Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [616,4,Mod(1,616)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(616, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("616.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 616 = 2^{3} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 616.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: | \(36.3451765635\) |
| 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: |
\( x^{6} - x^{5} - 94x^{4} + 161x^{3} + 533x^{2} - 384x - 468 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{6} - x^{5} - 94x^{4} + 161x^{3} + 533x^{2} - 384x - 468 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -13\nu^{5} + 558\nu^{4} + 1098\nu^{3} - 51127\nu^{2} + 40754\nu + 149353 ) / 4879 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 81\nu^{5} - 99\nu^{4} - 7592\nu^{3} + 14186\nu^{2} + 37310\nu - 14458 ) / 4879 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -895\nu^{5} + 1636\nu^{4} + 81598\nu^{3} - 221318\nu^{2} - 182819\nu + 713910 ) / 43911 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 409\nu^{5} - 1042\nu^{4} - 36046\nu^{3} + 121565\nu^{2} - 26404\nu - 158778 ) / 14637 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 232\nu^{5} - 361\nu^{4} - 21418\nu^{3} + 49856\nu^{2} + 99179\nu - 120576 ) / 6273 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} - \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 4\beta_{4} - 11\beta_{3} - 9\beta_{2} + 128 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 77\beta_{5} + 29\beta_{4} + 83\beta_{3} - 117\beta_{2} + 9\beta _1 - 177 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 21\beta_{5} - 354\beta_{4} - 1059\beta_{3} - 745\beta_{2} + 38\beta _1 + 10410 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6607\beta_{5} + 2986\beta_{4} + 7951\beta_{3} - 9599\beta_{2} + 890\beta _1 - 25570 ) / 4 \)
|
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 |
|
0 | −7.54674 | 0 | 5.43104 | 0 | 7.00000 | 0 | 29.9532 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.39012 | 0 | −17.9615 | 0 | 7.00000 | 0 | 13.8336 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.761718 | 0 | 15.6409 | 0 | 7.00000 | 0 | −26.4198 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.18857 | 0 | 2.34292 | 0 | 7.00000 | 0 | −22.2102 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.18220 | 0 | −10.8550 | 0 | 7.00000 | 0 | −9.50917 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.32780 | 0 | −8.59838 | 0 | 7.00000 | 0 | 42.3523 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 616.4.a.h | ✓ | 6 |
| 4.b | odd | 2 | 1 | 1232.4.a.ba | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1232.4.a.ba | 6 | 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\):
|
\( T_{3}^{6} - 95T_{3}^{4} + 10T_{3}^{3} + 2000T_{3}^{2} - 2200T_{3} - 2800 \)
|
|
\( T_{5}^{6} + 14T_{5}^{5} - 299T_{5}^{4} - 3864T_{5}^{3} + 12768T_{5}^{2} + 137056T_{5} - 333648 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 95 T^{4} + \cdots - 2800 \)
|
| $5$ |
\( T^{6} + 14 T^{5} + \cdots - 333648 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} + 70 T^{5} + \cdots + 187342848 \)
|
| $17$ |
\( T^{6} + 102 T^{5} + \cdots + 131616128 \)
|
| $19$ |
\( T^{6} + \cdots + 271652713984 \)
|
| $23$ |
\( T^{6} + \cdots - 1351057464576 \)
|
| $29$ |
\( T^{6} + \cdots - 25321485504 \)
|
| $31$ |
\( T^{6} + \cdots + 91303824924816 \)
|
| $37$ |
\( T^{6} + \cdots - 21235930324496 \)
|
| $41$ |
\( T^{6} + \cdots - 81213123421568 \)
|
| $43$ |
\( T^{6} + \cdots - 1651816972288 \)
|
| $47$ |
\( T^{6} + \cdots - 7070543379456 \)
|
| $53$ |
\( T^{6} + \cdots - 10489649778496 \)
|
| $59$ |
\( T^{6} + \cdots + 493861378249872 \)
|
| $61$ |
\( T^{6} + \cdots + 633825171314432 \)
|
| $67$ |
\( T^{6} + \cdots - 14\!\cdots\!88 \)
|
| $71$ |
\( T^{6} + \cdots + 83591455802624 \)
|
| $73$ |
\( T^{6} + \cdots - 61\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots + 89\!\cdots\!88 \)
|
| $83$ |
\( T^{6} + \cdots + 40\!\cdots\!36 \)
|
| $89$ |
\( T^{6} + \cdots + 23\!\cdots\!32 \)
|
| $97$ |
\( T^{6} + \cdots - 41\!\cdots\!96 \)
|
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