Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,10,Mod(7,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.c (of order \(3\), 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: | \(9.27064505095\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 2898 x^{8} - 21854 x^{7} + 6449040 x^{6} - 50448078 x^{5} + 5768656201 x^{4} + \cdots + 352749700583424 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{16}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 2898 x^{8} - 21854 x^{7} + 6449040 x^{6} - 50448078 x^{5} + 5768656201 x^{4} + \cdots + 352749700583424 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 83\!\cdots\!17 \nu^{9} + \cdots + 29\!\cdots\!48 ) / 30\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!39 \nu^{9} + \cdots + 46\!\cdots\!72 ) / 40\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 70\!\cdots\!27 \nu^{9} + \cdots + 13\!\cdots\!24 ) / 36\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 31\!\cdots\!39 \nu^{9} + \cdots - 17\!\cdots\!64 ) / 92\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 78\!\cdots\!81 \nu^{9} + \cdots + 54\!\cdots\!84 ) / 18\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 82\!\cdots\!75 \nu^{9} + \cdots - 27\!\cdots\!56 ) / 23\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 25\!\cdots\!75 \nu^{9} + \cdots + 13\!\cdots\!24 ) / 58\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 18\!\cdots\!51 \nu^{9} + \cdots - 38\!\cdots\!32 ) / 18\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 43\!\cdots\!23 \nu^{9} + \cdots + 15\!\cdots\!84 ) / 12\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} + 2\beta_{8} - 10\beta_{7} - 30\beta_{4} - 8\beta_{3} - 3\beta _1 - 1 ) / 540 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{9} - 40 \beta_{8} + 20 \beta_{7} - 2 \beta_{6} + 29 \beta_{5} - 914 \beta_{4} - 944 \beta_{3} + \cdots + 18 ) / 270 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 96 \beta_{8} + 347 \beta_{6} - 326 \beta_{5} - 1702 \beta_{4} + 2658 \beta_{3} + 3308 \beta_{2} + \cdots + 706505 ) / 108 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 16829 \beta_{9} + 293042 \beta_{8} - 116830 \beta_{7} + 187920 \beta_{5} + 147390 \beta_{4} + \cdots - 972746731 ) / 540 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1561187 \beta_{9} - 2103430 \beta_{8} + 14438180 \beta_{7} - 1561187 \beta_{6} - 936961 \beta_{5} + \cdots - 3977352 ) / 270 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 31852428 \beta_{8} - 11278237 \beta_{6} - 112094246 \beta_{5} + 1315504250 \beta_{4} + \cdots + 327326266877 ) / 108 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 5407476301 \beta_{9} + 4193595722 \beta_{8} - 51966546850 \beta_{7} + 10500582960 \beta_{5} + \cdots - 12082813764301 ) / 540 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 64541879098 \beta_{9} - 410374471180 \beta_{8} + 255913091120 \beta_{7} + 64541879098 \beta_{6} + \cdots - 183245360922 ) / 270 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2971759649088 \beta_{8} + 1856822491319 \beta_{6} - 1645151435318 \beta_{5} - 47506203919774 \beta_{4} + \cdots + 42\!\cdots\!37 ) / 108 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−8.00000 | + | 13.8564i | −134.603 | − | 39.5618i | −128.000 | − | 221.703i | −344.989 | − | 597.539i | 1625.01 | − | 1548.61i | −131.308 | + | 227.432i | 4096.00 | 16552.7 | + | 10650.3i | 11039.7 | ||||||||||||||||||||||||||||||||||
| 7.2 | −8.00000 | + | 13.8564i | −59.7337 | + | 126.944i | −128.000 | − | 221.703i | 1263.22 | + | 2187.96i | −1281.12 | − | 1843.25i | −3518.90 | + | 6094.92i | 4096.00 | −12546.8 | − | 15165.7i | −40423.0 | |||||||||||||||||||||||||||||||||||
| 7.3 | −8.00000 | + | 13.8564i | 52.6359 | + | 130.048i | −128.000 | − | 221.703i | −1087.16 | − | 1883.02i | −2223.08 | − | 311.039i | 711.591 | − | 1232.51i | 4096.00 | −14141.9 | + | 13690.4i | 34789.2 | |||||||||||||||||||||||||||||||||||
| 7.4 | −8.00000 | + | 13.8564i | 80.0855 | − | 115.192i | −128.000 | − | 221.703i | −490.401 | − | 849.399i | 955.470 | + | 2031.24i | −4936.51 | + | 8550.28i | 4096.00 | −6855.62 | − | 18450.5i | 15692.8 | |||||||||||||||||||||||||||||||||||
| 7.5 | −8.00000 | + | 13.8564i | 139.615 | + | 13.8094i | −128.000 | − | 221.703i | 744.835 | + | 1290.09i | −1308.27 | + | 1824.08i | 4649.63 | − | 8053.39i | 4096.00 | 19301.6 | + | 3855.99i | −23834.7 | |||||||||||||||||||||||||||||||||||
| 13.1 | −8.00000 | − | 13.8564i | −134.603 | + | 39.5618i | −128.000 | + | 221.703i | −344.989 | + | 597.539i | 1625.01 | + | 1548.61i | −131.308 | − | 227.432i | 4096.00 | 16552.7 | − | 10650.3i | 11039.7 | |||||||||||||||||||||||||||||||||||
| 13.2 | −8.00000 | − | 13.8564i | −59.7337 | − | 126.944i | −128.000 | + | 221.703i | 1263.22 | − | 2187.96i | −1281.12 | + | 1843.25i | −3518.90 | − | 6094.92i | 4096.00 | −12546.8 | + | 15165.7i | −40423.0 | |||||||||||||||||||||||||||||||||||
| 13.3 | −8.00000 | − | 13.8564i | 52.6359 | − | 130.048i | −128.000 | + | 221.703i | −1087.16 | + | 1883.02i | −2223.08 | + | 311.039i | 711.591 | + | 1232.51i | 4096.00 | −14141.9 | − | 13690.4i | 34789.2 | |||||||||||||||||||||||||||||||||||
| 13.4 | −8.00000 | − | 13.8564i | 80.0855 | + | 115.192i | −128.000 | + | 221.703i | −490.401 | + | 849.399i | 955.470 | − | 2031.24i | −4936.51 | − | 8550.28i | 4096.00 | −6855.62 | + | 18450.5i | 15692.8 | |||||||||||||||||||||||||||||||||||
| 13.5 | −8.00000 | − | 13.8564i | 139.615 | − | 13.8094i | −128.000 | + | 221.703i | 744.835 | − | 1290.09i | −1308.27 | − | 1824.08i | 4649.63 | + | 8053.39i | 4096.00 | 19301.6 | − | 3855.99i | −23834.7 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.10.c.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 54.10.c.b | 10 | ||
| 9.c | even | 3 | 1 | inner | 18.10.c.b | ✓ | 10 |
| 9.c | even | 3 | 1 | 162.10.a.j | 5 | ||
| 9.d | odd | 6 | 1 | 54.10.c.b | 10 | ||
| 9.d | odd | 6 | 1 | 162.10.a.i | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.10.c.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 18.10.c.b | ✓ | 10 | 9.c | even | 3 | 1 | inner |
| 54.10.c.b | 10 | 3.b | odd | 2 | 1 | ||
| 54.10.c.b | 10 | 9.d | odd | 6 | 1 | ||
| 162.10.a.i | 5 | 9.d | odd | 6 | 1 | ||
| 162.10.a.j | 5 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 171 T_{5}^{9} + 7398486 T_{5}^{8} + 3989961585 T_{5}^{7} + 43762573973430 T_{5}^{6} + \cdots + 30\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 256)^{5} \)
|
| $3$ |
\( T^{10} + \cdots + 29\!\cdots\!43 \)
|
| $5$ |
\( T^{10} + \cdots + 30\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 58\!\cdots\!96 \)
|
| $11$ |
\( T^{10} + \cdots + 41\!\cdots\!41 \)
|
| $13$ |
\( T^{10} + \cdots + 19\!\cdots\!84 \)
|
| $17$ |
\( (T^{5} + \cdots - 17\!\cdots\!40)^{2} \)
|
| $19$ |
\( (T^{5} + \cdots + 21\!\cdots\!04)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 74\!\cdots\!36 \)
|
| $29$ |
\( T^{10} + \cdots + 13\!\cdots\!24 \)
|
| $31$ |
\( T^{10} + \cdots + 55\!\cdots\!16 \)
|
| $37$ |
\( (T^{5} + \cdots + 80\!\cdots\!08)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 23\!\cdots\!89 \)
|
| $43$ |
\( T^{10} + \cdots + 25\!\cdots\!29 \)
|
| $47$ |
\( T^{10} + \cdots + 60\!\cdots\!24 \)
|
| $53$ |
\( (T^{5} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 70\!\cdots\!61 \)
|
| $61$ |
\( T^{10} + \cdots + 14\!\cdots\!44 \)
|
| $67$ |
\( T^{10} + \cdots + 56\!\cdots\!25 \)
|
| $71$ |
\( (T^{5} + \cdots - 62\!\cdots\!56)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots - 36\!\cdots\!96)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 91\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 31\!\cdots\!76 \)
|
| $89$ |
\( (T^{5} + \cdots - 16\!\cdots\!84)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 19\!\cdots\!25 \)
|
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