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Properties

Label 539.4.a.p
Level $539$
Weight $4$
Character orbit 539.a
Self dual yes
Analytic conductor $31.802$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(31.8020294931\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 126 x^{16} + 468 x^{15} + 6142 x^{14} - 20892 x^{13} - 150088 x^{12} + 458732 x^{11} + \cdots - 676352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 7^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{7} + 5) q^{4} + \beta_{8} q^{5} + (\beta_{9} + \beta_{8} + \cdots - \beta_{3}) q^{6}+ \cdots + (\beta_{4} - 3 \beta_1 + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{7} + 5) q^{4} + \beta_{8} q^{5} + (\beta_{9} + \beta_{8} + \cdots - \beta_{3}) q^{6}+ \cdots + ( - 11 \beta_{4} + 33 \beta_1 - 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} + 88 q^{4} + 84 q^{8} + 198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{2} + 88 q^{4} + 84 q^{8} + 198 q^{9} - 198 q^{11} + 148 q^{15} + 744 q^{16} + 776 q^{18} - 44 q^{22} - 200 q^{23} + 966 q^{25} + 532 q^{29} + 1908 q^{30} + 1196 q^{32} + 500 q^{36} + 1232 q^{37} + 256 q^{39} + 420 q^{43} - 968 q^{44} + 1608 q^{46} - 2160 q^{50} + 2688 q^{51} + 1136 q^{53} + 1728 q^{57} + 5244 q^{58} + 1316 q^{60} + 4984 q^{64} + 6092 q^{65} - 12 q^{67} - 268 q^{71} + 9168 q^{72} + 5432 q^{74} - 4112 q^{78} + 3064 q^{79} + 1382 q^{81} + 6508 q^{85} - 2132 q^{86} - 924 q^{88} - 4736 q^{92} + 3924 q^{93} - 1424 q^{95} - 2178 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 126 x^{16} + 468 x^{15} + 6142 x^{14} - 20892 x^{13} - 150088 x^{12} + 458732 x^{11} + \cdots - 676352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 51\!\cdots\!64 \nu^{17} + \cdots + 15\!\cdots\!52 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!98 \nu^{17} + \cdots + 51\!\cdots\!20 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!81 \nu^{17} + \cdots - 16\!\cdots\!68 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!87 \nu^{17} + \cdots - 20\!\cdots\!80 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!35 \nu^{17} + \cdots - 35\!\cdots\!60 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!48 \nu^{17} + \cdots - 10\!\cdots\!64 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!07 \nu^{17} + \cdots + 54\!\cdots\!60 ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!25 \nu^{17} + \cdots + 55\!\cdots\!64 ) / 78\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!46 \nu^{17} + \cdots - 58\!\cdots\!56 ) / 18\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!49 \nu^{17} + \cdots - 23\!\cdots\!28 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!91 \nu^{17} + \cdots - 24\!\cdots\!48 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 49\!\cdots\!01 \nu^{17} + \cdots + 18\!\cdots\!00 ) / 37\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51\!\cdots\!16 \nu^{17} + \cdots - 21\!\cdots\!72 ) / 37\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!62 \nu^{17} + \cdots + 53\!\cdots\!04 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!81 \nu^{17} + \cdots + 46\!\cdots\!40 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 26\!\cdots\!38 \nu^{17} + \cdots - 16\!\cdots\!96 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 53\!\cdots\!67 \nu^{17} + \cdots + 50\!\cdots\!28 ) / 10\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 7\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - 2\beta_{10} + 2\beta_{9} - 7\beta_{7} - 2\beta_{6} - 2\beta_{3} - 2\beta_{2} + 105 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 7 \beta_{7} - 35 \beta_{6} - 7 \beta_{5} - 6 \beta_{3} + \cdots + 21 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{17} + 7 \beta_{16} + 7 \beta_{15} + 14 \beta_{14} + 28 \beta_{12} + 116 \beta_{11} + \cdots + 3136 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21 \beta_{17} - 35 \beta_{16} + 35 \beta_{15} + 70 \beta_{13} + 70 \beta_{12} + 330 \beta_{11} + \cdots + 1470 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 546 \beta_{17} + 448 \beta_{16} + 462 \beta_{15} + 1092 \beta_{14} + 168 \beta_{13} + 1834 \beta_{12} + \cdots + 117019 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2086 \beta_{17} - 3206 \beta_{16} + 2982 \beta_{15} + 112 \beta_{14} + 5586 \beta_{13} + 5096 \beta_{12} + \cdots + 89803 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 33509 \beta_{17} + 21497 \beta_{16} + 23653 \beta_{15} + 65814 \beta_{14} + 15680 \beta_{13} + \cdots + 4815594 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 147455 \beta_{17} - 206941 \beta_{16} + 177597 \beta_{15} + 18228 \beta_{14} + 330666 \beta_{13} + \cdots + 5240116 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1881208 \beta_{17} + 913766 \beta_{16} + 1129968 \beta_{15} + 3565324 \beta_{14} + 1050504 \beta_{13} + \cdots + 208275781 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9035516 \beta_{17} - 11632236 \beta_{16} + 9357740 \beta_{15} + 1839852 \beta_{14} + 17688594 \beta_{13} + \cdots + 296898021 ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 100613583 \beta_{17} + 35957019 \beta_{16} + 53277595 \beta_{15} + 182757022 \beta_{14} + \cdots + 9275397936 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 512878905 \beta_{17} - 609339927 \beta_{16} + 468912199 \beta_{15} + 146389656 \beta_{14} + \cdots + 16425998666 ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5222194950 \beta_{17} + 1317714524 \beta_{16} + 2527991634 \beta_{15} + 9082345636 \beta_{14} + \cdots + 421139003719 ) / 7 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 27817191682 \beta_{17} - 30658651730 \beta_{16} + 22991907442 \beta_{15} + 10131423288 \beta_{14} + \cdots + 891497485487 ) / 7 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 37976605615 \beta_{17} + 6284283115 \beta_{16} + 17332804407 \beta_{15} + 63336425434 \beta_{14} + \cdots + 2769681860714 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 1465935778755 \beta_{17} - 1505486592113 \beta_{16} + 1117050654321 \beta_{15} + \cdots + 47651719254832 ) / 7 \) 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
−6.62496
−3.79654
−5.55877
−2.73034
−1.02380
−3.85222
0.116349
−2.71208
−0.994836
1.83359
2.76512
−0.0633051
1.78381
4.61224
5.93807
3.10964
4.18480
7.01323
−5.21075 −3.42758 19.1519 20.8741 17.8603 0 −58.1099 −15.2517 −108.770
1.2 −5.21075 3.42758 19.1519 −20.8741 −17.8603 0 −58.1099 −15.2517 108.770
1.3 −4.14455 −0.938496 9.17732 −11.5029 3.88965 0 −4.87946 −26.1192 47.6744
1.4 −4.14455 0.938496 9.17732 11.5029 −3.88965 0 −4.87946 −26.1192 −47.6744
1.5 −2.43801 −9.50045 −2.05610 −6.32245 23.1622 0 24.5169 63.2586 15.4142
1.6 −2.43801 9.50045 −2.05610 6.32245 −23.1622 0 24.5169 63.2586 −15.4142
1.7 −1.29786 −2.66092 −6.31555 13.7735 3.45351 0 18.5796 −19.9195 −17.8762
1.8 −1.29786 2.66092 −6.31555 −13.7735 −3.45351 0 18.5796 −19.9195 17.8762
1.9 0.419378 −8.45634 −7.82412 13.3596 −3.54640 0 −6.63629 44.5096 5.60272
1.10 0.419378 8.45634 −7.82412 −13.3596 3.54640 0 −6.63629 44.5096 −5.60272
1.11 1.35091 −2.03197 −6.17505 −8.08054 −2.74501 0 −19.1492 −22.8711 −10.9161
1.12 1.35091 2.03197 −6.17505 8.08054 2.74501 0 −19.1492 −22.8711 10.9161
1.13 3.19802 −6.65939 2.22735 −17.7078 −21.2969 0 −18.4611 17.3474 −56.6300
1.14 3.19802 6.65939 2.22735 17.7078 21.2969 0 −18.4611 17.3474 56.6300
1.15 4.52385 −7.98947 12.4652 −15.2329 −36.1432 0 20.2001 36.8316 −68.9115
1.16 4.52385 7.98947 12.4652 15.2329 36.1432 0 20.2001 36.8316 68.9115
1.17 5.59902 −6.94365 23.3490 4.57049 −38.8776 0 85.9393 21.2143 25.5903
1.18 5.59902 6.94365 23.3490 −4.57049 38.8776 0 85.9393 21.2143 −25.5903
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.4.a.p ✓ 18
7.b odd 2 1 inner 539.4.a.p ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.4.a.p ✓ 18 1.a even 1 1 trivial
539.4.a.p ✓ 18 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{9} - 2T_{2}^{8} - 56T_{2}^{7} + 90T_{2}^{6} + 971T_{2}^{5} - 1132T_{2}^{4} - 5524T_{2}^{3} + 3648T_{2}^{2} + 6972T_{2} - 3136 \) Copy content Toggle raw display
\( T_{3}^{18} - 342 T_{3}^{16} + 47566 T_{3}^{14} - 3454476 T_{3}^{12} + 139724841 T_{3}^{10} + \cdots - 266485041152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{9} - 2 T^{8} + \cdots - 3136)^{2} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots - 266485041152 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 77\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( (T + 11)^{18} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 90\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 77\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 22\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 18\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 63\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 25\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 97\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 46\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 62\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 15\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots + 71\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 85\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 55\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 37\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 19\!\cdots\!72 \) Copy content Toggle raw display
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