Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,11,Mod(5,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.5");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.d (of order \(6\), degree \(2\), 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: | \(11.4364305481\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 10 x^{19} + 63819 x^{18} - 574086 x^{17} + 1685636151 x^{16} - 13472077884 x^{15} + \cdots + 42\!\cdots\!17 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{46}\cdot 3^{37} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{20} - 10 x^{19} + 63819 x^{18} - 574086 x^{17} + 1685636151 x^{16} - 13472077884 x^{15} + \cdots + 42\!\cdots\!17 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 53\!\cdots\!04 \nu^{19} + \cdots + 96\!\cdots\!63 ) / 21\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 42\!\cdots\!57 \nu^{19} + \cdots - 33\!\cdots\!02 ) / 66\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 42\!\cdots\!57 \nu^{19} + \cdots - 35\!\cdots\!49 ) / 66\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!72 \nu^{19} + \cdots - 64\!\cdots\!73 ) / 13\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\!\cdots\!03 \nu^{19} + \cdots + 78\!\cdots\!76 ) / 13\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 25\!\cdots\!50 \nu^{19} + \cdots + 89\!\cdots\!70 ) / 41\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 30\!\cdots\!53 \nu^{19} + \cdots - 36\!\cdots\!26 ) / 41\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 30\!\cdots\!54 \nu^{19} + \cdots + 18\!\cdots\!29 ) / 41\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 40\!\cdots\!62 \nu^{19} + \cdots - 20\!\cdots\!14 ) / 41\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 47\!\cdots\!51 \nu^{19} + \cdots + 98\!\cdots\!78 ) / 41\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 19\!\cdots\!99 \nu^{19} + \cdots - 13\!\cdots\!28 ) / 17\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 25\!\cdots\!54 \nu^{19} + \cdots - 30\!\cdots\!25 ) / 20\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 95\!\cdots\!07 \nu^{19} + \cdots + 36\!\cdots\!31 ) / 41\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 55\!\cdots\!53 \nu^{19} + \cdots + 18\!\cdots\!64 ) / 17\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 14\!\cdots\!25 \nu^{19} + \cdots + 68\!\cdots\!34 ) / 41\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 20\!\cdots\!83 \nu^{19} + \cdots + 38\!\cdots\!87 ) / 41\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 26\!\cdots\!05 \nu^{19} + \cdots + 46\!\cdots\!86 ) / 41\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 54\!\cdots\!34 \nu^{19} + \cdots - 50\!\cdots\!91 ) / 41\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 80\!\cdots\!05 \nu^{19} + \cdots + 40\!\cdots\!69 ) / 41\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 10 \beta_{17} - 30 \beta_{14} + 8 \beta_{13} - 10 \beta_{12} + 8 \beta_{11} + 2 \beta_{10} + \cdots + 1492 ) / 3888 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 316 \beta_{17} + 648 \beta_{15} - 516 \beta_{14} + 170 \beta_{13} - 46 \beta_{12} - 262 \beta_{11} + \cdots - 12396926 ) / 1944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 14256 \beta_{19} - 14256 \beta_{18} - 104222 \beta_{17} + 6480 \beta_{16} - 1296 \beta_{15} + \cdots - 253049042 ) / 3888 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 115020 \beta_{19} + 100764 \beta_{18} - 2498350 \beta_{17} + 3240 \beta_{16} - 4524012 \beta_{15} + \cdots + 67970418806 ) / 972 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 270059184 \beta_{19} + 272217024 \beta_{18} + 1229184508 \beta_{17} - 456229584 \beta_{16} + \cdots + 5748673158226 ) / 3888 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5926077288 \beta_{19} - 5112591696 \beta_{18} + 71672748418 \beta_{17} - 684360576 \beta_{16} + \cdots - 16\!\cdots\!38 ) / 1944 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4534948582416 \beta_{19} - 4612226817744 \beta_{18} - 15407050889594 \beta_{17} + \cdots - 10\!\cdots\!20 ) / 3888 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 29605866477210 \beta_{19} + 25031329693974 \beta_{18} - 250500745288382 \beta_{17} + \cdots + 55\!\cdots\!23 ) / 486 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 73\!\cdots\!36 \beta_{19} + \cdots + 18\!\cdots\!78 ) / 3888 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 21\!\cdots\!48 \beta_{19} + \cdots - 30\!\cdots\!44 ) / 1944 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 11\!\cdots\!20 \beta_{19} + \cdots - 31\!\cdots\!06 ) / 3888 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 19\!\cdots\!16 \beta_{19} + \cdots + 21\!\cdots\!34 ) / 972 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 18\!\cdots\!40 \beta_{19} + \cdots + 50\!\cdots\!56 ) / 3888 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 65\!\cdots\!56 \beta_{19} + \cdots - 61\!\cdots\!02 ) / 1944 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 29\!\cdots\!64 \beta_{19} + \cdots - 80\!\cdots\!42 ) / 3888 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 27\!\cdots\!90 \beta_{19} + \cdots + 22\!\cdots\!81 ) / 486 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 47\!\cdots\!52 \beta_{19} + \cdots + 12\!\cdots\!50 ) / 3888 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 18\!\cdots\!16 \beta_{19} + \cdots - 13\!\cdots\!90 ) / 1944 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 74\!\cdots\!32 \beta_{19} + \cdots - 20\!\cdots\!92 ) / 3888 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−19.5959 | + | 11.3137i | −237.011 | − | 53.6159i | 256.000 | − | 443.405i | −1517.40 | − | 876.072i | 5251.05 | − | 1630.82i | −13322.5 | − | 23075.2i | 11585.2i | 53299.7 | + | 25415.2i | 39646.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −19.5959 | + | 11.3137i | −196.987 | + | 142.285i | 256.000 | − | 443.405i | 3091.89 | + | 1785.10i | 2250.37 | − | 5016.87i | 12604.4 | + | 21831.4i | 11585.2i | 18558.8 | − | 56056.7i | −80784.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −19.5959 | + | 11.3137i | 16.6955 | − | 242.426i | 256.000 | − | 443.405i | 2723.51 | + | 1572.42i | 2415.57 | + | 4939.44i | −1768.70 | − | 3063.49i | 11585.2i | −58491.5 | − | 8094.85i | −71159.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −19.5959 | + | 11.3137i | 57.7167 | + | 236.046i | 256.000 | − | 443.405i | −2305.51 | − | 1331.09i | −3801.57 | − | 3972.55i | −3513.89 | − | 6086.24i | 11585.2i | −52386.6 | + | 27247.6i | 60238.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | −19.5959 | + | 11.3137i | 208.763 | − | 124.366i | 256.000 | − | 443.405i | −4471.98 | − | 2581.90i | −2683.86 | + | 4798.96i | 7939.56 | + | 13751.7i | 11585.2i | 28115.1 | − | 51926.2i | 116843. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 19.5959 | − | 11.3137i | −230.699 | + | 76.3332i | 256.000 | − | 443.405i | 1617.35 | + | 933.780i | −3657.16 | + | 4105.89i | −504.464 | − | 873.757i | − | 11585.2i | 47395.5 | − | 35220.0i | 42258.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 19.5959 | − | 11.3137i | −120.535 | − | 210.998i | 256.000 | − | 443.405i | −1022.89 | − | 590.567i | −4749.17 | − | 2771.00i | −2684.05 | − | 4648.92i | − | 11585.2i | −29991.5 | + | 50865.5i | −26726.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 19.5959 | − | 11.3137i | 1.63905 | + | 242.994i | 256.000 | − | 443.405i | −1947.00 | − | 1124.10i | 2781.29 | + | 4743.16i | 14864.4 | + | 25746.0i | − | 11585.2i | −59043.6 | + | 796.560i | −50871.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.9 | 19.5959 | − | 11.3137i | 223.765 | − | 94.7529i | 256.000 | − | 443.405i | 3162.64 | + | 1825.95i | 3312.88 | − | 4388.38i | 5852.43 | + | 10136.7i | − | 11585.2i | 41092.8 | − | 42404.8i | 82633.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.10 | 19.5959 | − | 11.3137i | 234.653 | + | 63.1409i | 256.000 | − | 443.405i | −4289.59 | − | 2476.60i | 5312.61 | − | 1417.50i | −13348.2 | − | 23119.8i | − | 11585.2i | 51075.4 | + | 29632.5i | −112078. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | −19.5959 | − | 11.3137i | −237.011 | + | 53.6159i | 256.000 | + | 443.405i | −1517.40 | + | 876.072i | 5251.05 | + | 1630.82i | −13322.5 | + | 23075.2i | − | 11585.2i | 53299.7 | − | 25415.2i | 39646.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −19.5959 | − | 11.3137i | −196.987 | − | 142.285i | 256.000 | + | 443.405i | 3091.89 | − | 1785.10i | 2250.37 | + | 5016.87i | 12604.4 | − | 21831.4i | − | 11585.2i | 18558.8 | + | 56056.7i | −80784.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −19.5959 | − | 11.3137i | 16.6955 | + | 242.426i | 256.000 | + | 443.405i | 2723.51 | − | 1572.42i | 2415.57 | − | 4939.44i | −1768.70 | + | 3063.49i | − | 11585.2i | −58491.5 | + | 8094.85i | −71159.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −19.5959 | − | 11.3137i | 57.7167 | − | 236.046i | 256.000 | + | 443.405i | −2305.51 | + | 1331.09i | −3801.57 | + | 3972.55i | −3513.89 | + | 6086.24i | − | 11585.2i | −52386.6 | − | 27247.6i | 60238.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | −19.5959 | − | 11.3137i | 208.763 | + | 124.366i | 256.000 | + | 443.405i | −4471.98 | + | 2581.90i | −2683.86 | − | 4798.96i | 7939.56 | − | 13751.7i | − | 11585.2i | 28115.1 | + | 51926.2i | 116843. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 19.5959 | + | 11.3137i | −230.699 | − | 76.3332i | 256.000 | + | 443.405i | 1617.35 | − | 933.780i | −3657.16 | − | 4105.89i | −504.464 | + | 873.757i | 11585.2i | 47395.5 | + | 35220.0i | 42258.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 19.5959 | + | 11.3137i | −120.535 | + | 210.998i | 256.000 | + | 443.405i | −1022.89 | + | 590.567i | −4749.17 | + | 2771.00i | −2684.05 | + | 4648.92i | 11585.2i | −29991.5 | − | 50865.5i | −26726.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 19.5959 | + | 11.3137i | 1.63905 | − | 242.994i | 256.000 | + | 443.405i | −1947.00 | + | 1124.10i | 2781.29 | − | 4743.16i | 14864.4 | − | 25746.0i | 11585.2i | −59043.6 | − | 796.560i | −50871.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.9 | 19.5959 | + | 11.3137i | 223.765 | + | 94.7529i | 256.000 | + | 443.405i | 3162.64 | − | 1825.95i | 3312.88 | + | 4388.38i | 5852.43 | − | 10136.7i | 11585.2i | 41092.8 | + | 42404.8i | 82633.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.10 | 19.5959 | + | 11.3137i | 234.653 | − | 63.1409i | 256.000 | + | 443.405i | −4289.59 | + | 2476.60i | 5312.61 | + | 1417.50i | −13348.2 | + | 23119.8i | 11585.2i | 51075.4 | − | 29632.5i | −112078. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.11.d.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 54.11.d.a | 20 | ||
| 9.c | even | 3 | 1 | 54.11.d.a | 20 | ||
| 9.c | even | 3 | 1 | 162.11.b.c | 20 | ||
| 9.d | odd | 6 | 1 | inner | 18.11.d.a | ✓ | 20 |
| 9.d | odd | 6 | 1 | 162.11.b.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.11.d.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 18.11.d.a | ✓ | 20 | 9.d | odd | 6 | 1 | inner |
| 54.11.d.a | 20 | 3.b | odd | 2 | 1 | ||
| 54.11.d.a | 20 | 9.c | even | 3 | 1 | ||
| 162.11.b.c | 20 | 9.c | even | 3 | 1 | ||
| 162.11.b.c | 20 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 512 T^{2} + 262144)^{5} \)
|
| $3$ |
\( T^{20} + \cdots + 51\!\cdots\!01 \)
|
| $5$ |
\( T^{20} + \cdots + 58\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 17\!\cdots\!00 \)
|
| $11$ |
\( T^{20} + \cdots + 18\!\cdots\!81 \)
|
| $13$ |
\( T^{20} + \cdots + 53\!\cdots\!56 \)
|
| $17$ |
\( T^{20} + \cdots + 75\!\cdots\!00 \)
|
| $19$ |
\( (T^{10} + \cdots + 33\!\cdots\!40)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 48\!\cdots\!56 \)
|
| $29$ |
\( T^{20} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{20} + \cdots + 34\!\cdots\!00 \)
|
| $37$ |
\( (T^{10} + \cdots + 64\!\cdots\!68)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 75\!\cdots\!25 \)
|
| $43$ |
\( T^{20} + \cdots + 12\!\cdots\!25 \)
|
| $47$ |
\( T^{20} + \cdots + 90\!\cdots\!16 \)
|
| $53$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 60\!\cdots\!25 \)
|
| $61$ |
\( T^{20} + \cdots + 35\!\cdots\!04 \)
|
| $67$ |
\( T^{20} + \cdots + 41\!\cdots\!25 \)
|
| $71$ |
\( T^{20} + \cdots + 47\!\cdots\!64 \)
|
| $73$ |
\( (T^{10} + \cdots + 20\!\cdots\!20)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 39\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 22\!\cdots\!84 \)
|
| $89$ |
\( T^{20} + \cdots + 20\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 45\!\cdots\!25 \)
|
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