Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,6,Mod(67,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.67");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.b (of order \(2\), degree \(1\), 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: | \(16.3591496209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 469x^{4} + 57076x^{2} + 1345600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 469x^{4} + 57076x^{2} + 1345600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 691\nu^{3} - 214140\nu ) / 214832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9\nu^{5} - 6219\nu^{3} - 1939716\nu ) / 1074160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{4} - 3981\nu^{2} - 163240 ) / 1852 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{4} + 3483\nu^{2} + 438700 ) / 1852 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 501\nu^{5} + 190889\nu^{3} + 15013796\nu ) / 1074160 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{2} - 9\beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_{4} + 3\beta_{3} - 2815 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 36\beta_{5} - 859\beta_{2} + 2061\beta_1 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -1327\beta_{4} - 1161\beta_{3} + 212005 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 24876\beta_{5} + 477131\beta_{2} - 515565\beta_1 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
4.00000 | − | 9.00000i | 16.0000 | − | 70.5281i | − | 36.0000i | − | 150.350i | 64.0000 | −81.0000 | − | 282.112i | |||||||||||||||||||||||||||||||
| 67.2 | 4.00000 | − | 9.00000i | 16.0000 | − | 44.2038i | − | 36.0000i | 44.7980i | 64.0000 | −81.0000 | − | 176.815i | |||||||||||||||||||||||||||||||||
| 67.3 | 4.00000 | − | 9.00000i | 16.0000 | 20.7319i | − | 36.0000i | 187.552i | 64.0000 | −81.0000 | 82.9276i | |||||||||||||||||||||||||||||||||||
| 67.4 | 4.00000 | 9.00000i | 16.0000 | − | 20.7319i | 36.0000i | − | 187.552i | 64.0000 | −81.0000 | − | 82.9276i | ||||||||||||||||||||||||||||||||||
| 67.5 | 4.00000 | 9.00000i | 16.0000 | 44.2038i | 36.0000i | − | 44.7980i | 64.0000 | −81.0000 | 176.815i | ||||||||||||||||||||||||||||||||||||
| 67.6 | 4.00000 | 9.00000i | 16.0000 | 70.5281i | 36.0000i | 150.350i | 64.0000 | −81.0000 | 282.112i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 102.6.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 306.6.b.c | 6 | ||
| 4.b | odd | 2 | 1 | 816.6.c.a | 6 | ||
| 17.b | even | 2 | 1 | inner | 102.6.b.a | ✓ | 6 |
| 51.c | odd | 2 | 1 | 306.6.b.c | 6 | ||
| 68.d | odd | 2 | 1 | 816.6.c.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.6.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 102.6.b.a | ✓ | 6 | 17.b | even | 2 | 1 | inner |
| 306.6.b.c | 6 | 3.b | odd | 2 | 1 | ||
| 306.6.b.c | 6 | 51.c | odd | 2 | 1 | ||
| 816.6.c.a | 6 | 4.b | odd | 2 | 1 | ||
| 816.6.c.a | 6 | 68.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 7358T_{5}^{4} + 12697313T_{5}^{2} + 4177553956 \)
acting on \(S_{6}^{\mathrm{new}}(102, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{6} \)
|
| $3$ |
\( (T^{2} + 81)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 4177553956 \)
|
| $7$ |
\( T^{6} + \cdots + 1595775297600 \)
|
| $11$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + 764 T^{2} + \cdots - 48551250)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!93 \)
|
| $19$ |
\( (T^{3} - 148 T^{2} + \cdots + 843244200)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 13\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots + 60\!\cdots\!24 \)
|
| $31$ |
\( T^{6} + \cdots + 60\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 36\!\cdots\!36 \)
|
| $43$ |
\( (T^{3} + 3592 T^{2} + \cdots - 542322204628)^{2} \)
|
| $47$ |
\( (T^{3} - 36380 T^{2} + \cdots - 871319502432)^{2} \)
|
| $53$ |
\( (T^{3} + \cdots - 13472766209592)^{2} \)
|
| $59$ |
\( (T^{3} + \cdots + 15151900621824)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 48\!\cdots\!04 \)
|
| $67$ |
\( (T^{3} + \cdots - 2740946401280)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 41\!\cdots\!44 \)
|
| $73$ |
\( T^{6} + \cdots + 25\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots + 59\!\cdots\!36 \)
|
| $83$ |
\( (T^{3} + \cdots + 37819260025056)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots + 935459124485832)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 25\!\cdots\!24 \)
|
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