Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1003,2,Mod(1,1003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, 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("1003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1003 = 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1003.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: | \(8.00899532273\) |
| 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: |
\( x^{16} - 6 x^{15} - 5 x^{14} + 90 x^{13} - 82 x^{12} - 456 x^{11} + 723 x^{10} + 951 x^{9} - 2105 x^{8} + \cdots - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 6 x^{15} - 5 x^{14} + 90 x^{13} - 82 x^{12} - 456 x^{11} + 723 x^{10} + 951 x^{9} - 2105 x^{8} + \cdots - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 21607 \nu^{15} + 206853 \nu^{14} - 210704 \nu^{13} - 2846302 \nu^{12} + 6450296 \nu^{11} + \cdots - 1629531 ) / 796522 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 46479 \nu^{15} + 448281 \nu^{14} - 494240 \nu^{13} - 6086430 \nu^{12} + 14620862 \nu^{11} + \cdots - 252191 ) / 796522 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 73855 \nu^{15} + 331382 \nu^{14} + 929938 \nu^{13} - 5685137 \nu^{12} - 2256821 \nu^{11} + \cdots + 621523 ) / 398261 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} + \cdots - 1684399 ) / 796522 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} + \cdots - 1684399 ) / 796522 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 171663 \nu^{15} - 912757 \nu^{14} - 1423874 \nu^{13} + 14140202 \nu^{12} - 4762104 \nu^{11} + \cdots + 623575 ) / 796522 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 252191 \nu^{15} + 1466667 \nu^{14} + 1709236 \nu^{13} - 23191430 \nu^{12} + 14593232 \nu^{11} + \cdots - 5345561 ) / 796522 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 174999 \nu^{15} - 976139 \nu^{14} - 1206377 \nu^{13} + 14819972 \nu^{12} - 8664781 \nu^{11} + \cdots - 317590 ) / 398261 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 454985 \nu^{15} - 2624923 \nu^{14} - 2947570 \nu^{13} + 40413834 \nu^{12} - 26534880 \nu^{11} + \cdots - 180195 ) / 796522 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 493455 \nu^{15} + 2751753 \nu^{14} + 3520236 \nu^{13} - 42280306 \nu^{12} + 22964484 \nu^{11} + \cdots - 1226399 ) / 796522 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 710147 \nu^{15} + 3849819 \nu^{14} + 5665410 \nu^{13} - 60313288 \nu^{12} + 25543250 \nu^{11} + \cdots + 303563 ) / 796522 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 360387 \nu^{15} - 1988496 \nu^{14} - 2756504 \nu^{13} + 31115804 \nu^{12} - 14693575 \nu^{11} + \cdots + 1585084 ) / 398261 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 488289 \nu^{15} + 2675091 \nu^{14} + 3747450 \nu^{13} - 41578626 \nu^{12} + 19291176 \nu^{11} + \cdots - 1545655 ) / 398261 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{14} - \beta_{12} - \beta_{9} + \beta_{4} - \beta_{3} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} + \beta_{10} + 10\beta_{7} - 9\beta_{6} - \beta_{5} - 2\beta_{3} - \beta_{2} + 41\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} - 10 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots + 88 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2 \beta_{15} - \beta_{14} + 9 \beta_{13} + 14 \beta_{10} - \beta_{9} - \beta_{8} + 81 \beta_{7} + \cdots - 38 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{15} - 107 \beta_{14} - 17 \beta_{13} - 91 \beta_{12} + 11 \beta_{11} + 4 \beta_{10} - 81 \beta_{9} + \cdots + 560 \)
|
| \(\nu^{9}\) | \(=\) |
\( 30 \beta_{15} - 19 \beta_{14} + 54 \beta_{13} + \beta_{12} - \beta_{11} + 137 \beta_{10} - 13 \beta_{9} + \cdots - 348 \)
|
| \(\nu^{10}\) | \(=\) |
\( 22 \beta_{15} - 865 \beta_{14} - 201 \beta_{13} - 688 \beta_{12} + 92 \beta_{11} + 67 \beta_{10} + \cdots + 3732 \)
|
| \(\nu^{11}\) | \(=\) |
\( 314 \beta_{15} - 237 \beta_{14} + 232 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + 1173 \beta_{10} + \cdots - 2828 \)
|
| \(\nu^{12}\) | \(=\) |
\( 299 \beta_{15} - 6703 \beta_{14} - 2027 \beta_{13} - 5027 \beta_{12} + 711 \beta_{11} + 754 \beta_{10} + \cdots + 25562 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2868 \beta_{15} - 2464 \beta_{14} + 304 \beta_{13} + 211 \beta_{12} - 87 \beta_{11} + 9428 \beta_{10} + \cdots - 21700 \)
|
| \(\nu^{14}\) | \(=\) |
\( 3294 \beta_{15} - 50861 \beta_{14} - 18715 \beta_{13} - 36261 \beta_{12} + 5341 \beta_{11} + \cdots + 178066 \)
|
| \(\nu^{15}\) | \(=\) |
\( 24544 \beta_{15} - 23195 \beta_{14} - 8241 \beta_{13} + 1770 \beta_{12} - 452 \beta_{11} + 73300 \beta_{10} + \cdots - 161242 \)
|
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 |
|
−2.74400 | −0.794276 | 5.52956 | −3.63969 | 2.17950 | −4.59580 | −9.68512 | −2.36913 | 9.98732 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.52760 | −0.131380 | 4.38874 | −1.05573 | 0.332076 | 3.96203 | −6.03776 | −2.98274 | 2.66845 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.33579 | 2.58736 | 3.45592 | −4.30332 | −6.04353 | 1.79793 | −3.40073 | 3.69442 | 10.0517 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.98811 | 1.88888 | 1.95260 | 1.43913 | −3.75530 | −4.79587 | 0.0942457 | 0.567851 | −2.86115 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.79724 | −2.09283 | 1.23006 | −2.26932 | 3.76130 | −0.408229 | 1.38377 | 1.37993 | 4.07850 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.62876 | −1.50871 | 0.652867 | 1.81397 | 2.45732 | −1.88099 | 2.19416 | −0.723806 | −2.95453 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.886486 | −3.35617 | −1.21414 | −3.79855 | 2.97520 | −4.56031 | 2.84929 | 8.26386 | 3.36736 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.431919 | 0.385290 | −1.81345 | −0.726091 | −0.166414 | 1.71462 | 1.64710 | −2.85155 | 0.313613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.377947 | −1.60628 | −1.85716 | 0.658559 | 0.607088 | −1.52100 | 1.45780 | −0.419863 | −0.248900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.0187805 | 1.61675 | −1.99965 | −1.12710 | 0.0303633 | −0.878091 | −0.0751153 | −0.386118 | −0.0211674 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.446420 | −3.01975 | −1.80071 | −2.85809 | −1.34808 | 4.83764 | −1.69671 | 6.11887 | −1.27591 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.896165 | 1.99385 | −1.19689 | −0.516029 | 1.78681 | −1.20609 | −2.86494 | 0.975426 | −0.462447 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.35393 | −1.79865 | −0.166881 | 1.77529 | −2.43525 | 2.41510 | −2.93380 | 0.235159 | 2.40362 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.38820 | 2.64276 | −0.0729129 | −3.88583 | 3.66866 | −3.47551 | −2.87761 | 3.98417 | −5.39429 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.94258 | −1.53769 | 1.77364 | 1.30751 | −2.98709 | −3.17249 | −0.439731 | −0.635509 | 2.53995 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.67178 | −2.26915 | 5.13841 | −3.81472 | −6.06267 | 0.767070 | 8.38515 | 2.14904 | −10.1921 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 1003.2.a.h | ✓ | 16 |
| 3.b | odd | 2 | 1 | 9027.2.a.n | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1003.2.a.h | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 9027.2.a.n | 16 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):
|
\( T_{2}^{16} + 6 T_{2}^{15} - 5 T_{2}^{14} - 90 T_{2}^{13} - 82 T_{2}^{12} + 456 T_{2}^{11} + 723 T_{2}^{10} + \cdots - 1 \)
|
|
\( T_{3}^{16} + 7 T_{3}^{15} - 8 T_{3}^{14} - 154 T_{3}^{13} - 157 T_{3}^{12} + 1228 T_{3}^{11} + \cdots + 540 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 6 T^{15} + \cdots - 1 \)
|
| $3$ |
\( T^{16} + 7 T^{15} + \cdots + 540 \)
|
| $5$ |
\( T^{16} + 21 T^{15} + \cdots - 10177 \)
|
| $7$ |
\( T^{16} + 11 T^{15} + \cdots + 150053 \)
|
| $11$ |
\( T^{16} + 7 T^{15} + \cdots - 1246272 \)
|
| $13$ |
\( T^{16} + 16 T^{15} + \cdots + 39156176 \)
|
| $17$ |
\( (T - 1)^{16} \)
|
| $19$ |
\( T^{16} - T^{15} + \cdots - 45241463 \)
|
| $23$ |
\( T^{16} + \cdots - 2243683088 \)
|
| $29$ |
\( T^{16} + \cdots - 18054172293 \)
|
| $31$ |
\( T^{16} + \cdots - 2089303136 \)
|
| $37$ |
\( T^{16} + 28 T^{15} + \cdots + 93871312 \)
|
| $41$ |
\( T^{16} + 31 T^{15} + \cdots - 4051605 \)
|
| $43$ |
\( T^{16} + \cdots - 9106747489312 \)
|
| $47$ |
\( T^{16} + \cdots - 484870704 \)
|
| $53$ |
\( T^{16} + \cdots + 2659210263 \)
|
| $59$ |
\( (T - 1)^{16} \)
|
| $61$ |
\( T^{16} + \cdots + 74808359200 \)
|
| $67$ |
\( T^{16} + \cdots - 26547138880 \)
|
| $71$ |
\( T^{16} + \cdots - 1232446234112 \)
|
| $73$ |
\( T^{16} + \cdots - 521350404720 \)
|
| $79$ |
\( T^{16} + \cdots + 364375937949 \)
|
| $83$ |
\( T^{16} + \cdots + 1273873429168 \)
|
| $89$ |
\( T^{16} + \cdots + 58933363171168 \)
|
| $97$ |
\( T^{16} + \cdots + 1403955611216 \)
|
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