Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1006,2,Mod(1,1006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1006, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1006 = 2 \cdot 503 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1006.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: | \(8.03295044334\) |
| 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: |
\( x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 75810347 \nu^{11} - 750391815 \nu^{10} + 1111655415 \nu^{9} + 8946386707 \nu^{8} + \cdots - 21112206658 ) / 7685328593 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 85937175 \nu^{11} - 183905264 \nu^{10} - 2549240479 \nu^{9} + 2711003021 \nu^{8} + \cdots - 7917809651 ) / 7685328593 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 106132083 \nu^{11} - 842730380 \nu^{10} + 89329233 \nu^{9} + 11763798682 \nu^{8} + \cdots - 34213508272 ) / 7685328593 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 146823889 \nu^{11} + 471833772 \nu^{10} + 3883703902 \nu^{9} - 8276421068 \nu^{8} + \cdots + 8817559953 ) / 7685328593 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 159236033 \nu^{11} + 320476659 \nu^{10} + 4957239696 \nu^{9} - 5371489680 \nu^{8} + \cdots + 3243266922 ) / 7685328593 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 194027177 \nu^{11} + 1001308266 \nu^{10} + 2733553715 \nu^{9} - 14135059823 \nu^{8} + \cdots - 485767656 ) / 7685328593 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 206991323 \nu^{11} - 573051401 \nu^{10} - 5843045880 \nu^{9} + 9800401503 \nu^{8} + \cdots + 11406893253 ) / 7685328593 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 210621704 \nu^{11} + 1410256576 \nu^{10} + 1550724992 \nu^{9} - 21641699784 \nu^{8} + \cdots - 34320821610 ) / 7685328593 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 246499891 \nu^{11} - 1029238772 \nu^{10} - 4971149628 \nu^{9} + 16303798652 \nu^{8} + \cdots + 24583412642 ) / 7685328593 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 566115680 \nu^{11} - 2117638831 \nu^{10} - 12926378732 \nu^{9} + 34046046658 \nu^{8} + \cdots + 37969929527 ) / 7685328593 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 678429148 \nu^{11} + 3136864086 \nu^{10} + 11829253654 \nu^{9} - 46947000260 \nu^{8} + \cdots - 915307019 ) / 7685328593 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{9} - 2\beta_{8} - 3\beta_{7} + \beta_{6} - 2\beta_{5} - 4\beta_{4} - \beta_{3} + 3\beta_{2} + \beta _1 + 12 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6 \beta_{11} - 2 \beta_{10} - 13 \beta_{9} - 18 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 12 \beta_{5} + \cdots + 28 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4 \beta_{11} - 14 \beta_{10} - 83 \beta_{9} - 78 \beta_{8} - 49 \beta_{7} + 21 \beta_{6} - 50 \beta_{5} + \cdots + 184 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 54 \beta_{11} - 80 \beta_{10} - 413 \beta_{9} - 442 \beta_{8} - 173 \beta_{7} + 121 \beta_{6} + \cdots + 744 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 114 \beta_{11} - 454 \beta_{10} - 2185 \beta_{9} - 2162 \beta_{8} - 1033 \beta_{7} + 533 \beta_{6} + \cdots + 4060 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 804 \beta_{11} - 2316 \beta_{10} - 11057 \beta_{9} - 11262 \beta_{8} - 4717 \beta_{7} + 2873 \beta_{6} + \cdots + 19366 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3012 \beta_{11} - 12194 \beta_{10} - 56555 \beta_{9} - 56556 \beta_{8} - 24987 \beta_{7} + 13785 \beta_{6} + \cdots + 100286 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 16918 \beta_{11} - 61660 \beta_{10} - 286719 \beta_{9} - 288812 \beta_{8} - 123013 \beta_{7} + \cdots + 500102 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 78302 \beta_{11} - 316360 \beta_{10} - 1456547 \beta_{9} - 1460328 \beta_{8} - 631427 \beta_{7} + \cdots + 2551408 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 409448 \beta_{11} - 1601440 \beta_{10} - 7385239 \beta_{9} - 7418958 \beta_{8} - 3174907 \beta_{7} + \cdots + 12876656 ) / 2 \)
|
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 | −3.27612 | 1.00000 | −1.07414 | 3.27612 | 4.58087 | −1.00000 | 7.73297 | 1.07414 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.15470 | 1.00000 | 4.23481 | 3.15470 | −1.79853 | −1.00000 | 6.95214 | −4.23481 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −3.12415 | 1.00000 | −0.847598 | 3.12415 | −2.65658 | −1.00000 | 6.76030 | 0.847598 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.67724 | 1.00000 | 1.77374 | 1.67724 | −3.87316 | −1.00000 | −0.186857 | −1.77374 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.36355 | 1.00000 | 2.23842 | 1.36355 | 4.41277 | −1.00000 | −1.14074 | −2.23842 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.05851 | 1.00000 | −3.62237 | 1.05851 | −4.42054 | −1.00000 | −1.87955 | 3.62237 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −0.163781 | 1.00000 | −2.50185 | 0.163781 | 0.0804128 | −1.00000 | −2.97318 | 2.50185 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0.656123 | 1.00000 | 4.07967 | −0.656123 | −0.827612 | −1.00000 | −2.56950 | −4.07967 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.99694 | 1.00000 | 2.93695 | −1.99694 | 3.10313 | −1.00000 | 0.987768 | −2.93695 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 2.49671 | 1.00000 | 2.79920 | −2.49671 | 1.31469 | −1.00000 | 3.23355 | −2.79920 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 2.73829 | 1.00000 | −1.48375 | −2.73829 | 2.26909 | −1.00000 | 4.49825 | 1.48375 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 2.92999 | 1.00000 | −1.53308 | −2.92999 | −4.18455 | −1.00000 | 5.58485 | 1.53308 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(503\) | \(-1\) |
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 | 1006.2.a.i | ✓ | 12 |
| 3.b | odd | 2 | 1 | 9054.2.a.bj | 12 | ||
| 4.b | odd | 2 | 1 | 8048.2.a.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1006.2.a.i | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 8048.2.a.r | 12 | 4.b | odd | 2 | 1 | ||
| 9054.2.a.bj | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\):
|
\( T_{3}^{12} + 3 T_{3}^{11} - 27 T_{3}^{10} - 78 T_{3}^{9} + 267 T_{3}^{8} + 746 T_{3}^{7} - 1120 T_{3}^{6} + \cdots - 336 \)
|
|
\( T_{5}^{12} - 7 T_{5}^{11} - 18 T_{5}^{10} + 195 T_{5}^{9} + 31 T_{5}^{8} - 1922 T_{5}^{7} + 584 T_{5}^{6} + \cdots + 10584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots - 336 \)
|
| $5$ |
\( T^{12} - 7 T^{11} + \cdots + 10584 \)
|
| $7$ |
\( T^{12} + 2 T^{11} + \cdots + 4263 \)
|
| $11$ |
\( T^{12} - 18 T^{11} + \cdots + 32144 \)
|
| $13$ |
\( T^{12} + 4 T^{11} + \cdots + 238928 \)
|
| $17$ |
\( T^{12} - 12 T^{11} + \cdots + 1179136 \)
|
| $19$ |
\( T^{12} - 169 T^{10} + \cdots - 14590592 \)
|
| $23$ |
\( T^{12} + 9 T^{11} + \cdots - 3244067 \)
|
| $29$ |
\( T^{12} + \cdots + 231936768 \)
|
| $31$ |
\( T^{12} + 11 T^{11} + \cdots - 23817472 \)
|
| $37$ |
\( T^{12} + 22 T^{11} + \cdots + 4886848 \)
|
| $41$ |
\( T^{12} - 32 T^{11} + \cdots - 417024 \)
|
| $43$ |
\( T^{12} + 8 T^{11} + \cdots + 3607792 \)
|
| $47$ |
\( T^{12} + \cdots + 107614057 \)
|
| $53$ |
\( T^{12} + \cdots + 395038848 \)
|
| $59$ |
\( T^{12} + \cdots - 26353645568 \)
|
| $61$ |
\( T^{12} + \cdots + 903709552 \)
|
| $67$ |
\( T^{12} + \cdots - 455670064 \)
|
| $71$ |
\( T^{12} - 50 T^{11} + \cdots - 257024 \)
|
| $73$ |
\( T^{12} + \cdots - 2309284344 \)
|
| $79$ |
\( T^{12} + 9 T^{11} + \cdots - 57869696 \)
|
| $83$ |
\( T^{12} + \cdots + 4260038928 \)
|
| $89$ |
\( T^{12} + \cdots + 117284352 \)
|
| $97$ |
\( T^{12} + \cdots - 26594882712 \)
|
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