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Properties

Label 531.6.a.e
Level $531$
Weight $6$
Character orbit 531.a
Self dual yes
Analytic conductor $85.164$
Analytic rank $0$
Dimension $13$
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] = [531,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(85.1638083207\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + \cdots - 6400833792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 19) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1 + 29) q^{7} + ( - \beta_{9} + \beta_{6} + \beta_{4} + \cdots - 10) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 19) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1 + 29) q^{7} + ( - \beta_{9} + \beta_{6} + \beta_{4} + \cdots - 10) q^{8}+ \cdots + (492 \beta_{12} + 298 \beta_{11} + \cdots + 9095) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8} + 137 q^{10} - 250 q^{11} + 1054 q^{13} + 575 q^{14} + 922 q^{16} - 271 q^{17} + 671 q^{19} + 5491 q^{20} + 1094 q^{22} - 3975 q^{23} + 15569 q^{25} - 4622 q^{26} + 21214 q^{28} + 10613 q^{29} + 25597 q^{31} - 15966 q^{32} + 31796 q^{34} - 6729 q^{35} + 17585 q^{37} - 34903 q^{38} + 31382 q^{40} - 12537 q^{41} + 26644 q^{43} - 6654 q^{44} + 149005 q^{46} - 52087 q^{47} + 95384 q^{49} - 121821 q^{50} + 263630 q^{52} - 20014 q^{53} + 120932 q^{55} - 126688 q^{56} + 86066 q^{58} - 45253 q^{59} - 11667 q^{61} - 164794 q^{62} + 151893 q^{64} + 28674 q^{65} + 1106 q^{67} + 4043 q^{68} + 56066 q^{70} - 21230 q^{71} + 81131 q^{73} - 102042 q^{74} + 73900 q^{76} + 104655 q^{77} - 13470 q^{79} + 191969 q^{80} + 79909 q^{82} + 76149 q^{83} + 10035 q^{85} + 321496 q^{86} - 276779 q^{88} + 190205 q^{89} + 80601 q^{91} - 45672 q^{92} + 36768 q^{94} - 9875 q^{95} + 160850 q^{97} + 116644 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + \cdots - 6400833792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39783793826895 \nu^{12} - 489453600148070 \nu^{11} + \cdots - 11\!\cdots\!32 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 826874692028041 \nu^{12} + \cdots + 13\!\cdots\!08 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 54340442593255 \nu^{12} - 52352870381654 \nu^{11} + \cdots - 18\!\cdots\!52 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{12} + \cdots + 10\!\cdots\!96 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!71 \nu^{12} + \cdots + 17\!\cdots\!20 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!75 \nu^{12} + \cdots - 70\!\cdots\!36 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!33 \nu^{12} + \cdots + 23\!\cdots\!20 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22\!\cdots\!49 \nu^{12} + \cdots + 29\!\cdots\!28 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!69 \nu^{12} + \cdots - 20\!\cdots\!84 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 33\!\cdots\!37 \nu^{12} + \cdots + 40\!\cdots\!96 ) / 33\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 77\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 5 \beta_{8} + \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + \cdots + 3946 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{12} + 28 \beta_{10} + 146 \beta_{9} - 2 \beta_{8} - 30 \beta_{7} - 124 \beta_{6} + \cdots + 2511 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 430 \beta_{12} - 176 \beta_{11} - 384 \beta_{10} + 201 \beta_{9} + 838 \beta_{8} - 70 \beta_{7} + \cdots + 362506 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1613 \beta_{12} + 413 \beta_{11} + 5482 \beta_{10} + 17480 \beta_{9} - 295 \beta_{8} - 6826 \beta_{7} + \cdots + 379332 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 44114 \beta_{12} - 26902 \beta_{11} - 49184 \beta_{10} + 49688 \beta_{9} + 111710 \beta_{8} + \cdots + 36059531 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 192940 \beta_{12} + 99996 \beta_{11} + 801488 \beta_{10} + 1993913 \beta_{9} - 23868 \beta_{8} + \cdots + 50738702 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4003155 \beta_{12} - 3748361 \beta_{11} - 5521578 \beta_{10} + 8462900 \beta_{9} + 13829037 \beta_{8} + \cdots + 3721822926 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21290390 \beta_{12} + 16375980 \beta_{11} + 105394716 \beta_{10} + 225028878 \beta_{9} + \cdots + 6455842751 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 338534002 \beta_{12} - 488092428 \beta_{11} - 585026064 \beta_{10} + 1236018841 \beta_{9} + \cdots + 391375714806 \) 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
10.6702
9.82137
6.57049
6.46075
5.29201
2.69937
1.05732
−3.96558
−6.06926
−6.29094
−6.40804
−9.47242
−10.3652
−10.6702 0 81.8525 61.2778 0 186.685 −531.935 0 −653.844
1.2 −9.82137 0 64.4594 −77.2018 0 −23.9833 −318.795 0 758.228
1.3 −6.57049 0 11.1714 5.02964 0 −195.140 136.854 0 −33.0472
1.4 −6.46075 0 9.74133 95.9365 0 79.3965 143.808 0 −619.822
1.5 −5.29201 0 −3.99466 −108.071 0 212.139 190.484 0 571.910
1.6 −2.69937 0 −24.7134 80.3454 0 −162.093 153.090 0 −216.882
1.7 −1.05732 0 −30.8821 −33.7931 0 −85.6226 66.4863 0 35.7300
1.8 3.96558 0 −16.2741 −45.8225 0 214.983 −191.435 0 −181.713
1.9 6.06926 0 4.83596 −26.0687 0 −152.673 −164.866 0 −158.218
1.10 6.29094 0 7.57589 −82.0909 0 51.5315 −153.651 0 −516.429
1.11 6.40804 0 9.06300 77.9071 0 112.353 −146.981 0 499.232
1.12 9.47242 0 57.7268 42.5216 0 243.042 243.695 0 402.782
1.13 10.3652 0 75.4381 24.0297 0 −107.618 450.246 0 249.073
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 531.6.a.e 13
3.b odd 2 1 177.6.a.d ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.d ✓ 13 3.b odd 2 1
531.6.a.e 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 331 T_{2}^{11} + 41 T_{2}^{10} + 41990 T_{2}^{9} - 7229 T_{2}^{8} - 2592364 T_{2}^{7} + \cdots + 6400833792 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots + 6400833792 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 52\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 63\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 31\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 43\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 17\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 50\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 77\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 71\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( (T + 3481)^{13} \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 58\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 85\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 87\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 16\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 79\!\cdots\!20 \) Copy content Toggle raw display
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