Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,15,Mod(19,126)]
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, 5]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.19");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.n (of order \(6\), 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: | \(156.654499871\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{19} - 6266655317 x^{18} - 51228207045822 x^{17} + \cdots + 97\!\cdots\!27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{76}\cdot 3^{22}\cdot 7^{14} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 2 x^{19} - 6266655317 x^{18} - 51228207045822 x^{17} + \cdots + 97\!\cdots\!27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 12\!\cdots\!22 \nu^{19} + \cdots - 23\!\cdots\!29 ) / 54\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\!\cdots\!08 \nu^{19} + \cdots + 11\!\cdots\!23 ) / 20\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 33\!\cdots\!71 \nu^{19} + \cdots + 36\!\cdots\!84 ) / 15\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18\!\cdots\!52 \nu^{19} + \cdots - 22\!\cdots\!50 ) / 73\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 19\!\cdots\!65 \nu^{19} + \cdots - 44\!\cdots\!67 ) / 73\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\!\cdots\!67 \nu^{19} + \cdots + 44\!\cdots\!61 ) / 73\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 42\!\cdots\!55 \nu^{19} + \cdots - 48\!\cdots\!04 ) / 98\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 42\!\cdots\!29 \nu^{19} + \cdots - 48\!\cdots\!97 ) / 98\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 67\!\cdots\!21 \nu^{19} + \cdots + 26\!\cdots\!33 ) / 73\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 36\!\cdots\!93 \nu^{19} + \cdots - 71\!\cdots\!09 ) / 36\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 37\!\cdots\!27 \nu^{19} + \cdots - 20\!\cdots\!29 ) / 36\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 10\!\cdots\!87 \nu^{19} + \cdots + 50\!\cdots\!49 ) / 10\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 13\!\cdots\!51 \nu^{19} + \cdots - 28\!\cdots\!31 ) / 73\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 73\!\cdots\!72 \nu^{19} + \cdots + 20\!\cdots\!95 ) / 36\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 17\!\cdots\!65 \nu^{19} + \cdots - 35\!\cdots\!61 ) / 73\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 93\!\cdots\!52 \nu^{19} + \cdots + 40\!\cdots\!42 ) / 36\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 14\!\cdots\!65 \nu^{19} + \cdots + 11\!\cdots\!21 ) / 36\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 29\!\cdots\!90 \nu^{19} + \cdots + 43\!\cdots\!82 ) / 73\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 42\!\cdots\!95 \nu^{19} + \cdots + 24\!\cdots\!69 ) / 73\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} - \beta_{7} + 6\beta_{2} + 48\beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 116 \beta_{19} + 513 \beta_{18} - 111 \beta_{17} - 227 \beta_{16} + 170 \beta_{15} + \cdots + 3759958308 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 23438577 \beta_{19} + 26811599 \beta_{18} - 38400480 \beta_{17} - 14965683 \beta_{16} + \cdots + 276727548711918 ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 592126472751 \beta_{19} + 3155216884942 \beta_{18} - 1193340106599 \beta_{17} + \cdots + 14\!\cdots\!10 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17\!\cdots\!51 \beta_{19} + \cdots + 59\!\cdots\!92 ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 12\!\cdots\!43 \beta_{19} + \cdots + 21\!\cdots\!22 ) / 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 12\!\cdots\!25 \beta_{19} + \cdots + 58\!\cdots\!90 ) / 18 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 26\!\cdots\!25 \beta_{19} + \cdots + 35\!\cdots\!72 ) / 18 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 13\!\cdots\!47 \beta_{19} + \cdots + 22\!\cdots\!06 ) / 36 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 57\!\cdots\!19 \beta_{19} + \cdots + 59\!\cdots\!30 ) / 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 38\!\cdots\!27 \beta_{19} + \cdots + 41\!\cdots\!78 ) / 36 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 59\!\cdots\!91 \beta_{19} + \cdots + 52\!\cdots\!79 ) / 9 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 45\!\cdots\!74 \beta_{19} + \cdots + 38\!\cdots\!05 ) / 18 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 24\!\cdots\!94 \beta_{19} + \cdots + 18\!\cdots\!72 ) / 18 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 19\!\cdots\!19 \beta_{19} + \cdots + 14\!\cdots\!22 ) / 36 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 48\!\cdots\!87 \beta_{19} + \cdots + 33\!\cdots\!98 ) / 18 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 40\!\cdots\!77 \beta_{19} + \cdots + 27\!\cdots\!36 ) / 36 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 94\!\cdots\!81 \beta_{19} + \cdots + 61\!\cdots\!62 ) / 18 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 40\!\cdots\!55 \beta_{19} + \cdots + 25\!\cdots\!14 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−45.2548 | + | 78.3837i | 0 | −4096.00 | − | 7094.48i | −106192. | − | 61309.7i | 0 | −789690. | − | 233694.i | 741455. | 0 | 9.61136e6 | − | 5.54912e6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −45.2548 | + | 78.3837i | 0 | −4096.00 | − | 7094.48i | −38553.6 | − | 22259.0i | 0 | 725749. | − | 389245.i | 741455. | 0 | 3.48948e6 | − | 2.01465e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | −45.2548 | + | 78.3837i | 0 | −4096.00 | − | 7094.48i | −25206.6 | − | 14553.0i | 0 | 823165. | − | 24943.0i | 741455. | 0 | 2.28144e6 | − | 1.31719e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | −45.2548 | + | 78.3837i | 0 | −4096.00 | − | 7094.48i | 67507.4 | + | 38975.4i | 0 | −346065. | + | 747303.i | 741455. | 0 | −6.11007e6 | + | 3.52765e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | −45.2548 | + | 78.3837i | 0 | −4096.00 | − | 7094.48i | 112471. | + | 64935.4i | 0 | −629509. | − | 530982.i | 741455. | 0 | −1.01797e7 | + | 5.87728e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.6 | 45.2548 | − | 78.3837i | 0 | −4096.00 | − | 7094.48i | −81056.3 | − | 46797.9i | 0 | 587107. | − | 577519.i | −741455. | 0 | −7.33638e6 | + | 4.23566e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.7 | 45.2548 | − | 78.3837i | 0 | −4096.00 | − | 7094.48i | −45757.7 | − | 26418.2i | 0 | −602075. | + | 561898.i | −741455. | 0 | −4.14152e6 | + | 2.39111e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.8 | 45.2548 | − | 78.3837i | 0 | −4096.00 | − | 7094.48i | −10170.7 | − | 5872.03i | 0 | 552858. | + | 610386.i | −741455. | 0 | −920543. | + | 531476.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.9 | 45.2548 | − | 78.3837i | 0 | −4096.00 | − | 7094.48i | −5108.24 | − | 2949.25i | 0 | −382860. | − | 729137.i | −741455. | 0 | −462345. | + | 266935.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.10 | 45.2548 | − | 78.3837i | 0 | −4096.00 | − | 7094.48i | 130389. | + | 75280.2i | 0 | 789127. | − | 235587.i | −741455. | 0 | 1.18015e7 | − | 6.81358e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.1 | −45.2548 | − | 78.3837i | 0 | −4096.00 | + | 7094.48i | −106192. | + | 61309.7i | 0 | −789690. | + | 233694.i | 741455. | 0 | 9.61136e6 | + | 5.54912e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | −45.2548 | − | 78.3837i | 0 | −4096.00 | + | 7094.48i | −38553.6 | + | 22259.0i | 0 | 725749. | + | 389245.i | 741455. | 0 | 3.48948e6 | + | 2.01465e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | −45.2548 | − | 78.3837i | 0 | −4096.00 | + | 7094.48i | −25206.6 | + | 14553.0i | 0 | 823165. | + | 24943.0i | 741455. | 0 | 2.28144e6 | + | 1.31719e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | −45.2548 | − | 78.3837i | 0 | −4096.00 | + | 7094.48i | 67507.4 | − | 38975.4i | 0 | −346065. | − | 747303.i | 741455. | 0 | −6.11007e6 | − | 3.52765e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | −45.2548 | − | 78.3837i | 0 | −4096.00 | + | 7094.48i | 112471. | − | 64935.4i | 0 | −629509. | + | 530982.i | 741455. | 0 | −1.01797e7 | − | 5.87728e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 45.2548 | + | 78.3837i | 0 | −4096.00 | + | 7094.48i | −81056.3 | + | 46797.9i | 0 | 587107. | + | 577519.i | −741455. | 0 | −7.33638e6 | − | 4.23566e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 45.2548 | + | 78.3837i | 0 | −4096.00 | + | 7094.48i | −45757.7 | + | 26418.2i | 0 | −602075. | − | 561898.i | −741455. | 0 | −4.14152e6 | − | 2.39111e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 45.2548 | + | 78.3837i | 0 | −4096.00 | + | 7094.48i | −10170.7 | + | 5872.03i | 0 | 552858. | − | 610386.i | −741455. | 0 | −920543. | − | 531476.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.9 | 45.2548 | + | 78.3837i | 0 | −4096.00 | + | 7094.48i | −5108.24 | + | 2949.25i | 0 | −382860. | + | 729137.i | −741455. | 0 | −462345. | − | 266935.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.10 | 45.2548 | + | 78.3837i | 0 | −4096.00 | + | 7094.48i | 130389. | − | 75280.2i | 0 | 789127. | + | 235587.i | −741455. | 0 | 1.18015e7 | + | 6.81358e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.15.n.b | 20 | |
| 3.b | odd | 2 | 1 | 14.15.d.a | ✓ | 20 | |
| 7.d | odd | 6 | 1 | inner | 126.15.n.b | 20 | |
| 21.c | even | 2 | 1 | 98.15.d.b | 20 | ||
| 21.g | even | 6 | 1 | 14.15.d.a | ✓ | 20 | |
| 21.g | even | 6 | 1 | 98.15.b.c | 20 | ||
| 21.h | odd | 6 | 1 | 98.15.b.c | 20 | ||
| 21.h | odd | 6 | 1 | 98.15.d.b | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.15.d.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 14.15.d.a | ✓ | 20 | 21.g | even | 6 | 1 | |
| 98.15.b.c | 20 | 21.g | even | 6 | 1 | ||
| 98.15.b.c | 20 | 21.h | odd | 6 | 1 | ||
| 98.15.d.b | 20 | 21.c | even | 2 | 1 | ||
| 98.15.d.b | 20 | 21.h | odd | 6 | 1 | ||
| 126.15.n.b | 20 | 1.a | even | 1 | 1 | trivial | |
| 126.15.n.b | 20 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{20} + 3354 T_{5}^{19} - 37594494789 T_{5}^{18} - 126104512257594 T_{5}^{17} + \cdots + 68\!\cdots\!25 \)
acting on \(S_{15}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 8192 T^{2} + 67108864)^{5} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 68\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 20\!\cdots\!01 \)
|
| $11$ |
\( T^{20} + \cdots + 29\!\cdots\!89 \)
|
| $13$ |
\( T^{20} + \cdots + 19\!\cdots\!16 \)
|
| $17$ |
\( T^{20} + \cdots + 57\!\cdots\!21 \)
|
| $19$ |
\( T^{20} + \cdots + 34\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 33\!\cdots\!49 \)
|
| $29$ |
\( (T^{10} + \cdots - 12\!\cdots\!48)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 17\!\cdots\!29 \)
|
| $37$ |
\( T^{20} + \cdots + 65\!\cdots\!81 \)
|
| $41$ |
\( T^{20} + \cdots + 13\!\cdots\!24 \)
|
| $43$ |
\( (T^{10} + \cdots + 15\!\cdots\!28)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 68\!\cdots\!89 \)
|
| $53$ |
\( T^{20} + \cdots + 68\!\cdots\!69 \)
|
| $59$ |
\( T^{20} + \cdots + 22\!\cdots\!49 \)
|
| $61$ |
\( T^{20} + \cdots + 14\!\cdots\!29 \)
|
| $67$ |
\( T^{20} + \cdots + 13\!\cdots\!21 \)
|
| $71$ |
\( (T^{10} + \cdots + 14\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 14\!\cdots\!21 \)
|
| $79$ |
\( T^{20} + \cdots + 24\!\cdots\!25 \)
|
| $83$ |
\( T^{20} + \cdots + 21\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 70\!\cdots\!81 \)
|
| $97$ |
\( T^{20} + \cdots + 18\!\cdots\!56 \)
|
| show more | |
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