Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,3,Mod(111,176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(176, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("176.111");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 176.d (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: | \(4.79565265274\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.85177589904.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 12x^{6} - 40x^{5} + 111x^{4} - 372x^{3} + 850x^{2} - 2292x + 2883 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 12x^{6} - 40x^{5} + 111x^{4} - 372x^{3} + 850x^{2} - 2292x + 2883 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -10\nu^{7} + 71\nu^{6} - 63\nu^{5} + 1148\nu^{4} - 1765\nu^{3} + 5634\nu^{2} - 20775\nu + 16461 ) / 10044 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{7} - 2\nu^{6} + 26\nu^{5} - 106\nu^{4} + 328\nu^{3} - 1072\nu^{2} + 1314\nu - 5898 ) / 729 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{7} - \nu^{6} - 5\nu^{5} + 46\nu^{4} + 29\nu^{3} - 392\nu^{2} + 3357\nu - 5757 ) / 2916 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -86\nu^{7} - 121\nu^{6} - 1757\nu^{5} + 238\nu^{4} - 10963\nu^{3} + 23020\nu^{2} - 40653\nu + 119505 ) / 22599 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -82\nu^{7} - 137\nu^{6} - 1087\nu^{5} + 1304\nu^{4} - 6413\nu^{3} + 10586\nu^{2} - 33831\nu + 91419 ) / 15066 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{7} - 13\nu^{6} - 89\nu^{5} + 190\nu^{4} - 127\nu^{3} + 2224\nu^{2} - 1359\nu + 6951 ) / 972 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 226\nu^{7} + 584\nu^{6} + 3910\nu^{5} - 314\nu^{4} + 22808\nu^{3} - 46778\nu^{2} + 129042\nu - 365397 ) / 22599 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} + \beta_{6} - 6\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} - 12 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 6\beta_{6} - 5\beta_{5} + 5\beta_{4} - 2\beta_{3} + 13\beta_{2} + 10\beta _1 + 62 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12\beta_{7} + \beta_{6} + 29\beta_{5} - 3\beta_{4} - 15\beta_{3} - 6\beta_{2} - 10\beta _1 - 35 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -94\beta_{7} - 25\beta_{6} - 102\beta_{5} - 179\beta_{4} + 95\beta_{3} - 88\beta_{2} - 300 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 104\beta_{7} + 15\beta_{6} - 226\beta_{5} + 505\beta_{4} + 191\beta_{3} + 260\beta_{2} + 416\beta _1 + 2074 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 705\beta_{7} - 2\beta_{6} + 1121\beta_{5} + 501\beta_{4} - 1770\beta_{3} - 321\beta_{2} - 970\beta _1 - 2540 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).
| \(n\) | \(111\) | \(133\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 111.1 |
|
0 | − | 4.78633i | 0 | −0.580131 | 0 | − | 5.89077i | 0 | −13.9090 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 111.2 | 0 | − | 3.99405i | 0 | −7.53671 | 0 | 2.62200i | 0 | −6.95241 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.3 | 0 | − | 3.12663i | 0 | 9.26496 | 0 | 10.9562i | 0 | −0.775834 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.4 | 0 | − | 0.602295i | 0 | −0.148115 | 0 | 9.07666i | 0 | 8.63724 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.5 | 0 | 0.602295i | 0 | −0.148115 | 0 | − | 9.07666i | 0 | 8.63724 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.6 | 0 | 3.12663i | 0 | 9.26496 | 0 | − | 10.9562i | 0 | −0.775834 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.7 | 0 | 3.99405i | 0 | −7.53671 | 0 | − | 2.62200i | 0 | −6.95241 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 111.8 | 0 | 4.78633i | 0 | −0.580131 | 0 | 5.89077i | 0 | −13.9090 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 176.3.d.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1584.3.l.e | 8 | ||
| 4.b | odd | 2 | 1 | inner | 176.3.d.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 704.3.d.b | 8 | ||
| 8.d | odd | 2 | 1 | 704.3.d.b | 8 | ||
| 12.b | even | 2 | 1 | 1584.3.l.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 176.3.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 176.3.d.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 704.3.d.b | 8 | 8.b | even | 2 | 1 | ||
| 704.3.d.b | 8 | 8.d | odd | 2 | 1 | ||
| 1584.3.l.e | 8 | 3.b | odd | 2 | 1 | ||
| 1584.3.l.e | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 49T_{3}^{6} + 763T_{3}^{4} + 3843T_{3}^{2} + 1296 \)
acting on \(S_{3}^{\mathrm{new}}(176, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 49 T^{6} + \cdots + 1296 \)
|
| $5$ |
\( (T^{4} - T^{3} - 71 T^{2} + \cdots - 6)^{2} \)
|
| $7$ |
\( T^{8} + 244 T^{6} + \cdots + 2359296 \)
|
| $11$ |
\( (T^{2} + 11)^{4} \)
|
| $13$ |
\( (T^{4} - 2 T^{3} + \cdots - 4608)^{2} \)
|
| $17$ |
\( (T^{4} + 20 T^{3} + \cdots - 14064)^{2} \)
|
| $19$ |
\( T^{8} + 640 T^{6} + \cdots + 147456 \)
|
| $23$ |
\( T^{8} + \cdots + 10771948944 \)
|
| $29$ |
\( (T^{4} + 18 T^{3} + \cdots + 755712)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 25945477776 \)
|
| $37$ |
\( (T^{4} - 19 T^{3} + \cdots + 343326)^{2} \)
|
| $41$ |
\( (T^{4} + 70 T^{3} + \cdots - 1029696)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 121981245318144 \)
|
| $47$ |
\( T^{8} + \cdots + 8122773602304 \)
|
| $53$ |
\( (T^{4} + 124 T^{3} + \cdots + 27696)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 951865604496 \)
|
| $61$ |
\( (T^{4} - 138 T^{3} + \cdots + 244864)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 32773655629584 \)
|
| $71$ |
\( T^{8} + 13297 T^{6} + \cdots + 118984464 \)
|
| $73$ |
\( (T^{4} + 66 T^{3} + \cdots - 16294016)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 10241638465536 \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $89$ |
\( (T^{4} - 119 T^{3} + \cdots + 3662814)^{2} \)
|
| $97$ |
\( (T^{4} + 147 T^{3} + \cdots - 11582)^{2} \)
|
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