Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1200,3,Mod(401,1200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1200.401");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1200.l (of order \(2\), degree \(1\), not 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: | \(32.6976317232\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.574198272.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 600) |
| 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
\( x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -14\nu^{5} - 65\nu^{4} + 488\nu^{3} + 557\nu^{2} + 480\nu - 9489 ) / 885 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{5} + 5\nu^{4} + 26\nu^{3} + 234\nu^{2} - 400\nu - 28 ) / 295 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{5} + 64\nu^{4} + 26\nu^{3} + 116\nu^{2} - 2052\nu - 795 ) / 177 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -24\nu^{5} + 15\nu^{4} + 78\nu^{3} + 702\nu^{2} - 20\nu - 84 ) / 295 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -22\nu^{5} - 60\nu^{4} - 76\nu^{3} + 1086\nu^{2} + 2440\nu + 3463 ) / 295 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 3\beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 7\beta_{2} + 4\beta _1 + 47 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 5\beta_{4} + 5\beta_{3} - 30\beta_{2} + 2\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 61\beta_{5} - 74\beta_{4} + 68\beta_{3} - 176\beta_{2} + 74\beta _1 + 345 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).
| \(n\) | \(401\) | \(577\) | \(751\) | \(901\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 401.1 |
|
0 | −2.56627 | − | 1.55378i | 0 | 0 | 0 | 1.34301 | 0 | 4.17150 | + | 7.97487i | 0 | ||||||||||||||||||||||||||||||||
| 401.2 | 0 | −2.56627 | + | 1.55378i | 0 | 0 | 0 | 1.34301 | 0 | 4.17150 | − | 7.97487i | 0 | |||||||||||||||||||||||||||||||||
| 401.3 | 0 | 1.78875 | − | 2.40839i | 0 | 0 | 0 | −12.2014 | 0 | −2.60072 | − | 8.61605i | 0 | |||||||||||||||||||||||||||||||||
| 401.4 | 0 | 1.78875 | + | 2.40839i | 0 | 0 | 0 | −12.2014 | 0 | −2.60072 | + | 8.61605i | 0 | |||||||||||||||||||||||||||||||||
| 401.5 | 0 | 2.77752 | − | 1.13375i | 0 | 0 | 0 | 5.85843 | 0 | 6.42922 | − | 6.29803i | 0 | |||||||||||||||||||||||||||||||||
| 401.6 | 0 | 2.77752 | + | 1.13375i | 0 | 0 | 0 | 5.85843 | 0 | 6.42922 | + | 6.29803i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1200.3.l.w | 6 | |
| 3.b | odd | 2 | 1 | inner | 1200.3.l.w | 6 | |
| 4.b | odd | 2 | 1 | 600.3.l.d | ✓ | 6 | |
| 5.b | even | 2 | 1 | 1200.3.l.v | 6 | ||
| 5.c | odd | 4 | 2 | 1200.3.c.l | 12 | ||
| 12.b | even | 2 | 1 | 600.3.l.d | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1200.3.l.v | 6 | ||
| 15.e | even | 4 | 2 | 1200.3.c.l | 12 | ||
| 20.d | odd | 2 | 1 | 600.3.l.e | yes | 6 | |
| 20.e | even | 4 | 2 | 600.3.c.c | 12 | ||
| 60.h | even | 2 | 1 | 600.3.l.e | yes | 6 | |
| 60.l | odd | 4 | 2 | 600.3.c.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.3.c.c | 12 | 20.e | even | 4 | 2 | ||
| 600.3.c.c | 12 | 60.l | odd | 4 | 2 | ||
| 600.3.l.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 600.3.l.d | ✓ | 6 | 12.b | even | 2 | 1 | |
| 600.3.l.e | yes | 6 | 20.d | odd | 2 | 1 | |
| 600.3.l.e | yes | 6 | 60.h | even | 2 | 1 | |
| 1200.3.c.l | 12 | 5.c | odd | 4 | 2 | ||
| 1200.3.c.l | 12 | 15.e | even | 4 | 2 | ||
| 1200.3.l.v | 6 | 5.b | even | 2 | 1 | ||
| 1200.3.l.v | 6 | 15.d | odd | 2 | 1 | ||
| 1200.3.l.w | 6 | 1.a | even | 1 | 1 | trivial | |
| 1200.3.l.w | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):
|
\( T_{7}^{3} + 5T_{7}^{2} - 80T_{7} + 96 \)
|
|
\( T_{11}^{6} + 246T_{11}^{4} + 15129T_{11}^{2} + 161312 \)
|
|
\( T_{13}^{3} - 13T_{13}^{2} - 32T_{13} + 540 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 729 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{3} + 5 T^{2} - 80 T + 96)^{2} \)
|
| $11$ |
\( T^{6} + 246 T^{4} + \cdots + 161312 \)
|
| $13$ |
\( (T^{3} - 13 T^{2} + \cdots + 540)^{2} \)
|
| $17$ |
\( T^{6} + 1174 T^{4} + \cdots + 9435168 \)
|
| $19$ |
\( (T^{3} + 25 T^{2} + \cdots - 877)^{2} \)
|
| $23$ |
\( T^{6} + 2680 T^{4} + \cdots + 423055872 \)
|
| $29$ |
\( T^{6} + 2616 T^{4} + \cdots + 37601792 \)
|
| $31$ |
\( (T^{3} - 57 T^{2} + \cdots + 1804)^{2} \)
|
| $37$ |
\( (T^{3} - 38 T^{2} + \cdots + 69696)^{2} \)
|
| $41$ |
\( T^{6} + 4806 T^{4} + \cdots + 22418208 \)
|
| $43$ |
\( (T^{3} + T^{2} + \cdots - 59628)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 187644280832 \)
|
| $53$ |
\( T^{6} + \cdots + 24652657152 \)
|
| $59$ |
\( T^{6} + \cdots + 91824979968 \)
|
| $61$ |
\( (T^{3} - 31 T^{2} + \cdots + 117664)^{2} \)
|
| $67$ |
\( (T^{3} + 211 T^{2} + \cdots + 324893)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 15890452992 \)
|
| $73$ |
\( (T^{3} - 36 T^{2} + \cdots + 210978)^{2} \)
|
| $79$ |
\( (T^{3} - 38 T^{2} + \cdots - 370080)^{2} \)
|
| $83$ |
\( T^{6} + 5206 T^{4} + \cdots + 1152 \)
|
| $89$ |
\( T^{6} + \cdots + 5739061248 \)
|
| $97$ |
\( (T^{3} + 235 T^{2} + \cdots - 613636)^{2} \)
|
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