Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.a (trivial) |
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: | yes |
| Analytic conductor: | \(14.4554679514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1163891200.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.51797 | −3.86039 | 22.4480 | 5.00000 | 21.3015 | 0 | −79.7239 | −12.0974 | −27.5899 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.88087 | 2.89052 | 0.299392 | 5.00000 | −8.32721 | 0 | 22.1844 | −18.6449 | −14.4043 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.369315 | −9.74070 | −7.86361 | 5.00000 | 3.59738 | 0 | 5.85867 | 67.8812 | −1.84657 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.644648 | 4.18687 | −7.58443 | 5.00000 | 2.69906 | 0 | −10.0465 | −9.47008 | 3.22324 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.65606 | 0.332888 | −5.25746 | 5.00000 | 0.551283 | 0 | −21.9552 | −26.8892 | 8.28031 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.46745 | −9.80920 | 11.9581 | 5.00000 | −43.8221 | 0 | 17.6824 | 69.2204 | 22.3372 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.a.o | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2205.4.a.bz | 6 | ||
| 5.b | even | 2 | 1 | 1225.4.a.bj | 6 | ||
| 7.b | odd | 2 | 1 | 245.4.a.p | yes | 6 | |
| 7.c | even | 3 | 2 | 245.4.e.q | 12 | ||
| 7.d | odd | 6 | 2 | 245.4.e.p | 12 | ||
| 21.c | even | 2 | 1 | 2205.4.a.ca | 6 | ||
| 35.c | odd | 2 | 1 | 1225.4.a.bi | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.4.a.o | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 245.4.a.p | yes | 6 | 7.b | odd | 2 | 1 | |
| 245.4.e.p | 12 | 7.d | odd | 6 | 2 | ||
| 245.4.e.q | 12 | 7.c | even | 3 | 2 | ||
| 1225.4.a.bi | 6 | 35.c | odd | 2 | 1 | ||
| 1225.4.a.bj | 6 | 5.b | even | 2 | 1 | ||
| 2205.4.a.bz | 6 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.ca | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 29T_{2}^{4} - 28T_{2}^{3} + 134T_{2}^{2} - 24T_{2} - 28 \)
|
|
\( T_{3}^{6} + 16T_{3}^{5} + 12T_{3}^{4} - 564T_{3}^{3} - 355T_{3}^{2} + 4644T_{3} - 1486 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots - 28 \)
|
| $3$ |
\( T^{6} + 16 T^{5} + \cdots - 1486 \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 9225436100 \)
|
| $13$ |
\( T^{6} + \cdots + 12513937372 \)
|
| $17$ |
\( T^{6} + \cdots - 3807091762 \)
|
| $19$ |
\( T^{6} + \cdots + 1617325344 \)
|
| $23$ |
\( T^{6} + 336 T^{5} + \cdots + 298897696 \)
|
| $29$ |
\( T^{6} + \cdots - 544215793700 \)
|
| $31$ |
\( T^{6} + \cdots + 16539893268192 \)
|
| $37$ |
\( T^{6} + \cdots + 271258136464 \)
|
| $41$ |
\( T^{6} + \cdots + 144691772208184 \)
|
| $43$ |
\( T^{6} + \cdots + 440374360000 \)
|
| $47$ |
\( T^{6} + \cdots + 251564448569400 \)
|
| $53$ |
\( T^{6} + \cdots + 590408333736048 \)
|
| $59$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots - 842000334839552 \)
|
| $67$ |
\( T^{6} + \cdots + 885207397312 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!88 \)
|
| $73$ |
\( T^{6} + \cdots - 85\!\cdots\!92 \)
|
| $79$ |
\( T^{6} + \cdots - 12\!\cdots\!68 \)
|
| $83$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 45\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots - 510868966648482 \)
|
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