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Properties

Label 531.4.a.h
Level $531$
Weight $4$
Character orbit 531.a
Self dual yes
Analytic conductor $31.330$
Analytic rank $1$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 72 x^{12} + 448 x^{11} + 1897 x^{10} - 12642 x^{9} - 21599 x^{8} + 167704 x^{7} + \cdots + 788240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} + (\beta_{5} - \beta_{2} - 2) q^{5} + ( - \beta_{11} + \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} + (\beta_{5} - \beta_{2} - 2) q^{5} + ( - \beta_{11} + \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{8} + (\beta_{11} - \beta_{8} + \beta_{6} + \cdots - 4) q^{10}+ \cdots + (32 \beta_{13} - 4 \beta_{12} + \cdots - 341) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 6 q^{2} + 68 q^{4} - 40 q^{5} - 26 q^{7} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 6 q^{2} + 68 q^{4} - 40 q^{5} - 26 q^{7} - 72 q^{8} + 2 q^{10} - 110 q^{11} + 104 q^{13} - 84 q^{14} + 268 q^{16} - 136 q^{17} - 104 q^{19} - 944 q^{20} + 136 q^{22} - 328 q^{23} + 438 q^{25} - 524 q^{26} - 256 q^{28} - 1052 q^{29} - 104 q^{31} - 1212 q^{32} - 504 q^{34} - 420 q^{35} - 208 q^{37} - 850 q^{38} + 974 q^{40} - 800 q^{41} + 224 q^{43} - 1320 q^{44} - 1110 q^{46} - 496 q^{47} + 660 q^{49} - 1410 q^{50} + 132 q^{52} - 1296 q^{53} + 812 q^{55} - 1104 q^{56} - 350 q^{58} + 826 q^{59} + 500 q^{61} - 1044 q^{62} + 1434 q^{64} - 2844 q^{65} + 368 q^{67} - 1108 q^{68} - 1186 q^{70} - 2268 q^{71} - 1252 q^{73} - 3328 q^{74} - 208 q^{76} - 2750 q^{77} - 1250 q^{79} - 6964 q^{80} + 282 q^{82} - 4878 q^{83} - 1200 q^{85} - 1628 q^{86} + 750 q^{88} - 5312 q^{89} + 836 q^{91} - 2598 q^{92} - 2112 q^{94} - 800 q^{95} + 20 q^{97} - 5370 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 72 x^{12} + 448 x^{11} + 1897 x^{10} - 12642 x^{9} - 21599 x^{8} + 167704 x^{7} + \cdots + 788240 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48158651061 \nu^{13} + 160984518439 \nu^{12} + 3089447119813 \nu^{11} + \cdots - 26\!\cdots\!64 ) / 85\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 98101980757 \nu^{13} + 752559234487 \nu^{12} + 6385824890149 \nu^{11} + \cdots + 58\!\cdots\!20 ) / 42\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 256455347477 \nu^{13} + 966986865415 \nu^{12} + 22073922081317 \nu^{11} + \cdots + 54\!\cdots\!44 ) / 85\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 364244244041 \nu^{13} - 1597744791555 \nu^{12} - 25417777875161 \nu^{11} + \cdots - 58\!\cdots\!36 ) / 85\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59045299415 \nu^{13} + 147839605173 \nu^{12} + 5696132108207 \nu^{11} + \cdots + 59\!\cdots\!52 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 297710377765 \nu^{13} - 1064435458295 \nu^{12} - 24062040339989 \nu^{11} + \cdots - 24\!\cdots\!84 ) / 42\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 151911786665 \nu^{13} - 688610860235 \nu^{12} - 11369416291705 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 21\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 222012754925 \nu^{13} + 1105004461047 \nu^{12} + 16554284399053 \nu^{11} + \cdots + 86\!\cdots\!52 ) / 21\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 983252880673 \nu^{13} + 3173060443211 \nu^{12} + 81637490506097 \nu^{11} + \cdots - 51\!\cdots\!52 ) / 85\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1435647457937 \nu^{13} - 4121584256027 \nu^{12} - 122383217750113 \nu^{11} + \cdots - 64\!\cdots\!00 ) / 85\!\cdots\!36 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 21\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \cdots + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - \beta_{8} - \beta_{6} + \cdots + 175 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 53 \beta_{13} - 93 \beta_{12} + 94 \beta_{11} - 37 \beta_{10} + 55 \beta_{9} - 5 \beta_{8} + \cdots + 5452 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 234 \beta_{13} - 254 \beta_{12} + 252 \beta_{11} + 84 \beta_{10} + 256 \beta_{9} - 76 \beta_{8} + \cdots + 6870 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2018 \beta_{13} - 3322 \beta_{12} + 3404 \beta_{11} - 1030 \beta_{10} + 2080 \beta_{9} + \cdots + 134770 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9416 \beta_{13} - 10956 \beta_{12} + 11224 \beta_{11} + 2636 \beta_{10} + 9670 \beta_{9} + \cdots + 238549 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 67841 \beta_{13} - 108134 \beta_{12} + 113258 \beta_{11} - 25209 \beta_{10} + 68543 \beta_{9} + \cdots + 3472729 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 328146 \beta_{13} - 405420 \beta_{12} + 431752 \beta_{11} + 78220 \beta_{10} + 323813 \beta_{9} + \cdots + 7838865 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2148735 \beta_{13} - 3378965 \beta_{12} + 3639250 \beta_{11} - 555315 \beta_{10} + 2118229 \beta_{9} + \cdots + 92008162 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10662248 \beta_{13} - 13895888 \beta_{12} + 15365712 \beta_{11} + 2367642 \beta_{10} + \cdots + 249688778 \) 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
5.48454
5.25648
4.36424
3.85721
2.60321
2.20648
1.22995
0.463973
−0.833302
−2.32571
−2.98646
−3.69971
−4.79842
−4.82248
−5.48454 0 22.0802 −20.7502 0 −25.4118 −77.2235 0 113.805
1.2 −5.25648 0 19.6306 0.267731 0 23.5436 −61.1357 0 −1.40732
1.3 −4.36424 0 11.0466 1.31432 0 −7.93813 −13.2961 0 −5.73602
1.4 −3.85721 0 6.87805 −16.6394 0 25.7698 4.32758 0 64.1818
1.5 −2.60321 0 −1.22329 17.5611 0 14.2607 24.0102 0 −45.7154
1.6 −2.20648 0 −3.13145 10.3295 0 −34.5805 24.5613 0 −22.7919
1.7 −1.22995 0 −6.48722 −17.0112 0 −2.89285 17.8186 0 20.9230
1.8 −0.463973 0 −7.78473 11.5856 0 −19.0196 7.32369 0 −5.37541
1.9 0.833302 0 −7.30561 −6.02682 0 21.5470 −12.7542 0 −5.02217
1.10 2.32571 0 −2.59106 4.88808 0 −10.9691 −24.6318 0 11.3683
1.11 2.98646 0 0.918934 −5.47207 0 23.2171 −21.1473 0 −16.3421
1.12 3.69971 0 5.68786 8.63489 0 −12.0210 −8.55425 0 31.9466
1.13 4.79842 0 15.0248 −19.9986 0 0.784016 33.7080 0 −95.9617
1.14 4.82248 0 15.2563 −8.68289 0 −22.2892 34.9935 0 −41.8731
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.h ✓ 14
3.b odd 2 1 531.4.a.i yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.4.a.h ✓ 14 1.a even 1 1 trivial
531.4.a.i yes 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 6 T_{2}^{13} - 72 T_{2}^{12} - 448 T_{2}^{11} + 1897 T_{2}^{10} + 12642 T_{2}^{9} + \cdots + 788240 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(531))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 6 T^{13} + \cdots + 788240 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 1049897448000 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 61\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 36\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 44\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 29\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( (T - 59)^{14} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 22\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 61\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 66\!\cdots\!80 \) Copy content Toggle raw display
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