Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [66,3,Mod(23,66)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("66.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 66.c (of order \(2\), degree \(1\), 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: | \(1.79836974478\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 5x^{6} - 50x^{5} - 34x^{4} + 586x^{3} - 431x^{2} - 1830x + 3051 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 5x^{6} - 50x^{5} - 34x^{4} + 586x^{3} - 431x^{2} - 1830x + 3051 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -191\nu^{7} - 1099\nu^{6} - 2132\nu^{5} - 5086\nu^{4} + 56140\nu^{3} + 38126\nu^{2} - 134277\nu + 131895 ) / 271512 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 335\nu^{7} - 389\nu^{6} + 1844\nu^{5} - 14798\nu^{4} - 24388\nu^{3} + 148702\nu^{2} - 60087\nu - 470151 ) / 135756 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 919\nu^{7} + 839\nu^{6} + 10732\nu^{5} - 23650\nu^{4} - 54836\nu^{3} + 48002\nu^{2} - 68163\nu - 710667 ) / 271512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} - 4\nu^{5} + 38\nu^{4} + 164\nu^{3} - 190\nu^{2} - 771\nu + 1053 ) / 216 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1481 \nu^{7} + 1177 \nu^{6} + 14636 \nu^{5} - 49646 \nu^{4} - 150052 \nu^{3} + 216598 \nu^{2} + \cdots - 311229 ) / 271512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 389\nu^{7} + 310\nu^{6} + 1736\nu^{5} - 11570\nu^{4} - 53962\nu^{3} + 150256\nu^{2} + 174363\nu - 630936 ) / 67878 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{7} - 3\beta_{6} + 6\beta_{4} - 6\beta_{3} + 9\beta_{2} - 2\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{7} - 17\beta_{6} + \beta_{5} + 11\beta_{4} + 26\beta_{3} - 35\beta_{2} + 27\beta _1 + 65 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{7} + 87\beta_{6} + 45\beta_{5} - 78\beta_{4} + 15\beta_{3} - 42\beta_{2} + 88\beta _1 - 363 \)
|
| \(\nu^{6}\) | \(=\) |
\( 250\beta_{7} + 100\beta_{6} + 46\beta_{5} + 50\beta_{4} - 622\beta_{3} + 568\beta_{2} - 540\beta _1 - 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( -90\beta_{7} - 1776\beta_{6} - 594\beta_{5} + 1854\beta_{4} + 756\beta_{3} - 444\beta_{2} + 115\beta _1 + 7710 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
− | 1.41421i | −2.78135 | + | 1.12433i | −2.00000 | − | 7.21302i | 1.59004 | + | 3.93342i | −12.6357 | 2.82843i | 6.47176 | − | 6.25430i | −10.2007 | ||||||||||||||||||||||||||||||||||
| 23.2 | − | 1.41421i | −1.56045 | + | 2.56222i | −2.00000 | 8.92927i | 3.62353 | + | 2.20682i | 5.69565 | 2.82843i | −4.12996 | − | 7.99646i | 12.6279 | ||||||||||||||||||||||||||||||||||||
| 23.3 | − | 1.41421i | 1.06045 | − | 2.80632i | −2.00000 | − | 1.80782i | −3.96874 | − | 1.49971i | 0.994762 | 2.82843i | −6.75087 | − | 5.95195i | −2.55664 | |||||||||||||||||||||||||||||||||||
| 23.4 | − | 1.41421i | 2.28135 | + | 1.94820i | −2.00000 | − | 5.56529i | 2.75516 | − | 3.22631i | 9.94529 | 2.82843i | 1.40907 | + | 8.88901i | −7.87050 | |||||||||||||||||||||||||||||||||||
| 23.5 | 1.41421i | −2.78135 | − | 1.12433i | −2.00000 | 7.21302i | 1.59004 | − | 3.93342i | −12.6357 | − | 2.82843i | 6.47176 | + | 6.25430i | −10.2007 | ||||||||||||||||||||||||||||||||||||
| 23.6 | 1.41421i | −1.56045 | − | 2.56222i | −2.00000 | − | 8.92927i | 3.62353 | − | 2.20682i | 5.69565 | − | 2.82843i | −4.12996 | + | 7.99646i | 12.6279 | |||||||||||||||||||||||||||||||||||
| 23.7 | 1.41421i | 1.06045 | + | 2.80632i | −2.00000 | 1.80782i | −3.96874 | + | 1.49971i | 0.994762 | − | 2.82843i | −6.75087 | + | 5.95195i | −2.55664 | ||||||||||||||||||||||||||||||||||||
| 23.8 | 1.41421i | 2.28135 | − | 1.94820i | −2.00000 | 5.56529i | 2.75516 | + | 3.22631i | 9.94529 | − | 2.82843i | 1.40907 | − | 8.88901i | −7.87050 | ||||||||||||||||||||||||||||||||||||
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 | 66.3.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 66.3.c.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 528.3.i.e | 8 | ||
| 11.b | odd | 2 | 1 | 726.3.c.e | 8 | ||
| 12.b | even | 2 | 1 | 528.3.i.e | 8 | ||
| 33.d | even | 2 | 1 | 726.3.c.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.3.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 66.3.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 528.3.i.e | 8 | 4.b | odd | 2 | 1 | ||
| 528.3.i.e | 8 | 12.b | even | 2 | 1 | ||
| 726.3.c.e | 8 | 11.b | odd | 2 | 1 | ||
| 726.3.c.e | 8 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(66, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( T^{8} + 2 T^{7} + \cdots + 6561 \)
|
| $5$ |
\( T^{8} + 166 T^{6} + \cdots + 419904 \)
|
| $7$ |
\( (T^{4} - 4 T^{3} + \cdots - 712)^{2} \)
|
| $11$ |
\( (T^{2} + 11)^{4} \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 10464)^{2} \)
|
| $17$ |
\( T^{8} + 460 T^{6} + \cdots + 419904 \)
|
| $19$ |
\( (T^{4} - 32 T^{3} + \cdots - 227952)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 193023150336 \)
|
| $29$ |
\( T^{8} + 1604 T^{6} + \cdots + 6718464 \)
|
| $31$ |
\( (T^{4} - 14 T^{3} + \cdots - 52488)^{2} \)
|
| $37$ |
\( (T^{4} - 2 T^{3} + \cdots - 288624)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 6249851136 \)
|
| $43$ |
\( (T^{4} + 60 T^{3} + \cdots + 38248)^{2} \)
|
| $47$ |
\( T^{8} + 4856 T^{6} + \cdots + 241864704 \)
|
| $53$ |
\( T^{8} + \cdots + 15359376162816 \)
|
| $59$ |
\( T^{8} + \cdots + 16592000995584 \)
|
| $61$ |
\( (T^{4} - 12 T^{3} + \cdots + 2535424)^{2} \)
|
| $67$ |
\( (T^{4} + 10 T^{3} + \cdots + 15215784)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 18067226309136 \)
|
| $73$ |
\( (T^{4} - 196 T^{3} + \cdots + 1310112)^{2} \)
|
| $79$ |
\( (T^{4} + 20 T^{3} + \cdots + 50633192)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 74341367156736 \)
|
| $89$ |
\( T^{8} + \cdots + 83339079192576 \)
|
| $97$ |
\( (T^{4} + 66 T^{3} + \cdots + 22039744)^{2} \)
|
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