Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,74,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, 74, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 74);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 74 \) |
| 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: | \(101.245192121\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 39\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{54}\cdot 3^{29}\cdot 5^{5}\cdot 7^{3}\cdot 11 \) |
| 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
\( x^{6} - 3 x^{5} + \cdots - 39\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( 144\nu^{2} - 95391235968\nu - 14983597329730497183768 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 33047624079 \nu^{5} + \cdots - 47\!\cdots\!50 ) / 49\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!21 \nu^{5} + \cdots + 36\!\cdots\!50 ) / 49\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15\!\cdots\!81 \nu^{5} + \cdots + 47\!\cdots\!50 ) / 49\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 7949269664\beta _1 + 14983597329778192801752 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 55 \beta_{5} + 74 \beta_{4} + 1014781 \beta_{3} - 1892102595 \beta_{2} + \cdots + 59\!\cdots\!52 ) / 864 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 219620373765 \beta_{5} - 514764825826 \beta_{4} + \cdots + 42\!\cdots\!76 ) / 2592 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 35\!\cdots\!55 \beta_{5} + \cdots + 31\!\cdots\!72 ) / 1944 \)
|
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 |
|
−1.58745e11 | −1.50095e17 | 1.57552e22 | −9.96266e24 | 2.38267e28 | −1.56497e30 | −1.00175e33 | 2.25284e34 | 1.58152e36 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.36355e10 | −1.50095e17 | −2.44984e21 | 3.90094e25 | 1.25532e28 | 1.23901e31 | 9.94808e32 | 2.25284e34 | −3.26257e36 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −6.16667e10 | −1.50095e17 | −5.64195e21 | −2.06554e25 | 9.25584e27 | −8.66374e30 | 9.30346e32 | 2.25284e34 | 1.27375e36 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 6.04972e10 | −1.50095e17 | −5.78483e21 | −3.51515e25 | −9.08030e27 | 2.01238e30 | −9.21345e32 | 2.25284e34 | −2.12656e36 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.29665e11 | −1.50095e17 | 7.36834e21 | 3.31991e25 | −1.94621e28 | 5.58168e30 | −2.69236e32 | 2.25284e34 | 4.30478e36 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.85141e11 | −1.50095e17 | 2.48326e22 | −2.18904e25 | −2.77887e28 | −1.06714e31 | 2.84892e33 | 2.25284e34 | −4.05282e36 | |||||||||||||||||||||||||||||||||||||
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 | 3.74.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.74.a.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.74.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.74.a.b | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 71256829788 T_{2}^{5} + \cdots - 11\!\cdots\!28 \)
acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 11\!\cdots\!28 \)
|
| $3$ |
\( (T + 15\!\cdots\!21)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 26\!\cdots\!92 \)
|
| $13$ |
\( T^{6} + \cdots + 93\!\cdots\!12 \)
|
| $17$ |
\( T^{6} + \cdots - 23\!\cdots\!76 \)
|
| $19$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 60\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 38\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 53\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots + 49\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 61\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 62\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!04 \)
|
| $71$ |
\( T^{6} + \cdots - 12\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 66\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 47\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 11\!\cdots\!44 \)
|
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