Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [802,2,Mod(1,802)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(802, 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("802.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 802 = 2 \cdot 401 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 802.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: | \(6.40400224211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 21x^{8} - 21x^{7} + 124x^{6} + 231x^{5} - 34x^{4} - 255x^{3} - 64x^{2} + 70x + 21 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 21x^{8} - 21x^{7} + 124x^{6} + 231x^{5} - 34x^{4} - 255x^{3} - 64x^{2} + 70x + 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{8} + 2\nu^{7} + 16\nu^{6} - 9\nu^{5} - 87\nu^{4} - 63\nu^{3} + 43\nu^{2} + 40\nu + 3 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 3\nu^{8} - 14\nu^{7} + 25\nu^{6} + 78\nu^{5} - 21\nu^{4} - 112\nu^{3} - 18\nu^{2} + 46\nu + 12 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{9} - 6\nu^{8} - 28\nu^{7} + 53\nu^{6} + 147\nu^{5} - 69\nu^{4} - 167\nu^{3} + 42\nu^{2} + 44\nu - 3 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{9} + 3\nu^{8} + 37\nu^{7} - 11\nu^{6} - 225\nu^{5} - 150\nu^{4} + 251\nu^{3} + 183\nu^{2} - 86\nu - 48 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{9} + 2\nu^{8} + 17\nu^{7} - 13\nu^{6} - 99\nu^{5} - 34\nu^{4} + 113\nu^{3} + 50\nu^{2} - 37\nu - 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{9} + 9\nu^{8} + 65\nu^{7} - 64\nu^{6} - 372\nu^{5} - 81\nu^{4} + 418\nu^{3} + 138\nu^{2} - 127\nu - 30 ) / 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -4\nu^{9} + 8\nu^{8} + 67\nu^{7} - 48\nu^{6} - 384\nu^{5} - 165\nu^{4} + 382\nu^{3} + 190\nu^{2} - 105\nu - 36 ) / 3 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4\nu^{9} - 8\nu^{8} - 67\nu^{7} + 48\nu^{6} + 384\nu^{5} + 168\nu^{4} - 385\nu^{3} - 217\nu^{2} + 99\nu + 51 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + \beta_{5} - \beta_{4} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} + 8\beta _1 + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + \beta_{8} - 11\beta_{7} + \beta_{6} + 11\beta_{5} - 11\beta_{4} - \beta_{3} + \beta_{2} + 19\beta _1 + 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{9} - 31\beta_{7} + 10\beta_{6} + 32\beta_{5} - 32\beta_{4} - 10\beta_{3} + 11\beta_{2} + 88\beta _1 + 142 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32 \beta_{9} + 9 \beta_{8} - 128 \beta_{7} + 20 \beta_{6} + 131 \beta_{5} - 130 \beta_{4} - 22 \beta_{3} + \cdots + 575 \)
|
| \(\nu^{7}\) | \(=\) |
\( 132 \beta_{9} + 4 \beta_{8} - 415 \beta_{7} + 105 \beta_{6} + 439 \beta_{5} - 435 \beta_{4} + \cdots + 1919 \)
|
| \(\nu^{8}\) | \(=\) |
\( 440 \beta_{9} + 65 \beta_{8} - 1559 \beta_{7} + 290 \beta_{6} + 1646 \beta_{5} - 1622 \beta_{4} + \cdots + 7148 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1664 \beta_{9} + 47 \beta_{8} - 5342 \beta_{7} + 1193 \beta_{6} + 5786 \beta_{5} - 5683 \beta_{4} + \cdots + 24974 \)
|
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 | −2.58946 | 1.00000 | 3.26712 | −2.58946 | 3.10643 | 1.00000 | 3.70532 | 3.26712 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.56111 | 1.00000 | 2.40916 | −2.56111 | −4.67943 | 1.00000 | 3.55929 | 2.40916 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.132044 | 1.00000 | −2.73980 | 0.132044 | 0.855852 | 1.00000 | −2.98256 | −2.73980 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.370738 | 1.00000 | 2.45307 | 0.370738 | 3.91627 | 1.00000 | −2.86255 | 2.45307 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.30765 | 1.00000 | 1.89527 | 1.30765 | 0.812864 | 1.00000 | −1.29005 | 1.89527 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.86947 | 1.00000 | 1.73243 | 1.86947 | −0.506935 | 1.00000 | 0.494911 | 1.73243 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.30342 | 1.00000 | −1.45664 | 2.30342 | 2.06054 | 1.00000 | 2.30576 | −1.45664 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.86155 | 1.00000 | −0.300236 | 2.86155 | 3.26795 | 1.00000 | 5.18846 | −0.300236 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 3.13730 | 1.00000 | −2.18627 | 3.13730 | −3.26176 | 1.00000 | 6.84263 | −2.18627 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 3.16841 | 1.00000 | 3.92590 | 3.16841 | −4.57179 | 1.00000 | 7.03881 | 3.92590 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(401\) | \(1\) |
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 | 802.2.a.f | ✓ | 10 |
| 3.b | odd | 2 | 1 | 7218.2.a.v | 10 | ||
| 4.b | odd | 2 | 1 | 6416.2.a.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 802.2.a.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 6416.2.a.i | 10 | 4.b | odd | 2 | 1 | ||
| 7218.2.a.v | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 10 T_{3}^{9} + 24 T_{3}^{8} + 69 T_{3}^{7} - 401 T_{3}^{6} + 390 T_{3}^{5} + 986 T_{3}^{4} + \cdots + 52 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(802))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} - 10 T^{9} + \cdots + 52 \)
|
| $5$ |
\( T^{10} - 9 T^{9} + \cdots + 652 \)
|
| $7$ |
\( T^{10} - T^{9} + \cdots + 2016 \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots - 336 \)
|
| $13$ |
\( T^{10} - 10 T^{9} + \cdots - 16464 \)
|
| $17$ |
\( T^{10} - 72 T^{8} + \cdots - 3456 \)
|
| $19$ |
\( T^{10} + 4 T^{9} + \cdots + 1068 \)
|
| $23$ |
\( T^{10} - 17 T^{9} + \cdots - 51912 \)
|
| $29$ |
\( T^{10} + 4 T^{9} + \cdots - 21786 \)
|
| $31$ |
\( T^{10} + 9 T^{9} + \cdots + 261072 \)
|
| $37$ |
\( T^{10} - 4 T^{9} + \cdots + 3400736 \)
|
| $41$ |
\( T^{10} - 10 T^{9} + \cdots + 486289 \)
|
| $43$ |
\( T^{10} + 14 T^{9} + \cdots + 281344 \)
|
| $47$ |
\( T^{10} - 21 T^{9} + \cdots - 7114464 \)
|
| $53$ |
\( T^{10} + 16 T^{9} + \cdots - 39242256 \)
|
| $59$ |
\( T^{10} - 10 T^{9} + \cdots + 1632 \)
|
| $61$ |
\( T^{10} - 20 T^{9} + \cdots + 30797872 \)
|
| $67$ |
\( T^{10} + 27 T^{9} + \cdots - 246782 \)
|
| $71$ |
\( T^{10} + \cdots - 315615109 \)
|
| $73$ |
\( T^{10} + 16 T^{9} + \cdots + 36597547 \)
|
| $79$ |
\( T^{10} + 12 T^{9} + \cdots + 89297539 \)
|
| $83$ |
\( T^{10} + T^{9} + \cdots + 21830928 \)
|
| $89$ |
\( T^{10} + 5 T^{9} + \cdots + 48093143 \)
|
| $97$ |
\( T^{10} + 15 T^{9} + \cdots - 10585056 \)
|
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