Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6024,2,Mod(1,6024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6024.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6024 = 2^{3} \cdot 3 \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6024.a (trivial) |
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: | yes |
| Analytic conductor: | \(48.1018821776\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 4 x^{10} - 14 x^{9} + 63 x^{8} + 37 x^{7} - 270 x^{6} + 39 x^{5} + 391 x^{4} - 161 x^{3} + \cdots + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 4 x^{10} - 14 x^{9} + 63 x^{8} + 37 x^{7} - 270 x^{6} + 39 x^{5} + 391 x^{4} - 161 x^{3} + \cdots + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 9 \nu^{10} + 10 \nu^{9} + 187 \nu^{8} - 85 \nu^{7} - 1271 \nu^{6} - 154 \nu^{5} + 3569 \nu^{4} + \cdots + 119 ) / 388 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9 \nu^{10} - 10 \nu^{9} - 187 \nu^{8} + 85 \nu^{7} + 1271 \nu^{6} + 154 \nu^{5} - 3569 \nu^{4} + \cdots - 1283 ) / 388 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5 \nu^{10} + 9 \nu^{9} - 138 \nu^{8} - 220 \nu^{7} + 1172 \nu^{6} + 1661 \nu^{5} - 3433 \nu^{4} + \cdots - 1089 ) / 388 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23 \nu^{10} + 39 \nu^{9} + 476 \nu^{8} - 584 \nu^{7} - 3034 \nu^{6} + 2313 \nu^{5} + 6327 \nu^{4} + \cdots - 1199 ) / 388 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10 \nu^{10} - 29 \nu^{9} - 174 \nu^{8} + 451 \nu^{7} + 870 \nu^{6} - 1891 \nu^{5} - 1405 \nu^{4} + \cdots + 420 ) / 194 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 53 \nu^{10} + 96 \nu^{9} + 1092 \nu^{8} - 1424 \nu^{7} - 7204 \nu^{6} + 5643 \nu^{5} + 18110 \nu^{4} + \cdots + 1550 ) / 388 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 67 \nu^{10} + 166 \nu^{9} + 1259 \nu^{8} - 2595 \nu^{7} - 7223 \nu^{6} + 11186 \nu^{5} + \cdots + 1889 ) / 388 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 55 \nu^{10} + 256 \nu^{9} + 632 \nu^{8} - 3990 \nu^{7} + 136 \nu^{6} + 16487 \nu^{5} - 11278 \nu^{4} + \cdots - 2012 ) / 388 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 75 \nu^{10} + 291 \nu^{9} + 1065 \nu^{8} - 4489 \nu^{7} - 3091 \nu^{6} + 18172 \nu^{5} - 894 \nu^{4} + \cdots - 784 ) / 388 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 93 \nu^{10} + 311 \nu^{9} + 1439 \nu^{8} - 4659 \nu^{7} - 5633 \nu^{6} + 17864 \nu^{5} + 6244 \nu^{4} + \cdots + 230 ) / 388 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{10} - \beta_{9} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{10} - \beta_{9} + 3\beta_{2} + \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4 \beta_{10} - 3 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{10} - 11\beta_{9} + 2\beta_{6} + 16\beta_{5} + 4\beta_{4} + 2\beta_{3} + 37\beta_{2} + 9\beta _1 + 65 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 81 \beta_{10} - 49 \beta_{9} - 26 \beta_{8} + 18 \beta_{7} - 42 \beta_{6} + 90 \beta_{5} + 70 \beta_{4} + \cdots + 143 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 80 \beta_{10} - 53 \beta_{9} - 4 \beta_{8} - 9 \beta_{7} + 21 \beta_{6} + 172 \beta_{5} + 63 \beta_{4} + \cdots + 395 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 905 \beta_{10} - 439 \beta_{9} - 316 \beta_{8} + 112 \beta_{7} - 374 \beta_{6} + 1542 \beta_{5} + \cdots + 2097 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2035 \beta_{10} - 993 \beta_{9} - 254 \beta_{8} - 424 \beta_{7} + 636 \beta_{6} + 5768 \beta_{5} + \cdots + 10301 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( 5351 \beta_{10} - 2066 \beta_{9} - 1946 \beta_{8} + 128 \beta_{7} - 1553 \beta_{6} + 11900 \beta_{5} + \cdots + 15142 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 26879 \beta_{10} - 9587 \beta_{9} - 5166 \beta_{8} - 7226 \beta_{7} + 8630 \beta_{6} + 88198 \beta_{5} + \cdots + 137379 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 1.00000 | 0 | −4.08590 | 0 | 0.0849314 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −2.61516 | 0 | −2.32126 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −2.26647 | 0 | 4.00999 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | −2.17318 | 0 | −0.691834 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | −1.09211 | 0 | −4.78335 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | −0.333380 | 0 | 0.980555 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.00000 | 0 | −0.00733351 | 0 | 2.44620 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.00000 | 0 | 1.46967 | 0 | −2.39079 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.00000 | 0 | 2.33707 | 0 | −2.41392 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.00000 | 0 | 2.72048 | 0 | −0.390068 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.00000 | 0 | 3.04631 | 0 | −4.53045 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(251\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6024.2.a.m | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6024.2.a.m | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):
|
\( T_{5}^{11} + 3 T_{5}^{10} - 25 T_{5}^{9} - 69 T_{5}^{8} + 213 T_{5}^{7} + 563 T_{5}^{6} - 692 T_{5}^{5} + \cdots + 4 \)
|
|
\( T_{7}^{11} + 10 T_{7}^{10} + 8 T_{7}^{9} - 199 T_{7}^{8} - 592 T_{7}^{7} + 451 T_{7}^{6} + 3454 T_{7}^{5} + \cdots + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( (T - 1)^{11} \)
|
| $5$ |
\( T^{11} + 3 T^{10} + \cdots + 4 \)
|
| $7$ |
\( T^{11} + 10 T^{10} + \cdots + 64 \)
|
| $11$ |
\( T^{11} + 7 T^{10} + \cdots - 56 \)
|
| $13$ |
\( T^{11} + T^{10} + \cdots - 392 \)
|
| $17$ |
\( T^{11} - T^{10} + \cdots - 544 \)
|
| $19$ |
\( T^{11} + 23 T^{10} + \cdots - 2153272 \)
|
| $23$ |
\( T^{11} + 4 T^{10} + \cdots - 493936 \)
|
| $29$ |
\( T^{11} - 9 T^{10} + \cdots + 30752 \)
|
| $31$ |
\( T^{11} + 24 T^{10} + \cdots - 981568 \)
|
| $37$ |
\( T^{11} + 12 T^{10} + \cdots + 7853152 \)
|
| $41$ |
\( T^{11} + 43 T^{10} + \cdots + 10631936 \)
|
| $43$ |
\( T^{11} + 21 T^{10} + \cdots + 23964664 \)
|
| $47$ |
\( T^{11} + 25 T^{10} + \cdots - 8126464 \)
|
| $53$ |
\( T^{11} - 8 T^{10} + \cdots - 50997472 \)
|
| $59$ |
\( T^{11} + 19 T^{10} + \cdots + 1451992 \)
|
| $61$ |
\( T^{11} - 9 T^{10} + \cdots - 12406912 \)
|
| $67$ |
\( T^{11} + 29 T^{10} + \cdots + 99502856 \)
|
| $71$ |
\( T^{11} + \cdots + 3580205696 \)
|
| $73$ |
\( T^{11} + \cdots + 124175131 \)
|
| $79$ |
\( T^{11} + 30 T^{10} + \cdots + 1184384 \)
|
| $83$ |
\( T^{11} + \cdots + 435096112 \)
|
| $89$ |
\( T^{11} + \cdots + 3606797536 \)
|
| $97$ |
\( T^{11} + T^{10} + \cdots - 8566592 \)
|
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