Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [357,2,Mod(205,357)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(357, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("357.205");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 357 = 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 357.i (of order \(3\), 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: | \(2.85065935216\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 10.0.5743021975227.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4x^{9} + 2x^{8} + 10x^{7} - 8x^{6} - 12x^{5} - 24x^{4} + 90x^{3} + 54x^{2} - 324x + 243 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{10} - 4x^{9} + 2x^{8} + 10x^{7} - 8x^{6} - 12x^{5} - 24x^{4} + 90x^{3} + 54x^{2} - 324x + 243 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{9} + 4\nu^{8} - 2\nu^{7} - 10\nu^{6} + 8\nu^{5} + 12\nu^{4} + 24\nu^{3} - 90\nu^{2} - 54\nu + 324 ) / 81 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{9} + 17\nu^{8} + 11\nu^{7} - 38\nu^{6} - 26\nu^{5} + 39\nu^{4} + 219\nu^{3} - 189\nu^{2} - 594\nu + 810 ) / 162 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17 \nu^{9} + 32 \nu^{8} + 38 \nu^{7} - 89 \nu^{6} - 71 \nu^{5} + 69 \nu^{4} + 552 \nu^{3} + \cdots + 1863 ) / 162 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13 \nu^{9} + 43 \nu^{8} + 19 \nu^{7} - 130 \nu^{6} - 22 \nu^{5} + 183 \nu^{4} + 483 \nu^{3} + \cdots + 2916 ) / 162 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{9} - 11\nu^{8} - 18\nu^{7} + 32\nu^{6} + 33\nu^{5} - 13\nu^{4} - 237\nu^{3} + 78\nu^{2} + 720\nu - 567 ) / 54 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{9} - 7\nu^{8} - 6\nu^{7} + 20\nu^{6} + 12\nu^{5} - 17\nu^{4} - 103\nu^{3} + 84\nu^{2} + 330\nu - 414 ) / 18 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -4\nu^{9} + 8\nu^{8} + 9\nu^{7} - 23\nu^{6} - 15\nu^{5} + 16\nu^{4} + 132\nu^{3} - 87\nu^{2} - 423\nu + 459 ) / 18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 44 \nu^{9} - 107 \nu^{8} - 89 \nu^{7} + 317 \nu^{6} + 158 \nu^{5} - 306 \nu^{4} - 1569 \nu^{3} + \cdots - 6642 ) / 162 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{8} - 4\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - 2\beta_{8} - 2\beta_{6} + 4\beta_{5} + 4\beta_{4} - \beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} + 4\beta_{8} + 4\beta_{7} - 4\beta_{6} + 4\beta_{5} - 6\beta_{4} + 2\beta_{3} + 4\beta_{2} + 2\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{8} + 8\beta_{7} + 2\beta_{6} + 16\beta_{4} + 10\beta_{3} + 4\beta_{2} - 13\beta _1 + 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3 \beta_{9} + 24 \beta_{8} + 21 \beta_{7} + 19 \beta_{6} - \beta_{5} - \beta_{4} + 8 \beta_{3} + \cdots + 17 \)
|
| \(\nu^{8}\) | \(=\) |
\( 8 \beta_{9} + 20 \beta_{8} + 4 \beta_{7} + 24 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} + 31 \beta_{3} + \cdots - 15 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 12 \beta_{9} + 12 \beta_{8} + 16 \beta_{7} - 24 \beta_{6} + 4 \beta_{5} - 84 \beta_{4} + 48 \beta_{3} + \cdots + 88 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(190\) | \(239\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 205.1 |
|
−1.19803 | + | 2.07505i | 0.500000 | + | 0.866025i | −1.87055 | − | 3.23989i | −2.18232 | + | 3.77988i | −2.39606 | −2.64467 | + | 0.0756600i | 4.17178 | −0.500000 | + | 0.866025i | −5.22896 | − | 9.05683i | ||||||||||||||||||||||||||||||||||
| 205.2 | −0.565575 | + | 0.979604i | 0.500000 | + | 0.866025i | 0.360251 | + | 0.623973i | 0.634991 | − | 1.09984i | −1.13115 | 2.62416 | − | 0.337337i | −3.07729 | −0.500000 | + | 0.866025i | 0.718269 | + | 1.24408i | |||||||||||||||||||||||||||||||||||
| 205.3 | 0.614831 | − | 1.06492i | 0.500000 | + | 0.866025i | 0.243965 | + | 0.422560i | 1.20973 | − | 2.09531i | 1.22966 | −1.25146 | + | 2.33106i | 3.05931 | −0.500000 | + | 0.866025i | −1.48756 | − | 2.57653i | |||||||||||||||||||||||||||||||||||
| 205.4 | 0.903768 | − | 1.56537i | 0.500000 | + | 0.866025i | −0.633595 | − | 1.09742i | −1.32773 | + | 2.29969i | 1.80754 | 0.398245 | + | 2.61561i | 1.32458 | −0.500000 | + | 0.866025i | 2.39991 | + | 4.15677i | |||||||||||||||||||||||||||||||||||
| 205.5 | 1.24500 | − | 2.15641i | 0.500000 | + | 0.866025i | −2.10007 | − | 3.63743i | 1.16532 | − | 2.01840i | 2.49001 | −1.62628 | − | 2.08692i | −5.47838 | −0.500000 | + | 0.866025i | −2.90166 | − | 5.02583i | |||||||||||||||||||||||||||||||||||
| 256.1 | −1.19803 | − | 2.07505i | 0.500000 | − | 0.866025i | −1.87055 | + | 3.23989i | −2.18232 | − | 3.77988i | −2.39606 | −2.64467 | − | 0.0756600i | 4.17178 | −0.500000 | − | 0.866025i | −5.22896 | + | 9.05683i | |||||||||||||||||||||||||||||||||||
| 256.2 | −0.565575 | − | 0.979604i | 0.500000 | − | 0.866025i | 0.360251 | − | 0.623973i | 0.634991 | + | 1.09984i | −1.13115 | 2.62416 | + | 0.337337i | −3.07729 | −0.500000 | − | 0.866025i | 0.718269 | − | 1.24408i | |||||||||||||||||||||||||||||||||||
| 256.3 | 0.614831 | + | 1.06492i | 0.500000 | − | 0.866025i | 0.243965 | − | 0.422560i | 1.20973 | + | 2.09531i | 1.22966 | −1.25146 | − | 2.33106i | 3.05931 | −0.500000 | − | 0.866025i | −1.48756 | + | 2.57653i | |||||||||||||||||||||||||||||||||||
| 256.4 | 0.903768 | + | 1.56537i | 0.500000 | − | 0.866025i | −0.633595 | + | 1.09742i | −1.32773 | − | 2.29969i | 1.80754 | 0.398245 | − | 2.61561i | 1.32458 | −0.500000 | − | 0.866025i | 2.39991 | − | 4.15677i | |||||||||||||||||||||||||||||||||||
| 256.5 | 1.24500 | + | 2.15641i | 0.500000 | − | 0.866025i | −2.10007 | + | 3.63743i | 1.16532 | + | 2.01840i | 2.49001 | −1.62628 | + | 2.08692i | −5.47838 | −0.500000 | − | 0.866025i | −2.90166 | + | 5.02583i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 357.2.i.f | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1071.2.i.g | 10 | ||
| 7.c | even | 3 | 1 | inner | 357.2.i.f | ✓ | 10 |
| 7.c | even | 3 | 1 | 2499.2.a.ba | 5 | ||
| 7.d | odd | 6 | 1 | 2499.2.a.bb | 5 | ||
| 21.g | even | 6 | 1 | 7497.2.a.bw | 5 | ||
| 21.h | odd | 6 | 1 | 1071.2.i.g | 10 | ||
| 21.h | odd | 6 | 1 | 7497.2.a.bv | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.i.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 357.2.i.f | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 1071.2.i.g | 10 | 3.b | odd | 2 | 1 | ||
| 1071.2.i.g | 10 | 21.h | odd | 6 | 1 | ||
| 2499.2.a.ba | 5 | 7.c | even | 3 | 1 | ||
| 2499.2.a.bb | 5 | 7.d | odd | 6 | 1 | ||
| 7497.2.a.bv | 5 | 21.h | odd | 6 | 1 | ||
| 7497.2.a.bw | 5 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 2 T_{2}^{9} + 11 T_{2}^{8} - 14 T_{2}^{7} + 70 T_{2}^{6} - 85 T_{2}^{5} + 215 T_{2}^{4} + \cdots + 225 \)
acting on \(S_{2}^{\mathrm{new}}(357, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 225 \)
|
| $3$ |
\( (T^{2} - T + 1)^{5} \)
|
| $5$ |
\( T^{10} + T^{9} + \cdots + 6889 \)
|
| $7$ |
\( T^{10} + 5 T^{9} + \cdots + 16807 \)
|
| $11$ |
\( T^{10} - 11 T^{9} + \cdots + 248004 \)
|
| $13$ |
\( (T^{5} - 7 T^{4} + \cdots - 189)^{2} \)
|
| $17$ |
\( (T^{2} + T + 1)^{5} \)
|
| $19$ |
\( T^{10} + 9 T^{9} + \cdots + 11664 \)
|
| $23$ |
\( T^{10} - 23 T^{9} + \cdots + 891136 \)
|
| $29$ |
\( (T^{5} + 18 T^{4} + \cdots - 6441)^{2} \)
|
| $31$ |
\( T^{10} + 9 T^{9} + \cdots + 25472209 \)
|
| $37$ |
\( T^{10} + \cdots + 155850256 \)
|
| $41$ |
\( (T^{5} - 3 T^{4} + \cdots + 315)^{2} \)
|
| $43$ |
\( (T^{5} - 12 T^{4} + \cdots + 972)^{2} \)
|
| $47$ |
\( T^{10} + 11 T^{9} + \cdots + 9 \)
|
| $53$ |
\( T^{10} - 3 T^{9} + \cdots + 1763584 \)
|
| $59$ |
\( T^{10} - 14 T^{9} + \cdots + 994009 \)
|
| $61$ |
\( T^{10} + 29 T^{9} + \cdots + 968256 \)
|
| $67$ |
\( T^{10} + \cdots + 252047376 \)
|
| $71$ |
\( (T^{5} + 19 T^{4} + \cdots - 108)^{2} \)
|
| $73$ |
\( T^{10} + 11 T^{9} + \cdots + 4334724 \)
|
| $79$ |
\( T^{10} + T^{9} + \cdots + 1327104 \)
|
| $83$ |
\( (T^{5} - 5 T^{4} + \cdots + 3701)^{2} \)
|
| $89$ |
\( T^{10} + 8 T^{9} + \cdots + 100 \)
|
| $97$ |
\( (T^{5} - 19 T^{4} + \cdots - 284264)^{2} \)
|
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