Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,2,Mod(139,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.139");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 414.e (of order \(3\), degree \(2\), 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: | \(3.30580664368\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 10.0.288778218147.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 658 \nu^{9} - 2394 \nu^{8} - 4352 \nu^{7} - 10326 \nu^{6} - 25351 \nu^{5} - 51907 \nu^{4} + \cdots - 98232 ) / 72795 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 999 \nu^{9} - 537 \nu^{8} - 4321 \nu^{7} - 4688 \nu^{6} - 34543 \nu^{5} - 20431 \nu^{4} + \cdots - 9081 ) / 72795 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 339 \nu^{9} - 1348 \nu^{8} + 4381 \nu^{7} - 7882 \nu^{6} + 19883 \nu^{5} - 36059 \nu^{4} + \cdots - 29709 ) / 24265 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1348 \nu^{9} + 3356 \nu^{8} - 10907 \nu^{7} + 16239 \nu^{6} - 49501 \nu^{5} + 89773 \nu^{4} + \cdots + 245043 ) / 72795 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2533 \nu^{9} - 1626 \nu^{8} + 17417 \nu^{7} + 3021 \nu^{6} + 84646 \nu^{5} + 17167 \nu^{4} + \cdots + 59532 ) / 72795 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2694 \nu^{9} - 6203 \nu^{8} + 26226 \nu^{7} - 34722 \nu^{6} + 133958 \nu^{5} - 159864 \nu^{4} + \cdots - 139464 ) / 72795 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + \cdots - 54156 ) / 72795 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} + \cdots + 11514 ) / 14559 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 6042 \nu^{9} - 8994 \nu^{8} + 41363 \nu^{7} - 39341 \nu^{6} + 193324 \nu^{5} - 179927 \nu^{4} + \cdots - 134607 ) / 72795 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{8} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} - 2\beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{9} + 2\beta_{8} + 9\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 15 \beta_{9} - 2 \beta_{8} - 36 \beta_{7} - 5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} + \cdots - 36 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{9} - 8\beta_{8} - 11\beta_{6} - 8\beta_{5} + 11\beta_{4} + 29\beta_{3} - 29\beta_{2} + 14\beta_1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{9} - 8\beta_{8} - 7\beta_{6} - \beta_{5} - 15\beta_{4} + 8\beta_{3} - 7\beta_{2} + 8\beta _1 + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -3\beta_{9} + 65\beta_{8} - 43\beta_{6} + 86\beta_{5} - 89\beta_{4} + 43\beta_{3} + 47\beta_{2} - 110\beta_1 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 198 \beta_{9} + 86 \beta_{8} + 666 \beta_{7} + 200 \beta_{6} - 226 \beta_{5} + 112 \beta_{4} + \cdots - 2 \beta_1 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( 47\beta_{9} - 58\beta_{8} + 126\beta_{6} - 105\beta_{5} + 68\beta_{4} - 268\beta_{3} + 126\beta_{2} + 58\beta _1 - 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(235\) |
| \(\chi(n)\) | \(\beta_{7}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 139.1 |
|
0.500000 | + | 0.866025i | −1.61720 | − | 0.620220i | −0.500000 | + | 0.866025i | 0.536235 | − | 0.928786i | −0.271473 | − | 1.71064i | −0.121951 | − | 0.211225i | −1.00000 | 2.23065 | + | 2.00604i | 1.07247 | ||||||||||||||||||||||||||||||||||
| 139.2 | 0.500000 | + | 0.866025i | −0.958787 | + | 1.44247i | −0.500000 | + | 0.866025i | 0.217761 | − | 0.377174i | −1.72861 | − | 0.109097i | 2.31474 | + | 4.00925i | −1.00000 | −1.16146 | − | 2.76605i | 0.435523 | |||||||||||||||||||||||||||||||||||
| 139.3 | 0.500000 | + | 0.866025i | −0.376855 | + | 1.69056i | −0.500000 | + | 0.866025i | −0.990153 | + | 1.71499i | −1.65249 | + | 0.518912i | 0.245502 | + | 0.425221i | −1.00000 | −2.71596 | − | 1.27419i | −1.98031 | |||||||||||||||||||||||||||||||||||
| 139.4 | 0.500000 | + | 0.866025i | −0.278072 | − | 1.70958i | −0.500000 | + | 0.866025i | −1.69714 | + | 2.93953i | 1.34151 | − | 1.09561i | −1.74607 | − | 3.02428i | −1.00000 | −2.84535 | + | 0.950775i | −3.39428 | |||||||||||||||||||||||||||||||||||
| 139.5 | 0.500000 | + | 0.866025i | 1.73091 | + | 0.0627999i | −0.500000 | + | 0.866025i | 1.43330 | − | 2.48254i | 0.811070 | + | 1.53041i | 1.80778 | + | 3.13117i | −1.00000 | 2.99211 | + | 0.217402i | 2.86660 | |||||||||||||||||||||||||||||||||||
| 277.1 | 0.500000 | − | 0.866025i | −1.61720 | + | 0.620220i | −0.500000 | − | 0.866025i | 0.536235 | + | 0.928786i | −0.271473 | + | 1.71064i | −0.121951 | + | 0.211225i | −1.00000 | 2.23065 | − | 2.00604i | 1.07247 | |||||||||||||||||||||||||||||||||||
| 277.2 | 0.500000 | − | 0.866025i | −0.958787 | − | 1.44247i | −0.500000 | − | 0.866025i | 0.217761 | + | 0.377174i | −1.72861 | + | 0.109097i | 2.31474 | − | 4.00925i | −1.00000 | −1.16146 | + | 2.76605i | 0.435523 | |||||||||||||||||||||||||||||||||||
| 277.3 | 0.500000 | − | 0.866025i | −0.376855 | − | 1.69056i | −0.500000 | − | 0.866025i | −0.990153 | − | 1.71499i | −1.65249 | − | 0.518912i | 0.245502 | − | 0.425221i | −1.00000 | −2.71596 | + | 1.27419i | −1.98031 | |||||||||||||||||||||||||||||||||||
| 277.4 | 0.500000 | − | 0.866025i | −0.278072 | + | 1.70958i | −0.500000 | − | 0.866025i | −1.69714 | − | 2.93953i | 1.34151 | + | 1.09561i | −1.74607 | + | 3.02428i | −1.00000 | −2.84535 | − | 0.950775i | −3.39428 | |||||||||||||||||||||||||||||||||||
| 277.5 | 0.500000 | − | 0.866025i | 1.73091 | − | 0.0627999i | −0.500000 | − | 0.866025i | 1.43330 | + | 2.48254i | 0.811070 | − | 1.53041i | 1.80778 | − | 3.13117i | −1.00000 | 2.99211 | − | 0.217402i | 2.86660 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 414.2.e.d | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1242.2.e.b | 10 | ||
| 9.c | even | 3 | 1 | inner | 414.2.e.d | ✓ | 10 |
| 9.c | even | 3 | 1 | 3726.2.a.r | 5 | ||
| 9.d | odd | 6 | 1 | 1242.2.e.b | 10 | ||
| 9.d | odd | 6 | 1 | 3726.2.a.u | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 414.2.e.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 414.2.e.d | ✓ | 10 | 9.c | even | 3 | 1 | inner |
| 1242.2.e.b | 10 | 3.b | odd | 2 | 1 | ||
| 1242.2.e.b | 10 | 9.d | odd | 6 | 1 | ||
| 3726.2.a.r | 5 | 9.c | even | 3 | 1 | ||
| 3726.2.a.u | 5 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + T_{5}^{9} + 13T_{5}^{8} - 2T_{5}^{7} + 124T_{5}^{6} + T_{5}^{5} + 334T_{5}^{4} - 341T_{5}^{3} + 580T_{5}^{2} - 225T_{5} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{5} \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots + 243 \)
|
| $5$ |
\( T^{10} + T^{9} + \cdots + 81 \)
|
| $7$ |
\( T^{10} - 5 T^{9} + \cdots + 49 \)
|
| $11$ |
\( T^{10} + 11 T^{9} + \cdots + 84681 \)
|
| $13$ |
\( T^{10} - 6 T^{9} + \cdots + 1234321 \)
|
| $17$ |
\( (T^{5} - T^{4} - 75 T^{3} + \cdots - 2037)^{2} \)
|
| $19$ |
\( (T^{5} + 3 T^{4} + \cdots - 857)^{2} \)
|
| $23$ |
\( (T^{2} + T + 1)^{5} \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots + 8590761 \)
|
| $31$ |
\( T^{10} - 4 T^{9} + \cdots + 3207681 \)
|
| $37$ |
\( (T^{5} + 14 T^{4} + \cdots + 19189)^{2} \)
|
| $41$ |
\( T^{10} + 24 T^{9} + \cdots + 38900169 \)
|
| $43$ |
\( T^{10} - 27 T^{9} + \cdots + 458329 \)
|
| $47$ |
\( T^{10} + \cdots + 1378265625 \)
|
| $53$ |
\( (T^{5} + 13 T^{4} + \cdots + 213)^{2} \)
|
| $59$ |
\( T^{10} + 9 T^{9} + \cdots + 26594649 \)
|
| $61$ |
\( T^{10} - 3 T^{9} + \cdots + 3374569 \)
|
| $67$ |
\( T^{10} - 5 T^{9} + \cdots + 49 \)
|
| $71$ |
\( (T^{5} - 27 T^{4} + \cdots + 21357)^{2} \)
|
| $73$ |
\( (T^{5} - 17 T^{4} + \cdots - 297)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 119968209 \)
|
| $83$ |
\( T^{10} + \cdots + 107557641 \)
|
| $89$ |
\( (T^{5} - 39 T^{4} + \cdots + 62667)^{2} \)
|
| $97$ |
\( T^{10} - 28 T^{9} + \cdots + 20007729 \)
|
| show more | |
| show less |