Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,8,Mod(49,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.49");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.i (of order \(3\), degree \(2\), not 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: | \(44.9834436697\) |
| 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: |
\( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{15} \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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
\( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 4058788905 ) / 119996640 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 3338809065 ) / 59998320 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3916394 \nu^{11} + 21540167 \nu^{10} - 1457064271 \nu^{9} + 6395237967 \nu^{8} + \cdots + 87187467749373 ) / 780837815928960 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8715547 \nu^{11} + 274872853 \nu^{10} - 5371798301 \nu^{9} + 101334640866 \nu^{8} + \cdots + 44\!\cdots\!32 ) / 390418907964480 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 39397984 \nu^{11} + 946466653 \nu^{10} + 7691846527 \nu^{9} + 467122386875 \nu^{8} + \cdots + 77\!\cdots\!09 ) / 780837815928960 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 39397984 \nu^{11} + 494870173 \nu^{10} - 14898530657 \nu^{9} + 130856702487 \nu^{8} + \cdots - 16\!\cdots\!07 ) / 780837815928960 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 39397984 \nu^{11} + 491616591 \nu^{10} - 14882262747 \nu^{9} + 128803692245 \nu^{8} + \cdots - 30\!\cdots\!37 ) / 780837815928960 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10227479 \nu^{11} + 161179154 \nu^{10} - 3614811022 \nu^{9} + 56619062412 \nu^{8} + \cdots + 38\!\cdots\!46 ) / 65069817994080 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 205349422 \nu^{11} + 2882244785 \nu^{10} + 62835655811 \nu^{9} + 1007320564173 \nu^{8} + \cdots + 48\!\cdots\!27 ) / 780837815928960 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1292509692 \nu^{11} + 7097415769 \nu^{10} - 454447599701 \nu^{9} + \cdots + 43\!\cdots\!69 ) / 780837815928960 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1416577970 \nu^{11} + 8246680315 \nu^{10} - 530017555415 \nu^{9} + \cdots + 39\!\cdots\!97 ) / 780837815928960 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} - \beta _1 - 714 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 36 \beta_{11} - 16 \beta_{10} + 48 \beta_{9} - 24 \beta_{8} - 200 \beta_{6} + 8 \beta_{5} - 292 \beta_{4} + \cdots - 6984 ) / 72 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 18 \beta_{11} - 8 \beta_{10} + 32 \beta_{9} + 26 \beta_{8} + 381 \beta_{7} - 675 \beta_{6} + \cdots + 103221 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2468 \beta_{11} + 984 \beta_{10} - 2184 \beta_{9} + 952 \beta_{8} + 16 \beta_{7} + 8112 \beta_{6} + \cdots + 480252 ) / 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 11196 \beta_{11} + 4468 \beta_{10} - 12876 \beta_{9} - 10104 \beta_{8} - 94320 \beta_{7} + \cdots - 21509748 ) / 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1180224 \beta_{11} - 411656 \beta_{10} + 809336 \beta_{9} - 367672 \beta_{8} - 44088 \beta_{7} + \cdots - 246596076 ) / 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 402130 \beta_{11} - 140700 \beta_{10} + 345036 \beta_{9} + 258646 \beta_{8} + 1969693 \beta_{7} + \cdots + 382567035 ) / 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 171131364 \beta_{11} + 54820952 \beta_{10} - 97990536 \beta_{9} + 51195912 \beta_{8} + \cdots + 38050923240 ) / 72 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 446002812 \beta_{11} + 143415236 \beta_{10} - 304901180 \beta_{9} - 218430056 \beta_{8} + \cdots - 246838018302 ) / 36 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 7847972884 \beta_{11} - 2415017640 \beta_{10} + 3954731352 \beta_{9} - 2467799936 \beta_{8} + \cdots - 1850187526692 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −45.7172 | + | 9.84574i | 0 | −145.304 | + | 251.673i | 0 | 555.940 | + | 962.916i | 0 | 1993.12 | − | 900.239i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −36.5784 | − | 29.1379i | 0 | 167.952 | − | 290.901i | 0 | −442.025 | − | 765.610i | 0 | 488.965 | + | 2131.64i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −21.6493 | + | 41.4525i | 0 | 32.6274 | − | 56.5123i | 0 | 118.194 | + | 204.717i | 0 | −1249.62 | − | 1794.83i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 22.8679 | + | 40.7929i | 0 | 47.9866 | − | 83.1153i | 0 | 189.000 | + | 327.358i | 0 | −1141.12 | + | 1865.70i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 33.3118 | − | 32.8226i | 0 | −246.026 | + | 426.130i | 0 | 382.311 | + | 662.182i | 0 | 32.3523 | − | 2186.76i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 35.7652 | − | 30.1306i | 0 | 52.7641 | − | 91.3900i | 0 | −761.419 | − | 1318.82i | 0 | 371.296 | − | 2155.25i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −45.7172 | − | 9.84574i | 0 | −145.304 | − | 251.673i | 0 | 555.940 | − | 962.916i | 0 | 1993.12 | + | 900.239i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −36.5784 | + | 29.1379i | 0 | 167.952 | + | 290.901i | 0 | −442.025 | + | 765.610i | 0 | 488.965 | − | 2131.64i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | −21.6493 | − | 41.4525i | 0 | 32.6274 | + | 56.5123i | 0 | 118.194 | − | 204.717i | 0 | −1249.62 | + | 1794.83i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | 22.8679 | − | 40.7929i | 0 | 47.9866 | + | 83.1153i | 0 | 189.000 | − | 327.358i | 0 | −1141.12 | − | 1865.70i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 33.3118 | + | 32.8226i | 0 | −246.026 | − | 426.130i | 0 | 382.311 | − | 662.182i | 0 | 32.3523 | + | 2186.76i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 35.7652 | + | 30.1306i | 0 | 52.7641 | + | 91.3900i | 0 | −761.419 | + | 1318.82i | 0 | 371.296 | + | 2155.25i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
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 | 144.8.i.c | 12 | |
| 3.b | odd | 2 | 1 | 432.8.i.c | 12 | ||
| 4.b | odd | 2 | 1 | 9.8.c.a | ✓ | 12 | |
| 9.c | even | 3 | 1 | inner | 144.8.i.c | 12 | |
| 9.d | odd | 6 | 1 | 432.8.i.c | 12 | ||
| 12.b | even | 2 | 1 | 27.8.c.a | 12 | ||
| 36.f | odd | 6 | 1 | 9.8.c.a | ✓ | 12 | |
| 36.f | odd | 6 | 1 | 81.8.a.e | 6 | ||
| 36.h | even | 6 | 1 | 27.8.c.a | 12 | ||
| 36.h | even | 6 | 1 | 81.8.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.8.c.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 9.8.c.a | ✓ | 12 | 36.f | odd | 6 | 1 | |
| 27.8.c.a | 12 | 12.b | even | 2 | 1 | ||
| 27.8.c.a | 12 | 36.h | even | 6 | 1 | ||
| 81.8.a.c | 6 | 36.h | even | 6 | 1 | ||
| 81.8.a.e | 6 | 36.f | odd | 6 | 1 | ||
| 144.8.i.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 144.8.i.c | 12 | 9.c | even | 3 | 1 | inner | |
| 432.8.i.c | 12 | 3.b | odd | 2 | 1 | ||
| 432.8.i.c | 12 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 180 T_{5}^{11} + 248202 T_{5}^{10} - 26226720 T_{5}^{9} + 37979473779 T_{5}^{8} + \cdots + 10\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 10\!\cdots\!09 \)
|
| $5$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 10\!\cdots\!04 \)
|
| $11$ |
\( T^{12} + \cdots + 83\!\cdots\!81 \)
|
| $13$ |
\( T^{12} + \cdots + 50\!\cdots\!64 \)
|
| $17$ |
\( (T^{6} + \cdots - 16\!\cdots\!76)^{2} \)
|
| $19$ |
\( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 73\!\cdots\!84 \)
|
| $29$ |
\( T^{12} + \cdots + 53\!\cdots\!44 \)
|
| $31$ |
\( T^{12} + \cdots + 23\!\cdots\!56 \)
|
| $37$ |
\( (T^{6} + \cdots + 17\!\cdots\!64)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 45\!\cdots\!61 \)
|
| $43$ |
\( T^{12} + \cdots + 27\!\cdots\!69 \)
|
| $47$ |
\( T^{12} + \cdots + 56\!\cdots\!04 \)
|
| $53$ |
\( (T^{6} + \cdots + 54\!\cdots\!44)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 74\!\cdots\!89 \)
|
| $61$ |
\( T^{12} + \cdots + 75\!\cdots\!96 \)
|
| $67$ |
\( T^{12} + \cdots + 21\!\cdots\!49 \)
|
| $71$ |
\( (T^{6} + \cdots + 81\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 34\!\cdots\!16)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 11\!\cdots\!44 \)
|
| $83$ |
\( T^{12} + \cdots + 58\!\cdots\!64 \)
|
| $89$ |
\( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 83\!\cdots\!69 \)
|
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