Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.67");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(6.01819482059\) |
| 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: |
\( x^{6} - 2x^{5} + 70x^{4} - 2020x^{3} + 4166x^{2} - 70868x + 906797 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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
\( x^{6} - 2x^{5} + 70x^{4} - 2020x^{3} + 4166x^{2} - 70868x + 906797 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -508\nu^{5} + 948\nu^{4} - 54674\nu^{3} + 1288214\nu^{2} + 3712934\nu + 53304875 ) / 3665391 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -265\nu^{5} - 839\nu^{4} - 27997\nu^{3} + 247511\nu^{2} + 143985\nu + 19472725 ) / 1354864 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -40989\nu^{5} - 58195\nu^{4} - 4358569\nu^{3} + 61068603\nu^{2} - 292019155\nu + 3561912673 ) / 205261896 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -288\nu^{5} - 9083\nu^{4} + 103690\nu^{3} - 106653\nu^{2} + 12341128\nu - 106505732 ) / 1221797 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 282029 \nu^{5} - 2840851 \nu^{4} - 43692401 \nu^{3} + 215147203 \nu^{2} + 1507590261 \nu + 34919138729 ) / 205261896 \)
|
| \(\nu\) | \(=\) |
\( ( -6\beta_{3} + 4\beta_{2} + 3\beta _1 + 3 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 2\beta_{4} + 48\beta_{3} - 98\beta_{2} + 36\beta _1 - 284 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -51\beta_{5} + 78\beta_{4} + 246\beta_{3} + 62\beta_{2} - 69\beta _1 + 11319 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -325\beta_{5} - 202\beta_{4} - 2372\beta_{3} + 4550\beta_{2} + 562\beta _1 + 5274 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11277\beta_{5} - 4454\beta_{4} + 38112\beta_{3} - 200478\beta_{2} + 37206\beta _1 - 627778 ) / 12 \)
|
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 |
|
2.00000 | − | 3.00000i | 4.00000 | − | 16.7450i | − | 6.00000i | 13.5188i | 8.00000 | −9.00000 | − | 33.4901i | ||||||||||||||||||||||||||||||||
| 67.2 | 2.00000 | − | 3.00000i | 4.00000 | 0.380147i | − | 6.00000i | − | 33.8566i | 8.00000 | −9.00000 | 0.760294i | ||||||||||||||||||||||||||||||||||
| 67.3 | 2.00000 | − | 3.00000i | 4.00000 | 21.3649i | − | 6.00000i | 22.3377i | 8.00000 | −9.00000 | 42.7298i | |||||||||||||||||||||||||||||||||||
| 67.4 | 2.00000 | 3.00000i | 4.00000 | − | 21.3649i | 6.00000i | − | 22.3377i | 8.00000 | −9.00000 | − | 42.7298i | ||||||||||||||||||||||||||||||||||
| 67.5 | 2.00000 | 3.00000i | 4.00000 | − | 0.380147i | 6.00000i | 33.8566i | 8.00000 | −9.00000 | − | 0.760294i | |||||||||||||||||||||||||||||||||||
| 67.6 | 2.00000 | 3.00000i | 4.00000 | 16.7450i | 6.00000i | − | 13.5188i | 8.00000 | −9.00000 | 33.4901i | ||||||||||||||||||||||||||||||||||||
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 | 102.4.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 306.4.b.f | 6 | ||
| 4.b | odd | 2 | 1 | 816.4.c.e | 6 | ||
| 17.b | even | 2 | 1 | inner | 102.4.b.b | ✓ | 6 |
| 17.c | even | 4 | 1 | 1734.4.a.u | 3 | ||
| 17.c | even | 4 | 1 | 1734.4.a.v | 3 | ||
| 51.c | odd | 2 | 1 | 306.4.b.f | 6 | ||
| 68.d | odd | 2 | 1 | 816.4.c.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.4.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 102.4.b.b | ✓ | 6 | 17.b | even | 2 | 1 | inner |
| 306.4.b.f | 6 | 3.b | odd | 2 | 1 | ||
| 306.4.b.f | 6 | 51.c | odd | 2 | 1 | ||
| 816.4.c.e | 6 | 4.b | odd | 2 | 1 | ||
| 816.4.c.e | 6 | 68.d | odd | 2 | 1 | ||
| 1734.4.a.u | 3 | 17.c | even | 4 | 1 | ||
| 1734.4.a.v | 3 | 17.c | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 737T_{5}^{4} + 128096T_{5}^{2} + 18496 \)
acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} + 737 T^{4} + \cdots + 18496 \)
|
| $7$ |
\( T^{6} + 1828 T^{4} + \cdots + 104530176 \)
|
| $11$ |
\( T^{6} + 3673 T^{4} + \cdots + 192432384 \)
|
| $13$ |
\( (T^{3} - 55 T^{2} + \cdots + 135924)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 118587876497 \)
|
| $19$ |
\( (T^{3} - 91 T^{2} + \cdots + 1650768)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 125809252416 \)
|
| $29$ |
\( T^{6} + \cdots + 27145767225600 \)
|
| $31$ |
\( T^{6} + \cdots + 12297870676224 \)
|
| $37$ |
\( T^{6} + \cdots + 1062746562816 \)
|
| $41$ |
\( T^{6} + \cdots + 36996368311296 \)
|
| $43$ |
\( (T^{3} - 41 T^{2} + \cdots - 19663504)^{2} \)
|
| $47$ |
\( (T^{3} - 646 T^{2} + \cdots + 46426752)^{2} \)
|
| $53$ |
\( (T^{3} - 258 T^{2} + \cdots + 23861736)^{2} \)
|
| $59$ |
\( (T^{3} - 288 T^{2} + \cdots + 66213504)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + 828 T^{2} + \cdots - 43345856)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 31710773837824 \)
|
| $73$ |
\( T^{6} + \cdots + 24\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots + 39\!\cdots\!56 \)
|
| $83$ |
\( (T^{3} - 418 T^{2} + \cdots + 11676384)^{2} \)
|
| $89$ |
\( (T^{3} + 658 T^{2} + \cdots - 221045832)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 49\!\cdots\!96 \)
|
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