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Properties

Label 567.4.a.j
Level $567$
Weight $4$
Character orbit 567.a
Self dual yes
Analytic conductor $33.454$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 48x^{7} + 105x^{6} + 762x^{5} - 1002x^{4} - 4551x^{3} + 2292x^{2} + 7044x - 1792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} + (\beta_{7} - 3) q^{5} + 7 q^{7} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} + (\beta_{7} - 3) q^{5} + 7 q^{7} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 9) q^{8} + ( - \beta_{8} - \beta_{7} + \beta_{4} + \cdots + 3) q^{10}+ \cdots + (49 \beta_1 - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 36 q^{4} - 24 q^{5} + 63 q^{7} - 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 36 q^{4} - 24 q^{5} + 63 q^{7} - 75 q^{8} - 111 q^{11} + 18 q^{13} - 42 q^{14} + 144 q^{16} - 273 q^{17} + 45 q^{19} - 402 q^{20} - 162 q^{22} - 312 q^{23} + 279 q^{25} + 51 q^{26} + 252 q^{28} - 378 q^{29} + 18 q^{31} - 891 q^{32} - 324 q^{34} - 168 q^{35} - 36 q^{37} - 147 q^{38} + 405 q^{40} - 477 q^{41} - 171 q^{43} - 948 q^{44} - 378 q^{46} - 654 q^{47} + 441 q^{49} - 429 q^{50} + 747 q^{52} - 948 q^{53} - 216 q^{55} - 525 q^{56} + 297 q^{58} - 957 q^{59} - 198 q^{61} - 300 q^{62} + 2385 q^{64} - 2478 q^{65} - 333 q^{67} - 1443 q^{68} - 2826 q^{71} + 153 q^{73} - 2100 q^{74} - 144 q^{76} - 777 q^{77} + 1152 q^{79} - 4209 q^{80} - 3024 q^{82} - 1890 q^{83} - 648 q^{85} - 3837 q^{86} - 2268 q^{88} - 1302 q^{89} + 126 q^{91} - 987 q^{92} + 324 q^{94} - 3144 q^{95} - 1737 q^{97} - 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 48x^{7} + 105x^{6} + 762x^{5} - 1002x^{4} - 4551x^{3} + 2292x^{2} + 7044x - 1792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 19\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{8} - 103 \nu^{7} - 38 \nu^{6} + 4027 \nu^{5} - 4588 \nu^{4} - 44126 \nu^{3} + 55051 \nu^{2} + \cdots - 111104 ) / 2544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9 \nu^{8} - 19 \nu^{7} + 594 \nu^{6} + 819 \nu^{5} - 11152 \nu^{4} - 10410 \nu^{3} + 60327 \nu^{2} + \cdots - 45408 ) / 2544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43 \nu^{8} + 239 \nu^{7} + 1142 \nu^{6} - 7111 \nu^{5} - 376 \nu^{4} + 40010 \nu^{3} + \cdots + 219488 ) / 7632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83 \nu^{8} + 343 \nu^{7} + 3358 \nu^{6} - 11951 \nu^{5} - 40424 \nu^{4} + 113170 \nu^{3} + \cdots - 28400 ) / 7632 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73 \nu^{8} + 317 \nu^{7} + 3122 \nu^{6} - 10741 \nu^{5} - 43768 \nu^{4} + 87566 \nu^{3} + \cdots - 199840 ) / 7632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} + \beta_{6} + 3\beta_{5} - 2\beta_{4} + 2\beta_{3} + 26\beta_{2} + 40\beta _1 + 212 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{8} - 10\beta_{7} + 5\beta_{6} + 5\beta_{5} - 2\beta_{4} + 39\beta_{3} + 43\beta_{2} + 481\beta _1 + 373 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24 \beta_{8} - 102 \beta_{7} + 36 \beta_{6} + 126 \beta_{5} - 84 \beta_{4} + 107 \beta_{3} + 674 \beta_{2} + \cdots + 4994 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 324 \beta_{8} - 538 \beta_{7} + 215 \beta_{6} + 297 \beta_{5} - 178 \beta_{4} + 1261 \beta_{3} + \cdots + 13168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1446 \beta_{8} - 4028 \beta_{7} + 1138 \beta_{6} + 4144 \beta_{5} - 2872 \beta_{4} + 4314 \beta_{3} + \cdots + 131334 \) Copy content Toggle raw display

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

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
−4.61376
−3.42464
−2.33229
−1.65333
0.244793
1.46237
3.43672
4.23506
5.64507
−5.61376 0 23.5143 −10.6294 0 7.00000 −87.0935 0 59.6707
1.2 −4.42464 0 11.5774 3.85415 0 7.00000 −15.8288 0 −17.0532
1.3 −3.33229 0 3.10414 −16.7471 0 7.00000 16.3144 0 55.8062
1.4 −2.65333 0 −0.959863 20.0600 0 7.00000 23.7734 0 −53.2258
1.5 −0.755207 0 −7.42966 −13.2492 0 7.00000 11.6526 0 10.0059
1.6 0.462367 0 −7.78622 4.52993 0 7.00000 −7.29902 0 2.09449
1.7 2.43672 0 −2.06240 −5.23523 0 7.00000 −24.5192 0 −12.7568
1.8 3.23506 0 2.46559 9.90222 0 7.00000 −17.9041 0 32.0342
1.9 4.64507 0 13.5767 −16.4854 0 7.00000 25.9042 0 −76.5757
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.j 9
3.b odd 2 1 567.4.a.k 9
9.c even 3 2 63.4.f.c ✓ 18
9.d odd 6 2 189.4.f.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.c ✓ 18 9.c even 3 2
189.4.f.c 18 9.d odd 6 2
567.4.a.j 9 1.a even 1 1 trivial
567.4.a.k 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 6T_{2}^{8} - 36T_{2}^{7} - 231T_{2}^{6} + 342T_{2}^{5} + 2619T_{2}^{4} - 603T_{2}^{3} - 9234T_{2}^{2} - 1944T_{2} + 2808 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(567))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 6 T^{8} + \cdots + 2808 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 24 T^{8} + \cdots + 705922182 \) Copy content Toggle raw display
$7$ \( (T - 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 101515621572 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 3228162266096 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 50\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 278256660857 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 74\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 17\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 75\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 15\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 84\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 73\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 58\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 10\!\cdots\!14 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 67\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 15\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 34\!\cdots\!08 \) Copy content Toggle raw display
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