Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [197,2,Mod(1,197)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, 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("197.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 197 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 197.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: | \(1.57305291982\) |
| 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: |
\( x^{10} - 15x^{8} - x^{7} + 78x^{6} + 7x^{5} - 165x^{4} - 15x^{3} + 123x^{2} + 9x - 26 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{10} - 15x^{8} - x^{7} + 78x^{6} + 7x^{5} - 165x^{4} - 15x^{3} + 123x^{2} + 9x - 26 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 30\nu^{4} - 36\nu^{3} - 27\nu^{2} + 7\nu + 6 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 10\nu^{6} - 3\nu^{5} + 26\nu^{4} + 14\nu^{3} - 11\nu^{2} - 9\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - 12\nu^{6} - \nu^{5} + 44\nu^{4} + 2\nu^{3} - 53\nu^{2} + 5\nu + 16 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - 14\nu^{7} + \nu^{6} + 64\nu^{5} - 12\nu^{4} - 107\nu^{3} + 29\nu^{2} + 44\nu - 8 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 12\nu^{7} + 11\nu^{6} + 47\nu^{5} - 46\nu^{4} - 67\nu^{3} + 80\nu^{2} + 21\nu - 30 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 10\nu^{5} + 8\nu^{4} + 27\nu^{3} - 18\nu^{2} - 14\nu + 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 12\nu^{7} - 9\nu^{6} - 47\nu^{5} + 24\nu^{4} + 67\nu^{3} - 18\nu^{2} - 25\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 2\beta_{6} - \beta_{4} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{9} + 8\beta_{8} + 8\beta_{7} + 10\beta_{6} + \beta_{5} - 9\beta_{4} + 6\beta_{3} + 10\beta_{2} + 19\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{9} + 11\beta_{8} + 2\beta_{7} + 22\beta_{6} - 11\beta_{4} + 46\beta_{2} + 2\beta _1 + 74 \)
|
| \(\nu^{7}\) | \(=\) |
\( 67 \beta_{9} + 57 \beta_{8} + 55 \beta_{7} + 79 \beta_{6} + 10 \beta_{5} - 66 \beta_{4} + 33 \beta_{3} + \cdots + 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( 107 \beta_{9} + 94 \beta_{8} + 30 \beta_{7} + 184 \beta_{6} + 3 \beta_{5} - 95 \beta_{4} + 4 \beta_{3} + \cdots + 417 \)
|
| \(\nu^{9}\) | \(=\) |
\( 469 \beta_{9} + 394 \beta_{8} + 363 \beta_{7} + 579 \beta_{6} + 76 \beta_{5} - 456 \beta_{4} + \cdots + 279 \)
|
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 |
|
−2.62911 | 0.0119479 | 4.91223 | 0.817206 | −0.0314124 | 2.54267 | −7.65657 | −2.99986 | −2.14853 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.94873 | 2.99027 | 1.79756 | 3.37189 | −5.82723 | −0.532209 | 0.394492 | 5.94168 | −6.57091 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.85218 | 2.60828 | 1.43059 | −3.95640 | −4.83102 | 3.45456 | 1.05466 | 3.80315 | 7.32799 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.931586 | −2.17005 | −1.13215 | −1.44818 | 2.02159 | 1.12960 | 2.91786 | 1.70911 | 1.34911 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.530276 | 0.899248 | −1.71881 | 3.03207 | −0.476849 | 0.743459 | 1.97199 | −2.19135 | −1.60783 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.669007 | 1.75170 | −1.55243 | 0.170148 | 1.17190 | 5.01834 | −2.37660 | 0.0684478 | 0.113830 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.896375 | 3.41221 | −1.19651 | −0.448431 | 3.05862 | −2.39168 | −2.86527 | 8.64317 | −0.401962 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.82964 | 0.661864 | 1.34760 | 2.39480 | 1.21098 | −2.47846 | −1.19366 | −2.56194 | 4.38163 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.22583 | −1.75215 | 2.95430 | 1.17649 | −3.89998 | 3.66330 | 2.12410 | 0.0700264 | 2.61867 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.27104 | 1.58668 | 3.15762 | −3.10959 | 3.60342 | −0.149594 | 2.62899 | −0.482441 | −7.06199 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(197\) | \( -1 \) |
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 | 197.2.a.c | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1773.2.a.f | 10 | ||
| 4.b | odd | 2 | 1 | 3152.2.a.m | 10 | ||
| 5.b | even | 2 | 1 | 4925.2.a.i | 10 | ||
| 7.b | odd | 2 | 1 | 9653.2.a.j | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 197.2.a.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1773.2.a.f | 10 | 3.b | odd | 2 | 1 | ||
| 3152.2.a.m | 10 | 4.b | odd | 2 | 1 | ||
| 4925.2.a.i | 10 | 5.b | even | 2 | 1 | ||
| 9653.2.a.j | 10 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 15T_{2}^{8} + T_{2}^{7} + 78T_{2}^{6} - 7T_{2}^{5} - 165T_{2}^{4} + 15T_{2}^{3} + 123T_{2}^{2} - 9T_{2} - 26 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(197))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 15 T^{8} + \cdots - 26 \)
|
| $3$ |
\( T^{10} - 10 T^{9} + \cdots + 2 \)
|
| $5$ |
\( T^{10} - 2 T^{9} + \cdots + 32 \)
|
| $7$ |
\( T^{10} - 11 T^{9} + \cdots + 64 \)
|
| $11$ |
\( T^{10} - 2 T^{9} + \cdots - 5906 \)
|
| $13$ |
\( T^{10} - 8 T^{9} + \cdots + 448 \)
|
| $17$ |
\( T^{10} + 3 T^{9} + \cdots - 11008 \)
|
| $19$ |
\( T^{10} - 17 T^{9} + \cdots - 26944 \)
|
| $23$ |
\( T^{10} + 4 T^{9} + \cdots - 55696 \)
|
| $29$ |
\( T^{10} + 9 T^{9} + \cdots - 1849 \)
|
| $31$ |
\( T^{10} - 20 T^{9} + \cdots - 1018 \)
|
| $37$ |
\( T^{10} - 12 T^{9} + \cdots - 1031837 \)
|
| $41$ |
\( T^{10} + 18 T^{9} + \cdots - 12249251 \)
|
| $43$ |
\( T^{10} - 11 T^{9} + \cdots + 958064 \)
|
| $47$ |
\( T^{10} - 5 T^{9} + \cdots - 6076144 \)
|
| $53$ |
\( T^{10} + 6 T^{9} + \cdots + 24986 \)
|
| $59$ |
\( T^{10} + T^{9} + \cdots - 2663552 \)
|
| $61$ |
\( T^{10} - 4 T^{9} + \cdots + 7550167 \)
|
| $67$ |
\( T^{10} + \cdots + 142552394 \)
|
| $71$ |
\( T^{10} + \cdots - 112062458 \)
|
| $73$ |
\( T^{10} - 25 T^{9} + \cdots + 28387712 \)
|
| $79$ |
\( T^{10} + T^{9} + \cdots - 29837000 \)
|
| $83$ |
\( T^{10} - 28 T^{9} + \cdots + 303296 \)
|
| $89$ |
\( T^{10} + 6 T^{9} + \cdots + 57477472 \)
|
| $97$ |
\( T^{10} + \cdots - 151216646 \)
|
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