Properties

Label 567.4.a.g
Level $567$
Weight $4$
Character orbit 567.a
Self dual yes
Analytic conductor $33.454$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,4,Mod(1,567)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

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: yes
Analytic conductor: \(33.4540829733\)
Analytic rank: \(1\)
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: \( x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + ( - \beta_{4} - 4) q^{5} - 7 q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + ( - \beta_{4} - 4) q^{5} - 7 q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{8} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 6 \beta_1 - 3) q^{10} + (\beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 2) q^{11} + ( - \beta_{7} + \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 7) q^{13} + 7 \beta_1 q^{14} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + 8 \beta_1 + 9) q^{16} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 21) q^{17} + (3 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 11 \beta_1 + 17) q^{19} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 8 \beta_{2} + 7 \beta_1 - 42) q^{20} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \cdots + 18) q^{22}+ \cdots - 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8} - 14 q^{10} - 24 q^{11} + 68 q^{13} + 21 q^{14} + 103 q^{16} - 168 q^{17} + 176 q^{19} - 330 q^{20} + 151 q^{22} - 228 q^{23} + 244 q^{25} - 795 q^{26} - 301 q^{28} - 618 q^{29} + 72 q^{31} - 786 q^{32} - 261 q^{34} + 210 q^{35} + 210 q^{37} - 1032 q^{38} - 375 q^{40} - 420 q^{41} - 2 q^{43} - 387 q^{44} + 402 q^{46} - 570 q^{47} + 392 q^{49} - 1110 q^{50} - 431 q^{52} - 528 q^{53} - 838 q^{55} + 42 q^{56} + 37 q^{58} + 150 q^{59} + 578 q^{61} - 1170 q^{62} - 112 q^{64} + 366 q^{65} - 898 q^{67} - 2526 q^{68} + 98 q^{70} - 882 q^{71} + 972 q^{73} + 222 q^{74} + 1423 q^{76} + 168 q^{77} - 158 q^{79} - 2475 q^{80} - 211 q^{82} - 2958 q^{83} - 774 q^{85} + 114 q^{86} + 1317 q^{88} - 4380 q^{89} - 476 q^{91} - 4629 q^{92} - 3234 q^{94} - 930 q^{95} - 60 q^{97} - 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 14\nu^{6} - 81\nu^{5} + 657\nu^{4} + 1851\nu^{3} - 8352\nu^{2} - 11632\nu + 24200 ) / 744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} + 4\nu^{6} - 681\nu^{5} + 171\nu^{4} + 10857\nu^{3} - 9066\nu^{2} - 52264\nu + 61640 ) / 744 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{7} - 19\nu^{6} - 369\nu^{5} + 699\nu^{4} + 4950\nu^{3} - 6738\nu^{2} - 17633\nu + 19504 ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 5\nu^{6} + 381\nu^{5} - 228\nu^{4} - 6354\nu^{3} + 3687\nu^{2} + 30460\nu - 25436 ) / 186 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{5} + 2\beta_{4} + 27\beta_{2} + 8\beta _1 + 257 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 2\beta_{6} + 4\beta_{5} - 4\beta_{4} + 35\beta_{3} + 32\beta_{2} + 453\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{7} - 4\beta_{6} + 86\beta_{5} + 46\beta_{4} + \beta_{3} + 687\beta_{2} + 342\beta _1 + 5775 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 185\beta_{7} + 106\beta_{6} + 214\beta_{5} - 250\beta_{4} + 998\beta_{3} + 972\beta_{2} + 10837\beta _1 + 3250 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.24375
4.56358
2.61577
1.59498
0.807372
−2.93948
−3.93417
−4.95181
−5.24375 0 19.4970 4.06017 0 −7.00000 −60.2873 0 −21.2906
1.2 −4.56358 0 12.8263 −20.7910 0 −7.00000 −22.0250 0 94.8813
1.3 −2.61577 0 −1.15774 13.5431 0 −7.00000 23.9546 0 −35.4255
1.4 −1.59498 0 −5.45602 2.55632 0 −7.00000 21.4622 0 −4.07730
1.5 −0.807372 0 −7.34815 −18.2289 0 −7.00000 12.3917 0 14.7175
1.6 2.93948 0 0.640533 2.56885 0 −7.00000 −21.6330 0 7.55109
1.7 3.93417 0 7.47771 2.43142 0 −7.00000 −2.05480 0 9.56561
1.8 4.95181 0 16.5205 −16.1400 0 −7.00000 42.1917 0 −79.9221
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.4.a.g 8
3.b odd 2 1 567.4.a.i 8
9.c even 3 2 189.4.f.b 16
9.d odd 6 2 63.4.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.b 16 9.d odd 6 2
189.4.f.b 16 9.c even 3 2
567.4.a.g 8 1.a even 1 1 trivial
567.4.a.i 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{7} - 49T_{2}^{6} - 138T_{2}^{5} + 708T_{2}^{4} + 1941T_{2}^{3} - 2506T_{2}^{2} - 8592T_{2} - 4616 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(567))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} - 49 T^{6} + \cdots - 4616 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 30 T^{7} - 172 T^{6} + \cdots - 5370473 \) Copy content Toggle raw display
$7$ \( (T + 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 24 T^{7} + \cdots + 4399098628 \) Copy content Toggle raw display
$13$ \( T^{8} - 68 T^{7} + \cdots - 8057410599908 \) Copy content Toggle raw display
$17$ \( T^{8} + 168 T^{7} + \cdots + 2073513126204 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 604815137888177 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 145237005548415 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 498541969683460 \) Copy content Toggle raw display
$31$ \( T^{8} - 72 T^{7} + \cdots - 21\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{8} - 210 T^{7} + \cdots + 71\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{8} + 420 T^{7} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + 570 T^{7} + \cdots - 26\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + 528 T^{7} + \cdots + 86\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{8} - 150 T^{7} + \cdots + 34\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{8} - 578 T^{7} + \cdots + 79\!\cdots\!35 \) Copy content Toggle raw display
$67$ \( T^{8} + 898 T^{7} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{8} + 882 T^{7} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( T^{8} - 972 T^{7} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + 158 T^{7} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{8} + 2958 T^{7} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{8} + 4380 T^{7} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{8} + 60 T^{7} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
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