Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [213,2,Mod(212,213)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(213, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("213.212");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 213 = 3 \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 213.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: | \(1.70081356305\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.18604960000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 20x^{6} + 138x^{4} + 385x^{2} + 361 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 20x^{6} + 138x^{4} + 385x^{2} + 361 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{7} + 79\nu^{5} + 585\nu^{3} + 1155\nu ) / 247 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{7} - 102\nu^{5} - 377\nu^{3} - 225\nu ) / 247 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{7} - 102\nu^{5} - 377\nu^{3} + 269\nu ) / 247 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} + 31\nu^{4} + 143\nu^{2} + 198 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 11\nu^{2} + 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{6} - 62\nu^{4} - 260\nu^{2} - 253 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 79\nu^{7} + 1257\nu^{5} + 6019\nu^{3} + 8679\nu ) / 247 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 2\beta_{4} - 11 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - 6\beta_{3} + 16\beta_{2} - 3\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11\beta_{6} + 2\beta_{5} - 22\beta_{4} + 73 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{7} + 21\beta_{3} - 78\beta_{2} + 25\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 99\beta_{6} - 31\beta_{5} + 211\beta_{4} - 543 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 121\beta_{7} - 321\beta_{3} + 1373\beta_{2} - 567\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/213\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(143\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 212.1 |
|
− | 1.61803i | −1.61803 | + | 0.618034i | −0.618034 | 1.61803i | 1.00000 | + | 2.61803i | − | 4.84481i | − | 2.23607i | 2.23607 | − | 2.00000i | 2.61803 | |||||||||||||||||||||||||||||||||
| 212.2 | − | 1.61803i | −1.61803 | + | 0.618034i | −0.618034 | 1.61803i | 1.00000 | + | 2.61803i | 4.84481i | − | 2.23607i | 2.23607 | − | 2.00000i | 2.61803 | |||||||||||||||||||||||||||||||||||
| 212.3 | − | 0.618034i | 0.618034 | + | 1.61803i | 1.61803 | 0.618034i | 1.00000 | − | 0.381966i | − | 3.81154i | − | 2.23607i | −2.23607 | + | 2.00000i | 0.381966 | ||||||||||||||||||||||||||||||||||
| 212.4 | − | 0.618034i | 0.618034 | + | 1.61803i | 1.61803 | 0.618034i | 1.00000 | − | 0.381966i | 3.81154i | − | 2.23607i | −2.23607 | + | 2.00000i | 0.381966 | |||||||||||||||||||||||||||||||||||
| 212.5 | 0.618034i | 0.618034 | − | 1.61803i | 1.61803 | − | 0.618034i | 1.00000 | + | 0.381966i | − | 3.81154i | 2.23607i | −2.23607 | − | 2.00000i | 0.381966 | |||||||||||||||||||||||||||||||||||
| 212.6 | 0.618034i | 0.618034 | − | 1.61803i | 1.61803 | − | 0.618034i | 1.00000 | + | 0.381966i | 3.81154i | 2.23607i | −2.23607 | − | 2.00000i | 0.381966 | ||||||||||||||||||||||||||||||||||||
| 212.7 | 1.61803i | −1.61803 | − | 0.618034i | −0.618034 | − | 1.61803i | 1.00000 | − | 2.61803i | − | 4.84481i | 2.23607i | 2.23607 | + | 2.00000i | 2.61803 | |||||||||||||||||||||||||||||||||||
| 212.8 | 1.61803i | −1.61803 | − | 0.618034i | −0.618034 | − | 1.61803i | 1.00000 | − | 2.61803i | 4.84481i | 2.23607i | 2.23607 | + | 2.00000i | 2.61803 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 71.b | odd | 2 | 1 | inner |
| 213.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 213.2.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 213.2.b.a | ✓ | 8 |
| 71.b | odd | 2 | 1 | inner | 213.2.b.a | ✓ | 8 |
| 213.b | even | 2 | 1 | inner | 213.2.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 213.2.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 213.2.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 213.2.b.a | ✓ | 8 | 71.b | odd | 2 | 1 | inner |
| 213.2.b.a | ✓ | 8 | 213.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(213, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $7$ |
\( (T^{4} + 38 T^{2} + 341)^{2} \)
|
| $11$ |
\( (T^{4} - 38 T^{2} + 341)^{2} \)
|
| $13$ |
\( (T^{4} + 47 T^{2} + 341)^{2} \)
|
| $17$ |
\( (T^{4} - 38 T^{2} + 341)^{2} \)
|
| $19$ |
\( (T^{2} - 5)^{4} \)
|
| $23$ |
\( (T^{4} - 47 T^{2} + 341)^{2} \)
|
| $29$ |
\( (T^{4} + 47 T^{2} + 1)^{2} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} + 9 T + 19)^{4} \)
|
| $41$ |
\( (T^{4} - 103 T^{2} + 341)^{2} \)
|
| $43$ |
\( (T^{2} - 3 T - 29)^{4} \)
|
| $47$ |
\( (T^{4} - 103 T^{2} + 341)^{2} \)
|
| $53$ |
\( (T^{4} - 67 T^{2} + 341)^{2} \)
|
| $59$ |
\( (T^{4} - 190 T^{2} + 8525)^{2} \)
|
| $61$ |
\( (T^{4} + 190 T^{2} + 8525)^{2} \)
|
| $67$ |
\( (T^{4} + 163 T^{2} + 341)^{2} \)
|
| $71$ |
\( T^{8} + 16 T^{6} + \cdots + 25411681 \)
|
| $73$ |
\( (T^{2} + 2 T - 124)^{4} \)
|
| $79$ |
\( (T^{2} - 15 T + 45)^{4} \)
|
| $83$ |
\( (T^{4} + 82 T^{2} + 961)^{2} \)
|
| $89$ |
\( (T^{4} + 402 T^{2} + 36481)^{2} \)
|
| $97$ |
\( (T^{4} + 103 T^{2} + 341)^{2} \)
|
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