Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [399,2,Mod(43,399)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(399, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("399.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 399 = 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 399.bo (of order \(9\), degree \(6\), 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: | \(3.18603104065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{9})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 50\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!19 ) / 27\!\cdots\!94 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 964152485141125 \nu^{17} + \cdots + 23\!\cdots\!56 ) / 45\!\cdots\!99 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\!\cdots\!03 \nu^{17} + \cdots + 65\!\cdots\!83 ) / 27\!\cdots\!94 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!07 \nu^{17} + \cdots - 98\!\cdots\!73 ) / 27\!\cdots\!94 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!63 \nu^{17} + \cdots - 12\!\cdots\!12 ) / 27\!\cdots\!94 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 51\!\cdots\!07 \nu^{17} + \cdots - 20\!\cdots\!08 ) / 27\!\cdots\!94 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 65\!\cdots\!83 \nu^{17} + \cdots + 31\!\cdots\!42 ) / 27\!\cdots\!94 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 69\!\cdots\!33 \nu^{17} + \cdots + 36\!\cdots\!53 ) / 27\!\cdots\!94 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 38\!\cdots\!93 \nu^{17} + \cdots + 28\!\cdots\!00 ) / 91\!\cdots\!98 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 23\!\cdots\!56 \nu^{17} + \cdots - 43\!\cdots\!34 ) / 45\!\cdots\!99 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 16\!\cdots\!56 \nu^{17} + \cdots - 18\!\cdots\!32 ) / 27\!\cdots\!94 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 17\!\cdots\!19 \nu^{17} + \cdots + 99\!\cdots\!80 ) / 27\!\cdots\!94 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 85\!\cdots\!37 \nu^{17} + \cdots + 71\!\cdots\!05 ) / 91\!\cdots\!98 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 47\!\cdots\!18 \nu^{17} + \cdots + 80\!\cdots\!89 ) / 47\!\cdots\!42 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 27\!\cdots\!72 \nu^{17} + \cdots - 16\!\cdots\!02 ) / 27\!\cdots\!94 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 27\!\cdots\!71 \nu^{17} + \cdots - 33\!\cdots\!00 ) / 27\!\cdots\!94 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{15} - \beta_{12} + 2\beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{9} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{17} + 6 \beta_{14} - 5 \beta_{13} - 9 \beta_{11} - \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + \cdots - 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} - 15 \beta_{16} - 6 \beta_{15} - 8 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 7 \beta_{9} + \cdots - 36 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{17} - 20 \beta_{16} - 34 \beta_{15} - 34 \beta_{14} + 25 \beta_{13} + 25 \beta_{12} + \cdots + 46 \)
|
| \(\nu^{7}\) | \(=\) |
\( 45 \beta_{17} - 36 \beta_{14} - 54 \beta_{13} - 61 \beta_{11} - 13 \beta_{10} + 45 \beta_{9} + 54 \beta_{8} + \cdots - 61 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 24 \beta_{17} + 162 \beta_{16} + 199 \beta_{15} + 22 \beta_{13} - 152 \beta_{12} + 248 \beta_{11} + \cdots + 127 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 403 \beta_{17} + 691 \beta_{16} + 233 \beta_{15} + 233 \beta_{14} + 343 \beta_{13} + 343 \beta_{12} + \cdots + 378 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 504 \beta_{17} + 1209 \beta_{14} - 873 \beta_{13} + 180 \beta_{12} - 1381 \beta_{11} + 206 \beta_{10} + \cdots - 1381 \)
|
| \(\nu^{11}\) | \(=\) |
\( 882 \beta_{17} - 4596 \beta_{16} - 1587 \beta_{15} + 28 \beta_{13} - 2149 \beta_{12} + 2247 \beta_{11} + \cdots - 8711 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5163 \beta_{17} - 8758 \beta_{16} - 7559 \beta_{15} - 7559 \beta_{14} + 3744 \beta_{13} + 3744 \beta_{12} + \cdots + 7859 \)
|
| \(\nu^{13}\) | \(=\) |
\( 12082 \beta_{17} - 11061 \beta_{14} - 13201 \beta_{13} + 246 \beta_{12} - 13048 \beta_{11} + \cdots - 13048 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 11326 \beta_{17} + 61866 \beta_{16} + 48222 \beta_{15} + 9220 \beta_{13} - 29552 \beta_{12} + \cdots + 65974 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 123308 \beta_{17} + 202564 \beta_{16} + 77600 \beta_{15} + 77600 \beta_{14} + 78734 \beta_{13} + \cdots + 74626 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 175850 \beta_{17} + 311931 \beta_{14} - 172759 \beta_{13} + 62490 \beta_{12} - 265344 \beta_{11} + \cdots - 265344 \)
|
| \(\nu^{17}\) | \(=\) |
\( 300884 \beta_{17} - 1347894 \beta_{16} - 543853 \beta_{15} + 9498 \beta_{13} - 487563 \beta_{12} + \cdots - 2318046 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(134\) | \(211\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{8} + \beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−2.16597 | + | 0.788347i | −0.173648 | + | 0.984808i | 2.53783 | − | 2.12949i | −0.626322 | − | 0.525547i | −0.400254 | − | 2.26995i | 0.500000 | + | 0.866025i | −1.51310 | + | 2.62076i | −0.939693 | − | 0.342020i | 1.77091 | + | 0.644557i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −0.0505900 | + | 0.0184132i | −0.173648 | + | 0.984808i | −1.52987 | + | 1.28371i | 1.59327 | + | 1.33691i | −0.00934865 | − | 0.0530188i | 0.500000 | + | 0.866025i | 0.107595 | − | 0.186361i | −0.939693 | − | 0.342020i | −0.105220 | − | 0.0382971i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | 1.71656 | − | 0.624775i | −0.173648 | + | 0.984808i | 1.02413 | − | 0.859347i | 1.38035 | + | 1.15825i | 0.317207 | + | 1.79897i | 0.500000 | + | 0.866025i | −0.605643 | + | 1.04900i | −0.939693 | − | 0.342020i | 3.09309 | + | 1.12579i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 85.1 | −0.451595 | − | 2.56112i | −0.766044 | + | 0.642788i | −4.47602 | + | 1.62914i | 1.34902 | + | 0.491004i | 1.99220 | + | 1.67165i | 0.500000 | − | 0.866025i | 3.59313 | + | 6.22348i | 0.173648 | − | 0.984808i | 0.648308 | − | 3.67674i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 85.2 | −0.124392 | − | 0.705462i | −0.766044 | + | 0.642788i | 1.39718 | − | 0.508532i | −0.780211 | − | 0.283974i | 0.548752 | + | 0.460458i | 0.500000 | − | 0.866025i | −1.24889 | − | 2.16315i | 0.173648 | − | 0.984808i | −0.103281 | + | 0.585733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 85.3 | 0.0759867 | + | 0.430942i | −0.766044 | + | 0.642788i | 1.69945 | − | 0.618549i | 2.96328 | + | 1.07855i | −0.335213 | − | 0.281277i | 0.500000 | − | 0.866025i | 0.833284 | + | 1.44329i | 0.173648 | − | 0.984808i | −0.239621 | + | 1.35896i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.1 | −0.451595 | + | 2.56112i | −0.766044 | − | 0.642788i | −4.47602 | − | 1.62914i | 1.34902 | − | 0.491004i | 1.99220 | − | 1.67165i | 0.500000 | + | 0.866025i | 3.59313 | − | 6.22348i | 0.173648 | + | 0.984808i | 0.648308 | + | 3.67674i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.2 | −0.124392 | + | 0.705462i | −0.766044 | − | 0.642788i | 1.39718 | + | 0.508532i | −0.780211 | + | 0.283974i | 0.548752 | − | 0.460458i | 0.500000 | + | 0.866025i | −1.24889 | + | 2.16315i | 0.173648 | + | 0.984808i | −0.103281 | − | 0.585733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.3 | 0.0759867 | − | 0.430942i | −0.766044 | − | 0.642788i | 1.69945 | + | 0.618549i | 2.96328 | − | 1.07855i | −0.335213 | + | 0.281277i | 0.500000 | + | 0.866025i | 0.833284 | − | 1.44329i | 0.173648 | + | 0.984808i | −0.239621 | − | 1.35896i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.1 | −2.16597 | − | 0.788347i | −0.173648 | − | 0.984808i | 2.53783 | + | 2.12949i | −0.626322 | + | 0.525547i | −0.400254 | + | 2.26995i | 0.500000 | − | 0.866025i | −1.51310 | − | 2.62076i | −0.939693 | + | 0.342020i | 1.77091 | − | 0.644557i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.2 | −0.0505900 | − | 0.0184132i | −0.173648 | − | 0.984808i | −1.52987 | − | 1.28371i | 1.59327 | − | 1.33691i | −0.00934865 | + | 0.0530188i | 0.500000 | − | 0.866025i | 0.107595 | + | 0.186361i | −0.939693 | + | 0.342020i | −0.105220 | + | 0.0382971i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 232.3 | 1.71656 | + | 0.624775i | −0.173648 | − | 0.984808i | 1.02413 | + | 0.859347i | 1.38035 | − | 1.15825i | 0.317207 | − | 1.79897i | 0.500000 | − | 0.866025i | −0.605643 | − | 1.04900i | −0.939693 | + | 0.342020i | 3.09309 | − | 1.12579i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.1 | −1.45949 | + | 1.22466i | 0.939693 | + | 0.342020i | 0.283031 | − | 1.60515i | 0.525672 | + | 2.98124i | −1.79033 | + | 0.651628i | 0.500000 | − | 0.866025i | −0.352551 | − | 0.610637i | 0.766044 | + | 0.642788i | −4.41822 | − | 3.70732i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.2 | −0.897543 | + | 0.753128i | 0.939693 | + | 0.342020i | −0.108915 | + | 0.617687i | −0.421473 | − | 2.39029i | −1.10100 | + | 0.400731i | 0.500000 | − | 0.866025i | −1.53910 | − | 2.66580i | 0.766044 | + | 0.642788i | 2.17849 | + | 1.82797i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 253.3 | 1.85704 | − | 1.55824i | 0.939693 | + | 0.342020i | 0.673180 | − | 3.81779i | 0.0164156 | + | 0.0930973i | 2.27799 | − | 0.829121i | 0.500000 | − | 0.866025i | −2.27472 | − | 3.93994i | 0.766044 | + | 0.642788i | 0.175552 | + | 0.147306i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 358.1 | −1.45949 | − | 1.22466i | 0.939693 | − | 0.342020i | 0.283031 | + | 1.60515i | 0.525672 | − | 2.98124i | −1.79033 | − | 0.651628i | 0.500000 | + | 0.866025i | −0.352551 | + | 0.610637i | 0.766044 | − | 0.642788i | −4.41822 | + | 3.70732i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 358.2 | −0.897543 | − | 0.753128i | 0.939693 | − | 0.342020i | −0.108915 | − | 0.617687i | −0.421473 | + | 2.39029i | −1.10100 | − | 0.400731i | 0.500000 | + | 0.866025i | −1.53910 | + | 2.66580i | 0.766044 | − | 0.642788i | 2.17849 | − | 1.82797i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 358.3 | 1.85704 | + | 1.55824i | 0.939693 | − | 0.342020i | 0.673180 | + | 3.81779i | 0.0164156 | − | 0.0930973i | 2.27799 | + | 0.829121i | 0.500000 | + | 0.866025i | −2.27472 | + | 3.93994i | 0.766044 | − | 0.642788i | 0.175552 | − | 0.147306i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 399.2.bo.b | ✓ | 18 |
| 19.e | even | 9 | 1 | inner | 399.2.bo.b | ✓ | 18 |
| 19.e | even | 9 | 1 | 7581.2.a.bg | 9 | ||
| 19.f | odd | 18 | 1 | 7581.2.a.bh | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.bo.b | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 399.2.bo.b | ✓ | 18 | 19.e | even | 9 | 1 | inner |
| 7581.2.a.bg | 9 | 19.e | even | 9 | 1 | ||
| 7581.2.a.bh | 9 | 19.f | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} + 3 T_{2}^{17} + 3 T_{2}^{16} - 2 T_{2}^{15} - 6 T_{2}^{14} + 54 T_{2}^{13} + 298 T_{2}^{12} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + 3 T^{17} + \cdots + 1 \)
|
| $3$ |
\( (T^{6} - T^{3} + 1)^{3} \)
|
| $5$ |
\( T^{18} - 12 T^{17} + \cdots + 64 \)
|
| $7$ |
\( (T^{2} - T + 1)^{9} \)
|
| $11$ |
\( T^{18} - 12 T^{17} + \cdots + 2809 \)
|
| $13$ |
\( T^{18} - 42 T^{16} + \cdots + 6713281 \)
|
| $17$ |
\( T^{18} + \cdots + 186622921 \)
|
| $19$ |
\( T^{18} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{18} + 18 T^{17} + \cdots + 14295961 \)
|
| $29$ |
\( T^{18} + \cdots + 184065882841 \)
|
| $31$ |
\( T^{18} + 27 T^{17} + \cdots + 90649441 \)
|
| $37$ |
\( (T^{9} - 108 T^{7} + \cdots - 937)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 8135859601 \)
|
| $43$ |
\( T^{18} + \cdots + 3524356664929 \)
|
| $47$ |
\( T^{18} + \cdots + 39689803729 \)
|
| $53$ |
\( T^{18} + \cdots + 1696029568489 \)
|
| $59$ |
\( T^{18} + \cdots + 14933084401 \)
|
| $61$ |
\( T^{18} + \cdots + 85\!\cdots\!41 \)
|
| $67$ |
\( T^{18} + \cdots + 10\!\cdots\!96 \)
|
| $71$ |
\( T^{18} + \cdots + 26484846273649 \)
|
| $73$ |
\( T^{18} + \cdots + 80577426579049 \)
|
| $79$ |
\( T^{18} + \cdots + 29\!\cdots\!44 \)
|
| $83$ |
\( T^{18} + \cdots + 17\!\cdots\!69 \)
|
| $89$ |
\( T^{18} + \cdots + 175843963220689 \)
|
| $97$ |
\( T^{18} + \cdots + 40179586755169 \)
|
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