Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,2,Mod(49,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.z (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: | \(2.07611045255\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{14} - 2\nu^{12} + 63\nu^{10} - 11\nu^{8} - 289\nu^{6} + 69\nu^{4} + 1674\nu^{2} - 24057 ) / 15552 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{15} - 46\nu^{13} + 57\nu^{11} + 515\nu^{9} + 697\nu^{7} - 7677\nu^{5} + 23166\nu^{3} - 28431\nu ) / 69984 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} - 10\nu^{13} + 27\nu^{11} - 7\nu^{9} - 5\nu^{7} - 519\nu^{5} + 2874\nu^{3} - 4509\nu ) / 2592 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{15} + 7\nu^{13} - 21\nu^{11} + 22\nu^{9} + 26\nu^{7} + 198\nu^{5} - 1701\nu^{3} + 7290\nu ) / 2187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43 \nu^{15} - 15 \nu^{14} - 142 \nu^{13} + 438 \nu^{12} + 465 \nu^{11} - 1917 \nu^{10} + \cdots + 433755 ) / 93312 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 43 \nu^{15} + 15 \nu^{14} - 142 \nu^{13} - 438 \nu^{12} + 465 \nu^{11} + 1917 \nu^{10} + \cdots - 433755 ) / 93312 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 41 \nu^{15} - 79 \nu^{14} + 218 \nu^{13} + 22 \nu^{12} - 459 \nu^{11} + 195 \nu^{10} + \cdots - 9477 ) / 93312 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{15} + 8\nu^{13} - 78\nu^{11} - 52\nu^{9} + 238\nu^{7} + 882\nu^{5} - 4779\nu^{3} + 13122\nu ) / 8748 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 179 \nu^{15} - 237 \nu^{14} - 610 \nu^{13} + 66 \nu^{12} + 1911 \nu^{11} + 585 \nu^{10} + \cdots - 28431 ) / 279936 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 47 \nu^{15} - 229 \nu^{14} - 38 \nu^{13} + 946 \nu^{12} - 51 \nu^{11} - 1263 \nu^{10} + \cdots + 373977 ) / 93312 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 47 \nu^{15} + 229 \nu^{14} - 38 \nu^{13} - 946 \nu^{12} - 51 \nu^{11} + 1263 \nu^{10} + \cdots - 373977 ) / 93312 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 107 \nu^{15} + 399 \nu^{14} + 686 \nu^{13} - 1902 \nu^{12} - 1401 \nu^{11} + 3357 \nu^{10} + \cdots - 789507 ) / 139968 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 47 \nu^{15} + 357 \nu^{14} - 38 \nu^{13} - 1842 \nu^{12} - 51 \nu^{11} + 3951 \nu^{10} + \cdots - 840537 ) / 93312 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 107 \nu^{15} - 399 \nu^{14} + 686 \nu^{13} + 1902 \nu^{12} - 1401 \nu^{11} - 3357 \nu^{10} + \cdots + 789507 ) / 139968 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 47 \nu^{15} - 511 \nu^{14} + 38 \nu^{13} + 2038 \nu^{12} + 51 \nu^{11} - 4413 \nu^{10} + \cdots + 1040283 ) / 93312 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{13} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + \cdots - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{11} - 4\beta_{10} + 6\beta_{8} + 8\beta_{6} + 8\beta_{5} - 5\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4 \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - 8 \beta_{11} + 3 \beta_{10} - \beta_{9} + \cdots - 14 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12 \beta_{14} + 12 \beta_{12} + 12 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} + \cdots + 11 \beta_{2} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 10 \beta_{15} - 22 \beta_{14} - 17 \beta_{13} + 22 \beta_{12} + 22 \beta_{11} + 51 \beta_{10} + \cdots + 7 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 6 \beta_{14} - 6 \beta_{12} - \beta_{11} - \beta_{10} + 15 \beta_{9} - 11 \beta_{8} + \cdots + 51 \beta_{2} \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 56 \beta_{15} + 32 \beta_{14} + 47 \beta_{13} - 32 \beta_{12} - 30 \beta_{11} + 73 \beta_{10} + \cdots + 443 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 96 \beta_{14} + 96 \beta_{12} + 80 \beta_{11} + 80 \beta_{10} + 152 \beta_{9} - 312 \beta_{8} + \cdots - 85 \beta_{2} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 72 \beta_{15} + 140 \beta_{14} - 60 \beta_{13} - 140 \beta_{12} + 84 \beta_{11} - 112 \beta_{10} + \cdots + 445 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 216 \beta_{14} + 216 \beta_{12} - 120 \beta_{11} - 120 \beta_{10} - 272 \beta_{9} - 552 \beta_{8} + \cdots + 239 \beta_{2} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 136 \beta_{15} - 488 \beta_{14} - 417 \beta_{13} + 488 \beta_{12} - 376 \beta_{11} - 257 \beta_{10} + \cdots + 2473 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 228 \beta_{14} - 228 \beta_{12} + 179 \beta_{11} + 179 \beta_{10} - 413 \beta_{9} + \cdots - 1344 \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(1 + \beta_{1}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −2.86320 | + | 1.65307i | 0 | 0.877236 | + | 2.05681i | 0 | −0.517063 | + | 0.895580i | 0 | 3.96529 | − | 6.86809i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −1.69686 | + | 0.979681i | 0 | −2.16188 | − | 0.571200i | 0 | 1.42836 | − | 2.47400i | 0 | 0.419550 | − | 0.726682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −1.28064 | + | 0.739379i | 0 | −0.494086 | − | 2.18080i | 0 | −1.56631 | + | 2.71292i | 0 | −0.406637 | + | 0.704315i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | −0.180812 | + | 0.104392i | 0 | 1.60081 | + | 1.56122i | 0 | 1.72890 | − | 2.99455i | 0 | −1.47820 | + | 2.56033i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 0.180812 | − | 0.104392i | 0 | −1.60081 | + | 1.56122i | 0 | −1.72890 | + | 2.99455i | 0 | −1.47820 | + | 2.56033i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 1.28064 | − | 0.739379i | 0 | 0.494086 | − | 2.18080i | 0 | 1.56631 | − | 2.71292i | 0 | −0.406637 | + | 0.704315i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 1.69686 | − | 0.979681i | 0 | 2.16188 | − | 0.571200i | 0 | −1.42836 | + | 2.47400i | 0 | 0.419550 | − | 0.726682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 2.86320 | − | 1.65307i | 0 | −0.877236 | + | 2.05681i | 0 | 0.517063 | − | 0.895580i | 0 | 3.96529 | − | 6.86809i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.1 | 0 | −2.86320 | − | 1.65307i | 0 | 0.877236 | − | 2.05681i | 0 | −0.517063 | − | 0.895580i | 0 | 3.96529 | + | 6.86809i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.2 | 0 | −1.69686 | − | 0.979681i | 0 | −2.16188 | + | 0.571200i | 0 | 1.42836 | + | 2.47400i | 0 | 0.419550 | + | 0.726682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.3 | 0 | −1.28064 | − | 0.739379i | 0 | −0.494086 | + | 2.18080i | 0 | −1.56631 | − | 2.71292i | 0 | −0.406637 | − | 0.704315i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.4 | 0 | −0.180812 | − | 0.104392i | 0 | 1.60081 | − | 1.56122i | 0 | 1.72890 | + | 2.99455i | 0 | −1.47820 | − | 2.56033i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.5 | 0 | 0.180812 | + | 0.104392i | 0 | −1.60081 | − | 1.56122i | 0 | −1.72890 | − | 2.99455i | 0 | −1.47820 | − | 2.56033i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.6 | 0 | 1.28064 | + | 0.739379i | 0 | 0.494086 | + | 2.18080i | 0 | 1.56631 | + | 2.71292i | 0 | −0.406637 | − | 0.704315i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.7 | 0 | 1.69686 | + | 0.979681i | 0 | 2.16188 | + | 0.571200i | 0 | −1.42836 | − | 2.47400i | 0 | 0.419550 | + | 0.726682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 69.8 | 0 | 2.86320 | + | 1.65307i | 0 | −0.877236 | − | 2.05681i | 0 | 0.517063 | + | 0.895580i | 0 | 3.96529 | + | 6.86809i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 65.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 260.2.z.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 2340.2.cr.a | 16 | ||
| 4.b | odd | 2 | 1 | 1040.2.df.d | 16 | ||
| 5.b | even | 2 | 1 | inner | 260.2.z.a | ✓ | 16 |
| 5.c | odd | 4 | 2 | 1300.2.y.e | 16 | ||
| 13.c | even | 3 | 1 | 3380.2.d.d | 16 | ||
| 13.e | even | 6 | 1 | inner | 260.2.z.a | ✓ | 16 |
| 13.e | even | 6 | 1 | 3380.2.d.d | 16 | ||
| 13.f | odd | 12 | 2 | 3380.2.c.e | 16 | ||
| 15.d | odd | 2 | 1 | 2340.2.cr.a | 16 | ||
| 20.d | odd | 2 | 1 | 1040.2.df.d | 16 | ||
| 39.h | odd | 6 | 1 | 2340.2.cr.a | 16 | ||
| 52.i | odd | 6 | 1 | 1040.2.df.d | 16 | ||
| 65.l | even | 6 | 1 | inner | 260.2.z.a | ✓ | 16 |
| 65.l | even | 6 | 1 | 3380.2.d.d | 16 | ||
| 65.n | even | 6 | 1 | 3380.2.d.d | 16 | ||
| 65.r | odd | 12 | 2 | 1300.2.y.e | 16 | ||
| 65.s | odd | 12 | 2 | 3380.2.c.e | 16 | ||
| 195.y | odd | 6 | 1 | 2340.2.cr.a | 16 | ||
| 260.w | odd | 6 | 1 | 1040.2.df.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.z.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 260.2.z.a | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 260.2.z.a | ✓ | 16 | 13.e | even | 6 | 1 | inner |
| 260.2.z.a | ✓ | 16 | 65.l | even | 6 | 1 | inner |
| 1040.2.df.d | 16 | 4.b | odd | 2 | 1 | ||
| 1040.2.df.d | 16 | 20.d | odd | 2 | 1 | ||
| 1040.2.df.d | 16 | 52.i | odd | 6 | 1 | ||
| 1040.2.df.d | 16 | 260.w | odd | 6 | 1 | ||
| 1300.2.y.e | 16 | 5.c | odd | 4 | 2 | ||
| 1300.2.y.e | 16 | 65.r | odd | 12 | 2 | ||
| 2340.2.cr.a | 16 | 3.b | odd | 2 | 1 | ||
| 2340.2.cr.a | 16 | 15.d | odd | 2 | 1 | ||
| 2340.2.cr.a | 16 | 39.h | odd | 6 | 1 | ||
| 2340.2.cr.a | 16 | 195.y | odd | 6 | 1 | ||
| 3380.2.c.e | 16 | 13.f | odd | 12 | 2 | ||
| 3380.2.c.e | 16 | 65.s | odd | 12 | 2 | ||
| 3380.2.d.d | 16 | 13.c | even | 3 | 1 | ||
| 3380.2.d.d | 16 | 13.e | even | 6 | 1 | ||
| 3380.2.d.d | 16 | 65.l | even | 6 | 1 | ||
| 3380.2.d.d | 16 | 65.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(260, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 17 T^{14} + \cdots + 16 \)
|
| $5$ |
\( T^{16} + 7 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 31 T^{14} + \cdots + 1048576 \)
|
| $11$ |
\( (T^{8} + 3 T^{7} + \cdots + 4096)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 815730721 \)
|
| $17$ |
\( T^{16} + \cdots + 2750058481 \)
|
| $19$ |
\( (T^{8} + 9 T^{7} + \cdots + 412164)^{2} \)
|
| $23$ |
\( T^{16} - 89 T^{14} + \cdots + 1336336 \)
|
| $29$ |
\( (T^{8} - 6 T^{7} + \cdots + 154449)^{2} \)
|
| $31$ |
\( (T^{8} + 108 T^{6} + \cdots + 2304)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 85662167761 \)
|
| $41$ |
\( (T^{8} + 24 T^{7} + \cdots + 1771561)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 287107358976 \)
|
| $47$ |
\( (T^{8} - 132 T^{6} + \cdots + 147456)^{2} \)
|
| $53$ |
\( (T^{8} + 191 T^{6} + \cdots + 30976)^{2} \)
|
| $59$ |
\( (T^{8} + 15 T^{7} + \cdots + 1936)^{2} \)
|
| $61$ |
\( (T^{8} + 14 T^{7} + \cdots + 1338649)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 722642807056 \)
|
| $71$ |
\( (T^{8} + 9 T^{7} + \cdots + 3104644)^{2} \)
|
| $73$ |
\( (T^{8} - 133 T^{6} + \cdots + 861184)^{2} \)
|
| $79$ |
\( (T^{4} + 4 T^{3} + \cdots + 4096)^{4} \)
|
| $83$ |
\( (T^{8} - 228 T^{6} + \cdots + 36864)^{2} \)
|
| $89$ |
\( (T^{8} - 15 T^{7} + \cdots + 1936)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 9124290174736 \)
|
| show more | |
| show less |