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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} - 2\nu^{9} - 10\nu^{8} + 17\nu^{7} + 32\nu^{6} - 45\nu^{5} - 35\nu^{4} + 54\nu^{3} + 14\nu^{2} - 37\nu - 6 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 55 \nu^{4} + 378 \nu^{3} + \cdots + 6 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 64 \nu^{4} + 387 \nu^{3} + \cdots - 30 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4 \nu^{10} + 11 \nu^{9} + 40 \nu^{8} - 107 \nu^{7} - 137 \nu^{6} + 339 \nu^{5} + 188 \nu^{4} + \cdots + 6 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4 \nu^{10} - 11 \nu^{9} - 40 \nu^{8} + 107 \nu^{7} + 137 \nu^{6} - 339 \nu^{5} - 188 \nu^{4} + 396 \nu^{3} + \cdots - 33 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7 \nu^{10} - 20 \nu^{9} - 61 \nu^{8} + 179 \nu^{7} + 152 \nu^{6} - 489 \nu^{5} - 53 \nu^{4} + 432 \nu^{3} + \cdots + 21 ) / 9 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2 \nu^{10} + 5 \nu^{9} + 20 \nu^{8} - 44 \nu^{7} - 73 \nu^{6} + 119 \nu^{5} + 125 \nu^{4} - 114 \nu^{3} + \cdots + 15 ) / 3 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 7 \nu^{10} - 14 \nu^{9} - 79 \nu^{8} + 128 \nu^{7} + 314 \nu^{6} - 360 \nu^{5} - 506 \nu^{4} + 342 \nu^{3} + \cdots - 42 ) / 9 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 16 \nu^{10} - 47 \nu^{9} - 142 \nu^{8} + 431 \nu^{7} + 386 \nu^{6} - 1236 \nu^{5} - 305 \nu^{4} + \cdots - 6 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} - \beta_{7} + 7\beta_{6} + 8\beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{10} + \beta_{9} + \beta_{8} - 9\beta_{7} + 9\beta_{6} + 18\beta_{5} - 8\beta_{4} + 2\beta_{3} + 21\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{10} + \beta_{9} + \beta_{8} - 12 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} - 18 \beta_{4} + \cdots + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( 53 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} - 65 \beta_{7} + 72 \beta_{6} + 134 \beta_{5} - 56 \beta_{4} + \cdots + 154 \)
|
| \(\nu^{8}\) | \(=\) |
\( 80 \beta_{10} + 14 \beta_{9} + 16 \beta_{8} - 107 \beta_{7} + 292 \beta_{6} + 386 \beta_{5} - 134 \beta_{4} + \cdots + 613 \)
|
| \(\nu^{9}\) | \(=\) |
\( 339 \beta_{10} + 80 \beta_{9} + 93 \beta_{8} - 448 \beta_{7} + 551 \beta_{6} + 960 \beta_{5} + \cdots + 1180 \)
|
| \(\nu^{10}\) | \(=\) |
\( 598 \beta_{10} + 143 \beta_{9} + 172 \beta_{8} - 863 \beta_{7} + 1940 \beta_{6} + 2678 \beta_{5} + \cdots + 4071 \)
|
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 |
|
−2.70371 | 1.00000 | 5.31003 | −2.83529 | −2.70371 | 2.54457 | −8.94935 | 1.00000 | 7.66580 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.39607 | 1.00000 | 3.74115 | −0.714370 | −2.39607 | −0.642303 | −4.17191 | 1.00000 | 1.71168 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.53948 | 1.00000 | 0.369988 | −0.258725 | −1.53948 | 4.37144 | 2.50937 | 1.00000 | 0.398300 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.36137 | 1.00000 | −0.146675 | −3.87776 | −1.36137 | −4.18708 | 2.92242 | 1.00000 | 5.27907 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.14470 | 1.00000 | −0.689651 | 1.33318 | −1.14470 | −3.52170 | 3.07886 | 1.00000 | −1.52610 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.536504 | 1.00000 | −1.71216 | −3.00877 | −0.536504 | 0.211107 | 1.99159 | 1.00000 | 1.61422 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.186189 | 1.00000 | −1.96533 | 1.94515 | 0.186189 | 0.0328573 | −0.738300 | 1.00000 | 0.362164 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.423080 | 1.00000 | −1.82100 | −1.36483 | 0.423080 | 0.865577 | −1.61659 | 1.00000 | −0.577432 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.35090 | 1.00000 | −0.175068 | −0.638467 | 1.35090 | −0.931245 | −2.93830 | 1.00000 | −0.862505 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.57504 | 1.00000 | 0.480756 | 0.664958 | 1.57504 | −4.15209 | −2.39287 | 1.00000 | 1.04734 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.14662 | 1.00000 | 2.60797 | −4.24506 | 2.14662 | 0.408872 | 1.30509 | 1.00000 | −9.11254 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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.a | ✓ | 11 |
| 3.b | odd | 2 | 1 | 6039.2.a.d | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2013.2.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 6039.2.a.d | 11 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 4 T_{2}^{10} - 6 T_{2}^{9} - 37 T_{2}^{8} - 2 T_{2}^{7} + 109 T_{2}^{6} + 55 T_{2}^{5} + \cdots - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 4 T^{10} + \cdots - 3 \)
|
| $3$ |
\( (T - 1)^{11} \)
|
| $5$ |
\( T^{11} + 13 T^{10} + \cdots - 39 \)
|
| $7$ |
\( T^{11} + 5 T^{10} + \cdots + 1 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} + 3 T^{10} + \cdots - 113383 \)
|
| $17$ |
\( T^{11} + 7 T^{10} + \cdots + 21027 \)
|
| $19$ |
\( T^{11} + 8 T^{10} + \cdots + 10163 \)
|
| $23$ |
\( T^{11} + 15 T^{10} + \cdots - 198891 \)
|
| $29$ |
\( T^{11} + 8 T^{10} + \cdots - 529149 \)
|
| $31$ |
\( T^{11} + 17 T^{10} + \cdots - 2875241 \)
|
| $37$ |
\( T^{11} + 10 T^{10} + \cdots - 7025731 \)
|
| $41$ |
\( T^{11} + \cdots - 793812771 \)
|
| $43$ |
\( T^{11} + 7 T^{10} + \cdots - 2670529 \)
|
| $47$ |
\( T^{11} + \cdots + 149717619 \)
|
| $53$ |
\( T^{11} + 18 T^{10} + \cdots + 4627299 \)
|
| $59$ |
\( T^{11} + 43 T^{10} + \cdots - 91481937 \)
|
| $61$ |
\( (T - 1)^{11} \)
|
| $67$ |
\( T^{11} + 30 T^{10} + \cdots + 1573457 \)
|
| $71$ |
\( T^{11} + \cdots - 28070149767 \)
|
| $73$ |
\( T^{11} + \cdots + 126285977 \)
|
| $79$ |
\( T^{11} - 17 T^{10} + \cdots - 15718043 \)
|
| $83$ |
\( T^{11} + \cdots + 18319432683 \)
|
| $89$ |
\( T^{11} + \cdots - 78347255019 \)
|
| $97$ |
\( T^{11} + \cdots - 7174782977 \)
|
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