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: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{6} - 13\nu^{5} - 180\nu^{4} + 2006\nu^{3} - 8748\nu^{2} - 59944\nu + 152160 ) / 12672 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 42 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{6} + 13\nu^{5} + 180\nu^{4} + 2218\nu^{3} + 8748\nu^{2} - 248408\nu - 177504 ) / 12672 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47\nu^{6} + 31\nu^{5} - 6340\nu^{4} - 2834\nu^{3} + 207908\nu^{2} + 110776\nu - 1043232 ) / 12672 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -51\nu^{6} + 221\nu^{5} + 7284\nu^{4} - 25654\nu^{3} - 248340\nu^{2} + 453032\nu + 1126176 ) / 12672 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 3\beta_{2} + 73\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} - 6\beta_{4} + 94\beta_{3} + 45\beta_{2} - 12\beta _1 + 3057 \)
|
| \(\nu^{5}\) | \(=\) |
\( 45\beta_{6} + 54\beta_{5} + 348\beta_{4} - 60\beta_{3} + 267\beta_{2} + 6004\beta _1 - 405 \)
|
| \(\nu^{6}\) | \(=\) |
\( 375\beta_{6} + 234\beta_{5} - 858\beta_{4} + 8296\beta_{3} + 6075\beta_{2} - 3534\beta _1 + 249405 \)
|
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 | −9.65209 | 0 | −3.08797 | 0 | 7.00000 | 0 | 66.1628 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.79047 | 0 | 16.8493 | 0 | 7.00000 | 0 | 6.52956 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.69688 | 0 | −14.1250 | 0 | 7.00000 | 0 | −19.7268 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.517294 | 0 | −7.39668 | 0 | 7.00000 | 0 | −26.7324 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.19046 | 0 | 8.53193 | 0 | 7.00000 | 0 | −16.8209 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.14310 | 0 | 18.9749 | 0 | 7.00000 | 0 | 39.3101 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 9.28858 | 0 | −8.74645 | 0 | 7.00000 | 0 | 59.2778 | 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.k | ✓ | 7 |
| 4.b | odd | 2 | 1 | 1232.4.a.bc | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.k | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1232.4.a.bc | 7 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 144T_{3}^{5} + 354T_{3}^{4} + 5172T_{3}^{3} - 6504T_{3}^{2} - 34432T_{3} + 18816 \)
|
|
\( T_{5}^{7} - 11T_{5}^{6} - 468T_{5}^{5} + 2820T_{5}^{4} + 72712T_{5}^{3} - 41856T_{5}^{2} - 3193632T_{5} - 7697280 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 18816 \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots - 7697280 \)
|
| $7$ |
\( (T - 7)^{7} \)
|
| $11$ |
\( (T - 11)^{7} \)
|
| $13$ |
\( T^{7} + \cdots + 175897990400 \)
|
| $17$ |
\( T^{7} + \cdots + 1907883940864 \)
|
| $19$ |
\( T^{7} + \cdots + 4643111424 \)
|
| $23$ |
\( T^{7} + \cdots - 604187873280 \)
|
| $29$ |
\( T^{7} + \cdots + 28\!\cdots\!12 \)
|
| $31$ |
\( T^{7} + \cdots + 328802478283264 \)
|
| $37$ |
\( T^{7} + \cdots + 97\!\cdots\!60 \)
|
| $41$ |
\( T^{7} + \cdots - 71\!\cdots\!00 \)
|
| $43$ |
\( T^{7} + \cdots - 12\!\cdots\!92 \)
|
| $47$ |
\( T^{7} + \cdots - 55\!\cdots\!16 \)
|
| $53$ |
\( T^{7} + \cdots + 46\!\cdots\!48 \)
|
| $59$ |
\( T^{7} + \cdots - 10\!\cdots\!36 \)
|
| $61$ |
\( T^{7} + \cdots - 50\!\cdots\!80 \)
|
| $67$ |
\( T^{7} + \cdots + 14\!\cdots\!16 \)
|
| $71$ |
\( T^{7} + \cdots + 12\!\cdots\!12 \)
|
| $73$ |
\( T^{7} + \cdots + 73\!\cdots\!28 \)
|
| $79$ |
\( T^{7} + \cdots + 46\!\cdots\!20 \)
|
| $83$ |
\( T^{7} + \cdots - 54\!\cdots\!72 \)
|
| $89$ |
\( T^{7} + \cdots + 27\!\cdots\!60 \)
|
| $97$ |
\( T^{7} + \cdots - 27\!\cdots\!16 \)
|
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