Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4815,2,Mod(1,4815)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4815, 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("4815.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4815 = 3^{2} \cdot 5 \cdot 107 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4815.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: | \(38.4479685732\) |
| 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: |
\( x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1605) |
| 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
\( x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} + 24 \nu^{10} - 4 \nu^{9} - 402 \nu^{8} - 101 \nu^{7} + 2336 \nu^{6} + 799 \nu^{5} + \cdots - 174 ) / 49 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{11} + 26 \nu^{10} + 61 \nu^{9} - 411 \nu^{8} - 481 \nu^{7} + 2204 \nu^{6} + 1817 \nu^{5} + \cdots - 66 ) / 49 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 27 \nu^{11} - 38 \nu^{10} - 451 \nu^{9} + 612 \nu^{8} + 2663 \nu^{7} - 3372 \nu^{6} - 6651 \nu^{5} + \cdots + 55 ) / 49 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24 \nu^{11} - 61 \nu^{10} - 390 \nu^{9} + 985 \nu^{8} + 2182 \nu^{7} - 5480 \nu^{6} - 4834 \nu^{5} + \cdots + 283 ) / 49 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5 \nu^{11} - 6 \nu^{10} - 83 \nu^{9} + 97 \nu^{8} + 482 \nu^{7} - 542 \nu^{6} - 1150 \nu^{5} + 1252 \nu^{4} + \cdots + 12 ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 22 \nu^{11} - 60 \nu^{10} - 382 \nu^{9} + 956 \nu^{8} + 2384 \nu^{7} - 5203 \nu^{6} - 6481 \nu^{5} + \cdots + 190 ) / 49 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5 \nu^{11} + 6 \nu^{10} + 83 \nu^{9} - 97 \nu^{8} - 482 \nu^{7} + 542 \nu^{6} + 1157 \nu^{5} + \cdots - 33 ) / 7 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 43 \nu^{11} + 95 \nu^{10} + 711 \nu^{9} - 1530 \nu^{8} - 4134 \nu^{7} + 8479 \nu^{6} + 10086 \nu^{5} + \cdots - 358 ) / 49 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{5} + 6\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + \beta_{8} + 9\beta_{3} - \beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{11} - 9 \beta_{10} - 9 \beta_{9} - \beta_{8} + 10 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} + \cdots + 94 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{11} + 12\beta_{10} - \beta_{9} + 12\beta_{8} - \beta_{7} - \beta_{5} + 68\beta_{3} - 14\beta_{2} + 167\beta _1 - 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( 79 \beta_{11} - 67 \beta_{10} - 67 \beta_{9} - 14 \beta_{8} + 79 \beta_{7} + 27 \beta_{6} - 131 \beta_{5} + \cdots + 577 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 18 \beta_{11} + 108 \beta_{10} - 14 \beta_{9} + 104 \beta_{8} - 16 \beta_{7} + \beta_{6} - 10 \beta_{5} + \cdots - 85 \)
|
| \(\nu^{10}\) | \(=\) |
\( 577 \beta_{11} - 473 \beta_{10} - 473 \beta_{9} - 136 \beta_{8} + 576 \beta_{7} + 257 \beta_{6} + \cdots + 3634 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 209 \beta_{11} + 873 \beta_{10} - 129 \beta_{9} + 801 \beta_{8} - 177 \beta_{7} + 18 \beta_{6} + \cdots - 971 \)
|
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 |
|
−2.51437 | 0 | 4.32208 | −1.00000 | 0 | −0.976551 | −5.83856 | 0 | 2.51437 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43355 | 0 | 3.92219 | −1.00000 | 0 | 4.14842 | −4.67775 | 0 | 2.43355 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.36308 | 0 | 3.58415 | −1.00000 | 0 | −1.44739 | −3.74348 | 0 | 2.36308 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.79305 | 0 | 1.21504 | −1.00000 | 0 | 0.475581 | 1.40748 | 0 | 1.79305 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.921208 | 0 | −1.15138 | −1.00000 | 0 | 3.92138 | 2.90307 | 0 | 0.921208 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.851790 | 0 | −1.27445 | −1.00000 | 0 | −0.870792 | 2.78915 | 0 | 0.851790 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.142275 | 0 | −1.97976 | −1.00000 | 0 | −1.55684 | 0.566220 | 0 | 0.142275 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.155707 | 0 | −1.97576 | −1.00000 | 0 | 3.91791 | −0.619053 | 0 | −0.155707 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.37578 | 0 | −0.107224 | −1.00000 | 0 | −1.29286 | −2.89908 | 0 | −1.37578 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.73551 | 0 | 1.01200 | −1.00000 | 0 | −4.69063 | −1.71469 | 0 | −1.73551 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.11086 | 0 | 2.45572 | −1.00000 | 0 | 2.18333 | 0.961964 | 0 | −2.11086 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.64148 | 0 | 4.97740 | −1.00000 | 0 | 3.18844 | 7.86473 | 0 | −2.64148 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(107\) | \(1\) |
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 | 4815.2.a.u | 12 | |
| 3.b | odd | 2 | 1 | 1605.2.a.n | ✓ | 12 | |
| 15.d | odd | 2 | 1 | 8025.2.a.bf | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1605.2.a.n | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 4815.2.a.u | 12 | 1.a | even | 1 | 1 | trivial | |
| 8025.2.a.bf | 12 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4815))\):
|
\( T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 49 T_{2}^{9} + 71 T_{2}^{8} + 278 T_{2}^{7} - 92 T_{2}^{6} + \cdots - 6 \)
|
|
\( T_{7}^{12} - 7 T_{7}^{11} - 22 T_{7}^{10} + 228 T_{7}^{9} - 24 T_{7}^{8} - 1958 T_{7}^{7} + 496 T_{7}^{6} + \cdots + 2452 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 3 T^{11} + \cdots - 6 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( T^{12} - 7 T^{11} + \cdots + 2452 \)
|
| $11$ |
\( T^{12} + 4 T^{11} + \cdots - 241932 \)
|
| $13$ |
\( T^{12} - 13 T^{11} + \cdots + 1536 \)
|
| $17$ |
\( T^{12} - 4 T^{11} + \cdots + 14528 \)
|
| $19$ |
\( T^{12} - 14 T^{11} + \cdots - 2744 \)
|
| $23$ |
\( T^{12} + 11 T^{11} + \cdots - 163584 \)
|
| $29$ |
\( T^{12} - 7 T^{11} + \cdots + 2900296 \)
|
| $31$ |
\( T^{12} - 4 T^{11} + \cdots + 1036544 \)
|
| $37$ |
\( T^{12} + \cdots + 2560288768 \)
|
| $41$ |
\( T^{12} + 13 T^{11} + \cdots - 2011068 \)
|
| $43$ |
\( T^{12} + \cdots - 449959160 \)
|
| $47$ |
\( T^{12} + \cdots - 243892224 \)
|
| $53$ |
\( T^{12} + \cdots - 6817093632 \)
|
| $59$ |
\( T^{12} + \cdots - 541472768 \)
|
| $61$ |
\( T^{12} - 7 T^{11} + \cdots - 14 \)
|
| $67$ |
\( T^{12} + \cdots + 4253150328 \)
|
| $71$ |
\( T^{12} + \cdots + 1745519872 \)
|
| $73$ |
\( T^{12} + \cdots - 133064496 \)
|
| $79$ |
\( T^{12} + \cdots + 4997668664 \)
|
| $83$ |
\( T^{12} + \cdots - 4744520192 \)
|
| $89$ |
\( T^{12} - 10 T^{11} + \cdots - 4477728 \)
|
| $97$ |
\( T^{12} + \cdots + 164588990820 \)
|
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