Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,3,Mod(577,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([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("1014.577");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.f (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: | \(27.6294988061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 78) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4081829 \nu^{7} + 125878448 \nu^{6} - 175318185 \nu^{5} - 167602931 \nu^{4} + \cdots - 561922481148 ) / 28390156409030 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 207487050 \nu^{7} + 9646046383 \nu^{6} + 263049796262 \nu^{5} + 452388400182 \nu^{4} + \cdots + 20\!\cdots\!94 ) / 965265317907020 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 186368592 \nu^{7} + 13656504841 \nu^{6} + 24877044110 \nu^{5} + 30851132578 \nu^{4} + \cdots + 132247184234574 ) / 50803437784580 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 99170 \nu^{7} - 140821 \nu^{6} + 140781 \nu^{5} + 5777701 \nu^{4} + 747057527 \nu^{3} + \cdots + 19380524092 ) / 23917570690 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5690069035 \nu^{7} + 3991178383 \nu^{6} - 3261499808 \nu^{5} - 237335849778 \nu^{4} + \cdots - 378059897144086 ) / 965265317907020 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 15163829403 \nu^{7} + 4995591495 \nu^{6} + 440871714386 \nu^{5} + 11542895672784 \nu^{4} + \cdots + 30\!\cdots\!86 ) / 965265317907020 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{6} - \beta_{4} + 3\beta_{3} + 52\beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} - 44\beta_{6} + 53\beta_{5} + 2\beta_{4} - 2\beta_{3} - 3\beta_{2} + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 95\beta_{7} + 251\beta_{6} - 5\beta_{5} - 156\beta_{3} + 5\beta _1 - 2612 \)
|
| \(\nu^{5}\) | \(=\) |
\( -161\beta_{7} - 161\beta_{6} - 161\beta_{4} + 4239\beta_{3} + 545\beta_{2} - 2863\beta _1 - 3533 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9054\beta_{6} + 706\beta_{5} + 6941\beta_{4} - 15995\beta_{3} - 149842\beta_{2} + 706\beta _1 + 9054 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9760\beta_{7} + 313762\beta_{6} - 156783\beta_{5} - 9760\beta_{4} + 9760\beta_{3} + 57535\beta_{2} - 57535 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 577.1 |
|
−1.00000 | + | 1.00000i | −1.73205 | − | 2.00000i | −6.39181 | + | 6.39181i | 1.73205 | − | 1.73205i | −6.75784 | − | 6.75784i | 2.00000 | + | 2.00000i | 3.00000 | − | 12.7836i | ||||||||||||||||||||||||||||||
| 577.2 | −1.00000 | + | 1.00000i | −1.73205 | − | 2.00000i | 4.02578 | − | 4.02578i | 1.73205 | − | 1.73205i | 3.65976 | + | 3.65976i | 2.00000 | + | 2.00000i | 3.00000 | 8.05157i | ||||||||||||||||||||||||||||||||
| 577.3 | −1.00000 | + | 1.00000i | 1.73205 | − | 2.00000i | −5.04651 | + | 5.04651i | −1.73205 | + | 1.73205i | −3.68049 | − | 3.68049i | 2.00000 | + | 2.00000i | 3.00000 | − | 10.0930i | |||||||||||||||||||||||||||||||
| 577.4 | −1.00000 | + | 1.00000i | 1.73205 | − | 2.00000i | 4.41254 | − | 4.41254i | −1.73205 | + | 1.73205i | 5.77857 | + | 5.77857i | 2.00000 | + | 2.00000i | 3.00000 | 8.82508i | ||||||||||||||||||||||||||||||||
| 775.1 | −1.00000 | − | 1.00000i | −1.73205 | 2.00000i | −6.39181 | − | 6.39181i | 1.73205 | + | 1.73205i | −6.75784 | + | 6.75784i | 2.00000 | − | 2.00000i | 3.00000 | 12.7836i | |||||||||||||||||||||||||||||||||
| 775.2 | −1.00000 | − | 1.00000i | −1.73205 | 2.00000i | 4.02578 | + | 4.02578i | 1.73205 | + | 1.73205i | 3.65976 | − | 3.65976i | 2.00000 | − | 2.00000i | 3.00000 | − | 8.05157i | ||||||||||||||||||||||||||||||||
| 775.3 | −1.00000 | − | 1.00000i | 1.73205 | 2.00000i | −5.04651 | − | 5.04651i | −1.73205 | − | 1.73205i | −3.68049 | + | 3.68049i | 2.00000 | − | 2.00000i | 3.00000 | 10.0930i | |||||||||||||||||||||||||||||||||
| 775.4 | −1.00000 | − | 1.00000i | 1.73205 | 2.00000i | 4.41254 | + | 4.41254i | −1.73205 | − | 1.73205i | 5.77857 | − | 5.77857i | 2.00000 | − | 2.00000i | 3.00000 | − | 8.82508i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.3.f.h | 8 | |
| 13.b | even | 2 | 1 | 1014.3.f.j | 8 | ||
| 13.c | even | 3 | 1 | 78.3.l.c | ✓ | 8 | |
| 13.d | odd | 4 | 1 | inner | 1014.3.f.h | 8 | |
| 13.d | odd | 4 | 1 | 1014.3.f.j | 8 | ||
| 13.f | odd | 12 | 1 | 78.3.l.c | ✓ | 8 | |
| 39.i | odd | 6 | 1 | 234.3.bb.d | 8 | ||
| 39.k | even | 12 | 1 | 234.3.bb.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.3.l.c | ✓ | 8 | 13.c | even | 3 | 1 | |
| 78.3.l.c | ✓ | 8 | 13.f | odd | 12 | 1 | |
| 234.3.bb.d | 8 | 39.i | odd | 6 | 1 | ||
| 234.3.bb.d | 8 | 39.k | even | 12 | 1 | ||
| 1014.3.f.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 1014.3.f.h | 8 | 13.d | odd | 4 | 1 | inner | |
| 1014.3.f.j | 8 | 13.b | even | 2 | 1 | ||
| 1014.3.f.j | 8 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} - 282T_{5}^{5} + 4065T_{5}^{4} + 11916T_{5}^{3} + 38088T_{5}^{2} - 632592T_{5} + 5253264 \)
|
|
\( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 154T_{7}^{5} + 6817T_{7}^{4} + 1672T_{7}^{3} + 1568T_{7}^{2} - 117824T_{7} + 4426816 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{4} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots + 5253264 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 4426816 \)
|
| $11$ |
\( T^{8} + 12 T^{7} + \cdots + 973440000 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 4567597056 \)
|
| $19$ |
\( T^{8} + 44 T^{7} + \cdots + 1763584 \)
|
| $23$ |
\( T^{8} + \cdots + 17831863296 \)
|
| $29$ |
\( (T^{4} + 36 T^{3} + \cdots + 220452)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 33544655104 \)
|
| $37$ |
\( T^{8} + \cdots + 10744151716 \)
|
| $41$ |
\( T^{8} + \cdots + 370617958656 \)
|
| $43$ |
\( T^{8} + \cdots + 1819196698176 \)
|
| $47$ |
\( T^{8} + \cdots + 20494380893184 \)
|
| $53$ |
\( (T^{4} - 42 T^{3} + \cdots - 109128)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 15611728564224 \)
|
| $61$ |
\( (T^{4} - 90 T^{3} + \cdots - 17180643)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 42600893548096 \)
|
| $71$ |
\( T^{8} + \cdots + 1623606027264 \)
|
| $73$ |
\( T^{8} + \cdots + 261324417601 \)
|
| $79$ |
\( (T^{4} + 48 T^{3} + \cdots - 8209344)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 13865554427904 \)
|
| $89$ |
\( T^{8} + \cdots + 29\!\cdots\!64 \)
|
| $97$ |
\( T^{8} + \cdots + 14\!\cdots\!96 \)
|
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