Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [186,3,Mod(37,186)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(186, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("186.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 186 = 2 \cdot 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 186.g (of order \(6\), 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: | \(5.06813291710\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.67783163904.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 10x^{6} - 8x^{5} + 66x^{4} + 40x^{3} - 324x^{2} + 136x + 1156 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 10x^{6} - 8x^{5} + 66x^{4} + 40x^{3} - 324x^{2} + 136x + 1156 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 331 \nu^{7} + 3060 \nu^{6} - 5836 \nu^{5} - 17548 \nu^{4} - 34086 \nu^{3} + 145098 \nu^{2} + \cdots - 398752 ) / 561442 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 500 \nu^{7} - 2805 \nu^{6} - 3300 \nu^{5} + 14513 \nu^{4} + 44220 \nu^{3} - 171930 \nu^{2} + \cdots + 392258 ) / 561442 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 891 \nu^{7} - 1819 \nu^{6} - 2140 \nu^{5} + 67257 \nu^{4} + 28676 \nu^{3} - 111494 \nu^{2} + \cdots + 618460 ) / 561442 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -80\nu^{7} - 23\nu^{6} + 528\nu^{5} + 320\nu^{4} - 470\nu^{3} + 1088\nu^{2} + 9248\nu - 29924 ) / 33026 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2169 \nu^{7} - 10965 \nu^{6} - 22336 \nu^{5} + 55017 \nu^{4} + 187014 \nu^{3} - 153110 \nu^{2} + \cdots + 1562538 ) / 561442 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2474 \nu^{7} + 1360 \nu^{6} - 24349 \nu^{5} - 28768 \nu^{4} + 157844 \nu^{3} + 106950 \nu^{2} + \cdots + 179248 ) / 561442 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 5\beta_{3} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{7} - \beta_{6} + 6\beta_{5} + \beta_{4} - 4\beta_{2} + 4\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -10\beta_{5} + 10\beta_{4} - 16\beta_{3} - 10\beta_{2} + 4\beta _1 - 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{6} - 50\beta_{3} - 70\beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( 56\beta_{7} - 70\beta_{6} - 42\beta_{5} + 70\beta_{4} - 56\beta_{2} + 56\beta _1 + 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( -476\beta_{5} + 14\beta_{4} - 462\beta_{3} - 476\beta_{2} + 92\beta _1 - 462 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).
| \(n\) | \(125\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−1.41421 | −1.50000 | + | 0.866025i | 2.00000 | −4.04497 | + | 7.00610i | 2.12132 | − | 1.22474i | −5.42756 | − | 9.40081i | −2.82843 | 1.50000 | − | 2.59808i | 5.72045 | − | 9.90812i | ||||||||||||||||||||||||||||||
| 37.2 | −1.41421 | −1.50000 | + | 0.866025i | 2.00000 | 0.509438 | − | 0.882372i | 2.12132 | − | 1.22474i | 1.01335 | + | 1.75517i | −2.82843 | 1.50000 | − | 2.59808i | −0.720453 | + | 1.24786i | |||||||||||||||||||||||||||||||
| 37.3 | 1.41421 | −1.50000 | + | 0.866025i | 2.00000 | 0.0172414 | − | 0.0298629i | −2.12132 | + | 1.22474i | 1.68272 | + | 2.91456i | 2.82843 | 1.50000 | − | 2.59808i | 0.0243830 | − | 0.0422326i | |||||||||||||||||||||||||||||||
| 37.4 | 1.41421 | −1.50000 | + | 0.866025i | 2.00000 | 3.51829 | − | 6.09386i | −2.12132 | + | 1.22474i | −3.26851 | − | 5.66123i | 2.82843 | 1.50000 | − | 2.59808i | 4.97562 | − | 8.61802i | |||||||||||||||||||||||||||||||
| 181.1 | −1.41421 | −1.50000 | − | 0.866025i | 2.00000 | −4.04497 | − | 7.00610i | 2.12132 | + | 1.22474i | −5.42756 | + | 9.40081i | −2.82843 | 1.50000 | + | 2.59808i | 5.72045 | + | 9.90812i | |||||||||||||||||||||||||||||||
| 181.2 | −1.41421 | −1.50000 | − | 0.866025i | 2.00000 | 0.509438 | + | 0.882372i | 2.12132 | + | 1.22474i | 1.01335 | − | 1.75517i | −2.82843 | 1.50000 | + | 2.59808i | −0.720453 | − | 1.24786i | |||||||||||||||||||||||||||||||
| 181.3 | 1.41421 | −1.50000 | − | 0.866025i | 2.00000 | 0.0172414 | + | 0.0298629i | −2.12132 | − | 1.22474i | 1.68272 | − | 2.91456i | 2.82843 | 1.50000 | + | 2.59808i | 0.0243830 | + | 0.0422326i | |||||||||||||||||||||||||||||||
| 181.4 | 1.41421 | −1.50000 | − | 0.866025i | 2.00000 | 3.51829 | + | 6.09386i | −2.12132 | − | 1.22474i | −3.26851 | + | 5.66123i | 2.82843 | 1.50000 | + | 2.59808i | 4.97562 | + | 8.61802i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.e | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 186.3.g.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 558.3.n.a | 8 | ||
| 31.e | odd | 6 | 1 | inner | 186.3.g.a | ✓ | 8 |
| 93.g | even | 6 | 1 | 558.3.n.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.3.g.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 186.3.g.a | ✓ | 8 | 31.e | odd | 6 | 1 | inner |
| 558.3.n.a | 8 | 3.b | odd | 2 | 1 | ||
| 558.3.n.a | 8 | 93.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 58T_{5}^{6} - 120T_{5}^{5} + 3366T_{5}^{4} - 3480T_{5}^{3} + 3484T_{5}^{2} - 120T_{5} + 4 \)
acting on \(S_{3}^{\mathrm{new}}(186, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{4} \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{4} \)
|
| $5$ |
\( T^{8} + 58 T^{6} + \cdots + 4 \)
|
| $7$ |
\( T^{8} + 12 T^{7} + \cdots + 234256 \)
|
| $11$ |
\( T^{8} - 24 T^{7} + \cdots + 196 \)
|
| $13$ |
\( T^{8} - 72 T^{7} + \cdots + 40666129 \)
|
| $17$ |
\( T^{8} + \cdots + 11989374016 \)
|
| $19$ |
\( T^{8} + \cdots + 1183841649 \)
|
| $23$ |
\( T^{8} + \cdots + 13685256256 \)
|
| $29$ |
\( T^{8} + 1660 T^{6} + \cdots + 122633476 \)
|
| $31$ |
\( T^{8} + \cdots + 852891037441 \)
|
| $37$ |
\( T^{8} - 120 T^{7} + \cdots + 466948881 \)
|
| $41$ |
\( T^{8} + \cdots + 1951312847236 \)
|
| $43$ |
\( T^{8} + \cdots + 1481086566001 \)
|
| $47$ |
\( (T^{4} + 220 T^{3} + \cdots + 2149822)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 12177880461124 \)
|
| $59$ |
\( T^{8} + \cdots + 132269136692224 \)
|
| $61$ |
\( T^{8} + \cdots + 4768475811856 \)
|
| $67$ |
\( T^{8} + \cdots + 3885519053584 \)
|
| $71$ |
\( T^{8} + \cdots + 141581631820864 \)
|
| $73$ |
\( T^{8} + \cdots + 888991530353521 \)
|
| $79$ |
\( T^{8} + \cdots + 17\!\cdots\!56 \)
|
| $83$ |
\( T^{8} + \cdots + 18694883242564 \)
|
| $89$ |
\( T^{8} + \cdots + 496524691071556 \)
|
| $97$ |
\( (T^{4} - 20 T^{3} + \cdots - 296087)^{2} \)
|
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