Properties

Label 539.6.a.e
Level $539$
Weight $6$
Character orbit 539.a
Self dual yes
Analytic conductor $86.447$
Analytic rank $1$
Dimension $3$
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] = [539,6,Mod(1,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 539.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: \(86.4468788792\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (\beta_{2} - 9 \beta_1 - 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (26 \beta_{2} - 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (\beta_{2} - 9 \beta_1 - 5) q^{5} + ( - 13 \beta_{2} - 10 \beta_1 + 72) q^{6} + (26 \beta_{2} - 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9} + (9 \beta_{2} - 42 \beta_1 + 152) q^{10} + 121 q^{11} + (112 \beta_{2} + 70 \beta_1 - 354) q^{12} + ( - 70 \beta_{2} - 18 \beta_1 - 156) q^{13} + (27 \beta_{2} + 85 \beta_1 + 523) q^{15} + ( - 164 \beta_{2} + 36 \beta_1 + 652) q^{16} + ( - 132 \beta_{2} + 60 \beta_1 - 382) q^{17} + (48 \beta_{2} + 158 \beta_1 - 1288) q^{18} + (60 \beta_{2} + 60 \beta_1 - 480) q^{19} + (168 \beta_{2} + 66 \beta_1 + 1138) q^{20} + 121 \beta_{2} q^{22} + ( - 21 \beta_{2} + 501 \beta_1 - 1189) q^{23} + ( - 526 \beta_{2} - 72 \beta_1 + 3940) q^{24} + (265 \beta_{2} + 255 \beta_1 - 104) q^{25} + (160 \beta_{2} + 348 \beta_1 - 4160) q^{26} + ( - 57 \beta_{2} + 329 \beta_1 + 887) q^{27} + ( - 182 \beta_{2} - 138 \beta_1 - 1096) q^{29} + (245 \beta_{2} + 178 \beta_1 + 824) q^{30} + (101 \beta_{2} - 69 \beta_1 + 1389) q^{31} + (404 \beta_{2} + 1128 \beta_1 - 4512) q^{32} + (121 \beta_{2} - 121 \beta_1 - 1331) q^{33} + (26 \beta_{2} + 1032 \beta_1 - 8784) q^{34} + ( - 1188 \beta_{2} - 8 \beta_1 + 1588) q^{36} + (937 \beta_{2} + 591 \beta_1 + 5711) q^{37} + ( - 840 \beta_{2} - 120 \beta_1 + 3120) q^{38} + (844 \beta_{2} + 1036 \beta_1 - 2532) q^{39} + (46 \beta_{2} + 600 \beta_1 + 4892) q^{40} + (1378 \beta_{2} - 282 \beta_1 - 1904) q^{41} + (1190 \beta_{2} - 1110 \beta_1 - 8366) q^{43} + ( - 484 \beta_{2} - 726 \beta_1 + 3630) q^{44} + ( - 496 \beta_{2} + 544 \beta_1 - 6334) q^{45} + ( - 2107 \beta_{2} + 2130 \beta_1 - 6312) q^{46} + (600 \beta_{2} + 1272 \beta_1 + 5320) q^{47} + (2604 \beta_{2} + 628 \beta_1 - 20564) q^{48} + ( - 1674 \beta_{2} - 570 \beta_1 + 13880) q^{50} + (1034 \beta_{2} + 1102 \beta_1 - 7942) q^{51} + ( - 3256 \beta_{2} + 1008 \beta_1 + 11432) q^{52} + ( - 476 \beta_{2} - 1620 \beta_1 + 17402) q^{53} + (457 \beta_{2} + 1658 \beta_1 - 6824) q^{54} + (121 \beta_{2} - 1089 \beta_1 - 605) q^{55} + ( - 1560 \beta_{2} - 720 \beta_1 + 6960) q^{57} + ( - 92 \beta_{2} + 540 \beta_1 - 9904) q^{58} + ( - 3141 \beta_{2} - 747 \beta_1 + 1495) q^{59} + ( - 1376 \beta_{2} - 3478 \beta_1 - 3326) q^{60} + (1466 \beta_{2} + 4038 \beta_1 - 7508) q^{61} + (1123 \beta_{2} - 882 \beta_1 + 6952) q^{62} + ( - 3136 \beta_{2} + 936 \beta_1 - 7096) q^{64} + ( - 264 \beta_{2} + 4848 \beta_1 - 4172) q^{65} + ( - 1573 \beta_{2} - 1210 \beta_1 + 8712) q^{66} + ( - 6575 \beta_{2} - 2721 \beta_1 - 15011) q^{67} + ( - 6728 \beta_{2} + 2052 \beta_1 + 3516) q^{68} + ( - 3421 \beta_{2} - 3611 \beta_1 - 10477) q^{69} + ( - 3935 \beta_{2} - 2673 \beta_1 + 13985) q^{71} + (4820 \beta_{2} + 2040 \beta_1 - 32360) q^{72} + ( - 5370 \beta_{2} + 3522 \beta_1 - 6316) q^{73} + (781 \beta_{2} - 3258 \beta_1 + 52184) q^{74} + ( - 4824 \beta_{2} - 5096 \beta_1 + 9004) q^{75} + (4800 \beta_{2} + 2640 \beta_1 - 35520) q^{76} + ( - 7980 \beta_{2} - 920 \beta_1 + 41968) q^{78} + (3902 \beta_{2} + 1362 \beta_1 + 41262) q^{79} + ( - 1868 \beta_{2} + 12 \beta_1 - 39564) q^{80} + (4600 \beta_{2} - 6280 \beta_1 - 26879) q^{81} + ( - 6852 \beta_{2} - 9396 \beta_1 + 88256) q^{82} + ( - 3150 \beta_{2} - 11250 \beta_1 + 51726) q^{83} + ( - 3310 \beta_{2} + 7302 \beta_1 - 37114) q^{85} + ( - 10906 \beta_{2} - 11580 \beta_1 + 84880) q^{86} + (1960 \beta_{2} + 4296 \beta_1 + 5024) q^{87} + (3146 \beta_{2} - 22748) q^{88} + (5167 \beta_{2} + 4569 \beta_1 + 34085) q^{89} + ( - 5438 \beta_{2} + 5152 \beta_1 - 36192) q^{90} + ( - 1472 \beta_{2} + 5130 \beta_1 - 113886) q^{92} + (421 \beta_{2} - 1709 \beta_1 - 4971) q^{93} + (376 \beta_{2} + 1488 \beta_1 + 24480) q^{94} + ( - 1680 \beta_{2} + 120 \beta_1 - 7440) q^{95} + ( - 15404 \beta_{2} - 10808 \beta_1 + 29088) q^{96} + ( - 123 \beta_{2} - 6429 \beta_1 - 1085) q^{97} + ( - 2299 \beta_{2} + 1331 \beta_1 - 726) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9} + 414 q^{10} + 363 q^{11} - 992 q^{12} - 486 q^{13} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} - 3706 q^{18} - 1380 q^{19} + 3480 q^{20} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} - 12132 q^{26} + 2990 q^{27} - 3426 q^{29} + 2650 q^{30} + 4098 q^{31} - 12408 q^{32} - 4114 q^{33} - 25320 q^{34} + 4756 q^{36} + 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 26208 q^{43} + 10164 q^{44} - 18458 q^{45} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 41070 q^{50} - 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 2904 q^{55} + 20160 q^{57} - 29172 q^{58} + 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} - 20352 q^{64} - 7668 q^{65} + 24926 q^{66} - 47754 q^{67} + 12600 q^{68} - 35042 q^{69} + 39282 q^{71} - 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 21916 q^{75} - 103920 q^{76} + 124984 q^{78} + 125148 q^{79} - 118680 q^{80} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} - 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 68244 q^{88} + 106824 q^{89} - 103424 q^{90} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} - 22200 q^{95} + 76456 q^{96} - 9684 q^{97} - 847 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x - 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 34 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + 3\beta _1 + 34 \) 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
−0.749680
8.04796
−6.29828
−10.3963 −20.6466 76.0833 −8.64919 214.649 0 −458.304 183.283 89.9197
1.2 2.20859 −16.8394 −27.1221 −75.2230 −37.1913 0 −130.577 40.5643 −166.137
1.3 8.18772 3.48600 35.0388 59.8722 28.5424 0 24.8808 −230.848 490.217
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(-1\)

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 539.6.a.e 3
7.b odd 2 1 11.6.a.b 3
21.c even 2 1 99.6.a.g 3
28.d even 2 1 176.6.a.i 3
35.c odd 2 1 275.6.a.b 3
35.f even 4 2 275.6.b.b 6
56.e even 2 1 704.6.a.t 3
56.h odd 2 1 704.6.a.q 3
77.b even 2 1 121.6.a.d 3
231.h odd 2 1 1089.6.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 7.b odd 2 1
99.6.a.g 3 21.c even 2 1
121.6.a.d 3 77.b even 2 1
176.6.a.i 3 28.d even 2 1
275.6.a.b 3 35.c odd 2 1
275.6.b.b 6 35.f even 4 2
539.6.a.e 3 1.a even 1 1 trivial
704.6.a.q 3 56.h odd 2 1
704.6.a.t 3 56.e even 2 1
1089.6.a.r 3 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{3} - 90T_{2} + 188 \) Copy content Toggle raw display
\( T_{3}^{3} + 34T_{3}^{2} + 217T_{3} - 1212 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 90T + 188 \) Copy content Toggle raw display
$3$ \( T^{3} + 34 T^{2} + \cdots - 1212 \) Copy content Toggle raw display
$5$ \( T^{3} + 24 T^{2} + \cdots - 38954 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 486 T^{2} + \cdots - 164136608 \) Copy content Toggle raw display
$17$ \( T^{3} + 1086 T^{2} + \cdots - 331752056 \) Copy content Toggle raw display
$19$ \( T^{3} + 1380 T^{2} + \cdots - 57024000 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 17004325928 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 4029189120 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 1094344400 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 541788167034 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 201929821568 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 2443875098544 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 70174939136 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 1850911309656 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 7759637437060 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 15233874751008 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 147288561330212 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 1290398551704 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 34539701265952 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 1279883216320 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 411597824719824 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 90320980174650 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 10221902527106 \) Copy content Toggle raw display
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