Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6012,2,Mod(1,6012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6012.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: | \(48.0060616952\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2004) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + \cdots - 575149 ) / 51607 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + \cdots - 255465 ) / 51607 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} + \cdots + 789049 ) / 51607 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} + \cdots + 1064431 ) / 51607 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} + \cdots + 875617 ) / 51607 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} + \cdots + 172141 ) / 51607 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + \cdots + 292783 ) / 51607 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{3} + 4\beta_{2} + 10\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{8} + 4\beta_{7} + \beta_{5} + 14\beta_{4} + 2\beta_{3} + 31\beta_{2} + 5\beta _1 + 79 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22\beta_{8} + 38\beta_{7} - 23\beta_{6} + 36\beta_{5} + 17\beta_{4} - 12\beta_{3} + 91\beta_{2} + 123\beta _1 + 91 \)
|
| \(\nu^{6}\) | \(=\) |
\( 115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + \cdots + 1059 \)
|
| \(\nu^{7}\) | \(=\) |
\( 431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + \cdots + 2087 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + \cdots + 15652 \)
|
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 | 0 | 0 | −4.21153 | 0 | −5.13720 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −4.19525 | 0 | 1.43344 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −3.42964 | 0 | 3.44225 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | −1.90781 | 0 | 2.81337 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | −1.52998 | 0 | −1.05249 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 0.619742 | 0 | 1.05844 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | 0.792043 | 0 | 3.80237 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0 | 0 | 1.69402 | 0 | −4.12928 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 0 | 0 | 3.16840 | 0 | −0.230890 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(167\) | \(1\) |
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 | 6012.2.a.h | 9 | |
| 3.b | odd | 2 | 1 | 2004.2.a.d | ✓ | 9 | |
| 12.b | even | 2 | 1 | 8016.2.a.bb | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2004.2.a.d | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 6012.2.a.h | 9 | 1.a | even | 1 | 1 | trivial | |
| 8016.2.a.bb | 9 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{9} + 9T_{5}^{8} + 7T_{5}^{7} - 126T_{5}^{6} - 259T_{5}^{5} + 405T_{5}^{4} + 964T_{5}^{3} - 506T_{5}^{2} - 856T_{5} + 466 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + 9 T^{8} + \cdots + 466 \)
|
| $7$ |
\( T^{9} - 2 T^{8} + \cdots - 288 \)
|
| $11$ |
\( T^{9} + 7 T^{8} + \cdots - 36 \)
|
| $13$ |
\( T^{9} - 6 T^{8} + \cdots - 5520 \)
|
| $17$ |
\( T^{9} + 7 T^{8} + \cdots + 1562 \)
|
| $19$ |
\( T^{9} - 2 T^{8} + \cdots + 64 \)
|
| $23$ |
\( T^{9} + 19 T^{8} + \cdots + 288512 \)
|
| $29$ |
\( T^{9} + 13 T^{8} + \cdots + 272720 \)
|
| $31$ |
\( T^{9} - 12 T^{8} + \cdots - 280000 \)
|
| $37$ |
\( T^{9} - 15 T^{8} + \cdots - 5712 \)
|
| $41$ |
\( T^{9} + 18 T^{8} + \cdots - 11178 \)
|
| $43$ |
\( T^{9} + 6 T^{8} + \cdots - 8589994 \)
|
| $47$ |
\( T^{9} + 25 T^{8} + \cdots - 51047340 \)
|
| $53$ |
\( T^{9} + 17 T^{8} + \cdots - 12400550 \)
|
| $59$ |
\( T^{9} + 3 T^{8} + \cdots - 881152 \)
|
| $61$ |
\( T^{9} - 14 T^{8} + \cdots - 4149788 \)
|
| $67$ |
\( T^{9} + 4 T^{8} + \cdots + 150554 \)
|
| $71$ |
\( T^{9} + 17 T^{8} + \cdots - 343184800 \)
|
| $73$ |
\( T^{9} + 20 T^{8} + \cdots + 45115056 \)
|
| $79$ |
\( T^{9} + 8 T^{8} + \cdots + 2054818 \)
|
| $83$ |
\( T^{9} - T^{8} + \cdots + 3972576 \)
|
| $89$ |
\( T^{9} + 36 T^{8} + \cdots + 93648 \)
|
| $97$ |
\( T^{9} - 31 T^{8} + \cdots - 481925792 \)
|
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