nu^{17} - 4536 \nu^{16} + 3745 \nu^{15} + 6715 \nu^{14} + \cdots - 834048 ) / 12032 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2063 \nu^{19} + 5548 \nu^{18} - 8558 \nu^{17} + 6034 \nu^{16} - 12009 \nu^{15} + 43174 \nu^{14} + \cdots - 1522176 ) / 24064 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 2533 \nu^{19} - 8086 \nu^{18} + 12506 \nu^{17} - 10170 \nu^{16} + 19999 \nu^{15} - 66768 \nu^{14} + \cdots + 1425920 ) / 24064 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 2627 \nu^{19} - 8274 \nu^{18} + 12694 \nu^{17} - 10358 \nu^{16} + 21033 \nu^{15} - 69400 \nu^{14} + \cdots + 1329664 ) / 24064 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 4409 \nu^{19} + 12844 \nu^{18} - 19954 \nu^{17} + 14734 \nu^{16} - 27839 \nu^{15} + \cdots - 3192320 ) / 24064 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 2235 \nu^{19} + 7237 \nu^{18} - 10526 \nu^{17} + 8404 \nu^{16} - 17707 \nu^{15} + 59253 \nu^{14} + \cdots - 1031680 ) / 12032 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{17} - \beta_{16} - \beta_{12} - \beta_{11} + \beta_{5} - \beta_{3} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{17} - 2 \beta_{15} + \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - \beta_{10} + \cdots - \beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{18} - \beta_{16} + 2 \beta_{15} + 3 \beta_{14} + \beta_{13} + 2 \beta_{12} + 3 \beta_{7} + \cdots - \beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{19} + 2 \beta_{16} - \beta_{14} + \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + \beta_{7} + \cdots - 10 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{19} + \beta_{18} - \beta_{17} + 4 \beta_{16} - 2 \beta_{15} - 5 \beta_{14} + 3 \beta_{13} + \cdots + 8 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{17} + 6 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 8 \beta_{12} - 8 \beta_{11} + \cdots + 4 \beta_1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4 \beta_{19} + 3 \beta_{17} + 3 \beta_{16} - 4 \beta_{15} - 12 \beta_{13} - 13 \beta_{12} + 3 \beta_{11} + \cdots + 2 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 12 \beta_{19} - 10 \beta_{16} + 11 \beta_{14} - 11 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + \cdots + 28 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 11 \beta_{18} + 12 \beta_{17} - 13 \beta_{16} - 16 \beta_{15} + 13 \beta_{14} + \beta_{13} - 12 \beta_{12} + \cdots - 40 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 12 \beta_{18} - 36 \beta_{15} + 11 \beta_{14} + 11 \beta_{13} - 15 \beta_{12} + 15 \beta_{11} + \cdots - 24 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 12 \beta_{19} - \beta_{18} - 35 \beta_{17} + 26 \beta_{15} + 47 \beta_{14} + 5 \beta_{13} + 51 \beta_{12} + \cdots - 2 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 6 \beta_{19} + 40 \beta_{16} + 20 \beta_{14} - 20 \beta_{13} - 46 \beta_{12} - 46 \beta_{11} + 11 \beta_{10} + \cdots + 10 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 34 \beta_{19} - 2 \beta_{18} + 3 \beta_{17} + 17 \beta_{16} + 6 \beta_{15} - 10 \beta_{14} + \cdots + 220 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 32 \beta_{18} + 90 \beta_{17} + 2 \beta_{15} + 75 \beta_{14} + 75 \beta_{13} + 3 \beta_{12} + \cdots + 164 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 36 \beta_{19} + 5 \beta_{18} + 118 \beta_{17} + 3 \beta_{16} - 106 \beta_{15} + 69 \beta_{14} + \cdots + 104 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 26 \beta_{19} - 70 \beta_{16} + 79 \beta_{14} - 79 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + \cdots + 2 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 218 \beta_{19} + 101 \beta_{18} + 285 \beta_{17} + 50 \beta_{16} - 306 \beta_{15} - 49 \beta_{14} + \cdots - 388 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 40 \beta_{18} + 86 \beta_{17} - 190 \beta_{15} - 248 \beta_{14} - 248 \beta_{13} + 2 \beta_{12} + \cdots - 608 \beta_1 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 396 \beta_{19} - 10 \beta_{18} - 113 \beta_{17} + 273 \beta_{16} + 332 \beta_{15} + 490 \beta_{14} + \cdots - 974 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(511\) | \(545\) | \(613\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
0 | −1.65165 | − | 0.521599i | 0 | − | 0.575425i | 0 | 0.301863i | 0 | 2.45587 | + | 1.72299i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.2 | 0 | −1.65165 | + | 0.521599i | 0 | 0.575425i | 0 | − | 0.301863i | 0 | 2.45587 | − | 1.72299i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.3 | 0 | −1.39392 | − | 1.02809i | 0 | 3.10097i | 0 | 1.09021i | 0 | 0.886042 | + | 2.86617i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.4 | 0 | −1.39392 | + | 1.02809i | 0 | − | 3.10097i | 0 | − | 1.09021i | 0 | 0.886042 | − | 2.86617i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.5 | 0 | −1.15103 | − | 1.29427i | 0 | − | 2.52706i | 0 | 3.98394i | 0 | −0.350258 | + | 2.97948i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.6 | 0 | −1.15103 | + | 1.29427i | 0 | 2.52706i | 0 | − | 3.98394i | 0 | −0.350258 | − | 2.97948i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.7 | 0 | −0.781448 | − | 1.54575i | 0 | − | 1.94238i | 0 | − | 3.97018i | 0 | −1.77868 | + | 2.41584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.8 | 0 | −0.781448 | + | 1.54575i | 0 | 1.94238i | 0 | 3.97018i | 0 | −1.77868 | − | 2.41584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.9 | 0 | −0.627306 | − | 1.61446i | 0 | 4.11023i | 0 | − | 2.66199i | 0 | −2.21297 | + | 2.02552i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.10 | 0 | −0.627306 | + | 1.61446i | 0 | − | 4.11023i | 0 | 2.66199i | 0 | −2.21297 | − | 2.02552i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.11 | 0 | 0.627306 | − | 1.61446i | 0 | − | 4.11023i | 0 | − | 2.66199i | 0 | −2.21297 | − | 2.02552i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.12 | 0 | 0.627306 | + | 1.61446i | 0 | 4.11023i | 0 | 2.66199i | 0 | −2.21297 | + | 2.02552i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.13 | 0 | 0.781448 | − | 1.54575i | 0 | 1.94238i | 0 | − | 3.97018i | 0 | −1.77868 | − | 2.41584i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.14 | 0 | 0.781448 | + | 1.54575i | 0 | − | 1.94238i | 0 | 3.97018i | 0 | −1.77868 | + | 2.41584i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.15 | 0 | 1.15103 | − | 1.29427i | 0 | 2.52706i | 0 | 3.98394i | 0 | −0.350258 | − | 2.97948i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.16 | 0 | 1.15103 | + | 1.29427i | 0 | − | 2.52706i | 0 | − | 3.98394i | 0 | −0.350258 | + | 2.97948i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.17 | 0 | 1.39392 | − | 1.02809i | 0 | − | 3.10097i | 0 | 1.09021i | 0 | 0.886042 | − | 2.86617i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.18 | 0 | 1.39392 | + | 1.02809i | 0 | 3.10097i | 0 | − | 1.09021i | 0 | 0.886042 | + | 2.86617i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.19 | 0 | 1.65165 | − | 0.521599i | 0 | 0.575425i | 0 | 0.301863i | 0 | 2.45587 | − | 1.72299i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.20 | 0 | 1.65165 | + | 0.521599i | 0 | − | 0.575425i | 0 | − | 0.301863i | 0 | 2.45587 | + | 1.72299i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 816.2.e.c | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 816.2.e.c | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 816.2.e.c | ✓ | 20 |
| 12.b | even | 2 | 1 | inner | 816.2.e.c | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 816.2.e.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 816.2.e.c | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 816.2.e.c | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 816.2.e.c | ✓ | 20 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 37T_{5}^{8} + 468T_{5}^{6} + 2440T_{5}^{4} + 4672T_{5}^{2} + 1296 \)
acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 2 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( (T^{10} + 37 T^{8} + \cdots + 1296)^{2} \)
|
| $7$ |
\( (T^{10} + 40 T^{8} + \cdots + 192)^{2} \)
|
| $11$ |
\( (T^{10} - 67 T^{8} + \cdots - 432)^{2} \)
|
| $13$ |
\( (T^{5} - 3 T^{4} - 20 T^{3} + \cdots - 64)^{4} \)
|
| $17$ |
\( (T^{2} + 1)^{10} \)
|
| $19$ |
\( (T^{10} + 163 T^{8} + \cdots + 277248)^{2} \)
|
| $23$ |
\( (T^{10} - 95 T^{8} + \cdots - 3888)^{2} \)
|
| $29$ |
\( (T^{10} + 144 T^{8} + \cdots + 9585216)^{2} \)
|
| $31$ |
\( (T^{10} + 220 T^{8} + \cdots + 6912)^{2} \)
|
| $37$ |
\( (T^{5} - 80 T^{3} + \cdots - 248)^{4} \)
|
| $41$ |
\( (T^{10} + 81 T^{8} + \cdots + 20736)^{2} \)
|
| $43$ |
\( (T^{10} + 187 T^{8} + \cdots + 5038848)^{2} \)
|
| $47$ |
\( (T^{10} - 136 T^{8} + \cdots - 442368)^{2} \)
|
| $53$ |
\( (T^{10} + 376 T^{8} + \cdots + 107495424)^{2} \)
|
| $59$ |
\( (T^{10} - 172 T^{8} + \cdots - 248832)^{2} \)
|
| $61$ |
\( (T^{5} + 12 T^{4} + \cdots + 7048)^{4} \)
|
| $67$ |
\( (T^{10} + 340 T^{8} + \cdots + 85675008)^{2} \)
|
| $71$ |
\( (T^{10} - 472 T^{8} + \cdots - 3817152)^{2} \)
|
| $73$ |
\( (T^{5} - 16 T^{4} + \cdots - 1152)^{4} \)
|
| $79$ |
\( (T^{10} + 40 T^{8} + \cdots + 192)^{2} \)
|
| $83$ |
\( (T^{10} - 544 T^{8} + \cdots - 977190912)^{2} \)
|
| $89$ |
\( (T^{10} + 508 T^{8} + \cdots + 746496)^{2} \)
|
| $97$ |
\( (T^{5} - 8 T^{4} + \cdots - 23488)^{4} \)
|
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