Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(5.62293045871\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 1336\nu^{5} + 539944\nu^{3} + 62785035\nu + 22961400 ) / 45922800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 184675260 \nu^{2} + \cdots - 19238373000 ) / 160729800 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 253559460 \nu^{2} + \cdots - 42245695800 ) / 160729800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 414289260 \nu^{2} + \cdots - 95929449000 ) / 53576600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 2081\nu^{4} + 677750\nu^{2} + 15435\nu + 67639185 ) / 5145 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 186 \nu^{7} + 3905 \nu^{6} + 189921 \nu^{5} + 4186160 \nu^{4} + 58431309 \nu^{3} + \cdots + 143583726000 ) / 20091225 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3841 \nu^{7} + 4053796 \nu^{5} + 10168620 \nu^{4} + 1353967864 \nu^{3} + 7171501260 \nu^{2} + \cdots + 1079543341800 ) / 160729800 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 10\beta_{3} + 7\beta_{2} ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{4} - 56\beta_{3} + 77\beta_{2} - 18036 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 91\beta_{7} - 378\beta_{6} + 189\beta_{5} - 327\beta_{4} + 4614\beta_{3} - 1946\beta_{2} - 19656\beta _1 + 9639 ) / 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 490\beta_{7} + 4921\beta_{4} + 32018\beta_{3} - 58051\beta_{2} + 6987168 ) / 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64218 \beta_{7} + 252504 \beta_{6} - 126252 \beta_{5} + 123001 \beta_{4} - 2199832 \beta_{3} + \cdots - 10857672 ) / 54 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -509845\beta_{7} + 138915\beta_{5} - 2755893\beta_{4} - 14260554\beta_{3} + 34254668\beta_{2} - 2984456799 ) / 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 36660344 \beta_{7} - 133246512 \beta_{6} + 66623256 \beta_{5} - 50552683 \beta_{4} + 1075524286 \beta_{3} + \cdots + 8061413976 ) / 54 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
4.00000 | − | 6.92820i | −46.0138 | + | 8.35049i | −32.0000 | − | 55.4256i | 103.252 | + | 178.837i | −126.201 | + | 352.195i | −808.604 | + | 1400.54i | −512.000 | 2047.54 | − | 768.476i | 1652.03 | ||||||||||||||||||||||||||||
| 7.2 | 4.00000 | − | 6.92820i | −11.3727 | + | 45.3615i | −32.0000 | − | 55.4256i | −247.107 | − | 428.002i | 268.783 | + | 260.238i | 648.241 | − | 1122.79i | −512.000 | −1928.33 | − | 1031.76i | −3953.72 | |||||||||||||||||||||||||||||
| 7.3 | 4.00000 | − | 6.92820i | 14.5665 | − | 44.4389i | −32.0000 | − | 55.4256i | −47.1766 | − | 81.7123i | −249.616 | − | 278.675i | −101.366 | + | 175.572i | −512.000 | −1762.64 | − | 1294.64i | −754.826 | |||||||||||||||||||||||||||||
| 7.4 | 4.00000 | − | 6.92820i | 36.8200 | + | 28.8321i | −32.0000 | − | 55.4256i | 218.032 | + | 377.643i | 347.034 | − | 139.768i | 239.730 | − | 415.224i | −512.000 | 524.422 | + | 2123.19i | 3488.51 | |||||||||||||||||||||||||||||
| 13.1 | 4.00000 | + | 6.92820i | −46.0138 | − | 8.35049i | −32.0000 | + | 55.4256i | 103.252 | − | 178.837i | −126.201 | − | 352.195i | −808.604 | − | 1400.54i | −512.000 | 2047.54 | + | 768.476i | 1652.03 | |||||||||||||||||||||||||||||
| 13.2 | 4.00000 | + | 6.92820i | −11.3727 | − | 45.3615i | −32.0000 | + | 55.4256i | −247.107 | + | 428.002i | 268.783 | − | 260.238i | 648.241 | + | 1122.79i | −512.000 | −1928.33 | + | 1031.76i | −3953.72 | |||||||||||||||||||||||||||||
| 13.3 | 4.00000 | + | 6.92820i | 14.5665 | + | 44.4389i | −32.0000 | + | 55.4256i | −47.1766 | + | 81.7123i | −249.616 | + | 278.675i | −101.366 | − | 175.572i | −512.000 | −1762.64 | + | 1294.64i | −754.826 | |||||||||||||||||||||||||||||
| 13.4 | 4.00000 | + | 6.92820i | 36.8200 | − | 28.8321i | −32.0000 | + | 55.4256i | 218.032 | − | 377.643i | 347.034 | + | 139.768i | 239.730 | + | 415.224i | −512.000 | 524.422 | − | 2123.19i | 3488.51 | |||||||||||||||||||||||||||||
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 | 18.8.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 54.8.c.b | 8 | ||
| 4.b | odd | 2 | 1 | 144.8.i.b | 8 | ||
| 9.c | even | 3 | 1 | inner | 18.8.c.b | ✓ | 8 |
| 9.c | even | 3 | 1 | 162.8.a.h | 4 | ||
| 9.d | odd | 6 | 1 | 54.8.c.b | 8 | ||
| 9.d | odd | 6 | 1 | 162.8.a.i | 4 | ||
| 12.b | even | 2 | 1 | 432.8.i.b | 8 | ||
| 36.f | odd | 6 | 1 | 144.8.i.b | 8 | ||
| 36.h | even | 6 | 1 | 432.8.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.8.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 18.8.c.b | ✓ | 8 | 9.c | even | 3 | 1 | inner |
| 54.8.c.b | 8 | 3.b | odd | 2 | 1 | ||
| 54.8.c.b | 8 | 9.d | odd | 6 | 1 | ||
| 144.8.i.b | 8 | 4.b | odd | 2 | 1 | ||
| 144.8.i.b | 8 | 36.f | odd | 6 | 1 | ||
| 162.8.a.h | 4 | 9.c | even | 3 | 1 | ||
| 162.8.a.i | 4 | 9.d | odd | 6 | 1 | ||
| 432.8.i.b | 8 | 12.b | even | 2 | 1 | ||
| 432.8.i.b | 8 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 54 T_{5}^{7} + 244431 T_{5}^{6} - 33030990 T_{5}^{5} + 55374420825 T_{5}^{4} + \cdots + 17\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 22876792454961 \)
|
| $5$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 41\!\cdots\!56 \)
|
| $11$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $13$ |
\( T^{8} + \cdots + 27\!\cdots\!04 \)
|
| $17$ |
\( (T^{4} + \cdots - 21\!\cdots\!28)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots - 57\!\cdots\!48)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 18\!\cdots\!76 \)
|
| $31$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( (T^{4} + \cdots + 14\!\cdots\!48)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 61\!\cdots\!61 \)
|
| $43$ |
\( T^{8} + \cdots + 14\!\cdots\!81 \)
|
| $47$ |
\( T^{8} + \cdots + 15\!\cdots\!16 \)
|
| $53$ |
\( (T^{4} + \cdots + 32\!\cdots\!80)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 73\!\cdots\!09 \)
|
| $61$ |
\( T^{8} + \cdots + 40\!\cdots\!24 \)
|
| $67$ |
\( T^{8} + \cdots + 77\!\cdots\!25 \)
|
| $71$ |
\( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 66\!\cdots\!64 \)
|
| $83$ |
\( T^{8} + \cdots + 72\!\cdots\!64 \)
|
| $89$ |
\( (T^{4} + \cdots - 25\!\cdots\!80)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 30\!\cdots\!25 \)
|
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