Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1620,3,Mod(973,1620)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1620.973");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1620.l (of order \(4\), degree \(2\), 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: | \(44.1418028264\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4 x^{9} - 14 x^{8} + 88 x^{7} - 1097 x^{6} + 70 x^{5} + 28027 x^{4} - 74480 x^{3} + \cdots + 6110945 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{10} - 4 x^{9} - 14 x^{8} + 88 x^{7} - 1097 x^{6} + 70 x^{5} + 28027 x^{4} - 74480 x^{3} + \cdots + 6110945 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 382567313 \nu^{9} - 3335562891 \nu^{8} + 12779831831 \nu^{7} - 19193099729 \nu^{6} + \cdots + 237826459748375 ) / 160047256575240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 29380567113 \nu^{9} + 196173281255 \nu^{8} - 1911328495623 \nu^{7} + \cdots + 72\!\cdots\!65 ) / 66\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 127366488911 \nu^{9} + 751826790585 \nu^{8} + 1456381105519 \nu^{7} + \cdots - 28\!\cdots\!45 ) / 20\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12754405663 \nu^{9} + 118983011830 \nu^{8} - 193531667023 \nu^{7} + \cdots - 18\!\cdots\!60 ) / 16\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 41983573 \nu^{9} - 421762191 \nu^{8} + 1229279611 \nu^{7} - 1827145549 \nu^{6} + \cdots + 54368553686995 ) / 3722029222680 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24291064846 \nu^{9} - 180195976135 \nu^{8} + 342488149366 \nu^{7} + 1369677272027 \nu^{6} + \cdots + 37\!\cdots\!45 ) / 16\!\cdots\!50 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 48629078779 \nu^{9} - 422864042640 \nu^{8} + 1084962170409 \nu^{7} + 667034191048 \nu^{6} + \cdots + 76\!\cdots\!80 ) / 16\!\cdots\!50 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 58367172759 \nu^{9} + 414239230240 \nu^{8} - 537518445089 \nu^{7} + \cdots - 94\!\cdots\!30 ) / 16\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 5 \beta_{9} + 5 \beta_{8} - 10 \beta_{7} - 4 \beta_{6} - 5 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} + \cdots - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 29 \beta_{9} - 25 \beta_{8} - 33 \beta_{7} - \beta_{6} - 61 \beta_{5} + 48 \beta_{4} - 14 \beta_{3} + \cdots + 416 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 102 \beta_{9} - 142 \beta_{8} - 205 \beta_{7} - 83 \beta_{6} - 121 \beta_{5} + 61 \beta_{4} + \cdots + 1483 \)
|
| \(\nu^{6}\) | \(=\) |
\( 107 \beta_{9} + 1093 \beta_{8} - 1767 \beta_{7} - 2753 \beta_{6} - 1595 \beta_{5} - 387 \beta_{4} + \cdots + 1346 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1497 \beta_{9} + 9968 \beta_{8} - 8740 \beta_{7} - 14732 \beta_{6} - 8061 \beta_{5} + 9166 \beta_{4} + \cdots + 28584 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 34987 \beta_{9} - 11848 \beta_{8} - 15903 \beta_{7} - 32418 \beta_{6} - 48460 \beta_{5} + \cdots + 558563 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 207604 \beta_{9} - 265069 \beta_{8} - 153690 \beta_{7} - 99666 \beta_{6} - 117432 \beta_{5} + \cdots + 2713781 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).
| \(n\) | \(811\) | \(1297\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 973.1 |
|
0 | 0 | 0 | −4.94990 | + | 0.706059i | 0 | −3.77932 | + | 3.77932i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||
| 973.2 | 0 | 0 | 0 | −2.00374 | + | 4.58094i | 0 | 2.75016 | − | 2.75016i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 973.3 | 0 | 0 | 0 | 0.510319 | − | 4.97389i | 0 | −3.40965 | + | 3.40965i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 973.4 | 0 | 0 | 0 | 4.46188 | + | 2.25646i | 0 | 5.12172 | − | 5.12172i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 973.5 | 0 | 0 | 0 | 4.98144 | + | 0.430428i | 0 | −5.68292 | + | 5.68292i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1297.1 | 0 | 0 | 0 | −4.94990 | − | 0.706059i | 0 | −3.77932 | − | 3.77932i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1297.2 | 0 | 0 | 0 | −2.00374 | − | 4.58094i | 0 | 2.75016 | + | 2.75016i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1297.3 | 0 | 0 | 0 | 0.510319 | + | 4.97389i | 0 | −3.40965 | − | 3.40965i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1297.4 | 0 | 0 | 0 | 4.46188 | − | 2.25646i | 0 | 5.12172 | + | 5.12172i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1297.5 | 0 | 0 | 0 | 4.98144 | − | 0.430428i | 0 | −5.68292 | − | 5.68292i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1620.3.l.d | yes | 10 |
| 3.b | odd | 2 | 1 | 1620.3.l.c | ✓ | 10 | |
| 5.c | odd | 4 | 1 | inner | 1620.3.l.d | yes | 10 |
| 15.e | even | 4 | 1 | 1620.3.l.c | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1620.3.l.c | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 1620.3.l.c | ✓ | 10 | 15.e | even | 4 | 1 | |
| 1620.3.l.d | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 1620.3.l.d | yes | 10 | 5.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):
|
\( T_{7}^{10} + 10 T_{7}^{9} + 50 T_{7}^{8} + 4 T_{7}^{7} + 3045 T_{7}^{6} + 29686 T_{7}^{5} + \cdots + 34047752 \)
|
|
\( T_{11}^{5} - 9T_{11}^{4} - 136T_{11}^{3} + 646T_{11}^{2} + 5184T_{11} - 3088 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 6 T^{9} + \cdots + 9765625 \)
|
| $7$ |
\( T^{10} + 10 T^{9} + \cdots + 34047752 \)
|
| $11$ |
\( (T^{5} - 9 T^{4} + \cdots - 3088)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 4050000000 \)
|
| $17$ |
\( T^{10} + \cdots + 107657280200 \)
|
| $19$ |
\( T^{10} + \cdots + 131384500900 \)
|
| $23$ |
\( T^{10} + \cdots + 15623804040968 \)
|
| $29$ |
\( T^{10} + \cdots + 136330563940624 \)
|
| $31$ |
\( (T^{5} + 31 T^{4} + \cdots + 5378960)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 209952000000 \)
|
| $41$ |
\( (T^{5} + 3 T^{4} + \cdots - 32467276)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 16200000000 \)
|
| $47$ |
\( T^{10} + \cdots + 5248800000000 \)
|
| $53$ |
\( T^{10} + \cdots + 9976812217352 \)
|
| $59$ |
\( T^{10} + \cdots + 53\!\cdots\!64 \)
|
| $61$ |
\( (T^{5} - 30 T^{4} + \cdots + 1015920000)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 19\!\cdots\!68 \)
|
| $71$ |
\( (T^{5} + 33 T^{4} + \cdots + 683092)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 19\!\cdots\!48 \)
|
| $79$ |
\( T^{10} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 54\!\cdots\!00 \)
|
| $89$ |
\( T^{10} + \cdots + 23\!\cdots\!64 \)
|
| $97$ |
\( T^{10} + \cdots + 10\!\cdots\!00 \)
|
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