Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,4,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.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: | \(109.684550701\) |
| 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: |
\( x^{18} - 2 x^{17} - 108 x^{16} + 212 x^{15} + 4721 x^{14} - 8963 x^{13} - 107626 x^{12} + 194656 x^{11} + \cdots + 9847296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 143) |
| 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
\( x^{18} - 2 x^{17} - 108 x^{16} + 212 x^{15} + 4721 x^{14} - 8963 x^{13} - 107626 x^{12} + 194656 x^{11} + \cdots + 9847296 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 20\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 87\!\cdots\!43 \nu^{17} + \cdots + 69\!\cdots\!48 ) / 38\!\cdots\!12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\!\cdots\!81 \nu^{17} + \cdots + 17\!\cdots\!96 ) / 77\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 21\!\cdots\!85 \nu^{17} + \cdots - 26\!\cdots\!64 ) / 77\!\cdots\!24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\!\cdots\!15 \nu^{17} + \cdots + 43\!\cdots\!04 ) / 38\!\cdots\!12 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 18\!\cdots\!99 \nu^{17} + \cdots + 10\!\cdots\!00 ) / 38\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 47\!\cdots\!53 \nu^{17} + \cdots - 37\!\cdots\!80 ) / 96\!\cdots\!28 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11\!\cdots\!97 \nu^{17} + \cdots + 16\!\cdots\!20 ) / 19\!\cdots\!56 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 55\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!60 ) / 77\!\cdots\!24 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 57\!\cdots\!87 \nu^{17} + \cdots + 75\!\cdots\!60 ) / 77\!\cdots\!24 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 19\!\cdots\!00 \nu^{17} + \cdots - 24\!\cdots\!40 ) / 24\!\cdots\!32 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 12\!\cdots\!95 \nu^{17} + \cdots + 58\!\cdots\!40 ) / 14\!\cdots\!12 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 20\!\cdots\!57 \nu^{17} + \cdots - 96\!\cdots\!72 ) / 14\!\cdots\!12 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 14\!\cdots\!05 \nu^{17} + \cdots + 58\!\cdots\!56 ) / 96\!\cdots\!28 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 63\!\cdots\!33 \nu^{17} + \cdots - 28\!\cdots\!44 ) / 38\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 20\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} + 2 \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 243 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} + \beta_{16} - 3 \beta_{13} + \beta_{12} - 5 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots - 54 \)
|
| \(\nu^{6}\) | \(=\) |
\( 43 \beta_{17} + 5 \beta_{16} - 9 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 3 \beta_{12} + 110 \beta_{11} + \cdots + 5694 \)
|
| \(\nu^{7}\) | \(=\) |
\( 58 \beta_{17} + 42 \beta_{16} + 11 \beta_{15} + 41 \beta_{14} - 146 \beta_{13} + 34 \beta_{12} + \cdots - 2429 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1392 \beta_{17} + 243 \beta_{16} - 505 \beta_{15} - 164 \beta_{14} + 397 \beta_{13} - 177 \beta_{12} + \cdots + 142456 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2365 \beta_{17} + 1423 \beta_{16} + 839 \beta_{15} + 2748 \beta_{14} - 5235 \beta_{13} + 811 \beta_{12} + \cdots - 95658 \)
|
| \(\nu^{10}\) | \(=\) |
\( 40683 \beta_{17} + 8386 \beta_{16} - 20201 \beta_{15} - 9310 \beta_{14} + 15992 \beta_{13} + \cdots + 3703901 \)
|
| \(\nu^{11}\) | \(=\) |
\( 84881 \beta_{17} + 45180 \beta_{16} + 42082 \beta_{15} + 125248 \beta_{14} - 168488 \beta_{13} + \cdots - 3497365 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1132592 \beta_{17} + 253357 \beta_{16} - 712064 \beta_{15} - 429353 \beta_{14} + 565631 \beta_{13} + \cdots + 98915292 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2858244 \beta_{17} + 1392248 \beta_{16} + 1758968 \beta_{15} + 4867680 \beta_{14} - 5175540 \beta_{13} + \cdots - 122138637 \)
|
| \(\nu^{14}\) | \(=\) |
\( 30685237 \beta_{17} + 7156458 \beta_{16} - 23629696 \beta_{15} - 17447418 \beta_{14} + 18790318 \beta_{13} + \cdots + 2695761351 \)
|
| \(\nu^{15}\) | \(=\) |
\( 92570193 \beta_{17} + 42169053 \beta_{16} + 66545956 \beta_{15} + 173975032 \beta_{14} + \cdots - 4134600078 \)
|
| \(\nu^{16}\) | \(=\) |
\( 817043035 \beta_{17} + 194236989 \beta_{16} - 758941021 \beta_{15} - 653984098 \beta_{14} + \cdots + 74662407118 \)
|
| \(\nu^{17}\) | \(=\) |
\( 2919002722 \beta_{17} + 1262961878 \beta_{16} + 2368607115 \beta_{15} + 5916221793 \beta_{14} + \cdots - 136855788777 \)
|
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 |
|
−5.22956 | 1.03843 | 19.3483 | −12.1782 | −5.43054 | −36.1879 | −59.3469 | −25.9217 | 63.6866 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.83238 | 9.74664 | 15.3519 | −8.44122 | −47.0995 | 12.5767 | −35.5272 | 67.9971 | 40.7912 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.83029 | −7.08819 | 15.3317 | 9.07266 | 34.2380 | 2.15508 | −35.4141 | 23.2424 | −43.8235 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.97618 | −1.00306 | 7.81003 | 0.345299 | 3.98837 | 17.7538 | 0.755338 | −25.9939 | −1.37297 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.37885 | 7.92479 | 3.41661 | 15.9579 | −26.7767 | −29.7230 | 15.4866 | 35.8023 | −53.9193 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.22320 | 2.89827 | −3.05739 | −19.1319 | −6.44342 | −4.29594 | 24.5828 | −18.6000 | 42.5339 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.18512 | −9.62593 | −3.22525 | −10.6961 | 21.0338 | 26.7047 | 24.5285 | 65.6585 | 23.3723 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.26989 | 3.96847 | −6.38737 | 4.99906 | −5.03953 | 32.3730 | 18.2704 | −11.2513 | −6.34827 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.468651 | −6.65460 | −7.78037 | −3.45502 | 3.11869 | −11.5900 | 7.39549 | 17.2837 | 1.61920 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.370518 | −0.548979 | −7.86272 | 11.6192 | 0.203406 | −19.9902 | 5.87742 | −26.6986 | −4.30511 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.901974 | 8.17246 | −7.18644 | −11.3696 | 7.37135 | −8.81530 | −13.6978 | 39.7891 | −10.2551 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.32803 | −4.42417 | −2.58028 | 12.1881 | −10.2996 | 11.2388 | −24.6312 | −7.42671 | 28.3743 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.37174 | −6.00967 | −2.37485 | −22.0017 | −14.2534 | −21.9243 | −24.6064 | 9.11608 | −52.1824 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.40428 | 7.21858 | −2.21943 | 9.84845 | 17.3555 | 8.26104 | −24.5704 | 25.1078 | 23.6784 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 3.98165 | −8.48796 | 7.85352 | 2.25841 | −33.7961 | 28.5605 | −0.583227 | 45.0454 | 8.99219 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 4.27424 | 4.33951 | 10.2691 | −0.295728 | 18.5481 | −11.5737 | 9.69883 | −8.16862 | −1.26401 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 4.98751 | −4.47034 | 16.8752 | 20.1605 | −22.2959 | −23.8481 | 44.2654 | −7.01607 | 100.551 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 5.51522 | 3.00574 | 22.4176 | −18.8802 | 16.5773 | 0.324979 | 79.5165 | −17.9655 | −104.128 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1859.4.a.j | 18 | |
| 13.b | even | 2 | 1 | 1859.4.a.k | 18 | ||
| 13.d | odd | 4 | 2 | 143.4.b.a | ✓ | 36 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.b.a | ✓ | 36 | 13.d | odd | 4 | 2 | |
| 1859.4.a.j | 18 | 1.a | even | 1 | 1 | trivial | |
| 1859.4.a.k | 18 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} + 2 T_{2}^{17} - 108 T_{2}^{16} - 212 T_{2}^{15} + 4721 T_{2}^{14} + 8963 T_{2}^{13} + \cdots + 9847296 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + 2 T^{17} + \cdots + 9847296 \)
|
| $3$ |
\( T^{18} + \cdots - 179055689728 \)
|
| $5$ |
\( T^{18} + \cdots - 16\!\cdots\!84 \)
|
| $7$ |
\( T^{18} + \cdots - 20\!\cdots\!40 \)
|
| $11$ |
\( (T + 11)^{18} \)
|
| $13$ |
\( T^{18} \)
|
| $17$ |
\( T^{18} + \cdots - 14\!\cdots\!00 \)
|
| $19$ |
\( T^{18} + \cdots + 34\!\cdots\!04 \)
|
| $23$ |
\( T^{18} + \cdots - 69\!\cdots\!76 \)
|
| $29$ |
\( T^{18} + \cdots - 19\!\cdots\!96 \)
|
| $31$ |
\( T^{18} + \cdots - 76\!\cdots\!12 \)
|
| $37$ |
\( T^{18} + \cdots + 66\!\cdots\!92 \)
|
| $41$ |
\( T^{18} + \cdots + 16\!\cdots\!24 \)
|
| $43$ |
\( T^{18} + \cdots + 26\!\cdots\!04 \)
|
| $47$ |
\( T^{18} + \cdots + 10\!\cdots\!64 \)
|
| $53$ |
\( T^{18} + \cdots + 26\!\cdots\!24 \)
|
| $59$ |
\( T^{18} + \cdots - 14\!\cdots\!36 \)
|
| $61$ |
\( T^{18} + \cdots - 22\!\cdots\!56 \)
|
| $67$ |
\( T^{18} + \cdots + 46\!\cdots\!36 \)
|
| $71$ |
\( T^{18} + \cdots - 39\!\cdots\!52 \)
|
| $73$ |
\( T^{18} + \cdots - 26\!\cdots\!84 \)
|
| $79$ |
\( T^{18} + \cdots + 38\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots - 82\!\cdots\!76 \)
|
| $89$ |
\( T^{18} + \cdots - 26\!\cdots\!00 \)
|
| $97$ |
\( T^{18} + \cdots + 46\!\cdots\!68 \)
|
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