Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,72,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, 72, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 72);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 72 \) |
| 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: | \(95.7738481683\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2 x^{4} + \cdots - 10\!\cdots\!54 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2 x^{4} + \cdots - 10\!\cdots\!54 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( 576\nu^{2} - 115512328608\nu - 2717540473422020841408 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 627183 \nu^{4} + 387649018384857 \nu^{3} + \cdots + 83\!\cdots\!22 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 89019 \nu^{4} + 254610810147501 \nu^{3} + \cdots + 20\!\cdots\!46 ) / 14\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 10 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 4813013692\beta _1 + 2717540473470150978328 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 15486 \beta_{4} - 30772 \beta_{3} + 408044456 \beta_{2} + \cdots + 20\!\cdots\!70 ) / 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 229717936820656 \beta_{4} + \cdots + 19\!\cdots\!48 ) / 5184 \)
|
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 |
|
−8.27630e10 | 5.00315e16 | 4.48853e21 | −1.31164e24 | −4.14076e27 | −1.62984e30 | −1.76066e32 | 2.50316e33 | 1.08555e35 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.66543e10 | 5.00315e16 | −1.65073e21 | 1.17208e25 | −1.33355e27 | −5.64464e29 | 1.06935e32 | 2.50316e33 | −3.12409e35 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.05621e10 | 5.00315e16 | −1.93838e21 | −6.43665e24 | −1.02875e27 | 4.31786e29 | 8.84080e31 | 2.50316e33 | 1.32351e35 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.18503e10 | 5.00315e16 | −6.09737e20 | 3.60424e24 | 2.09383e27 | −2.88017e29 | −1.24334e32 | 2.50316e33 | 1.50838e35 | ||||||||||||||||||||||||||||||||||
| 1.5 | 6.30778e10 | 5.00315e16 | 1.61762e21 | −6.15414e24 | 3.15588e27 | 7.89316e29 | −4.69021e31 | 2.50316e33 | −3.88189e35 | ||||||||||||||||||||||||||||||||||
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 | 3.72.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9.72.a.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.72.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9.72.a.a | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 25051277688 T_{2}^{4} + \cdots + 11\!\cdots\!68 \)
acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 11\!\cdots\!68 \)
|
| $3$ |
\( (T - 50\!\cdots\!07)^{5} \)
|
| $5$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 90\!\cdots\!00 \)
|
| $11$ |
\( T^{5} + \cdots + 30\!\cdots\!56 \)
|
| $13$ |
\( T^{5} + \cdots + 41\!\cdots\!04 \)
|
| $17$ |
\( T^{5} + \cdots + 89\!\cdots\!56 \)
|
| $19$ |
\( T^{5} + \cdots - 38\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 50\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 68\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 31\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 36\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots - 23\!\cdots\!72 \)
|
| $47$ |
\( T^{5} + \cdots + 63\!\cdots\!64 \)
|
| $53$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 44\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 54\!\cdots\!88 \)
|
| $67$ |
\( T^{5} + \cdots - 23\!\cdots\!44 \)
|
| $71$ |
\( T^{5} + \cdots + 63\!\cdots\!28 \)
|
| $73$ |
\( T^{5} + \cdots - 30\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 26\!\cdots\!64 \)
|
| $89$ |
\( T^{5} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 22\!\cdots\!44 \)
|
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