Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(846,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.846");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.b (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: | \(6.76332905120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.77720518656.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 161x^{4} - 220x^{3} + 232x^{2} - 132x + 33 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 161x^{4} - 220x^{3} + 232x^{2} - 132x + 33 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{6} - 6\nu^{5} + 47\nu^{4} - 84\nu^{3} + 239\nu^{2} - 198\nu + 138 ) / 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{7} + 11\nu^{6} + 38\nu^{5} + 599\nu^{4} - 1864\nu^{3} + 5303\nu^{2} - 12048\nu + 7545 ) / 1755 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -35\nu^{7} - 53\nu^{6} - 371\nu^{5} - 1748\nu^{4} + 901\nu^{3} - 8756\nu^{2} + 4272\nu - 4827 ) / 1755 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{7} + 5\nu^{6} - 121\nu^{5} + 17\nu^{4} - 475\nu^{3} + 464\nu^{2} - 1041\nu + 1866 ) / 351 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14\nu^{7} - 49\nu^{6} + 359\nu^{5} - 775\nu^{4} + 2237\nu^{3} - 2605\nu^{2} + 3135\nu - 1158 ) / 351 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -188\nu^{7} + 658\nu^{6} - 4520\nu^{5} + 9655\nu^{4} - 24524\nu^{3} + 27460\nu^{2} - 26103\nu + 8781 ) / 1755 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -376\nu^{7} + 1316\nu^{6} - 9040\nu^{5} + 19310\nu^{4} - 49048\nu^{3} + 54920\nu^{2} - 48696\nu + 15807 ) / 1755 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + 2\beta_{4} - 2\beta_{3} - 2\beta _1 - 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{7} + 21\beta_{6} - 2\beta_{5} + 3\beta_{4} - 3\beta_{3} - 6\beta_{2} - 11 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -23\beta_{7} + 44\beta_{6} - 20\beta_{4} + 28\beta_{3} - 12\beta_{2} + 44\beta _1 + 87 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 116\beta_{7} - 207\beta_{6} + 68\beta_{5} - 55\beta_{4} + 75\beta_{3} + 80\beta_{2} + 60\beta _1 + 186 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 203\beta_{7} - 366\beta_{6} + 60\beta_{5} + 96\beta_{4} - 160\beta_{3} + 135\beta_{2} - 285\beta _1 - 456 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 1107 \beta_{7} + 1902 \beta_{6} - 954 \beta_{5} + 868 \beta_{4} - 1386 \beta_{3} - 812 \beta_{2} + \cdots - 3121 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(365\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 846.1 |
|
− | 1.93185i | 0 | −1.73205 | − | 3.31662i | 0 | 2.34521 | + | 1.22474i | − | 0.517638i | 3.00000 | −6.40723 | |||||||||||||||||||||||||||||||||||||
| 846.2 | − | 1.93185i | 0 | −1.73205 | 3.31662i | 0 | −2.34521 | + | 1.22474i | − | 0.517638i | 3.00000 | 6.40723 | |||||||||||||||||||||||||||||||||||||||
| 846.3 | − | 0.517638i | 0 | 1.73205 | − | 3.31662i | 0 | −2.34521 | + | 1.22474i | − | 1.93185i | 3.00000 | −1.71681 | ||||||||||||||||||||||||||||||||||||||
| 846.4 | − | 0.517638i | 0 | 1.73205 | 3.31662i | 0 | 2.34521 | + | 1.22474i | − | 1.93185i | 3.00000 | 1.71681 | |||||||||||||||||||||||||||||||||||||||
| 846.5 | 0.517638i | 0 | 1.73205 | − | 3.31662i | 0 | 2.34521 | − | 1.22474i | 1.93185i | 3.00000 | 1.71681 | ||||||||||||||||||||||||||||||||||||||||
| 846.6 | 0.517638i | 0 | 1.73205 | 3.31662i | 0 | −2.34521 | − | 1.22474i | 1.93185i | 3.00000 | −1.71681 | |||||||||||||||||||||||||||||||||||||||||
| 846.7 | 1.93185i | 0 | −1.73205 | − | 3.31662i | 0 | −2.34521 | − | 1.22474i | 0.517638i | 3.00000 | 6.40723 | ||||||||||||||||||||||||||||||||||||||||
| 846.8 | 1.93185i | 0 | −1.73205 | 3.31662i | 0 | 2.34521 | − | 1.22474i | 0.517638i | 3.00000 | −6.40723 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 77.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 847.2.b.d | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 847.2.b.d | ✓ | 8 |
| 11.b | odd | 2 | 1 | inner | 847.2.b.d | ✓ | 8 |
| 11.c | even | 5 | 4 | 847.2.l.k | 32 | ||
| 11.d | odd | 10 | 4 | 847.2.l.k | 32 | ||
| 77.b | even | 2 | 1 | inner | 847.2.b.d | ✓ | 8 |
| 77.j | odd | 10 | 4 | 847.2.l.k | 32 | ||
| 77.l | even | 10 | 4 | 847.2.l.k | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.2.b.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 847.2.b.d | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 847.2.b.d | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 847.2.b.d | ✓ | 8 | 77.b | even | 2 | 1 | inner |
| 847.2.l.k | 32 | 11.c | even | 5 | 4 | ||
| 847.2.l.k | 32 | 11.d | odd | 10 | 4 | ||
| 847.2.l.k | 32 | 77.j | odd | 10 | 4 | ||
| 847.2.l.k | 32 | 77.l | even | 10 | 4 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 4 T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} + 11)^{4} \)
|
| $7$ |
\( (T^{4} - 8 T^{2} + 49)^{2} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{4} - 44 T^{2} + 121)^{2} \)
|
| $17$ |
\( (T^{4} - 44 T^{2} + 121)^{2} \)
|
| $19$ |
\( (T^{2} - 22)^{4} \)
|
| $23$ |
\( (T^{2} + 6 T + 6)^{4} \)
|
| $29$ |
\( (T^{4} + 52 T^{2} + 529)^{2} \)
|
| $31$ |
\( (T^{4} + 88 T^{2} + 484)^{2} \)
|
| $37$ |
\( (T^{2} + 10 T + 13)^{4} \)
|
| $41$ |
\( (T^{4} - 44 T^{2} + 121)^{2} \)
|
| $43$ |
\( (T^{4} + 112 T^{2} + 2704)^{2} \)
|
| $47$ |
\( (T^{4} + 88 T^{2} + 484)^{2} \)
|
| $53$ |
\( (T^{2} + 6 T - 39)^{4} \)
|
| $59$ |
\( (T^{2} + 44)^{4} \)
|
| $61$ |
\( (T^{2} - 66)^{4} \)
|
| $67$ |
\( (T^{2} + 2 T - 74)^{4} \)
|
| $71$ |
\( (T^{2} + 8 T + 4)^{4} \)
|
| $73$ |
\( (T^{2} - 22)^{4} \)
|
| $79$ |
\( (T^{4} + 84 T^{2} + 36)^{2} \)
|
| $83$ |
\( (T^{2} - 88)^{4} \)
|
| $89$ |
\( (T^{2} + 33)^{4} \)
|
| $97$ |
\( (T^{4} + 286 T^{2} + 14641)^{2} \)
|
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