Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2013,2,Mod(1,2013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, 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("2013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2013 = 3 \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2013.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: | \(16.0738859269\) |
| 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: |
\( x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 18 \nu^{11} + 63 \nu^{10} + 341 \nu^{9} - 1448 \nu^{8} - 920 \nu^{7} + 7681 \nu^{6} - 1364 \nu^{5} + \cdots - 2408 ) / 313 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14 \nu^{11} - 49 \nu^{10} + 13 \nu^{9} - 404 \nu^{8} + 646 \nu^{7} + 4494 \nu^{6} - 5060 \nu^{5} + \cdots - 2370 ) / 313 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 41 \nu^{11} - 13 \nu^{10} + 1281 \nu^{9} - 1142 \nu^{8} - 8599 \nu^{7} + 6871 \nu^{6} + 23359 \nu^{5} + \cdots + 2201 ) / 313 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 110 \nu^{11} + 385 \nu^{10} + 1284 \nu^{9} - 4293 \nu^{8} - 5970 \nu^{7} + 15396 \nu^{6} + \cdots + 65 ) / 313 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 98 \nu^{11} - 656 \nu^{10} + 404 \nu^{9} + 4997 \nu^{8} - 7059 \nu^{7} - 13927 \nu^{6} + 22798 \nu^{5} + \cdots + 3442 ) / 313 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 121 \nu^{11} - 580 \nu^{10} - 536 \nu^{9} + 4691 \nu^{8} + 620 \nu^{7} - 13117 \nu^{6} - 1925 \nu^{5} + \cdots - 854 ) / 313 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 134 \nu^{11} + 782 \nu^{10} - 35 \nu^{9} - 6015 \nu^{8} + 4593 \nu^{7} + 16769 \nu^{6} + \cdots - 1059 ) / 313 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 154 \nu^{11} - 539 \nu^{10} - 1735 \nu^{9} + 5572 \nu^{8} + 8984 \nu^{7} - 20052 \nu^{6} - 24047 \nu^{5} + \cdots - 404 ) / 313 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 236 \nu^{11} + 1139 \nu^{10} + 1167 \nu^{9} - 10047 \nu^{8} - 1142 \nu^{7} + 32229 \nu^{6} + \cdots - 1767 ) / 313 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 3\beta_{2} + 4\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14 \beta_{11} + 13 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} - 12 \beta_{6} + 16 \beta_{5} + \cdots + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 46 \beta_{11} + 42 \beta_{10} - 48 \beta_{9} - 22 \beta_{8} - 20 \beta_{7} - 40 \beta_{6} + 58 \beta_{5} + \cdots + 132 \)
|
| \(\nu^{7}\) | \(=\) |
\( 166 \beta_{11} + 148 \beta_{10} - 177 \beta_{9} - 75 \beta_{8} - 75 \beta_{7} - 137 \beta_{6} + \cdots + 364 \)
|
| \(\nu^{8}\) | \(=\) |
\( 549 \beta_{11} + 483 \beta_{10} - 606 \beta_{9} - 237 \beta_{8} - 287 \beta_{7} - 458 \beta_{6} + \cdots + 1192 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1852 \beta_{11} + 1613 \beta_{10} - 2093 \beta_{9} - 797 \beta_{8} - 1020 \beta_{7} - 1526 \beta_{6} + \cdots + 3611 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6095 \beta_{11} + 5266 \beta_{10} - 7064 \beta_{9} - 2636 \beta_{8} - 3600 \beta_{7} - 5065 \beta_{6} + \cdots + 11557 \)
|
| \(\nu^{11}\) | \(=\) |
\( 20133 \beta_{11} + 17302 \beta_{10} - 23796 \beta_{9} - 8850 \beta_{8} - 12387 \beta_{7} - 16746 \beta_{6} + \cdots + 36321 \)
|
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 |
|
−2.73351 | 1.00000 | 5.47208 | −3.64694 | −2.73351 | −3.96128 | −9.49098 | 1.00000 | 9.96895 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.62825 | 1.00000 | 4.90768 | 2.26825 | −2.62825 | −0.677820 | −7.64209 | 1.00000 | −5.96152 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.31792 | 1.00000 | 3.37276 | −2.47951 | −2.31792 | 3.26859 | −3.18195 | 1.00000 | 5.74732 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.03538 | 1.00000 | 2.14278 | 3.64320 | −2.03538 | −4.68983 | −0.290605 | 1.00000 | −7.41529 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.45633 | 1.00000 | 0.120890 | −0.271377 | −1.45633 | −0.415334 | 2.73660 | 1.00000 | 0.395214 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.19907 | 1.00000 | −0.562235 | 0.908382 | −1.19907 | −2.10536 | 3.07229 | 1.00000 | −1.08921 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.141429 | 1.00000 | −1.98000 | −4.38024 | −0.141429 | 3.47902 | 0.562887 | 1.00000 | 0.619492 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.109182 | 1.00000 | −1.98808 | 0.0420221 | 0.109182 | −3.35497 | −0.435426 | 1.00000 | 0.00458805 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.235537 | 1.00000 | −1.94452 | 2.04419 | 0.235537 | −0.723848 | −0.929081 | 1.00000 | 0.481482 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.22712 | 1.00000 | −0.494178 | −1.92536 | 1.22712 | 2.05062 | −3.06065 | 1.00000 | −2.36265 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.66112 | 1.00000 | 0.759316 | −1.47319 | 1.66112 | −2.81455 | −2.06092 | 1.00000 | −2.44715 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.27893 | 1.00000 | 3.19351 | −1.72942 | 2.27893 | −5.05524 | 2.71992 | 1.00000 | −3.94122 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
| \(61\) | \(1\) |
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 | 2013.2.a.c | ✓ | 12 |
| 3.b | odd | 2 | 1 | 6039.2.a.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2013.2.a.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 6039.2.a.f | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 7 T_{2}^{11} + 6 T_{2}^{10} - 57 T_{2}^{9} - 123 T_{2}^{8} + 97 T_{2}^{7} + 414 T_{2}^{6} + \cdots - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 7 T^{11} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} + 7 T^{11} + \cdots - 34 \)
|
| $7$ |
\( T^{12} + 15 T^{11} + \cdots - 8872 \)
|
| $11$ |
\( (T - 1)^{12} \)
|
| $13$ |
\( T^{12} + 11 T^{11} + \cdots - 3268306 \)
|
| $17$ |
\( T^{12} + 33 T^{11} + \cdots - 6367574 \)
|
| $19$ |
\( T^{12} + 24 T^{11} + \cdots + 483158 \)
|
| $23$ |
\( T^{12} + \cdots + 287946976 \)
|
| $29$ |
\( T^{12} + 16 T^{11} + \cdots - 19062934 \)
|
| $31$ |
\( T^{12} - T^{11} + \cdots - 65898142 \)
|
| $37$ |
\( T^{12} + 6 T^{11} + \cdots - 82268290 \)
|
| $41$ |
\( T^{12} + \cdots + 201360448 \)
|
| $43$ |
\( T^{12} + \cdots + 5092595932 \)
|
| $47$ |
\( T^{12} + 18 T^{11} + \cdots - 31072 \)
|
| $53$ |
\( T^{12} + \cdots - 452172932 \)
|
| $59$ |
\( T^{12} + \cdots + 598610932 \)
|
| $61$ |
\( (T + 1)^{12} \)
|
| $67$ |
\( T^{12} - 355 T^{10} + \cdots + 2931146 \)
|
| $71$ |
\( T^{12} + \cdots - 533942480 \)
|
| $73$ |
\( T^{12} + \cdots + 102223264294 \)
|
| $79$ |
\( T^{12} + \cdots - 63864138140 \)
|
| $83$ |
\( T^{12} + \cdots + 18840952084 \)
|
| $89$ |
\( T^{12} + \cdots + 1198522060 \)
|
| $97$ |
\( T^{12} + \cdots - 250165313954 \)
|
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