Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [34,2,Mod(9,34)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("34.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 34 = 2 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 34.d (of order \(8\), 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: | \(0.271491366872\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
−0.707107 | + | 0.707107i | 0.707107 | + | 0.292893i | − | 1.00000i | −0.585786 | + | 1.41421i | −0.707107 | + | 0.292893i | −1.41421 | − | 3.41421i | 0.707107 | + | 0.707107i | −1.70711 | − | 1.70711i | −0.585786 | − | 1.41421i | |||||||||||||
| 15.1 | 0.707107 | + | 0.707107i | −0.707107 | − | 1.70711i | 1.00000i | −3.41421 | + | 1.41421i | 0.707107 | − | 1.70711i | 1.41421 | + | 0.585786i | −0.707107 | + | 0.707107i | −0.292893 | + | 0.292893i | −3.41421 | − | 1.41421i | |||||||||||||||
| 19.1 | −0.707107 | − | 0.707107i | 0.707107 | − | 0.292893i | 1.00000i | −0.585786 | − | 1.41421i | −0.707107 | − | 0.292893i | −1.41421 | + | 3.41421i | 0.707107 | − | 0.707107i | −1.70711 | + | 1.70711i | −0.585786 | + | 1.41421i | |||||||||||||||
| 25.1 | 0.707107 | − | 0.707107i | −0.707107 | + | 1.70711i | − | 1.00000i | −3.41421 | − | 1.41421i | 0.707107 | + | 1.70711i | 1.41421 | − | 0.585786i | −0.707107 | − | 0.707107i | −0.292893 | − | 0.292893i | −3.41421 | + | 1.41421i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.d | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 34.2.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 306.2.l.c | 4 | ||
| 4.b | odd | 2 | 1 | 272.2.v.b | 4 | ||
| 5.b | even | 2 | 1 | 850.2.l.a | 4 | ||
| 5.c | odd | 4 | 1 | 850.2.o.a | 4 | ||
| 5.c | odd | 4 | 1 | 850.2.o.b | 4 | ||
| 17.b | even | 2 | 1 | 578.2.d.b | 4 | ||
| 17.c | even | 4 | 1 | 578.2.d.a | 4 | ||
| 17.c | even | 4 | 1 | 578.2.d.c | 4 | ||
| 17.d | even | 8 | 1 | inner | 34.2.d.a | ✓ | 4 |
| 17.d | even | 8 | 1 | 578.2.d.a | 4 | ||
| 17.d | even | 8 | 1 | 578.2.d.b | 4 | ||
| 17.d | even | 8 | 1 | 578.2.d.c | 4 | ||
| 17.e | odd | 16 | 2 | 578.2.a.i | 4 | ||
| 17.e | odd | 16 | 2 | 578.2.b.d | 4 | ||
| 17.e | odd | 16 | 4 | 578.2.c.f | 8 | ||
| 51.g | odd | 8 | 1 | 306.2.l.c | 4 | ||
| 51.i | even | 16 | 2 | 5202.2.a.bw | 4 | ||
| 68.g | odd | 8 | 1 | 272.2.v.b | 4 | ||
| 68.i | even | 16 | 2 | 4624.2.a.bn | 4 | ||
| 85.k | odd | 8 | 1 | 850.2.o.a | 4 | ||
| 85.m | even | 8 | 1 | 850.2.l.a | 4 | ||
| 85.n | odd | 8 | 1 | 850.2.o.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.2.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 34.2.d.a | ✓ | 4 | 17.d | even | 8 | 1 | inner |
| 272.2.v.b | 4 | 4.b | odd | 2 | 1 | ||
| 272.2.v.b | 4 | 68.g | odd | 8 | 1 | ||
| 306.2.l.c | 4 | 3.b | odd | 2 | 1 | ||
| 306.2.l.c | 4 | 51.g | odd | 8 | 1 | ||
| 578.2.a.i | 4 | 17.e | odd | 16 | 2 | ||
| 578.2.b.d | 4 | 17.e | odd | 16 | 2 | ||
| 578.2.c.f | 8 | 17.e | odd | 16 | 4 | ||
| 578.2.d.a | 4 | 17.c | even | 4 | 1 | ||
| 578.2.d.a | 4 | 17.d | even | 8 | 1 | ||
| 578.2.d.b | 4 | 17.b | even | 2 | 1 | ||
| 578.2.d.b | 4 | 17.d | even | 8 | 1 | ||
| 578.2.d.c | 4 | 17.c | even | 4 | 1 | ||
| 578.2.d.c | 4 | 17.d | even | 8 | 1 | ||
| 850.2.l.a | 4 | 5.b | even | 2 | 1 | ||
| 850.2.l.a | 4 | 85.m | even | 8 | 1 | ||
| 850.2.o.a | 4 | 5.c | odd | 4 | 1 | ||
| 850.2.o.a | 4 | 85.k | odd | 8 | 1 | ||
| 850.2.o.b | 4 | 5.c | odd | 4 | 1 | ||
| 850.2.o.b | 4 | 85.n | odd | 8 | 1 | ||
| 4624.2.a.bn | 4 | 68.i | even | 16 | 2 | ||
| 5202.2.a.bw | 4 | 51.i | even | 16 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(34, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 1 \)
|
| $3$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 32 \)
|
| $7$ |
\( T^{4} + 8 T^{2} + \cdots + 32 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $13$ |
\( T^{4} + 24T^{2} + 16 \)
|
| $17$ |
\( T^{4} + 16T^{2} + 289 \)
|
| $19$ |
\( T^{4} - 8 T^{3} + \cdots + 16 \)
|
| $23$ |
\( T^{4} + 16 T^{3} + \cdots + 512 \)
|
| $29$ |
\( T^{4} + 8 T^{2} + \cdots + 32 \)
|
| $31$ |
\( T^{4} + 8 T^{2} + \cdots + 32 \)
|
| $37$ |
\( T^{4} + 8 T^{3} + \cdots + 128 \)
|
| $41$ |
\( T^{4} - 16 T^{3} + \cdots + 4802 \)
|
| $43$ |
\( T^{4} - 12 T^{3} + \cdots + 324 \)
|
| $47$ |
\( T^{4} + 96T^{2} + 256 \)
|
| $53$ |
\( T^{4} - 8 T^{3} + \cdots + 16 \)
|
| $59$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
|
| $61$ |
\( T^{4} - 16 T^{3} + \cdots + 512 \)
|
| $67$ |
\( (T^{2} + 12 T + 34)^{2} \)
|
| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 128 \)
|
| $73$ |
\( T^{4} + 98 T^{2} + \cdots + 4802 \)
|
| $79$ |
\( T^{4} + 8 T^{3} + \cdots + 128 \)
|
| $83$ |
\( T^{4} - 12 T^{3} + \cdots + 196 \)
|
| $89$ |
\( T^{4} + 228T^{2} + 196 \)
|
| $97$ |
\( T^{4} + 12 T^{3} + \cdots + 162 \)
|
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