Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5994,2,Mod(1,5994)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5994, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5994.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5994 = 2 \cdot 3^{4} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5994.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: | \(47.8623309716\) |
| 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: |
\( x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 666) |
| 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
\( x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 12122 \nu^{9} + 3878 \nu^{8} - 319687 \nu^{7} - 111802 \nu^{6} + 2840680 \nu^{5} + 934458 \nu^{4} + \cdots - 997761 ) / 272073 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 26903 \nu^{9} - 4417 \nu^{8} + 686777 \nu^{7} + 167612 \nu^{6} - 5914326 \nu^{5} - 1567558 \nu^{4} + \cdots + 163557 ) / 272073 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 41434 \nu^{9} - 836 \nu^{8} + 1069090 \nu^{7} + 114325 \nu^{6} - 9388827 \nu^{5} + \cdots + 2748360 ) / 272073 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 47656 \nu^{9} + 1827 \nu^{8} + 1222032 \nu^{7} + 118233 \nu^{6} - 10625383 \nu^{5} + \cdots + 3520176 ) / 272073 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 60657 \nu^{9} - 5654 \nu^{8} + 1574170 \nu^{7} + 307810 \nu^{6} - 13868663 \nu^{5} + \cdots + 3609258 ) / 272073 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 21496 \nu^{9} - 2373 \nu^{8} + 554947 \nu^{7} + 118506 \nu^{6} - 4849246 \nu^{5} - 1242004 \nu^{4} + \cdots + 672259 ) / 90691 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 113902 \nu^{9} - 12498 \nu^{8} + 2951775 \nu^{7} + 629301 \nu^{6} - 25978321 \nu^{5} + \cdots + 3991668 ) / 272073 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 53070 \nu^{9} - 3960 \nu^{8} + 1379192 \nu^{7} + 236054 \nu^{6} - 12163529 \nu^{5} + \cdots + 2694093 ) / 90691 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 163381 \nu^{9} + 810 \nu^{8} - 4214799 \nu^{7} - 530595 \nu^{6} + 36878434 \nu^{5} + 6849019 \nu^{4} + \cdots - 7890852 ) / 272073 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta _1 - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 3\beta _1 + 17 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{9} + 2\beta_{8} + 7\beta_{7} - \beta_{6} - 19\beta_{5} + 11\beta_{4} + \beta_{3} - 4\beta_{2} - 11\beta _1 - 6 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{9} + 14\beta_{8} - \beta_{7} + 4\beta_{6} - 12\beta_{5} - 11\beta_{3} + 8\beta_{2} + 35\beta _1 + 137 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 42 \beta_{9} + 33 \beta_{8} + 50 \beta_{7} + 34 \beta_{6} - 190 \beta_{5} + 116 \beta_{4} + 6 \beta_{3} + \cdots - 35 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 93 \beta_{9} + 158 \beta_{8} + 66 \beta_{6} - 152 \beta_{5} + 67 \beta_{4} - 125 \beta_{3} + 53 \beta_{2} + \cdots + 1237 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 480 \beta_{9} + 439 \beta_{8} + 386 \beta_{7} + 568 \beta_{6} - 1961 \beta_{5} + 1195 \beta_{4} + \cdots - 168 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1092 \beta_{9} + 1711 \beta_{8} + 121 \beta_{7} + 812 \beta_{6} - 1935 \beta_{5} + 1146 \beta_{4} + \cdots + 11752 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 5214 \beta_{9} + 5400 \beta_{8} + 3229 \beta_{7} + 7028 \beta_{6} - 20582 \beta_{5} + 12289 \beta_{4} + \cdots - 163 ) / 3 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.69049 | 0 | 0.539128 | −1.00000 | 0 | 3.69049 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −3.49123 | 0 | 2.86107 | −1.00000 | 0 | 3.49123 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −3.28277 | 0 | −3.13291 | −1.00000 | 0 | 3.28277 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −1.43355 | 0 | 0.697985 | −1.00000 | 0 | 1.43355 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | −0.0738674 | 0 | 4.67803 | −1.00000 | 0 | 0.0738674 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0 | 1.00000 | 0.363318 | 0 | −5.03822 | −1.00000 | 0 | −0.363318 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0 | 1.00000 | 0.949474 | 0 | −4.04317 | −1.00000 | 0 | −0.949474 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0 | 1.00000 | 2.84378 | 0 | −0.483762 | −1.00000 | 0 | −2.84378 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 0 | 1.00000 | 3.33861 | 0 | 0.195962 | −1.00000 | 0 | −3.33861 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 0 | 1.00000 | 3.47672 | 0 | 4.72589 | −1.00000 | 0 | −3.47672 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 5994.2.a.ba | 10 | |
| 3.b | odd | 2 | 1 | 5994.2.a.bb | 10 | ||
| 9.c | even | 3 | 2 | 1998.2.e.e | 20 | ||
| 9.d | odd | 6 | 2 | 666.2.e.e | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 666.2.e.e | ✓ | 20 | 9.d | odd | 6 | 2 | |
| 1998.2.e.e | 20 | 9.c | even | 3 | 2 | ||
| 5994.2.a.ba | 10 | 1.a | even | 1 | 1 | trivial | |
| 5994.2.a.bb | 10 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5994))\):
|
\( T_{5}^{10} + T_{5}^{9} - 35 T_{5}^{8} - 26 T_{5}^{7} + 424 T_{5}^{6} + 196 T_{5}^{5} - 1958 T_{5}^{4} + \cdots - 51 \)
|
|
\( T_{11}^{10} + 3 T_{11}^{9} - 67 T_{11}^{8} - 204 T_{11}^{7} + 1466 T_{11}^{6} + 4343 T_{11}^{5} + \cdots - 5039 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + T^{9} + \cdots - 51 \)
|
| $7$ |
\( T^{10} - T^{9} + \cdots + 144 \)
|
| $11$ |
\( T^{10} + 3 T^{9} + \cdots - 5039 \)
|
| $13$ |
\( T^{10} - 12 T^{9} + \cdots + 169 \)
|
| $17$ |
\( T^{10} + 12 T^{9} + \cdots + 769872 \)
|
| $19$ |
\( T^{10} - 24 T^{9} + \cdots + 7248 \)
|
| $23$ |
\( T^{10} + 3 T^{9} + \cdots - 40937 \)
|
| $29$ |
\( T^{10} + 4 T^{9} + \cdots - 254741 \)
|
| $31$ |
\( T^{10} - 11 T^{9} + \cdots + 2288361 \)
|
| $37$ |
\( (T + 1)^{10} \)
|
| $41$ |
\( T^{10} - 3 T^{9} + \cdots - 107747 \)
|
| $43$ |
\( T^{10} - 6 T^{9} + \cdots + 340464 \)
|
| $47$ |
\( T^{10} + 8 T^{9} + \cdots - 10100112 \)
|
| $53$ |
\( T^{10} - 10 T^{9} + \cdots + 1011376 \)
|
| $59$ |
\( T^{10} + 10 T^{9} + \cdots + 14068272 \)
|
| $61$ |
\( T^{10} - 2 T^{9} + \cdots - 1169519 \)
|
| $67$ |
\( T^{10} - 7 T^{9} + \cdots + 1275319 \)
|
| $71$ |
\( T^{10} + 15 T^{9} + \cdots + 11950032 \)
|
| $73$ |
\( T^{10} - 3 T^{9} + \cdots - 15936 \)
|
| $79$ |
\( T^{10} + \cdots - 118547631 \)
|
| $83$ |
\( T^{10} - 23 T^{9} + \cdots - 69498213 \)
|
| $89$ |
\( T^{10} + 21 T^{9} + \cdots - 3312 \)
|
| $97$ |
\( T^{10} + \cdots + 18102071296 \)
|
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