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Properties

Label 567.2.e.d
Level $567$
Weight $2$
Character orbit 567.e
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(163,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.e (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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{3} - \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 5 q^{4} + 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 5 q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + 7 q^{10} + 5 q^{11} - 10 q^{13} + 23 q^{14} + q^{16} + 6 q^{17} + 8 q^{19} + 16 q^{20} - 14 q^{22} - 12 q^{23} - 8 q^{25} + q^{26} + 5 q^{28} + 20 q^{29} + 18 q^{31} - 10 q^{32} - 23 q^{35} - 20 q^{38} + 18 q^{40} + 10 q^{41} - 14 q^{43} + 13 q^{44} - 12 q^{46} - 21 q^{47} + 17 q^{49} - 76 q^{50} + 25 q^{52} - 12 q^{53} - 52 q^{55} - 39 q^{56} + 7 q^{58} + 6 q^{59} + 20 q^{61} - 36 q^{62} - 46 q^{64} + 8 q^{65} + 5 q^{67} + 51 q^{68} + 17 q^{70} + 18 q^{71} + 6 q^{73} + 10 q^{76} + 37 q^{77} + 10 q^{79} - 2 q^{80} + 35 q^{82} - 18 q^{83} - 18 q^{85} - 22 q^{86} - 18 q^{88} - 22 q^{89} + 13 q^{91} + 72 q^{92} + 15 q^{94} - 16 q^{95} - 18 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} + \cdots + 88979 ) / 8102 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.373419 − 0.0835272i
2.11692 − 0.978886i
0.0512865 + 1.21608i
−1.54162 − 1.88572i
0.373419 + 0.0835272i
2.11692 + 0.978886i
0.0512865 − 1.21608i
−1.54162 + 1.88572i
−1.18584 + 2.05393i 0 −1.81242 − 3.13920i 0.614373 − 1.06412i 0 −1.66774 − 2.05393i 3.85358 0 1.45709 + 2.52376i
163.2 −0.186423 + 0.322894i 0 0.930493 + 1.61166i 0.710717 − 1.23100i 0 2.62597 − 0.322894i −1.43955 0 0.264988 + 0.458972i
163.3 0.768262 − 1.33067i 0 −0.180452 − 0.312552i 1.57880 − 2.73457i 0 −2.28677 + 1.33067i 2.51851 0 −2.42587 − 4.20173i
163.4 1.10400 − 1.91218i 0 −1.43762 − 2.49004i −1.90389 + 3.29764i 0 1.82854 + 1.91218i −1.93254 0 4.20379 + 7.28117i
487.1 −1.18584 − 2.05393i 0 −1.81242 + 3.13920i 0.614373 + 1.06412i 0 −1.66774 + 2.05393i 3.85358 0 1.45709 − 2.52376i
487.2 −0.186423 − 0.322894i 0 0.930493 − 1.61166i 0.710717 + 1.23100i 0 2.62597 + 0.322894i −1.43955 0 0.264988 − 0.458972i
487.3 0.768262 + 1.33067i 0 −0.180452 + 0.312552i 1.57880 + 2.73457i 0 −2.28677 − 1.33067i 2.51851 0 −2.42587 + 4.20173i
487.4 1.10400 + 1.91218i 0 −1.43762 + 2.49004i −1.90389 − 3.29764i 0 1.82854 − 1.91218i −1.93254 0 4.20379 − 7.28117i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.4
Significant digits:
Format:

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 567.2.e.d yes 8
3.b odd 2 1 567.2.e.c ✓ 8
7.c even 3 1 inner 567.2.e.d yes 8
7.c even 3 1 3969.2.a.s 4
7.d odd 6 1 3969.2.a.t 4
9.c even 3 1 567.2.g.k 8
9.c even 3 1 567.2.h.j 8
9.d odd 6 1 567.2.g.j 8
9.d odd 6 1 567.2.h.k 8
21.g even 6 1 3969.2.a.w 4
21.h odd 6 1 567.2.e.c ✓ 8
21.h odd 6 1 3969.2.a.x 4
63.g even 3 1 567.2.h.j 8
63.h even 3 1 567.2.g.k 8
63.j odd 6 1 567.2.g.j 8
63.n odd 6 1 567.2.h.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c ✓ 8 3.b odd 2 1
567.2.e.c ✓ 8 21.h odd 6 1
567.2.e.d yes 8 1.a even 1 1 trivial
567.2.e.d yes 8 7.c even 3 1 inner
567.2.g.j 8 9.d odd 6 1
567.2.g.j 8 63.j odd 6 1
567.2.g.k 8 9.c even 3 1
567.2.g.k 8 63.h even 3 1
567.2.h.j 8 9.c even 3 1
567.2.h.j 8 63.g even 3 1
567.2.h.k 8 9.d odd 6 1
567.2.h.k 8 63.n odd 6 1
3969.2.a.s 4 7.c even 3 1
3969.2.a.t 4 7.d odd 6 1
3969.2.a.w 4 21.g even 6 1
3969.2.a.x 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} - 6T_{2}^{5} + 39T_{2}^{4} - 30T_{2}^{3} + 54T_{2}^{2} + 18T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 62001 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} - 20 T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + \cdots + 321489 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{3} + \cdots + 63)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 18 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$37$ \( T^{8} + 78 T^{6} + \cdots + 904401 \) Copy content Toggle raw display
$41$ \( (T^{4} - 5 T^{3} + \cdots + 441)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 7 T^{3} - 30 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 21 T^{7} + \cdots + 194481 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 1087849 \) Copy content Toggle raw display
$67$ \( T^{8} - 5 T^{7} + \cdots + 124609 \) Copy content Toggle raw display
$71$ \( (T^{4} - 9 T^{3} + \cdots + 243)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + \cdots + 5239521 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$83$ \( (T^{4} + 9 T^{3} + \cdots - 5103)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 22 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$97$ \( (T^{4} + 9 T^{3} + \cdots + 2877)^{2} \) Copy content Toggle raw display
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