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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 656x^{6} - 4002x^{5} + 415806x^{4} - 1312656x^{3} + 13535681x^{2} + 29074530x + 211120900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{12} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 656x^{6} - 4002x^{5} + 415806x^{4} - 1312656x^{3} + 13535681x^{2} + 29074530x + 211120900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 23536797184 \nu^{7} - 165138738920 \nu^{6} + 14918813246784 \nu^{5} - 198867552689163 \nu^{4} + \cdots + 63\!\cdots\!70 ) / 22\!\cdots\!70 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 1260823104009 \nu^{7} + 91808414090335 \nu^{6} + \cdots + 87\!\cdots\!00 ) / 15\!\cdots\!90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5722877147621 \nu^{7} + 307292194094355 \nu^{6} + 265826875762626 \nu^{5} + \cdots + 46\!\cdots\!40 ) / 15\!\cdots\!90 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23377649944 \nu^{7} + 12723630563 \nu^{6} + 14817937671669 \nu^{5} - 46778677537944 \nu^{4} + \cdots + 86\!\cdots\!76 ) / 19\!\cdots\!54 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4938956104 \nu^{7} - 15835102535 \nu^{6} - 3130560338079 \nu^{5} + 9882851164104 \nu^{4} + \cdots - 27\!\cdots\!08 ) / 21\!\cdots\!06 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 300936617039716 \nu^{7} + \cdots - 16\!\cdots\!00 ) / 69\!\cdots\!05 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 108905371390689 \nu^{7} - 253317645659990 \nu^{6} + \cdots - 50\!\cdots\!15 ) / 77\!\cdots\!45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 6\beta_{5} - 2\beta_{2} - 1 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 81\beta_{6} + 31\beta_{5} + 81\beta_{4} + 14\beta_{3} - 116\beta_{2} - 35456\beta _1 + 13 ) / 108 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1072\beta_{7} + 6045\beta_{5} - 81\beta_{4} - 1323\beta_{3} + 6113\beta_{2} + 1323\beta _1 + 162415 ) / 108 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6265\beta_{7} - 26568\beta_{6} - 38246\beta_{5} - 328\beta_{3} + 12070\beta_{2} + 10840684\beta _1 - 10846949 ) / 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 162655 \beta_{7} + 215217 \beta_{6} + 916407 \beta_{5} + 215217 \beta_{4} + 702186 \beta_{3} + \cdots + 864841 ) / 108 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 1176852 \beta_{7} + 4774643 \beta_{5} - 3760263 \beta_{4} - 1186457 \beta_{3} + 5913075 \beta_{2} + \cdots + 1559220337 ) / 12 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 286592518 \beta_{7} - 123165279 \beta_{6} - 1617813114 \beta_{5} + 50870997 \beta_{3} + \cdots - 79007600251 ) / 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | −128.927 | + | 55.3251i | −128.000 | − | 221.703i | 897.758 | + | 1554.96i | −264.808 | + | 2229.06i | 5590.83 | − | 9683.61i | −4096.00 | 13561.3 | − | 14265.8i | 28728.3 | ||||||||||||||||||||||||||||
| 7.2 | 8.00000 | − | 13.8564i | −70.4011 | − | 121.354i | −128.000 | − | 221.703i | −306.282 | − | 530.497i | −2244.73 | + | 4.67750i | −369.801 | + | 640.513i | −4096.00 | −9770.38 | + | 17086.8i | −9801.04 | |||||||||||||||||||||||||||||
| 7.3 | 8.00000 | − | 13.8564i | 18.5508 | + | 139.064i | −128.000 | − | 221.703i | 53.8967 | + | 93.3519i | 2075.34 | + | 855.466i | −3803.97 | + | 6588.66i | −4096.00 | −18994.7 | + | 5159.51i | 1724.70 | |||||||||||||||||||||||||||||
| 7.4 | 8.00000 | − | 13.8564i | 140.277 | + | 2.30842i | −128.000 | − | 221.703i | −559.872 | − | 969.728i | 1154.20 | − | 1925.27i | 2150.43 | − | 3724.66i | −4096.00 | 19672.3 | + | 647.638i | −17915.9 | |||||||||||||||||||||||||||||
| 13.1 | 8.00000 | + | 13.8564i | −128.927 | − | 55.3251i | −128.000 | + | 221.703i | 897.758 | − | 1554.96i | −264.808 | − | 2229.06i | 5590.83 | + | 9683.61i | −4096.00 | 13561.3 | + | 14265.8i | 28728.3 | |||||||||||||||||||||||||||||
| 13.2 | 8.00000 | + | 13.8564i | −70.4011 | + | 121.354i | −128.000 | + | 221.703i | −306.282 | + | 530.497i | −2244.73 | − | 4.67750i | −369.801 | − | 640.513i | −4096.00 | −9770.38 | − | 17086.8i | −9801.04 | |||||||||||||||||||||||||||||
| 13.3 | 8.00000 | + | 13.8564i | 18.5508 | − | 139.064i | −128.000 | + | 221.703i | 53.8967 | − | 93.3519i | 2075.34 | − | 855.466i | −3803.97 | − | 6588.66i | −4096.00 | −18994.7 | − | 5159.51i | 1724.70 | |||||||||||||||||||||||||||||
| 13.4 | 8.00000 | + | 13.8564i | 140.277 | − | 2.30842i | −128.000 | + | 221.703i | −559.872 | + | 969.728i | 1154.20 | + | 1925.27i | 2150.43 | + | 3724.66i | −4096.00 | 19672.3 | − | 647.638i | −17915.9 | |||||||||||||||||||||||||||||
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.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 54.10.c.a | 8 | ||
| 9.c | even | 3 | 1 | inner | 18.10.c.a | ✓ | 8 |
| 9.c | even | 3 | 1 | 162.10.a.e | 4 | ||
| 9.d | odd | 6 | 1 | 54.10.c.a | 8 | ||
| 9.d | odd | 6 | 1 | 162.10.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.10.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 18.10.c.a | ✓ | 8 | 9.c | even | 3 | 1 | inner |
| 54.10.c.a | 8 | 3.b | odd | 2 | 1 | ||
| 54.10.c.a | 8 | 9.d | odd | 6 | 1 | ||
| 162.10.a.e | 4 | 9.c | even | 3 | 1 | ||
| 162.10.a.h | 4 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 171 T_{5}^{7} + 2446902 T_{5}^{6} + 2353883409 T_{5}^{5} + 5546419468182 T_{5}^{4} + \cdots + 17\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 15\!\cdots\!21 \)
|
| $5$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 73\!\cdots\!76 \)
|
| $11$ |
\( T^{8} + \cdots + 50\!\cdots\!25 \)
|
| $13$ |
\( T^{8} + \cdots + 52\!\cdots\!24 \)
|
| $17$ |
\( (T^{4} + \cdots + 74\!\cdots\!76)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots + 41\!\cdots\!40)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 39\!\cdots\!56 \)
|
| $31$ |
\( T^{8} + \cdots + 93\!\cdots\!00 \)
|
| $37$ |
\( (T^{4} + \cdots + 26\!\cdots\!88)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 48\!\cdots\!09 \)
|
| $43$ |
\( T^{8} + \cdots + 34\!\cdots\!01 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $53$ |
\( (T^{4} + \cdots - 33\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 90\!\cdots\!49 \)
|
| $61$ |
\( T^{8} + \cdots + 84\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!25 \)
|
| $71$ |
\( (T^{4} + \cdots + 29\!\cdots\!36)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots - 16\!\cdots\!52)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 33\!\cdots\!24 \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots - 30\!\cdots\!20)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 20\!\cdots\!41 \)
|
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