Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [170,2,Mod(21,170)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(170, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("170.21");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 170 = 2 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 170.h (of order \(4\), degree \(2\), 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: | \(1.35745683436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.23045668864.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} - 96\nu^{5} - 505\nu^{3} - 618\nu ) / 272 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} - 17\nu^{6} - 52\nu^{5} - 340\nu^{4} - 406\nu^{3} - 2057\nu^{2} - 900\nu - 3706 ) / 272 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 17\nu^{6} + 52\nu^{5} - 340\nu^{4} + 406\nu^{3} - 2057\nu^{2} + 900\nu - 3706 ) / 272 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 20\nu^{4} - 113\nu^{2} - 162 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 20\nu^{5} - 4\nu^{4} - 121\nu^{3} - 52\nu^{2} - 218\nu - 136 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 20\nu^{5} + 4\nu^{4} - 121\nu^{3} + 52\nu^{2} - 218\nu + 136 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 6\beta_{2} - 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{6} - 13\beta_{5} + 13\beta_{4} + 13\beta_{3} + 57 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{7} + 15\beta_{6} + 25\beta_{4} - 25\beta_{3} - 82\beta_{2} + 57\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -40\beta_{7} + 40\beta_{6} + 139\beta_{5} - 147\beta_{4} - 147\beta_{3} - 511 \)
|
| \(\nu^{7}\) | \(=\) |
\( -187\beta_{7} - 187\beta_{6} - 379\beta_{4} + 379\beta_{3} + 914\beta_{2} - 511\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).
| \(n\) | \(71\) | \(137\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.1 |
|
− | 1.00000i | −2.31113 | + | 2.31113i | −1.00000 | 0.707107 | − | 0.707107i | 2.31113 | + | 2.31113i | −3.26843 | − | 3.26843i | 1.00000i | − | 7.68265i | −0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||
| 21.2 | − | 1.00000i | −1.20140 | + | 1.20140i | −1.00000 | −0.707107 | + | 0.707107i | 1.20140 | + | 1.20140i | 1.69904 | + | 1.69904i | 1.00000i | 0.113256i | 0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||
| 21.3 | − | 1.00000i | 1.60402 | − | 1.60402i | −1.00000 | 0.707107 | − | 0.707107i | −1.60402 | − | 1.60402i | 2.26843 | + | 2.26843i | 1.00000i | − | 2.14578i | −0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||
| 21.4 | − | 1.00000i | 1.90851 | − | 1.90851i | −1.00000 | −0.707107 | + | 0.707107i | −1.90851 | − | 1.90851i | −2.69904 | − | 2.69904i | 1.00000i | − | 4.28483i | 0.707107 | + | 0.707107i | |||||||||||||||||||||||||||||||
| 81.1 | 1.00000i | −2.31113 | − | 2.31113i | −1.00000 | 0.707107 | + | 0.707107i | 2.31113 | − | 2.31113i | −3.26843 | + | 3.26843i | − | 1.00000i | 7.68265i | −0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||
| 81.2 | 1.00000i | −1.20140 | − | 1.20140i | −1.00000 | −0.707107 | − | 0.707107i | 1.20140 | − | 1.20140i | 1.69904 | − | 1.69904i | − | 1.00000i | − | 0.113256i | 0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||
| 81.3 | 1.00000i | 1.60402 | + | 1.60402i | −1.00000 | 0.707107 | + | 0.707107i | −1.60402 | + | 1.60402i | 2.26843 | − | 2.26843i | − | 1.00000i | 2.14578i | −0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||
| 81.4 | 1.00000i | 1.90851 | + | 1.90851i | −1.00000 | −0.707107 | − | 0.707107i | −1.90851 | + | 1.90851i | −2.69904 | + | 2.69904i | − | 1.00000i | 4.28483i | 0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 170.2.h.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1530.2.q.g | 8 | ||
| 4.b | odd | 2 | 1 | 1360.2.bt.b | 8 | ||
| 5.b | even | 2 | 1 | 850.2.h.n | 8 | ||
| 5.c | odd | 4 | 1 | 850.2.g.i | 8 | ||
| 5.c | odd | 4 | 1 | 850.2.g.l | 8 | ||
| 17.c | even | 4 | 1 | inner | 170.2.h.b | ✓ | 8 |
| 17.d | even | 8 | 1 | 2890.2.a.bd | 4 | ||
| 17.d | even | 8 | 1 | 2890.2.a.be | 4 | ||
| 17.d | even | 8 | 2 | 2890.2.b.o | 8 | ||
| 51.f | odd | 4 | 1 | 1530.2.q.g | 8 | ||
| 68.f | odd | 4 | 1 | 1360.2.bt.b | 8 | ||
| 85.f | odd | 4 | 1 | 850.2.g.l | 8 | ||
| 85.i | odd | 4 | 1 | 850.2.g.i | 8 | ||
| 85.j | even | 4 | 1 | 850.2.h.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.2.h.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 170.2.h.b | ✓ | 8 | 17.c | even | 4 | 1 | inner |
| 850.2.g.i | 8 | 5.c | odd | 4 | 1 | ||
| 850.2.g.i | 8 | 85.i | odd | 4 | 1 | ||
| 850.2.g.l | 8 | 5.c | odd | 4 | 1 | ||
| 850.2.g.l | 8 | 85.f | odd | 4 | 1 | ||
| 850.2.h.n | 8 | 5.b | even | 2 | 1 | ||
| 850.2.h.n | 8 | 85.j | even | 4 | 1 | ||
| 1360.2.bt.b | 8 | 4.b | odd | 2 | 1 | ||
| 1360.2.bt.b | 8 | 68.f | odd | 4 | 1 | ||
| 1530.2.q.g | 8 | 3.b | odd | 2 | 1 | ||
| 1530.2.q.g | 8 | 51.f | odd | 4 | 1 | ||
| 2890.2.a.bd | 4 | 17.d | even | 8 | 1 | ||
| 2890.2.a.be | 4 | 17.d | even | 8 | 1 | ||
| 2890.2.b.o | 8 | 17.d | even | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{5} + 101T_{3}^{4} - 52T_{3}^{3} + 8T_{3}^{2} + 136T_{3} + 1156 \)
acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} - 4 T^{5} + \cdots + 1156 \)
|
| $5$ |
\( (T^{4} + 1)^{2} \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 18496 \)
|
| $11$ |
\( (T^{4} - 4 T^{3} + \cdots + 196)^{2} \)
|
| $13$ |
\( (T^{4} + 6 T^{3} - 3 T^{2} + \cdots - 4)^{2} \)
|
| $17$ |
\( T^{8} + 8 T^{7} + \cdots + 83521 \)
|
| $19$ |
\( T^{8} + 86 T^{6} + \cdots + 50176 \)
|
| $23$ |
\( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $29$ |
\( T^{8} - 24 T^{7} + \cdots + 4624 \)
|
| $31$ |
\( T^{8} - 4 T^{7} + \cdots + 4624 \)
|
| $37$ |
\( T^{8} + 20 T^{7} + \cdots + 506944 \)
|
| $41$ |
\( T^{8} + 5256 T^{4} + 5626384 \)
|
| $43$ |
\( T^{8} + 116 T^{6} + \cdots + 3136 \)
|
| $47$ |
\( (T^{4} - 10 T^{3} + \cdots - 392)^{2} \)
|
| $53$ |
\( T^{8} + 58 T^{6} + \cdots + 784 \)
|
| $59$ |
\( T^{8} + 222 T^{6} + \cdots + 80656 \)
|
| $61$ |
\( T^{8} + 16 T^{7} + \cdots + 1547536 \)
|
| $67$ |
\( (T^{4} + 8 T^{3} + \cdots + 752)^{2} \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots + 1817104 \)
|
| $73$ |
\( T^{8} - 28 T^{7} + \cdots + 30316036 \)
|
| $79$ |
\( T^{8} - 8 T^{7} + \cdots + 3625216 \)
|
| $83$ |
\( T^{8} + 364 T^{6} + \cdots + 23116864 \)
|
| $89$ |
\( (T^{4} + 22 T^{3} + \cdots - 32368)^{2} \)
|
| $97$ |
\( T^{8} - 20 T^{7} + \cdots + 15984004 \)
|
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