Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5687,2,Mod(1,5687)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5687, 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("5687.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5687 = 11^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5687.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: | \(45.4109236295\) |
| 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: |
\( x^{12} - 17x^{10} - x^{9} + 106x^{8} + 11x^{7} - 294x^{6} - 41x^{5} + 341x^{4} + 56x^{3} - 113x^{2} - 23x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{12} - 17x^{10} - x^{9} + 106x^{8} + 11x^{7} - 294x^{6} - 41x^{5} + 341x^{4} + 56x^{3} - 113x^{2} - 23x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8 \nu^{11} - 68 \nu^{10} + 116 \nu^{9} + 994 \nu^{8} - 490 \nu^{7} - 5090 \nu^{6} + 379 \nu^{5} + \cdots + 641 ) / 279 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5 \nu^{11} - 27 \nu^{10} + 57 \nu^{9} + 381 \nu^{8} - 159 \nu^{7} - 1794 \nu^{6} - 46 \nu^{5} + \cdots + 265 ) / 93 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7 \nu^{11} + 13 \nu^{10} - 55 \nu^{9} - 149 \nu^{8} - 106 \nu^{7} + 385 \nu^{6} + 1540 \nu^{5} + \cdots + 404 ) / 93 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 22 \nu^{11} + \nu^{10} - 412 \nu^{9} + 10 \nu^{8} + 2789 \nu^{7} - 278 \nu^{6} - 8180 \nu^{5} + \cdots + 911 ) / 279 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 19 \nu^{11} - 9 \nu^{10} - 291 \nu^{9} + 96 \nu^{8} + 1590 \nu^{7} - 288 \nu^{6} - 3694 \nu^{5} + \cdots + 16 ) / 93 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 52 \nu^{11} + 23 \nu^{10} + 847 \nu^{9} - 328 \nu^{8} - 5045 \nu^{7} + 1697 \nu^{6} + 13298 \nu^{5} + \cdots + 28 ) / 279 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 56 \nu^{11} - 11 \nu^{10} + 905 \nu^{9} + 169 \nu^{8} - 5290 \nu^{7} - 848 \nu^{6} + 13627 \nu^{5} + \cdots + 488 ) / 279 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 56 \nu^{11} + 11 \nu^{10} - 905 \nu^{9} - 169 \nu^{8} + 5290 \nu^{7} + 848 \nu^{6} - 13627 \nu^{5} + \cdots - 488 ) / 279 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 26 \nu^{11} - 4 \nu^{10} + 439 \nu^{9} + 53 \nu^{8} - 2693 \nu^{7} - 190 \nu^{6} + 7269 \nu^{5} + \cdots + 324 ) / 93 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{6} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} + 7\beta_{10} + 8\beta_{9} - 2\beta_{8} + \beta_{6} - \beta_{3} + 2\beta_{2} + 26\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{11} + 12 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + 8 \beta_{6} + \beta_{4} - \beta_{3} + \cdots + 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{11} + 45 \beta_{10} + 52 \beta_{9} - 23 \beta_{8} - \beta_{7} + 12 \beta_{6} - \beta_{5} + \cdots + 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( 77 \beta_{11} + 104 \beta_{10} + 16 \beta_{9} - 15 \beta_{8} - 14 \beta_{7} + 53 \beta_{6} + \beta_{5} + \cdots + 326 \)
|
| \(\nu^{9}\) | \(=\) |
\( 132 \beta_{11} + 290 \beta_{10} + 321 \beta_{9} - 190 \beta_{8} - 13 \beta_{7} + 103 \beta_{6} + \cdots + 33 \)
|
| \(\nu^{10}\) | \(=\) |
\( 543 \beta_{11} + 791 \beta_{10} + 167 \beta_{9} - 151 \beta_{8} - 131 \beta_{7} + 340 \beta_{6} + \cdots + 1766 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1061 \beta_{11} + 1890 \beta_{10} + 1959 \beta_{9} - 1385 \beta_{8} - 117 \beta_{7} + 779 \beta_{6} + \cdots + 369 \)
|
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 |
|
−2.35979 | −1.18778 | 3.56860 | −3.66906 | 2.80291 | −2.11239 | −3.70157 | −1.58918 | 8.65820 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24083 | 0.846532 | 3.02130 | −0.0513841 | −1.89693 | −0.00517342 | −2.28856 | −2.28338 | 0.115143 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.81794 | 0.692635 | 1.30490 | 1.38476 | −1.25917 | 3.78391 | 1.26366 | −2.52026 | −2.51741 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.47803 | −2.20877 | 0.184568 | 3.29004 | 3.26462 | 0.804614 | 2.68326 | 1.87865 | −4.86277 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.672770 | −1.30215 | −1.54738 | −2.20663 | 0.876050 | 3.69097 | 2.38657 | −1.30440 | 1.48456 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.254788 | −2.27857 | −1.93508 | 2.20767 | 0.580554 | −1.18928 | 1.00261 | 2.19190 | −0.562489 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.0369284 | 2.38745 | −1.99864 | −0.621200 | 0.0881648 | −2.84760 | −0.147663 | 2.69994 | −0.0229399 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.743165 | 0.180683 | −1.44771 | 0.736338 | 0.134278 | 0.405140 | −2.56221 | −2.96735 | 0.547220 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.40748 | 1.11907 | −0.0190137 | 3.38052 | 1.57506 | −1.09403 | −2.84171 | −1.74768 | 4.75800 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.01412 | 0.517517 | 2.05669 | −1.57949 | 1.04234 | 2.84635 | 0.114176 | −2.73218 | −3.18128 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.05800 | 2.45219 | 2.23535 | −2.73137 | 5.04660 | −2.52887 | 0.484357 | 3.01322 | −5.62115 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.56445 | −3.21881 | 4.57641 | 0.859801 | −8.25448 | −0.753643 | 6.60708 | 7.36072 | 2.20492 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(47\) | \( -1 \) |
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 | 5687.2.a.w | yes | 12 |
| 11.b | odd | 2 | 1 | 5687.2.a.v | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5687.2.a.v | ✓ | 12 | 11.b | odd | 2 | 1 | |
| 5687.2.a.w | yes | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5687))\):
|
\( T_{2}^{12} - 17 T_{2}^{10} - T_{2}^{9} + 106 T_{2}^{8} + 11 T_{2}^{7} - 294 T_{2}^{6} - 41 T_{2}^{5} + \cdots + 1 \)
|
|
\( T_{3}^{12} + 2 T_{3}^{11} - 17 T_{3}^{10} - 27 T_{3}^{9} + 104 T_{3}^{8} + 114 T_{3}^{7} - 286 T_{3}^{6} + \cdots - 9 \)
|
|
\( T_{5}^{12} - T_{5}^{11} - 29 T_{5}^{10} + 28 T_{5}^{9} + 286 T_{5}^{8} - 257 T_{5}^{7} - 1141 T_{5}^{6} + \cdots + 24 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 17 T^{10} + \cdots + 1 \)
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| $3$ |
\( T^{12} + 2 T^{11} + \cdots - 9 \)
|
| $5$ |
\( T^{12} - T^{11} + \cdots + 24 \)
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| $7$ |
\( T^{12} - T^{11} + \cdots - 1 \)
|
| $11$ |
\( T^{12} \)
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| $13$ |
\( T^{12} - T^{11} + \cdots + 3867 \)
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| $17$ |
\( T^{12} + 11 T^{11} + \cdots - 33 \)
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| $19$ |
\( T^{12} - T^{11} + \cdots + 130687 \)
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| $23$ |
\( T^{12} + 3 T^{11} + \cdots + 3109 \)
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| $29$ |
\( T^{12} + 16 T^{11} + \cdots + 347 \)
|
| $31$ |
\( T^{12} + 9 T^{11} + \cdots + 93645099 \)
|
| $37$ |
\( T^{12} + \cdots - 1217680728 \)
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| $41$ |
\( T^{12} + \cdots - 388802311 \)
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| $43$ |
\( T^{12} + 11 T^{11} + \cdots + 692967 \)
|
| $47$ |
\( (T - 1)^{12} \)
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| $53$ |
\( T^{12} + 41 T^{11} + \cdots + 276361 \)
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| $59$ |
\( T^{12} + 6 T^{11} + \cdots - 26568 \)
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| $61$ |
\( T^{12} + \cdots - 557189701 \)
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| $67$ |
\( T^{12} + \cdots + 7511912221 \)
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| $71$ |
\( T^{12} + 16 T^{11} + \cdots - 631989 \)
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| $73$ |
\( T^{12} + \cdots + 226636749 \)
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| $79$ |
\( T^{12} + \cdots + 165890259 \)
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| $83$ |
\( T^{12} + \cdots + 256341005239 \)
|
| $89$ |
\( T^{12} + \cdots - 5523003853 \)
|
| $97$ |
\( T^{12} + \cdots + 305563131 \)
|
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