Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8004,2,Mod(1,8004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8004.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8004.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: | \(63.9122617778\) |
| 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: |
\( x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 8071 \nu^{11} + 19357 \nu^{10} + 231201 \nu^{9} - 479516 \nu^{8} - 2165385 \nu^{7} + \cdots + 770552 ) / 1181592 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 58412 \nu^{11} + 150230 \nu^{10} + 1862577 \nu^{9} - 3812431 \nu^{8} - 21650127 \nu^{7} + \cdots + 11033668 ) / 4135572 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 237136 \nu^{11} + 958345 \nu^{10} + 6749571 \nu^{9} - 26540003 \nu^{8} - 67159668 \nu^{7} + \cdots - 69925000 ) / 16542288 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 643238 \nu^{11} - 1717169 \nu^{10} - 20607159 \nu^{9} + 44452597 \nu^{8} + 241331934 \nu^{7} + \cdots - 32135128 ) / 16542288 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 227217 \nu^{11} - 656309 \nu^{10} - 7166323 \nu^{9} + 17182508 \nu^{8} + 82610957 \nu^{7} + \cdots - 13046896 ) / 5514096 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 756551 \nu^{11} - 1925114 \nu^{10} - 24168090 \nu^{9} + 49854049 \nu^{8} + 280180599 \nu^{7} + \cdots - 37571512 ) / 16542288 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 324494 \nu^{11} - 886849 \nu^{10} - 10152007 \nu^{9} + 23105733 \nu^{8} + 114673742 \nu^{7} + \cdots + 21253000 ) / 5514096 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1038679 \nu^{11} - 2548933 \nu^{10} - 33488751 \nu^{9} + 66191846 \nu^{8} + 390702759 \nu^{7} + \cdots - 33296528 ) / 16542288 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 545408 \nu^{11} + 1592693 \nu^{10} + 17173041 \nu^{9} - 42407701 \nu^{8} - 196470678 \nu^{7} + \cdots - 35157824 ) / 8271144 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1186027 \nu^{11} - 3327328 \nu^{10} - 37462464 \nu^{9} + 87445547 \nu^{8} + 429697791 \nu^{7} + \cdots + 36924904 ) / 16542288 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} - \beta_{8} + \beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} + 10\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 16 \beta_{11} + \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 4 \beta_{6} + 32 \beta_{5} - 16 \beta_{4} + \cdots + 68 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 23 \beta_{11} + 22 \beta_{10} + 20 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} + 32 \beta_{5} + \cdots + 88 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 264 \beta_{11} + \beta_{10} + 32 \beta_{9} - 133 \beta_{8} + 131 \beta_{7} - 97 \beta_{6} + \cdots + 902 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 502 \beta_{11} + 362 \beta_{10} + 340 \beta_{9} - 58 \beta_{8} + 120 \beta_{7} - 264 \beta_{6} + \cdots + 1682 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4357 \beta_{11} + 94 \beta_{10} + 724 \beta_{9} - 1744 \beta_{8} + 1742 \beta_{7} - 1889 \beta_{6} + \cdots + 13128 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 10242 \beta_{11} + 5512 \beta_{10} + 5601 \beta_{9} - 1365 \beta_{8} + 2655 \beta_{7} - 5768 \beta_{6} + \cdots + 31070 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 72074 \beta_{11} + 3343 \beta_{10} + 14471 \beta_{9} - 24328 \beta_{8} + 25276 \beta_{7} + \cdots + 202616 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 197916 \beta_{11} + 82860 \beta_{10} + 92278 \beta_{9} - 29324 \beta_{8} + 52846 \beta_{7} + \cdots + 564322 \)
|
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 | −1.00000 | 0 | −4.15588 | 0 | −1.11441 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −3.12492 | 0 | 4.33538 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −3.06377 | 0 | 3.59303 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −2.52462 | 0 | −2.76852 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | −0.425546 | 0 | −4.01837 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | −0.305886 | 0 | −0.140026 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 0.0891245 | 0 | 2.52085 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 0.413592 | 0 | 2.75070 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | −1.00000 | 0 | 1.42583 | 0 | −1.49015 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | −1.00000 | 0 | 2.48147 | 0 | −0.0448560 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | −1.00000 | 0 | 2.64977 | 0 | −1.93393 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | −1.00000 | 0 | 3.54084 | 0 | 2.31030 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
| \(29\) | \(1\) |
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 | 8004.2.a.g | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8004.2.a.g | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):
|
\( T_{5}^{12} + 3 T_{5}^{11} - 31 T_{5}^{10} - 80 T_{5}^{9} + 347 T_{5}^{8} + 697 T_{5}^{7} - 1714 T_{5}^{6} + \cdots + 16 \)
|
|
\( T_{7}^{12} - 4 T_{7}^{11} - 33 T_{7}^{10} + 127 T_{7}^{9} + 382 T_{7}^{8} - 1320 T_{7}^{7} - 2177 T_{7}^{6} + \cdots - 56 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( T^{12} + 3 T^{11} + \cdots + 16 \)
|
| $7$ |
\( T^{12} - 4 T^{11} + \cdots - 56 \)
|
| $11$ |
\( T^{12} + 5 T^{11} + \cdots - 448 \)
|
| $13$ |
\( T^{12} + 6 T^{11} + \cdots - 280 \)
|
| $17$ |
\( T^{12} + 7 T^{11} + \cdots - 113154 \)
|
| $19$ |
\( T^{12} + 3 T^{11} + \cdots + 57724 \)
|
| $23$ |
\( (T - 1)^{12} \)
|
| $29$ |
\( (T + 1)^{12} \)
|
| $31$ |
\( T^{12} + \cdots - 151516160 \)
|
| $37$ |
\( T^{12} + 20 T^{11} + \cdots + 5033728 \)
|
| $41$ |
\( T^{12} + 3 T^{11} + \cdots + 31140960 \)
|
| $43$ |
\( T^{12} + \cdots + 139228272 \)
|
| $47$ |
\( T^{12} + \cdots - 238331376 \)
|
| $53$ |
\( T^{12} + 3 T^{11} + \cdots + 12000192 \)
|
| $59$ |
\( T^{12} + \cdots + 491629824 \)
|
| $61$ |
\( T^{12} + \cdots - 509206528 \)
|
| $67$ |
\( T^{12} + 9 T^{11} + \cdots + 2048 \)
|
| $71$ |
\( T^{12} - 7 T^{11} + \cdots - 68516352 \)
|
| $73$ |
\( T^{12} + 9 T^{11} + \cdots + 30528 \)
|
| $79$ |
\( T^{12} - 14 T^{11} + \cdots - 33564676 \)
|
| $83$ |
\( T^{12} + \cdots + 407213568 \)
|
| $89$ |
\( T^{12} + 22 T^{11} + \cdots - 79609014 \)
|
| $97$ |
\( T^{12} + \cdots + 9231761664 \)
|
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