Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,2,Mod(7,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.t (of order \(12\), degree \(4\), 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: | \(5.19027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4 \nu^{15} - 8 \nu^{14} + 145 \nu^{13} - 221 \nu^{12} + 2010 \nu^{11} - 2295 \nu^{10} + 13435 \nu^{9} + \cdots + 756 ) / 4968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 53 \nu^{15} - 138 \nu^{14} + 1490 \nu^{13} - 3657 \nu^{12} + 15489 \nu^{11} - 35397 \nu^{10} + \cdots - 9315 ) / 14904 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37\nu^{14} + 979\nu^{12} + 9381\nu^{10} + 41836\nu^{8} + 89761\nu^{6} + 83688\nu^{4} + 26658\nu^{2} + 4887 ) / 2484 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 89 \nu^{15} - 132 \nu^{14} + 2381 \nu^{13} - 3750 \nu^{12} + 23229 \nu^{11} - 39834 \nu^{10} + \cdots - 35964 ) / 14904 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 44 \nu^{15} - 37 \nu^{14} + 1250 \nu^{13} - 979 \nu^{12} + 13278 \nu^{11} - 9381 \nu^{10} + \cdots - 2403 ) / 4968 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 89 \nu^{15} + 132 \nu^{14} + 2381 \nu^{13} + 3750 \nu^{12} + 23229 \nu^{11} + 39834 \nu^{10} + \cdots + 35964 ) / 14904 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29 \nu^{15} - 81 \nu^{14} + 758 \nu^{13} - 2229 \nu^{12} + 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 61 \nu^{15} - 77 \nu^{14} + 1711 \nu^{13} - 2015 \nu^{12} + 17991 \nu^{11} - 18924 \nu^{10} + \cdots + 7587 ) / 4968 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 29 \nu^{15} - 81 \nu^{14} - 758 \nu^{13} - 2229 \nu^{12} - 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 73 \nu^{15} - 38 \nu^{14} - 1939 \nu^{13} - 998 \nu^{12} - 18639 \nu^{11} - 9504 \nu^{10} + \cdots - 3861 ) / 4968 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( -2\nu^{14} - 53\nu^{12} - 510\nu^{10} - 2303\nu^{8} - 5132\nu^{6} - 5331\nu^{4} - 2142\nu^{2} + 36\nu - 162 ) / 72 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 19 \nu^{15} - 23 \nu^{14} - 522 \nu^{13} - 598 \nu^{12} - 5327 \nu^{11} - 5543 \nu^{10} + \cdots + 2277 ) / 1656 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 77 \nu^{15} - 46 \nu^{14} - 2084 \nu^{13} - 1219 \nu^{12} - 20649 \nu^{11} - 11799 \nu^{10} + \cdots - 3105 ) / 4968 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 2\nu^{15} + 54\nu^{13} + 535\nu^{11} + 2522\nu^{9} + 5991\nu^{7} + 6892\nu^{5} + 3345\nu^{3} + 450\nu + 36 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 5 \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} + \cdots + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( 38 \beta_{15} - 24 \beta_{14} + 21 \beta_{13} + 15 \beta_{11} + 17 \beta_{10} - 21 \beta_{9} - 17 \beta_{8} + \cdots - 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 22 \beta_{14} - 27 \beta_{13} + 62 \beta_{12} - 40 \beta_{11} + 88 \beta_{10} - 27 \beta_{9} + \cdots - 259 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 524 \beta_{15} + 265 \beta_{14} - 293 \beta_{13} - 182 \beta_{11} - 222 \beta_{10} + 293 \beta_{9} + \cdots + 128 \)
|
| \(\nu^{8}\) | \(=\) |
\( 234 \beta_{14} + 316 \beta_{13} - 792 \beta_{12} + 546 \beta_{11} - 923 \beta_{10} + 316 \beta_{9} + \cdots + 2713 \)
|
| \(\nu^{9}\) | \(=\) |
\( 6506 \beta_{15} - 2942 \beta_{14} + 3637 \beta_{13} + 2101 \beta_{11} + 2695 \beta_{10} - 3637 \beta_{9} + \cdots - 1749 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2594 \beta_{14} - 3605 \beta_{13} + 9514 \beta_{12} - 6680 \beta_{11} + 10101 \beta_{10} + \cdots - 29935 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 77366 \beta_{15} + 33081 \beta_{14} - 43200 \beta_{13} - 23991 \beta_{11} - 31763 \beta_{10} + \cdots + 21658 \)
|
| \(\nu^{12}\) | \(=\) |
\( 29416 \beta_{14} + 41016 \beta_{13} - 111416 \beta_{12} + 78628 \beta_{11} - 113122 \beta_{10} + \cdots + 337525 \)
|
| \(\nu^{13}\) | \(=\) |
\( 902396 \beta_{15} - 375304 \beta_{14} + 503534 \beta_{13} + 273782 \beta_{11} + 369198 \beta_{10} + \cdots - 257408 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 336384 \beta_{14} - 467572 \beta_{13} + 1289976 \beta_{12} - 911652 \beta_{11} + 1281923 \beta_{10} + \cdots - 3840844 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 10431278 \beta_{15} + 4279454 \beta_{14} - 5818687 \beta_{13} - 3128719 \beta_{11} - 4262047 \beta_{10} + \cdots + 3002670 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-\beta_{5} - \beta_{7}\) | \(\beta_{7}\) |
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 |
|
0.866025 | + | 0.500000i | −0.531582 | − | 1.98389i | 0.500000 | + | 0.866025i | 0 | 0.531582 | − | 1.98389i | 1.54650 | + | 2.67861i | 1.00000i | −1.05516 | + | 0.609199i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | 0.866025 | + | 0.500000i | −0.0746104 | − | 0.278450i | 0.500000 | + | 0.866025i | 0 | 0.0746104 | − | 0.278450i | −2.39698 | − | 4.15170i | 1.00000i | 2.52611 | − | 1.45845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 0.866025 | + | 0.500000i | 0.272763 | + | 1.01796i | 0.500000 | + | 0.866025i | 0 | −0.272763 | + | 1.01796i | 0.303090 | + | 0.524968i | 1.00000i | 1.63622 | − | 0.944675i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 0.866025 | + | 0.500000i | 0.333429 | + | 1.24438i | 0.500000 | + | 0.866025i | 0 | −0.333429 | + | 1.24438i | 1.27944 | + | 2.21606i | 1.00000i | 1.16078 | − | 0.670177i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.1 | 0.866025 | − | 0.500000i | −0.531582 | + | 1.98389i | 0.500000 | − | 0.866025i | 0 | 0.531582 | + | 1.98389i | 1.54650 | − | 2.67861i | − | 1.00000i | −1.05516 | − | 0.609199i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.2 | 0.866025 | − | 0.500000i | −0.0746104 | + | 0.278450i | 0.500000 | − | 0.866025i | 0 | 0.0746104 | + | 0.278450i | −2.39698 | + | 4.15170i | − | 1.00000i | 2.52611 | + | 1.45845i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.3 | 0.866025 | − | 0.500000i | 0.272763 | − | 1.01796i | 0.500000 | − | 0.866025i | 0 | −0.272763 | − | 1.01796i | 0.303090 | − | 0.524968i | − | 1.00000i | 1.63622 | + | 0.944675i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.4 | 0.866025 | − | 0.500000i | 0.333429 | − | 1.24438i | 0.500000 | − | 0.866025i | 0 | −0.333429 | − | 1.24438i | 1.27944 | − | 2.21606i | − | 1.00000i | 1.16078 | + | 0.670177i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 557.1 | −0.866025 | + | 0.500000i | −2.30455 | − | 0.617503i | 0.500000 | − | 0.866025i | 0 | 2.30455 | − | 0.617503i | −1.46897 | + | 2.54433i | 1.00000i | 2.33157 | + | 1.34613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 557.2 | −0.866025 | + | 0.500000i | −1.71844 | − | 0.460454i | 0.500000 | − | 0.866025i | 0 | 1.71844 | − | 0.460454i | −0.386889 | + | 0.670111i | 1.00000i | 0.142931 | + | 0.0825211i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 557.3 | −0.866025 | + | 0.500000i | 0.752511 | + | 0.201635i | 0.500000 | − | 0.866025i | 0 | −0.752511 | + | 0.201635i | 1.21153 | − | 2.09844i | 1.00000i | −2.07246 | − | 1.19654i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 557.4 | −0.866025 | + | 0.500000i | 3.27048 | + | 0.876322i | 0.500000 | − | 0.866025i | 0 | −3.27048 | + | 0.876322i | −2.08773 | + | 3.61605i | 1.00000i | 7.33001 | + | 4.23198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 643.1 | −0.866025 | − | 0.500000i | −2.30455 | + | 0.617503i | 0.500000 | + | 0.866025i | 0 | 2.30455 | + | 0.617503i | −1.46897 | − | 2.54433i | − | 1.00000i | 2.33157 | − | 1.34613i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 643.2 | −0.866025 | − | 0.500000i | −1.71844 | + | 0.460454i | 0.500000 | + | 0.866025i | 0 | 1.71844 | + | 0.460454i | −0.386889 | − | 0.670111i | − | 1.00000i | 0.142931 | − | 0.0825211i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 643.3 | −0.866025 | − | 0.500000i | 0.752511 | − | 0.201635i | 0.500000 | + | 0.866025i | 0 | −0.752511 | − | 0.201635i | 1.21153 | + | 2.09844i | − | 1.00000i | −2.07246 | + | 1.19654i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 643.4 | −0.866025 | − | 0.500000i | 3.27048 | − | 0.876322i | 0.500000 | + | 0.866025i | 0 | −3.27048 | − | 0.876322i | −2.08773 | − | 3.61605i | − | 1.00000i | 7.33001 | − | 4.23198i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.t | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 650.2.t.f | ✓ | 16 |
| 5.b | even | 2 | 1 | 650.2.t.h | yes | 16 | |
| 5.c | odd | 4 | 1 | 650.2.w.f | yes | 16 | |
| 5.c | odd | 4 | 1 | 650.2.w.h | yes | 16 | |
| 13.f | odd | 12 | 1 | 650.2.w.h | yes | 16 | |
| 65.o | even | 12 | 1 | 650.2.t.h | yes | 16 | |
| 65.s | odd | 12 | 1 | 650.2.w.f | yes | 16 | |
| 65.t | even | 12 | 1 | inner | 650.2.t.f | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.2.t.f | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 650.2.t.f | ✓ | 16 | 65.t | even | 12 | 1 | inner |
| 650.2.t.h | yes | 16 | 5.b | even | 2 | 1 | |
| 650.2.t.h | yes | 16 | 65.o | even | 12 | 1 | |
| 650.2.w.f | yes | 16 | 5.c | odd | 4 | 1 | |
| 650.2.w.f | yes | 16 | 65.s | odd | 12 | 1 | |
| 650.2.w.h | yes | 16 | 5.c | odd | 4 | 1 | |
| 650.2.w.h | yes | 16 | 13.f | odd | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 12 T_{3}^{14} - 12 T_{3}^{13} + 23 T_{3}^{12} + 120 T_{3}^{11} + 372 T_{3}^{10} + 312 T_{3}^{9} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{16} - 12 T^{14} + \cdots + 81 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + 4 T^{15} + \cdots + 279841 \)
|
| $11$ |
\( T^{16} + 4 T^{15} + \cdots + 81 \)
|
| $13$ |
\( T^{16} + \cdots + 815730721 \)
|
| $17$ |
\( T^{16} - 8 T^{15} + \cdots + 9199089 \)
|
| $19$ |
\( T^{16} + 16 T^{15} + \cdots + 81 \)
|
| $23$ |
\( T^{16} - 4 T^{15} + \cdots + 11881809 \)
|
| $29$ |
\( T^{16} + \cdots + 7001170929 \)
|
| $31$ |
\( T^{16} + 8 T^{15} + \cdots + 20736 \)
|
| $37$ |
\( T^{16} - 20 T^{15} + \cdots + 187489 \)
|
| $41$ |
\( T^{16} + \cdots + 5749430625 \)
|
| $43$ |
\( T^{16} + \cdots + 45270647361 \)
|
| $47$ |
\( (T^{8} + 16 T^{7} + \cdots - 2124351)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 178024081041 \)
|
| $59$ |
\( T^{16} - 24 T^{15} + \cdots + 3470769 \)
|
| $61$ |
\( T^{16} + 20 T^{15} + \cdots + 42849 \)
|
| $67$ |
\( T^{16} + \cdots + 36051756129 \)
|
| $71$ |
\( T^{16} + \cdots + 10\!\cdots\!64 \)
|
| $73$ |
\( T^{16} + \cdots + 8100000000 \)
|
| $79$ |
\( T^{16} + \cdots + 211086032481 \)
|
| $83$ |
\( (T^{8} - 40 T^{7} + \cdots - 419175)^{2} \)
|
| $89$ |
\( T^{16} + \cdots + 709755295760289 \)
|
| $97$ |
\( T^{16} + \cdots + 45\!\cdots\!89 \)
|
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