Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,6,Mod(33,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, 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("68.33");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 68.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: | \(10.9060997473\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1278x^{6} + 512312x^{4} + 62430336x^{2} + 5018112 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 1278x^{6} + 512312x^{4} + 62430336x^{2} + 5018112 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -49\nu^{6} - 35766\nu^{4} - 4885688\nu^{2} + 78889968 ) / 449712 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{7} + 12618\nu^{5} + 4474216\nu^{3} + 483682176\nu ) / 5396544 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{6} - 12618\nu^{4} - 3799648\nu^{2} - 147522456 ) / 224856 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23\nu^{7} + 29790\nu^{5} + 11937616\nu^{3} + 1412587584\nu ) / 3297888 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2053\nu^{7} + 2641158\nu^{5} + 1071763448\nu^{3} + 133416416448\nu ) / 59361984 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 320 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{7} - 14\beta_{6} - 3\beta_{4} - 477\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -49\beta_{5} - 589\beta_{3} + 22\beta_{2} + 152473 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2274\beta_{7} + 11580\beta_{6} - 1038\beta_{4} + 243790\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35766\beta_{5} + 330214\beta_{3} - 25236\beta_{2} - 77776310 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1388244\beta_{7} - 7588856\beta_{6} + 2901516\beta_{4} - 129602124\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | − | 24.1287i | 0 | 67.0341i | 0 | 203.357i | 0 | −339.196 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | − | 21.5762i | 0 | − | 102.247i | 0 | − | 16.6326i | 0 | −222.531 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | − | 15.1721i | 0 | 33.2091i | 0 | − | 141.607i | 0 | 12.8076 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | − | 0.283606i | 0 | − | 25.4002i | 0 | 121.962i | 0 | 242.920 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 0.283606i | 0 | 25.4002i | 0 | − | 121.962i | 0 | 242.920 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 15.1721i | 0 | − | 33.2091i | 0 | 141.607i | 0 | 12.8076 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 21.5762i | 0 | 102.247i | 0 | 16.6326i | 0 | −222.531 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 24.1287i | 0 | − | 67.0341i | 0 | − | 203.357i | 0 | −339.196 | 0 | |||||||||||||||||||||||||||||||||||||||||
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 | 68.6.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 612.6.b.d | 8 | ||
| 4.b | odd | 2 | 1 | 272.6.b.d | 8 | ||
| 17.b | even | 2 | 1 | inner | 68.6.b.a | ✓ | 8 |
| 51.c | odd | 2 | 1 | 612.6.b.d | 8 | ||
| 68.d | odd | 2 | 1 | 272.6.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.6.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 68.6.b.a | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 272.6.b.d | 8 | 4.b | odd | 2 | 1 | ||
| 272.6.b.d | 8 | 68.d | odd | 2 | 1 | ||
| 612.6.b.d | 8 | 3.b | odd | 2 | 1 | ||
| 612.6.b.d | 8 | 51.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(68, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 1278 T^{6} + \cdots + 5018112 \)
|
| $5$ |
\( T^{8} + \cdots + 33425581088768 \)
|
| $7$ |
\( T^{8} + \cdots + 34\!\cdots\!88 \)
|
| $11$ |
\( T^{8} + \cdots + 21\!\cdots\!32 \)
|
| $13$ |
\( (T^{4} - 606 T^{3} + \cdots - 58758829536)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 40\!\cdots\!01 \)
|
| $19$ |
\( (T^{4} + \cdots + 17328502278912)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 25\!\cdots\!72 \)
|
| $29$ |
\( T^{8} + \cdots + 27\!\cdots\!32 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!88 \)
|
| $37$ |
\( T^{8} + \cdots + 10\!\cdots\!72 \)
|
| $41$ |
\( T^{8} + \cdots + 21\!\cdots\!72 \)
|
| $43$ |
\( (T^{4} + \cdots - 21\!\cdots\!36)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots - 13\!\cdots\!72)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots - 62\!\cdots\!72)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots - 60\!\cdots\!20)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 37\!\cdots\!72 \)
|
| $67$ |
\( (T^{4} + \cdots - 70\!\cdots\!32)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 82\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 28\!\cdots\!88 \)
|
| $83$ |
\( (T^{4} + \cdots - 29\!\cdots\!60)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots - 15\!\cdots\!12)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 19\!\cdots\!00 \)
|
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