Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1027,2,Mod(1,1027)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1027, 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("1027.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1027 = 13 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1027.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: | \(8.20063628759\) |
| Analytic rank: | \(1\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 7 x^{16} - 2 x^{15} + 100 x^{14} - 81 x^{13} - 614 x^{12} + 617 x^{11} + 2112 x^{10} + \cdots - 19 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 7 x^{16} - 2 x^{15} + 100 x^{14} - 81 x^{13} - 614 x^{12} + 617 x^{11} + 2112 x^{10} + \cdots - 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2755839 \nu^{16} + 46813507 \nu^{15} - 214759083 \nu^{14} - 98797598 \nu^{13} + \cdots + 1389051357 ) / 97981891 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4342081 \nu^{16} - 48771736 \nu^{15} + 136834313 \nu^{14} + 336708740 \nu^{13} - 2095670996 \nu^{12} + \cdots - 680023891 ) / 97981891 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7274157 \nu^{16} + 79162327 \nu^{15} - 218034873 \nu^{14} - 490244374 \nu^{13} + \cdots + 742769882 ) / 97981891 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9369504 \nu^{16} - 82549402 \nu^{15} + 117182625 \nu^{14} + 829671053 \nu^{13} - 2318863727 \nu^{12} + \cdots - 132558183 ) / 97981891 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1774903 \nu^{16} + 14793227 \nu^{15} - 14985222 \nu^{14} - 166350324 \nu^{13} + \cdots + 62194020 ) / 13997413 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 12547049 \nu^{16} - 84393615 \nu^{15} - 50743966 \nu^{14} + 1267309744 \nu^{13} + \cdots + 164775245 ) / 97981891 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 17119759 \nu^{16} - 142542768 \nu^{15} + 156428468 \nu^{14} + 1493234881 \nu^{13} + \cdots - 20371585 ) / 97981891 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 22814646 \nu^{16} + 184367046 \nu^{15} - 170671207 \nu^{14} - 1962518490 \nu^{13} + \cdots - 296208356 ) / 97981891 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 24332597 \nu^{16} + 189738562 \nu^{15} - 117225675 \nu^{14} - 2226005163 \nu^{13} + \cdots - 72088281 ) / 97981891 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 36878451 \nu^{16} - 310495895 \nu^{15} + 352611546 \nu^{14} + 3297112842 \nu^{13} + \cdots - 576874694 ) / 97981891 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 48786727 \nu^{16} - 396681868 \nu^{15} + 364940667 \nu^{14} + 4358665737 \nu^{13} + \cdots + 126535509 ) / 97981891 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 49840568 \nu^{16} - 425316300 \nu^{15} + 531548136 \nu^{14} + 4333645094 \nu^{13} + \cdots - 1293208865 ) / 97981891 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 51575618 \nu^{16} + 422547162 \nu^{15} - 400503376 \nu^{14} - 4679614064 \nu^{13} + \cdots + 836036927 ) / 97981891 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 63470207 \nu^{16} + 516408867 \nu^{15} - 471475116 \nu^{14} - 5718841393 \nu^{13} + \cdots + 720546076 ) / 97981891 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{7} + 2\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{16} - 3 \beta_{14} + 13 \beta_{13} - 12 \beta_{12} + 12 \beta_{11} - 3 \beta_{10} + \cdots + 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8 \beta_{16} + \beta_{15} - 14 \beta_{14} + 42 \beta_{13} - 35 \beta_{12} + 34 \beta_{11} + \cdots + 141 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 39 \beta_{16} + 4 \beta_{15} - 45 \beta_{14} + 146 \beta_{13} - 137 \beta_{12} + 133 \beta_{11} + \cdots + 392 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 140 \beta_{16} + 20 \beta_{15} - 162 \beta_{14} + 472 \beta_{13} - 437 \beta_{12} + 415 \beta_{11} + \cdots + 1227 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 518 \beta_{16} + 73 \beta_{15} - 527 \beta_{14} + 1549 \beta_{13} - 1507 \beta_{12} + 1422 \beta_{11} + \cdots + 3628 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1750 \beta_{16} + 274 \beta_{15} - 1766 \beta_{14} + 4973 \beta_{13} - 4870 \beta_{12} + 4523 \beta_{11} + \cdots + 11179 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 5905 \beta_{16} + 933 \beta_{15} - 5735 \beta_{14} + 16002 \beta_{13} - 16043 \beta_{12} + \cdots + 33879 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 19257 \beta_{16} + 3158 \beta_{15} - 18756 \beta_{14} + 51093 \beta_{13} - 51660 \beta_{12} + \cdots + 104275 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 62424 \beta_{16} + 10265 \beta_{15} - 60694 \beta_{14} + 163070 \beta_{13} - 166923 \beta_{12} + \cdots + 319649 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 199377 \beta_{16} + 33016 \beta_{15} - 196622 \beta_{14} + 518801 \beta_{13} - 534554 \beta_{12} + \cdots + 986205 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 633527 \beta_{16} + 104031 \beta_{15} - 634445 \beta_{14} + 1649255 \beta_{13} - 1711096 \beta_{12} + \cdots + 3041862 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 1998661 \beta_{16} + 324713 \beta_{15} - 2046513 \beta_{14} + 5234918 \beta_{13} - 5455253 \beta_{12} + \cdots + 9411476 \)
|
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 |
|
−2.81628 | −1.87092 | 5.93142 | −4.19212 | 5.26902 | −4.14869 | −11.0720 | 0.500332 | 11.8062 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.79100 | 2.37319 | 5.78968 | 1.14176 | −6.62358 | −1.84356 | −10.5770 | 2.63205 | −3.18664 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.47793 | 0.734180 | 4.14014 | −3.26147 | −1.81925 | 4.56218 | −5.30311 | −2.46098 | 8.08169 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.29652 | −2.24801 | 3.27399 | 0.721699 | 5.16259 | −1.57122 | −2.92574 | 2.05354 | −1.65739 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.21139 | 2.59042 | 2.89025 | −3.58657 | −5.72843 | −0.815783 | −1.96869 | 3.71027 | 7.93132 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.05754 | −0.874259 | 2.23347 | 2.05589 | 1.79882 | 4.62683 | −0.480367 | −2.23567 | −4.23008 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.61010 | 0.110270 | 0.592426 | −0.517371 | −0.177546 | −3.40939 | 2.26634 | −2.98784 | 0.833019 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.17977 | −1.77573 | −0.608136 | −1.41364 | 2.09496 | 1.49298 | 3.07701 | 0.153211 | 1.66777 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.687867 | 1.60450 | −1.52684 | −0.800684 | −1.10369 | 1.49371 | 2.42600 | −0.425565 | 0.550764 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.598074 | 1.23387 | −1.64231 | 1.41184 | −0.737947 | −3.08803 | 2.17837 | −1.47756 | −0.844386 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.0761313 | −3.11927 | −1.99420 | −4.34769 | 0.237474 | 2.22949 | 0.304084 | 6.72982 | 0.330995 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.507489 | 2.92830 | −1.74245 | −3.16217 | 1.48608 | −1.78516 | −1.89925 | 5.57491 | −1.60477 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.982906 | −1.33498 | −1.03390 | 0.566968 | −1.31216 | 2.10768 | −2.98203 | −1.21783 | 0.557276 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.39982 | 0.133762 | −0.0405098 | 1.74187 | 0.187243 | −3.04714 | −2.85634 | −2.98211 | 2.43830 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.75236 | −0.334620 | 1.07076 | −2.68096 | −0.586374 | 2.17338 | −1.62837 | −2.88803 | −4.69801 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 1.99848 | −3.07644 | 1.99392 | −0.823717 | −6.14821 | 3.38816 | −0.0121547 | 6.46450 | −1.64618 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.16155 | 0.925721 | 2.67231 | −3.85364 | 2.00099 | −3.36546 | 1.45322 | −2.14304 | −8.32984 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
| \(79\) | \(-1\) |
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 | 1027.2.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 9243.2.a.k | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1027.2.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 9243.2.a.k | 17 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} + 10 T_{2}^{16} + 22 T_{2}^{15} - 90 T_{2}^{14} - 431 T_{2}^{13} - 29 T_{2}^{12} + 2401 T_{2}^{11} + \cdots + 64 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1027))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} + 10 T^{16} + \cdots + 64 \)
|
| $3$ |
\( T^{17} + 2 T^{16} + \cdots - 10 \)
|
| $5$ |
\( T^{17} + 21 T^{16} + \cdots + 7936 \)
|
| $7$ |
\( T^{17} + T^{16} + \cdots + 3077791 \)
|
| $11$ |
\( T^{17} + 24 T^{16} + \cdots - 7238 \)
|
| $13$ |
\( (T - 1)^{17} \)
|
| $17$ |
\( T^{17} + \cdots + 373355648 \)
|
| $19$ |
\( T^{17} + 3 T^{16} + \cdots + 25018870 \)
|
| $23$ |
\( T^{17} + 12 T^{16} + \cdots + 43606783 \)
|
| $29$ |
\( T^{17} + \cdots + 19041741218 \)
|
| $31$ |
\( T^{17} + \cdots + 46825655932 \)
|
| $37$ |
\( T^{17} + \cdots + 26671877122 \)
|
| $41$ |
\( T^{17} + \cdots - 54059538709 \)
|
| $43$ |
\( T^{17} + \cdots + 370395058 \)
|
| $47$ |
\( T^{17} + \cdots + 24713102772595 \)
|
| $53$ |
\( T^{17} + \cdots - 1351406202826 \)
|
| $59$ |
\( T^{17} + \cdots - 11521023295780 \)
|
| $61$ |
\( T^{17} + \cdots + 172825213715594 \)
|
| $67$ |
\( T^{17} + \cdots - 37015480375142 \)
|
| $71$ |
\( T^{17} + \cdots + 94054589802176 \)
|
| $73$ |
\( T^{17} + \cdots - 683835459056 \)
|
| $79$ |
\( (T - 1)^{17} \)
|
| $83$ |
\( T^{17} + \cdots + 1487472137478 \)
|
| $89$ |
\( T^{17} + \cdots - 22669145935744 \)
|
| $97$ |
\( T^{17} + \cdots - 560983388614 \)
|
| show more | |
| show less |