Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [538,4,Mod(1,538)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(538, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("538.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 538 = 2 \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 538.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: | \(31.7430275831\) |
| Analytic rank: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 8 x^{17} - 299 x^{16} + 2191 x^{15} + 37840 x^{14} - 247598 x^{13} - 2647415 x^{12} + \cdots - 88752454191 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 8 x^{17} - 299 x^{16} + 2191 x^{15} + 37840 x^{14} - 247598 x^{13} - 2647415 x^{12} + \cdots - 88752454191 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 37\!\cdots\!82 \nu^{17} + \cdots - 16\!\cdots\!21 ) / 46\!\cdots\!83 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 86\!\cdots\!20 \nu^{17} + \cdots + 14\!\cdots\!08 ) / 46\!\cdots\!83 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 96\!\cdots\!65 \nu^{17} + \cdots + 94\!\cdots\!71 ) / 46\!\cdots\!83 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 48\!\cdots\!24 \nu^{17} + \cdots - 17\!\cdots\!28 ) / 15\!\cdots\!61 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!14 \nu^{17} + \cdots + 19\!\cdots\!59 ) / 46\!\cdots\!83 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16\!\cdots\!65 \nu^{17} + \cdots - 20\!\cdots\!69 ) / 46\!\cdots\!83 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15\!\cdots\!22 \nu^{17} + \cdots - 62\!\cdots\!20 ) / 22\!\cdots\!23 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11\!\cdots\!32 \nu^{17} + \cdots - 56\!\cdots\!66 ) / 15\!\cdots\!61 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 35\!\cdots\!73 \nu^{17} + \cdots - 18\!\cdots\!03 ) / 46\!\cdots\!83 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 46\!\cdots\!70 \nu^{17} + \cdots - 20\!\cdots\!82 ) / 46\!\cdots\!83 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 51\!\cdots\!50 \nu^{17} + \cdots - 37\!\cdots\!75 ) / 46\!\cdots\!83 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 79\!\cdots\!02 \nu^{17} + \cdots + 19\!\cdots\!58 ) / 46\!\cdots\!83 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 96\!\cdots\!80 \nu^{17} + \cdots + 30\!\cdots\!23 ) / 46\!\cdots\!83 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 16\!\cdots\!54 \nu^{17} + \cdots + 22\!\cdots\!82 ) / 66\!\cdots\!69 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 46\!\cdots\!87 \nu^{17} + \cdots - 81\!\cdots\!80 ) / 15\!\cdots\!61 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{16} + 2 \beta_{15} + \beta_{13} + 2 \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{16} + 12 \beta_{15} - 11 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 12 \beta_{11} - 2 \beta_{10} + \cdots + 2009 \)
|
| \(\nu^{5}\) | \(=\) |
\( 47 \beta_{17} - 133 \beta_{16} + 286 \beta_{15} + 6 \beta_{14} + 107 \beta_{13} + 223 \beta_{12} + \cdots + 4826 \)
|
| \(\nu^{6}\) | \(=\) |
\( 158 \beta_{17} + 133 \beta_{16} + 1782 \beta_{15} - 633 \beta_{14} - 841 \beta_{13} + 869 \beta_{12} + \cdots + 138658 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9205 \beta_{17} - 12730 \beta_{16} + 31332 \beta_{15} + 5725 \beta_{14} + 8089 \beta_{13} + \cdots + 563989 \)
|
| \(\nu^{8}\) | \(=\) |
\( 50776 \beta_{17} - 2710 \beta_{16} + 210012 \beta_{15} + 31605 \beta_{14} - 92792 \beta_{13} + \cdots + 11033021 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1320636 \beta_{17} - 1100820 \beta_{16} + 3192796 \beta_{15} + 1351853 \beta_{14} + 451009 \beta_{13} + \cdots + 62818591 \)
|
| \(\nu^{10}\) | \(=\) |
\( 9323774 \beta_{17} - 1250327 \beta_{16} + 23238464 \beta_{15} + 13429221 \beta_{14} - 9740489 \beta_{13} + \cdots + 967374607 \)
|
| \(\nu^{11}\) | \(=\) |
\( 167290991 \beta_{17} - 91756884 \beta_{16} + 319661318 \beta_{15} + 220119148 \beta_{14} + \cdots + 6832350713 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1360355421 \beta_{17} - 180173382 \beta_{16} + 2518806481 \beta_{15} + 2314681562 \beta_{14} + \cdots + 90603121880 \)
|
| \(\nu^{13}\) | \(=\) |
\( 19877318550 \beta_{17} - 7575755222 \beta_{16} + 32080146507 \beta_{15} + 30234474849 \beta_{14} + \cdots + 734094023256 \)
|
| \(\nu^{14}\) | \(=\) |
\( 176217896590 \beta_{17} - 21155732137 \beta_{16} + 270839867619 \beta_{15} + 319656866206 \beta_{14} + \cdots + 8868534584464 \)
|
| \(\nu^{15}\) | \(=\) |
\( 2276358256426 \beta_{17} - 629209803248 \beta_{16} + 3250104415442 \beta_{15} + 3786996551451 \beta_{14} + \cdots + 78401514071171 \)
|
| \(\nu^{16}\) | \(=\) |
\( 21311919742425 \beta_{17} - 2305631492651 \beta_{16} + 29012500268375 \beta_{15} + \cdots + 893718141516642 \)
|
| \(\nu^{17}\) | \(=\) |
\( 254842936006909 \beta_{17} - 53128324367238 \beta_{16} + 332901825837843 \beta_{15} + \cdots + 83\!\cdots\!99 \)
|
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 |
|
−2.00000 | −9.38330 | 4.00000 | 10.4994 | 18.7666 | −17.4188 | −8.00000 | 61.0463 | −20.9988 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −8.34766 | 4.00000 | −3.24714 | 16.6953 | −29.7231 | −8.00000 | 42.6835 | 6.49427 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −8.16036 | 4.00000 | 14.6446 | 16.3207 | 25.1719 | −8.00000 | 39.5915 | −29.2892 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −7.12493 | 4.00000 | −16.6917 | 14.2499 | 6.60478 | −8.00000 | 23.7646 | 33.3834 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −5.67980 | 4.00000 | −21.2339 | 11.3596 | 18.4502 | −8.00000 | 5.26017 | 42.4677 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | −5.46707 | 4.00000 | 18.5547 | 10.9341 | 0.683479 | −8.00000 | 2.88890 | −37.1094 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | −5.22760 | 4.00000 | −15.6593 | 10.4552 | −28.8654 | −8.00000 | 0.327799 | 31.3186 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | −3.60255 | 4.00000 | 4.79286 | 7.20510 | 32.0745 | −8.00000 | −14.0216 | −9.58572 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.00000 | −1.23043 | 4.00000 | −2.13428 | 2.46085 | 8.15129 | −8.00000 | −25.4860 | 4.26856 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −2.00000 | −0.762997 | 4.00000 | −1.03810 | 1.52599 | −6.09255 | −8.00000 | −26.4178 | 2.07620 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −2.00000 | 3.10721 | 4.00000 | 10.1855 | −6.21442 | −30.6558 | −8.00000 | −17.3453 | −20.3709 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −2.00000 | 3.70639 | 4.00000 | −12.9085 | −7.41277 | 17.6758 | −8.00000 | −13.2627 | 25.8169 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | −2.00000 | 4.20658 | 4.00000 | 9.50054 | −8.41316 | 12.2916 | −8.00000 | −9.30467 | −19.0011 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | −2.00000 | 4.83544 | 4.00000 | 5.46105 | −9.67088 | −8.19157 | −8.00000 | −3.61854 | −10.9221 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | −2.00000 | 5.70481 | 4.00000 | −16.8654 | −11.4096 | 3.72165 | −8.00000 | 5.54487 | 33.7307 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | −2.00000 | 6.35951 | 4.00000 | 6.17843 | −12.7190 | −27.3928 | −8.00000 | 13.4434 | −12.3569 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | −2.00000 | 7.73705 | 4.00000 | 1.13148 | −15.4741 | −14.8617 | −8.00000 | 32.8619 | −2.26297 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | −2.00000 | 9.32972 | 4.00000 | −17.1703 | −18.6594 | 29.3764 | −8.00000 | 60.0437 | 34.3406 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(269\) | \( -1 \) |
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 | 538.4.a.c | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.4.a.c | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{18} + 10 T_{3}^{17} - 282 T_{3}^{16} - 2865 T_{3}^{15} + 32445 T_{3}^{14} + 334305 T_{3}^{13} + \cdots + 1533857738490 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(538))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{18} \)
|
| $3$ |
\( T^{18} + \cdots + 1533857738490 \)
|
| $5$ |
\( T^{18} + \cdots - 75\!\cdots\!31 \)
|
| $7$ |
\( T^{18} + \cdots + 12\!\cdots\!50 \)
|
| $11$ |
\( T^{18} + \cdots + 36\!\cdots\!20 \)
|
| $13$ |
\( T^{18} + \cdots - 41\!\cdots\!46 \)
|
| $17$ |
\( T^{18} + \cdots - 92\!\cdots\!32 \)
|
| $19$ |
\( T^{18} + \cdots - 55\!\cdots\!40 \)
|
| $23$ |
\( T^{18} + \cdots - 52\!\cdots\!00 \)
|
| $29$ |
\( T^{18} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( T^{18} + \cdots + 10\!\cdots\!32 \)
|
| $37$ |
\( T^{18} + \cdots - 34\!\cdots\!47 \)
|
| $41$ |
\( T^{18} + \cdots + 62\!\cdots\!80 \)
|
| $43$ |
\( T^{18} + \cdots - 23\!\cdots\!92 \)
|
| $47$ |
\( T^{18} + \cdots - 45\!\cdots\!16 \)
|
| $53$ |
\( T^{18} + \cdots - 42\!\cdots\!60 \)
|
| $59$ |
\( T^{18} + \cdots + 52\!\cdots\!00 \)
|
| $61$ |
\( T^{18} + \cdots + 36\!\cdots\!23 \)
|
| $67$ |
\( T^{18} + \cdots - 33\!\cdots\!24 \)
|
| $71$ |
\( T^{18} + \cdots - 15\!\cdots\!24 \)
|
| $73$ |
\( T^{18} + \cdots - 57\!\cdots\!90 \)
|
| $79$ |
\( T^{18} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots - 33\!\cdots\!36 \)
|
| $89$ |
\( T^{18} + \cdots + 14\!\cdots\!85 \)
|
| $97$ |
\( T^{18} + \cdots - 68\!\cdots\!48 \)
|
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