Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(13.8301772501\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} + 6588 x^{8} - 41138 x^{7} + 37278682 x^{6} - 116750682 x^{5} + 41137230429 x^{4} + \cdots + 14\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{21}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} + 6588 x^{8} - 41138 x^{7} + 37278682 x^{6} - 116750682 x^{5} + 41137230429 x^{4} + \cdots + 14\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 34\!\cdots\!31 \nu^{9} + \cdots + 26\!\cdots\!00 ) / 79\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21\!\cdots\!89 \nu^{9} + \cdots + 26\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 35\!\cdots\!17 \nu^{9} + \cdots + 44\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37\!\cdots\!97 \nu^{9} + \cdots + 16\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14\!\cdots\!91 \nu^{9} + \cdots - 24\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!35 \nu^{9} + \cdots - 36\!\cdots\!00 ) / 29\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!47 \nu^{9} + \cdots - 63\!\cdots\!00 ) / 18\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 76\!\cdots\!87 \nu^{9} + \cdots - 19\!\cdots\!00 ) / 18\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 10\!\cdots\!41 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 16\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} - \beta_{7} - 23\beta_{5} - 2\beta_{4} - 2\beta_{3} + 21\beta_{2} - 182\beta _1 - 1 ) / 486 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 81 \beta_{9} - 152 \beta_{8} - 172 \beta_{7} - 81 \beta_{6} - 950 \beta_{5} - 173 \beta_{4} + \cdots - 2560843 ) / 972 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 36752 \beta_{8} + 36752 \beta_{7} + 14175 \beta_{6} + 816028 \beta_{5} + 30061 \beta_{4} + \cdots + 16600739 ) / 1944 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 534033 \beta_{9} + 1950308 \beta_{8} + 2062999 \beta_{7} + 4220204 \beta_{5} - 265462 \beta_{4} + \cdots + 3429235 ) / 1944 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 48329460 \beta_{9} + 53604062 \beta_{8} - 89053307 \beta_{7} - 48329460 \beta_{6} + \cdots - 138430104731 ) / 972 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2673400828 \beta_{8} + 2673400828 \beta_{7} + 2715451047 \beta_{6} + 72690702644 \beta_{5} + \cdots + 127442243309971 ) / 1944 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 564733890135 \beta_{9} + 504300799396 \beta_{8} - 6398302591 \beta_{7} - 19610857669388 \beta_{5} + \cdots - 3237431517091 ) / 1944 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 8317395337374 \beta_{9} - 22280902230698 \beta_{8} - 40355535292723 \beta_{7} + \cdots - 34\!\cdots\!77 ) / 972 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 57\!\cdots\!48 \beta_{8} + \cdots + 16\!\cdots\!61 ) / 1944 \)
|
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 |
|
−16.0000 | + | 27.7128i | −358.924 | − | 219.819i | −512.000 | − | 886.810i | 5838.03 | + | 10111.8i | 11834.6 | − | 6429.70i | 21599.0 | − | 37410.6i | 32768.0 | 80506.3 | + | 157797.i | −373634. | ||||||||||||||||||||||||||||||||||
| 7.2 | −16.0000 | + | 27.7128i | −203.072 | + | 368.658i | −512.000 | − | 886.810i | −1430.68 | − | 2478.00i | −6967.39 | − | 11526.2i | 20759.6 | − | 35956.6i | 32768.0 | −94670.3 | − | 149728.i | 91563.3 | |||||||||||||||||||||||||||||||||||
| 7.3 | −16.0000 | + | 27.7128i | −172.782 | − | 383.788i | −512.000 | − | 886.810i | −5374.28 | − | 9308.52i | 13400.4 | + | 1352.34i | 1182.09 | − | 2047.45i | 32768.0 | −117440. | + | 132623.i | 343954. | |||||||||||||||||||||||||||||||||||
| 7.4 | −16.0000 | + | 27.7128i | 291.335 | − | 303.761i | −512.000 | − | 886.810i | 2559.55 | + | 4433.28i | 3756.70 | + | 12933.9i | −2007.35 | + | 3476.83i | 32768.0 | −7394.27 | − | 176993.i | −163811. | |||||||||||||||||||||||||||||||||||
| 7.5 | −16.0000 | + | 27.7128i | 321.943 | + | 271.108i | −512.000 | − | 886.810i | 459.373 | + | 795.658i | −12664.3 | + | 4584.21i | −19269.3 | + | 33375.5i | 32768.0 | 30147.6 | + | 174563.i | −29399.9 | |||||||||||||||||||||||||||||||||||
| 13.1 | −16.0000 | − | 27.7128i | −358.924 | + | 219.819i | −512.000 | + | 886.810i | 5838.03 | − | 10111.8i | 11834.6 | + | 6429.70i | 21599.0 | + | 37410.6i | 32768.0 | 80506.3 | − | 157797.i | −373634. | |||||||||||||||||||||||||||||||||||
| 13.2 | −16.0000 | − | 27.7128i | −203.072 | − | 368.658i | −512.000 | + | 886.810i | −1430.68 | + | 2478.00i | −6967.39 | + | 11526.2i | 20759.6 | + | 35956.6i | 32768.0 | −94670.3 | + | 149728.i | 91563.3 | |||||||||||||||||||||||||||||||||||
| 13.3 | −16.0000 | − | 27.7128i | −172.782 | + | 383.788i | −512.000 | + | 886.810i | −5374.28 | + | 9308.52i | 13400.4 | − | 1352.34i | 1182.09 | + | 2047.45i | 32768.0 | −117440. | − | 132623.i | 343954. | |||||||||||||||||||||||||||||||||||
| 13.4 | −16.0000 | − | 27.7128i | 291.335 | + | 303.761i | −512.000 | + | 886.810i | 2559.55 | − | 4433.28i | 3756.70 | − | 12933.9i | −2007.35 | − | 3476.83i | 32768.0 | −7394.27 | + | 176993.i | −163811. | |||||||||||||||||||||||||||||||||||
| 13.5 | −16.0000 | − | 27.7128i | 321.943 | − | 271.108i | −512.000 | + | 886.810i | 459.373 | − | 795.658i | −12664.3 | − | 4584.21i | −19269.3 | − | 33375.5i | 32768.0 | 30147.6 | − | 174563.i | −29399.9 | |||||||||||||||||||||||||||||||||||
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 | 18.12.c.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 54.12.c.a | 10 | ||
| 9.c | even | 3 | 1 | inner | 18.12.c.a | ✓ | 10 |
| 9.c | even | 3 | 1 | 162.12.a.f | 5 | ||
| 9.d | odd | 6 | 1 | 54.12.c.a | 10 | ||
| 9.d | odd | 6 | 1 | 162.12.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.12.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 18.12.c.a | ✓ | 10 | 9.c | even | 3 | 1 | inner |
| 54.12.c.a | 10 | 3.b | odd | 2 | 1 | ||
| 54.12.c.a | 10 | 9.d | odd | 6 | 1 | ||
| 162.12.a.e | 5 | 9.d | odd | 6 | 1 | ||
| 162.12.a.f | 5 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 4104 T_{5}^{9} + 151970526 T_{5}^{8} - 292980026160 T_{5}^{7} + \cdots + 28\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{5} \)
|
| $3$ |
\( T^{10} + \cdots + 17\!\cdots\!07 \)
|
| $5$ |
\( T^{10} + \cdots + 28\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 43\!\cdots\!44 \)
|
| $11$ |
\( T^{10} + \cdots + 76\!\cdots\!25 \)
|
| $13$ |
\( T^{10} + \cdots + 15\!\cdots\!24 \)
|
| $17$ |
\( (T^{5} + \cdots + 85\!\cdots\!40)^{2} \)
|
| $19$ |
\( (T^{5} + \cdots + 90\!\cdots\!72)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots + 93\!\cdots\!44 \)
|
| $31$ |
\( T^{10} + \cdots + 56\!\cdots\!00 \)
|
| $37$ |
\( (T^{5} + \cdots - 11\!\cdots\!08)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 71\!\cdots\!89 \)
|
| $43$ |
\( T^{10} + \cdots + 97\!\cdots\!61 \)
|
| $47$ |
\( T^{10} + \cdots + 13\!\cdots\!36 \)
|
| $53$ |
\( (T^{5} + \cdots - 30\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 35\!\cdots\!69 \)
|
| $61$ |
\( T^{10} + \cdots + 73\!\cdots\!96 \)
|
| $67$ |
\( T^{10} + \cdots + 15\!\cdots\!25 \)
|
| $71$ |
\( (T^{5} + \cdots + 89\!\cdots\!48)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots - 19\!\cdots\!32)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 61\!\cdots\!56 \)
|
| $89$ |
\( (T^{5} + \cdots + 68\!\cdots\!60)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 43\!\cdots\!25 \)
|
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