Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(117,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("370.117");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 370.h (of order \(4\), 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: | \(2.95446487479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 3x^{8} - 8x^{7} - 26x^{6} + 12x^{5} + 24x^{4} + 166x^{3} + 113x^{2} - 152x + 160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{10} + 3x^{8} - 8x^{7} - 26x^{6} + 12x^{5} + 24x^{4} + 166x^{3} + 113x^{2} - 152x + 160 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 105 \nu^{9} + 3868 \nu^{8} - 17854 \nu^{7} + 34822 \nu^{6} - 123604 \nu^{5} + 73205 \nu^{4} + \cdots - 881880 ) / 887851 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4830421 \nu^{9} + 27279422 \nu^{8} + 39766871 \nu^{7} + 36230609 \nu^{6} - 248412067 \nu^{5} + \cdots - 1481360957 ) / 5161077863 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4830421 \nu^{9} - 27279422 \nu^{8} - 39766871 \nu^{7} - 36230609 \nu^{6} + 248412067 \nu^{5} + \cdots + 1481360957 ) / 5161077863 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12470333 \nu^{9} - 79772050 \nu^{8} - 195227485 \nu^{7} + 26574182 \nu^{6} + \cdots - 18775197608 ) / 10322155726 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 40939147 \nu^{9} + 281415368 \nu^{8} - 506483273 \nu^{7} + 1711086296 \nu^{6} + \cdots + 1358343716 ) / 20644311452 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 22047 \nu^{9} + 420 \nu^{8} + 81613 \nu^{7} - 247792 \nu^{6} - 433934 \nu^{5} - 229852 \nu^{4} + \cdots + 10748 ) / 3551404 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 78590417 \nu^{9} - 53270874 \nu^{8} - 432258509 \nu^{7} + 565223046 \nu^{6} + \cdots - 13883295960 ) / 10322155726 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 106312323 \nu^{9} - 59569470 \nu^{8} - 234376097 \nu^{7} + 621147016 \nu^{6} + \cdots + 27078850384 ) / 10322155726 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 151550159 \nu^{9} + 32728460 \nu^{8} - 412250275 \nu^{7} + 1324404682 \nu^{6} + \cdots + 25888295120 ) / 10322155726 \)
|
| \(\nu\) | \(=\) |
\( \beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{9} + 4\beta_{7} - 5\beta_{6} + \beta_{5} - 5\beta_{4} + 2\beta_{3} + \beta_{2} - 3\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - 7\beta_{8} + 2\beta_{7} - 5\beta_{6} + 3\beta_{5} + 4\beta_{4} + 3\beta_{3} + 4\beta_{2} - 11\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{9} + 12 \beta_{8} - 9 \beta_{7} + 16 \beta_{6} + 4 \beta_{5} + 17 \beta_{4} + 17 \beta_{3} + \cdots - 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 44 \beta_{9} + 54 \beta_{8} - 11 \beta_{7} - 41 \beta_{6} - 11 \beta_{5} - 30 \beta_{4} - 5 \beta_{3} + \cdots - 44 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 79 \beta_{9} - 72 \beta_{8} + 70 \beta_{7} - 221 \beta_{6} + 27 \beta_{5} - 123 \beta_{4} - 18 \beta_{3} + \cdots + 78 \)
|
| \(\nu^{8}\) | \(=\) |
\( 201 \beta_{9} - 271 \beta_{8} - 9 \beta_{7} + 144 \beta_{6} + 208 \beta_{5} + 307 \beta_{4} + 103 \beta_{3} + \cdots + 313 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 7 \beta_{9} + 854 \beta_{8} - 688 \beta_{7} + 777 \beta_{6} + 33 \beta_{5} + 675 \beta_{4} + \cdots - 942 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(297\) |
| \(\chi(n)\) | \(-\beta_{6}\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 117.1 |
|
1.00000 | −1.70160 | − | 1.70160i | 1.00000 | 1.73245 | − | 1.41373i | −1.70160 | − | 1.70160i | 2.82745 | + | 2.82745i | 1.00000 | 2.79087i | 1.73245 | − | 1.41373i | ||||||||||||||||||||||||||||||||||||||
| 117.2 | 1.00000 | −1.05122 | − | 1.05122i | 1.00000 | −2.15147 | + | 0.609231i | −1.05122 | − | 1.05122i | −1.21846 | − | 1.21846i | 1.00000 | − | 0.789858i | −2.15147 | + | 0.609231i | ||||||||||||||||||||||||||||||||||||||
| 117.3 | 1.00000 | −0.215231 | − | 0.215231i | 1.00000 | 2.21058 | + | 0.336630i | −0.215231 | − | 0.215231i | −0.673260 | − | 0.673260i | 1.00000 | − | 2.90735i | 2.21058 | + | 0.336630i | ||||||||||||||||||||||||||||||||||||||
| 117.4 | 1.00000 | 1.78347 | + | 1.78347i | 1.00000 | −2.22195 | − | 0.250846i | 1.78347 | + | 1.78347i | 0.501691 | + | 0.501691i | 1.00000 | 3.36152i | −2.22195 | − | 0.250846i | |||||||||||||||||||||||||||||||||||||||
| 117.5 | 1.00000 | 2.18458 | + | 2.18458i | 1.00000 | 1.43040 | + | 1.71871i | 2.18458 | + | 2.18458i | −3.43742 | − | 3.43742i | 1.00000 | 6.54482i | 1.43040 | + | 1.71871i | |||||||||||||||||||||||||||||||||||||||
| 253.1 | 1.00000 | −1.70160 | + | 1.70160i | 1.00000 | 1.73245 | + | 1.41373i | −1.70160 | + | 1.70160i | 2.82745 | − | 2.82745i | 1.00000 | − | 2.79087i | 1.73245 | + | 1.41373i | ||||||||||||||||||||||||||||||||||||||
| 253.2 | 1.00000 | −1.05122 | + | 1.05122i | 1.00000 | −2.15147 | − | 0.609231i | −1.05122 | + | 1.05122i | −1.21846 | + | 1.21846i | 1.00000 | 0.789858i | −2.15147 | − | 0.609231i | |||||||||||||||||||||||||||||||||||||||
| 253.3 | 1.00000 | −0.215231 | + | 0.215231i | 1.00000 | 2.21058 | − | 0.336630i | −0.215231 | + | 0.215231i | −0.673260 | + | 0.673260i | 1.00000 | 2.90735i | 2.21058 | − | 0.336630i | |||||||||||||||||||||||||||||||||||||||
| 253.4 | 1.00000 | 1.78347 | − | 1.78347i | 1.00000 | −2.22195 | + | 0.250846i | 1.78347 | − | 1.78347i | 0.501691 | − | 0.501691i | 1.00000 | − | 3.36152i | −2.22195 | + | 0.250846i | ||||||||||||||||||||||||||||||||||||||
| 253.5 | 1.00000 | 2.18458 | − | 2.18458i | 1.00000 | 1.43040 | − | 1.71871i | 2.18458 | − | 2.18458i | −3.43742 | + | 3.43742i | 1.00000 | − | 6.54482i | 1.43040 | − | 1.71871i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 185.k | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 370.2.h.d | yes | 10 |
| 5.c | odd | 4 | 1 | 370.2.g.d | ✓ | 10 | |
| 37.d | odd | 4 | 1 | 370.2.g.d | ✓ | 10 | |
| 185.k | even | 4 | 1 | inner | 370.2.h.d | yes | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.g.d | ✓ | 10 | 5.c | odd | 4 | 1 | |
| 370.2.g.d | ✓ | 10 | 37.d | odd | 4 | 1 | |
| 370.2.h.d | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 370.2.h.d | yes | 10 | 185.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 2T_{3}^{9} + 2T_{3}^{8} + 12T_{3}^{7} + 59T_{3}^{6} - 60T_{3}^{5} + 74T_{3}^{4} + 442T_{3}^{3} + 961T_{3}^{2} + 372T_{3} + 72 \)
acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} - 2 T^{9} + \cdots + 72 \)
|
| $5$ |
\( T^{10} - 2 T^{9} + \cdots + 3125 \)
|
| $7$ |
\( T^{10} + 4 T^{9} + \cdots + 512 \)
|
| $11$ |
\( T^{10} + 82 T^{8} + \cdots + 266256 \)
|
| $13$ |
\( (T^{5} + 6 T^{4} - T^{3} + \cdots - 4)^{2} \)
|
| $17$ |
\( T^{10} + 92 T^{8} + \cdots + 118336 \)
|
| $19$ |
\( T^{10} - 8 T^{9} + \cdots + 12800 \)
|
| $23$ |
\( (T^{5} - 2 T^{4} + \cdots - 500)^{2} \)
|
| $29$ |
\( T^{10} + 32 T^{9} + \cdots + 4222418 \)
|
| $31$ |
\( T^{10} + 26 T^{9} + \cdots + 10952 \)
|
| $37$ |
\( T^{10} + 2 T^{9} + \cdots + 69343957 \)
|
| $41$ |
\( T^{10} + 130 T^{8} + \cdots + 57600 \)
|
| $43$ |
\( (T^{5} - 6 T^{4} + \cdots - 592)^{2} \)
|
| $47$ |
\( T^{10} - 48 T^{9} + \cdots + 18678272 \)
|
| $53$ |
\( T^{10} + 2 T^{9} + \cdots + 4310048 \)
|
| $59$ |
\( T^{10} + \cdots + 1340377088 \)
|
| $61$ |
\( T^{10} + 24 T^{9} + \cdots + 4418 \)
|
| $67$ |
\( T^{10} + 10 T^{9} + \cdots + 18072072 \)
|
| $71$ |
\( (T^{5} + 8 T^{4} + \cdots - 576)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 369430562 \)
|
| $79$ |
\( T^{10} + 2 T^{9} + \cdots + 23970888 \)
|
| $83$ |
\( T^{10} - 8 T^{9} + \cdots + 46851200 \)
|
| $89$ |
\( T^{10} + 2 T^{9} + \cdots + 3591200 \)
|
| $97$ |
\( T^{10} + 48 T^{8} + \cdots + 2304 \)
|
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