Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [47,12,Mod(1,47)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("47.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 47 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 47.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: | \(36.1121294863\) |
| 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: |
\( x^{18} - 5 x^{17} - 26312 x^{16} + 112440 x^{15} + 283223176 x^{14} - 1063247008 x^{13} + \cdots - 59\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{23}\cdot 3^{5} \) |
| 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
\( x^{18} - 5 x^{17} - 26312 x^{16} + 112440 x^{15} + 283223176 x^{14} - 1063247008 x^{13} + \cdots - 59\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 27\!\cdots\!51 \nu^{17} + \cdots + 64\!\cdots\!92 ) / 14\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\!\cdots\!51 \nu^{17} + \cdots - 68\!\cdots\!92 ) / 14\!\cdots\!16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 41\!\cdots\!61 \nu^{17} + \cdots - 96\!\cdots\!48 ) / 64\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 24\!\cdots\!83 \nu^{17} + \cdots - 44\!\cdots\!60 ) / 10\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 22\!\cdots\!37 \nu^{17} + \cdots + 55\!\cdots\!12 ) / 64\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 31\!\cdots\!93 \nu^{17} + \cdots + 72\!\cdots\!80 ) / 64\!\cdots\!36 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 67\!\cdots\!43 \nu^{17} + \cdots - 16\!\cdots\!20 ) / 10\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10\!\cdots\!03 \nu^{17} + \cdots - 22\!\cdots\!40 ) / 10\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12\!\cdots\!09 \nu^{17} + \cdots + 30\!\cdots\!40 ) / 10\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 76\!\cdots\!99 \nu^{17} + \cdots - 18\!\cdots\!92 ) / 58\!\cdots\!76 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 29\!\cdots\!73 \nu^{17} + \cdots - 68\!\cdots\!80 ) / 16\!\cdots\!40 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 16\!\cdots\!31 \nu^{17} + \cdots + 39\!\cdots\!60 ) / 80\!\cdots\!20 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 24\!\cdots\!31 \nu^{17} + \cdots + 57\!\cdots\!80 ) / 10\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 31\!\cdots\!59 \nu^{17} + \cdots + 73\!\cdots\!20 ) / 10\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 34\!\cdots\!91 \nu^{17} + \cdots - 81\!\cdots\!40 ) / 10\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 39\!\cdots\!73 \nu^{17} + \cdots - 93\!\cdots\!60 ) / 98\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2925 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{7} + 2\beta_{6} - 4\beta_{4} + 5\beta_{3} - 2\beta_{2} + 4787\beta _1 + 1865 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 21 \beta_{17} + 38 \beta_{16} - \beta_{15} + 4 \beta_{14} + 3 \beta_{13} - 8 \beta_{12} + \cdots + 14004970 \)
|
| \(\nu^{5}\) | \(=\) |
\( 125 \beta_{17} - 434 \beta_{16} - 2489 \beta_{15} + 1800 \beta_{14} + \beta_{13} - 756 \beta_{12} + \cdots + 34567686 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 176647 \beta_{17} + 418070 \beta_{16} - 33165 \beta_{15} + 66392 \beta_{14} + 54621 \beta_{13} + \cdots + 76745219638 \)
|
| \(\nu^{7}\) | \(=\) |
\( 561613 \beta_{17} - 3047346 \beta_{16} - 30167745 \beta_{15} + 20062104 \beta_{14} + 270817 \beta_{13} + \cdots + 448202576342 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1236168295 \beta_{17} + 3495121206 \beta_{16} - 558827181 \beta_{15} + 671253592 \beta_{14} + \cdots + 449463084771998 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3339373275 \beta_{17} - 8448198786 \beta_{16} - 272532213865 \beta_{15} + 165272790776 \beta_{14} + \cdots + 46\!\cdots\!54 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 8342637921215 \beta_{17} + 26692507998374 \beta_{16} - 6572188330165 \beta_{15} + \cdots + 27\!\cdots\!54 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 84525120080659 \beta_{17} + 71460278770990 \beta_{16} + \cdots + 43\!\cdots\!34 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 56\!\cdots\!27 \beta_{17} + \cdots + 17\!\cdots\!38 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 10\!\cdots\!43 \beta_{17} + \cdots + 36\!\cdots\!34 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 38\!\cdots\!71 \beta_{17} + \cdots + 11\!\cdots\!14 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 97\!\cdots\!47 \beta_{17} + \cdots + 30\!\cdots\!74 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 26\!\cdots\!43 \beta_{17} + \cdots + 74\!\cdots\!66 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 85\!\cdots\!67 \beta_{17} + \cdots + 24\!\cdots\!54 \)
|
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 |
|
−87.3249 | −37.8064 | 5577.65 | 6811.29 | 3301.44 | 70830.0 | −308226. | −175718. | −594795. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −77.8439 | −399.773 | 4011.68 | 4588.56 | 31119.9 | −41827.3 | −152860. | −17328.2 | −357191. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −66.8877 | −28.4948 | 2425.96 | −11134.2 | 1905.95 | 26041.5 | −25280.7 | −176335. | 744743. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −63.5911 | 659.263 | 1995.83 | −248.237 | −41923.3 | 33939.5 | 3317.62 | 257481. | 15785.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −55.7801 | 182.610 | 1063.42 | 13060.1 | −10186.0 | −19355.1 | 54919.8 | −143801. | −728492. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −36.9731 | −398.879 | −680.992 | −7977.72 | 14747.8 | −20638.4 | 100899. | −18042.6 | 294961. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −34.7749 | 652.864 | −838.704 | −9282.22 | −22703.3 | 2890.09 | 100385. | 249084. | 322789. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −20.8107 | −787.695 | −1614.91 | −4473.09 | 16392.5 | −81871.1 | 76227.8 | 443316. | 93088.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −5.12951 | −237.386 | −2021.69 | 5234.83 | 1217.67 | 3457.15 | 20875.5 | −120795. | −26852.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.188144 | −489.098 | −2047.96 | −10054.8 | −92.0206 | 65019.5 | −770.630 | 62069.7 | −1891.75 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 23.3430 | −768.011 | −1503.10 | 1254.36 | −17927.7 | 43688.7 | −82893.4 | 412694. | 29280.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 23.3513 | 253.607 | −1502.71 | 2547.55 | 5922.07 | 37727.3 | −82914.0 | −112830. | 59488.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 30.6155 | 633.376 | −1110.69 | −1280.84 | 19391.1 | −38082.7 | −96704.9 | 224018. | −39213.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 48.4504 | 208.371 | 299.444 | 8128.67 | 10095.6 | −34048.1 | −84718.3 | −133729. | 393837. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 65.2549 | −2.37739 | 2210.20 | −5959.99 | −155.136 | 43997.2 | 10584.2 | −177141. | −388918. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 66.0577 | −690.504 | 2315.62 | 6211.10 | −45613.1 | −13464.5 | 17678.7 | 299649. | 410291. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 72.6564 | 484.922 | 3230.95 | −12337.1 | 35232.7 | −69037.6 | 85948.6 | 58002.6 | −896370. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 78.1986 | −226.989 | 4067.03 | 3445.84 | −17750.2 | −74182.1 | 157885. | −125623. | 269460. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(47\) | \(-1\) |
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 | 47.12.a.a | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.12.a.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} + 41 T_{2}^{17} - 25530 T_{2}^{16} - 945176 T_{2}^{15} + 267296376 T_{2}^{14} + \cdots - 82\!\cdots\!52 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(47))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots - 82\!\cdots\!52 \)
|
| $3$ |
\( T^{18} + \cdots - 57\!\cdots\!04 \)
|
| $5$ |
\( T^{18} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{18} + \cdots - 36\!\cdots\!76 \)
|
| $11$ |
\( T^{18} + \cdots - 89\!\cdots\!80 \)
|
| $13$ |
\( T^{18} + \cdots + 16\!\cdots\!84 \)
|
| $17$ |
\( T^{18} + \cdots - 23\!\cdots\!40 \)
|
| $19$ |
\( T^{18} + \cdots + 12\!\cdots\!36 \)
|
| $23$ |
\( T^{18} + \cdots + 22\!\cdots\!24 \)
|
| $29$ |
\( T^{18} + \cdots + 30\!\cdots\!40 \)
|
| $31$ |
\( T^{18} + \cdots + 32\!\cdots\!72 \)
|
| $37$ |
\( T^{18} + \cdots - 51\!\cdots\!80 \)
|
| $41$ |
\( T^{18} + \cdots + 20\!\cdots\!84 \)
|
| $43$ |
\( T^{18} + \cdots + 12\!\cdots\!00 \)
|
| $47$ |
\( (T - 229345007)^{18} \)
|
| $53$ |
\( T^{18} + \cdots + 74\!\cdots\!44 \)
|
| $59$ |
\( T^{18} + \cdots + 66\!\cdots\!56 \)
|
| $61$ |
\( T^{18} + \cdots - 15\!\cdots\!00 \)
|
| $67$ |
\( T^{18} + \cdots + 88\!\cdots\!16 \)
|
| $71$ |
\( T^{18} + \cdots - 99\!\cdots\!00 \)
|
| $73$ |
\( T^{18} + \cdots + 93\!\cdots\!00 \)
|
| $79$ |
\( T^{18} + \cdots + 60\!\cdots\!60 \)
|
| $83$ |
\( T^{18} + \cdots + 27\!\cdots\!68 \)
|
| $89$ |
\( T^{18} + \cdots - 11\!\cdots\!20 \)
|
| $97$ |
\( T^{18} + \cdots + 93\!\cdots\!20 \)
|
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