Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [182,2,Mod(9,182)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(182, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("182.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 182 = 2 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 182.h (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: | \(1.45327731679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 10.0.23207289578928.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{9} - \nu^{8} - 21\nu^{7} + 35\nu^{6} + 68\nu^{5} - 111\nu^{4} - 201\nu^{3} + 351\nu^{2} + 405\nu - 891 ) / 81 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{9} + 2\nu^{8} + 45\nu^{7} - 28\nu^{6} - 136\nu^{5} + 99\nu^{4} + 393\nu^{3} - 342\nu^{2} - 675\nu + 567 ) / 81 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + 2\nu^{7} - 8\nu^{6} - 5\nu^{5} + 26\nu^{4} + 13\nu^{3} - 69\nu^{2} - 24\nu + 162 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} - 2\nu^{7} + 8\nu^{6} + 5\nu^{5} - 26\nu^{4} - 13\nu^{3} + 78\nu^{2} + 33\nu - 171 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{9} + 20\nu^{8} + 15\nu^{7} - 88\nu^{6} - 37\nu^{5} + 276\nu^{4} + 87\nu^{3} - 783\nu^{2} + 108\nu + 972 ) / 81 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8\nu^{9} + 5\nu^{8} - 30\nu^{7} - 31\nu^{6} + 92\nu^{5} + 96\nu^{4} - 273\nu^{3} - 243\nu^{2} + 459\nu + 729 ) / 81 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8\nu^{9} - 4\nu^{8} - 57\nu^{7} + 59\nu^{6} + 164\nu^{5} - 174\nu^{4} - 480\nu^{3} + 567\nu^{2} + 891\nu - 1215 ) / 81 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -6\nu^{9} + 2\nu^{8} + 35\nu^{7} - 24\nu^{6} - 109\nu^{5} + 77\nu^{4} + 306\nu^{3} - 273\nu^{2} - 531\nu + 432 ) / 27 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 23 \nu^{9} + 37 \nu^{8} + 141 \nu^{7} - 245 \nu^{6} - 410 \nu^{5} + 780 \nu^{4} + 1119 \nu^{3} + \cdots + 3969 ) / 81 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{2} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \cdots + 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{9} - 2\beta_{8} + 9\beta_{7} - 10\beta_{6} + \beta_{5} - 2\beta_{4} + 9\beta_{3} + 5\beta_{2} - 6\beta _1 - 7 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{9} - 19 \beta_{8} - 12 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} + \cdots + 1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{9} - 8\beta_{8} + 24\beta_{7} - 13\beta_{6} + \beta_{5} - 17\beta_{4} + 35\beta_{2} - 3\beta _1 + 41 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 19 \beta_{9} - 28 \beta_{8} - 21 \beta_{7} + \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 23 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 49 \beta_{9} + 46 \beta_{8} - 21 \beta_{7} + 41 \beta_{6} + 64 \beta_{5} - 32 \beta_{4} - 12 \beta_{3} + \cdots + 185 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 23 \beta_{9} + 29 \beta_{8} - 45 \beta_{7} + 151 \beta_{6} + 2 \beta_{5} + 50 \beta_{4} + 3 \beta_{3} + \cdots - 104 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(157\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
−0.500000 | − | 0.866025i | −2.39949 | −0.500000 | + | 0.866025i | −0.561520 | + | 0.972581i | 1.19975 | + | 2.07802i | 2.64005 | + | 0.173642i | 1.00000 | 2.75756 | 1.12304 | ||||||||||||||||||||||||||||||||||||||
| 9.2 | −0.500000 | − | 0.866025i | −1.44440 | −0.500000 | + | 0.866025i | 0.451759 | − | 0.782469i | 0.722201 | + | 1.25089i | −0.686411 | − | 2.55516i | 1.00000 | −0.913706 | −0.903518 | |||||||||||||||||||||||||||||||||||||||
| 9.3 | −0.500000 | − | 0.866025i | 0.160199 | −0.500000 | + | 0.866025i | −1.66520 | + | 2.88422i | −0.0800993 | − | 0.138736i | −0.402064 | + | 2.61502i | 1.00000 | −2.97434 | 3.33041 | |||||||||||||||||||||||||||||||||||||||
| 9.4 | −0.500000 | − | 0.866025i | 2.05726 | −0.500000 | + | 0.866025i | 1.69274 | − | 2.93192i | −1.02863 | − | 1.78164i | −2.60522 | + | 0.461340i | 1.00000 | 1.23231 | −3.38549 | |||||||||||||||||||||||||||||||||||||||
| 9.5 | −0.500000 | − | 0.866025i | 2.62644 | −0.500000 | + | 0.866025i | −0.917780 | + | 1.58964i | −1.31322 | − | 2.27456i | 1.05365 | − | 2.42690i | 1.00000 | 3.89817 | 1.83556 | |||||||||||||||||||||||||||||||||||||||
| 81.1 | −0.500000 | + | 0.866025i | −2.39949 | −0.500000 | − | 0.866025i | −0.561520 | − | 0.972581i | 1.19975 | − | 2.07802i | 2.64005 | − | 0.173642i | 1.00000 | 2.75756 | 1.12304 | |||||||||||||||||||||||||||||||||||||||
| 81.2 | −0.500000 | + | 0.866025i | −1.44440 | −0.500000 | − | 0.866025i | 0.451759 | + | 0.782469i | 0.722201 | − | 1.25089i | −0.686411 | + | 2.55516i | 1.00000 | −0.913706 | −0.903518 | |||||||||||||||||||||||||||||||||||||||
| 81.3 | −0.500000 | + | 0.866025i | 0.160199 | −0.500000 | − | 0.866025i | −1.66520 | − | 2.88422i | −0.0800993 | + | 0.138736i | −0.402064 | − | 2.61502i | 1.00000 | −2.97434 | 3.33041 | |||||||||||||||||||||||||||||||||||||||
| 81.4 | −0.500000 | + | 0.866025i | 2.05726 | −0.500000 | − | 0.866025i | 1.69274 | + | 2.93192i | −1.02863 | + | 1.78164i | −2.60522 | − | 0.461340i | 1.00000 | 1.23231 | −3.38549 | |||||||||||||||||||||||||||||||||||||||
| 81.5 | −0.500000 | + | 0.866025i | 2.62644 | −0.500000 | − | 0.866025i | −0.917780 | − | 1.58964i | −1.31322 | + | 2.27456i | 1.05365 | + | 2.42690i | 1.00000 | 3.89817 | 1.83556 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.g | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 182.2.h.d | yes | 10 |
| 3.b | odd | 2 | 1 | 1638.2.p.k | 10 | ||
| 7.b | odd | 2 | 1 | 1274.2.h.s | 10 | ||
| 7.c | even | 3 | 1 | 182.2.e.d | ✓ | 10 | |
| 7.c | even | 3 | 1 | 1274.2.g.p | 10 | ||
| 7.d | odd | 6 | 1 | 1274.2.e.s | 10 | ||
| 7.d | odd | 6 | 1 | 1274.2.g.q | 10 | ||
| 13.c | even | 3 | 1 | 182.2.e.d | ✓ | 10 | |
| 21.h | odd | 6 | 1 | 1638.2.m.j | 10 | ||
| 39.i | odd | 6 | 1 | 1638.2.m.j | 10 | ||
| 91.g | even | 3 | 1 | inner | 182.2.h.d | yes | 10 |
| 91.h | even | 3 | 1 | 1274.2.g.p | 10 | ||
| 91.m | odd | 6 | 1 | 1274.2.h.s | 10 | ||
| 91.n | odd | 6 | 1 | 1274.2.e.s | 10 | ||
| 91.v | odd | 6 | 1 | 1274.2.g.q | 10 | ||
| 273.bm | odd | 6 | 1 | 1638.2.p.k | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.2.e.d | ✓ | 10 | 7.c | even | 3 | 1 | |
| 182.2.e.d | ✓ | 10 | 13.c | even | 3 | 1 | |
| 182.2.h.d | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 182.2.h.d | yes | 10 | 91.g | even | 3 | 1 | inner |
| 1274.2.e.s | 10 | 7.d | odd | 6 | 1 | ||
| 1274.2.e.s | 10 | 91.n | odd | 6 | 1 | ||
| 1274.2.g.p | 10 | 7.c | even | 3 | 1 | ||
| 1274.2.g.p | 10 | 91.h | even | 3 | 1 | ||
| 1274.2.g.q | 10 | 7.d | odd | 6 | 1 | ||
| 1274.2.g.q | 10 | 91.v | odd | 6 | 1 | ||
| 1274.2.h.s | 10 | 7.b | odd | 2 | 1 | ||
| 1274.2.h.s | 10 | 91.m | odd | 6 | 1 | ||
| 1638.2.m.j | 10 | 21.h | odd | 6 | 1 | ||
| 1638.2.m.j | 10 | 39.i | odd | 6 | 1 | ||
| 1638.2.p.k | 10 | 3.b | odd | 2 | 1 | ||
| 1638.2.p.k | 10 | 273.bm | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(182, [\chi])\):
|
\( T_{3}^{5} - T_{3}^{4} - 9T_{3}^{3} + 6T_{3}^{2} + 18T_{3} - 3 \)
|
|
\( T_{5}^{10} + 2 T_{5}^{9} + 16 T_{5}^{8} + 26 T_{5}^{7} + 187 T_{5}^{6} + 293 T_{5}^{5} + 667 T_{5}^{4} + \cdots + 441 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{5} \)
|
| $3$ |
\( (T^{5} - T^{4} - 9 T^{3} + \cdots - 3)^{2} \)
|
| $5$ |
\( T^{10} + 2 T^{9} + \cdots + 441 \)
|
| $7$ |
\( T^{10} + 4 T^{8} + \cdots + 16807 \)
|
| $11$ |
\( (T^{5} - 4 T^{4} - 24 T^{3} + \cdots + 9)^{2} \)
|
| $13$ |
\( T^{10} + 6 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( T^{10} + 3 T^{9} + \cdots + 6718464 \)
|
| $19$ |
\( (T^{5} - 5 T^{4} + \cdots - 7089)^{2} \)
|
| $23$ |
\( T^{10} - 4 T^{9} + \cdots + 144 \)
|
| $29$ |
\( T^{10} - T^{9} + \cdots + 103041 \)
|
| $31$ |
\( T^{10} + 21 T^{9} + \cdots + 32228329 \)
|
| $37$ |
\( T^{10} - 10 T^{9} + \cdots + 144 \)
|
| $41$ |
\( T^{10} + 9 T^{9} + \cdots + 7733961 \)
|
| $43$ |
\( T^{10} - 9 T^{9} + \cdots + 6538249 \)
|
| $47$ |
\( T^{10} + 13 T^{9} + \cdots + 9529569 \)
|
| $53$ |
\( T^{10} + 15 T^{9} + \cdots + 5948721 \)
|
| $59$ |
\( T^{10} - T^{9} + \cdots + 52012944 \)
|
| $61$ |
\( (T^{5} - 15 T^{4} + \cdots + 22213)^{2} \)
|
| $67$ |
\( (T^{5} - 220 T^{3} + \cdots - 4703)^{2} \)
|
| $71$ |
\( T^{10} + \cdots + 1069878681 \)
|
| $73$ |
\( T^{10} + 4 T^{9} + \cdots + 41744521 \)
|
| $79$ |
\( T^{10} + T^{9} + \cdots + 66049 \)
|
| $83$ |
\( (T^{5} - 48 T^{4} + \cdots - 20412)^{2} \)
|
| $89$ |
\( T^{10} + 9 T^{9} + \cdots + 44836416 \)
|
| $97$ |
\( T^{10} + 19 T^{9} + \cdots + 42523441 \)
|
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