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Properties

Label 567.2.h.l
Level $567$
Weight $2$
Character orbit 567.h
Analytic conductor $4.528$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(298,567)]
 
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([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.h (of order \(3\), degree \(2\), not 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{15} q^{2} + ( - \beta_{13} + \beta_{12} + \beta_{9} + \cdots + 1) q^{4}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{15} q^{2} + ( - \beta_{13} + \beta_{12} + \beta_{9} + \cdots + 1) q^{4}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 6 q^{7} - 14 q^{10} - 6 q^{13} + 12 q^{16} - 24 q^{19} - 2 q^{22} - 26 q^{28} + 40 q^{31} + 4 q^{37} - 36 q^{40} - 10 q^{43} + 36 q^{46} + 10 q^{49} - 34 q^{52} + 8 q^{55} + 22 q^{58} + 72 q^{61} + 76 q^{64} - 36 q^{67} - 46 q^{70} - 32 q^{73} - 58 q^{76} - 64 q^{79} + 2 q^{82} - 30 q^{85} + 72 q^{88} - 22 q^{91} + 108 q^{94} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 14x^{12} - 39x^{10} + 77x^{8} - 156x^{6} + 224x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{15} - 96\nu^{13} + 122\nu^{11} - 469\nu^{9} + 1099\nu^{7} - 872\nu^{5} + 2304\nu^{3} - 1984\nu ) / 2688 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} - 8\nu^{13} - 54\nu^{11} + 63\nu^{9} - 329\nu^{7} + 744\nu^{5} - 424\nu^{3} + 2112\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{14} + 72\nu^{12} - 214\nu^{10} + 539\nu^{8} - 749\nu^{6} + 1648\nu^{4} - 2064\nu^{2} + 2048 ) / 1344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{15} - 9\nu^{13} + 32\nu^{11} - 112\nu^{9} + 301\nu^{7} - 269\nu^{5} + 804\nu^{3} - 592\nu ) / 672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{14} + 9\nu^{12} - 32\nu^{10} + 112\nu^{8} - 301\nu^{6} + 269\nu^{4} - 804\nu^{2} + 592 ) / 336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{15} - 45\nu^{13} + 104\nu^{11} - 322\nu^{9} + 679\nu^{7} - 1037\nu^{5} + 1752\nu^{3} - 496\nu ) / 672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} + 4\nu^{12} - 14\nu^{10} + 39\nu^{8} - 77\nu^{6} + 156\nu^{4} - 160\nu^{2} + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -23\nu^{14} + 12\nu^{12} - 178\nu^{10} + 161\nu^{8} - 539\nu^{6} + 1348\nu^{4} - 1296\nu^{2} + 3776 ) / 1344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\nu^{14} - 48\nu^{12} + 166\nu^{10} - 371\nu^{8} + 413\nu^{6} - 856\nu^{4} + 144\nu^{2} - 320 ) / 1344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -37\nu^{15} + 12\nu^{13} - 38\nu^{11} - 77\nu^{9} + 791\nu^{7} - 1564\nu^{5} + 4416\nu^{3} - 4736\nu ) / 2688 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -19\nu^{15} + 114\nu^{13} - 410\nu^{11} + 889\nu^{9} - 2065\nu^{7} + 3146\nu^{5} - 4416\nu^{3} + 5632\nu ) / 1344 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -19\nu^{14} + 72\nu^{12} - 74\nu^{10} + 301\nu^{8} - 91\nu^{6} + 80\nu^{4} - 384\nu^{2} - 2432 ) / 672 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\nu^{14} - 96\nu^{12} + 346\nu^{10} - 665\nu^{8} + 1463\nu^{6} - 2608\nu^{4} + 2808\nu^{2} - 4448 ) / 672 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -19\nu^{15} + 44\nu^{13} - 186\nu^{11} + 357\nu^{9} - 1015\nu^{7} + 1732\nu^{5} - 2064\nu^{3} + 3392\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -37\nu^{15} + 138\nu^{13} - 374\nu^{11} + 1015\nu^{9} - 1771\nu^{7} + 2930\nu^{5} - 3648\nu^{3} + 2656\nu ) / 1344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{10} + 2\beta_{6} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} - \beta_{12} - 2\beta_{9} - 3\beta_{8} - \beta_{5} - \beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} - \beta_{11} + 2\beta_{10} - \beta_{6} + \beta_{4} + 2\beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{14} - \beta_{11} - \beta_{10} - 4\beta_{6} + 7\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{13} + \beta_{12} + 2\beta_{9} + 6\beta_{8} - 2\beta_{5} + 10\beta_{3} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{15} - 6\beta_{14} + 6\beta_{11} - 10\beta_{10} + 8\beta_{6} + 9\beta_{4} - 3\beta_{2} + 8\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{13} - \beta_{12} - 6\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{5} + 4\beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 25\beta_{15} - 15\beta_{14} - 9\beta_{11} - 7\beta_{10} + 17\beta_{6} - 21\beta_{4} + 6\beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5\beta_{13} + 11\beta_{12} + 19\beta_{9} - 9\beta_{8} + 33\beta_{7} - 4\beta_{5} - 31\beta_{3} + 58 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30\beta_{15} + 14\beta_{14} - 58\beta_{11} - 13\beta_{10} + 5\beta_{6} - 14\beta_{4} + 14\beta_{2} - 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -14\beta_{13} + 11\beta_{12} + 29\beta_{9} - 12\beta_{8} + \beta_{7} - 13\beta_{5} - 14\beta_{3} + 29 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 21 \beta_{15} + 17 \beta_{14} + 17 \beta_{11} - 19 \beta_{10} - 10 \beta_{6} + 142 \beta_{4} + \cdots - 108 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -35\beta_{13} - 56\beta_{12} + 14\beta_{9} - 117\beta_{8} - 57\beta_{7} - 20\beta_{5} + 124\beta_{3} - 85 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -68\beta_{15} - 183\beta_{14} + 135\beta_{11} - \beta_{10} - 79\beta_{6} + 120\beta_{4} + 57\beta_{2} - 52\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\beta_{9}\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
−1.14160 + 0.834713i
−1.41264 + 0.0667052i
0.776749 + 1.18180i
1.04779 + 0.949812i
−1.04779 − 0.949812i
−0.776749 − 1.18180i
1.41264 − 0.0667052i
1.14160 − 0.834713i
−1.14160 − 0.834713i
−1.41264 − 0.0667052i
0.776749 − 1.18180i
1.04779 − 0.949812i
−1.04779 + 0.949812i
−0.776749 + 1.18180i
1.41264 + 0.0667052i
1.14160 + 0.834713i
−2.58737 0 4.69447 1.14160 − 1.97731i 0 −0.369922 + 2.61976i −6.97158 0 −2.95374 + 5.11603i
298.2 −1.52818 0 0.335323 1.41264 − 2.44676i 0 −2.61442 + 0.405935i 2.54392 0 −2.15876 + 3.73909i
298.3 −1.27020 0 −0.386601 −0.776749 + 1.34537i 0 −1.15207 − 2.38175i 3.03145 0 0.986623 − 1.70888i
298.4 −0.597336 0 −1.64319 −1.04779 + 1.81482i 0 2.63641 + 0.222079i 2.17621 0 0.625881 − 1.08406i
298.5 0.597336 0 −1.64319 1.04779 − 1.81482i 0 2.63641 + 0.222079i −2.17621 0 0.625881 − 1.08406i
298.6 1.27020 0 −0.386601 0.776749 − 1.34537i 0 −1.15207 − 2.38175i −3.03145 0 0.986623 − 1.70888i
298.7 1.52818 0 0.335323 −1.41264 + 2.44676i 0 −2.61442 + 0.405935i −2.54392 0 −2.15876 + 3.73909i
298.8 2.58737 0 4.69447 −1.14160 + 1.97731i 0 −0.369922 + 2.61976i 6.97158 0 −2.95374 + 5.11603i
352.1 −2.58737 0 4.69447 1.14160 + 1.97731i 0 −0.369922 − 2.61976i −6.97158 0 −2.95374 − 5.11603i
352.2 −1.52818 0 0.335323 1.41264 + 2.44676i 0 −2.61442 − 0.405935i 2.54392 0 −2.15876 − 3.73909i
352.3 −1.27020 0 −0.386601 −0.776749 − 1.34537i 0 −1.15207 + 2.38175i 3.03145 0 0.986623 + 1.70888i
352.4 −0.597336 0 −1.64319 −1.04779 − 1.81482i 0 2.63641 − 0.222079i 2.17621 0 0.625881 + 1.08406i
352.5 0.597336 0 −1.64319 1.04779 + 1.81482i 0 2.63641 − 0.222079i −2.17621 0 0.625881 + 1.08406i
352.6 1.27020 0 −0.386601 0.776749 + 1.34537i 0 −1.15207 + 2.38175i −3.03145 0 0.986623 + 1.70888i
352.7 1.52818 0 0.335323 −1.41264 − 2.44676i 0 −2.61442 − 0.405935i −2.54392 0 −2.15876 − 3.73909i
352.8 2.58737 0 4.69447 −1.14160 − 1.97731i 0 −0.369922 − 2.61976i 6.97158 0 −2.95374 − 5.11603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.h even 3 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.h.l 16
3.b odd 2 1 inner 567.2.h.l 16
7.c even 3 1 567.2.g.l 16
9.c even 3 1 567.2.e.g ✓ 16
9.c even 3 1 567.2.g.l 16
9.d odd 6 1 567.2.e.g ✓ 16
9.d odd 6 1 567.2.g.l 16
21.h odd 6 1 567.2.g.l 16
63.g even 3 1 567.2.e.g ✓ 16
63.h even 3 1 inner 567.2.h.l 16
63.h even 3 1 3969.2.a.bg 8
63.i even 6 1 3969.2.a.bf 8
63.j odd 6 1 inner 567.2.h.l 16
63.j odd 6 1 3969.2.a.bg 8
63.n odd 6 1 567.2.e.g ✓ 16
63.t odd 6 1 3969.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.g ✓ 16 9.c even 3 1
567.2.e.g ✓ 16 9.d odd 6 1
567.2.e.g ✓ 16 63.g even 3 1
567.2.e.g ✓ 16 63.n odd 6 1
567.2.g.l 16 7.c even 3 1
567.2.g.l 16 9.c even 3 1
567.2.g.l 16 9.d odd 6 1
567.2.g.l 16 21.h odd 6 1
567.2.h.l 16 1.a even 1 1 trivial
567.2.h.l 16 3.b odd 2 1 inner
567.2.h.l 16 63.h even 3 1 inner
567.2.h.l 16 63.j odd 6 1 inner
3969.2.a.bf 8 63.i even 6 1
3969.2.a.bf 8 63.t odd 6 1
3969.2.a.bg 8 63.h even 3 1
3969.2.a.bg 8 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{8} - 11T_{2}^{6} + 34T_{2}^{4} - 36T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{8} + 3T_{13}^{7} + 41T_{13}^{6} + 108T_{13}^{5} + 1281T_{13}^{4} + 2970T_{13}^{3} + 11972T_{13}^{2} - 4998T_{13} + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 11 T^{6} + 34 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 20 T^{14} + \cdots + 194481 \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 35 T^{14} + \cdots + 10556001 \) Copy content Toggle raw display
$13$ \( (T^{8} + 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 54 T^{14} + \cdots + 15752961 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 625681)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 449920319121 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 5554571841 \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots - 651)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + \cdots + 729)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 25344958401 \) Copy content Toggle raw display
$43$ \( (T^{8} + 5 T^{7} + \cdots + 100489)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 189 T^{6} + \cdots + 321489)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 860013262161 \) Copy content Toggle raw display
$59$ \( (T^{8} - 60 T^{6} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{3} + \cdots - 3269)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9 T^{3} + \cdots - 2699)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 357 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 441)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{3} + \cdots - 71)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + 111 T^{14} + \cdots + 15752961 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( (T^{8} + 7 T^{7} + \cdots + 974169)^{2} \) Copy content Toggle raw display
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