Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6724,2,Mod(1,6724)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6724, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("6724.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6724 = 2^{2} \cdot 41^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6724.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: | \(53.6914103191\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} - 25 x^{10} + 25 x^{9} + 218 x^{8} - 237 x^{7} - 774 x^{6} + 973 x^{5} + 939 x^{4} + \cdots - 71 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 25 x^{10} + 25 x^{9} + 218 x^{8} - 237 x^{7} - 774 x^{6} + 973 x^{5} + 939 x^{4} + \cdots - 71 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 26695 \nu^{11} - 25070 \nu^{10} + 634043 \nu^{9} + 580313 \nu^{8} - 5083063 \nu^{7} + \cdots - 543782 ) / 229329 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39959 \nu^{11} - 27380 \nu^{10} + 960295 \nu^{9} + 625836 \nu^{8} - 7824592 \nu^{7} + \cdots - 1513673 ) / 229329 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 67120 \nu^{11} - 27273 \nu^{10} + 1638950 \nu^{9} + 625393 \nu^{8} - 13730346 \nu^{7} + \cdots - 3554847 ) / 229329 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 90200 \nu^{11} - 33690 \nu^{10} + 2204800 \nu^{9} + 780080 \nu^{8} - 18517014 \nu^{7} + \cdots - 4243443 ) / 229329 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 123890 \nu^{11} - 50200 \nu^{10} + 3035080 \nu^{9} + 1146586 \nu^{8} - 25553732 \nu^{7} + \cdots - 6404200 ) / 229329 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 63149 \nu^{11} + 25448 \nu^{10} - 1543128 \nu^{9} - 593785 \nu^{8} + 12948501 \nu^{7} + \cdots + 3084270 ) / 76443 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 68568 \nu^{11} - 24390 \nu^{10} + 1680898 \nu^{9} + 561066 \nu^{8} - 14178876 \nu^{7} + \cdots - 4194922 ) / 76443 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 72761 \nu^{11} + 29750 \nu^{10} - 1779491 \nu^{9} - 682800 \nu^{8} + 14945935 \nu^{7} + \cdots + 3723240 ) / 76443 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 218837 \nu^{11} - 87971 \nu^{10} + 5360998 \nu^{9} + 2041914 \nu^{8} - 45145474 \nu^{7} + \cdots - 10320089 ) / 229329 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 609241 \nu^{11} + 233404 \nu^{10} - 14911958 \nu^{9} - 5391219 \nu^{8} + 125452877 \nu^{7} + \cdots + 31567903 ) / 229329 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{9} - 2\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{11} + 2\beta_{9} + 9\beta_{8} - 10\beta_{7} + 13\beta_{6} + 13\beta_{5} - 10\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13 \beta_{11} - 13 \beta_{10} - 11 \beta_{9} + 3 \beta_{8} - 26 \beta_{7} + 27 \beta_{6} + 15 \beta_{5} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 93 \beta_{11} - 7 \beta_{10} + 35 \beta_{9} + 82 \beta_{8} - 102 \beta_{7} + 149 \beta_{6} + 132 \beta_{5} + \cdots + 224 \)
|
| \(\nu^{7}\) | \(=\) |
\( 152 \beta_{11} - 145 \beta_{10} - 100 \beta_{9} + 61 \beta_{8} - 301 \beta_{7} + 324 \beta_{6} + \cdots + 96 \)
|
| \(\nu^{8}\) | \(=\) |
\( 886 \beta_{11} - 149 \beta_{10} + 454 \beta_{9} + 778 \beta_{8} - 1083 \beta_{7} + 1659 \beta_{6} + \cdots + 2048 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1744 \beta_{11} - 1569 \beta_{10} - 821 \beta_{9} + 907 \beta_{8} - 3410 \beta_{7} + 3835 \beta_{6} + \cdots + 1950 \)
|
| \(\nu^{10}\) | \(=\) |
\( 8773 \beta_{11} - 2298 \beta_{10} + 5261 \beta_{9} + 7680 \beta_{8} - 11819 \beta_{7} + 18348 \beta_{6} + \cdots + 19847 \)
|
| \(\nu^{11}\) | \(=\) |
\( 19930 \beta_{11} - 16938 \beta_{10} - 6026 \beta_{9} + 12002 \beta_{8} - 38461 \beta_{7} + 45212 \beta_{6} + \cdots + 28720 \)
|
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 | −3.37117 | 0 | −1.44144 | 0 | −0.739469 | 0 | 8.36476 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.70965 | 0 | 3.82887 | 0 | 3.41537 | 0 | 4.34219 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.56405 | 0 | 2.04294 | 0 | 2.86728 | 0 | −0.553751 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.40210 | 0 | −3.06396 | 0 | 2.19179 | 0 | −1.03412 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.36836 | 0 | −2.59238 | 0 | −4.39794 | 0 | −1.12759 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.785733 | 0 | 4.14319 | 0 | −4.76933 | 0 | −2.38262 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.164503 | 0 | −0.137342 | 0 | 2.79610 | 0 | −2.97294 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.679749 | 0 | 1.57397 | 0 | −2.78868 | 0 | −2.53794 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.41933 | 0 | 1.64085 | 0 | −1.58761 | 0 | −0.985494 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.51946 | 0 | −1.76248 | 0 | −2.02824 | 0 | 3.34770 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.76006 | 0 | −3.33839 | 0 | 3.45340 | 0 | 4.61791 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 2.98695 | 0 | −2.89384 | 0 | 1.58733 | 0 | 5.92190 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 6724.2.a.h | ✓ | 12 |
| 41.b | even | 2 | 1 | 6724.2.a.i | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6724.2.a.h | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 6724.2.a.i | yes | 12 | 41.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + T_{3}^{11} - 25 T_{3}^{10} - 25 T_{3}^{9} + 218 T_{3}^{8} + 237 T_{3}^{7} - 774 T_{3}^{6} + \cdots - 71 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + T^{11} + \cdots - 71 \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots - 2241 \)
|
| $7$ |
\( T^{12} - 52 T^{10} + \cdots + 45821 \)
|
| $11$ |
\( T^{12} + 9 T^{11} + \cdots - 29457 \)
|
| $13$ |
\( T^{12} + 7 T^{11} + \cdots - 83 \)
|
| $17$ |
\( T^{12} + 12 T^{11} + \cdots - 481923 \)
|
| $19$ |
\( T^{12} + 10 T^{11} + \cdots + 101933 \)
|
| $23$ |
\( T^{12} - 6 T^{11} + \cdots + 1705509 \)
|
| $29$ |
\( T^{12} + 19 T^{11} + \cdots - 70982379 \)
|
| $31$ |
\( T^{12} - 11 T^{11} + \cdots + 340873 \)
|
| $37$ |
\( T^{12} + \cdots - 1621659593 \)
|
| $41$ |
\( T^{12} \)
|
| $43$ |
\( T^{12} - 5 T^{11} + \cdots + 89850683 \)
|
| $47$ |
\( T^{12} + 11 T^{11} + \cdots + 89434449 \)
|
| $53$ |
\( T^{12} + \cdots + 124722639 \)
|
| $59$ |
\( T^{12} + \cdots + 2965999653 \)
|
| $61$ |
\( T^{12} + \cdots + 100138221 \)
|
| $67$ |
\( T^{12} + \cdots + 1580802847 \)
|
| $71$ |
\( T^{12} + \cdots - 601907571 \)
|
| $73$ |
\( T^{12} + \cdots + 138657499 \)
|
| $79$ |
\( T^{12} + \cdots - 2760151549 \)
|
| $83$ |
\( T^{12} + \cdots + 12737691261 \)
|
| $89$ |
\( T^{12} + \cdots - 262033234875 \)
|
| $97$ |
\( T^{12} + \cdots + 543049109 \)
|
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