Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,10,Mod(37,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (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: | \(64.8945153566\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 373x^{4} - 756x^{3} + 139129x^{2} - 140994x + 142884 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 5^{2}\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 373x^{4} - 756x^{3} + 139129x^{2} - 140994x + 142884 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 373\nu^{3} - 378\nu^{2} + 139129\nu - 140994 ) / 140994 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 49\nu^{4} + 266\nu^{3} + 18277\nu^{2} - 18522\nu + 4443711 ) / 1865 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 373\nu^{5} - 378\nu^{4} + 144799\nu^{3} - 493479\nu^{2} + 58946707\nu - 89811315 ) / 50355 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8705\nu^{5} - 7938\nu^{4} - 3067037\nu^{3} + 5264406\nu^{2} - 1237726121\nu + 423069318 ) / 704970 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4061\nu^{5} + 1481237\nu^{3} - 3837960\nu^{2} + 552501401\nu - 559907586 ) / 234990 \)
|
| \(\nu\) | \(=\) |
\( ( -8\beta_{5} - 7\beta_{4} + 7\beta_{3} - \beta_{2} - 5\beta _1 + 2 ) / 1680 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -83\beta_{5} - 19\beta_{4} - 38\beta_{3} - 19\beta_{2} + 208846\beta _1 - 19 ) / 840 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2611\beta_{5} + 5222\beta_{4} + 2611\beta_{3} + 2984\beta_{2} + 2611\beta _1 + 636905 ) / 1680 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2236\beta_{5} - 841\beta_{4} + 841\beta_{3} + 3077\beta_{2} - 7790759\beta _1 - 7789918 ) / 84 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 76381\beta_{5} - 988267\beta_{4} - 1976534\beta_{3} - 988267\beta_{2} + 394479238\beta _1 - 988267 ) / 1680 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−8.00000 | + | 13.8564i | 0 | −128.000 | − | 221.703i | −1070.48 | + | 1854.13i | 0 | 1163.76 | − | 6244.94i | 4096.00 | 0 | −17127.7 | − | 29666.1i | ||||||||||||||||||||||||||
| 37.2 | −8.00000 | + | 13.8564i | 0 | −128.000 | − | 221.703i | −236.226 | + | 409.156i | 0 | −844.647 | + | 6296.04i | 4096.00 | 0 | −3779.62 | − | 6546.49i | |||||||||||||||||||||||||||
| 37.3 | −8.00000 | + | 13.8564i | 0 | −128.000 | − | 221.703i | 1126.21 | − | 1950.65i | 0 | 5935.39 | − | 2263.80i | 4096.00 | 0 | 18019.3 | + | 31210.4i | |||||||||||||||||||||||||||
| 109.1 | −8.00000 | − | 13.8564i | 0 | −128.000 | + | 221.703i | −1070.48 | − | 1854.13i | 0 | 1163.76 | + | 6244.94i | 4096.00 | 0 | −17127.7 | + | 29666.1i | |||||||||||||||||||||||||||
| 109.2 | −8.00000 | − | 13.8564i | 0 | −128.000 | + | 221.703i | −236.226 | − | 409.156i | 0 | −844.647 | − | 6296.04i | 4096.00 | 0 | −3779.62 | + | 6546.49i | |||||||||||||||||||||||||||
| 109.3 | −8.00000 | − | 13.8564i | 0 | −128.000 | + | 221.703i | 1126.21 | + | 1950.65i | 0 | 5935.39 | + | 2263.80i | 4096.00 | 0 | 18019.3 | − | 31210.4i | |||||||||||||||||||||||||||
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 | 126.10.g.d | 6 | |
| 3.b | odd | 2 | 1 | 42.10.e.d | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 126.10.g.d | 6 | |
| 21.g | even | 6 | 1 | 294.10.a.u | 3 | ||
| 21.h | odd | 6 | 1 | 42.10.e.d | ✓ | 6 | |
| 21.h | odd | 6 | 1 | 294.10.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.10.e.d | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 42.10.e.d | ✓ | 6 | 21.h | odd | 6 | 1 | |
| 126.10.g.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 126.10.g.d | 6 | 7.c | even | 3 | 1 | inner | |
| 294.10.a.r | 3 | 21.h | odd | 6 | 1 | ||
| 294.10.a.u | 3 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 361 T_{5}^{5} + 5005321 T_{5}^{4} + 2796780000 T_{5}^{3} + 24588101227500 T_{5}^{2} + \cdots + 51\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 256)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 51\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 65\!\cdots\!43 \)
|
| $11$ |
\( T^{6} + \cdots + 68\!\cdots\!96 \)
|
| $13$ |
\( (T^{3} + \cdots + 54191577720804)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!44 \)
|
| $23$ |
\( T^{6} + \cdots + 59\!\cdots\!04 \)
|
| $29$ |
\( (T^{3} + \cdots + 80\!\cdots\!72)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 15\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 23\!\cdots\!16 \)
|
| $41$ |
\( (T^{3} + \cdots - 93\!\cdots\!24)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots - 24\!\cdots\!42)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 28\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots + 93\!\cdots\!16 \)
|
| $61$ |
\( T^{6} + \cdots + 78\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + \cdots - 12\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 48\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots + 30\!\cdots\!09 \)
|
| $83$ |
\( (T^{3} + \cdots + 15\!\cdots\!28)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 73\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 17\!\cdots\!04)^{2} \)
|
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