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: | 10.0.578281160704.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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
\( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{8} - 2\nu^{6} + 2\nu^{5} + 3\nu^{4} + 6\nu^{3} - 2\nu^{2} + 8\nu - 16 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} + 2\nu^{6} - 2\nu^{5} - 3\nu^{4} - 6\nu^{3} - 6\nu^{2} + 16 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} + 6\nu^{7} + 2\nu^{6} + 9\nu^{5} + 8\nu^{4} - 6\nu^{3} - 28\nu^{2} - 8\nu - 80 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} - 2\nu^{8} - 6\nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} + 14\nu^{3} + 12\nu^{2} + 8\nu + 32 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} + 6\nu^{8} - 2\nu^{7} + 2\nu^{6} - 7\nu^{5} - 16\nu^{4} - 14\nu^{3} + 4\nu^{2} - 24\nu + 80 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{9} + 8\nu^{8} - 2\nu^{7} + 6\nu^{6} + \nu^{5} - 14\nu^{4} - 26\nu^{3} + 16\nu^{2} - 72\nu + 96 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{9} - 3\nu^{8} + 4\nu^{7} - 2\nu^{6} + 5\nu^{4} + 6\nu^{3} - 22\nu^{2} + 24\nu - 64 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} + 4\nu^{8} + 2\nu^{7} + 2\nu^{6} - 9\nu^{5} - 22\nu^{4} - 14\nu^{3} - 24\nu^{2} + 8\nu + 48 ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -5\nu^{9} + 2\nu^{7} + 10\nu^{6} + 23\nu^{5} + 22\nu^{4} - 6\nu^{3} - 88\nu - 64 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - 4\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{8} + \beta_{7} - \beta_{6} + 4\beta_{5} - \beta_{4} + 3\beta_{2} + \beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{8} - 5\beta_{7} - 3\beta_{6} - \beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 + 2 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4\beta_{9} + 4\beta_{8} - 3\beta_{7} - 7\beta_{6} - 3\beta_{4} - 4\beta_{3} + \beta_{2} - 3\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2\beta_{9} + 2\beta_{8} + 3\beta_{7} + \beta_{6} - 4\beta_{5} - 7\beta_{4} - 8\beta_{3} - 3\beta_{2} - \beta _1 + 18 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 4 \beta_{9} - 4 \beta_{8} - \beta_{7} - 7 \beta_{6} + 12 \beta_{5} - 11 \beta_{4} - 8 \beta_{3} + \cdots - 4 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -2\beta_{8} - 7\beta_{7} - 13\beta_{6} + 16\beta_{5} + \beta_{4} + 12\beta_{3} - 7\beta_{2} - 7\beta _1 + 6 ) / 2 \)
|
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 |
|
−1.23417 | − | 0.690521i | − | 1.81026i | 1.04636 | + | 1.70444i | − | 0.282461i | −1.25002 | + | 2.23417i | 3.84939 | −0.114433 | − | 2.82611i | −0.277041 | −0.195045 | + | 0.348605i | ||||||||||||||||||||||||||||||||||||
| 485.2 | −1.23417 | + | 0.690521i | 1.81026i | 1.04636 | − | 1.70444i | 0.282461i | −1.25002 | − | 2.23417i | 3.84939 | −0.114433 | + | 2.82611i | −0.277041 | −0.195045 | − | 0.348605i | |||||||||||||||||||||||||||||||||||||||
| 485.3 | −0.739098 | − | 1.20571i | 2.35300i | −0.907469 | + | 1.78227i | − | 4.16794i | 2.83703 | − | 1.73910i | −0.933222 | 2.81961 | − | 0.223131i | −2.53661 | −5.02532 | + | 3.08051i | ||||||||||||||||||||||||||||||||||||||
| 485.4 | −0.739098 | + | 1.20571i | − | 2.35300i | −0.907469 | − | 1.78227i | 4.16794i | 2.83703 | + | 1.73910i | −0.933222 | 2.81961 | + | 0.223131i | −2.53661 | −5.02532 | − | 3.08051i | ||||||||||||||||||||||||||||||||||||||
| 485.5 | 0.246440 | − | 1.39258i | − | 3.05779i | −1.87853 | − | 0.686372i | − | 0.699283i | −4.25820 | − | 0.753560i | −3.27803 | −1.41877 | + | 2.44685i | −6.35006 | −0.973805 | − | 0.172331i | |||||||||||||||||||||||||||||||||||||
| 485.6 | 0.246440 | + | 1.39258i | 3.05779i | −1.87853 | + | 0.686372i | 0.699283i | −4.25820 | + | 0.753560i | −3.27803 | −1.41877 | − | 2.44685i | −6.35006 | −0.973805 | + | 0.172331i | |||||||||||||||||||||||||||||||||||||||
| 485.7 | 0.428185 | − | 1.34783i | 1.33544i | −1.63332 | − | 1.15424i | 1.93119i | 1.79995 | + | 0.571815i | 1.83930 | −2.25509 | + | 1.70721i | 1.21660 | 2.60293 | + | 0.826906i | |||||||||||||||||||||||||||||||||||||||
| 485.8 | 0.428185 | + | 1.34783i | − | 1.33544i | −1.63332 | + | 1.15424i | − | 1.93119i | 1.79995 | − | 0.571815i | 1.83930 | −2.25509 | − | 1.70721i | 1.21660 | 2.60293 | − | 0.826906i | |||||||||||||||||||||||||||||||||||||
| 485.9 | 1.29865 | − | 0.559929i | − | 0.229967i | 1.37296 | − | 1.45430i | − | 2.51595i | −0.128765 | − | 0.298645i | −1.47743 | 0.968683 | − | 2.65738i | 2.94712 | −1.40875 | − | 3.26733i | |||||||||||||||||||||||||||||||||||||
| 485.10 | 1.29865 | + | 0.559929i | 0.229967i | 1.37296 | + | 1.45430i | 2.51595i | −0.128765 | + | 0.298645i | −1.47743 | 0.968683 | + | 2.65738i | 2.94712 | −1.40875 | + | 3.26733i | |||||||||||||||||||||||||||||||||||||||
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.d | 10 | |
| 4.b | odd | 2 | 1 | 3872.2.c.f | 10 | ||
| 8.b | even | 2 | 1 | inner | 968.2.c.d | 10 | |
| 8.d | odd | 2 | 1 | 3872.2.c.f | 10 | ||
| 11.b | odd | 2 | 1 | 88.2.c.a | ✓ | 10 | |
| 11.c | even | 5 | 4 | 968.2.o.h | 40 | ||
| 11.d | odd | 10 | 4 | 968.2.o.g | 40 | ||
| 33.d | even | 2 | 1 | 792.2.f.g | 10 | ||
| 44.c | even | 2 | 1 | 352.2.c.a | 10 | ||
| 88.b | odd | 2 | 1 | 88.2.c.a | ✓ | 10 | |
| 88.g | even | 2 | 1 | 352.2.c.a | 10 | ||
| 88.o | even | 10 | 4 | 968.2.o.h | 40 | ||
| 88.p | odd | 10 | 4 | 968.2.o.g | 40 | ||
| 132.d | odd | 2 | 1 | 3168.2.f.g | 10 | ||
| 176.i | even | 4 | 1 | 2816.2.a.p | 5 | ||
| 176.i | even | 4 | 1 | 2816.2.a.q | 5 | ||
| 176.l | odd | 4 | 1 | 2816.2.a.o | 5 | ||
| 176.l | odd | 4 | 1 | 2816.2.a.r | 5 | ||
| 264.m | even | 2 | 1 | 792.2.f.g | 10 | ||
| 264.p | odd | 2 | 1 | 3168.2.f.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.2.c.a | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 88.2.c.a | ✓ | 10 | 88.b | odd | 2 | 1 | |
| 352.2.c.a | 10 | 44.c | even | 2 | 1 | ||
| 352.2.c.a | 10 | 88.g | even | 2 | 1 | ||
| 792.2.f.g | 10 | 33.d | even | 2 | 1 | ||
| 792.2.f.g | 10 | 264.m | even | 2 | 1 | ||
| 968.2.c.d | 10 | 1.a | even | 1 | 1 | trivial | |
| 968.2.c.d | 10 | 8.b | even | 2 | 1 | inner | |
| 968.2.o.g | 40 | 11.d | odd | 10 | 4 | ||
| 968.2.o.g | 40 | 88.p | odd | 10 | 4 | ||
| 968.2.o.h | 40 | 11.c | even | 5 | 4 | ||
| 968.2.o.h | 40 | 88.o | even | 10 | 4 | ||
| 2816.2.a.o | 5 | 176.l | odd | 4 | 1 | ||
| 2816.2.a.p | 5 | 176.i | even | 4 | 1 | ||
| 2816.2.a.q | 5 | 176.i | even | 4 | 1 | ||
| 2816.2.a.r | 5 | 176.l | odd | 4 | 1 | ||
| 3168.2.f.g | 10 | 132.d | odd | 2 | 1 | ||
| 3168.2.f.g | 10 | 264.p | odd | 2 | 1 | ||
| 3872.2.c.f | 10 | 4.b | odd | 2 | 1 | ||
| 3872.2.c.f | 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 \(S_{2}^{\mathrm{new}}(968, [\chi])\):
|
\( T_{3}^{10} + 20T_{3}^{8} + 134T_{3}^{6} + 356T_{3}^{4} + 321T_{3}^{2} + 16 \)
|
|
\( T_{5}^{10} + 28T_{5}^{8} + 214T_{5}^{6} + 524T_{5}^{4} + 241T_{5}^{2} + 16 \)
|
|
\( T_{7}^{5} - 16T_{7}^{3} - 8T_{7}^{2} + 40T_{7} + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 2 T^{8} + \cdots + 32 \)
|
| $3$ |
\( T^{10} + 20 T^{8} + \cdots + 16 \)
|
| $5$ |
\( T^{10} + 28 T^{8} + \cdots + 16 \)
|
| $7$ |
\( (T^{5} - 16 T^{3} + \cdots + 32)^{2} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} + 52 T^{8} + \cdots + 256 \)
|
| $17$ |
\( (T^{5} - 2 T^{4} + \cdots - 464)^{2} \)
|
| $19$ |
\( T^{10} + 96 T^{8} + \cdots + 262144 \)
|
| $23$ |
\( (T^{5} + 6 T^{4} + \cdots + 314)^{2} \)
|
| $29$ |
\( T^{10} + 132 T^{8} + \cdots + 262144 \)
|
| $31$ |
\( (T^{5} + 2 T^{4} + \cdots - 226)^{2} \)
|
| $37$ |
\( T^{10} + 140 T^{8} + \cdots + 179776 \)
|
| $41$ |
\( (T^{5} + 2 T^{4} + \cdots + 464)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 100962304 \)
|
| $47$ |
\( (T^{5} + 2 T^{4} + \cdots - 224)^{2} \)
|
| $53$ |
\( T^{10} + 448 T^{8} + \cdots + 44302336 \)
|
| $59$ |
\( T^{10} + 148 T^{8} + \cdots + 183184 \)
|
| $61$ |
\( T^{10} + 260 T^{8} + \cdots + 65536 \)
|
| $67$ |
\( T^{10} + 180 T^{8} + \cdots + 446224 \)
|
| $71$ |
\( (T^{5} + 6 T^{4} + \cdots + 83746)^{2} \)
|
| $73$ |
\( (T^{5} - 2 T^{4} + \cdots - 12752)^{2} \)
|
| $79$ |
\( (T^{5} + 8 T^{4} + \cdots - 11008)^{2} \)
|
| $83$ |
\( T^{10} + 400 T^{8} + \cdots + 802816 \)
|
| $89$ |
\( (T^{5} + 2 T^{4} + \cdots + 3566)^{2} \)
|
| $97$ |
\( (T^{5} + 10 T^{4} + \cdots + 20462)^{2} \)
|
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