Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [46,11,Mod(45,46)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46.45");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 46.b (of order \(2\), degree \(1\), 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: | \(29.2264336230\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 120620418 x^{18} + 14359423740 x^{17} + \cdots + 24\!\cdots\!12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | multiple of \( 2^{62}\cdot 3^{4}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} + 120620418 x^{18} + 14359423740 x^{17} + \cdots + 24\!\cdots\!12 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 26\!\cdots\!22 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 51\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 98\!\cdots\!75 \nu^{19} + \cdots + 57\!\cdots\!72 ) / 29\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 56\!\cdots\!77 \nu^{19} + \cdots - 22\!\cdots\!64 ) / 14\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 51\!\cdots\!47 \nu^{19} + \cdots - 45\!\cdots\!28 ) / 37\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!43 \nu^{19} + \cdots + 18\!\cdots\!52 ) / 10\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19\!\cdots\!93 \nu^{19} + \cdots + 20\!\cdots\!36 ) / 14\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 43\!\cdots\!91 \nu^{19} + \cdots + 85\!\cdots\!40 ) / 14\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 92\!\cdots\!99 \nu^{19} + \cdots - 16\!\cdots\!24 ) / 10\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11\!\cdots\!53 \nu^{19} + \cdots + 76\!\cdots\!28 ) / 10\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 21\!\cdots\!94 \nu^{19} + \cdots + 50\!\cdots\!80 ) / 14\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 77\!\cdots\!43 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 50\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 57\!\cdots\!69 \nu^{19} + \cdots + 79\!\cdots\!36 ) / 25\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 35\!\cdots\!69 \nu^{19} + \cdots + 45\!\cdots\!16 ) / 12\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 25\!\cdots\!97 \nu^{19} + \cdots + 62\!\cdots\!64 ) / 74\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 56\!\cdots\!95 \nu^{19} + \cdots - 56\!\cdots\!08 ) / 10\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 86\!\cdots\!35 \nu^{19} + \cdots + 20\!\cdots\!32 ) / 14\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 81\!\cdots\!39 \nu^{19} + \cdots - 13\!\cdots\!76 ) / 10\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 13\!\cdots\!15 \nu^{19} + \cdots + 41\!\cdots\!76 ) / 10\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 14\!\cdots\!87 \nu^{19} + \cdots + 39\!\cdots\!96 ) / 10\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2 \beta_{18} + \beta_{17} + \beta_{16} + 4 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \cdots - 12060212 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 81 \beta_{19} - 81 \beta_{18} - 17443 \beta_{17} + 8550 \beta_{16} + 15981 \beta_{15} + \cdots - 2218701337 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 3050809 \beta_{19} - 61737913 \beta_{18} - 33686228 \beta_{17} - 2022431 \beta_{16} + \cdots + 249555470342379 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3293454927 \beta_{19} + 5049468135 \beta_{18} + 671098803882 \beta_{17} - 335747076570 \beta_{16} + \cdots + 88\!\cdots\!17 \)
|
| \(\nu^{6}\) | \(=\) |
\( 157215249582401 \beta_{19} + \cdots - 59\!\cdots\!67 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 11\!\cdots\!81 \beta_{19} + \cdots - 29\!\cdots\!29 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 60\!\cdots\!45 \beta_{19} + \cdots + 15\!\cdots\!83 \)
|
| \(\nu^{9}\) | \(=\) |
\( 40\!\cdots\!15 \beta_{19} + \cdots + 87\!\cdots\!69 \)
|
| \(\nu^{10}\) | \(=\) |
\( 21\!\cdots\!17 \beta_{19} + \cdots - 41\!\cdots\!55 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 13\!\cdots\!61 \beta_{19} + \cdots - 23\!\cdots\!61 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 70\!\cdots\!45 \beta_{19} + \cdots + 11\!\cdots\!71 \)
|
| \(\nu^{13}\) | \(=\) |
\( 47\!\cdots\!99 \beta_{19} + \cdots + 56\!\cdots\!25 \)
|
| \(\nu^{14}\) | \(=\) |
\( 23\!\cdots\!45 \beta_{19} + \cdots - 34\!\cdots\!11 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 16\!\cdots\!41 \beta_{19} + \cdots - 10\!\cdots\!65 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 74\!\cdots\!89 \beta_{19} + \cdots + 10\!\cdots\!99 \)
|
| \(\nu^{17}\) | \(=\) |
\( 54\!\cdots\!03 \beta_{19} + \cdots + 11\!\cdots\!69 \)
|
| \(\nu^{18}\) | \(=\) |
\( 23\!\cdots\!65 \beta_{19} + \cdots - 31\!\cdots\!19 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 18\!\cdots\!69 \beta_{19} + \cdots + 33\!\cdots\!15 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/46\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 45.1 |
|
−22.6274 | −356.088 | 512.000 | − | 2602.68i | 8057.35 | − | 4637.41i | −11585.2 | 67749.5 | 58892.0i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.2 | −22.6274 | −356.088 | 512.000 | 2602.68i | 8057.35 | 4637.41i | −11585.2 | 67749.5 | − | 58892.0i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.3 | −22.6274 | −127.392 | 512.000 | − | 3722.70i | 2882.54 | − | 27422.0i | −11585.2 | −42820.4 | 84235.2i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.4 | −22.6274 | −127.392 | 512.000 | 3722.70i | 2882.54 | 27422.0i | −11585.2 | −42820.4 | − | 84235.2i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.5 | −22.6274 | 29.9811 | 512.000 | − | 5626.40i | −678.395 | 23833.7i | −11585.2 | −58150.1 | 127311.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.6 | −22.6274 | 29.9811 | 512.000 | 5626.40i | −678.395 | − | 23833.7i | −11585.2 | −58150.1 | − | 127311.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.7 | −22.6274 | 191.103 | 512.000 | − | 1180.32i | −4324.16 | − | 17666.5i | −11585.2 | −22528.8 | 26707.5i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.8 | −22.6274 | 191.103 | 512.000 | 1180.32i | −4324.16 | 17666.5i | −11585.2 | −22528.8 | − | 26707.5i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.9 | −22.6274 | 365.746 | 512.000 | − | 1225.89i | −8275.89 | 7881.79i | −11585.2 | 74721.1 | 27738.6i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.10 | −22.6274 | 365.746 | 512.000 | 1225.89i | −8275.89 | − | 7881.79i | −11585.2 | 74721.1 | − | 27738.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.11 | 22.6274 | −464.105 | 512.000 | − | 4984.71i | −10501.5 | − | 20466.0i | 11585.2 | 156344. | − | 112791.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.12 | 22.6274 | −464.105 | 512.000 | 4984.71i | −10501.5 | 20466.0i | 11585.2 | 156344. | 112791.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.13 | 22.6274 | −209.515 | 512.000 | − | 1077.37i | −4740.79 | 6799.51i | 11585.2 | −15152.3 | − | 24378.1i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.14 | 22.6274 | −209.515 | 512.000 | 1077.37i | −4740.79 | − | 6799.51i | 11585.2 | −15152.3 | 24378.1i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.15 | 22.6274 | −90.1332 | 512.000 | − | 4281.32i | −2039.48 | 14153.3i | 11585.2 | −50925.0 | − | 96875.3i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.16 | 22.6274 | −90.1332 | 512.000 | 4281.32i | −2039.48 | − | 14153.3i | 11585.2 | −50925.0 | 96875.3i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.17 | 22.6274 | 224.180 | 512.000 | − | 1828.23i | 5072.62 | − | 19992.6i | 11585.2 | −8792.19 | − | 41368.1i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.18 | 22.6274 | 224.180 | 512.000 | 1828.23i | 5072.62 | 19992.6i | 11585.2 | −8792.19 | 41368.1i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.19 | 22.6274 | 374.223 | 512.000 | − | 4304.29i | 8467.70 | 13952.6i | 11585.2 | 80993.8 | − | 97395.0i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.20 | 22.6274 | 374.223 | 512.000 | 4304.29i | 8467.70 | − | 13952.6i | 11585.2 | 80993.8 | 97395.0i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 46.11.b.a | ✓ | 20 |
| 23.b | odd | 2 | 1 | inner | 46.11.b.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.11.b.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 46.11.b.a | ✓ | 20 | 23.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(46, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 512)^{10} \)
|
| $3$ |
\( (T^{10} + \cdots - 69\!\cdots\!76)^{2} \)
|
| $5$ |
\( T^{20} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 53\!\cdots\!84 \)
|
| $11$ |
\( T^{20} + \cdots + 13\!\cdots\!84 \)
|
| $13$ |
\( (T^{10} + \cdots - 55\!\cdots\!12)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 39\!\cdots\!00 \)
|
| $19$ |
\( T^{20} + \cdots + 76\!\cdots\!16 \)
|
| $23$ |
\( T^{20} + \cdots + 14\!\cdots\!01 \)
|
| $29$ |
\( (T^{10} + \cdots - 50\!\cdots\!36)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots - 15\!\cdots\!48)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 27\!\cdots\!24 \)
|
| $41$ |
\( (T^{10} + \cdots + 18\!\cdots\!28)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 18\!\cdots\!16 \)
|
| $47$ |
\( (T^{10} + \cdots - 47\!\cdots\!84)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 40\!\cdots\!76 \)
|
| $59$ |
\( (T^{10} + \cdots + 19\!\cdots\!44)^{2} \)
|
| $61$ |
\( T^{20} + \cdots + 50\!\cdots\!64 \)
|
| $67$ |
\( T^{20} + \cdots + 70\!\cdots\!84 \)
|
| $71$ |
\( (T^{10} + \cdots - 82\!\cdots\!52)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots - 13\!\cdots\!16)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 38\!\cdots\!44 \)
|
| $83$ |
\( T^{20} + \cdots + 53\!\cdots\!00 \)
|
| $89$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 23\!\cdots\!24 \)
|
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