Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8046,2,Mod(1,8046)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8046, 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("8046.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8046 = 2 \cdot 3^{3} \cdot 149 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8046.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: | \(64.2476334663\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24748733 \nu^{11} + 373555153 \nu^{10} - 2862765831 \nu^{9} - 8579109025 \nu^{8} + \cdots - 135751602546 ) / 24466366011 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 142183724 \nu^{11} + 838314338 \nu^{10} + 2550850410 \nu^{9} - 22317146501 \nu^{8} + \cdots - 100388062134 ) / 24466366011 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 743676686 \nu^{11} + 3071881331 \nu^{10} + 19932549972 \nu^{9} - 83789564729 \nu^{8} + \cdots + 65302848768 ) / 24466366011 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 260555422 \nu^{11} - 1008357826 \nu^{10} - 7145245871 \nu^{9} + 27363214940 \nu^{8} + \cdots + 915421136 ) / 8155455337 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 791457988 \nu^{11} - 2873219941 \nu^{10} - 22855402146 \nu^{9} + 79587640096 \nu^{8} + \cdots - 29055232476 ) / 24466366011 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 905897063 \nu^{11} - 3376570442 \nu^{10} - 25430053989 \nu^{9} + 92407418951 \nu^{8} + \cdots - 7334114235 ) / 24466366011 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 303077175 \nu^{11} + 1063333243 \nu^{10} + 8694135210 \nu^{9} - 28833755646 \nu^{8} + \cdots + 44724727668 ) / 8155455337 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 329344029 \nu^{11} + 1140734420 \nu^{10} + 9678326681 \nu^{9} - 31578991963 \nu^{8} + \cdots + 17837198367 ) / 8155455337 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 353735657 \nu^{11} + 1292153510 \nu^{10} + 10017640191 \nu^{9} - 35438802905 \nu^{8} + \cdots - 27070193104 ) / 8155455337 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 701406468 \nu^{11} - 2905422747 \nu^{10} - 18475689520 \nu^{9} + 78645513257 \nu^{8} + \cdots - 1717939244 ) / 8155455337 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + 2\beta_{10} + \beta_{9} + 3\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{3} + 10\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 3 \beta_{11} + 16 \beta_{10} + 3 \beta_{9} - 14 \beta_{8} + \beta_{7} + 20 \beta_{6} - 6 \beta_{5} + \cdots + 68 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 19 \beta_{11} + 43 \beta_{10} + 23 \beta_{9} - 3 \beta_{8} + 50 \beta_{7} + 35 \beta_{6} + 45 \beta_{5} + \cdots + 31 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 62 \beta_{11} + 258 \beta_{10} + 85 \beta_{9} - 184 \beta_{8} + 55 \beta_{7} + 335 \beta_{6} + \cdots + 846 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 313 \beta_{11} + 786 \beta_{10} + 465 \beta_{9} - 93 \beta_{8} + 795 \beta_{7} + 780 \beta_{6} + \cdots + 576 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1051 \beta_{11} + 4141 \beta_{10} + 1808 \beta_{9} - 2429 \beta_{8} + 1441 \beta_{7} + 5432 \beta_{6} + \cdots + 11067 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4900 \beta_{11} + 13689 \beta_{10} + 8697 \beta_{9} - 2068 \beta_{8} + 12789 \beta_{7} + 14973 \beta_{6} + \cdots + 11256 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 16949 \beta_{11} + 66601 \beta_{10} + 34473 \beta_{9} - 32858 \beta_{8} + 30539 \beta_{7} + \cdots + 151126 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 75864 \beta_{11} + 233383 \beta_{10} + 155287 \beta_{9} - 40971 \beta_{8} + 207800 \beta_{7} + \cdots + 215351 \)
|
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 | −4.06732 | 0 | 1.57160 | 1.00000 | 0 | −4.06732 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −3.27568 | 0 | 4.39969 | 1.00000 | 0 | −3.27568 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −2.95427 | 0 | 0.794893 | 1.00000 | 0 | −2.95427 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | −2.60206 | 0 | −4.35416 | 1.00000 | 0 | −2.60206 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | −0.761360 | 0 | 1.23946 | 1.00000 | 0 | −0.761360 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 0.00512154 | 0 | −2.98814 | 1.00000 | 0 | 0.00512154 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0 | 1.00000 | 0.539392 | 0 | 0.739124 | 1.00000 | 0 | 0.539392 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0 | 1.00000 | 0.815716 | 0 | −4.13057 | 1.00000 | 0 | 0.815716 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 0 | 1.00000 | 1.11280 | 0 | −0.851837 | 1.00000 | 0 | 1.11280 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 0 | 1.00000 | 1.28984 | 0 | 3.20787 | 1.00000 | 0 | 1.28984 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 0 | 1.00000 | 3.43402 | 0 | −1.90397 | 1.00000 | 0 | 3.43402 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.00000 | 0 | 1.00000 | 3.46379 | 0 | −3.72396 | 1.00000 | 0 | 3.46379 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(149\) | \(-1\) |
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 | 8046.2.a.n | yes | 12 |
| 3.b | odd | 2 | 1 | 8046.2.a.k | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8046.2.a.k | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 8046.2.a.n | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):
|
\( T_{5}^{12} + 3 T_{5}^{11} - 31 T_{5}^{10} - 82 T_{5}^{9} + 334 T_{5}^{8} + 684 T_{5}^{7} - 1561 T_{5}^{6} + \cdots - 3 \)
|
|
\( T_{11}^{12} + 14 T_{11}^{11} + 11 T_{11}^{10} - 638 T_{11}^{9} - 2421 T_{11}^{8} + 7069 T_{11}^{7} + \cdots + 142707 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + 3 T^{11} + \cdots - 3 \)
|
| $7$ |
\( T^{12} + 6 T^{11} + \cdots + 5243 \)
|
| $11$ |
\( T^{12} + 14 T^{11} + \cdots + 142707 \)
|
| $13$ |
\( T^{12} + 3 T^{11} + \cdots - 3213 \)
|
| $17$ |
\( T^{12} + 8 T^{11} + \cdots - 2605641 \)
|
| $19$ |
\( T^{12} + 4 T^{11} + \cdots + 25091771 \)
|
| $23$ |
\( T^{12} + 13 T^{11} + \cdots - 27922527 \)
|
| $29$ |
\( T^{12} + 23 T^{11} + \cdots + 14936033 \)
|
| $31$ |
\( T^{12} + \cdots - 875298829 \)
|
| $37$ |
\( T^{12} + 19 T^{11} + \cdots - 67701 \)
|
| $41$ |
\( T^{12} + \cdots - 157154904 \)
|
| $43$ |
\( T^{12} + \cdots - 13302491399 \)
|
| $47$ |
\( T^{12} + \cdots + 695675223 \)
|
| $53$ |
\( T^{12} + \cdots - 930648429 \)
|
| $59$ |
\( T^{12} + \cdots - 124731861 \)
|
| $61$ |
\( T^{12} + \cdots + 571805313 \)
|
| $67$ |
\( T^{12} + \cdots - 116275971 \)
|
| $71$ |
\( T^{12} + 15 T^{11} + \cdots - 21949329 \)
|
| $73$ |
\( T^{12} + \cdots + 2330721897 \)
|
| $79$ |
\( T^{12} + \cdots - 2551227479 \)
|
| $83$ |
\( T^{12} + \cdots + 1485381849 \)
|
| $89$ |
\( T^{12} + \cdots + 342597297 \)
|
| $97$ |
\( T^{12} + \cdots + 1044093953679 \)
|
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