Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,6,Mod(81,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.81");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.i (of order \(3\), degree \(2\), not 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: | \(33.3598345211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 13) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 174777 \nu^{7} - 1087478 \nu^{6} + 11735389 \nu^{5} - 5093863 \nu^{4} + 780153066 \nu^{3} + \cdots + 180172599494 ) / 5480995715 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots - 207840771072 ) / 526175588640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots - 207840771072 ) / 526175588640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1278708 \nu^{7} - 699108 \nu^{6} - 85858756 \nu^{5} + 37267852 \nu^{4} - 6036679024 \nu^{3} + \cdots - 18421442051 ) / 5480995715 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 163476431 \nu^{7} + 338880779 \nu^{6} - 12327003362 \nu^{5} + 33519877559 \nu^{4} + \cdots + 1093886512128 ) / 263087794320 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1424846745 \nu^{3} + \cdots + 6608032024 ) / 337292044 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 87076937 \nu^{7} + 166612553 \nu^{6} - 6060637334 \nu^{5} + 17508240653 \nu^{4} + \cdots + 537815172096 ) / 40475045280 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{7} - \beta_{4} - 141\beta_{3} + \beta_{2} + 4\beta _1 - 137 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -16\beta_{6} - 65\beta_{4} + 95 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 276\beta_{7} - 16\beta_{5} + 9149\beta_{3} - 33\beta_{2} ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 128 \beta_{7} + 1120 \beta_{6} - 1120 \beta_{5} + 4321 \beta_{4} - 11743 \beta_{3} - 4321 \beta_{2} + \cdots - 11615 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 608\beta_{6} - 63\beta_{4} - 18532\beta _1 + 597065 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 18176\beta_{7} + 74736\beta_{5} + 1109599\beta_{3} + 288513\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −12.4354 | − | 21.5388i | 0 | 44.5576 | 0 | 27.1222 | − | 46.9770i | 0 | −187.780 | + | 325.244i | 0 | ||||||||||||||||||||||||||||||||||||
| 81.2 | 0 | −2.52313 | − | 4.37020i | 0 | −9.82751 | 0 | −17.3506 | + | 30.0522i | 0 | 108.768 | − | 188.391i | 0 | |||||||||||||||||||||||||||||||||||||
| 81.3 | 0 | 1.64205 | + | 2.84411i | 0 | −58.6393 | 0 | −61.9973 | + | 107.382i | 0 | 116.107 | − | 201.104i | 0 | |||||||||||||||||||||||||||||||||||||
| 81.4 | 0 | 9.31652 | + | 16.1367i | 0 | 13.9092 | 0 | 18.2258 | − | 31.5679i | 0 | −52.0950 | + | 90.2311i | 0 | |||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −12.4354 | + | 21.5388i | 0 | 44.5576 | 0 | 27.1222 | + | 46.9770i | 0 | −187.780 | − | 325.244i | 0 | |||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −2.52313 | + | 4.37020i | 0 | −9.82751 | 0 | −17.3506 | − | 30.0522i | 0 | 108.768 | + | 188.391i | 0 | |||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | 1.64205 | − | 2.84411i | 0 | −58.6393 | 0 | −61.9973 | − | 107.382i | 0 | 116.107 | + | 201.104i | 0 | |||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | 9.31652 | − | 16.1367i | 0 | 13.9092 | 0 | 18.2258 | + | 31.5679i | 0 | −52.0950 | − | 90.2311i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.6.i.b | 8 | |
| 4.b | odd | 2 | 1 | 13.6.c.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 117.6.g.b | 8 | ||
| 13.c | even | 3 | 1 | inner | 208.6.i.b | 8 | |
| 52.i | odd | 6 | 1 | 169.6.a.c | 4 | ||
| 52.j | odd | 6 | 1 | 13.6.c.a | ✓ | 8 | |
| 52.j | odd | 6 | 1 | 169.6.a.d | 4 | ||
| 52.l | even | 12 | 2 | 169.6.b.c | 8 | ||
| 156.p | even | 6 | 1 | 117.6.g.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.c.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 13.6.c.a | ✓ | 8 | 52.j | odd | 6 | 1 | |
| 117.6.g.b | 8 | 12.b | even | 2 | 1 | ||
| 117.6.g.b | 8 | 156.p | even | 6 | 1 | ||
| 169.6.a.c | 4 | 52.i | odd | 6 | 1 | ||
| 169.6.a.d | 4 | 52.j | odd | 6 | 1 | ||
| 169.6.b.c | 8 | 52.l | even | 12 | 2 | ||
| 208.6.i.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 208.6.i.b | 8 | 13.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 8 T_{3}^{7} + 533 T_{3}^{6} - 1912 T_{3}^{5} + 219641 T_{3}^{4} + 308600 T_{3}^{3} + \cdots + 58982400 \)
acting on \(S_{6}^{\mathrm{new}}(208, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 8 T^{7} + \cdots + 58982400 \)
|
| $5$ |
\( (T^{4} + 10 T^{3} + \cdots + 357156)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 72383069214976 \)
|
| $11$ |
\( T^{8} + \cdots + 40\!\cdots\!56 \)
|
| $13$ |
\( T^{8} + \cdots + 19\!\cdots\!01 \)
|
| $17$ |
\( T^{8} + \cdots + 14\!\cdots\!09 \)
|
| $19$ |
\( T^{8} + \cdots + 15\!\cdots\!96 \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!16 \)
|
| $29$ |
\( T^{8} + \cdots + 18\!\cdots\!61 \)
|
| $31$ |
\( (T^{4} + \cdots + 4693845913600)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 28\!\cdots\!09 \)
|
| $41$ |
\( T^{8} + \cdots + 96\!\cdots\!41 \)
|
| $43$ |
\( T^{8} + \cdots + 63\!\cdots\!04 \)
|
| $47$ |
\( (T^{4} + \cdots + 24\!\cdots\!92)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots - 38\!\cdots\!36)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 21\!\cdots\!44 \)
|
| $61$ |
\( T^{8} + \cdots + 13\!\cdots\!61 \)
|
| $67$ |
\( T^{8} + \cdots + 69\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots + 47\!\cdots\!56 \)
|
| $73$ |
\( (T^{4} + \cdots - 34\!\cdots\!48)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 43\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots - 30\!\cdots\!40)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 91\!\cdots\!36 \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
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