Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(116,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.116");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.e (of order \(3\), degree \(2\), not 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: | \(14.4554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.5567659200.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 17x^{4} - 28x^{3} + 289x^{2} - 238x + 196 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 17x^{4} - 28x^{3} + 289x^{2} - 238x + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 14 ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{3} - 14\nu^{2} + 289\nu ) / 238 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} + 17\nu^{2} - 14\nu + 173 ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{5} - 173\nu^{3} + 392\nu^{2} - 2941\nu + 2422 ) / 238 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 11\beta_{3} - \beta_{2} - \beta _1 - 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 17\beta_{2} + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( -17\beta_{5} + 17\beta_{4} - 187\beta_{3} + 31\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{5} + 392\beta_{3} - 303\beta_{2} - 303\beta _1 - 392 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 116.1 |
|
−1.74283 | − | 3.01866i | −0.425119 | + | 0.736328i | −2.07488 | + | 3.59380i | −2.50000 | − | 4.33013i | 2.96363 | 0 | −13.4206 | 13.1385 | + | 22.7566i | −8.71413 | + | 15.0933i | ||||||||||||||||||||||||
| 116.2 | 0.930543 | + | 1.61175i | −4.76818 | + | 8.25873i | 2.26818 | − | 3.92860i | −2.50000 | − | 4.33013i | −17.7480 | 0 | 23.3312 | −31.9711 | − | 55.3755i | 4.65272 | − | 8.05874i | |||||||||||||||||||||||||
| 116.3 | 2.31228 | + | 4.00499i | 4.19330 | − | 7.26300i | −6.69330 | + | 11.5931i | −2.50000 | − | 4.33013i | 38.7844 | 0 | −24.9107 | −21.6675 | − | 37.5292i | 11.5614 | − | 20.0250i | |||||||||||||||||||||||||
| 226.1 | −1.74283 | + | 3.01866i | −0.425119 | − | 0.736328i | −2.07488 | − | 3.59380i | −2.50000 | + | 4.33013i | 2.96363 | 0 | −13.4206 | 13.1385 | − | 22.7566i | −8.71413 | − | 15.0933i | |||||||||||||||||||||||||
| 226.2 | 0.930543 | − | 1.61175i | −4.76818 | − | 8.25873i | 2.26818 | + | 3.92860i | −2.50000 | + | 4.33013i | −17.7480 | 0 | 23.3312 | −31.9711 | + | 55.3755i | 4.65272 | + | 8.05874i | |||||||||||||||||||||||||
| 226.3 | 2.31228 | − | 4.00499i | 4.19330 | + | 7.26300i | −6.69330 | − | 11.5931i | −2.50000 | + | 4.33013i | 38.7844 | 0 | −24.9107 | −21.6675 | + | 37.5292i | 11.5614 | + | 20.0250i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.e.m | 6 | |
| 7.b | odd | 2 | 1 | 245.4.e.n | 6 | ||
| 7.c | even | 3 | 1 | 35.4.a.c | ✓ | 3 | |
| 7.c | even | 3 | 1 | inner | 245.4.e.m | 6 | |
| 7.d | odd | 6 | 1 | 245.4.a.l | 3 | ||
| 7.d | odd | 6 | 1 | 245.4.e.n | 6 | ||
| 21.g | even | 6 | 1 | 2205.4.a.bm | 3 | ||
| 21.h | odd | 6 | 1 | 315.4.a.p | 3 | ||
| 28.g | odd | 6 | 1 | 560.4.a.u | 3 | ||
| 35.i | odd | 6 | 1 | 1225.4.a.y | 3 | ||
| 35.j | even | 6 | 1 | 175.4.a.f | 3 | ||
| 35.l | odd | 12 | 2 | 175.4.b.e | 6 | ||
| 56.k | odd | 6 | 1 | 2240.4.a.bv | 3 | ||
| 56.p | even | 6 | 1 | 2240.4.a.bt | 3 | ||
| 105.o | odd | 6 | 1 | 1575.4.a.ba | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.c | ✓ | 3 | 7.c | even | 3 | 1 | |
| 175.4.a.f | 3 | 35.j | even | 6 | 1 | ||
| 175.4.b.e | 6 | 35.l | odd | 12 | 2 | ||
| 245.4.a.l | 3 | 7.d | odd | 6 | 1 | ||
| 245.4.e.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 245.4.e.m | 6 | 7.c | even | 3 | 1 | inner | |
| 245.4.e.n | 6 | 7.b | odd | 2 | 1 | ||
| 245.4.e.n | 6 | 7.d | odd | 6 | 1 | ||
| 315.4.a.p | 3 | 21.h | odd | 6 | 1 | ||
| 560.4.a.u | 3 | 28.g | odd | 6 | 1 | ||
| 1225.4.a.y | 3 | 35.i | odd | 6 | 1 | ||
| 1575.4.a.ba | 3 | 105.o | odd | 6 | 1 | ||
| 2205.4.a.bm | 3 | 21.g | even | 6 | 1 | ||
| 2240.4.a.bt | 3 | 56.p | even | 6 | 1 | ||
| 2240.4.a.bv | 3 | 56.k | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{6} - 3T_{2}^{5} + 23T_{2}^{4} - 18T_{2}^{3} + 286T_{2}^{2} - 420T_{2} + 900 \)
|
|
\( T_{3}^{6} + 2T_{3}^{5} + 83T_{3}^{4} - 22T_{3}^{3} + 6377T_{3}^{2} + 5372T_{3} + 4624 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 900 \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 4624 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 74 T^{5} + \cdots + 59166864 \)
|
| $13$ |
\( (T^{3} - 44 T^{2} + \cdots - 44870)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 3215570436 \)
|
| $19$ |
\( T^{6} + 168 T^{5} + \cdots + 824838400 \)
|
| $23$ |
\( T^{6} + \cdots + 8905319424 \)
|
| $29$ |
\( (T^{3} - 332 T^{2} + \cdots + 2565450)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 2517630976 \)
|
| $37$ |
\( T^{6} - 54 T^{5} + \cdots + 662341696 \)
|
| $41$ |
\( (T^{3} - 362 T^{2} + \cdots - 1536192)^{2} \)
|
| $43$ |
\( (T^{3} + 16 T^{2} + \cdots - 1524560)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 24690086463744 \)
|
| $53$ |
\( T^{6} + \cdots + 81\!\cdots\!96 \)
|
| $59$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 14\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} + 136 T^{2} + \cdots + 15575040)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 23\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + 1660 T^{2} + \cdots - 42727104)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 40443494630400 \)
|
| $97$ |
\( (T^{3} - 100 T^{2} + \cdots - 1978018)^{2} \)
|
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