Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,2,Mod(31,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.g (of order \(11\), degree \(10\), minimal) |
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: | no |
| Analytic conductor: | \(1.83655924649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{11})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} + \cdots + 529 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} + \cdots + 529 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 25\!\cdots\!20 \nu^{19} + \cdots + 57\!\cdots\!44 ) / 14\!\cdots\!87 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 68\!\cdots\!51 \nu^{19} + \cdots - 28\!\cdots\!27 ) / 14\!\cdots\!87 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!36 \nu^{19} + \cdots - 61\!\cdots\!32 ) / 14\!\cdots\!87 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 64\!\cdots\!88 \nu^{19} + \cdots - 88\!\cdots\!56 ) / 14\!\cdots\!87 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 87\!\cdots\!15 \nu^{19} + \cdots + 23\!\cdots\!08 ) / 14\!\cdots\!87 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!91 \nu^{19} + \cdots + 24\!\cdots\!54 ) / 14\!\cdots\!87 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16\!\cdots\!64 \nu^{19} + \cdots + 56\!\cdots\!03 ) / 14\!\cdots\!87 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 76\!\cdots\!98 \nu^{19} + \cdots - 94\!\cdots\!93 ) / 61\!\cdots\!69 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 21\!\cdots\!27 \nu^{19} + \cdots - 39\!\cdots\!82 ) / 14\!\cdots\!87 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 22\!\cdots\!84 \nu^{19} + \cdots - 32\!\cdots\!98 ) / 14\!\cdots\!87 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 22\!\cdots\!71 \nu^{19} + \cdots - 58\!\cdots\!01 ) / 14\!\cdots\!87 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 28\!\cdots\!65 \nu^{19} + \cdots + 30\!\cdots\!95 ) / 14\!\cdots\!87 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 29\!\cdots\!15 \nu^{19} + \cdots - 61\!\cdots\!61 ) / 14\!\cdots\!87 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 40\!\cdots\!38 \nu^{19} + \cdots + 34\!\cdots\!17 ) / 14\!\cdots\!87 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 17\!\cdots\!17 \nu^{19} + \cdots + 94\!\cdots\!20 ) / 61\!\cdots\!69 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 56\!\cdots\!79 \nu^{19} + \cdots - 13\!\cdots\!22 ) / 14\!\cdots\!87 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 72\!\cdots\!74 \nu^{19} + \cdots - 38\!\cdots\!34 ) / 14\!\cdots\!87 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 14\!\cdots\!30 \nu^{19} + \cdots - 16\!\cdots\!18 ) / 14\!\cdots\!87 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + \cdots + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{19} + \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} + \cdots - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18 \beta_{19} - 2 \beta_{18} + 25 \beta_{16} + 12 \beta_{15} - 27 \beta_{14} + 18 \beta_{13} - 15 \beta_{12} + \cdots - 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 29 \beta_{19} - 79 \beta_{18} + 52 \beta_{17} - 70 \beta_{16} - 27 \beta_{15} - 32 \beta_{14} + \cdots - 62 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 56 \beta_{19} - 239 \beta_{18} + 282 \beta_{17} - 369 \beta_{16} - 239 \beta_{15} + 37 \beta_{14} + \cdots - 89 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1081 \beta_{19} - 361 \beta_{18} + 535 \beta_{17} - 1933 \beta_{16} - 1081 \beta_{15} + 1400 \beta_{14} + \cdots + 852 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3798 \beta_{19} + 2438 \beta_{18} - 1058 \beta_{17} - 1773 \beta_{16} - 1058 \beta_{15} + 4464 \beta_{14} + \cdots + 4464 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 5135 \beta_{19} + 16473 \beta_{18} - 15916 \beta_{17} + 15193 \beta_{16} + 8789 \beta_{15} + \cdots + 14706 \)
|
| \(\nu^{10}\) | \(=\) |
\( 36734 \beta_{19} + 50703 \beta_{18} - 59591 \beta_{17} + 118069 \beta_{16} + 66327 \beta_{15} + \cdots - 9938 \)
|
| \(\nu^{11}\) | \(=\) |
\( 256857 \beta_{19} - 76135 \beta_{17} + 344581 \beta_{16} + 196934 \beta_{15} - 318507 \beta_{14} + \cdots - 232313 \)
|
| \(\nu^{12}\) | \(=\) |
\( 778490 \beta_{19} - 778490 \beta_{18} + 560354 \beta_{17} - 972564 \beta_{14} + 654831 \beta_{13} + \cdots - 1153146 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 3979257 \beta_{18} + 3979257 \beta_{17} - 5344858 \beta_{16} - 3042920 \beta_{15} - 1672508 \beta_{13} + \cdots - 2111735 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 11985926 \beta_{19} - 8599271 \beta_{18} + 11985926 \beta_{17} - 27688350 \beta_{16} - 15700939 \beta_{15} + \cdots + 6433964 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 61492387 \beta_{19} + 17642007 \beta_{18} - 58907654 \beta_{16} - 33525393 \beta_{15} + \cdots + 66536352 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 132340428 \beta_{19} + 221906105 \beta_{18} - 184749586 \beta_{17} + 120271015 \beta_{16} + \cdots + 242001342 \)
|
| \(\nu^{17}\) | \(=\) |
\( 271144644 \beta_{19} + 873784229 \beta_{18} - 949293604 \beta_{17} + 1539365880 \beta_{16} + \cdots + 217095417 \)
|
| \(\nu^{18}\) | \(=\) |
\( 3426020339 \beta_{19} + 1059630164 \beta_{18} - 2039347103 \beta_{17} + 6028541248 \beta_{16} + \cdots - 2602520909 \)
|
| \(\nu^{19}\) | \(=\) |
\( 13471243579 \beta_{19} - 7998279224 \beta_{18} + 4175877645 \beta_{17} + 7359415616 \beta_{16} + \cdots - 16796005904 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(51\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0.142315 | − | 0.989821i | −1.10832 | − | 2.42687i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | −2.55990 | + | 0.751655i | −0.986597 | − | 0.634047i | −0.415415 | + | 0.909632i | −2.69677 | + | 3.11224i | −0.841254 | + | 0.540641i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0.142315 | − | 0.989821i | 1.40549 | + | 3.07760i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 3.24630 | − | 0.953198i | 2.97617 | + | 1.91267i | −0.415415 | + | 0.909632i | −5.53162 | + | 6.38383i | −0.841254 | + | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | 0.959493 | + | 0.281733i | −0.880272 | + | 1.01589i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | −1.13082 | + | 0.726736i | 0.394126 | + | 0.863015i | 0.654861 | + | 0.755750i | 0.169795 | + | 1.18095i | −0.415415 | + | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | 0.959493 | + | 0.281733i | 1.48208 | − | 1.71041i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 1.90392 | − | 1.22358i | −0.119262 | − | 0.261148i | 0.654861 | + | 0.755750i | −0.302001 | − | 2.10046i | −0.415415 | + | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.1 | −0.841254 | − | 0.540641i | −0.420808 | − | 2.92679i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −1.22833 | + | 2.68968i | 2.73530 | − | 3.15671i | 0.142315 | − | 0.989821i | −5.51052 | + | 1.61803i | 0.654861 | + | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.2 | −0.841254 | − | 0.540641i | 0.0390478 | + | 0.271583i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | 0.113980 | − | 0.249581i | −0.365070 | + | 0.421313i | 0.142315 | − | 0.989821i | 2.80625 | − | 0.823988i | 0.654861 | + | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.1 | −0.841254 | + | 0.540641i | −0.420808 | + | 2.92679i | 0.415415 | − | 0.909632i | −0.959493 | + | 0.281733i | −1.22833 | − | 2.68968i | 2.73530 | + | 3.15671i | 0.142315 | + | 0.989821i | −5.51052 | − | 1.61803i | 0.654861 | − | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.2 | −0.841254 | + | 0.540641i | 0.0390478 | − | 0.271583i | 0.415415 | − | 0.909632i | −0.959493 | + | 0.281733i | 0.113980 | + | 0.249581i | −0.365070 | − | 0.421313i | 0.142315 | + | 0.989821i | 2.80625 | + | 0.823988i | 0.654861 | − | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.1 | 0.959493 | − | 0.281733i | −0.880272 | − | 1.01589i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | −1.13082 | − | 0.726736i | 0.394126 | − | 0.863015i | 0.654861 | − | 0.755750i | 0.169795 | − | 1.18095i | −0.415415 | − | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | 0.959493 | − | 0.281733i | 1.48208 | + | 1.71041i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | 1.90392 | + | 1.22358i | −0.119262 | + | 0.261148i | 0.654861 | − | 0.755750i | −0.302001 | + | 2.10046i | −0.415415 | − | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.1 | −0.415415 | + | 0.909632i | −1.50807 | − | 0.442808i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | 1.02927 | − | 1.18784i | −0.440465 | + | 3.06350i | 0.959493 | − | 0.281733i | −0.445574 | − | 0.286353i | 0.142315 | + | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | −0.415415 | + | 0.909632i | −0.248602 | − | 0.0729960i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | 0.169672 | − | 0.195812i | 0.663794 | − | 4.61679i | 0.959493 | − | 0.281733i | −2.46729 | − | 1.58563i | 0.142315 | + | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.1 | 0.654861 | − | 0.755750i | −1.71555 | + | 1.10252i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | −0.290219 | + | 2.01852i | 3.10381 | − | 0.911362i | −0.841254 | − | 0.540641i | 0.481321 | − | 1.05395i | 0.959493 | + | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.2 | 0.654861 | − | 0.755750i | 1.45499 | − | 0.935068i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | 0.246141 | − | 1.71195i | 1.53819 | − | 0.451652i | −0.841254 | − | 0.540641i | −0.00358844 | + | 0.00785758i | 0.959493 | + | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 141.1 | 0.142315 | + | 0.989821i | −1.10832 | + | 2.42687i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | −2.55990 | − | 0.751655i | −0.986597 | + | 0.634047i | −0.415415 | − | 0.909632i | −2.69677 | − | 3.11224i | −0.841254 | − | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 141.2 | 0.142315 | + | 0.989821i | 1.40549 | − | 3.07760i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 3.24630 | + | 0.953198i | 2.97617 | − | 1.91267i | −0.415415 | − | 0.909632i | −5.53162 | − | 6.38383i | −0.841254 | − | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.1 | 0.654861 | + | 0.755750i | −1.71555 | − | 1.10252i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | −0.290219 | − | 2.01852i | 3.10381 | + | 0.911362i | −0.841254 | + | 0.540641i | 0.481321 | + | 1.05395i | 0.959493 | − | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.2 | 0.654861 | + | 0.755750i | 1.45499 | + | 0.935068i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | 0.246141 | + | 1.71195i | 1.53819 | + | 0.451652i | −0.841254 | + | 0.540641i | −0.00358844 | − | 0.00785758i | 0.959493 | − | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 211.1 | −0.415415 | − | 0.909632i | −1.50807 | + | 0.442808i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | 1.02927 | + | 1.18784i | −0.440465 | − | 3.06350i | 0.959493 | + | 0.281733i | −0.445574 | + | 0.286353i | 0.142315 | − | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 211.2 | −0.415415 | − | 0.909632i | −0.248602 | + | 0.0729960i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | 0.169672 | + | 0.195812i | 0.663794 | + | 4.61679i | 0.959493 | + | 0.281733i | −2.46729 | + | 1.58563i | 0.142315 | − | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.c | even | 11 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 230.2.g.b | ✓ | 20 |
| 23.c | even | 11 | 1 | inner | 230.2.g.b | ✓ | 20 |
| 23.c | even | 11 | 1 | 5290.2.a.bj | 10 | ||
| 23.d | odd | 22 | 1 | 5290.2.a.bi | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.2.g.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 230.2.g.b | ✓ | 20 | 23.c | even | 11 | 1 | inner |
| 5290.2.a.bi | 10 | 23.d | odd | 22 | 1 | ||
| 5290.2.a.bj | 10 | 23.c | even | 11 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 3 T_{3}^{19} + 21 T_{3}^{18} + 66 T_{3}^{17} + 252 T_{3}^{16} + 701 T_{3}^{15} + \cdots + 1024 \)
acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{20} + 3 T^{19} + \cdots + 1024 \)
|
| $5$ |
\( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{20} - 19 T^{19} + \cdots + 38809 \)
|
| $11$ |
\( T^{20} - 7 T^{19} + \cdots + 2374681 \)
|
| $13$ |
\( T^{20} + \cdots + 13416820561 \)
|
| $17$ |
\( T^{20} + \cdots + 605553664 \)
|
| $19$ |
\( T^{20} + \cdots + 392951329 \)
|
| $23$ |
\( T^{20} + \cdots + 41426511213649 \)
|
| $29$ |
\( T^{20} + \cdots + 3969466491904 \)
|
| $31$ |
\( T^{20} + 17 T^{19} + \cdots + 1024 \)
|
| $37$ |
\( T^{20} + \cdots + 7650363233041 \)
|
| $41$ |
\( T^{20} + \cdots + 1804635361 \)
|
| $43$ |
\( T^{20} + \cdots + 23760372736 \)
|
| $47$ |
\( (T^{10} - 21 T^{9} + \cdots + 130098649)^{2} \)
|
| $53$ |
\( T^{20} - 19 T^{19} + \cdots + 7447441 \)
|
| $59$ |
\( T^{20} + \cdots + 27292338778849 \)
|
| $61$ |
\( T^{20} + \cdots + 22\!\cdots\!04 \)
|
| $67$ |
\( T^{20} + \cdots + 246200702534656 \)
|
| $71$ |
\( T^{20} + \cdots + 95950331339776 \)
|
| $73$ |
\( T^{20} + \cdots + 859836921496576 \)
|
| $79$ |
\( T^{20} + \cdots + 90\!\cdots\!76 \)
|
| $83$ |
\( T^{20} + \cdots + 160435495936 \)
|
| $89$ |
\( T^{20} + \cdots + 20\!\cdots\!09 \)
|
| $97$ |
\( T^{20} + \cdots + 286047133533184 \)
|
| show more | |
| show less |