Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,17,Mod(55,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 17, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.55");
S:= CuspForms(chi, 17);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 17 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.d (of order \(2\), degree \(1\), 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: | \(102.264462630\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 144842045980 x^{6} - 11543882633160 x^{5} + \cdots + 75\!\cdots\!16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3^{8}\cdot 7^{6} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 144842045980 x^{6} - 11543882633160 x^{5} + \cdots + 75\!\cdots\!16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11\!\cdots\!30 \nu^{7} + \cdots - 52\!\cdots\!00 ) / 14\!\cdots\!01 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\!\cdots\!30 \nu^{7} + \cdots - 52\!\cdots\!00 ) / 14\!\cdots\!01 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43\!\cdots\!70 \nu^{7} + \cdots + 75\!\cdots\!20 ) / 14\!\cdots\!01 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!12 \nu^{7} + \cdots - 58\!\cdots\!40 ) / 14\!\cdots\!01 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!18 \nu^{7} + \cdots + 12\!\cdots\!04 ) / 14\!\cdots\!35 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 43\!\cdots\!64 \nu^{7} + \cdots + 12\!\cdots\!86 ) / 43\!\cdots\!05 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 69\!\cdots\!76 \nu^{7} + \cdots + 65\!\cdots\!12 ) / 62\!\cdots\!15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10 \beta_{7} - 20 \beta_{6} - 110 \beta_{5} + 85935 \beta_{4} + 2231511 \beta_{3} - 148 \beta_{2} + \cdots - 144842045980 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 1338470160 \beta_{7} - 2910485250 \beta_{6} + 315576220145 \beta_{5} - 1482536299 \beta_{4} + \cdots + 17315823949740 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 202889946280 \beta_{7} + 1718706350180 \beta_{6} - 88244111482740 \beta_{5} + \cdots + 10\!\cdots\!04 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 23\!\cdots\!00 \beta_{7} + \cdots - 48\!\cdots\!00 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 93\!\cdots\!10 \beta_{7} + \cdots - 47\!\cdots\!60 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 10\!\cdots\!90 \beta_{7} + \cdots + 33\!\cdots\!80 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−270.026 | 0 | 7378.00 | − | 234861.i | 0 | −4.98661e6 | − | 2.89252e6i | 1.57042e7 | 0 | 6.34186e7i | |||||||||||||||||||||||||||||||||||||||
| 55.2 | −270.026 | 0 | 7378.00 | 234861.i | 0 | −4.98661e6 | + | 2.89252e6i | 1.57042e7 | 0 | − | 6.34186e7i | ||||||||||||||||||||||||||||||||||||||||
| 55.3 | −40.3121 | 0 | −63910.9 | − | 623292.i | 0 | 5.20719e6 | − | 2.47347e6i | 5.21828e6 | 0 | 2.51262e7i | ||||||||||||||||||||||||||||||||||||||||
| 55.4 | −40.3121 | 0 | −63910.9 | 623292.i | 0 | 5.20719e6 | + | 2.47347e6i | 5.21828e6 | 0 | − | 2.51262e7i | ||||||||||||||||||||||||||||||||||||||||
| 55.5 | 174.041 | 0 | −35245.9 | − | 367486.i | 0 | −2.61707e6 | + | 5.13652e6i | −1.75401e7 | 0 | − | 6.39575e7i | |||||||||||||||||||||||||||||||||||||||
| 55.6 | 174.041 | 0 | −35245.9 | 367486.i | 0 | −2.61707e6 | − | 5.13652e6i | −1.75401e7 | 0 | 6.39575e7i | |||||||||||||||||||||||||||||||||||||||||
| 55.7 | 408.297 | 0 | 101171. | − | 25884.6i | 0 | 879246. | + | 5.69736e6i | 1.45496e7 | 0 | − | 1.05686e7i | |||||||||||||||||||||||||||||||||||||||
| 55.8 | 408.297 | 0 | 101171. | 25884.6i | 0 | 879246. | − | 5.69736e6i | 1.45496e7 | 0 | 1.05686e7i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.17.d.c | 8 | |
| 3.b | odd | 2 | 1 | 7.17.b.b | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 63.17.d.c | 8 | |
| 12.b | even | 2 | 1 | 112.17.c.b | 8 | ||
| 21.c | even | 2 | 1 | 7.17.b.b | ✓ | 8 | |
| 84.h | odd | 2 | 1 | 112.17.c.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.17.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 7.17.b.b | ✓ | 8 | 21.c | even | 2 | 1 | |
| 63.17.d.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 63.17.d.c | 8 | 7.b | odd | 2 | 1 | inner | |
| 112.17.c.b | 8 | 12.b | even | 2 | 1 | ||
| 112.17.c.b | 8 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 272T_{2}^{3} - 98776T_{2}^{2} + 15713792T_{2} + 773514240 \)
acting on \(S_{17}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 272 T^{3} + \cdots + 773514240)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 12\!\cdots\!01 \)
|
| $11$ |
\( (T^{4} + \cdots - 13\!\cdots\!36)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
| $17$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 56\!\cdots\!00 \)
|
| $23$ |
\( (T^{4} + \cdots + 23\!\cdots\!60)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 96\!\cdots\!56)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 39\!\cdots\!00 \)
|
| $37$ |
\( (T^{4} + \cdots + 34\!\cdots\!60)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 41\!\cdots\!00 \)
|
| $43$ |
\( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 50\!\cdots\!00 \)
|
| $53$ |
\( (T^{4} + \cdots + 22\!\cdots\!80)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 38\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 73\!\cdots\!00 \)
|
| $67$ |
\( (T^{4} + \cdots + 43\!\cdots\!20)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 12\!\cdots\!16)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 32\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + \cdots + 87\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 69\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
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