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: | \(0\) |
| 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: |
\( x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 39753790711 \nu^{11} - 198200510956 \nu^{10} - 967064971419 \nu^{9} + 5714838266656 \nu^{8} + \cdots + 17827377683725 ) / 7247830497003 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 80534503868 \nu^{11} - 351696237482 \nu^{10} - 2108871784872 \nu^{9} + 10131881046614 \nu^{8} + \cdots + 42821455278854 ) / 7247830497003 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 82493402629 \nu^{11} + 263830494832 \nu^{10} + 2411694496521 \nu^{9} + \cdots - 37905558611308 ) / 7247830497003 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 111684601157 \nu^{11} - 439380667640 \nu^{10} - 3018114108333 \nu^{9} + \cdots + 78764454042977 ) / 7247830497003 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 126477824479 \nu^{11} - 465007218553 \nu^{10} - 3547935397485 \nu^{9} + \cdots + 52979135676436 ) / 7247830497003 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 233966606153 \nu^{11} + 906289859315 \nu^{10} + 6340426420320 \nu^{9} + \cdots - 116176930526963 ) / 7247830497003 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 264188216893 \nu^{11} + 1052591859745 \nu^{10} + 7122140689275 \nu^{9} + \cdots - 75969812453317 ) / 7247830497003 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 265780092172 \nu^{11} - 1045158248329 \nu^{10} - 7226529730506 \nu^{9} + \cdots + 135370428068599 ) / 7247830497003 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 116959146335 \nu^{11} + 438173460485 \nu^{10} + 3253152757098 \nu^{9} + \cdots - 57643004413031 ) / 2415943499001 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 376299976007 \nu^{11} - 1405118384054 \nu^{10} - 10441796596398 \nu^{9} + \cdots + 171474493496885 ) / 7247830497003 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{3} - 3\beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{11} + 17 \beta_{10} - 12 \beta_{8} + 15 \beta_{7} + 18 \beta_{6} + 19 \beta_{5} - 13 \beta_{4} + \cdots + 61 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 22 \beta_{11} - 15 \beta_{10} - 6 \beta_{9} - 18 \beta_{8} - \beta_{7} - 16 \beta_{6} + 9 \beta_{5} + \cdots + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( 55 \beta_{11} + 270 \beta_{10} - 4 \beta_{9} - 149 \beta_{8} + 211 \beta_{7} + 292 \beta_{6} + \cdots + 742 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 334 \beta_{11} - 165 \beta_{10} + 18 \beta_{9} - 258 \beta_{8} - 31 \beta_{7} - 224 \beta_{6} + \cdots + 437 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1032 \beta_{11} + 4150 \beta_{10} - 61 \beta_{9} - 1902 \beta_{8} + 2902 \beta_{7} + 4487 \beta_{6} + \cdots + 9899 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4527 \beta_{11} - 1300 \beta_{10} + 1303 \beta_{9} - 3491 \beta_{8} - 706 \beta_{7} - 2872 \beta_{6} + \cdots + 8796 \)
|
| \(\nu^{10}\) | \(=\) |
\( 17114 \beta_{11} + 62852 \beta_{10} - 375 \beta_{9} - 24898 \beta_{8} + 39584 \beta_{7} + 67054 \beta_{6} + \cdots + 138910 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 58540 \beta_{11} - 1621 \beta_{10} + 28961 \beta_{9} - 46940 \beta_{8} - 13406 \beta_{7} + \cdots + 158591 \)
|
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 | −3.72799 | 0 | −2.68331 | 1.00000 | 0 | −3.72799 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −2.78478 | 0 | 2.18432 | 1.00000 | 0 | −2.78478 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −2.38357 | 0 | 4.24377 | 1.00000 | 0 | −2.38357 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | −1.09224 | 0 | −1.25253 | 1.00000 | 0 | −1.09224 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | −0.302744 | 0 | −1.13869 | 1.00000 | 0 | −0.302744 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 0.521020 | 0 | −3.80450 | 1.00000 | 0 | 0.521020 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0 | 1.00000 | 1.20508 | 0 | 4.22185 | 1.00000 | 0 | 1.20508 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0 | 1.00000 | 1.33540 | 0 | 2.26685 | 1.00000 | 0 | 1.33540 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 0 | 1.00000 | 2.27289 | 0 | 3.85060 | 1.00000 | 0 | 2.27289 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 0 | 1.00000 | 2.74383 | 0 | 0.881706 | 1.00000 | 0 | 2.74383 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 0 | 1.00000 | 3.31930 | 0 | −3.41918 | 1.00000 | 0 | 3.31930 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.00000 | 0 | 1.00000 | 3.89381 | 0 | 0.649105 | 1.00000 | 0 | 3.89381 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.p | yes | 12 |
| 3.b | odd | 2 | 1 | 8046.2.a.i | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8046.2.a.i | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 8046.2.a.p | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):
|
\( T_{5}^{12} - 5 T_{5}^{11} - 23 T_{5}^{10} + 142 T_{5}^{9} + 104 T_{5}^{8} - 1302 T_{5}^{7} + 607 T_{5}^{6} + \cdots - 553 \)
|
|
\( T_{11}^{12} - 6 T_{11}^{11} - 49 T_{11}^{10} + 326 T_{11}^{9} + 679 T_{11}^{8} - 6139 T_{11}^{7} + \cdots + 11955 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 5 T^{11} + \cdots - 553 \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots - 9733 \)
|
| $11$ |
\( T^{12} - 6 T^{11} + \cdots + 11955 \)
|
| $13$ |
\( T^{12} - 3 T^{11} + \cdots - 52011 \)
|
| $17$ |
\( T^{12} - 6 T^{11} + \cdots - 2500715 \)
|
| $19$ |
\( T^{12} - 8 T^{11} + \cdots - 25933 \)
|
| $23$ |
\( T^{12} - 11 T^{11} + \cdots + 198773 \)
|
| $29$ |
\( T^{12} - 29 T^{11} + \cdots + 2060575 \)
|
| $31$ |
\( T^{12} - 2 T^{11} + \cdots - 9 \)
|
| $37$ |
\( T^{12} + \cdots - 8082289269 \)
|
| $41$ |
\( T^{12} - 22 T^{11} + \cdots + 790584 \)
|
| $43$ |
\( T^{12} + \cdots - 128007485 \)
|
| $47$ |
\( T^{12} + \cdots + 196700481 \)
|
| $53$ |
\( T^{12} - 12 T^{11} + \cdots + 16414009 \)
|
| $59$ |
\( T^{12} - 34 T^{11} + \cdots + 61641315 \)
|
| $61$ |
\( T^{12} + 4 T^{11} + \cdots + 59168457 \)
|
| $67$ |
\( T^{12} + \cdots - 347750655 \)
|
| $71$ |
\( T^{12} - 21 T^{11} + \cdots - 21896973 \)
|
| $73$ |
\( T^{12} + \cdots + 148306725 \)
|
| $79$ |
\( T^{12} + \cdots - 1339385557 \)
|
| $83$ |
\( T^{12} - 10 T^{11} + \cdots + 1274877 \)
|
| $89$ |
\( T^{12} + \cdots + 204074257 \)
|
| $97$ |
\( T^{12} + \cdots + 297139217 \)
|
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