Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,4,Mod(239,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.239");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.o (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: | \(14.1604584014\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 84x^{6} + 2444x^{4} - 3696x^{2} + 115600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{2} \) |
| 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
\( x^{8} - 84x^{6} + 2444x^{4} - 3696x^{2} + 115600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 33 \nu^{7} + 323 \nu^{6} + 3520 \nu^{5} - 32300 \nu^{4} - 136752 \nu^{3} + 950912 \nu^{2} + \cdots - 1485800 ) / 1421200 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} + 320\nu^{5} - 12432\nu^{3} + 99080\nu ) / 64600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21\nu^{7} - 85\nu^{6} + 1594\nu^{5} + 4760\nu^{4} - 41804\nu^{3} - 3400\nu^{2} - 213424\nu - 2765560 ) / 284240 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\nu^{7} - 1594\nu^{5} + 41804\nu^{3} + 213424\nu ) / 142120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 273 \nu^{7} + 2244 \nu^{6} + 24275 \nu^{5} - 168300 \nu^{4} - 792162 \nu^{3} + 5035536 \nu^{2} + \cdots - 7741800 ) / 710600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -27\nu^{7} + 2370\nu^{5} - 76188\nu^{3} + 624480\nu ) / 37400 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 84\nu^{5} - 2104\nu^{3} - 10584\nu ) / 680 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 3\beta_{4} - 6\beta_{2} ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{5} + 6\beta_{4} + 12\beta_{3} + 5\beta_{2} - 12\beta _1 + 126 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{7} + 13\beta_{6} + 105\beta_{4} - 153\beta_{2} ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{6} + 22\beta_{5} + 28\beta_{4} + 56\beta_{3} + 105\beta_{2} - 232\beta _1 + 542 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 315\beta_{7} + 104\beta_{6} + 3282\beta_{4} - 1149\beta_{2} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1828\beta_{6} + 3656\beta_{5} - 432\beta_{4} - 864\beta_{3} + 17540\beta_{2} - 38736\beta _1 - 9072 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 14952\beta_{7} - 7586\beta_{6} + 157290\beta_{4} + 80316\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
0 | −2.44949 | − | 4.58258i | 0 | −10.5357 | − | 3.74166i | 0 | 19.5959 | 0 | −15.0000 | + | 22.4499i | 0 | ||||||||||||||||||||||||||||||||||||
| 239.2 | 0 | −2.44949 | − | 4.58258i | 0 | 10.5357 | − | 3.74166i | 0 | 19.5959 | 0 | −15.0000 | + | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.3 | 0 | −2.44949 | + | 4.58258i | 0 | −10.5357 | + | 3.74166i | 0 | 19.5959 | 0 | −15.0000 | − | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.4 | 0 | −2.44949 | + | 4.58258i | 0 | 10.5357 | + | 3.74166i | 0 | 19.5959 | 0 | −15.0000 | − | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.5 | 0 | 2.44949 | − | 4.58258i | 0 | −10.5357 | + | 3.74166i | 0 | −19.5959 | 0 | −15.0000 | − | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.6 | 0 | 2.44949 | − | 4.58258i | 0 | 10.5357 | + | 3.74166i | 0 | −19.5959 | 0 | −15.0000 | − | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.7 | 0 | 2.44949 | + | 4.58258i | 0 | −10.5357 | − | 3.74166i | 0 | −19.5959 | 0 | −15.0000 | + | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
| 239.8 | 0 | 2.44949 | + | 4.58258i | 0 | 10.5357 | − | 3.74166i | 0 | −19.5959 | 0 | −15.0000 | + | 22.4499i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 20.d | odd | 2 | 1 | inner |
| 60.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.4.o.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 15.d | odd | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 20.d | odd | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| 60.h | even | 2 | 1 | inner | 240.4.o.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.4.o.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 240.4.o.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
| 240.4.o.b | ✓ | 8 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 384 \)
acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 30 T^{2} + 729)^{2} \)
|
| $5$ |
\( (T^{4} - 194 T^{2} + 15625)^{2} \)
|
| $7$ |
\( (T^{2} - 384)^{4} \)
|
| $11$ |
\( (T^{2} - 2664)^{4} \)
|
| $13$ |
\( (T^{2} + 6216)^{4} \)
|
| $17$ |
\( (T^{2} - 444)^{4} \)
|
| $19$ |
\( (T^{2} + 9324)^{4} \)
|
| $23$ |
\( (T^{2} + 10164)^{4} \)
|
| $29$ |
\( (T^{2} + 35000)^{4} \)
|
| $31$ |
\( (T^{2} + 83916)^{4} \)
|
| $37$ |
\( (T^{2} + 6216)^{4} \)
|
| $41$ |
\( (T^{2} + 118496)^{4} \)
|
| $43$ |
\( (T^{2} - 10584)^{4} \)
|
| $47$ |
\( (T^{2} + 2100)^{4} \)
|
| $53$ |
\( (T^{2} - 195804)^{4} \)
|
| $59$ |
\( (T^{2} - 130536)^{4} \)
|
| $61$ |
\( (T + 602)^{8} \)
|
| $67$ |
\( (T^{2} - 424536)^{4} \)
|
| $71$ |
\( (T^{2} - 95904)^{4} \)
|
| $73$ |
\( (T^{2} + 24864)^{4} \)
|
| $79$ |
\( (T^{2} + 9324)^{4} \)
|
| $83$ |
\( (T^{2} + 961716)^{4} \)
|
| $89$ |
\( (T^{2} + 224)^{4} \)
|
| $97$ |
\( T^{8} \)
|
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