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Properties

Label 531.4.a.e
Level $531$
Weight $4$
Character orbit 531.a
Self dual yes
Analytic conductor $31.330$
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] = [531,4,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(31.3300142130\)
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: \( x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 177)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{6} - 5) q^{5} + (\beta_{7} + 7) q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{6} - 5) q^{5} + (\beta_{7} + 7) q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{8} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{10}+ \cdots + ( - 68 \beta_{7} + 84 \beta_{6} + \cdots + 104) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 16\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 5\nu^{6} + 30\nu^{5} - 135\nu^{4} - 247\nu^{3} + 842\nu^{2} + 294\nu - 748 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 5\nu^{6} - 30\nu^{5} + 137\nu^{4} + 243\nu^{3} - 886\nu^{2} - 256\nu + 876 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 26\nu^{6} + 145\nu^{5} - 707\nu^{4} - 1110\nu^{3} + 4513\nu^{2} + 898\nu - 4128 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 15\nu^{6} - 88\nu^{5} + 403\nu^{4} + 689\nu^{3} - 2520\nu^{2} - 648\nu + 2248 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 18\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 26\beta_{2} + 39\beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 2\beta_{5} + 8\beta_{4} + 28\beta_{3} + 75\beta_{2} + 387\beta _1 + 554 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{7} + 16\beta_{6} + 74\beta_{5} + 84\beta_{4} + 79\beta_{3} + 654\beta_{2} + 1202\beta _1 + 5068 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 220\beta_{7} + 80\beta_{6} + 160\beta_{5} + 386\beta_{4} + 718\beta_{3} + 2358\beta_{2} + 9045\beta _1 + 17078 \) 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
−4.15242
−3.08481
−3.06139
−1.04902
1.03574
2.26905
4.77847
5.26439
−5.15242 0 18.5474 −1.69104 0 −11.0943 −54.3449 0 8.71294
1.2 −4.08481 0 8.68564 7.45529 0 34.0237 −2.80073 0 −30.4534
1.3 −4.06139 0 8.49485 −16.2722 0 6.77038 −2.00979 0 66.0875
1.4 −2.04902 0 −3.80150 −16.1855 0 −1.13960 24.1816 0 33.1645
1.5 0.0357401 0 −7.99872 7.80970 0 4.90513 −0.571796 0 0.279120
1.6 1.26905 0 −6.38952 −14.8820 0 22.4903 −18.2610 0 −18.8859
1.7 3.77847 0 6.27681 5.74283 0 −24.4226 −6.51103 0 21.6991
1.8 4.26439 0 10.1850 −13.9771 0 21.4669 9.31764 0 −59.6039
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

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 531.4.a.e 8
3.b odd 2 1 177.4.a.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.d ✓ 8 3.b odd 2 1
531.4.a.e 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6T_{2}^{7} - 31T_{2}^{6} - 209T_{2}^{5} + 214T_{2}^{4} + 2015T_{2}^{3} + 336T_{2}^{2} - 3596T_{2} + 128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{7} + \cdots + 128 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 42 T^{7} + \cdots - 30976256 \) Copy content Toggle raw display
$7$ \( T^{8} - 53 T^{7} + \cdots - 168443392 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 229585463488 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1833054248 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 75388834904 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 40302801741824 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 7187882575616 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 17\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 40\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 99\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 29\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T + 59)^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 50\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 14\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 24\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 31\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 98\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
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