Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1078,4,Mod(1,1078)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1078, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1078.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.a (trivial) |
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: | yes |
| Analytic conductor: | \(63.6040589862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 230x^{8} + 17873x^{6} - 545770x^{4} + 5445376x^{2} - 43808 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 230x^{8} + 17873x^{6} - 545770x^{4} + 5445376x^{2} - 43808 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4721\nu^{8} - 1391658\nu^{6} + 120394953\nu^{4} - 3030111822\nu^{2} + 7739172184 ) / 389990832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -75\nu^{8} + 17102\nu^{6} - 1149155\nu^{4} + 14635346\nu^{2} + 270146792 ) / 5908952 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7883\nu^{8} - 1167246\nu^{6} + 24664899\nu^{4} + 1314490806\nu^{2} - 15483720200 ) / 389990832 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -52593\nu^{8} + 10574458\nu^{6} - 654564281\nu^{4} + 12765213022\nu^{2} - 27001502744 ) / 389990832 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5625\nu^{9} + 1282650\nu^{7} - 98004529\nu^{5} + 2899881310\nu^{3} - 29338733688\nu ) / 1749049792 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 984827\nu^{9} - 236227150\nu^{7} + 19467724451\nu^{5} - 642646389466\nu^{3} + 7010357053992\nu ) / 57718643136 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5625\nu^{9} + 1282650\nu^{7} - 98004529\nu^{5} + 2899881310\nu^{3} - 27589683896\nu ) / 249864256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 90075\nu^{9} - 20539502\nu^{7} + 1581039523\nu^{5} - 47996335442\nu^{3} + 483331077016\nu ) / 655893672 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 9899777 \nu^{9} + 2269073482 \nu^{7} - 175562724281 \nu^{5} + 5327112199534 \nu^{3} - 52409766336120 \nu ) / 57718643136 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 7\beta_{5} ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} - 7\beta_{2} - 4\beta _1 + 320 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -28\beta_{9} - 21\beta_{8} + 138\beta_{7} - 28\beta_{6} - 518\beta_{5} ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -42\beta_{4} - 369\beta_{3} - 665\beta_{2} - 549\beta _1 + 23739 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3745\beta_{9} - 2415\beta_{8} + 17298\beta_{7} - 3745\beta_{6} - 44352\beta_{5} ) / 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( -915\beta_{4} - 6127\beta_{3} - 8976\beta_{2} - 9374\beta _1 + 289779 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -417592\beta_{9} - 235221\beta_{8} + 1989212\beta_{7} - 452242\beta_{6} - 4097226\beta_{5} ) / 7 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -816984\beta_{4} - 4516269\beta_{3} - 6055665\beta_{2} - 7331391\beta _1 + 186468679 ) / 7 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -44407839\beta_{9} - 22386273\beta_{8} + 218137834\beta_{7} - 52308963\beta_{6} - 394243696\beta_{5} ) / 7 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −9.70883 | 4.00000 | −11.6647 | −19.4177 | 0 | 8.00000 | 67.2615 | −23.3293 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −8.71608 | 4.00000 | −0.961480 | −17.4322 | 0 | 8.00000 | 48.9700 | −1.92296 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −7.62780 | 4.00000 | 20.3575 | −15.2556 | 0 | 8.00000 | 31.1833 | 40.7151 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −3.05343 | 4.00000 | −16.7923 | −6.10685 | 0 | 8.00000 | −17.6766 | −33.5847 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −1.50394 | 4.00000 | −10.8903 | −3.00789 | 0 | 8.00000 | −24.7382 | −21.7807 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 1.50394 | 4.00000 | 10.8903 | 3.00789 | 0 | 8.00000 | −24.7382 | 21.7807 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 3.05343 | 4.00000 | 16.7923 | 6.10685 | 0 | 8.00000 | −17.6766 | 33.5847 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 7.62780 | 4.00000 | −20.3575 | 15.2556 | 0 | 8.00000 | 31.1833 | −40.7151 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.00000 | 8.71608 | 4.00000 | 0.961480 | 17.4322 | 0 | 8.00000 | 48.9700 | 1.92296 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.00000 | 9.70883 | 4.00000 | 11.6647 | 19.4177 | 0 | 8.00000 | 67.2615 | 23.3293 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1078.4.a.bd | ✓ | 10 |
| 7.b | odd | 2 | 1 | inner | 1078.4.a.bd | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1078.4.a.bd | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1078.4.a.bd | ✓ | 10 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 240T_{3}^{8} + 19733T_{3}^{6} - 619180T_{3}^{4} + 5186916T_{3}^{2} - 8786432 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1078))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{10} \)
|
| $3$ |
\( T^{10} - 240 T^{8} + \cdots - 8786432 \)
|
| $5$ |
\( T^{10} + \cdots - 1743333152 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T - 11)^{10} \)
|
| $13$ |
\( T^{10} + \cdots - 411157405632512 \)
|
| $17$ |
\( T^{10} + \cdots - 55\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots - 42448455729152 \)
|
| $23$ |
\( (T^{5} - 134 T^{4} + \cdots - 501769016)^{2} \)
|
| $29$ |
\( (T^{5} - 102832 T^{3} + \cdots - 234175564800)^{2} \)
|
| $31$ |
\( T^{10} + \cdots - 43\!\cdots\!28 \)
|
| $37$ |
\( (T^{5} + 120 T^{4} + \cdots - 510113042984)^{2} \)
|
| $41$ |
\( T^{10} + \cdots - 14\!\cdots\!08 \)
|
| $43$ |
\( (T^{5} - 294 T^{4} + \cdots - 20279860736)^{2} \)
|
| $47$ |
\( T^{10} + \cdots - 53\!\cdots\!48 \)
|
| $53$ |
\( (T^{5} + \cdots + 3184882953216)^{2} \)
|
| $59$ |
\( T^{10} + \cdots - 19\!\cdots\!28 \)
|
| $61$ |
\( T^{10} + \cdots - 11\!\cdots\!92 \)
|
| $67$ |
\( (T^{5} - 944 T^{4} + \cdots + 260505346960)^{2} \)
|
| $71$ |
\( (T^{5} - 308 T^{4} + \cdots + 796433467040)^{2} \)
|
| $73$ |
\( T^{10} + \cdots - 71\!\cdots\!68 \)
|
| $79$ |
\( (T^{5} + \cdots + 32519776481024)^{2} \)
|
| $83$ |
\( T^{10} + \cdots - 57\!\cdots\!92 \)
|
| $89$ |
\( T^{10} + \cdots - 12\!\cdots\!48 \)
|
| $97$ |
\( T^{10} + \cdots - 18\!\cdots\!08 \)
|
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