Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [520,2,Mod(49,520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(520, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("520.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 520 = 2^{3} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 520.bz (of order \(6\), 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: | \(4.15222090511\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 38 x^{18} + 603 x^{16} + 5164 x^{14} + 25699 x^{12} + 74306 x^{10} + 117541 x^{8} + 89376 x^{6} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{20} + 38 x^{18} + 603 x^{16} + 5164 x^{14} + 25699 x^{12} + 74306 x^{10} + 117541 x^{8} + 89376 x^{6} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3 \nu^{18} + 133 \nu^{16} + 2390 \nu^{14} + 22490 \nu^{12} + 118435 \nu^{10} + 340505 \nu^{8} + \cdots + 1856 ) / 14336 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2 \nu^{19} + 11 \nu^{18} + 62 \nu^{17} + 333 \nu^{16} + 740 \nu^{15} + 3942 \nu^{14} + 3996 \nu^{13} + \cdots + 320 ) / 2048 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2 \nu^{19} + 11 \nu^{18} - 62 \nu^{17} + 333 \nu^{16} - 740 \nu^{15} + 3942 \nu^{14} - 3996 \nu^{13} + \cdots + 320 ) / 2048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9 \nu^{18} - 175 \nu^{16} - 226 \nu^{14} + 17874 \nu^{12} + 168183 \nu^{10} + 612341 \nu^{8} + \cdots - 41408 ) / 7168 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 29 \nu^{19} + 1099 \nu^{17} + 17354 \nu^{15} + 147366 \nu^{13} + 722781 \nu^{11} + 2036439 \nu^{9} + \cdots + 7168 ) / 14336 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 38 \nu^{19} - 95 \nu^{18} + 1274 \nu^{17} - 3129 \nu^{16} + 17580 \nu^{15} - 41934 \nu^{14} + \cdots + 22464 ) / 14336 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 38 \nu^{19} + 95 \nu^{18} + 1274 \nu^{17} + 3129 \nu^{16} + 17580 \nu^{15} + 41934 \nu^{14} + \cdots - 22464 ) / 14336 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 43 \nu^{19} + 114 \nu^{18} + 1421 \nu^{17} + 3934 \nu^{16} + 19174 \nu^{15} + 56100 \nu^{14} + \cdots + 135040 ) / 14336 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{19} - 12 \nu^{18} + 31 \nu^{17} - 404 \nu^{16} + 370 \nu^{15} - 5576 \nu^{14} + 1998 \nu^{13} + \cdots - 6400 ) / 1024 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{19} + 12 \nu^{18} + 31 \nu^{17} + 404 \nu^{16} + 370 \nu^{15} + 5576 \nu^{14} + 1998 \nu^{13} + \cdots + 6400 ) / 1024 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 9 \nu^{19} - 12 \nu^{18} + 315 \nu^{17} - 448 \nu^{16} + 4566 \nu^{15} - 6956 \nu^{14} + 35466 \nu^{13} + \cdots - 256 ) / 1792 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 125 \nu^{18} - 4235 \nu^{16} - 58890 \nu^{14} - 433382 \nu^{12} - 1813501 \nu^{10} - 4307927 \nu^{8} + \cdots - 46272 ) / 7168 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 114 \nu^{19} - 9 \nu^{18} + 4270 \nu^{17} - 175 \nu^{16} + 67076 \nu^{15} - 226 \nu^{14} + \cdots - 41408 ) / 14336 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 197 \nu^{19} + 9 \nu^{18} - 7203 \nu^{17} + 175 \nu^{16} - 109754 \nu^{15} + 226 \nu^{14} + \cdots + 70080 ) / 14336 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 212 \nu^{19} + 159 \nu^{18} - 7868 \nu^{17} + 5481 \nu^{16} - 121928 \nu^{15} + 77838 \nu^{14} + \cdots + 12352 ) / 14336 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 114 \nu^{19} - 41 \nu^{18} + 4214 \nu^{17} - 1631 \nu^{16} + 64836 \nu^{15} - 27026 \nu^{14} + \cdots - 44480 ) / 7168 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 114 \nu^{19} + 41 \nu^{18} + 4214 \nu^{17} + 1631 \nu^{16} + 64836 \nu^{15} + 27026 \nu^{14} + \cdots + 44480 ) / 7168 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 248 \nu^{19} - 229 \nu^{18} + 9576 \nu^{17} - 7539 \nu^{16} + 154416 \nu^{15} - 101050 \nu^{14} + \cdots - 65216 ) / 14336 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{16} + \beta_{15} + \beta_{11} - \beta_{6} + \beta_{5} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{4} - \beta_{3} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8 \beta_{16} - 8 \beta_{15} - \beta_{13} - 8 \beta_{11} - 2 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} + \cdots + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{18} - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - 10 \beta_{11} + \cdots + 40 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 61 \beta_{16} + 61 \beta_{15} + 11 \beta_{13} + \beta_{12} + 61 \beta_{11} - \beta_{9} + 25 \beta_{8} + \cdots - 171 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8 \beta_{19} + 10 \beta_{18} + 10 \beta_{17} + 32 \beta_{16} + 28 \beta_{15} + 24 \beta_{14} - 4 \beta_{13} + \cdots - 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2 \beta_{18} + 2 \beta_{17} + 468 \beta_{16} - 468 \beta_{15} - 91 \beta_{13} - 22 \beta_{12} + \cdots + 1231 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 164 \beta_{19} - 69 \beta_{18} - 69 \beta_{17} - 378 \beta_{16} - 294 \beta_{15} - 210 \beta_{14} + \cdots + 22 \)
|
| \(\nu^{10}\) | \(=\) |
\( 40 \beta_{18} - 40 \beta_{17} - 3637 \beta_{16} + 3637 \beta_{15} + 675 \beta_{13} + 309 \beta_{12} + \cdots - 9199 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2240 \beta_{19} + 366 \beta_{18} + 366 \beta_{17} + 3944 \beta_{16} + 2780 \beta_{15} + 1592 \beta_{14} + \cdots - 126 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 538 \beta_{18} + 538 \beta_{17} + 28628 \beta_{16} - 28628 \beta_{15} - 4731 \beta_{13} + \cdots + 70711 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 25692 \beta_{19} - 1133 \beta_{18} - 1133 \beta_{17} - 38442 \beta_{16} - 25038 \beta_{15} + \cdots - 14 \)
|
| \(\nu^{14}\) | \(=\) |
\( 6144 \beta_{18} - 6144 \beta_{17} - 227909 \beta_{16} + 227909 \beta_{15} + 31983 \beta_{13} + \cdots - 555379 \)
|
| \(\nu^{15}\) | \(=\) |
\( 267928 \beta_{19} - 5954 \beta_{18} - 5954 \beta_{17} + 359096 \beta_{16} + 219924 \beta_{15} + \cdots + 11774 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 64378 \beta_{18} + 64378 \beta_{17} + 1832020 \beta_{16} - 1832020 \beta_{15} - 210191 \beta_{13} + \cdots + 4434835 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 2635972 \beta_{19} + 166707 \beta_{18} + 166707 \beta_{17} - 3260730 \beta_{16} - 1905654 \beta_{15} + \cdots - 205842 \)
|
| \(\nu^{18}\) | \(=\) |
\( 640448 \beta_{18} - 640448 \beta_{17} - 14846597 \beta_{16} + 14846597 \beta_{15} + 1343779 \beta_{13} + \cdots - 35868607 \)
|
| \(\nu^{19}\) | \(=\) |
\( 24959968 \beta_{19} - 2258546 \beta_{18} - 2258546 \beta_{17} + 29035592 \beta_{16} + 16387228 \beta_{15} + \cdots + 2632946 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(261\) | \(391\) | \(417\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −2.15263 | + | 1.24282i | 0 | 1.11620 | − | 1.93755i | 0 | −0.605315 | + | 1.04844i | 0 | 1.58921 | − | 2.75260i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −2.14226 | + | 1.23683i | 0 | 0.0719752 | + | 2.23491i | 0 | 2.50730 | − | 4.34278i | 0 | 1.55952 | − | 2.70116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −1.90586 | + | 1.10035i | 0 | −1.55967 | + | 1.60232i | 0 | −2.10630 | + | 3.64823i | 0 | 0.921538 | − | 1.59615i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | −0.608617 | + | 0.351385i | 0 | 2.17567 | − | 0.516210i | 0 | 1.29083 | − | 2.23579i | 0 | −1.25306 | + | 2.17036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | −0.335258 | + | 0.193561i | 0 | −1.58463 | − | 1.57764i | 0 | −0.725987 | + | 1.25745i | 0 | −1.42507 | + | 2.46829i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | −0.140145 | + | 0.0809130i | 0 | −1.96139 | − | 1.07375i | 0 | 1.99111 | − | 3.44870i | 0 | −1.48691 | + | 2.57540i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 0.777287 | − | 0.448767i | 0 | −0.772469 | + | 2.09840i | 0 | 0.0340458 | − | 0.0589691i | 0 | −1.09722 | + | 1.90044i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 1.64972 | − | 0.952464i | 0 | 1.83918 | + | 1.27178i | 0 | −0.763696 | + | 1.32276i | 0 | 0.314374 | − | 0.544512i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 0 | 2.33549 | − | 1.34839i | 0 | 1.34188 | − | 1.78868i | 0 | 0.567133 | − | 0.982304i | 0 | 2.13633 | − | 3.70023i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 0 | 2.52228 | − | 1.45624i | 0 | −2.16675 | + | 0.552440i | 0 | 0.310874 | − | 0.538450i | 0 | 2.74128 | − | 4.74803i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.1 | 0 | −2.15263 | − | 1.24282i | 0 | 1.11620 | + | 1.93755i | 0 | −0.605315 | − | 1.04844i | 0 | 1.58921 | + | 2.75260i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.2 | 0 | −2.14226 | − | 1.23683i | 0 | 0.0719752 | − | 2.23491i | 0 | 2.50730 | + | 4.34278i | 0 | 1.55952 | + | 2.70116i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.3 | 0 | −1.90586 | − | 1.10035i | 0 | −1.55967 | − | 1.60232i | 0 | −2.10630 | − | 3.64823i | 0 | 0.921538 | + | 1.59615i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.4 | 0 | −0.608617 | − | 0.351385i | 0 | 2.17567 | + | 0.516210i | 0 | 1.29083 | + | 2.23579i | 0 | −1.25306 | − | 2.17036i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.5 | 0 | −0.335258 | − | 0.193561i | 0 | −1.58463 | + | 1.57764i | 0 | −0.725987 | − | 1.25745i | 0 | −1.42507 | − | 2.46829i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.6 | 0 | −0.140145 | − | 0.0809130i | 0 | −1.96139 | + | 1.07375i | 0 | 1.99111 | + | 3.44870i | 0 | −1.48691 | − | 2.57540i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.7 | 0 | 0.777287 | + | 0.448767i | 0 | −0.772469 | − | 2.09840i | 0 | 0.0340458 | + | 0.0589691i | 0 | −1.09722 | − | 1.90044i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.8 | 0 | 1.64972 | + | 0.952464i | 0 | 1.83918 | − | 1.27178i | 0 | −0.763696 | − | 1.32276i | 0 | 0.314374 | + | 0.544512i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.9 | 0 | 2.33549 | + | 1.34839i | 0 | 1.34188 | + | 1.78868i | 0 | 0.567133 | + | 0.982304i | 0 | 2.13633 | + | 3.70023i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 329.10 | 0 | 2.52228 | + | 1.45624i | 0 | −2.16675 | − | 0.552440i | 0 | 0.310874 | + | 0.538450i | 0 | 2.74128 | + | 4.74803i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 520.2.bz.a | ✓ | 20 |
| 4.b | odd | 2 | 1 | 1040.2.df.e | 20 | ||
| 5.b | even | 2 | 1 | 520.2.bz.b | yes | 20 | |
| 13.e | even | 6 | 1 | 520.2.bz.b | yes | 20 | |
| 20.d | odd | 2 | 1 | 1040.2.df.f | 20 | ||
| 52.i | odd | 6 | 1 | 1040.2.df.f | 20 | ||
| 65.l | even | 6 | 1 | inner | 520.2.bz.a | ✓ | 20 |
| 260.w | odd | 6 | 1 | 1040.2.df.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.bz.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 520.2.bz.a | ✓ | 20 | 65.l | even | 6 | 1 | inner |
| 520.2.bz.b | yes | 20 | 5.b | even | 2 | 1 | |
| 520.2.bz.b | yes | 20 | 13.e | even | 6 | 1 | |
| 1040.2.df.e | 20 | 4.b | odd | 2 | 1 | ||
| 1040.2.df.e | 20 | 260.w | odd | 6 | 1 | ||
| 1040.2.df.f | 20 | 20.d | odd | 2 | 1 | ||
| 1040.2.df.f | 20 | 52.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 19 T_{3}^{18} + 240 T_{3}^{16} + 48 T_{3}^{15} - 1733 T_{3}^{14} - 540 T_{3}^{13} + \cdots + 64 \)
acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} - 19 T^{18} + \cdots + 64 \)
|
| $5$ |
\( T^{20} + 3 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} - 5 T^{19} + \cdots + 784 \)
|
| $11$ |
\( T^{20} - 3 T^{19} + \cdots + 6210064 \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{20} - 15 T^{19} + \cdots + 515524 \)
|
| $19$ |
\( T^{20} - 3 T^{19} + \cdots + 85264 \)
|
| $23$ |
\( T^{20} + \cdots + 3152584904704 \)
|
| $29$ |
\( T^{20} + \cdots + 698588385856 \)
|
| $31$ |
\( T^{20} + \cdots + 1588971175936 \)
|
| $37$ |
\( T^{20} + \cdots + 7227026785969 \)
|
| $41$ |
\( T^{20} + \cdots + 167683698064 \)
|
| $43$ |
\( T^{20} + \cdots + 18872560193536 \)
|
| $47$ |
\( (T^{10} + 13 T^{9} + \cdots + 3162112)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 246397075456 \)
|
| $59$ |
\( T^{20} + \cdots + 53508873880576 \)
|
| $61$ |
\( T^{20} + \cdots + 11\!\cdots\!84 \)
|
| $67$ |
\( T^{20} + \cdots + 11\!\cdots\!56 \)
|
| $71$ |
\( T^{20} + \cdots + 140234100834304 \)
|
| $73$ |
\( (T^{10} - 23 T^{9} + \cdots + 4040704)^{2} \)
|
| $79$ |
\( (T^{10} + 14 T^{9} + \cdots + 2916352)^{2} \)
|
| $83$ |
\( (T^{10} + 16 T^{9} + \cdots + 604143616)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 16\!\cdots\!76 \)
|
| $97$ |
\( T^{20} + \cdots + 31013805552016 \)
|
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