Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,9,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, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.5");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| 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: | \(7.33281498110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + \cdots + 19\!\cdots\!29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{22} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + \cdots + 19\!\cdots\!29 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 34\!\cdots\!96 \nu^{15} + \cdots + 63\!\cdots\!32 ) / 19\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 81\!\cdots\!41 \nu^{15} + \cdots - 11\!\cdots\!72 ) / 37\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 38\!\cdots\!49 \nu^{15} + \cdots - 62\!\cdots\!12 ) / 12\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 47\!\cdots\!81 \nu^{15} + \cdots + 18\!\cdots\!72 ) / 87\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 47\!\cdots\!81 \nu^{15} + \cdots + 19\!\cdots\!43 ) / 87\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 42\!\cdots\!02 \nu^{15} + \cdots - 27\!\cdots\!59 ) / 12\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 47\!\cdots\!09 \nu^{15} + \cdots + 62\!\cdots\!98 ) / 62\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 54\!\cdots\!84 \nu^{15} + \cdots + 26\!\cdots\!27 ) / 37\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 99\!\cdots\!79 \nu^{15} + \cdots - 36\!\cdots\!53 ) / 37\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 20\!\cdots\!20 \nu^{15} + \cdots - 76\!\cdots\!19 ) / 74\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\!\cdots\!21 \nu^{15} + \cdots - 56\!\cdots\!48 ) / 37\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 13\!\cdots\!19 \nu^{15} + \cdots + 54\!\cdots\!65 ) / 24\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 21\!\cdots\!81 \nu^{15} + \cdots - 33\!\cdots\!92 ) / 37\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 37\!\cdots\!96 \nu^{15} + \cdots - 30\!\cdots\!27 ) / 62\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 64\!\cdots\!87 \nu^{15} + \cdots - 59\!\cdots\!74 ) / 74\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{15} - 3 \beta_{14} - 8 \beta_{11} + \beta_{7} + 34 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} + \cdots + 206 ) / 432 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 20 \beta_{15} - 59 \beta_{14} + 92 \beta_{13} - 38 \beta_{12} - 58 \beta_{11} + 32 \beta_{8} + \cdots - 146924 ) / 216 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3672 \beta_{15} + 5635 \beta_{14} + 1302 \beta_{13} + 262 \beta_{12} + 9944 \beta_{11} + \cdots - 1593186 ) / 432 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 22378 \beta_{15} + 34620 \beta_{14} - 71740 \beta_{13} + 28764 \beta_{12} + 47418 \beta_{11} + \cdots + 94798366 ) / 108 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 6032470 \beta_{15} - 9352216 \beta_{14} - 1843886 \beta_{13} - 600646 \beta_{12} + \cdots + 2860174232 ) / 432 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 81622314 \beta_{15} - 86641573 \beta_{14} + 215082324 \beta_{13} - 81614740 \beta_{12} + \cdots - 279461384412 ) / 216 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 10020101254 \beta_{15} + 15541382265 \beta_{14} + 1962435236 \beta_{13} + 1432177632 \beta_{12} + \cdots - 4875666715538 ) / 432 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 35633514154 \beta_{15} + 29773148095 \beta_{14} - 82097391760 \beta_{13} + 29468779843 \beta_{12} + \cdots + 106744244912947 ) / 54 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 16589997986976 \beta_{15} - 25883898471449 \beta_{14} - 1650338438634 \beta_{13} + \cdots + 83\!\cdots\!86 ) / 432 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 244807683855094 \beta_{15} - 177684355620726 \beta_{14} + 511525743001360 \beta_{13} + \cdots - 66\!\cdots\!96 ) / 216 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 27\!\cdots\!38 \beta_{15} + \cdots - 14\!\cdots\!12 ) / 432 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 20\!\cdots\!82 \beta_{15} + \cdots + 51\!\cdots\!56 ) / 108 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 44\!\cdots\!86 \beta_{15} + \cdots + 25\!\cdots\!90 ) / 432 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 70\!\cdots\!80 \beta_{15} + \cdots - 16\!\cdots\!04 ) / 216 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 71\!\cdots\!24 \beta_{15} + \cdots - 43\!\cdots\!94 ) / 432 \)
|
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 |
|
−9.79796 | + | 5.65685i | −69.4086 | + | 41.7547i | 64.0000 | − | 110.851i | −719.139 | − | 415.195i | 443.862 | − | 801.745i | 1343.41 | + | 2326.85i | 1448.15i | 3074.10 | − | 5796.26i | 9394.80 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −9.79796 | + | 5.65685i | −24.1444 | − | 77.3178i | 64.0000 | − | 110.851i | −69.6596 | − | 40.2180i | 673.941 | + | 620.976i | 370.849 | + | 642.329i | 1448.15i | −5395.10 | + | 3733.58i | 910.029 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −9.79796 | + | 5.65685i | 22.8219 | + | 77.7185i | 64.0000 | − | 110.851i | 683.343 | + | 394.528i | −663.250 | − | 632.382i | 89.7132 | + | 155.388i | 1448.15i | −5519.32 | + | 3547.36i | −8927.15 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −9.79796 | + | 5.65685i | 77.7361 | − | 22.7616i | 64.0000 | − | 110.851i | −115.044 | − | 66.4206i | −632.896 | + | 662.760i | −1060.32 | − | 1836.53i | 1448.15i | 5524.82 | − | 3538.80i | 1502.93 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | 9.79796 | − | 5.65685i | −47.1686 | + | 65.8492i | 64.0000 | − | 110.851i | −476.096 | − | 274.874i | −89.6569 | + | 912.014i | −1631.36 | − | 2825.60i | − | 1448.15i | −2111.24 | − | 6212.04i | −6219.69 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 9.79796 | − | 5.65685i | 7.38176 | − | 80.6629i | 64.0000 | − | 110.851i | 935.250 | + | 539.967i | −383.972 | − | 832.090i | −2056.09 | − | 3561.25i | − | 1448.15i | −6452.02 | − | 1190.87i | 12218.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 9.79796 | − | 5.65685i | 36.7596 | − | 72.1785i | 64.0000 | − | 110.851i | −944.709 | − | 545.428i | −48.1341 | − | 915.145i | 1250.70 | + | 2166.27i | − | 1448.15i | −3858.46 | − | 5306.50i | −12341.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 9.79796 | − | 5.65685i | 59.0222 | + | 55.4742i | 64.0000 | − | 110.851i | 265.055 | + | 153.030i | 892.106 | + | 209.654i | 770.107 | + | 1333.86i | − | 1448.15i | 406.233 | + | 6548.41i | 3462.67 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | −9.79796 | − | 5.65685i | −69.4086 | − | 41.7547i | 64.0000 | + | 110.851i | −719.139 | + | 415.195i | 443.862 | + | 801.745i | 1343.41 | − | 2326.85i | − | 1448.15i | 3074.10 | + | 5796.26i | 9394.80 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −9.79796 | − | 5.65685i | −24.1444 | + | 77.3178i | 64.0000 | + | 110.851i | −69.6596 | + | 40.2180i | 673.941 | − | 620.976i | 370.849 | − | 642.329i | − | 1448.15i | −5395.10 | − | 3733.58i | 910.029 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −9.79796 | − | 5.65685i | 22.8219 | − | 77.7185i | 64.0000 | + | 110.851i | 683.343 | − | 394.528i | −663.250 | + | 632.382i | 89.7132 | − | 155.388i | − | 1448.15i | −5519.32 | − | 3547.36i | −8927.15 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −9.79796 | − | 5.65685i | 77.7361 | + | 22.7616i | 64.0000 | + | 110.851i | −115.044 | + | 66.4206i | −632.896 | − | 662.760i | −1060.32 | + | 1836.53i | − | 1448.15i | 5524.82 | + | 3538.80i | 1502.93 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 9.79796 | + | 5.65685i | −47.1686 | − | 65.8492i | 64.0000 | + | 110.851i | −476.096 | + | 274.874i | −89.6569 | − | 912.014i | −1631.36 | + | 2825.60i | 1448.15i | −2111.24 | + | 6212.04i | −6219.69 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 9.79796 | + | 5.65685i | 7.38176 | + | 80.6629i | 64.0000 | + | 110.851i | 935.250 | − | 539.967i | −383.972 | + | 832.090i | −2056.09 | + | 3561.25i | 1448.15i | −6452.02 | + | 1190.87i | 12218.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 9.79796 | + | 5.65685i | 36.7596 | + | 72.1785i | 64.0000 | + | 110.851i | −944.709 | + | 545.428i | −48.1341 | + | 915.145i | 1250.70 | − | 2166.27i | 1448.15i | −3858.46 | + | 5306.50i | −12341.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 9.79796 | + | 5.65685i | 59.0222 | − | 55.4742i | 64.0000 | + | 110.851i | 265.055 | − | 153.030i | 892.106 | − | 209.654i | 770.107 | − | 1333.86i | 1448.15i | 406.233 | − | 6548.41i | 3462.67 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 18.9.d.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 54.9.d.a | 16 | ||
| 4.b | odd | 2 | 1 | 144.9.q.b | 16 | ||
| 9.c | even | 3 | 1 | 54.9.d.a | 16 | ||
| 9.c | even | 3 | 1 | 162.9.b.c | 16 | ||
| 9.d | odd | 6 | 1 | inner | 18.9.d.a | ✓ | 16 |
| 9.d | odd | 6 | 1 | 162.9.b.c | 16 | ||
| 12.b | even | 2 | 1 | 432.9.q.c | 16 | ||
| 36.f | odd | 6 | 1 | 432.9.q.c | 16 | ||
| 36.h | even | 6 | 1 | 144.9.q.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.9.d.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 18.9.d.a | ✓ | 16 | 9.d | odd | 6 | 1 | inner |
| 54.9.d.a | 16 | 3.b | odd | 2 | 1 | ||
| 54.9.d.a | 16 | 9.c | even | 3 | 1 | ||
| 144.9.q.b | 16 | 4.b | odd | 2 | 1 | ||
| 144.9.q.b | 16 | 36.h | even | 6 | 1 | ||
| 162.9.b.c | 16 | 9.c | even | 3 | 1 | ||
| 162.9.b.c | 16 | 9.d | odd | 6 | 1 | ||
| 432.9.q.c | 16 | 12.b | even | 2 | 1 | ||
| 432.9.q.c | 16 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 128 T^{2} + 16384)^{4} \)
|
| $3$ |
\( T^{16} + \cdots + 34\!\cdots\!81 \)
|
| $5$ |
\( T^{16} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( T^{16} + \cdots + 15\!\cdots\!00 \)
|
| $11$ |
\( T^{16} + \cdots + 52\!\cdots\!49 \)
|
| $13$ |
\( T^{16} + \cdots + 22\!\cdots\!16 \)
|
| $17$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
|
| $19$ |
\( (T^{8} + \cdots - 11\!\cdots\!40)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 12\!\cdots\!24 \)
|
| $29$ |
\( T^{16} + \cdots + 73\!\cdots\!00 \)
|
| $31$ |
\( T^{16} + \cdots + 20\!\cdots\!00 \)
|
| $37$ |
\( (T^{8} + \cdots - 17\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 24\!\cdots\!25 \)
|
| $43$ |
\( T^{16} + \cdots + 23\!\cdots\!25 \)
|
| $47$ |
\( T^{16} + \cdots + 18\!\cdots\!44 \)
|
| $53$ |
\( T^{16} + \cdots + 73\!\cdots\!00 \)
|
| $59$ |
\( T^{16} + \cdots + 23\!\cdots\!25 \)
|
| $61$ |
\( T^{16} + \cdots + 11\!\cdots\!16 \)
|
| $67$ |
\( T^{16} + \cdots + 63\!\cdots\!25 \)
|
| $71$ |
\( T^{16} + \cdots + 26\!\cdots\!64 \)
|
| $73$ |
\( (T^{8} + \cdots + 79\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 27\!\cdots\!84 \)
|
| $89$ |
\( T^{16} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 23\!\cdots\!25 \)
|
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