Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,41,Mod(2,3)]
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([1]))
N = Newforms(chi, 41, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 41);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 41 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(30.4026855589\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 249659905422 x^{10} + \cdots + 12\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{63}\cdot 3^{98}\cdot 5^{6}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 249659905422 x^{10} + \cdots + 12\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 55\!\cdots\!75 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 80\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 94\!\cdots\!75 \nu^{11} + \cdots + 57\!\cdots\!00 ) / 40\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!75 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 36\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 69\!\cdots\!27 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 50\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 74\!\cdots\!83 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31\!\cdots\!75 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 33\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 28\!\cdots\!51 \nu^{11} + \cdots + 38\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 77\!\cdots\!39 \nu^{11} + \cdots - 30\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 90\!\cdots\!53 \nu^{11} + \cdots - 91\!\cdots\!00 ) / 40\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 62\!\cdots\!61 \nu^{11} + \cdots - 32\!\cdots\!00 ) / 66\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 34\beta_{2} - 10\beta _1 - 1497959432532 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{8} + 50 \beta_{7} - 46 \beta_{5} + 520 \beta_{4} + 18779 \beta_{3} - 90796669 \beta_{2} - 2828274708871 \beta_1 ) / 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6233 \beta_{11} - 24932 \beta_{10} + 45981 \beta_{9} + 39748 \beta_{8} - 4824637 \beta_{7} + \cdots + 21\!\cdots\!04 ) / 648 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2441271912 \beta_{11} - 4882543824 \beta_{10} - 7643741864 \beta_{9} - 635297265967 \beta_{8} + \cdots + 11\!\cdots\!13 \beta_1 ) / 972 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 13\!\cdots\!37 \beta_{11} + \cdots - 29\!\cdots\!84 ) / 972 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 68\!\cdots\!76 \beta_{11} + \cdots - 18\!\cdots\!77 \beta_1 ) / 1458 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 25\!\cdots\!81 \beta_{11} + \cdots + 45\!\cdots\!64 ) / 1458 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 47\!\cdots\!60 \beta_{11} + \cdots + 98\!\cdots\!27 \beta_1 ) / 729 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 14\!\cdots\!55 \beta_{11} + \cdots - 24\!\cdots\!48 ) / 729 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 59\!\cdots\!72 \beta_{11} + \cdots - 10\!\cdots\!50 \beta_1 ) / 729 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
− | 2.02257e6i | 8.90215e8 | + | 3.37123e9i | −2.99130e12 | − | 2.53853e13i | 6.81856e15 | − | 1.80053e15i | −1.01945e17 | 3.82628e18i | −1.05727e19 | + | 6.00224e18i | −5.13437e19 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2.2 | − | 1.47013e6i | 1.53875e9 | − | 3.12889e9i | −1.06178e12 | − | 3.18557e13i | −4.59988e15 | − | 2.26216e15i | 7.85404e14 | − | 5.54756e16i | −7.42219e18 | − | 9.62912e18i | −4.68322e19 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2.3 | − | 1.37915e6i | −3.48296e9 | − | 1.63234e8i | −8.02549e11 | 4.92196e13i | −2.25124e14 | + | 4.80353e15i | 5.39353e16 | − | 4.09556e17i | 1.21044e19 | + | 1.13707e18i | 6.78814e19 | |||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | − | 762842.i | 3.30429e9 | + | 1.11325e9i | 5.17583e11 | 7.70076e13i | 8.49232e14 | − | 2.52065e15i | −3.01966e15 | − | 1.23359e18i | 9.67903e18 | + | 7.35699e18i | 5.87447e19 | |||||||||||||||||||||||||||||||||||||||||||||||
| 2.5 | − | 489884.i | −3.83248e8 | + | 3.46566e9i | 8.59526e11 | − | 1.65087e14i | 1.69777e15 | + | 1.87747e14i | 1.22230e17 | − | 9.59700e17i | −1.18639e19 | − | 2.65641e18i | −8.08734e19 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2.6 | − | 108093.i | −2.05308e9 | − | 2.81825e9i | 1.08783e12 | − | 1.13503e14i | −3.04633e14 | + | 2.21924e14i | −1.19157e17 | − | 2.36436e17i | −3.72735e18 | + | 1.15722e19i | −1.22689e19 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2.7 | 108093.i | −2.05308e9 | + | 2.81825e9i | 1.08783e12 | 1.13503e14i | −3.04633e14 | − | 2.21924e14i | −1.19157e17 | 2.36436e17i | −3.72735e18 | − | 1.15722e19i | −1.22689e19 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 2.8 | 489884.i | −3.83248e8 | − | 3.46566e9i | 8.59526e11 | 1.65087e14i | 1.69777e15 | − | 1.87747e14i | 1.22230e17 | 9.59700e17i | −1.18639e19 | + | 2.65641e18i | −8.08734e19 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 2.9 | 762842.i | 3.30429e9 | − | 1.11325e9i | 5.17583e11 | − | 7.70076e13i | 8.49232e14 | + | 2.52065e15i | −3.01966e15 | 1.23359e18i | 9.67903e18 | − | 7.35699e18i | 5.87447e19 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 2.10 | 1.37915e6i | −3.48296e9 | + | 1.63234e8i | −8.02549e11 | − | 4.92196e13i | −2.25124e14 | − | 4.80353e15i | 5.39353e16 | 4.09556e17i | 1.21044e19 | − | 1.13707e18i | 6.78814e19 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 2.11 | 1.47013e6i | 1.53875e9 | + | 3.12889e9i | −1.06178e12 | 3.18557e13i | −4.59988e15 | + | 2.26216e15i | 7.85404e14 | 5.54756e16i | −7.42219e18 | + | 9.62912e18i | −4.68322e19 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 2.12 | 2.02257e6i | 8.90215e8 | − | 3.37123e9i | −2.99130e12 | 2.53853e13i | 6.81856e15 | + | 1.80053e15i | −1.01945e17 | − | 3.82628e18i | −1.05727e19 | − | 6.00224e18i | −5.13437e19 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3.41.b.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 3.41.b.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.41.b.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 3.41.b.a | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{41}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 27\!\cdots\!00 \)
|
| $3$ |
\( T^{12} + \cdots + 32\!\cdots\!01 \)
|
| $5$ |
\( T^{12} + \cdots + 32\!\cdots\!00 \)
|
| $7$ |
\( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{12} + \cdots + 86\!\cdots\!00 \)
|
| $13$ |
\( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $19$ |
\( (T^{6} + \cdots + 12\!\cdots\!16)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{12} + \cdots + 98\!\cdots\!00 \)
|
| $31$ |
\( (T^{6} + \cdots + 70\!\cdots\!24)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 62\!\cdots\!00 \)
|
| $43$ |
\( (T^{6} + \cdots + 86\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 87\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 30\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 19\!\cdots\!00 \)
|
| $61$ |
\( (T^{6} + \cdots + 38\!\cdots\!44)^{2} \)
|
| $67$ |
\( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $73$ |
\( (T^{6} + \cdots - 52\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots - 39\!\cdots\!24)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( (T^{6} + \cdots + 76\!\cdots\!00)^{2} \)
|
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