Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,4,Mod(1,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, 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("644.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 644.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: | \(37.9972300437\) |
| 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: |
\( x^{10} - 2 x^{9} - 190 x^{8} + 254 x^{7} + 11484 x^{6} - 12654 x^{5} - 247874 x^{4} + 342554 x^{3} + \cdots + 2134836 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| 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
\( x^{10} - 2 x^{9} - 190 x^{8} + 254 x^{7} + 11484 x^{6} - 12654 x^{5} - 247874 x^{4} + 342554 x^{3} + \cdots + 2134836 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 90606705301 \nu^{9} + 33907497579 \nu^{8} + 17267975983567 \nu^{7} + \cdots + 12\!\cdots\!12 ) / 80697363210756 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47239702730 \nu^{9} + 17443896603 \nu^{8} + 8996765229323 \nu^{7} + \cdots + 59\!\cdots\!70 ) / 40348681605378 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 47239702730 \nu^{9} + 17443896603 \nu^{8} + 8996765229323 \nu^{7} + \cdots + 60\!\cdots\!34 ) / 40348681605378 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 48446913703 \nu^{9} - 1469562090 \nu^{8} - 9241910434102 \nu^{7} - 5723325138036 \nu^{6} + \cdots - 55\!\cdots\!00 ) / 40348681605378 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 48776070074 \nu^{9} + 32568044091 \nu^{8} + 9315941846207 \nu^{7} + \cdots + 77\!\cdots\!94 ) / 40348681605378 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 58026810283 \nu^{9} - 24513067371 \nu^{8} - 11052695886523 \nu^{7} - 2786077271283 \nu^{6} + \cdots - 77\!\cdots\!20 ) / 26899121070252 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 101257162021 \nu^{9} - 41566477491 \nu^{8} - 19311664700755 \nu^{7} - 4883509733481 \nu^{6} + \cdots - 13\!\cdots\!72 ) / 40348681605378 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 78064981527 \nu^{9} - 31080615077 \nu^{8} - 14895781411941 \nu^{7} - 3969585926389 \nu^{6} + \cdots - 10\!\cdots\!44 ) / 26899121070252 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 3\beta_{8} - \beta_{7} + 3\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 70\beta _1 + 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8 \beta_{9} - 2 \beta_{8} - 16 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} - 121 \beta_{4} + 90 \beta_{3} + \cdots + 2677 \)
|
| \(\nu^{5}\) | \(=\) |
\( 182 \beta_{9} + 320 \beta_{8} - 176 \beta_{7} + 399 \beta_{6} - 287 \beta_{5} - 277 \beta_{4} + \cdots + 5855 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1276 \beta_{9} + 337 \beta_{8} - 2742 \beta_{7} + 1166 \beta_{6} - 1467 \beta_{5} - 12634 \beta_{4} + \cdots + 227045 \)
|
| \(\nu^{7}\) | \(=\) |
\( 22969 \beta_{9} + 31321 \beta_{8} - 23994 \beta_{7} + 45512 \beta_{6} - 34602 \beta_{5} - 47239 \beta_{4} + \cdots + 858344 \)
|
| \(\nu^{8}\) | \(=\) |
\( 167380 \beta_{9} + 92155 \beta_{8} - 354291 \beta_{7} + 195980 \beta_{6} - 214065 \beta_{5} + \cdots + 21134361 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2629950 \beta_{9} + 3104296 \beta_{8} - 2998490 \beta_{7} + 4959068 \beta_{6} - 3923206 \beta_{5} + \cdots + 110531160 \)
|
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 | −9.52645 | 0 | 20.4581 | 0 | −7.00000 | 0 | 63.7533 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.71795 | 0 | 4.00427 | 0 | −7.00000 | 0 | 18.1309 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.92389 | 0 | −8.10077 | 0 | −7.00000 | 0 | −11.6031 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.746046 | 0 | −13.6244 | 0 | −7.00000 | 0 | −26.4434 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.635759 | 0 | 17.8869 | 0 | −7.00000 | 0 | −26.5958 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.295360 | 0 | −6.25374 | 0 | −7.00000 | 0 | −26.9128 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 5.04470 | 0 | 6.15911 | 0 | −7.00000 | 0 | −1.55096 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 6.24694 | 0 | −22.1406 | 0 | −7.00000 | 0 | 12.0243 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 8.54358 | 0 | 1.17568 | 0 | −7.00000 | 0 | 45.9928 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 10.0102 | 0 | 20.4354 | 0 | −7.00000 | 0 | 73.2048 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 644.4.a.d | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.4.a.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 8 T_{3}^{9} - 163 T_{3}^{8} + 1218 T_{3}^{7} + 7984 T_{3}^{6} - 50944 T_{3}^{5} + \cdots + 94816 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(644))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 8 T^{9} + \cdots + 94816 \)
|
| $5$ |
\( T^{10} + \cdots + 3313535232 \)
|
| $7$ |
\( (T + 7)^{10} \)
|
| $11$ |
\( T^{10} + \cdots - 24909452522496 \)
|
| $13$ |
\( T^{10} + \cdots + 38\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots - 74\!\cdots\!64 \)
|
| $19$ |
\( T^{10} + \cdots + 87\!\cdots\!84 \)
|
| $23$ |
\( (T - 23)^{10} \)
|
| $29$ |
\( T^{10} + \cdots - 56\!\cdots\!76 \)
|
| $31$ |
\( T^{10} + \cdots + 78\!\cdots\!36 \)
|
| $37$ |
\( T^{10} + \cdots - 12\!\cdots\!64 \)
|
| $41$ |
\( T^{10} + \cdots - 35\!\cdots\!76 \)
|
| $43$ |
\( T^{10} + \cdots - 54\!\cdots\!72 \)
|
| $47$ |
\( T^{10} + \cdots + 48\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots - 15\!\cdots\!44 \)
|
| $59$ |
\( T^{10} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 43\!\cdots\!48 \)
|
| $67$ |
\( T^{10} + \cdots - 38\!\cdots\!28 \)
|
| $71$ |
\( T^{10} + \cdots + 26\!\cdots\!64 \)
|
| $73$ |
\( T^{10} + \cdots + 26\!\cdots\!32 \)
|
| $79$ |
\( T^{10} + \cdots + 43\!\cdots\!68 \)
|
| $83$ |
\( T^{10} + \cdots + 28\!\cdots\!64 \)
|
| $89$ |
\( T^{10} + \cdots + 43\!\cdots\!44 \)
|
| $97$ |
\( T^{10} + \cdots + 19\!\cdots\!64 \)
|
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