Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,26,Mod(47,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.47");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.c (of order \(2\), degree \(1\), 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: | \(190.078454377\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 15431099652 x^{14} - 108017697424 x^{13} + \cdots + 94\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{126}\cdot 3^{88}\cdot 5^{6}\cdot 17^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 15431099652 x^{14} - 108017697424 x^{13} + \cdots + 94\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 16\!\cdots\!20 \nu^{14} + \cdots + 50\!\cdots\!00 ) / 22\!\cdots\!05 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 32768 \nu^{15} - 245760 \nu^{14} + 505646273273856 \nu^{13} + \cdots - 42\!\cdots\!19 ) / 21\!\cdots\!01 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60\!\cdots\!60 \nu^{14} + \cdots + 65\!\cdots\!00 ) / 22\!\cdots\!05 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 85\!\cdots\!20 \nu^{14} + \cdots - 72\!\cdots\!20 ) / 20\!\cdots\!55 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!16 \nu^{14} + \cdots - 51\!\cdots\!40 ) / 22\!\cdots\!05 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 52854784 \nu^{15} + 396410880 \nu^{14} + \cdots - 12\!\cdots\!02 ) / 72\!\cdots\!67 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 78\!\cdots\!60 \nu^{14} + \cdots - 83\!\cdots\!20 ) / 28\!\cdots\!65 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!12 \nu^{15} + \cdots + 26\!\cdots\!83 ) / 27\!\cdots\!63 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 15\!\cdots\!72 \nu^{14} + \cdots - 24\!\cdots\!60 ) / 22\!\cdots\!05 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 84\!\cdots\!44 \nu^{14} + \cdots - 21\!\cdots\!20 ) / 73\!\cdots\!35 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 33\!\cdots\!76 \nu^{15} + \cdots + 19\!\cdots\!25 ) / 48\!\cdots\!95 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14\!\cdots\!24 \nu^{15} + \cdots - 54\!\cdots\!20 ) / 53\!\cdots\!65 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 18\!\cdots\!92 \nu^{15} + \cdots - 21\!\cdots\!20 ) / 13\!\cdots\!95 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 76\!\cdots\!68 \nu^{15} + \cdots - 10\!\cdots\!45 ) / 53\!\cdots\!65 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 15\!\cdots\!72 \nu^{15} + \cdots + 26\!\cdots\!90 ) / 53\!\cdots\!65 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 4839\beta_{2} + 177147 ) / 354294 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} + 9\beta_{4} + 3042\beta_{3} + 9678\beta_{2} + 1620495\beta _1 - 1366786502192070 ) / 708588 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 1215 \beta_{15} + 19683 \beta_{14} + 59049 \beta_{13} - 209223 \beta_{12} + \cdots - 13\!\cdots\!68 ) / 472392 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 14580 \beta_{15} + 236196 \beta_{14} + 708588 \beta_{13} - 2510676 \beta_{12} + \cdots - 23\!\cdots\!40 ) / 2834352 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 954192750199875 \beta_{15} + 707515830596133 \beta_{14} + \cdots - 11\!\cdots\!48 ) / 5668704 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 954192750175575 \beta_{15} + 707515830202473 \beta_{14} + \cdots + 18\!\cdots\!84 ) / 1889568 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 29\!\cdots\!76 \beta_{15} + \cdots + 39\!\cdots\!12 ) / 11337408 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 23\!\cdots\!48 \beta_{15} + \cdots + 11\!\cdots\!12 ) / 22674816 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 74\!\cdots\!09 \beta_{15} + \cdots + 11\!\cdots\!52 ) / 5038848 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 33\!\cdots\!25 \beta_{15} + \cdots - 26\!\cdots\!40 ) / 45349632 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 26\!\cdots\!71 \beta_{15} + \cdots - 28\!\cdots\!84 ) / 90699264 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 53\!\cdots\!16 \beta_{15} + \cdots + 92\!\cdots\!44 ) / 30233088 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 87\!\cdots\!07 \beta_{15} + \cdots + 35\!\cdots\!00 ) / 181398528 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 61\!\cdots\!19 \beta_{15} + \cdots + 35\!\cdots\!12 ) / 181398528 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 12\!\cdots\!43 \beta_{15} + \cdots + 17\!\cdots\!92 ) / 120932352 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −918401. | − | 61867.2i | 0 | 5.51778e8i | 0 | − | 2.28210e10i | 0 | 8.39634e11 | + | 1.13638e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −918401. | + | 61867.2i | 0 | − | 5.51778e8i | 0 | 2.28210e10i | 0 | 8.39634e11 | − | 1.13638e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.3 | 0 | −557706. | − | 732293.i | 0 | − | 6.95529e8i | 0 | − | 4.07376e10i | 0 | −2.25217e11 | + | 8.16808e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.4 | 0 | −557706. | + | 732293.i | 0 | 6.95529e8i | 0 | 4.07376e10i | 0 | −2.25217e11 | − | 8.16808e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.5 | 0 | −541329. | − | 744481.i | 0 | 1.04042e8i | 0 | 6.51098e10i | 0 | −2.61214e11 | + | 8.06018e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.6 | 0 | −541329. | + | 744481.i | 0 | − | 1.04042e8i | 0 | − | 6.51098e10i | 0 | −2.61214e11 | − | 8.06018e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.7 | 0 | −328882. | − | 859724.i | 0 | 5.80022e8i | 0 | − | 8.51800e9i | 0 | −6.30962e11 | + | 5.65496e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.8 | 0 | −328882. | + | 859724.i | 0 | − | 5.80022e8i | 0 | 8.51800e9i | 0 | −6.30962e11 | − | 5.65496e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.9 | 0 | 328882. | − | 859724.i | 0 | − | 5.80022e8i | 0 | − | 8.51800e9i | 0 | −6.30962e11 | − | 5.65496e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.10 | 0 | 328882. | + | 859724.i | 0 | 5.80022e8i | 0 | 8.51800e9i | 0 | −6.30962e11 | + | 5.65496e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.11 | 0 | 541329. | − | 744481.i | 0 | − | 1.04042e8i | 0 | 6.51098e10i | 0 | −2.61214e11 | − | 8.06018e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.12 | 0 | 541329. | + | 744481.i | 0 | 1.04042e8i | 0 | − | 6.51098e10i | 0 | −2.61214e11 | + | 8.06018e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.13 | 0 | 557706. | − | 732293.i | 0 | 6.95529e8i | 0 | − | 4.07376e10i | 0 | −2.25217e11 | − | 8.16808e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.14 | 0 | 557706. | + | 732293.i | 0 | − | 6.95529e8i | 0 | 4.07376e10i | 0 | −2.25217e11 | + | 8.16808e11i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.15 | 0 | 918401. | − | 61867.2i | 0 | − | 5.51778e8i | 0 | − | 2.28210e10i | 0 | 8.39634e11 | − | 1.13638e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.16 | 0 | 918401. | + | 61867.2i | 0 | 5.51778e8i | 0 | 2.28210e10i | 0 | 8.39634e11 | + | 1.13638e11i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.26.c.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 48.26.c.b | ✓ | 16 |
| 4.b | odd | 2 | 1 | inner | 48.26.c.b | ✓ | 16 |
| 12.b | even | 2 | 1 | inner | 48.26.c.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.26.c.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 48.26.c.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 48.26.c.b | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
| 48.26.c.b | ✓ | 16 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + \cdots + 53\!\cdots\!00 \)
acting on \(S_{26}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} + \cdots + 26\!\cdots\!01 \)
|
| $5$ |
\( (T^{8} + \cdots + 53\!\cdots\!00)^{2} \)
|
| $7$ |
\( (T^{8} + \cdots + 26\!\cdots\!84)^{2} \)
|
| $11$ |
\( (T^{8} + \cdots + 98\!\cdots\!00)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 24\!\cdots\!00)^{4} \)
|
| $17$ |
\( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $19$ |
\( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \)
|
| $23$ |
\( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 18\!\cdots\!00)^{4} \)
|
| $41$ |
\( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 56\!\cdots\!04)^{2} \)
|
| $47$ |
\( (T^{8} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{8} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 78\!\cdots\!36)^{4} \)
|
| $67$ |
\( (T^{8} + \cdots + 37\!\cdots\!64)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots + 34\!\cdots\!00)^{4} \)
|
| $79$ |
\( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{8} + \cdots + 73\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{8} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 18\!\cdots\!00)^{4} \)
|
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