Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
| Analytic rank: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 893) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} - 5 \nu^{14} - 7 \nu^{13} + 71 \nu^{12} - 42 \nu^{11} - 332 \nu^{10} + 459 \nu^{9} + 560 \nu^{8} + \cdots - 82 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{15} + 5 \nu^{14} + 6 \nu^{13} - 69 \nu^{12} + 59 \nu^{11} + 301 \nu^{10} - 569 \nu^{9} + \cdots + 125 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2 \nu^{15} + 7 \nu^{14} + 29 \nu^{13} - 116 \nu^{12} - 140 \nu^{11} + 715 \nu^{10} + 245 \nu^{9} + \cdots - 89 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5 \nu^{15} - 21 \nu^{14} - 60 \nu^{13} + 331 \nu^{12} + 166 \nu^{11} - 1891 \nu^{10} + 321 \nu^{9} + \cdots + 112 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7 \nu^{15} - 24 \nu^{14} - 105 \nu^{13} + 397 \nu^{12} + 558 \nu^{11} - 2444 \nu^{10} - 1357 \nu^{9} + \cdots + 400 ) / 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 7 \nu^{15} + 24 \nu^{14} + 105 \nu^{13} - 397 \nu^{12} - 558 \nu^{11} + 2444 \nu^{10} + 1357 \nu^{9} + \cdots - 418 ) / 3 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8 \nu^{15} - 27 \nu^{14} - 123 \nu^{13} + 451 \nu^{12} + 685 \nu^{11} - 2815 \nu^{10} - 1821 \nu^{9} + \cdots + 493 ) / 3 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10 \nu^{15} - 39 \nu^{14} - 131 \nu^{13} + 624 \nu^{12} + 501 \nu^{11} - 3649 \nu^{10} - 283 \nu^{9} + \cdots + 268 ) / 3 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11 \nu^{15} - 41 \nu^{14} - 153 \nu^{13} + 666 \nu^{12} + 689 \nu^{11} - 3987 \nu^{10} - 1074 \nu^{9} + \cdots + 433 ) / 3 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 12 \nu^{15} - 46 \nu^{14} - 159 \nu^{13} + 735 \nu^{12} + 630 \nu^{11} - 4288 \nu^{10} - 505 \nu^{9} + \cdots + 308 ) / 3 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 15 \nu^{15} - 50 \nu^{14} - 230 \nu^{13} + 832 \nu^{12} + 1273 \nu^{11} - 5164 \nu^{10} - 3336 \nu^{9} + \cdots + 864 ) / 3 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 14 \nu^{15} + 45 \nu^{14} + 225 \nu^{13} - 767 \nu^{12} - 1344 \nu^{11} + 4924 \nu^{10} + 3941 \nu^{9} + \cdots - 1055 ) / 3 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 26 \nu^{15} + 93 \nu^{14} + 375 \nu^{13} - 1521 \nu^{12} - 1841 \nu^{11} + 9205 \nu^{10} + \cdots - 1299 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{11} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} - \beta_{7} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{14} - \beta_{13} + 8 \beta_{12} - 8 \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 9 \beta_{8} + \cdots + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{15} + 10 \beta_{14} - 10 \beta_{13} + 57 \beta_{12} - 55 \beta_{11} - \beta_{10} + 19 \beta_{9} + \cdots + 19 \)
|
| \(\nu^{8}\) | \(=\) |
\( 12 \beta_{15} - 12 \beta_{14} + 15 \beta_{13} + 12 \beta_{12} + 15 \beta_{11} - 28 \beta_{10} + \cdots + 520 \)
|
| \(\nu^{9}\) | \(=\) |
\( 14 \beta_{15} + 78 \beta_{14} - 70 \beta_{13} + 392 \beta_{12} - 362 \beta_{11} - 18 \beta_{10} + \cdots + 132 \)
|
| \(\nu^{10}\) | \(=\) |
\( 104 \beta_{15} - 102 \beta_{14} + 157 \beta_{13} + 103 \beta_{12} + 153 \beta_{11} - 271 \beta_{10} + \cdots + 3231 \)
|
| \(\nu^{11}\) | \(=\) |
\( 135 \beta_{15} + 562 \beta_{14} - 419 \beta_{13} + 2653 \beta_{12} - 2344 \beta_{11} - 210 \beta_{10} + \cdots + 805 \)
|
| \(\nu^{12}\) | \(=\) |
\( 804 \beta_{15} - 763 \beta_{14} + 1408 \beta_{13} + 782 \beta_{12} + 1331 \beta_{11} - 2266 \beta_{10} + \cdots + 20410 \)
|
| \(\nu^{13}\) | \(=\) |
\( 1125 \beta_{15} + 3911 \beta_{14} - 2262 \beta_{13} + 17817 \beta_{12} - 15089 \beta_{11} - 2024 \beta_{10} + \cdots + 4547 \)
|
| \(\nu^{14}\) | \(=\) |
\( 5905 \beta_{15} - 5381 \beta_{14} + 11585 \beta_{13} + 5620 \beta_{12} + 10646 \beta_{11} + \cdots + 130369 \)
|
| \(\nu^{15}\) | \(=\) |
\( 8722 \beta_{15} + 26735 \beta_{14} - 11053 \beta_{13} + 119217 \beta_{12} - 97026 \beta_{11} + \cdots + 24228 \)
|
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 |
|
−2.57826 | 0 | 4.64740 | 3.07310 | 0 | −1.66345 | −6.82569 | 0 | −7.92325 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.97762 | 0 | 1.91096 | 2.84370 | 0 | −3.61101 | 0.176083 | 0 | −5.62374 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.60180 | 0 | 0.565755 | −0.438857 | 0 | 0.237690 | 2.29737 | 0 | 0.702960 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.04691 | 0 | −0.903974 | −3.78031 | 0 | −0.354270 | 3.04021 | 0 | 3.95766 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.880178 | 0 | −1.22529 | −0.815813 | 0 | −4.34804 | 2.83883 | 0 | 0.718060 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.788915 | 0 | −1.37761 | 2.33649 | 0 | 4.65174 | 2.66465 | 0 | −1.84329 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.601130 | 0 | −1.63864 | 0.0865384 | 0 | 1.51992 | 2.18730 | 0 | −0.0520209 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.469922 | 0 | −1.77917 | −0.709772 | 0 | −4.48221 | −1.77592 | 0 | −0.333538 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.489985 | 0 | −1.75991 | 2.01274 | 0 | 0.633596 | −1.84230 | 0 | 0.986215 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.640778 | 0 | −1.58940 | −3.12262 | 0 | −2.58301 | −2.30001 | 0 | −2.00091 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.22346 | 0 | −0.503155 | −1.49488 | 0 | 1.92629 | −3.06250 | 0 | −1.82892 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.38504 | 0 | −0.0816767 | 2.56770 | 0 | 0.393019 | −2.88320 | 0 | 3.55636 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.88684 | 0 | 1.56016 | 2.70381 | 0 | 1.30906 | −0.829905 | 0 | 5.10166 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.38924 | 0 | 3.70845 | −0.579664 | 0 | −3.82951 | 4.08190 | 0 | −1.38495 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.39915 | 0 | 3.75590 | −2.64565 | 0 | 2.83947 | 4.21266 | 0 | −6.34730 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.59041 | 0 | 4.71020 | −1.03652 | 0 | −1.63930 | 7.02052 | 0 | −2.68500 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
| \(47\) | \(-1\) |
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 | 8037.2.a.n | 16 | |
| 3.b | odd | 2 | 1 | 893.2.a.b | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 893.2.a.b | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 8037.2.a.n | 16 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
|
\( T_{2}^{16} - 4 T_{2}^{15} - 13 T_{2}^{14} + 65 T_{2}^{13} + 47 T_{2}^{12} - 390 T_{2}^{11} + 4 T_{2}^{10} + \cdots - 25 \)
|
|
\( T_{5}^{16} - T_{5}^{15} - 38 T_{5}^{14} + 38 T_{5}^{13} + 559 T_{5}^{12} - 481 T_{5}^{11} - 4109 T_{5}^{10} + \cdots - 176 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 4 T^{15} + \cdots - 25 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - T^{15} + \cdots - 176 \)
|
| $7$ |
\( T^{16} + 9 T^{15} + \cdots + 2015 \)
|
| $11$ |
\( T^{16} - 103 T^{14} + \cdots - 2032 \)
|
| $13$ |
\( T^{16} + \cdots + 157573520 \)
|
| $17$ |
\( T^{16} - 8 T^{15} + \cdots - 1388995 \)
|
| $19$ |
\( (T + 1)^{16} \)
|
| $23$ |
\( T^{16} + \cdots + 2887806976 \)
|
| $29$ |
\( T^{16} + \cdots + 106094320720 \)
|
| $31$ |
\( T^{16} + \cdots - 2972038480 \)
|
| $37$ |
\( T^{16} + \cdots + 6410122727 \)
|
| $41$ |
\( T^{16} + \cdots - 409610608 \)
|
| $43$ |
\( T^{16} + \cdots + 19033721168 \)
|
| $47$ |
\( (T - 1)^{16} \)
|
| $53$ |
\( T^{16} + \cdots + 1967116789 \)
|
| $59$ |
\( T^{16} - 32 T^{15} + \cdots + 2445115 \)
|
| $61$ |
\( T^{16} + \cdots - 68101260581 \)
|
| $67$ |
\( T^{16} + \cdots + 197528409232 \)
|
| $71$ |
\( T^{16} + \cdots + 1049581589 \)
|
| $73$ |
\( T^{16} + \cdots - 117807304912 \)
|
| $79$ |
\( T^{16} + \cdots - 89847608413511 \)
|
| $83$ |
\( T^{16} + \cdots + 2890416745216 \)
|
| $89$ |
\( T^{16} + \cdots - 70\!\cdots\!31 \)
|
| $97$ |
\( T^{16} + \cdots - 404975905643 \)
|
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