Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [567,2,Mod(566,567)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("567.566");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 567.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: | \(4.52751779461\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 63) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -49\nu^{10} + 259\nu^{8} - 1369\nu^{6} + 861\nu^{4} - 252\nu^{2} - 7266 ) / 4299 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -280\nu^{10} + 1480\nu^{8} - 8437\nu^{6} + 13518\nu^{4} - 35832\nu^{2} + 5769 ) / 12897 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -730\nu^{11} + 5701\nu^{9} - 30748\nu^{7} + 76698\nu^{5} - 122898\nu^{3} + 66168\nu ) / 12897 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -161\nu^{10} + 851\nu^{8} - 3884\nu^{6} + 2829\nu^{4} - 828\nu^{2} - 6678 ) / 4299 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 494\nu^{11} - 3430\nu^{9} + 18130\nu^{7} - 38978\nu^{5} + 64569\nu^{3} - 34836\nu ) / 4299 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -592\nu^{11} + 3948\nu^{9} - 20868\nu^{7} + 40700\nu^{5} - 65073\nu^{3} + 3108\nu ) / 4299 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -296\nu^{10} + 1974\nu^{8} - 10434\nu^{6} + 20350\nu^{4} - 34686\nu^{2} + 5853 ) / 4299 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -680\nu^{11} + 5232\nu^{9} - 28269\nu^{7} + 68245\nu^{5} - 110358\nu^{3} + 59457\nu ) / 4299 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1100\nu^{11} - 7452\nu^{9} + 38775\nu^{7} - 75625\nu^{5} + 112518\nu^{3} - 5775\nu ) / 4299 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -416\nu^{11} + 2813\nu^{9} - 14664\nu^{7} + 28600\nu^{5} - 41854\nu^{3} + 2184\nu ) / 1433 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -2237\nu^{10} + 15509\nu^{8} - 81362\nu^{6} + 167049\nu^{4} - 249792\nu^{2} + 42912 ) / 12897 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - 3\beta_{3} ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{11} - 3\beta_{7} + \beta_{2} + \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{10} + 4\beta_{9} - \beta_{6} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{11} - 12\beta_{7} + \beta_{4} + 7\beta_{2} - 5\beta _1 - 7 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16\beta_{10} + 19\beta_{9} - 19\beta_{8} + 2\beta_{6} - 2\beta_{5} + 48\beta_{3} ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7\beta_{4} - 23\beta _1 - 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -67\beta_{10} - 88\beta_{9} - 88\beta_{8} - 23\beta_{6} - 23\beta_{5} + 201\beta_{3} ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -67\beta_{11} + 222\beta_{7} + 37\beta_{4} - 178\beta_{2} - 104\beta _1 - 118 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -289\beta_{10} - 400\beta_{9} - 134\beta_{6} ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -289\beta_{11} + 978\beta_{7} - 178\beta_{4} - 823\beta_{2} + 467\beta _1 + 511 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( -1267\beta_{10} - 1801\beta_{9} + 1801\beta_{8} - 668\beta_{6} + 668\beta_{5} - 3801\beta_{3} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(407\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 566.1 |
|
− | 2.27639i | 0 | −3.18194 | −1.43429 | 0 | 0.239123 | + | 2.63492i | 2.69056i | 0 | 3.26499i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.2 | − | 2.27639i | 0 | −3.18194 | 1.43429 | 0 | 0.239123 | − | 2.63492i | 2.69056i | 0 | − | 3.26499i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.3 | − | 1.18593i | 0 | 0.593579 | −2.83797 | 0 | 2.46050 | + | 0.972582i | − | 3.07579i | 0 | 3.36562i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.4 | − | 1.18593i | 0 | 0.593579 | 2.83797 | 0 | 2.46050 | − | 0.972582i | − | 3.07579i | 0 | − | 3.36562i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.5 | − | 0.641589i | 0 | 1.58836 | −2.21105 | 0 | −1.69963 | − | 2.02763i | − | 2.30225i | 0 | 1.41858i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.6 | − | 0.641589i | 0 | 1.58836 | 2.21105 | 0 | −1.69963 | + | 2.02763i | − | 2.30225i | 0 | − | 1.41858i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.7 | 0.641589i | 0 | 1.58836 | −2.21105 | 0 | −1.69963 | + | 2.02763i | 2.30225i | 0 | − | 1.41858i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.8 | 0.641589i | 0 | 1.58836 | 2.21105 | 0 | −1.69963 | − | 2.02763i | 2.30225i | 0 | 1.41858i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.9 | 1.18593i | 0 | 0.593579 | −2.83797 | 0 | 2.46050 | − | 0.972582i | 3.07579i | 0 | − | 3.36562i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.10 | 1.18593i | 0 | 0.593579 | 2.83797 | 0 | 2.46050 | + | 0.972582i | 3.07579i | 0 | 3.36562i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.11 | 2.27639i | 0 | −3.18194 | −1.43429 | 0 | 0.239123 | − | 2.63492i | − | 2.69056i | 0 | − | 3.26499i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 566.12 | 2.27639i | 0 | −3.18194 | 1.43429 | 0 | 0.239123 | + | 2.63492i | − | 2.69056i | 0 | 3.26499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{4} + 10T_{2}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + 7 T^{4} + 10 T^{2} + 3)^{2} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{6} - 15 T^{4} + \cdots - 81)^{2} \)
|
| $7$ |
\( (T^{6} - 2 T^{5} + \cdots + 343)^{2} \)
|
| $11$ |
\( (T^{6} + 22 T^{4} + 121 T^{2} + 3)^{2} \)
|
| $13$ |
\( (T^{6} + 45 T^{4} + \cdots + 1323)^{2} \)
|
| $17$ |
\( (T^{6} - 54 T^{4} + \cdots - 729)^{2} \)
|
| $19$ |
\( (T^{6} + 63 T^{4} + \cdots + 2187)^{2} \)
|
| $23$ |
\( (T^{6} + 22 T^{4} + \cdots + 27)^{2} \)
|
| $29$ |
\( (T^{6} + 31 T^{4} + \cdots + 243)^{2} \)
|
| $31$ |
\( (T^{6} + 129 T^{4} + \cdots + 27)^{2} \)
|
| $37$ |
\( (T^{3} + T^{2} - 4 T - 1)^{4} \)
|
| $41$ |
\( (T^{6} - 72 T^{4} + \cdots - 729)^{2} \)
|
| $43$ |
\( (T^{3} - 5 T^{2} + \cdots + 197)^{4} \)
|
| $47$ |
\( (T^{6} - 93 T^{4} + \cdots - 81)^{2} \)
|
| $53$ |
\( (T^{6} + 118 T^{4} + \cdots + 20667)^{2} \)
|
| $59$ |
\( (T^{6} - 219 T^{4} + \cdots - 136161)^{2} \)
|
| $61$ |
\( (T^{6} + 24 T^{4} + \cdots + 243)^{2} \)
|
| $67$ |
\( (T^{3} + 6 T^{2} - 15 T + 7)^{4} \)
|
| $71$ |
\( (T^{6} + 163 T^{4} + \cdots + 363)^{2} \)
|
| $73$ |
\( (T^{6} + 279 T^{4} + \cdots + 2187)^{2} \)
|
| $79$ |
\( (T^{3} - 3 T^{2} - 24 T + 79)^{4} \)
|
| $83$ |
\( (T^{6} - 234 T^{4} + \cdots - 363609)^{2} \)
|
| $89$ |
\( (T^{6} - 324 T^{4} + \cdots - 531441)^{2} \)
|
| $97$ |
\( (T^{6} + 69 T^{4} + \cdots + 3267)^{2} \)
|
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