Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,68,Mod(1,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 68, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 68);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 68 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(85.2871055790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3 x^{5} + \cdots - 80\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{46}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11^{2}\cdot 13\cdot 17 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3 x^{5} + \cdots - 80\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( 36\nu^{2} - 22243446972\nu - 219189618955593367272 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 547656442731 \nu^{5} + \cdots - 43\!\cdots\!40 ) / 50\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\!\cdots\!21 \nu^{5} + \cdots + 25\!\cdots\!40 ) / 35\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 35\!\cdots\!79 \nu^{5} + \cdots + 11\!\cdots\!12 ) / 70\!\cdots\!16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 3707241162\beta _1 + 219189618966715090758 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 141 \beta_{5} + 110 \beta_{4} + 2166905 \beta_{3} + 9413817317 \beta_{2} + \cdots + 81\!\cdots\!68 ) / 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 215683758605 \beta_{5} + 626554370414 \beta_{4} + \cdots + 38\!\cdots\!64 ) / 648 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17\!\cdots\!61 \beta_{5} + \cdots + 17\!\cdots\!72 ) / 1944 \)
|
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 |
|
−2.07551e10 | −5.55906e15 | 2.83199e20 | −8.85856e22 | 1.15379e26 | −1.53665e28 | −2.81491e30 | 3.09032e31 | 1.83860e33 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −9.88316e9 | −5.55906e15 | −4.98972e19 | −3.20918e23 | 5.49411e25 | 2.08311e28 | 1.95164e30 | 3.09032e31 | 3.17168e33 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.54666e9 | −5.55906e15 | −1.34995e20 | 1.66367e23 | 1.97161e25 | −1.26991e28 | 1.00218e30 | 3.09032e31 | −5.90048e32 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.17747e10 | −5.55906e15 | −8.92992e18 | 8.66624e22 | −6.54564e25 | 3.38994e28 | −1.84279e30 | 3.09032e31 | 1.02043e33 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.54594e10 | −5.55906e15 | 9.14202e19 | −4.50800e23 | −8.59399e25 | −2.44076e28 | −8.68106e29 | 3.09032e31 | −6.96912e33 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.06861e10 | −5.55906e15 | 2.80340e20 | 4.26985e23 | −1.14995e26 | −3.03250e28 | 2.74642e30 | 3.09032e31 | 8.83265e33 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
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 | 3.68.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.68.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.68.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.68.a.c | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 13735355166 T_{2}^{5} + \cdots - 27\!\cdots\!52 \)
acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 27\!\cdots\!52 \)
|
| $3$ |
\( (T + 55\!\cdots\!23)^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 78\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 92\!\cdots\!52 \)
|
| $13$ |
\( T^{6} + \cdots - 66\!\cdots\!12 \)
|
| $17$ |
\( T^{6} + \cdots - 21\!\cdots\!84 \)
|
| $19$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 79\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 50\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 59\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 64\!\cdots\!84 \)
|
| $47$ |
\( T^{6} + \cdots - 88\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 12\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 39\!\cdots\!68 \)
|
| $89$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 74\!\cdots\!44 \)
|
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