Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,6,Mod(25,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, 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("78.25");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.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: | \(12.5099379454\) |
| 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: |
\( x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{2} \) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11164\nu^{5} + 33517\nu^{4} + 136007\nu^{3} - 12505719\nu^{2} + 1388809431\nu - 7854226236 ) / 1807933527 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1108\nu^{5} - 131752\nu^{4} - 404420\nu^{3} + 1632084\nu^{2} + 4068576\nu - 9481422087 ) / 35449677 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 383264 \nu^{5} + 1222625 \nu^{4} - 21601613 \nu^{3} - 1608343917 \nu^{2} + 37929979503 \nu - 221669454492 ) / 1807933527 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8786\nu^{5} - 276878\nu^{4} - 3206890\nu^{3} + 12941778\nu^{2} + 32262192\nu - 10639840401 ) / 35449677 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 307633 \nu^{5} + 917590 \nu^{4} + 5937014 \nu^{3} - 1150320390 \nu^{2} + 39082121901 \nu - 220426926804 ) / 1205289018 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} - 4\beta_{3} - 2\beta_{2} + 2\beta _1 + 15 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -8\beta_{5} - \beta_{3} + 365\beta_1 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1388\beta_{5} - 365\beta_{4} - 1460\beta_{3} + 694\beta_{2} - 18386\beta _1 + 71025 ) / 48 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 554\beta_{4} - 4393\beta_{2} - 1008681 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -586204\beta_{5} - 126607\beta_{4} + 506428\beta_{3} + 293102\beta_{2} + 11019394\beta _1 + 43151439 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
− | 4.00000i | −9.00000 | −16.0000 | − | 37.1158i | 36.0000i | − | 176.407i | 64.0000i | 81.0000 | −148.463 | |||||||||||||||||||||||||||||||||
| 25.2 | − | 4.00000i | −9.00000 | −16.0000 | − | 9.73803i | 36.0000i | 105.184i | 64.0000i | 81.0000 | −38.9521 | |||||||||||||||||||||||||||||||||||
| 25.3 | − | 4.00000i | −9.00000 | −16.0000 | 86.8538i | 36.0000i | − | 98.7774i | 64.0000i | 81.0000 | 347.415 | |||||||||||||||||||||||||||||||||||
| 25.4 | 4.00000i | −9.00000 | −16.0000 | − | 86.8538i | − | 36.0000i | 98.7774i | − | 64.0000i | 81.0000 | 347.415 | ||||||||||||||||||||||||||||||||||
| 25.5 | 4.00000i | −9.00000 | −16.0000 | 9.73803i | − | 36.0000i | − | 105.184i | − | 64.0000i | 81.0000 | −38.9521 | ||||||||||||||||||||||||||||||||||
| 25.6 | 4.00000i | −9.00000 | −16.0000 | 37.1158i | − | 36.0000i | 176.407i | − | 64.0000i | 81.0000 | −148.463 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.6.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 234.6.b.c | 6 | ||
| 4.b | odd | 2 | 1 | 624.6.c.d | 6 | ||
| 13.b | even | 2 | 1 | inner | 78.6.b.a | ✓ | 6 |
| 13.d | odd | 4 | 1 | 1014.6.a.o | 3 | ||
| 13.d | odd | 4 | 1 | 1014.6.a.q | 3 | ||
| 39.d | odd | 2 | 1 | 234.6.b.c | 6 | ||
| 52.b | odd | 2 | 1 | 624.6.c.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.6.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 78.6.b.a | ✓ | 6 | 13.b | even | 2 | 1 | inner |
| 234.6.b.c | 6 | 3.b | odd | 2 | 1 | ||
| 234.6.b.c | 6 | 39.d | odd | 2 | 1 | ||
| 624.6.c.d | 6 | 4.b | odd | 2 | 1 | ||
| 624.6.c.d | 6 | 52.b | odd | 2 | 1 | ||
| 1014.6.a.o | 3 | 13.d | odd | 4 | 1 | ||
| 1014.6.a.q | 3 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 9016T_{5}^{4} + 11237904T_{5}^{2} + 985457664 \)
acting on \(S_{6}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{3} \)
|
| $3$ |
\( (T + 9)^{6} \)
|
| $5$ |
\( T^{6} + 9016 T^{4} + \cdots + 985457664 \)
|
| $7$ |
\( T^{6} + \cdots + 3359273140224 \)
|
| $11$ |
\( T^{6} + \cdots + 871026613185600 \)
|
| $13$ |
\( T^{6} + \cdots + 51\!\cdots\!57 \)
|
| $17$ |
\( (T^{3} + 418 T^{2} + \cdots - 1783218312)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 224220196832256 \)
|
| $23$ |
\( (T^{3} + 208 T^{2} + \cdots + 2602266624)^{2} \)
|
| $29$ |
\( (T^{3} - 9394 T^{2} + \cdots - 2546571960)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 15\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 1729964364544)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!24 \)
|
| $53$ |
\( (T^{3} + \cdots - 26960052479832)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $61$ |
\( (T^{3} + \cdots + 17380383329800)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 94\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + \cdots - 35835411156480)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 35\!\cdots\!16 \)
|
| $97$ |
\( T^{6} + \cdots + 30\!\cdots\!64 \)
|
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