Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,14,Mod(7,18)]
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([4]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.c (of order \(3\), 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: | \(19.3015672113\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 70059 x^{10} + 5065444 x^{9} + 4093250893 x^{8} + 217797867390 x^{7} + \cdots + 18\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{31} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} + 70059 x^{10} + 5065444 x^{9} + 4093250893 x^{8} + 217797867390 x^{7} + \cdots + 18\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 31\!\cdots\!37 \nu^{11} + \cdots + 57\!\cdots\!00 ) / 45\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13\!\cdots\!16 \nu^{11} + \cdots + 27\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 18\!\cdots\!95 \nu^{11} + \cdots - 31\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 29\!\cdots\!72 \nu^{11} + \cdots - 21\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 22\!\cdots\!24 \nu^{11} + \cdots + 13\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!62 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\!\cdots\!65 \nu^{11} + \cdots + 13\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 35\!\cdots\!56 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 61\!\cdots\!69 \nu^{11} + \cdots + 96\!\cdots\!00 ) / 53\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 81\!\cdots\!64 \nu^{11} + \cdots - 15\!\cdots\!00 ) / 53\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 12\!\cdots\!59 \nu^{11} + \cdots - 21\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 10\beta_{3} - 33\beta_{2} - 162\beta _1 + 162 ) / 486 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 81 \beta_{11} + 81 \beta_{10} - 106 \beta_{7} - 162 \beta_{6} + 162 \beta_{5} + \cdots - 22698468 \beta_1 ) / 972 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5751 \beta_{9} + 17874 \beta_{8} - 183020 \beta_{7} - 26551 \beta_{5} + 46872 \beta_{4} + \cdots - 2529901836 ) / 1944 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 10535454 \beta_{11} - 10206729 \beta_{10} - 10206729 \beta_{9} + 10535454 \beta_{8} + \cdots - 2137791233160 ) / 1944 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1628036820 \beta_{11} - 27029295 \beta_{10} + 11980274819 \beta_{7} - 2865310200 \beta_{6} + \cdots + 224679868319448 \beta_1 ) / 1944 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 142445079873 \beta_{9} - 163547918073 \beta_{8} + 455835407831 \beta_{7} - 72697977623 \beta_{5} + \cdots + 29\!\cdots\!89 ) / 486 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 30006560167236 \beta_{11} + 6503591548161 \beta_{10} + 6503591548161 \beta_{9} + \cdots + 43\!\cdots\!94 ) / 486 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 20\!\cdots\!17 \beta_{11} + \cdots - 35\!\cdots\!00 \beta_1 ) / 972 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 28\!\cdots\!65 \beta_{9} + \cdots - 12\!\cdots\!48 ) / 1944 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 25\!\cdots\!22 \beta_{11} + \cdots - 43\!\cdots\!76 ) / 1944 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 56\!\cdots\!64 \beta_{11} + \cdots + 86\!\cdots\!16 \beta_1 ) / 1944 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
32.0000 | − | 55.4256i | −908.101 | + | 877.312i | −2048.00 | − | 3547.24i | −10927.3 | − | 18926.7i | 19566.3 | + | 78406.0i | −146006. | + | 252890.i | −262144. | 54970.7 | − | 1.59338e6i | −1.39870e6 | ||||||||||||||||||||||||||||||||||||||||
| 7.2 | 32.0000 | − | 55.4256i | −474.924 | − | 1169.94i | −2048.00 | − | 3547.24i | −27506.8 | − | 47643.2i | −80042.5 | − | 11115.3i | 207952. | − | 360184.i | −262144. | −1.14322e6 | + | 1.11127e6i | −3.52087e6 | |||||||||||||||||||||||||||||||||||||||||
| 7.3 | 32.0000 | − | 55.4256i | 89.9347 | + | 1259.46i | −2048.00 | − | 3547.24i | 25611.0 | + | 44359.5i | 72684.2 | + | 35318.0i | 190239. | − | 329504.i | −262144. | −1.57815e6 | + | 226538.i | 3.27820e6 | |||||||||||||||||||||||||||||||||||||||||
| 7.4 | 32.0000 | − | 55.4256i | 93.1775 | − | 1259.22i | −2048.00 | − | 3547.24i | 10005.6 | + | 17330.2i | −66811.5 | − | 45459.5i | −202059. | + | 349976.i | −262144. | −1.57696e6 | − | 234662.i | 1.28072e6 | |||||||||||||||||||||||||||||||||||||||||
| 7.5 | 32.0000 | − | 55.4256i | 875.085 | + | 910.247i | −2048.00 | − | 3547.24i | −25441.5 | − | 44066.0i | 78453.7 | − | 19374.2i | −46634.7 | + | 80773.6i | −262144. | −62776.4 | + | 1.59309e6i | −3.25652e6 | |||||||||||||||||||||||||||||||||||||||||
| 7.6 | 32.0000 | − | 55.4256i | 1242.83 | − | 222.942i | −2048.00 | − | 3547.24i | 10007.1 | + | 17332.8i | 27413.8 | − | 76018.6i | 73479.4 | − | 127270.i | −262144. | 1.49492e6 | − | 554158.i | 1.28091e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.1 | 32.0000 | + | 55.4256i | −908.101 | − | 877.312i | −2048.00 | + | 3547.24i | −10927.3 | + | 18926.7i | 19566.3 | − | 78406.0i | −146006. | − | 252890.i | −262144. | 54970.7 | + | 1.59338e6i | −1.39870e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.2 | 32.0000 | + | 55.4256i | −474.924 | + | 1169.94i | −2048.00 | + | 3547.24i | −27506.8 | + | 47643.2i | −80042.5 | + | 11115.3i | 207952. | + | 360184.i | −262144. | −1.14322e6 | − | 1.11127e6i | −3.52087e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.3 | 32.0000 | + | 55.4256i | 89.9347 | − | 1259.46i | −2048.00 | + | 3547.24i | 25611.0 | − | 44359.5i | 72684.2 | − | 35318.0i | 190239. | + | 329504.i | −262144. | −1.57815e6 | − | 226538.i | 3.27820e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.4 | 32.0000 | + | 55.4256i | 93.1775 | + | 1259.22i | −2048.00 | + | 3547.24i | 10005.6 | − | 17330.2i | −66811.5 | + | 45459.5i | −202059. | − | 349976.i | −262144. | −1.57696e6 | + | 234662.i | 1.28072e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.5 | 32.0000 | + | 55.4256i | 875.085 | − | 910.247i | −2048.00 | + | 3547.24i | −25441.5 | + | 44066.0i | 78453.7 | + | 19374.2i | −46634.7 | − | 80773.6i | −262144. | −62776.4 | − | 1.59309e6i | −3.25652e6 | |||||||||||||||||||||||||||||||||||||||||
| 13.6 | 32.0000 | + | 55.4256i | 1242.83 | + | 222.942i | −2048.00 | + | 3547.24i | 10007.1 | − | 17332.8i | 27413.8 | + | 76018.6i | 73479.4 | + | 127270.i | −262144. | 1.49492e6 | + | 554158.i | 1.28091e6 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.14.c.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 54.14.c.a | 12 | ||
| 9.c | even | 3 | 1 | inner | 18.14.c.a | ✓ | 12 |
| 9.c | even | 3 | 1 | 162.14.a.e | 6 | ||
| 9.d | odd | 6 | 1 | 54.14.c.a | 12 | ||
| 9.d | odd | 6 | 1 | 162.14.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.14.c.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 18.14.c.a | ✓ | 12 | 9.c | even | 3 | 1 | inner |
| 54.14.c.a | 12 | 3.b | odd | 2 | 1 | ||
| 54.14.c.a | 12 | 9.d | odd | 6 | 1 | ||
| 162.14.a.e | 6 | 9.c | even | 3 | 1 | ||
| 162.14.a.h | 6 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 36504 T_{5}^{11} + 5425226046 T_{5}^{10} + 76323554469360 T_{5}^{9} + \cdots + 15\!\cdots\!00 \)
acting on \(S_{14}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 64 T + 4096)^{6} \)
|
| $3$ |
\( T^{12} + \cdots + 16\!\cdots\!89 \)
|
| $5$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 65\!\cdots\!04 \)
|
| $11$ |
\( T^{12} + \cdots + 50\!\cdots\!25 \)
|
| $13$ |
\( T^{12} + \cdots + 11\!\cdots\!24 \)
|
| $17$ |
\( (T^{6} + \cdots - 15\!\cdots\!60)^{2} \)
|
| $19$ |
\( (T^{6} + \cdots - 57\!\cdots\!52)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 30\!\cdots\!00 \)
|
| $29$ |
\( T^{12} + \cdots + 80\!\cdots\!84 \)
|
| $31$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( (T^{6} + \cdots - 15\!\cdots\!96)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 15\!\cdots\!29 \)
|
| $43$ |
\( T^{12} + \cdots + 64\!\cdots\!69 \)
|
| $47$ |
\( T^{12} + \cdots + 40\!\cdots\!04 \)
|
| $53$ |
\( (T^{6} + \cdots + 87\!\cdots\!80)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 18\!\cdots\!61 \)
|
| $61$ |
\( T^{12} + \cdots + 17\!\cdots\!04 \)
|
| $67$ |
\( T^{12} + \cdots + 65\!\cdots\!25 \)
|
| $71$ |
\( (T^{6} + \cdots + 70\!\cdots\!76)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots - 18\!\cdots\!56)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 49\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 27\!\cdots\!76 \)
|
| $89$ |
\( (T^{6} + \cdots + 46\!\cdots\!80)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 11\!\cdots\!25 \)
|
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