Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,2,Mod(71,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([6, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(2.79476407074\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\Q(\zeta_{15})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{15}^{2} + \zeta_{15} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{15}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{15}^{5} + \zeta_{15} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{15}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{15}^{7} - \zeta_{15}^{2} \)
|
| \(\beta_{6}\) | \(=\) |
\( \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15} \)
|
| \(\beta_{7}\) | \(=\) |
\( \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{5} + \zeta_{15}^{4} + 2\zeta_{15} - 1 \)
|
| \(\zeta_{15}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 5 \)
|
| \(\zeta_{15}^{2}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 4\beta _1 - 1 ) / 5 \)
|
| \(\zeta_{15}^{3}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{15}^{4}\) | \(=\) |
\( ( \beta_{7} - 4\beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 5 \)
|
| \(\zeta_{15}^{5}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} - \beta _1 - 1 ) / 5 \)
|
| \(\zeta_{15}^{6}\) | \(=\) |
\( \beta_{4} \)
|
| \(\zeta_{15}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - 4\beta_{5} - 2\beta_{4} + \beta_{3} - 4\beta _1 + 1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 71.1 |
|
0.309017 | − | 0.951057i | −0.169131 | − | 0.122881i | −0.809017 | − | 0.587785i | 1.11803 | − | 1.93649i | −0.169131 | + | 0.122881i | 1.00000 | −0.809017 | + | 0.587785i | −0.913545 | − | 2.81160i | −1.49622 | − | 1.66172i | ||||||||||||||||||||||||||
| 71.2 | 0.309017 | − | 0.951057i | 1.47815 | + | 1.07394i | −0.809017 | − | 0.587785i | 1.11803 | + | 1.93649i | 1.47815 | − | 1.07394i | 1.00000 | −0.809017 | + | 0.587785i | 0.104528 | + | 0.321706i | 2.18720 | − | 0.464905i | |||||||||||||||||||||||||||
| 141.1 | −0.809017 | − | 0.587785i | −0.413545 | − | 1.27276i | 0.309017 | + | 0.951057i | −1.11803 | + | 1.93649i | −0.413545 | + | 1.27276i | 1.00000 | 0.309017 | − | 0.951057i | 0.978148 | − | 0.710666i | 2.04275 | − | 0.909491i | |||||||||||||||||||||||||||
| 141.2 | −0.809017 | − | 0.587785i | 0.604528 | + | 1.86055i | 0.309017 | + | 0.951057i | −1.11803 | − | 1.93649i | 0.604528 | − | 1.86055i | 1.00000 | 0.309017 | − | 0.951057i | −0.669131 | + | 0.486152i | −0.233733 | + | 2.22382i | |||||||||||||||||||||||||||
| 211.1 | −0.809017 | + | 0.587785i | −0.413545 | + | 1.27276i | 0.309017 | − | 0.951057i | −1.11803 | − | 1.93649i | −0.413545 | − | 1.27276i | 1.00000 | 0.309017 | + | 0.951057i | 0.978148 | + | 0.710666i | 2.04275 | + | 0.909491i | |||||||||||||||||||||||||||
| 211.2 | −0.809017 | + | 0.587785i | 0.604528 | − | 1.86055i | 0.309017 | − | 0.951057i | −1.11803 | + | 1.93649i | 0.604528 | + | 1.86055i | 1.00000 | 0.309017 | + | 0.951057i | −0.669131 | − | 0.486152i | −0.233733 | − | 2.22382i | |||||||||||||||||||||||||||
| 281.1 | 0.309017 | + | 0.951057i | −0.169131 | + | 0.122881i | −0.809017 | + | 0.587785i | 1.11803 | + | 1.93649i | −0.169131 | − | 0.122881i | 1.00000 | −0.809017 | − | 0.587785i | −0.913545 | + | 2.81160i | −1.49622 | + | 1.66172i | |||||||||||||||||||||||||||
| 281.2 | 0.309017 | + | 0.951057i | 1.47815 | − | 1.07394i | −0.809017 | + | 0.587785i | 1.11803 | − | 1.93649i | 1.47815 | + | 1.07394i | 1.00000 | −0.809017 | − | 0.587785i | 0.104528 | − | 0.321706i | 2.18720 | + | 0.464905i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.2.h.a | ✓ | 8 |
| 25.d | even | 5 | 1 | inner | 350.2.h.a | ✓ | 8 |
| 25.d | even | 5 | 1 | 8750.2.a.j | 4 | ||
| 25.e | even | 10 | 1 | 8750.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.2.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 350.2.h.a | ✓ | 8 | 25.d | even | 5 | 1 | inner |
| 8750.2.a.e | 4 | 25.e | even | 10 | 1 | ||
| 8750.2.a.j | 4 | 25.d | even | 5 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3T_{3}^{7} + 8T_{3}^{6} - 11T_{3}^{5} + 15T_{3}^{4} - 11T_{3}^{3} + 18T_{3}^{2} + 7T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - 3 T^{7} + \cdots + 1 \)
|
| $5$ |
\( (T^{4} + 5 T^{2} + 25)^{2} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 841 \)
|
| $13$ |
\( T^{8} - 11 T^{7} + \cdots + 841 \)
|
| $17$ |
\( T^{8} - 14 T^{7} + \cdots + 3721 \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots + 841 \)
|
| $23$ |
\( T^{8} - 15 T^{7} + \cdots + 25 \)
|
| $29$ |
\( T^{8} + 8 T^{7} + \cdots + 1 \)
|
| $31$ |
\( T^{8} + 7 T^{7} + \cdots + 961 \)
|
| $37$ |
\( T^{8} - 14 T^{7} + \cdots + 961 \)
|
| $41$ |
\( T^{8} - 3 T^{7} + \cdots + 4626801 \)
|
| $43$ |
\( (T^{4} + 3 T^{3} + \cdots + 241)^{2} \)
|
| $47$ |
\( T^{8} - 2 T^{7} + \cdots + 11095561 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 22801 \)
|
| $59$ |
\( T^{8} - 11 T^{7} + \cdots + 841 \)
|
| $61$ |
\( T^{8} + 125 T^{6} + \cdots + 15625 \)
|
| $67$ |
\( T^{8} - 14 T^{7} + \cdots + 3481 \)
|
| $71$ |
\( T^{8} - 23 T^{7} + \cdots + 29149201 \)
|
| $73$ |
\( T^{8} + 5 T^{7} + \cdots + 5736025 \)
|
| $79$ |
\( T^{8} - 2 T^{7} + \cdots + 130321 \)
|
| $83$ |
\( T^{8} + 27 T^{7} + \cdots + 11895601 \)
|
| $89$ |
\( T^{8} + 15 T^{7} + \cdots + 157628025 \)
|
| $97$ |
\( T^{8} - 40 T^{7} + \cdots + 49632025 \)
|
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