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Properties

Label 567.3.r.e
Level $567$
Weight $3$
Character orbit 567.r
Analytic conductor $15.450$
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] = [567,3,Mod(134,567)]
 
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([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 567.r (of order \(6\), 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: \(15.4496309892\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 24 x^{13} + 53 x^{12} - 204 x^{11} + 558 x^{10} - 774 x^{9} + 828 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 189)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + (\beta_{15} - \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{14} + 2 \beta_{13} + \cdots + 4 \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + (\beta_{15} - \beta_{7} + \beta_{5} + \cdots + 1) q^{4}+ \cdots + (7 \beta_{12} - 7 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 104 q^{10} - 36 q^{13} - 132 q^{16} + 24 q^{19} + 136 q^{22} + 108 q^{25} + 112 q^{28} + 28 q^{31} + 12 q^{34} - 8 q^{37} - 336 q^{40} + 152 q^{43} + 216 q^{46} - 56 q^{49} + 272 q^{52} + 392 q^{55} + 220 q^{58} - 180 q^{61} - 1400 q^{64} + 132 q^{67} - 196 q^{70} + 544 q^{73} - 544 q^{76} - 316 q^{79} + 56 q^{82} + 228 q^{85} - 112 q^{91} + 348 q^{94} + 364 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 24 x^{13} + 53 x^{12} - 204 x^{11} + 558 x^{10} - 774 x^{9} + 828 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5308 \nu^{15} - 1848807 \nu^{14} + 8059386 \nu^{13} - 17584291 \nu^{12} + 5248684 \nu^{11} + \cdots + 38277515 ) / 14998056 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3206 \nu^{15} - 422441 \nu^{14} + 2408918 \nu^{13} - 6570244 \nu^{12} + 7086702 \nu^{11} + \cdots - 4114364 ) / 6452652 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 719244081 \nu^{15} - 10263744652 \nu^{14} + 49240655732 \nu^{13} - 118092098582 \nu^{12} + \cdots - 123429157870 ) / 28578795708 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 853452506 \nu^{15} + 15054837795 \nu^{14} - 58596227883 \nu^{13} + 114461139710 \nu^{12} + \cdots + 181687905857 ) / 28578795708 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 877063124 \nu^{15} + 8923960935 \nu^{14} - 31513860022 \nu^{13} + 54735175967 \nu^{12} + \cdots + 153008860269 ) / 19052530472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5005454969 \nu^{15} + 14679186004 \nu^{14} - 10201343903 \nu^{13} - 83157377070 \nu^{12} + \cdots - 282887453916 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 73162655 \nu^{15} - 438519228 \nu^{14} + 1353003173 \nu^{13} - 1932210504 \nu^{12} + \cdots - 2423267822 ) / 554928072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7841528487 \nu^{15} + 44264728654 \nu^{14} - 126490914953 \nu^{13} + 150936930568 \nu^{12} + \cdots + 335194841220 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4897642729 \nu^{15} - 30544559928 \nu^{14} + 93237868224 \nu^{13} - 128883450994 \nu^{12} + \cdots - 276537357084 ) / 28578795708 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13165626409 \nu^{15} + 49323905421 \nu^{14} - 86065005573 \nu^{13} - 61631144017 \nu^{12} + \cdots - 1592795783 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7468266179 \nu^{15} - 54786723414 \nu^{14} + 185122155581 \nu^{13} - 306214414221 \nu^{12} + \cdots - 290951585753 ) / 28578795708 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20500959619 \nu^{15} - 118193842303 \nu^{14} + 340178914019 \nu^{13} - 408134366227 \nu^{12} + \cdots + 27506980635 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 30636828786 \nu^{15} + 174033298487 \nu^{14} - 499838872186 \nu^{13} + 597009520147 \nu^{12} + \cdots - 50539468661 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32381797470 \nu^{15} + 165536397445 \nu^{14} - 412924491002 \nu^{13} + 305403330677 \nu^{12} + \cdots - 20655655043 ) / 57157591416 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 41598552575 \nu^{15} + 236973390875 \nu^{14} - 680460125303 \nu^{13} + 821277659479 \nu^{12} + \cdots + 302641281419 ) / 57157591416 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} - 3 \beta_{9} + \cdots + 8 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 18 \beta_{12} + 4 \beta_{11} - 9 \beta_{9} - 18 \beta_{8} + \cdots + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} - 3 \beta_{14} + 30 \beta_{13} + 39 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 18 \beta_{9} + \cdots - 10 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} + 13 \beta_{13} + 14 \beta_{12} - 2 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} - 18 \beta_{6} + \cdots - 104 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 41 \beta_{15} + 28 \beta_{14} - 117 \beta_{13} - 108 \beta_{12} - 55 \beta_{11} - 9 \beta_{10} + \cdots - 628 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 122 \beta_{15} + 9 \beta_{14} - 369 \beta_{13} - 540 \beta_{12} - 196 \beta_{11} + 369 \beta_{9} + \cdots - 415 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 443 \beta_{15} + 115 \beta_{14} - 2535 \beta_{13} - 3303 \beta_{12} - 667 \beta_{11} + 89 \beta_{10} + \cdots + 1244 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 105 \beta_{14} - 1309 \beta_{13} - 1646 \beta_{12} + 210 \beta_{10} - 1309 \beta_{9} - 1646 \beta_{8} + \cdots + 6608 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4540 \beta_{15} - 1743 \beta_{14} + 7260 \beta_{13} + 8421 \beta_{12} + 6986 \beta_{11} + 974 \beta_{10} + \cdots + 59786 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21950 \beta_{15} - 3873 \beta_{14} + 65385 \beta_{13} + 89334 \beta_{12} + 34084 \beta_{11} + \cdots + 91168 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 44935 \beta_{15} - 5764 \beta_{14} + 226227 \beta_{13} + 298860 \beta_{12} + 69479 \beta_{11} + \cdots - 140356 ) / 18 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1448 \beta_{14} + 20008 \beta_{13} + 26180 \beta_{12} - 2896 \beta_{10} + 20008 \beta_{9} + 26180 \beta_{8} + \cdots - 91619 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 436607 \beta_{15} + 148967 \beta_{14} - 594141 \beta_{13} - 758013 \beta_{12} - 675865 \beta_{11} + \cdots - 5486968 ) / 18 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1962430 \beta_{15} + 391251 \beta_{14} - 5931783 \beta_{13} - 7942950 \beta_{12} - 3039548 \beta_{11} + \cdots - 8667572 ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4202492 \beta_{15} + 414573 \beta_{14} - 20476482 \beta_{13} - 27159009 \beta_{12} - 6507286 \beta_{11} + \cdots + 13654214 ) / 18 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
0.479789 − 0.479789i
2.18316 − 2.18316i
0.762341 + 0.762341i
0.806936 − 0.806936i
−1.58982 − 1.58982i
−1.10386 + 1.10386i
1.42260 + 1.42260i
0.0388520 + 0.0388520i
0.479789 + 0.479789i
2.18316 + 2.18316i
0.762341 − 0.762341i
0.806936 + 0.806936i
−1.58982 + 1.58982i
−1.10386 − 1.10386i
1.42260 − 1.42260i
0.0388520 − 0.0388520i
−3.40185 + 1.96406i 0 5.71505 − 9.89876i 6.11763 + 3.53202i 0 1.32288 + 2.29129i 29.1863i 0 −27.7484
134.2 −1.97223 + 1.13867i 0 0.593126 − 1.02732i −4.28273 − 2.47263i 0 −1.32288 − 2.29129i − 6.40785i 0 11.2620
134.3 −0.803773 + 0.464058i 0 −1.56930 + 2.71811i 3.51956 + 2.03202i 0 1.32288 + 2.29129i − 6.62545i 0 −3.77190
134.4 −0.625847 + 0.361333i 0 −1.73888 + 3.01182i 6.88081 + 3.97263i 0 −1.32288 − 2.29129i − 5.40392i 0 −5.74177
134.5 0.625847 − 0.361333i 0 −1.73888 + 3.01182i −6.88081 − 3.97263i 0 −1.32288 − 2.29129i 5.40392i 0 −5.74177
134.6 0.803773 − 0.464058i 0 −1.56930 + 2.71811i −3.51956 − 2.03202i 0 1.32288 + 2.29129i 6.62545i 0 −3.77190
134.7 1.97223 − 1.13867i 0 0.593126 − 1.02732i 4.28273 + 2.47263i 0 −1.32288 − 2.29129i 6.40785i 0 11.2620
134.8 3.40185 − 1.96406i 0 5.71505 − 9.89876i −6.11763 − 3.53202i 0 1.32288 + 2.29129i − 29.1863i 0 −27.7484
512.1 −3.40185 − 1.96406i 0 5.71505 + 9.89876i 6.11763 − 3.53202i 0 1.32288 − 2.29129i − 29.1863i 0 −27.7484
512.2 −1.97223 − 1.13867i 0 0.593126 + 1.02732i −4.28273 + 2.47263i 0 −1.32288 + 2.29129i 6.40785i 0 11.2620
512.3 −0.803773 − 0.464058i 0 −1.56930 − 2.71811i 3.51956 − 2.03202i 0 1.32288 − 2.29129i 6.62545i 0 −3.77190
512.4 −0.625847 − 0.361333i 0 −1.73888 − 3.01182i 6.88081 − 3.97263i 0 −1.32288 + 2.29129i 5.40392i 0 −5.74177
512.5 0.625847 + 0.361333i 0 −1.73888 − 3.01182i −6.88081 + 3.97263i 0 −1.32288 + 2.29129i − 5.40392i 0 −5.74177
512.6 0.803773 + 0.464058i 0 −1.56930 − 2.71811i −3.51956 + 2.03202i 0 1.32288 − 2.29129i − 6.62545i 0 −3.77190
512.7 1.97223 + 1.13867i 0 0.593126 + 1.02732i 4.28273 − 2.47263i 0 −1.32288 + 2.29129i − 6.40785i 0 11.2620
512.8 3.40185 + 1.96406i 0 5.71505 + 9.89876i −6.11763 + 3.53202i 0 1.32288 − 2.29129i 29.1863i 0 −27.7484
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.3.r.e 16
3.b odd 2 1 inner 567.3.r.e 16
9.c even 3 1 189.3.b.c ✓ 8
9.c even 3 1 inner 567.3.r.e 16
9.d odd 6 1 189.3.b.c ✓ 8
9.d odd 6 1 inner 567.3.r.e 16
36.f odd 6 1 3024.3.d.j 8
36.h even 6 1 3024.3.d.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.c ✓ 8 9.c even 3 1
189.3.b.c ✓ 8 9.d odd 6 1
567.3.r.e 16 1.a even 1 1 trivial
567.3.r.e 16 3.b odd 2 1 inner
567.3.r.e 16 9.c even 3 1 inner
567.3.r.e 16 9.d odd 6 1 inner
3024.3.d.j 8 36.f odd 6 1
3024.3.d.j 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{16} - 22T_{2}^{14} + 375T_{2}^{12} - 2158T_{2}^{10} + 9205T_{2}^{8} - 11496T_{2}^{6} + 10476T_{2}^{4} - 4320T_{2}^{2} + 1296 \) Copy content Toggle raw display
\( T_{5}^{16} - 154 T_{5}^{14} + 15531 T_{5}^{12} - 911050 T_{5}^{10} + 38814961 T_{5}^{8} + \cdots + 1618961043456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 22 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 1618961043456 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{8} + 18 T^{7} + \cdots + 710435716)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 876 T^{6} + \cdots + 57562569)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 42256)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 1064977956)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 814032)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 8642606308561)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + 12198 T^{6} + \cdots + 52968101904)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( (T^{8} + 90 T^{7} + \cdots + 277501382656)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 605048616325696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 14742 T^{6} + \cdots + 259415011584)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 136 T^{3} + \cdots + 9014688)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11865737630224)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 4824840688704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 35\!\cdots\!84)^{2} \) Copy content Toggle raw display
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