Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,16,Mod(67,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not 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: | \(139.839634998\) |
| 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: |
\( x^{10} - x^{9} + 7979102 x^{8} - 3342530557 x^{7} + 48610066963550 x^{6} + \cdots + 51\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{4}\cdot 5^{4}\cdot 7^{8} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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
\( x^{10} - x^{9} + 7979102 x^{8} - 3342530557 x^{7} + 48610066963550 x^{6} + \cdots + 51\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 74\!\cdots\!37 \nu^{9} + \cdots + 78\!\cdots\!36 ) / 79\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\!\cdots\!43 \nu^{9} + \cdots + 72\!\cdots\!80 ) / 75\!\cdots\!96 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 20\!\cdots\!67 \nu^{9} + \cdots + 13\!\cdots\!76 ) / 26\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!47 \nu^{9} + \cdots + 11\!\cdots\!60 ) / 28\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 51\!\cdots\!71 \nu^{9} + \cdots + 34\!\cdots\!12 ) / 49\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!53 \nu^{9} + \cdots - 35\!\cdots\!76 ) / 31\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 98\!\cdots\!31 \nu^{9} + \cdots + 14\!\cdots\!20 ) / 22\!\cdots\!88 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!59 \nu^{9} + \cdots - 11\!\cdots\!96 ) / 10\!\cdots\!28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 24\!\cdots\!39 \nu^{9} + \cdots + 17\!\cdots\!92 ) / 39\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - \beta_{6} - 62\beta_{5} + 266\beta_{3} - 266\beta_{2} - 1189\beta _1 + 1189 ) / 5880 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 3093 \beta_{9} - 407 \beta_{8} + 3093 \beta_{7} - 407 \beta_{6} + 14946 \beta_{5} + \cdots - 18766843283 \beta_1 ) / 5880 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -893659\beta_{8} - 501091\beta_{7} + 33558626\beta_{4} + 213913526\beta_{2} + 840299797337 ) / 840 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13945090965 \beta_{9} + 2314811519 \beta_{6} - 97674552618 \beta_{5} + 8802557323630 \beta_{3} + \cdots - 78\!\cdots\!87 ) / 5880 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 18097746929989 \beta_{9} + 34176992980549 \beta_{8} + 18097746929989 \beta_{7} + \cdots - 36\!\cdots\!79 \beta_1 ) / 5880 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 259732276705309 \beta_{8} + \cdots + 10\!\cdots\!33 ) / 168 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 92\!\cdots\!77 \beta_{9} + \cdots - 19\!\cdots\!55 ) / 5880 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 30\!\cdots\!21 \beta_{9} + \cdots - 16\!\cdots\!75 \beta_1 ) / 5880 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 12\!\cdots\!47 \beta_{8} + \cdots + 14\!\cdots\!21 ) / 840 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
64.0000 | + | 110.851i | −3194.86 | + | 5533.66i | −8192.00 | + | 14189.0i | 99492.8 | + | 172327.i | −817884. | 0 | −2.09715e6 | −1.32398e7 | − | 2.29320e7i | −1.27351e7 | + | 2.20578e7i | ||||||||||||||||||||||||||||||||||||
| 67.2 | 64.0000 | + | 110.851i | −2378.65 | + | 4119.95i | −8192.00 | + | 14189.0i | −110581. | − | 191531.i | −608935. | 0 | −2.09715e6 | −4.14154e6 | − | 7.17335e6i | 1.41543e7 | − | 2.45160e7i | |||||||||||||||||||||||||||||||||||||
| 67.3 | 64.0000 | + | 110.851i | −350.525 | + | 607.127i | −8192.00 | + | 14189.0i | 6451.18 | + | 11173.8i | −89734.4 | 0 | −2.09715e6 | 6.92872e6 | + | 1.20009e7i | −825752. | + | 1.43024e6i | |||||||||||||||||||||||||||||||||||||
| 67.4 | 64.0000 | + | 110.851i | 1325.39 | − | 2295.64i | −8192.00 | + | 14189.0i | −56213.5 | − | 97364.6i | 339300. | 0 | −2.09715e6 | 3.66114e6 | + | 6.34128e6i | 7.19532e6 | − | 1.24627e7i | |||||||||||||||||||||||||||||||||||||
| 67.5 | 64.0000 | + | 110.851i | 3401.15 | − | 5890.97i | −8192.00 | + | 14189.0i | 55365.7 | + | 95896.2i | 870695. | 0 | −2.09715e6 | −1.59612e7 | − | 2.76456e7i | −7.08681e6 | + | 1.22747e7i | |||||||||||||||||||||||||||||||||||||
| 79.1 | 64.0000 | − | 110.851i | −3194.86 | − | 5533.66i | −8192.00 | − | 14189.0i | 99492.8 | − | 172327.i | −817884. | 0 | −2.09715e6 | −1.32398e7 | + | 2.29320e7i | −1.27351e7 | − | 2.20578e7i | |||||||||||||||||||||||||||||||||||||
| 79.2 | 64.0000 | − | 110.851i | −2378.65 | − | 4119.95i | −8192.00 | − | 14189.0i | −110581. | + | 191531.i | −608935. | 0 | −2.09715e6 | −4.14154e6 | + | 7.17335e6i | 1.41543e7 | + | 2.45160e7i | |||||||||||||||||||||||||||||||||||||
| 79.3 | 64.0000 | − | 110.851i | −350.525 | − | 607.127i | −8192.00 | − | 14189.0i | 6451.18 | − | 11173.8i | −89734.4 | 0 | −2.09715e6 | 6.92872e6 | − | 1.20009e7i | −825752. | − | 1.43024e6i | |||||||||||||||||||||||||||||||||||||
| 79.4 | 64.0000 | − | 110.851i | 1325.39 | + | 2295.64i | −8192.00 | − | 14189.0i | −56213.5 | + | 97364.6i | 339300. | 0 | −2.09715e6 | 3.66114e6 | − | 6.34128e6i | 7.19532e6 | + | 1.24627e7i | |||||||||||||||||||||||||||||||||||||
| 79.5 | 64.0000 | − | 110.851i | 3401.15 | + | 5890.97i | −8192.00 | − | 14189.0i | 55365.7 | − | 95896.2i | 870695. | 0 | −2.09715e6 | −1.59612e7 | + | 2.76456e7i | −7.08681e6 | − | 1.22747e7i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.16.c.n | 10 | |
| 7.b | odd | 2 | 1 | 14.16.c.b | ✓ | 10 | |
| 7.c | even | 3 | 1 | 98.16.a.i | 5 | ||
| 7.c | even | 3 | 1 | inner | 98.16.c.n | 10 | |
| 7.d | odd | 6 | 1 | 14.16.c.b | ✓ | 10 | |
| 7.d | odd | 6 | 1 | 98.16.a.h | 5 | ||
| 21.c | even | 2 | 1 | 126.16.g.c | 10 | ||
| 21.g | even | 6 | 1 | 126.16.g.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.16.c.b | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 14.16.c.b | ✓ | 10 | 7.d | odd | 6 | 1 | |
| 98.16.a.h | 5 | 7.d | odd | 6 | 1 | ||
| 98.16.a.i | 5 | 7.c | even | 3 | 1 | ||
| 98.16.c.n | 10 | 1.a | even | 1 | 1 | trivial | |
| 98.16.c.n | 10 | 7.c | even | 3 | 1 | inner | |
| 126.16.g.c | 10 | 21.c | even | 2 | 1 | ||
| 126.16.g.c | 10 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 2395 T_{3}^{9} + 61492987 T_{3}^{8} + 119018122470 T_{3}^{7} + \cdots + 14\!\cdots\!25 \)
acting on \(S_{16}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 128 T + 16384)^{5} \)
|
| $3$ |
\( T^{10} + \cdots + 14\!\cdots\!25 \)
|
| $5$ |
\( T^{10} + \cdots + 49\!\cdots\!25 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 53\!\cdots\!25 \)
|
| $13$ |
\( (T^{5} + \cdots - 13\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 10\!\cdots\!01 \)
|
| $19$ |
\( T^{10} + \cdots + 21\!\cdots\!25 \)
|
| $23$ |
\( T^{10} + \cdots + 13\!\cdots\!01 \)
|
| $29$ |
\( (T^{5} + \cdots + 49\!\cdots\!36)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 19\!\cdots\!25 \)
|
| $37$ |
\( T^{10} + \cdots + 31\!\cdots\!09 \)
|
| $41$ |
\( (T^{5} + \cdots + 90\!\cdots\!28)^{2} \)
|
| $43$ |
\( (T^{5} + \cdots + 28\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 23\!\cdots\!61 \)
|
| $53$ |
\( T^{10} + \cdots + 14\!\cdots\!49 \)
|
| $59$ |
\( T^{10} + \cdots + 29\!\cdots\!25 \)
|
| $61$ |
\( T^{10} + \cdots + 34\!\cdots\!61 \)
|
| $67$ |
\( T^{10} + \cdots + 11\!\cdots\!81 \)
|
| $71$ |
\( (T^{5} + \cdots - 30\!\cdots\!76)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 13\!\cdots\!25 \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!25 \)
|
| $83$ |
\( (T^{5} + \cdots - 36\!\cdots\!32)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 19\!\cdots\!29 \)
|
| $97$ |
\( (T^{5} + \cdots - 22\!\cdots\!00)^{2} \)
|
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