Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [220,2,Mod(131,220)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 220 = 2^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 220.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: | \(1.75670884447\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.31116960000.7 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 14x^{5} + 53x^{4} - 134x^{3} - 218x^{2} + 288x + 904 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - 4x^{7} + 14x^{5} + 53x^{4} - 134x^{3} - 218x^{2} + 288x + 904 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} - 2\nu^{4} - \nu^{3} - 31\nu^{2} + 32\nu - 4 ) / 66 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 53\nu^{7} + 1568\nu^{6} - 6230\nu^{5} - 1872\nu^{4} + 24919\nu^{3} + 115828\nu^{2} - 221174\nu - 346324 ) / 132264 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -106\nu^{7} + 371\nu^{6} + 1939\nu^{5} - 5775\nu^{4} - 13265\nu^{3} + 25858\nu^{2} + 164794\nu - 86908 ) / 132264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -143\nu^{7} + 1252\nu^{6} - 3382\nu^{5} - 5196\nu^{4} + 11867\nu^{3} + 38372\nu^{2} - 52846\nu - 256484 ) / 132264 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 98\nu^{7} - 343\nu^{6} + 79\nu^{5} + 660\nu^{4} + 6025\nu^{3} - 9869\nu^{2} - 2000\nu + 19208 ) / 33066 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 143\nu^{7} - 250\nu^{6} + 376\nu^{5} - 3822\nu^{4} + 11179\nu^{3} + 3712\nu^{2} - 1262\nu - 92212 ) / 44088 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -215\nu^{7} + 1003\nu^{6} - 250\nu^{5} - 2634\nu^{4} - 4195\nu^{3} + 17945\nu^{2} + 39560\nu - 41138 ) / 33066 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{7} - 2\beta_{6} + 8\beta_{5} - \beta_{4} + 3\beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{7} - 8\beta_{6} + 15\beta_{5} - 16\beta_{4} + 8\beta_{2} + \beta _1 - 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( 25\beta_{7} + 21\beta_{6} - 6\beta_{5} - 76\beta_{4} - 30\beta_{3} + 15\beta_{2} + 5\beta _1 - 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( 60\beta_{7} + 96\beta_{6} - 55\beta_{5} - 148\beta_{4} - 120\beta_{3} - 36\beta_{2} - 87\beta _1 - 158 \)
|
| \(\nu^{7}\) | \(=\) |
\( -35\beta_{7} + 385\beta_{6} - 322\beta_{5} - 378\beta_{4} - 174\beta_{3} - 175\beta_{2} - 399\beta _1 - 381 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(111\) | \(177\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 131.1 |
|
−1.32288 | − | 0.500000i | − | 3.25937i | 1.50000 | + | 1.32288i | −1.00000 | −1.62968 | + | 4.31174i | −3.25937 | −1.32288 | − | 2.50000i | −7.62348 | 1.32288 | + | 0.500000i | |||||||||||||||||||||||||||||||
| 131.2 | −1.32288 | − | 0.500000i | 0.613616i | 1.50000 | + | 1.32288i | −1.00000 | 0.306808 | − | 0.811738i | 0.613616 | −1.32288 | − | 2.50000i | 2.62348 | 1.32288 | + | 0.500000i | |||||||||||||||||||||||||||||||||
| 131.3 | −1.32288 | + | 0.500000i | − | 0.613616i | 1.50000 | − | 1.32288i | −1.00000 | 0.306808 | + | 0.811738i | 0.613616 | −1.32288 | + | 2.50000i | 2.62348 | 1.32288 | − | 0.500000i | ||||||||||||||||||||||||||||||||
| 131.4 | −1.32288 | + | 0.500000i | 3.25937i | 1.50000 | − | 1.32288i | −1.00000 | −1.62968 | − | 4.31174i | −3.25937 | −1.32288 | + | 2.50000i | −7.62348 | 1.32288 | − | 0.500000i | |||||||||||||||||||||||||||||||||
| 131.5 | 1.32288 | − | 0.500000i | − | 0.613616i | 1.50000 | − | 1.32288i | −1.00000 | −0.306808 | − | 0.811738i | −0.613616 | 1.32288 | − | 2.50000i | 2.62348 | −1.32288 | + | 0.500000i | ||||||||||||||||||||||||||||||||
| 131.6 | 1.32288 | − | 0.500000i | 3.25937i | 1.50000 | − | 1.32288i | −1.00000 | 1.62968 | + | 4.31174i | 3.25937 | 1.32288 | − | 2.50000i | −7.62348 | −1.32288 | + | 0.500000i | |||||||||||||||||||||||||||||||||
| 131.7 | 1.32288 | + | 0.500000i | − | 3.25937i | 1.50000 | + | 1.32288i | −1.00000 | 1.62968 | − | 4.31174i | 3.25937 | 1.32288 | + | 2.50000i | −7.62348 | −1.32288 | − | 0.500000i | ||||||||||||||||||||||||||||||||
| 131.8 | 1.32288 | + | 0.500000i | 0.613616i | 1.50000 | + | 1.32288i | −1.00000 | −0.306808 | + | 0.811738i | −0.613616 | 1.32288 | + | 2.50000i | 2.62348 | −1.32288 | − | 0.500000i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 44.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 220.2.d.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 220.2.d.c | ✓ | 8 |
| 8.b | even | 2 | 1 | 3520.2.f.i | 8 | ||
| 8.d | odd | 2 | 1 | 3520.2.f.i | 8 | ||
| 11.b | odd | 2 | 1 | inner | 220.2.d.c | ✓ | 8 |
| 44.c | even | 2 | 1 | inner | 220.2.d.c | ✓ | 8 |
| 88.b | odd | 2 | 1 | 3520.2.f.i | 8 | ||
| 88.g | even | 2 | 1 | 3520.2.f.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.2.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 220.2.d.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 220.2.d.c | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 220.2.d.c | ✓ | 8 | 44.c | even | 2 | 1 | inner |
| 3520.2.f.i | 8 | 8.b | even | 2 | 1 | ||
| 3520.2.f.i | 8 | 8.d | odd | 2 | 1 | ||
| 3520.2.f.i | 8 | 88.b | odd | 2 | 1 | ||
| 3520.2.f.i | 8 | 88.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\):
|
\( T_{3}^{4} + 11T_{3}^{2} + 4 \)
|
|
\( T_{7}^{4} - 11T_{7}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 3 T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{4} + 11 T^{2} + 4)^{2} \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( (T^{4} - 11 T^{2} + 4)^{2} \)
|
| $11$ |
\( T^{8} - 178 T^{4} + 14641 \)
|
| $13$ |
\( (T^{2} + 4)^{4} \)
|
| $17$ |
\( (T^{4} + 53 T^{2} + 676)^{2} \)
|
| $19$ |
\( (T^{4} - 71 T^{2} + 1024)^{2} \)
|
| $23$ |
\( (T^{4} + 44 T^{2} + 64)^{2} \)
|
| $29$ |
\( (T^{4} + 65 T^{2} + 400)^{2} \)
|
| $31$ |
\( (T^{4} + 11 T^{2} + 4)^{2} \)
|
| $37$ |
\( (T^{2} - T - 26)^{4} \)
|
| $41$ |
\( (T^{2} + 64)^{4} \)
|
| $43$ |
\( (T^{4} - 44 T^{2} + 64)^{2} \)
|
| $47$ |
\( (T^{4} + 44 T^{2} + 64)^{2} \)
|
| $53$ |
\( (T^{2} + 9 T - 6)^{4} \)
|
| $59$ |
\( (T^{4} + 156 T^{2} + 2304)^{2} \)
|
| $61$ |
\( (T^{4} + 137 T^{2} + 256)^{2} \)
|
| $67$ |
\( (T^{2} + 60)^{4} \)
|
| $71$ |
\( (T^{4} + 99 T^{2} + 324)^{2} \)
|
| $73$ |
\( (T^{2} + 4)^{4} \)
|
| $79$ |
\( (T^{4} - 176 T^{2} + 1024)^{2} \)
|
| $83$ |
\( (T^{2} - 60)^{4} \)
|
| $89$ |
\( (T^{2} + 7 T - 14)^{4} \)
|
| $97$ |
\( (T - 2)^{8} \)
|
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