Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1617,4,Mod(1,1617)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1617.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1617.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: | \(95.4060884793\) |
| 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: |
\( x^{7} - 3x^{6} - 38x^{5} + 73x^{4} + 383x^{3} - 256x^{2} - 676x - 224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{7} - 3x^{6} - 38x^{5} + 73x^{4} + 383x^{3} - 256x^{2} - 676x - 224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 36\nu^{4} - 73\nu^{3} - 311\nu^{2} + 290\nu + 240 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} - 11\nu^{5} - 108\nu^{4} + 291\nu^{3} + 1003\nu^{2} - 1414\nu - 1328 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{6} - 11\nu^{5} - 108\nu^{4} + 315\nu^{3} + 955\nu^{2} - 1870\nu - 1064 ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - 13\nu^{5} - 96\nu^{4} + 339\nu^{3} + 725\nu^{2} - 1598\nu - 676 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{2} + 21\beta _1 + 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - 6\beta_{4} - 3\beta_{3} + 33\beta_{2} + 41\beta _1 + 269 \)
|
| \(\nu^{5}\) | \(=\) |
\( 36\beta_{5} - 48\beta_{4} - 18\beta_{3} + 107\beta_{2} + 519\beta _1 + 584 \)
|
| \(\nu^{6}\) | \(=\) |
\( 36\beta_{6} + 107\beta_{5} - 287\beta_{4} - 174\beta_{3} + 1052\beta_{2} + 1479\beta _1 + 6995 \)
|
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 |
|
−5.54076 | −3.00000 | 22.7000 | 15.6321 | 16.6223 | 0 | −81.4494 | 9.00000 | −86.6138 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.02575 | −3.00000 | 8.20667 | 2.56007 | 12.0773 | 0 | −0.831993 | 9.00000 | −10.3062 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79271 | −3.00000 | −4.78619 | −15.7758 | 5.37813 | 0 | 22.9219 | 9.00000 | 28.2814 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.46203 | −3.00000 | −5.86248 | 13.7188 | 4.38608 | 0 | 20.2673 | 9.00000 | −20.0572 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.816662 | −3.00000 | −7.33306 | −2.99797 | −2.44999 | 0 | −12.5219 | 9.00000 | −2.44833 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.28176 | −3.00000 | 2.76992 | −6.75357 | −9.84527 | 0 | −17.1638 | 9.00000 | −22.1636 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.72283 | −3.00000 | 14.3051 | 13.6164 | −14.1685 | 0 | 29.7779 | 9.00000 | 64.3077 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1617.4.a.q | ✓ | 7 |
| 7.b | odd | 2 | 1 | 1617.4.a.r | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1617.4.a.q | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1617.4.a.r | yes | 7 | 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):
|
\( T_{2}^{7} + 4T_{2}^{6} - 35T_{2}^{5} - 127T_{2}^{4} + 270T_{2}^{3} + 927T_{2}^{2} + 52T_{2} - 740 \)
|
|
\( T_{5}^{7} - 20T_{5}^{6} - 264T_{5}^{5} + 6386T_{5}^{4} + 4315T_{5}^{3} - 359910T_{5}^{2} - 133484T_{5} + 2387792 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 4 T^{6} + \cdots - 740 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( T^{7} - 20 T^{6} + \cdots + 2387792 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( (T - 11)^{7} \)
|
| $13$ |
\( T^{7} + \cdots - 115171354880 \)
|
| $17$ |
\( T^{7} + \cdots - 370778627072 \)
|
| $19$ |
\( T^{7} + \cdots + 534703012144 \)
|
| $23$ |
\( T^{7} + \cdots - 76625986977792 \)
|
| $29$ |
\( T^{7} + \cdots + 47\!\cdots\!60 \)
|
| $31$ |
\( T^{7} + \cdots + 37\!\cdots\!56 \)
|
| $37$ |
\( T^{7} + \cdots - 68\!\cdots\!12 \)
|
| $41$ |
\( T^{7} + \cdots - 10\!\cdots\!08 \)
|
| $43$ |
\( T^{7} + \cdots - 23\!\cdots\!84 \)
|
| $47$ |
\( T^{7} + \cdots + 92\!\cdots\!20 \)
|
| $53$ |
\( T^{7} + \cdots - 16\!\cdots\!52 \)
|
| $59$ |
\( T^{7} + \cdots - 22\!\cdots\!44 \)
|
| $61$ |
\( T^{7} + \cdots + 39\!\cdots\!40 \)
|
| $67$ |
\( T^{7} + \cdots + 15\!\cdots\!20 \)
|
| $71$ |
\( T^{7} + \cdots - 10\!\cdots\!32 \)
|
| $73$ |
\( T^{7} + \cdots - 39\!\cdots\!60 \)
|
| $79$ |
\( T^{7} + \cdots + 13\!\cdots\!24 \)
|
| $83$ |
\( T^{7} + \cdots + 16\!\cdots\!40 \)
|
| $89$ |
\( T^{7} + \cdots + 11\!\cdots\!96 \)
|
| $97$ |
\( T^{7} + \cdots - 92\!\cdots\!68 \)
|
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