Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, 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("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 89x^{4} - 48x^{3} + 2191x^{2} + 1536x - 11379 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - 89x^{4} - 48x^{3} + 2191x^{2} + 1536x - 11379 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -19\nu^{5} + 112\nu^{4} + 904\nu^{3} - 4924\nu^{2} - 6771\nu + 35580 ) / 4818 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 41\nu^{4} - 284\nu^{3} + 2264\nu^{2} + 3603\nu - 23529 ) / 438 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 112\nu^{5} - 787\nu^{4} - 5836\nu^{3} + 34858\nu^{2} + 64764\nu - 221019 ) / 4818 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -310\nu^{5} + 1447\nu^{4} + 18046\nu^{3} - 62842\nu^{2} - 228642\nu + 431031 ) / 4818 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - 3\beta_{2} + \beta _1 + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - 34\beta_{2} + 40\beta _1 + 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} - 72\beta_{4} + 46\beta_{3} - 226\beta_{2} + 82\beta _1 + 1195 \)
|
| \(\nu^{5}\) | \(=\) |
\( 24\beta_{5} - 308\beta_{4} + 12\beta_{3} - 2426\beta_{2} + 1771\beta _1 + 2284 \)
|
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 |
|
−2.00000 | −6.65835 | 4.00000 | 5.00000 | 13.3167 | 35.3037 | −8.00000 | 17.3336 | −10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −4.06354 | 4.00000 | 5.00000 | 8.12707 | −21.1775 | −8.00000 | −10.4877 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.42105 | 4.00000 | 5.00000 | 6.84209 | −14.1242 | −8.00000 | −15.2964 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 4.91463 | 4.00000 | 5.00000 | −9.82925 | 24.2966 | −8.00000 | −2.84644 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 6.79368 | 4.00000 | 5.00000 | −13.5874 | −4.53778 | −8.00000 | 19.1541 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 8.43462 | 4.00000 | 5.00000 | −16.8692 | −31.7609 | −8.00000 | 44.1429 | −10.0000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 1210.4.a.be | ✓ | 6 |
| 11.b | odd | 2 | 1 | 1210.4.a.bh | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1210.4.a.be | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1210.4.a.bh | yes | 6 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{6} - 6T_{3}^{5} - 89T_{3}^{4} + 396T_{3}^{3} + 2575T_{3}^{2} - 5718T_{3} - 26067 \)
|
|
\( T_{7}^{6} + 12T_{7}^{5} - 1685T_{7}^{4} - 25428T_{7}^{3} + 572095T_{7}^{2} + 11107872T_{7} + 36977649 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 6 T^{5} + \cdots - 26067 \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} + 12 T^{5} + \cdots + 36977649 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 1057625712 \)
|
| $17$ |
\( T^{6} + \cdots + 5036638464 \)
|
| $19$ |
\( T^{6} + \cdots - 46360242864 \)
|
| $23$ |
\( T^{6} + \cdots - 11909286576 \)
|
| $29$ |
\( T^{6} + \cdots - 1468611837696 \)
|
| $31$ |
\( T^{6} + \cdots - 3479849006192 \)
|
| $37$ |
\( T^{6} + \cdots - 67599713102592 \)
|
| $41$ |
\( T^{6} + \cdots + 34780406452341 \)
|
| $43$ |
\( T^{6} + \cdots + 177447392859825 \)
|
| $47$ |
\( T^{6} + \cdots + 173132641512189 \)
|
| $53$ |
\( T^{6} + \cdots + 53735042884176 \)
|
| $59$ |
\( T^{6} + \cdots + 447730786263312 \)
|
| $61$ |
\( T^{6} + \cdots + 362073865275837 \)
|
| $67$ |
\( T^{6} + \cdots - 36782553732467 \)
|
| $71$ |
\( T^{6} + \cdots - 14\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots - 542110887493632 \)
|
| $79$ |
\( T^{6} + \cdots - 13\!\cdots\!52 \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!12 \)
|
| $89$ |
\( T^{6} + \cdots - 12\!\cdots\!83 \)
|
| $97$ |
\( T^{6} + \cdots - 72\!\cdots\!48 \)
|
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