Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,2,Mod(101,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 310.h (of order \(5\), degree \(4\), 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: | \(2.47536246266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.484000000.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 10\beta_{3} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( -38\beta_{7} - 75\beta_{6} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(251\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
0.309017 | − | 0.951057i | −0.771681 | − | 2.37499i | −0.809017 | − | 0.587785i | −1.00000 | −2.49721 | −0.271681 | − | 0.197388i | −0.809017 | + | 0.587785i | −2.61803 | + | 1.90211i | −0.309017 | + | 0.951057i | ||||||||||||||||||||||||||||
| 101.2 | 0.309017 | − | 0.951057i | 0.771681 | + | 2.37499i | −0.809017 | − | 0.587785i | −1.00000 | 2.49721 | 1.27168 | + | 0.923930i | −0.809017 | + | 0.587785i | −2.61803 | + | 1.90211i | −0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||
| 171.1 | −0.809017 | + | 0.587785i | −1.07448 | − | 0.780656i | 0.309017 | − | 0.951057i | −1.00000 | 1.32813 | −0.574481 | + | 1.76807i | 0.309017 | + | 0.951057i | −0.381966 | − | 1.17557i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||
| 171.2 | −0.809017 | + | 0.587785i | 1.07448 | + | 0.780656i | 0.309017 | − | 0.951057i | −1.00000 | −1.32813 | 1.57448 | − | 4.84575i | 0.309017 | + | 0.951057i | −0.381966 | − | 1.17557i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||
| 221.1 | 0.309017 | + | 0.951057i | −0.771681 | + | 2.37499i | −0.809017 | + | 0.587785i | −1.00000 | −2.49721 | −0.271681 | + | 0.197388i | −0.809017 | − | 0.587785i | −2.61803 | − | 1.90211i | −0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||
| 221.2 | 0.309017 | + | 0.951057i | 0.771681 | − | 2.37499i | −0.809017 | + | 0.587785i | −1.00000 | 2.49721 | 1.27168 | − | 0.923930i | −0.809017 | − | 0.587785i | −2.61803 | − | 1.90211i | −0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||
| 281.1 | −0.809017 | − | 0.587785i | −1.07448 | + | 0.780656i | 0.309017 | + | 0.951057i | −1.00000 | 1.32813 | −0.574481 | − | 1.76807i | 0.309017 | − | 0.951057i | −0.381966 | + | 1.17557i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||
| 281.2 | −0.809017 | − | 0.587785i | 1.07448 | − | 0.780656i | 0.309017 | + | 0.951057i | −1.00000 | −1.32813 | 1.57448 | + | 4.84575i | 0.309017 | − | 0.951057i | −0.381966 | + | 1.17557i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 310.2.h.c | ✓ | 8 |
| 31.d | even | 5 | 1 | inner | 310.2.h.c | ✓ | 8 |
| 31.d | even | 5 | 1 | 9610.2.a.bp | 4 | ||
| 31.f | odd | 10 | 1 | 9610.2.a.bo | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.2.h.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 310.2.h.c | ✓ | 8 | 31.d | even | 5 | 1 | inner |
| 9610.2.a.bo | 4 | 31.f | odd | 10 | 1 | ||
| 9610.2.a.bp | 4 | 31.d | even | 5 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 9T_{3}^{6} + 31T_{3}^{4} - 11T_{3}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(310, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + 9 T^{6} + \cdots + 121 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 25 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 21025 \)
|
| $13$ |
\( T^{8} - 14 T^{7} + \cdots + 3025 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 21025 \)
|
| $19$ |
\( (T^{4} - 4 T^{3} + 16 T^{2} + \cdots + 16)^{2} \)
|
| $23$ |
\( T^{8} + 10 T^{7} + \cdots + 212521 \)
|
| $29$ |
\( T^{8} + 12 T^{7} + \cdots + 121 \)
|
| $31$ |
\( T^{8} + 10 T^{7} + \cdots + 923521 \)
|
| $37$ |
\( (T^{4} - 10 T^{3} + 22 T^{2} + \cdots + 1)^{2} \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 5755201 \)
|
| $43$ |
\( T^{8} - 8 T^{7} + \cdots + 23814400 \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 15625 \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 6400 \)
|
| $59$ |
\( (T^{4} + 18 T^{3} + \cdots + 1296)^{2} \)
|
| $61$ |
\( (T^{4} + 2 T^{3} + \cdots + 275)^{2} \)
|
| $67$ |
\( (T^{4} - 24 T^{3} + \cdots - 3461)^{2} \)
|
| $71$ |
\( T^{8} - 10 T^{7} + \cdots + 49350625 \)
|
| $73$ |
\( T^{8} - 28 T^{7} + \cdots + 4946176 \)
|
| $79$ |
\( T^{8} + 4 T^{7} + \cdots + 19321 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 17161 \)
|
| $89$ |
\( T^{8} - 4 T^{7} + \cdots + 256 \)
|
| $97$ |
\( T^{8} - 12 T^{7} + \cdots + 5382400 \)
|
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