Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,4,Mod(13,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("68.13");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 68.e (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: | \(4.01212988039\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.27793984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{5} + \nu^{4} + 49\nu^{3} - 7\nu^{2} + 342\nu - 98 ) / 171 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 342 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -41\nu^{5} - 14\nu^{4} - 2\nu^{3} + 98\nu^{2} - 1995\nu + 4 ) / 57 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -122\nu^{5} + 7\nu^{4} + \nu^{3} + 293\nu^{2} - 5985\nu + 1708 ) / 171 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 10\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 7\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} - 7\beta_{4} - \beta_{2} + \beta _1 - 70 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{4} - 24\beta_{3} - 49\beta _1 + 24 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0 | −2.58966 | − | 2.58966i | 0 | 8.91820 | + | 8.91820i | 0 | 12.7365 | − | 12.7365i | 0 | − | 13.5873i | 0 | |||||||||||||||||||||||||||||
| 13.2 | 0 | 0.712590 | + | 0.712590i | 0 | −13.5887 | − | 13.5887i | 0 | 12.5335 | − | 12.5335i | 0 | − | 25.9844i | 0 | ||||||||||||||||||||||||||||||
| 13.3 | 0 | 4.87707 | + | 4.87707i | 0 | 4.67048 | + | 4.67048i | 0 | −3.27000 | + | 3.27000i | 0 | 20.5717i | 0 | |||||||||||||||||||||||||||||||
| 21.1 | 0 | −2.58966 | + | 2.58966i | 0 | 8.91820 | − | 8.91820i | 0 | 12.7365 | + | 12.7365i | 0 | 13.5873i | 0 | |||||||||||||||||||||||||||||||
| 21.2 | 0 | 0.712590 | − | 0.712590i | 0 | −13.5887 | + | 13.5887i | 0 | 12.5335 | + | 12.5335i | 0 | 25.9844i | 0 | |||||||||||||||||||||||||||||||
| 21.3 | 0 | 4.87707 | − | 4.87707i | 0 | 4.67048 | − | 4.67048i | 0 | −3.27000 | − | 3.27000i | 0 | − | 20.5717i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 68.4.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 612.4.k.b | 6 | ||
| 4.b | odd | 2 | 1 | 272.4.o.c | 6 | ||
| 17.c | even | 4 | 1 | inner | 68.4.e.b | ✓ | 6 |
| 17.d | even | 8 | 2 | 1156.4.a.i | 6 | ||
| 17.d | even | 8 | 2 | 1156.4.b.f | 6 | ||
| 51.f | odd | 4 | 1 | 612.4.k.b | 6 | ||
| 68.f | odd | 4 | 1 | 272.4.o.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.4.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 68.4.e.b | ✓ | 6 | 17.c | even | 4 | 1 | inner |
| 272.4.o.c | 6 | 4.b | odd | 2 | 1 | ||
| 272.4.o.c | 6 | 68.f | odd | 4 | 1 | ||
| 612.4.k.b | 6 | 3.b | odd | 2 | 1 | ||
| 612.4.k.b | 6 | 51.f | odd | 4 | 1 | ||
| 1156.4.a.i | 6 | 17.d | even | 8 | 2 | ||
| 1156.4.b.f | 6 | 17.d | even | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 6T_{3}^{5} + 18T_{3}^{4} + 96T_{3}^{3} + 484T_{3}^{2} - 792T_{3} + 648 \)
acting on \(S_{4}^{\mathrm{new}}(68, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 6 T^{5} + \cdots + 648 \)
|
| $5$ |
\( T^{6} - 2264 T^{3} + \cdots + 2562848 \)
|
| $7$ |
\( T^{6} - 44 T^{5} + \cdots + 2179872 \)
|
| $11$ |
\( T^{6} - 58 T^{5} + \cdots + 143888648 \)
|
| $13$ |
\( (T^{3} - 46 T^{2} + \cdots + 3896)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 118587876497 \)
|
| $19$ |
\( T^{6} + \cdots + 37636000000 \)
|
| $23$ |
\( T^{6} + 32 T^{5} + \cdots + 32772608 \)
|
| $29$ |
\( T^{6} + \cdots + 56691570507552 \)
|
| $31$ |
\( T^{6} + \cdots + 276333674528 \)
|
| $37$ |
\( T^{6} + \cdots + 294101549568 \)
|
| $41$ |
\( T^{6} + \cdots + 332174590800392 \)
|
| $43$ |
\( T^{6} + \cdots + 295872199276096 \)
|
| $47$ |
\( (T^{3} - 664 T^{2} + \cdots + 42870784)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 25883510957056 \)
|
| $59$ |
\( T^{6} + \cdots + 50\!\cdots\!44 \)
|
| $61$ |
\( T^{6} + \cdots + 105473055616128 \)
|
| $67$ |
\( (T^{3} - 350 T^{2} + \cdots + 63674784)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!68 \)
|
| $73$ |
\( T^{6} + \cdots + 61\!\cdots\!28 \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!68 \)
|
| $83$ |
\( T^{6} + \cdots + 141781602355264 \)
|
| $89$ |
\( (T^{3} - 1620 T^{2} + \cdots + 31482432)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 41\!\cdots\!68 \)
|
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