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: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} + 1731382 x^{12} + 1165594103293 x^{10} + \cdots + 31\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{37} \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} + 1731382 x^{12} + 1165594103293 x^{10} + \cdots + 31\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 24\!\cdots\!17 \nu^{13} + \cdots + 36\!\cdots\!00 ) / 72\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 57\!\cdots\!59 \nu^{13} + \cdots + 64\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 92\!\cdots\!89 \nu^{13} + \cdots + 80\!\cdots\!00 ) / 77\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 92\!\cdots\!63 \nu^{13} + \cdots - 85\!\cdots\!00 ) / 77\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 98\!\cdots\!22 \nu^{13} + \cdots + 19\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 58\!\cdots\!91 \nu^{13} + \cdots + 12\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11\!\cdots\!91 \nu^{13} + \cdots + 18\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 39\!\cdots\!83 \nu^{13} + \cdots - 31\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 41\!\cdots\!35 \nu^{13} + \cdots - 22\!\cdots\!00 ) / 48\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 96\!\cdots\!88 \nu^{13} + \cdots - 41\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 59\!\cdots\!73 \nu^{13} + \cdots - 72\!\cdots\!00 ) / 85\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 15\!\cdots\!45 \nu^{13} + \cdots + 40\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 11\!\cdots\!17 \nu^{13} + \cdots - 20\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 178\beta_{4} + 275\beta_{3} - 2\beta_{2} + 665\beta _1 - 300 ) / 1458 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 54 \beta_{13} + 162 \beta_{12} - 54 \beta_{11} + 65 \beta_{9} + 184 \beta_{8} - 301 \beta_{6} + \cdots - 721213946 ) / 2916 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 21006 \beta_{13} - 56619 \beta_{12} - 52056 \beta_{11} + 73062 \beta_{10} - 335365 \beta_{9} + \cdots - 14266706156 ) / 5832 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 80378298 \beta_{13} - 225282141 \beta_{12} + 48672792 \beta_{11} - 63411012 \beta_{10} + \cdots + 555287116900364 ) / 5832 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 21414951936 \beta_{13} + 36674382069 \beta_{12} + 44650685538 \beta_{11} - 66065637474 \beta_{10} + \cdots + 42\!\cdots\!48 ) / 5832 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 24146296285329 \beta_{13} + 65562548136120 \beta_{12} - 10393614845595 \beta_{11} + \cdots - 12\!\cdots\!42 ) / 2916 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 84\!\cdots\!97 \beta_{13} + \cdots - 19\!\cdots\!58 ) / 2916 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 13\!\cdots\!28 \beta_{13} + \cdots + 61\!\cdots\!02 ) / 2916 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 11\!\cdots\!22 \beta_{13} + \cdots + 27\!\cdots\!72 ) / 5832 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 15\!\cdots\!42 \beta_{13} + \cdots - 63\!\cdots\!68 ) / 5832 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 73\!\cdots\!60 \beta_{13} + \cdots - 17\!\cdots\!00 ) / 5832 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 42\!\cdots\!83 \beta_{13} + \cdots + 16\!\cdots\!98 ) / 2916 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 22\!\cdots\!67 \beta_{13} + \cdots + 55\!\cdots\!78 ) / 2916 \)
|
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 | −1213.16 | + | 350.092i | −2048.00 | − | 3547.24i | −32012.7 | − | 55447.6i | 19417.1 | − | 78443.1i | 1161.05 | − | 2011.00i | 262144. | 1.34919e6 | − | 849436.i | 4.09763e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −32.0000 | + | 55.4256i | −1204.78 | + | 377.914i | −2048.00 | − | 3547.24i | 19823.4 | + | 34335.2i | 17606.9 | − | 78869.2i | 102668. | − | 177826.i | 262144. | 1.30868e6 | − | 910610.i | −2.53740e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −32.0000 | + | 55.4256i | −635.542 | − | 1091.06i | −2048.00 | − | 3547.24i | 7551.80 | + | 13080.1i | 80810.0 | − | 311.434i | −207431. | + | 359282.i | 262144. | −786496. | + | 1.38683e6i | −966631. | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −32.0000 | + | 55.4256i | −40.1816 | + | 1262.03i | −2048.00 | − | 3547.24i | 10473.7 | + | 18141.0i | −68662.7 | − | 42611.9i | −105988. | + | 183576.i | 262144. | −1.59109e6 | − | 101420.i | −1.34063e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −32.0000 | + | 55.4256i | 528.794 | − | 1146.60i | −2048.00 | − | 3547.24i | −6173.64 | − | 10693.1i | 46629.8 | + | 66000.0i | 177750. | − | 307872.i | 262144. | −1.03508e6 | − | 1.21263e6i | 790225. | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | −32.0000 | + | 55.4256i | 1051.06 | + | 699.706i | −2048.00 | − | 3547.24i | 6861.84 | + | 11885.1i | −72415.7 | + | 35865.2i | 261610. | − | 453121.i | 262144. | 615145. | + | 1.47087e6i | −878316. | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | −32.0000 | + | 55.4256i | 1260.31 | + | 77.0655i | −2048.00 | − | 3547.24i | −24776.5 | − | 42914.1i | −44601.4 | + | 67387.4i | −212830. | + | 368632.i | 262144. | 1.58244e6 | + | 194253.i | 3.17139e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.1 | −32.0000 | − | 55.4256i | −1213.16 | − | 350.092i | −2048.00 | + | 3547.24i | −32012.7 | + | 55447.6i | 19417.1 | + | 78443.1i | 1161.05 | + | 2011.00i | 262144. | 1.34919e6 | + | 849436.i | 4.09763e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −32.0000 | − | 55.4256i | −1204.78 | − | 377.914i | −2048.00 | + | 3547.24i | 19823.4 | − | 34335.2i | 17606.9 | + | 78869.2i | 102668. | + | 177826.i | 262144. | 1.30868e6 | + | 910610.i | −2.53740e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | −32.0000 | − | 55.4256i | −635.542 | + | 1091.06i | −2048.00 | + | 3547.24i | 7551.80 | − | 13080.1i | 80810.0 | + | 311.434i | −207431. | − | 359282.i | 262144. | −786496. | − | 1.38683e6i | −966631. | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | −32.0000 | − | 55.4256i | −40.1816 | − | 1262.03i | −2048.00 | + | 3547.24i | 10473.7 | − | 18141.0i | −68662.7 | + | 42611.9i | −105988. | − | 183576.i | 262144. | −1.59109e6 | + | 101420.i | −1.34063e6 | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | −32.0000 | − | 55.4256i | 528.794 | + | 1146.60i | −2048.00 | + | 3547.24i | −6173.64 | + | 10693.1i | 46629.8 | − | 66000.0i | 177750. | + | 307872.i | 262144. | −1.03508e6 | + | 1.21263e6i | 790225. | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | −32.0000 | − | 55.4256i | 1051.06 | − | 699.706i | −2048.00 | + | 3547.24i | 6861.84 | − | 11885.1i | −72415.7 | − | 35865.2i | 261610. | + | 453121.i | 262144. | 615145. | − | 1.47087e6i | −878316. | |||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | −32.0000 | − | 55.4256i | 1260.31 | − | 77.0655i | −2048.00 | + | 3547.24i | −24776.5 | + | 42914.1i | −44601.4 | − | 67387.4i | −212830. | − | 368632.i | 262144. | 1.58244e6 | − | 194253.i | 3.17139e6 | |||||||||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 14 |
| 3.b | odd | 2 | 1 | 54.14.c.b | 14 | ||
| 9.c | even | 3 | 1 | inner | 18.14.c.b | ✓ | 14 |
| 9.c | even | 3 | 1 | 162.14.a.j | 7 | ||
| 9.d | odd | 6 | 1 | 54.14.c.b | 14 | ||
| 9.d | odd | 6 | 1 | 162.14.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.14.c.b | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 18.14.c.b | ✓ | 14 | 9.c | even | 3 | 1 | inner |
| 54.14.c.b | 14 | 3.b | odd | 2 | 1 | ||
| 54.14.c.b | 14 | 9.d | odd | 6 | 1 | ||
| 162.14.a.i | 7 | 9.d | odd | 6 | 1 | ||
| 162.14.a.j | 7 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} + 36504 T_{5}^{13} + 5233435542 T_{5}^{12} - 30822279726816 T_{5}^{11} + \cdots + 45\!\cdots\!00 \)
acting on \(S_{14}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 64 T + 4096)^{7} \)
|
| $3$ |
\( T^{14} + \cdots + 26\!\cdots\!47 \)
|
| $5$ |
\( T^{14} + \cdots + 45\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots + 11\!\cdots\!64 \)
|
| $11$ |
\( T^{14} + \cdots + 17\!\cdots\!25 \)
|
| $13$ |
\( T^{14} + \cdots + 43\!\cdots\!56 \)
|
| $17$ |
\( (T^{7} + \cdots - 43\!\cdots\!32)^{2} \)
|
| $19$ |
\( (T^{7} + \cdots + 22\!\cdots\!20)^{2} \)
|
| $23$ |
\( T^{14} + \cdots + 40\!\cdots\!00 \)
|
| $29$ |
\( T^{14} + \cdots + 54\!\cdots\!96 \)
|
| $31$ |
\( T^{14} + \cdots + 34\!\cdots\!00 \)
|
| $37$ |
\( (T^{7} + \cdots - 30\!\cdots\!68)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 50\!\cdots\!89 \)
|
| $43$ |
\( T^{14} + \cdots + 72\!\cdots\!01 \)
|
| $47$ |
\( T^{14} + \cdots + 56\!\cdots\!00 \)
|
| $53$ |
\( (T^{7} + \cdots + 14\!\cdots\!80)^{2} \)
|
| $59$ |
\( T^{14} + \cdots + 35\!\cdots\!41 \)
|
| $61$ |
\( T^{14} + \cdots + 54\!\cdots\!00 \)
|
| $67$ |
\( T^{14} + \cdots + 25\!\cdots\!25 \)
|
| $71$ |
\( (T^{7} + \cdots + 34\!\cdots\!32)^{2} \)
|
| $73$ |
\( (T^{7} + \cdots + 44\!\cdots\!68)^{2} \)
|
| $79$ |
\( T^{14} + \cdots + 44\!\cdots\!44 \)
|
| $83$ |
\( T^{14} + \cdots + 45\!\cdots\!84 \)
|
| $89$ |
\( (T^{7} + \cdots + 44\!\cdots\!60)^{2} \)
|
| $97$ |
\( T^{14} + \cdots + 14\!\cdots\!25 \)
|
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