Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [968,2,Mod(485,968)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(968, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("968.485");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 968 = 2^{3} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 968.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: | \(7.72951891566\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.237305000762368.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{8} + 2x^{6} - 4x^{5} + 4x^{4} - 8x^{2} + 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{8} + 2x^{6} - 4x^{5} + 4x^{4} - 8x^{2} + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - \nu^{6} + 2\nu^{4} - 4\nu^{3} + 4\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + \nu^{6} - 4\nu^{4} - 4\nu^{3} + 12\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - \nu^{7} + 2\nu^{5} - 4\nu^{4} + 4\nu^{3} - 8\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + \nu^{5} - \nu^{4} - 4\nu^{2} + 8\nu - 8 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - \nu^{7} + 4\nu^{6} + 2\nu^{5} + 4\nu^{3} - 8\nu + 16 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} - \nu^{7} + 4\nu^{6} - 6\nu^{5} + 12\nu^{3} - 24\nu + 32 ) / 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} + \nu^{7} - 2\nu^{4} + 8\nu^{3} - 8\nu^{2} + 8\nu + 8 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + \nu^{7} - 3\nu^{6} + 2\nu^{5} + 4\nu^{4} - 4\nu^{3} + 4\nu^{2} + 8\nu - 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{9} + 2\beta_{7} + 2\beta_{4} - 2\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{9} + \beta_{8} + 2\beta_{6} - \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{9} - 2\beta_{7} + 4\beta_{6} + 2\beta_{4} - 2\beta_{2} - \beta _1 + 4 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{9} - \beta_{8} + 6\beta_{6} + \beta_{5} - 6\beta_{4} + \beta_{3} - 2\beta_{2} - 7 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 4 \beta_{3} + \cdots + 8 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( \beta_{9} - 7\beta_{8} + 8\beta_{7} + 2\beta_{6} + 7\beta_{5} + 6\beta_{4} - \beta_{3} + 2\beta_{2} + 4\beta _1 + 7 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -6\beta_{9} + 8\beta_{8} - 2\beta_{7} + 4\beta_{6} + 2\beta_{4} + 22\beta_{2} - \beta _1 - 4 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(849\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
−1.36273 | − | 0.378114i | 3.25849i | 1.71406 | + | 1.03053i | − | 1.14481i | 1.23208 | − | 4.44043i | −3.12590 | −1.94614 | − | 2.05245i | −7.61774 | −0.432867 | + | 1.56006i | |||||||||||||||||||||||||||||||||||||
| 485.2 | −1.36273 | + | 0.378114i | − | 3.25849i | 1.71406 | − | 1.03053i | 1.14481i | 1.23208 | + | 4.44043i | −3.12590 | −1.94614 | + | 2.05245i | −7.61774 | −0.432867 | − | 1.56006i | ||||||||||||||||||||||||||||||||||||||
| 485.3 | −0.921630 | − | 1.07266i | − | 1.02292i | −0.301198 | + | 1.97719i | − | 1.25086i | −1.09725 | + | 0.942755i | −0.0747365 | 2.39845 | − | 1.49915i | 1.95363 | −1.34174 | + | 1.15283i | |||||||||||||||||||||||||||||||||||||
| 485.4 | −0.921630 | + | 1.07266i | 1.02292i | −0.301198 | − | 1.97719i | 1.25086i | −1.09725 | − | 0.942755i | −0.0747365 | 2.39845 | + | 1.49915i | 1.95363 | −1.34174 | − | 1.15283i | |||||||||||||||||||||||||||||||||||||||
| 485.5 | −0.0720143 | − | 1.41238i | 1.81186i | −1.98963 | + | 0.203423i | 3.73541i | 2.55904 | − | 0.130480i | 1.19046 | 0.430592 | + | 2.79546i | −0.282844 | 5.27582 | − | 0.269003i | |||||||||||||||||||||||||||||||||||||||
| 485.6 | −0.0720143 | + | 1.41238i | − | 1.81186i | −1.98963 | − | 0.203423i | − | 3.73541i | 2.55904 | + | 0.130480i | 1.19046 | 0.430592 | − | 2.79546i | −0.282844 | 5.27582 | + | 0.269003i | |||||||||||||||||||||||||||||||||||||
| 485.7 | 1.10029 | − | 0.888466i | 2.08484i | 0.421256 | − | 1.95513i | − | 3.17217i | 1.85231 | + | 2.29392i | 4.92845 | −1.27357 | − | 2.52548i | −1.34656 | −2.81837 | − | 3.49030i | ||||||||||||||||||||||||||||||||||||||
| 485.8 | 1.10029 | + | 0.888466i | − | 2.08484i | 0.421256 | + | 1.95513i | 3.17217i | 1.85231 | − | 2.29392i | 4.92845 | −1.27357 | + | 2.52548i | −1.34656 | −2.81837 | + | 3.49030i | ||||||||||||||||||||||||||||||||||||||
| 485.9 | 1.25609 | − | 0.649804i | − | 0.840529i | 1.15551 | − | 1.63242i | 2.02702i | −0.546179 | − | 1.05578i | −2.91827 | 0.390669 | − | 2.80132i | 2.29351 | 1.31716 | + | 2.54611i | ||||||||||||||||||||||||||||||||||||||
| 485.10 | 1.25609 | + | 0.649804i | 0.840529i | 1.15551 | + | 1.63242i | − | 2.02702i | −0.546179 | + | 1.05578i | −2.91827 | 0.390669 | + | 2.80132i | 2.29351 | 1.31716 | − | 2.54611i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 968.2.c.f | yes | 10 |
| 4.b | odd | 2 | 1 | 3872.2.c.e | 10 | ||
| 8.b | even | 2 | 1 | inner | 968.2.c.f | yes | 10 |
| 8.d | odd | 2 | 1 | 3872.2.c.e | 10 | ||
| 11.b | odd | 2 | 1 | 968.2.c.e | ✓ | 10 | |
| 11.c | even | 5 | 4 | 968.2.o.e | 40 | ||
| 11.d | odd | 10 | 4 | 968.2.o.f | 40 | ||
| 44.c | even | 2 | 1 | 3872.2.c.d | 10 | ||
| 88.b | odd | 2 | 1 | 968.2.c.e | ✓ | 10 | |
| 88.g | even | 2 | 1 | 3872.2.c.d | 10 | ||
| 88.o | even | 10 | 4 | 968.2.o.e | 40 | ||
| 88.p | odd | 10 | 4 | 968.2.o.f | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 968.2.c.e | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 968.2.c.e | ✓ | 10 | 88.b | odd | 2 | 1 | |
| 968.2.c.f | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 968.2.c.f | yes | 10 | 8.b | even | 2 | 1 | inner |
| 968.2.o.e | 40 | 11.c | even | 5 | 4 | ||
| 968.2.o.e | 40 | 88.o | even | 10 | 4 | ||
| 968.2.o.f | 40 | 11.d | odd | 10 | 4 | ||
| 968.2.o.f | 40 | 88.p | odd | 10 | 4 | ||
| 3872.2.c.d | 10 | 44.c | even | 2 | 1 | ||
| 3872.2.c.d | 10 | 88.g | even | 2 | 1 | ||
| 3872.2.c.e | 10 | 4.b | odd | 2 | 1 | ||
| 3872.2.c.e | 10 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):
|
\( T_{3}^{10} + 20T_{3}^{8} + 128T_{3}^{6} + 332T_{3}^{4} + 336T_{3}^{2} + 112 \)
|
|
\( T_{5}^{10} + 31T_{5}^{8} + 322T_{5}^{6} + 1322T_{5}^{4} + 2149T_{5}^{2} + 1183 \)
|
|
\( T_{7}^{5} - 22T_{7}^{3} - 22T_{7}^{2} + 52T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - T^{8} + \cdots + 32 \)
|
| $3$ |
\( T^{10} + 20 T^{8} + \cdots + 112 \)
|
| $5$ |
\( T^{10} + 31 T^{8} + \cdots + 1183 \)
|
| $7$ |
\( (T^{5} - 22 T^{3} - 22 T^{2} + \cdots + 4)^{2} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} + 67 T^{8} + \cdots + 7 \)
|
| $17$ |
\( (T^{5} - T^{4} - 50 T^{3} + \cdots - 853)^{2} \)
|
| $19$ |
\( T^{10} + 60 T^{8} + \cdots + 32368 \)
|
| $23$ |
\( (T^{5} + 6 T^{4} + \cdots + 164)^{2} \)
|
| $29$ |
\( T^{10} + 123 T^{8} + \cdots + 199927 \)
|
| $31$ |
\( (T^{5} - 10 T^{4} + \cdots + 1268)^{2} \)
|
| $37$ |
\( T^{10} + 227 T^{8} + \cdots + 2716903 \)
|
| $41$ |
\( (T^{5} + T^{4} - 110 T^{3} + \cdots - 971)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 340652032 \)
|
| $47$ |
\( (T^{5} - 16 T^{4} + \cdots + 7228)^{2} \)
|
| $53$ |
\( T^{10} + 223 T^{8} + \cdots + 10607527 \)
|
| $59$ |
\( T^{10} + \cdots + 598678528 \)
|
| $61$ |
\( T^{10} + 80 T^{8} + \cdots + 114688 \)
|
| $67$ |
\( T^{10} + \cdots + 191909872 \)
|
| $71$ |
\( (T^{5} + 6 T^{4} + \cdots + 544)^{2} \)
|
| $73$ |
\( (T^{5} + 14 T^{4} + \cdots + 28736)^{2} \)
|
| $79$ |
\( (T^{5} + 4 T^{4} + \cdots - 6668)^{2} \)
|
| $83$ |
\( T^{10} + 124 T^{8} + \cdots + 1330672 \)
|
| $89$ |
\( (T^{5} - T^{4} - 62 T^{3} + \cdots + 1271)^{2} \)
|
| $97$ |
\( (T^{5} - 5 T^{4} + \cdots + 707)^{2} \)
|
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