Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1694,4,Mod(1,1694)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, 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("1694.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1694.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: | \(99.9492355497\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 2 x^{9} - 170 x^{8} + 272 x^{7} + 9867 x^{6} - 15814 x^{5} - 224781 x^{4} + 465188 x^{3} + \cdots + 3321791 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 11^{3} \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{10} - 2 x^{9} - 170 x^{8} + 272 x^{7} + 9867 x^{6} - 15814 x^{5} - 224781 x^{4} + 465188 x^{3} + \cdots + 3321791 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 44799065 \nu^{9} - 14631545 \nu^{8} - 7801434643 \nu^{7} + 158477885 \nu^{6} + \cdots - 111907769880071 ) / 3161544431232 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24980323 \nu^{9} + 35711543 \nu^{8} - 3646119781 \nu^{7} - 10120856899 \nu^{6} + \cdots + 37559042092869 ) / 1580772215616 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 79484875 \nu^{9} + 37188124 \nu^{8} + 14586029225 \nu^{7} - 5780951622 \nu^{6} + \cdots + 323786795924240 ) / 790386107808 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 159109795 \nu^{9} - 97920145 \nu^{8} - 26371734681 \nu^{7} + 1419049273 \nu^{6} + \cdots - 320033809213855 ) / 1580772215616 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 223995325 \nu^{9} + 73157725 \nu^{8} + 39007173215 \nu^{7} - 792389425 \nu^{6} + \cdots + 551634988322275 ) / 1580772215616 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 72796701 \nu^{9} - 84516735 \nu^{8} + 13082024263 \nu^{7} + 14891557899 \nu^{6} + \cdots + 174594723031527 ) / 287413130112 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 120368305 \nu^{9} + 58037051 \nu^{8} - 20938213643 \nu^{7} - 13807471791 \nu^{6} + \cdots - 254886474357083 ) / 287413130112 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1519726595 \nu^{9} - 934763903 \nu^{8} - 258588133633 \nu^{7} + 44631953043 \nu^{6} + \cdots - 33\!\cdots\!29 ) / 3161544431232 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1638052369 \nu^{9} + 830840093 \nu^{8} + 279985627843 \nu^{7} - 26344728273 \nu^{6} + \cdots + 36\!\cdots\!27 ) / 3161544431232 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 10\beta _1 + 5 ) / 11 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{9} + 7\beta_{8} - 2\beta_{6} + \beta_{5} + 11\beta_{4} - 11\beta_{2} + 376 ) / 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 42 \beta_{9} + 18 \beta_{8} - 2 \beta_{6} - 20 \beta_{5} + 44 \beta_{4} + 22 \beta_{3} - 44 \beta_{2} + \cdots + 403 ) / 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 810 \beta_{9} + 630 \beta_{8} - 136 \beta_{6} - 107 \beta_{5} + 704 \beta_{4} + 66 \beta_{3} + \cdots + 20821 ) / 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5109 \beta_{9} + 2441 \beta_{8} - 44 \beta_{7} - 226 \beta_{6} - 4527 \beta_{5} + 4719 \beta_{4} + \cdots + 55038 ) / 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 71028 \beta_{9} + 51812 \beta_{8} + 2244 \beta_{7} - 6720 \beta_{6} - 25556 \beta_{5} + 51128 \beta_{4} + \cdots + 1361495 ) / 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 522132 \beta_{9} + 268192 \beta_{8} + 5324 \beta_{7} - 14052 \beta_{6} - 475319 \beta_{5} + \cdots + 5850369 ) / 11 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6217845 \beta_{9} + 4249479 \beta_{8} + 350196 \beta_{7} - 287874 \beta_{6} - 3214971 \beta_{5} + \cdots + 97918700 ) / 11 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 49755246 \beta_{9} + 26766678 \beta_{8} + 1488036 \beta_{7} - 532002 \beta_{6} - 44097536 \beta_{5} + \cdots + 561814151 ) / 11 \)
|
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 | −9.23021 | 4.00000 | 21.4359 | −18.4604 | −7.00000 | 8.00000 | 58.1968 | 42.8717 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −8.67752 | 4.00000 | −19.3608 | −17.3550 | −7.00000 | 8.00000 | 48.2993 | −38.7215 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −5.72443 | 4.00000 | −20.2096 | −11.4489 | −7.00000 | 8.00000 | 5.76908 | −40.4193 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −3.78041 | 4.00000 | −0.302686 | −7.56082 | −7.00000 | 8.00000 | −12.7085 | −0.605371 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −0.193520 | 4.00000 | 11.6336 | −0.387040 | −7.00000 | 8.00000 | −26.9626 | 23.2671 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 2.21074 | 4.00000 | 0.348917 | 4.42147 | −7.00000 | 8.00000 | −22.1126 | 0.697834 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 3.06983 | 4.00000 | −15.2781 | 6.13965 | −7.00000 | 8.00000 | −17.5762 | −30.5562 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 3.66464 | 4.00000 | 10.5070 | 7.32929 | −7.00000 | 8.00000 | −13.5704 | 21.0140 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.00000 | 5.70519 | 4.00000 | −1.86828 | 11.4104 | −7.00000 | 8.00000 | 5.54914 | −3.73657 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.00000 | 9.95570 | 4.00000 | −13.9059 | 19.9114 | −7.00000 | 8.00000 | 72.1159 | −27.8118 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1694.4.a.bk | 10 | |
| 11.b | odd | 2 | 1 | 1694.4.a.bh | 10 | ||
| 11.c | even | 5 | 2 | 154.4.f.f | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.f.f | ✓ | 20 | 11.c | even | 5 | 2 | |
| 1694.4.a.bh | 10 | 11.b | odd | 2 | 1 | ||
| 1694.4.a.bk | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):
|
\( T_{3}^{10} + 3 T_{3}^{9} - 179 T_{3}^{8} - 406 T_{3}^{7} + 10088 T_{3}^{6} + 9244 T_{3}^{5} + \cdots - 473836 \)
|
|
\( T_{5}^{10} + 27 T_{5}^{9} - 595 T_{5}^{8} - 19563 T_{5}^{7} + 55214 T_{5}^{6} + 3767732 T_{5}^{5} + \cdots + 42977264 \)
|
|
\( T_{13}^{10} + 78 T_{13}^{9} - 8717 T_{13}^{8} - 497359 T_{13}^{7} + 36329851 T_{13}^{6} + \cdots + 56\!\cdots\!96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{10} \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots - 473836 \)
|
| $5$ |
\( T^{10} + 27 T^{9} + \cdots + 42977264 \)
|
| $7$ |
\( (T + 7)^{10} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 56\!\cdots\!96 \)
|
| $17$ |
\( T^{10} + \cdots + 16\!\cdots\!96 \)
|
| $19$ |
\( T^{10} + \cdots + 42\!\cdots\!24 \)
|
| $23$ |
\( T^{10} + \cdots - 30\!\cdots\!56 \)
|
| $29$ |
\( T^{10} + \cdots + 29\!\cdots\!24 \)
|
| $31$ |
\( T^{10} + \cdots - 90\!\cdots\!04 \)
|
| $37$ |
\( T^{10} + \cdots - 24\!\cdots\!56 \)
|
| $41$ |
\( T^{10} + \cdots + 27\!\cdots\!16 \)
|
| $43$ |
\( T^{10} + \cdots - 10\!\cdots\!89 \)
|
| $47$ |
\( T^{10} + \cdots - 55\!\cdots\!84 \)
|
| $53$ |
\( T^{10} + \cdots - 37\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots + 30\!\cdots\!76 \)
|
| $61$ |
\( T^{10} + \cdots - 32\!\cdots\!44 \)
|
| $67$ |
\( T^{10} + \cdots - 51\!\cdots\!59 \)
|
| $71$ |
\( T^{10} + \cdots - 33\!\cdots\!64 \)
|
| $73$ |
\( T^{10} + \cdots - 13\!\cdots\!64 \)
|
| $79$ |
\( T^{10} + \cdots - 54\!\cdots\!16 \)
|
| $83$ |
\( T^{10} + \cdots + 90\!\cdots\!76 \)
|
| $89$ |
\( T^{10} + \cdots + 18\!\cdots\!96 \)
|
| $97$ |
\( T^{10} + \cdots - 12\!\cdots\!00 \)
|
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