Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,10,Mod(1,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(162.236288392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3 x^{9} - 3667 x^{8} + 7323 x^{7} + 4338847 x^{6} - 6510663 x^{5} - 1903644413 x^{4} + \cdots - 2925217654100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{12}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3 x^{9} - 3667 x^{8} + 7323 x^{7} + 4338847 x^{6} - 6510663 x^{5} - 1903644413 x^{4} + \cdots - 2925217654100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 734 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu^{2} - 1300\nu + 2221 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1102135 \nu^{9} + 1584867766 \nu^{8} - 34410507895 \nu^{7} - 5319688467150 \nu^{6} + \cdots + 13\!\cdots\!44 ) / 183178893482496 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7427897 \nu^{9} - 67925338 \nu^{8} + 22243759721 \nu^{7} + 373369732658 \nu^{6} + \cdots - 34\!\cdots\!12 ) / 183178893482496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 18172435 \nu^{9} + 240505438 \nu^{8} - 63081423971 \nu^{7} - 897412069094 \nu^{6} + \cdots + 29\!\cdots\!24 ) / 183178893482496 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19198605 \nu^{9} + 1258794766 \nu^{8} + 54472412109 \nu^{7} - 3977836407750 \nu^{6} + \cdots + 10\!\cdots\!16 ) / 183178893482496 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7550281 \nu^{9} - 39296650 \nu^{8} - 25372278649 \nu^{7} + 89512194086 \nu^{6} + \cdots + 68\!\cdots\!24 ) / 22897361685312 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 125147307 \nu^{9} - 2023391842 \nu^{8} - 415941351051 \nu^{7} + 6156770978442 \nu^{6} + \cdots - 70\!\cdots\!80 ) / 366357786964992 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 734 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 1304\beta _1 + 715 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - 2\beta_{8} + \beta_{7} - 2\beta_{5} + 2\beta_{3} + 1837\beta_{2} + 4006\beta _1 + 957060 \)
|
| \(\nu^{5}\) | \(=\) |
\( 20 \beta_{9} - 4 \beta_{8} + 22 \beta_{7} - 45 \beta_{6} - 31 \beta_{5} + 2299 \beta_{3} + \cdots + 2905976 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5934 \beta_{9} - 5382 \beta_{8} + 3615 \beta_{7} - 729 \beta_{6} - 4941 \beta_{5} - 252 \beta_{4} + \cdots + 1471481855 \)
|
| \(\nu^{7}\) | \(=\) |
\( 58500 \beta_{9} - 6900 \beta_{8} + 76554 \beta_{7} - 140310 \beta_{6} - 105630 \beta_{5} + \cdots + 7304887899 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13199502 \beta_{9} - 11296590 \beta_{8} + 8979135 \beta_{7} - 2503980 \beta_{6} - 10132674 \beta_{5} + \cdots + 2426673958588 \)
|
| \(\nu^{9}\) | \(=\) |
\( 140684604 \beta_{9} - 14781228 \beta_{8} + 193590282 \beta_{7} - 324333711 \beta_{6} + \cdots + 16513930227194 \)
|
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 |
|
−42.2473 | 0 | 1272.83 | 625.000 | 0 | −2401.00 | −32143.0 | 0 | −26404.5 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −33.0884 | 0 | 582.845 | 625.000 | 0 | −2401.00 | −2344.15 | 0 | −20680.3 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −21.2570 | 0 | −60.1407 | 625.000 | 0 | −2401.00 | 12162.0 | 0 | −13285.6 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −12.1091 | 0 | −365.369 | 625.000 | 0 | −2401.00 | 10624.2 | 0 | −7568.19 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −5.43866 | 0 | −482.421 | 625.000 | 0 | −2401.00 | 5408.32 | 0 | −3399.16 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.95949 | 0 | −503.241 | 625.000 | 0 | −2401.00 | −3004.60 | 0 | 1849.68 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 20.3422 | 0 | −98.1948 | 625.000 | 0 | −2401.00 | −12412.7 | 0 | 12713.9 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 21.3576 | 0 | −55.8526 | 625.000 | 0 | −2401.00 | −12128.0 | 0 | 13348.5 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 34.0061 | 0 | 644.414 | 625.000 | 0 | −2401.00 | 4502.90 | 0 | 21253.8 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 42.4751 | 0 | 1292.13 | 625.000 | 0 | −2401.00 | 33136.1 | 0 | 26546.9 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 315.10.a.q | yes | 10 |
| 3.b | odd | 2 | 1 | 315.10.a.p | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.10.a.p | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 315.10.a.q | yes | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 7 T_{2}^{9} - 3649 T_{2}^{8} + 22001 T_{2}^{7} + 4287390 T_{2}^{6} - 19470724 T_{2}^{5} + \cdots - 3634454155264 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 3634454155264 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( (T - 625)^{10} \)
|
| $7$ |
\( (T + 2401)^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 79\!\cdots\!00 \)
|
| $13$ |
\( T^{10} + \cdots - 29\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots - 17\!\cdots\!72 \)
|
| $19$ |
\( T^{10} + \cdots + 24\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots - 37\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots - 20\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots - 26\!\cdots\!12 \)
|
| $37$ |
\( T^{10} + \cdots + 14\!\cdots\!96 \)
|
| $41$ |
\( T^{10} + \cdots - 21\!\cdots\!72 \)
|
| $43$ |
\( T^{10} + \cdots - 13\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots - 15\!\cdots\!56 \)
|
| $53$ |
\( T^{10} + \cdots + 30\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots - 77\!\cdots\!56 \)
|
| $61$ |
\( T^{10} + \cdots + 29\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots + 17\!\cdots\!28 \)
|
| $71$ |
\( T^{10} + \cdots + 20\!\cdots\!48 \)
|
| $73$ |
\( T^{10} + \cdots - 44\!\cdots\!16 \)
|
| $79$ |
\( T^{10} + \cdots + 24\!\cdots\!28 \)
|
| $83$ |
\( T^{10} + \cdots + 43\!\cdots\!00 \)
|
| $89$ |
\( T^{10} + \cdots - 35\!\cdots\!48 \)
|
| $97$ |
\( T^{10} + \cdots - 87\!\cdots\!52 \)
|
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